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Excess Noise in One Dimensional Quantum Nanowires

HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Device fabrication and mountin...
 Device operation and DC charac...
 Noise measurements
 Noise in silicon nanowires
 Sequential ablation of carbon...
 Lorentzian type of noise in carbon...
 Energy band simulation of carbon...
 Thermally activated 1/F noise in...
 Conclusions
 Appendices
 References
 Biographical sketch
 

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EXCESS NOISE IN ONE DIME NSIONAL QUANTUM NANOWIRES By SHAHED REZA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by SHAHED REZA

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Dedicated to My mother Nurunahar Tahera And my father ATM Naderuzzaman Talukdar

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iv ACKNOWLEDGMENTS I would like to thank my wife, Misty Sabi na, for her support through all my years of graduate school and re search. Without her limitle ss patience and constant encouragement, this dissertation would have been impossible. Also there is our little bundle of joy, Ronan Reza, whose smile at the end of a long day at work gave me the energy to continue my str uggle well past midnight and start afresh the next day. My parents have been the key people to in fluence my life and I am indeed grateful to them for all that they have done to help make me what I am today. Over all these years, in every difficult period of time they were the source of support and comfort. I consider every achievement in my life theirs as well. My work would have been impossible wit hout the guidance of Dr. Gijs Bosman. I was introduced to this topic of noise through one of his le ctures back in 1998 and have been interested in ever since. It is from him that I learned the numerous tricks of the trade, learned literally how to extract inform ation from noise. My heartfelt thanks go to him for his constant patience and trust in me and for all of his advi ce about my research and professional career. Thanks go to Quye n T. Huynh, for her help in performing the ablation experiments. I would like to thank Dr. Andrew G. Ri nzler and his student Dr. Jennifer SippelOakley for giving us the carbon nanotubes devices that I have used in my experiments. I sincerely appreciate their pati ence at the beginning of the co llaboration when I was just learning how to make the connections to the devices without damaging them.

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v Also, I appreciate Dr. M. Saif Islam at UC Davis and his colleagues Theodore I. Kamins and R. Stanley Williams at HP labs for providing us with the silicon nanowire devices. Special thanks go to Dr. Saif Isla m for the fruitful discussions during the analysis phase, which led to the succ essful modeling of the data. I have to thank Dr. Charles Saylor, Dr. Randy Duensing, Sathya Vijayakumar, and Dr. Mark Limkeman at Invivo Diagnosti c Imaging, for their constant support and thought provoking discussions on multip le aspects of my research. I am grateful to and Dr. Jing Guo and Dr. Ant Ural for allowing me to attend their lectures on carbon nanotubes and countless helpful discussions and advice. And finally I would like to thank my committee members Dr Scott E. Thompson and Dr. Kirk J. Ziegler for making the time to serve on my supervisory committee with their busy schedule.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................x ii CHAPTER 1 INTRODUCTION........................................................................................................1 2 DEVICE FABRICATION AND MOUNTING.........................................................11 Fabrication of Carbon Nanotubes...............................................................................11 Fabrication of Silicon Nanowires...............................................................................12 Mounting of Devices..................................................................................................18 3 DEVICE OPERATION AND DC CHARACTERISTICS........................................21 DC Characteristics of Carbon Nanotubes...................................................................21 Electronic Structure and Pr operties of Carbon Nanotube...................................21 DC Characteristics and Measurements................................................................25 DC Characteristics of Silicon Nanowires...................................................................30 Resistance Measurements....................................................................................30 Resistance Model.................................................................................................33 4 NOISE MEASUREMENTS.......................................................................................39 Noise Measurement System.......................................................................................39 Noise Characterization of the LNA............................................................................39 Instrument Control and Other Measurement Issues...................................................41 Setup for Low Temperature Measurement.................................................................44 Overview of the Observed Noise Spectra...................................................................45 5 NOISE IN SILICON NANOWIRES..........................................................................50 Introduction.................................................................................................................50 Noise Model................................................................................................................50 Summary and Conclusion...........................................................................................59

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vii 6 SEQUENTIAL ABLATION OF CARBON NANOTUBES.....................................61 Setup for the Sequential Ablation Technique.............................................................61 Result and Analysis....................................................................................................62 Details of the Observed Annealing Effect..................................................................71 Summary and Conclusions.........................................................................................73 7 LORENTZIAN TYPE OF NO ISE IN CARBON NANOTUBES.............................77 Introduction to the Lorentzian Noise Component......................................................77 Results and Analysis...................................................................................................80 Effect of Gate Voltage................................................................................................86 8 ENERGY BAND SIMULATION OF CARBON NANOTUBES.............................88 Introduction.................................................................................................................88 Self Consistent Solution of Char ge Density and Poissons Equation.........................90 Calculation of Charge Density............................................................................90 Calculation of Potential.......................................................................................94 Self Consistent Solution Method.........................................................................96 Results and Discussions..............................................................................................98 9 THERMALLY ACTIVATED 1/F NOISE IN CARBON NANOTUBES...............103 Introduction...............................................................................................................103 Calculation of the Distribu tion of Activation Energies............................................104 Physical Location of the Trap Centers......................................................................109 10 CONCLUSIONS......................................................................................................114 Summary of Results and Conclusions......................................................................114 Future Work..............................................................................................................116 APPENDIX A EQUIVALENT NOISE SOURCE S OF BROOKDEAL-5004 LNA.......................118 B FLOW CHART OF THE INST RUMENT CONTROL PROGRAM......................120 LIST OF REFERENCES.................................................................................................122 BIOGRAPHICAL SKETCH...........................................................................................127

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viii LIST OF TABLES Table page 2-1 Measured nanowire dimensions from SEM................................................................17 3-1 Device resistance......................................................................................................... 35 5-1 Relative noise magnitude............................................................................................52 6-1 Noise and conductance data of samples A, B and C...................................................76 7-1 Activation energies calcu lated using equation (7-2)...................................................83

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ix LIST OF FIGURES Figure page 1-1 Carbon nanotube and the planer graphene sheet........................................................2 1-2 Scanning electron microscope (SEM) image of silicon nanowires connecting two electrodes.............................................................................................................9 2-1 Cross-section of the devi ce showing different layers...............................................13 2-3 SEM image of suspended carbon nanotubes............................................................15 2-4 Illustration of fabricatio n steps for silicon nanowires..............................................16 2-5 Mounting of the device on a TO-8 package.............................................................19 3-1 Nanotube formation fr om graphene sheet................................................................22 3-2 Formation of Zigzag and armc hair nanotube from graphene...................................23 3-3 E-k diagram of graphene in the first Brillouin zone using the -band nearestneighbor tight-binding model...................................................................................24 3-5 Measured drain current vs. voltage ch aracteristics at di fferent gate bias.................28 3-6 Measured dynamic conductance and the ca lculated components vs. gate bias........31 3-7 Measured dynamic conductance vs. gate voltage bias at 77K and 300K.................32 3-8 Calculated vs. measured resist ance, including contact resistance............................38 4-1 Noise measurement setup.........................................................................................40 4-2 Setup for the noise characteriza tion of the low noise amplifier...............................42 4-3 Voltage noise spectral density at the input side of Brookdeal-5004 LNA for different source resistance........................................................................................43 4-4 A typical plot of the measured current noise spectral density..................................46 4-5 Plot depicting the relationship be tween the 1/f noise and device current................48

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x 5-1 1/f noise coefficient A vs. effective resistivity ......................................................51 5-2 Circuit representation of the noise model................................................................53 5-4 Contact and bulk noise components calculated from measured data.......................58 6-1 Pulsed bias experimental set-up for sequential ablation of metallic CNTs..............63 6-2 The measured low bias conductances of metallic and semiconducting CNTs in sample A...................................................................................................................65 6-3 Calculated conductances of the ab lated metallic CNTs for sample A.....................67 6-4 Measured At/Rt and AtGt 2 at Vg = +14V for sample A, plotted as a function of the number of CNT ablation attempts performed on the sample.............................69 6-5 Calculated A and A/R for the ablated metallic CNTs for sample A.........................70 6-6 Calculated A vs. R values for the ablated CNTs.......................................................72 7-2 A diagram describing the shift of Lo rentzian spectra with temperature..................79 7-3 Lorentzian characteristic times for device A and B.................................................81 7-4 A plot of the Lorentzian Plateau value vs. characteristic time for device B at Vg = 0 and 14V..............................................................................................................84 7-5 A plot of the Lorentzian Plateau value and characterist ic time at different gate voltages for device A................................................................................................85 8-1 Explanation of the charge cal culation using equation (8-1).....................................89 8-2 Density of state plots calculated using (8-4)............................................................92 8-3 A plot of the calculated density of carriers per unit length for metallic and semiconducting CNTs at T = 300K..........................................................................93 8-4 Flow-chart for the iterative solution scheme............................................................99 8-5 The plot of Ei vs. position at Vg = 0 and +14 V for a metallic CNT......................101 8-6 Band-bending at Vg = +14 V for a semiconducting CNT......................................102 9-1 Measured 1/f Noise coefficient over temperature for sample A............................105 9-2 Distributions of activation energies for sample A and B.......................................107 9-3 Measured and calcula ted frequency exponent vs. T for sample A.....................108

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xi 9-4 D(E) vs. T plot calculated for Vg = 0 and +14V.....................................................110 9-5 The difference of Ei due to the change in Vg = 0V to +14V vs. position...............112

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EXCESS NOISE IN ONE DIME NSIONAL QUANTUM NANOWIRES By Shahed Reza December 2006 Chair: Gijs Bosman Major Department: Electrical and Computer Engineering. Silicon nanowires and carbon nanotubes are two promising novel devices in the nanotechnology area. A study of the current-vol tage and the low frequency excess noise properties of these devices is presented. From the silicon nanowire current-voltage and noise characteristics measured at room temperature in the linear regime of de vice operation, the bulk a nd contact resistance contributions are extracted and modeled. The ex cess noise observed at low frequencies is interpreted in terms of bulk and contact noise contributions, with the former comparable, in terms of Hooge parameter values, to the low noise leve ls observed in high quality silicon devices. The contact noise is significan t in some devices and is attributed to the impinging end of the bridging nanowires. The charge transport and noise propert ies of three terminal, gated devices containing multiple single-wall metallic and semiconducting carbon nanotubes were measured at room temperature. A method to separate contributions from the metallic and semiconducting carbon nanotubes by sequential abla tion using high voltage pulsed bias is

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xiii presented. The relative low frequency excess noi se of the metallic tubes was observed to be two orders of magnitude lower th an that of the semiconductor tubes. The low frequency noise of single-walled carbon nanotubes is st udied over the 77K to 300K temperature range. Lorentzian shaped spectra along with 1/f noise spectra have been observed. From the Lorentzian noise components, a range of thermal activation energies from 0.08 to 0.51 eV for the associ ated fluctuation mechanisms is obtained. From the 1/f noise spectra, a distribution of activation energies of fluctuation processes ranging from 0.2 to 0.7eV is derived. These fi ndings indicate that the observed noise spectra are caused by number fluctuations. Us ing simulation results and the observed gate dependence of the noise producing activation energy distribution, the physical origin of the observed noise phenomena was s hown to be the contact region.

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1 CHAPTER 1 INTRODUCTION In the last decades of the 20th century there has been a remarkable progress in electronics. The technology that has been the vehicle of this revolution is the complementary metal oxide silicon field eff ect transistor (CMOSFET or CMOS). Since the introduction of this tec hnology the basic device remained mostly unchanged, but the device size gradually became smaller. This miniaturization enabled semiconductor industries to pack more devices in a chip, thus increasing th e functionality, speed and cost reduction. Over the last three decades this miniaturization trend continued in a very consistent manner, roughly every three y ears quadrupling the p acking density, obeying Moores law. There are strong indications that the limit of the miniaturization is rapidly approaching. The current state of the art MO S device has a feature size of 20nm and the physical limit for scaling is expected to be ~ 10nm. Below this feature size the error in the lithography and the uncertainty of the doping profile will be a great obstacle for manufacturing high performance devices [1]. Unless a new technology is discovered, the semiconductor industry faces the danger of becoming stagnant. This necessitates the research for device structures that may ta ke the place of the current CMOS technology. Several new devices have been propose d. Among them, ferromagnetic field effect transistors, single electron transistors, mol ecular transistors, sili con nanowires and carbon nanotubes are currently in the forefront of research.

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2 Figure 1-1. Carbon nanotube and the planer graphene sheet (Image courtesy of Dr. Ant Ural, Dept. of ECE, University of Florida).

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3 Carbon nanotubes (CNT) were discovered by Iijima in 1991 [2]. The arrangement of atoms in a CNT is the same as the atomic arrangement in graphite. A CNT can be described as a graphene sheet rolled on to itself and forming a hollow tube [3] (see Fig. 11). The ends of the nanotube can either be open or capped by the so-called Fullerenes, a half sphere structure [4]. CNTs can be either single walled or multi-walled. The diameter of a single wall CNT generally is between 0.7 to 10nm and its length can vary from nm to ms depending on the growth conditions. Because of the small diameter CNTs operate as one-dimensional quantum structures. Furthermore, the CNTs can be grown and the critical feature size, the diameter, is determ ined by the growth process and thus, unlike in MOS technology, does not depend on lithography. This is a significant advantage of CNTs over the current CMOS technology. The carbon nanotube is a unique device. First, the CNT can be either metallic or semiconductor, so it can potentially be used both as the device and the interconnects. Furthermore, the band-gap of the semiconducti ng tube depends on the diameter. This can potentially eliminate the need for comple x band-gap engineering and there are many possible ways this property can be utilized. CNT is a very good conductor of heat and current; the current dens ity of CNTs can be orders of magnitude higher than copper [5]. The carbon nanotube is a very stable structure having a tensile strength of about 60 times greater than steel [6]. Despite these very attractive properties, there remain several significant obstacles before the CNT tec hnology can be considered for mainstream electronics. The fabrication method of CNT is still fa r from perfect. During the CNT growth process a mixture of CNTs of different leng th, diameter and orientation is formed. The

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4 process to selectively grow a CN T with a certain feature size at a precise location is still unknown. Another problem, first reported by Collins et al. [7], is high excess noise. One would expect lower excess noise levels since the atoms in a graphene structure are well ordered and properly terminate d. On the contrary the 1/f lik e noise in CNTs was observed to be extremely high. Compared to a carbon film resistor, which is generally considered too noisy for many applications, CNT is seve ral orders of magnitude more noisy. Unless a way to reduce this noise is found, the appli cability of CNTs in mainstream electronics will be severely limited. Collins et al. [7] observed that both single-wall and multi-wall CNTs are equally noisy. The observed excess noise obeys the well-known expression for 1/f type noise, f I A f Sdc f 2 / 1) ( (1-1) where Idc is the dc current level, f is the frequency, and A and are constants. The exponent was found to be approximately equal to one as expected for 1/f type noise. The parameter A represents the relative magnitude of the 1/f noise. Additionally, Collins el al. established the following empirical relationship between the 1/f noise coefficient A and the resistance of the device R 1110 / R AS (1-2) Recently, Snow et al. [8] reported an additional de pendence in (1-2) for a two dimensional network devices or mats, 3 1 1110 9 / L R A S (1-3) where L is the device sample length in m and which may be longer than the length of the constituent CNTs. A reduction in 1/f noise with device length is expected for an

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5 electronic system where 1/f noise varies inve rsely with the number of carriers, but the fact that A/R is inversely proportional to 1/L1.3 cannot be explained from the number of carriers; normally one would expect a 1/L2 relationship. Snow et al. also reported on 1/f noise gate voltage dependence. The noise ma gnitude is minimum at 0V gate bias and maximum at the gate voltage where the device inverts from p-type to n-type mode of operation. Furthermore, it was reported that at positive gate voltage the resistivity fluctuation S is related to the resistivity by 6 3 S. (1-4) This type of power law dependence is characte ristic of percolating systems. And in fact the two-dimensional mats used by Snow et al. resemble a percolation system [9] where the device consists of many interconn ected CNTs. Unlike a conventional onedimensional device, where a carrier flows from one contact to the other ballistically through one CNT, in these devices the carriers flow through the multiple intersecting network of CNTs. This can re sult in a length dependent tran sport properties and it is the likely cause of the 1/L1.3 dependence shown in (1-3). In an attempt to exclude possible noise from the contact, Collins et al. [7] conducted a four-point probe measurement, but did not observe a change in the measured noise between a twoand fourpoint probe experiments. Henc e, they concluded that the source of the noise is not the contacts. However, now it is understood that a four-probe technique cannot be used for carbon nanotubes, because the contacts ar e an integral part of the mesoscopic CNT device and the addition of another set of contacts completely changes the device itself [10]. Hence, the result of the four-probe measurement does not rule out contact as a potential sour ce of 1/f noise. Recently Kingrey et al. [11] reported

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6 the effect of annealing and passivation studies. It was obser ved that after annealing, the noise magnitude of a CNT at room temperatur e decreased but it was still substantial and varied widely across the temperature range of 80 to 450K. In an attempt to prevent absorption of species back on to the CNT afte r heating, two different types of passivation layers, SiO2 and Polymethylmethacrylate, were trie d. In both cases no significant change in noise characteristics was observed. In the same study the presence of Lorentzian spectra along with 1/f noise was reported. The presence of Lorentzian spectr a is also reported by Tarkiainen et al. [12]. The observed Lorentzian spectra can be expr essed by the following expression [13], 2 21 ) 0 ( ) ( L LS f S (1-5) where SL(0) is the plateau value of the Lorentzian, is the angular frequency and is a characteristic time. This type of noise sp ectra is associated with a two energy level system. The characteristic time of the observe d spectra was found to be a function of the bias voltage. Most importantly, the noise char acteristics were obser ved to be dependent on the direction in which the bias was appl ied, which may correspond to the location of the trap centers. The exact mechanism of the 1/f noise phenomena in general is still unclear. However, two widely accepted models for this phenomenon exist. The first one proposed by McWhorter [14] was successful in expl aining the noise in MOSFETs. This model assumes that the noise is caused by carrier number fluctuations. The second model, proposed by Hooge [15], postula tes that the 1/f noise phenom ena originates in the bulk because of mobility fluctuation of carri ers. Hooges model requires the exponent in (11) to be exactly 1; however McWhorter s model does not have this restriction.

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7 Commonly, the Hooges parameter H is used as a figure of merit to compare the 1/f noise levels between devices and technologies even if the source is of McWhorter type. This figure of merit H is related to the 1/f noise coefficient A by N AH (1-6) where N is the number of carriers associated with the fluctuation process. Ishigami et al. [16] in their recent publicati on addressed the issue of which mechanism is responsible for the observed 1/f noise. They established that the 1/f noise parameter A is inversely proportional to the number of carriers N Since e V V L c Nth g g (where cg, L, Vg and Vth are gate capacitance, device length, gate voltage and gate threshold voltage respectively), it follows that th gV V A 1 (1-7) Based on (1-7) Ishigami et al. concluded that the noise is of the mobility fluctuation origin. Their argument is based on a bulk MO SFET analogy where equation (1-7) is true only if the noise has a mobility fluctuation origin. For a number fluctuation origin one would expect 21th gV V A relationship. Although this statement is shown to be true for MOSFETs, it is not clear whet her it would be applicable for a CNT device, because, unlike a MOSFET where the device operation is controlled by the modulation of the channel conductance using gate voltage, fo r a CNT device the control is achieved by modulating the Schottky barriers formed at th e contact region. In addition, CNT devices typically have a very significant contact re sistance which dominates the current voltage characteristic, and this further complicates the analogy with a si mple MOSFET 1/f noise model.

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8 Ishigami et al. estimated the value of H to be (9.3 0.4)10-3, which is much lower than the previously reported value of ~ 0.2 by Collins et al. [7]. This is also supported by a recent publication by Lin et al. [17]. Their estimate for H is 210-3. Lin el al. also established that A is inversely proportional to the device length, which is consistent with the N A 1 behavior. The findings of Lin et al. further suggested that 1/f noise in CNTs is not substantially affect ed by the acoustic phonon scattering or ionized impurity scattering. To understand th e effect of contact and bulk Lin et al. added a third gate to modulate the bulk region of the device. The 1/f noise level in this modified device also showed similar dependence on device resi stance as a conventional device. In this case the measured relative noi se magnitude supported the N A 1 model established for a conventional device. In summary, the CNT exhibits thermal, shot and low frequency excess noise. The thermal and shot noise have been characteri zed and modeled satisfa ctorily [18]. The 1/f noise magnitude in a CNT is several orders of magnitude higher than in the conventional silicon devices and the origin and mechanism of this noise component are still unclear. Now the silicon (Si) nanowire devices used in our study will be introduced. Silicon nanowire (SNW) is another leading candida te in the nanotechnology research arena. Similar to CNTs, these devices have a large le ngth over diameter ratio. SNWs are grown; thus, like a CNT, the need for lithography is eliminated. The major advantage of SNW over CNT technology is the constituent material Si is the material of choice for the mainstream semiconductor industry. As a resu lt, integrating SNW in currently available processes would be easier than the CNT technology.

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9 Figure 1-2. Scanning electron microscope (S EM) image of silicon nanowires connecting two electrodes.

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10 Currently available SNWs typically have a larger radius than the CNTs. The typical SNW radius is on the order of ~50nm, and the ra dius of a typical CNTs is on the order of ~1nm. Although there has been theoretical work done on the noise of nanowires [19-22], no experimental work has been reported on th e noise in SNW. The result of our study on Si nanowires presented in this manuscript will help fill the void. In this dissertation, the result of a stu dy of the low frequency excess noise of CNT and Si nanowires is presented. Chapter 2 de scribes the fabrication methods and sample geometry used in this study. In chap ter 3 the details of device operation, DC measurements and in chapter 4 the deta ils of noise measurement techniques are discussed. In chapter 5 the results and anal ysis of the noise measurements on SNWs and the possible source of the observed noise are di scussed. In chapter 6 a sequential ablation technique to determine the characteristics of individual CNTs is presented. The analysis of the thermally activated Lorentzian component of noise is discussed in chapter 7. In chapter 8 the details of CNT simulation techniqu e are discussed. Finally, the analysis of the thermally activated 1/f noise in CNTs and the conclusions resulting from the dissertation work are presented in chapter 9 and chapter 10, respectively.

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11 CHAPTER 2 DEVICE FABRICATION AND MOUNTING Fabrication of Carbon Nanotubes The growth method for the devices used in this study is the chemical vapor deposition (CVD) growth method [23]. Ther e are several other methods for growing carbon nanotubes; among them the arc discha rge method, [24] and laser ablation method are popular [25]. In the CVD growth method, a substrate with a catalys t (typically Fe, Co or Ni) is placed in a furnace and a hydrocarbon and hydrogen gas flow are added. The hydrocarbon gas acts as the source of car bon. At a high temperature (between 500C and 1000C) the hydrocarbon, usually methane, is catalytically decom posed and CNTs are formed. The diameter of the grown CNT is a pproximately equal to the diameter of the catalyst particles used [26]. The CNT can grow outwards from the catalyst particle while the catalyst particle is attached to the s ubstrate or the nanotube grows in between the catalyst particle and the substrate, with the catalyst particle traveling on the tip of the nanotube. A localized growth of nanotube can be obtained by cont rolling the placement of the catalyst. In the case of the devices used in this study, the nanotubes were grown on a silicon substrate with a top oxide layer, typically 600nm thick (Fig. 2-1). The catalyst, 10mg/ml Fe(NO3)3H2O particles, was dispersed over the en tire surface of the substrate so the nanotube growth occurred over the entire surface. The gas flow rate for both hydrogen and methane was 200sccm [27]. Optical lithography was used to pattern Cr/Pd (sputtered on to the wafer, 5nm/45nm thick respectively) 500m long electrodes spaced 1m apart

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12 (the Pd likely overcoats the edge of the Cr la yer so that electrical contact to the nanotubes is via the Pd; see Fig. 2-2). Fabrication of Silicon Nanowires Silicon nanobridges were grown between el ectrically isolated electrodes formed from the top silicon layer of (110)-oriented silicon-on-insulator (SO I) substrates [28]. Approximately 1nm Au was deposited on the (111)-oriented sides of the electrodes and annealed in a H2 ambient at 670C to form nanoscale Au-Si alloy cataly st islands. The structure was then exposed to a mixture of 15sccm SiH4, 60sccm HCl, and 30sccm B2H6 (100ppm in H2) in a H2 ambient at 680C and a total pr essure of 1.3kPa for 30min to grow nanowires bridging between electrodes wi th a separation of 10m or less. Note that B2H6 is added to provide p-type dopants in the form of boron. It was found that the doping concentration can be controlled by increasing the partial pressure of B2H6. Since the dopant is incorporated dur ing the growth process as oppos ed to adding it later using ion implantation, it is expected that the la ttice structure of the device will be relatively defect free. As will be shown later, the noi se measurement results support this idea. Before nanowire growth, reactive ion etching was used to remove Au catalyst from all areas of the substrate other than the sidewall s. This helped to suppress the uncatalyzed growth of Si between electrodes, ensuring good electrical isolation. Highlights of the fabrication process for the bridging nanowires are illustrated in Fig. 2-4. The dimensions of the Si nanowires used in our experime nts were measured using a scanning electron microscope (SEM) and are presented in Table 2-1.

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13 SiO2 600nm Carbon Nanotube Gate Si Pd/Cr SiO2 600nm SiO2 600nm Carbon Nanotube Gate contact Si Si Pd/Cr Pd/Cr Source Drain Figure 2-1. Cross-section of the device showing different layers.

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14 500 m 1 m Pd/Cr contact Pd/Cr contact Carbon nanotubes Figure 2-2. Top view of the device show ing the nanotubes and the electrode layout.

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15 1.5 m Electrode Carbon nanotube Figure 2-3. SEM image of suspended carbon nanotubes.

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16 (d) Si substrate SiO2 La y er ( a ) Si Electrodes Angled catalyst deposition ( b ) Au Catal y st ( c ) SNW g rowth [111] direction Figure 2-4. Illustration of fabrication steps for silicon na nowires; (a) Etching to form electrodes on a SOI substrate (b) Angled deposition of Au catalyst particles and (c) Nanowire growth in [111] direction. SEM image of multiple nanowires bridging across the gap between the Si electrodes shown in (d) [28].

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17 TABLE 2-1 Measured nanowire dimensions from SEM Device Wire 1 Wire 2 Wire 3 Wire 4 Wire 5 Length (cm) Radius (cm) Length (cm) Radius (cm) Length (cm) Radius (cm) Length (cm) Radius (cm) Length (cm) Radius (cm) a 7.5 10-4 6.8 10-6 6.6 10-46.2 10-6 b 4. 10-4 5.8 10-6 4.7 10-44.8 10-6 c 8.3 10-4 8.0 10-6 Wafer 1 d 7.0 10-4 6.8 10-6 a 3.0 10-4 6.7 10-6 3.3 10-47.5 10-6 b 3.4 10-4 3.8 10-6 3.6 10-41.1 10-53.0 10-45.4 10-6 c 6.0 10-4 7.5 10-6 d 6.3 10-4 4.2 10-6 6.5 10-45.0 10-66.4 10-48.3 10-66.0 10-4 6.7 10-6 9.0 10-44.7 10-6e 8.4 10-4 4.6 10-6 1.1 10-36.0 10-6 f 1.0 10-3 5.0 10-6 g 1.0 10-3 6.3 10-6 8.5 10-44.2 10-6 h 1.0 10-3 4.8 10-6 1.0 10-34.3 10-6 Wafer 2 i 1.5 10-3 4.7 10-6 1.1 10-35.6 10-6

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18 Mounting of Devices The silicon substrate containing the de vices was mounted on a standard TO-8 package (Fig. 2-5). First a 50 mil alumina substrate with top and bottom gold-plated surfaces was attached to the TO-8 package using silver loaded conductive epoxy. Then the silicon substrate containing the devices wa s attached on the alumina substrate using the conductive epoxy. This way the silicon s ubstrate is isolated from the body of the package and thus can be used as the back gate. Next the electrodes and the back gate were bonded to the pins of the TO -8 package using a wedge bonder. Some of the CNT devices were fabricated on Pd only electrodes. For these devices it was not possible to bond to the electrode using the available bonders. Because of the softness of Pd, the Au bond wire would not attach to Pd. Hence, conductive epoxy was used to connect the bond wire to the Pd electrode. First a bon d was made on the pin of the TO-8 package, and then the bond wire was cut at an appropriate length so that the wire reached the surface of the el ectrode. A minute amount of epoxy was then dispensed on the electrode using a fine needle and then th e bond wire was carefully pushed into this drop of epoxy using a fine set of tweezers. The epoxy was then cured by baking in an oven at 120C for 8 to 12 hours. During our first attempt to measure the CNT devices it was discovered that these devices are extremely sensitive to static electricity. For a noise measurement the pins of the TO-8 package need to be soldered to the measuring equipment. But the soldering process and occasionally the mere handling was enough to destroy these highly sensitive devices.

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19 Alumina Substrate Conductive Epoxy TO-8 Package Pin Sample Wire bond Access to back gate Alumina Substrate Alumina Substrate Conductive Epoxy Conductive Epoxy TO-8 Package Pin TO-8 Package Pin Sample Sample Wire bond Access to back gate Wire bond Access to back gate Figure 2-5. Mounting of the device on a TO-8 package.

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20 To eliminate this problem all of the pins of the TO-8 package were soldered to the body of the package using copper wires before attach ing the substrates to the package. In this way both ends of a nanotube in the sample always remain shorted together thus preventing build-up of static charge that may damage the device. After the pins were soldered to the measurement system the wi re connecting the pins to the body of the package was removed and measurements were made. The SNW devices did not show a static sensitivity hence, did not requi re the procedure described above.

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21 CHAPTER 3 DEVICE OPERATION AND DC CHARACTERISTICS DC Characteristics of Carbon Nanotubes Electronic Structure and Propert ies of Carbon Nanotube A graphene sheet is a two dimensional structure of carbon atoms arranged in a honeycomb like formation, one carbon atom at each vertex of the hexagon. A CNT has the same basic structure of graphene but in stead of a planer structure, it has a hollow tubular shape. The physical, chemical and el ectrical properties of a CNT depend on how the CNT was formed from the basic graphene structure. Consider the graphene lattice presented in Fig. 3-1, a CNT can be formed by cutting along the dotted lines joining them together forming a cylindrical shape. The vector represented by A B is called the chiral vector C. The length of the chiral vector defines the circumference of the CNT. The chiral vect or can be expressed in terms of the lattice vectors a1 and a2, mathematically, 2 1ma na Ch (3-1) where n and m are integers and different combinations of n and m yield CNTs of different chirality (the length and the angle of the chir al vector). The chirality uniquely defines a particular type of CNT (n,m) [29], for example zigzag (n,0) and armchair (n,n) CNTs. The formation of these two types of CNTs are explained in Fig. 3-2. Typically, the length of a CNT is on the order of m and the diameter is on the order of nm. Because of the small diameter the CN T operates as a one-dimensional device.

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22 a1a2 a1a2 Circumference vector: C = n a1+ m a2 • • (chiral angle) C ( 4 2 ) ( 4 2 ) A B(a) (b) Figure 3-1. Nanotube formation from graphene sheet; (a) de finition of the unit vectors and (b) the directions of the chiral ve ctor for a (4,2) CNT is shown. (Image courtesy of Dr. Ant Ural, Dept. of ECE, University of Florida).

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23 ( n 0 ) zigzag ( n 0 ) zigzag(a) ( n n ) armchair ( n n ) armchair ( n n ) armchair (b) • • A ( 3 3 ) ( 3 3 ) • • A ( 4 0 ) ( 4 0 ) Figure 3-2. Formation of Zigzag and armchair nanotube from graphene. (Image courtesy of Dr. Ant Ural, Dept. of EC E, University of Florida)

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24 E (eV)K MK conduction valence E (eV)K MK E (eV) E (eV)K MK conduction valence(a) (b) Figure 3-3. E-k diagram of graphene in the first Brillouin zone using the -band nearestneighbor tight-binding model; (a) in 3-di mension, and (b) 2D representation showing the conduction and valence bands (Image courtesy of Dr. Ant Ural, Dept. of ECE, University of Florida).

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25 The energy dispersion relation (E-k plot) of a CNT can be obtained from the E-k plot of graphene shown in Fig. 3-3. The E-k plot for a CNT can only be subsets of the two dimensional plot for graphene, because of the quantization in the circumferential direction [30]. These subsets or the energy ba nd are given by slicing the E-k surface for graphene with vertical period ic parallel planes with a constant spacing. The spacing of these planes is determined by the chirality of the CNT [18]. If, for a CNT, one of these planes intersects the K points, where the conduction band intersects the valence band, then for that particular CNT, the band gap is zero, i.e. the CNT is metallic. This can only happen if, |n m| = multiple of 3 (3-2) If the planes do not intersect the K points, the CNT is semiconducting. For example, the armchair (n,n) CNT is always metallic, but the zigzag (n,0) CNT is metallic if n is a multiple of 3, semiconducting otherwise. Fo r the same reason, of a randomly grown collection of a total of n CNTs, n/3 are expected to be metallic and the rest semiconducting. The energy band-gap of a semiconducting CNT is related to the diameter of the CNT by [18] t C C gd a t E (3-3) where t is the nearest neighbor tig ht binding overlap energy, aC-C is the nearest neighbor atomic distance and dt is the diameter of the tube. DC Characteristics and Measurements As mentioned before, the conduction and vale nce band of a metal lic CNT intersects and as a result the resistance of a metallic CNT is independent of the gate bias. The resistance of a metallic CNT can be written as

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26 bulk contact quantum metallicR R R R (3-4) Rquantum is the lower limit of resistance for a quantum resistance, equals to 12.9 kper sub-band[31]. Rcontact and Rbulk are the contact and bulk resi stances due to non-idealities present in the device. So, the lower limit fo r the resistance of metallic CNTs is 12.9 k. The characteristics of a semiconducting CN T are more complicated as the current conduction is a function of the gate bias. The operation of a semiconducting CNT also depends on the nature of the source and the drain metal contact pa ds. Typically metals with hi gh work function such as Ti and Pd are used for these contacts. In this case at a low drain-source bias (Vds) the Fermi level of the contacts lines up close to the valence band. As a result the potential barrier for the holes becomes small but the barrier for the electrons becomes large and hole conduction dominates via tunneling. For this reason a typical CN T FET device (CNFET) resembles the characteristics of a conventional p-channel MOSFET. If a metal with a smaller work function is used, then the Fermi level aligns somewhere between the conduction band (Ec) and valence band (Ev), depending on the work function of the metal. Fig. 3-3 shows a case where the Ferm i level is aligned at the middle of the band gap. In this case at a gate voltage clos e to 0V, both electron and holes experience a potential barrier so conduction is negligible (see Fig. 3-4(a )). With increasingly negative gate bias, the width of the ba rrier for the holes decreases and increasing number of holes can tunnel through (see Fig. 3-4(b)). At a su fficiently high negative bias the barrier is completely removed and the current through the devices becomes independent of gate bias.

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27 (b) Source Drain EC EV qVds qVds Source Drain EC EV (a) Figure 3-4. Qualitative response of the na notube conduction and va lence band at (a) a gate voltage below threshold voltage so that the CNT is off and (b ) at a gate voltage above threshold voltage so that the CNT is on [32]

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28 -4 10-4-3 10-4-2 10-4-1 10-40 1 10-42 10-43 10-44 10-4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.10.20.30.4 0.5 Vds (V) Ids (A) -15V -10V -5V 0V 5V 10V 15V 20V 25V 30V Vgs (Volt) Figure 3-5. Measured drain current vs. voltage characteristics at different gate bias

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29 Because of the symmetry of the Fermi level a lignment, electron transport can be achieved with a sufficiently high positive gate, which makes this particular CNT an ambipolar device. The Ids vs. Vds plot at different gate bias for a typical device used in our study is presented in Fig. 3-5. The plot is almost linear, but with higher Vds, sub-linear characteristics were observed in most devices due to phonon dispersion in few cases super-linear characteristics were observed as well [33]. From th e devices studied, each contained a randomly grown matrix of me tallic and semiconducting CNTs between the drain and source Pd metal contacts. Fig. 35 indicates p-type ch annel operation and it shows an on-state at Vg = 0V gate bias as expected ba sed on the earlier discussion. The device becomes more conductive at increasing negative gate bias and saturates at Vg = – 10V gate bias. With increasing positive gate bias the conduction decreases because the barrier for hole tunneling becomes increas ingly wider. This device does not show ambipolar transport because th e gate bias required to lower the barr ier enough for electron transport cannot be achieved. A high gate bias is required because the Fermi level is pinned close to the valence band edge and the gate oxide layer in these devices was thick. The metallic CNTs do not respond to the gate bias, so at a high positive gate bias, when the semiconducting CNTs are off the measured current It is the total current through the metallic CNTs Im. With Im known, at any gate bias the current through the semiconducting CNTs Is, can be calculated from m t sI I I (3-5)

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30 Alternatively, the total c onductance of the semiconducting CNTs can be obtained by subtracting the total con ductance of the metallic CNTs (measured when the semiconducting CNTs are off) from the measured device conductance. Plots of the total dynamic (ac) conductance Gac and the metallic and semiconducting components are presented in Fig. 3-6. The DC characteristics were found to be a weak function of temperature for the temperature range of our experiment 77K to 300K. A plot of the ac conductances at 77K and 300K is shown in Fig. 3-7. DC Characteristics of Silicon Nanowires Resistance Measurements The SNW devices used in this study have a much larger radius than the CNT devices described earlier, around ~50 nm. As a result, unlike CNTs, these devices do not operate in the quantum domain. The resistance of these devices can be characterized by equation (3-4) with Rquantum 0. Since, the calculation of the bulk component of the resistance is relatively straightforward, it opens up an opportunity to model the total resistance in terms of bulk and contact co mponents. The details of method employed for this modeling is presented next. The I-V characteristics of the SNW de vices were measured using an HP4145B semiconductor parameter analyzer. The device s were found to be very linear over the voltage range of 5V.

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31 Gac (S) 0 1 10-42 10-43 10-44 10-4 5 10-46 10-47 10-4-15 -10 -5 0 5 10 15 20 25 30 Vg (V) Total Metallic Semiconducting Figure 3-6. Measured dynamic conductance and the calculated components vs. gate bias.

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32 3.0 10-43.5 10-44.0 10-44.5 10-45.0 10-45.5 10-46.0 10-46.5 10-4-15 -10 -5 0 5 10 15Vg (V) T = 77K T = 300K G ac (S) Figure 3-7. Measured dynamic conductance vs. gate voltage bias at 77K and 300K.

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33 The device resistance Rm calculated from the slope of th e I-V plot (see Table 3-1), is the parallel combination of the re sistances of the bridging wires (Ri) in the device, i.e., N i i mR R, 11 1 (3-6) where N is the number of wires in the device. The total resistance of an individual wire is the sum of bulk resistance Rbi and contact resistance, Rci i.e. ci bi iR R R (3-7) The bulk resistance of a wire is related to the bulk resistivity b by, 2 i i b bir l R (3-8) where ri and li are the radius and length of wire i, respectively. The effective resistivity of a device is calculated from the measured resistances and the dimensions of the wires measured from SEM images using N i i i ml r R, 1 2 (3-9) From (3-7) (3-9), N i ci i m bR R R, 11 (3-10) So the effective resistivity calculated using (3-10) is equal to the bulk resistivity b only in the absence of contact resistance ( Rci 0) and greater otherwise. Resistance Model Bulk resistivity was calculated from the devices with the lowest resistivity and noise. As these devices have the lowest contact resistance, the resistivity calculated using

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34 (3-10) and neglecting contact re sistance gives the best estimate of the bulk resistivity. The carrier density p is related to the bulk resistivity by the following expression; p qp 1 (3-11) where p is the hole mobility and q is the electron charge. The corresponding carrier densit ies calculated using the re sistivity versus impurityconcentration relationship for bulk Si at 300K [34], are 5 1018 cm-3 and 1.3 1018 cm-3 for wafers 1 and 2, respectively. Cui et al. [35] reported that the carri er mobility in highly doped silicon nanowires is comparable to that observed in bulk silicon. Consequently, the above listed values are assumed good estimat es of the nanowire carrier densities. The bulk resistances of all other devices were calculated using these carrier concentrations. Note that dopant fluctuation is ignored in our analysis. This is a reasonable assumption given the large number of dopant atoms per wi re. Also the physical diameters of the nanowire are used in the calculation. Due to the presence of surf ace charge, the surface region of the nanowire is expected to be depl eted, and as a result the effective diameter becomes less than the physical diameter. The depletion width depends on the surface charge density and the number of traps fille d. Our initial estimate of surface charge density is 2 1012 q/cm-2. The calculation for the worst-case scenario shows that the resistance model presented here and the noi se model (to be presented in chapter 5) remains valid. Since the surface charge density is not well characterized at this time, the analysis using the physical diameters of the nanowires is presented.

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35 TABLE 3-1 Device resistance Rm ( ) Device Number of wires a 2 1.76 105 b 2 3.11 105 c 1 1.38 105 Wafer 1 d 1 2.47 105 a 2 5.55 104 b 3 4.85 104 c 1 1.89 105 d 5 7.47 104 e 2 5.08 105 f 1 6.30 105 g 2 1.76 105 h 2 2.64 105 Wafer 2 i 2 3.93 105

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36 From the devices containing only one nanowire, the contact resistance was obtained from b m cR R R (3-12) It was observed that the c ontact resistance is inversel y proportional to the crosssectional area with a proportionality constant Kc of 1.69 10-5 -cm2, i.e. 2 i c cir K R (3-13) This model was applied to all other devices to calculate the contact resistance. The minimum and maximum contact resistances obtained from (3-13) for individual nanowires were 48.9 k and 383 k respectively. The total resistance was calculated by combining the calculated contact and the bul k resistances. A plot of the measured resistance and the resistance calculated using the above model is shown in Fig. 3-8. The plot shows good agreement between the measur ed and the calculated resistance for all devices except for two devices. Our model suggests a common mechanism for the contact resistance in all devices, most likely resulting from the interface be tween the impinging end of the nanowire and the sidewall. The base end of the nanowire is connected epitaxially to the silicon sidewall, and thus the contact resistance on this side s hould be negligible. On the impinging side however, the nanowire makes c ontact to the silicon electrode through the pinholes of the native oxide [36] so the contact resistance on th is side is expected to be dominant. It is possible for the actual contac t area to be different from the wire crosssection, because the nanowire has to burro w through a native oxide layer. However, the good fit of the model indicates that for al l but two of the nanowires, the impinging

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37 contacts are very uniform. A closer SEM ex amination of these two devices showed a nanowire with a contact area much smaller than the cross-section of the nanowire, which may explain why these two are different from the other devices. These two devices were not used for subsequent noise modeling.

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38 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 Measured Resistance (105 ) Calculated Resistance (105 ) Wafer 1 Wafer 2 Slope 1 Line Figure 3-8. Calculated vs. measured resi stance, including contact resistance. The uncertainties in the calculated resistan ce due to the measurement uncertainties are shown. The uncertainties for the meas ured resistance are too small to be displayed in the plot. The devices ha ving only one nanowire are marked with a circle.

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39 CHAPTER 4 NOISE MEASUREMENTS Noise Measurement System A typical noise measurement system is shown in Fig. 4-1. The bias circuit shown is required to measure excess noise. The bias circuit needs to have lower noise than the device under test (DUT). Often the noise generated by the DUT is lower than the detectable range of the spectrum analyzer, fo r this reason the low noise amplifier (LNA) is used to amplify and bring the noise signa l within the dynamic ra nge of the spectrum analyzer. Brookdeal-5004, a commercially available LNA was used in this study. HP3561A, a low frequency spectrum analyzer (S A) was used to acquire the time domain data and calculate the voltage noise spectral density [37]. The computer workstation was used as an instrument controller for th e SA and for importing the data for further processing. Noise Characterization of the LNA The noise characteristics of th e LNA is of the highest importance as it is at the front of the signal chain [38]. For this reason the LNA was characterized for noise performance in the beginning of the study. The noise of th e LNA can be adequately described in terms of a voltage and current noise s ource at the input node as shown in Fig. 4-2. If a resistor Rs is connected to the input, th e voltage noise spectra l density measured at the output of the LNA is given by [38] ] 4 [2 2 vn s in s B veff voS R S TR k A S (4-1)

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40 LNA Workstation CN T Bias CircuitHP 3561A Instrument Control and Data Import LNA Workstation CN T Bias CircuitHP 3561A Instrument Control and Data Import Figure 4-1. Noise measurement setup.

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41 where ) (i s i v veffR R R A A Ri, Svn and Sin are the effective gain, input resistance, equivalent voltage and current noise sources of the LNA respectively. The current noise spectral density for the resistance Rs is given by [37] s B siR T k S / 4 (4-2) From (4-1) referring to the input side of the LNA; vn s in s B veff voS R S TR k A S 2 24. (4-3) According to (4-3) a log-log plot of 2 veff voA Svs. Rs should show three distinct regions (i) vn veff voS A S 2, a constant for 24s in s B vnR S TR k S (ii) a slope ~1 region for s B vnTR k S 4 and s B s inTR k R S 42and (iii) a slope ~2 region for vn s B s inS TR k R S 42. One such plot is presented in Fig. 4-3. Th e plot indicates the different regions clearly however in region (iii) the eff ect of RC roll-off at higher fr equency is visible. If the device resistance is within region (ii) a mean ingful noise measurement can be made. If it is in region (i), the equivale nt voltage noise of the LNA do minates and if in region (iii) the current noise of the LNA dominates. Regi ons (i) and (iii) are onl y visible at certain frequencies, this indicates that the equivalent noise sources are thus frequency dependent. The extracted spectral densities Svn and Sin using (4-3) are presented in appendix A. Instrument Control and Other Measurement Issues As mentioned in the previous section the LNA is used for amplification, as the noise signal is typically too sm all for the SA to detect dire ctly. But a situation can arise when the noise signal is la rge enough to saturate the L NA and that will result in an incorrect noise reading.

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42 R s R i is Sis vn Svn in Sin A v Svo Figure 4-2. Setup for the noise characterization of the low noise amplifier.

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43 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 101 102 103 104 105 106 107 108 Resistance ( )Input reflected voltage spectral density (V2/Hz) 10 Hz 50 Hz 100 Hz 500 Hz 1 kHz 5 kHz 10 kHz 4KTR Line ( i ) ( ii ) ( iii ) Figure 4-3: Voltage noise spectral density at the input side of Brookdeal-5004 LNA for different source resistance.

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44 To prevent this situation, an analog oscill oscope was connected in parallel to the SA to monitor the output of the LNA for c lipping during noise meas urement. Also, even if the LNA does not saturate, a random high noise spike can overload the SA input during a noise measurement and again cause an incorr ect reading. An instrument control routine was written in the HPVEE graphical programming language to continuously check for an overload condition during measurement a nd stop the measurement if overload is detected. Additionally an option to select the attenuation level for the internal attenuator of the SA is added to the code that allo wed precise control of the noise magnitude presented to the SA input. The SA collects a se t of noise data for a certain time span, and then calculates the spectral noise density in the frequency domain. The calculated spectral noise density for the first 10% of the bandwidth is considered not accurate due to finite averaging time. So a control routine is wr itten to start with the specified minimum frequency span and repeat the measurement fo r every decade of frequency range up to the maximum specified frequency and discarding th e first 10% of the data for each decade. A flow chart of the instrument control program is presented in appendix B. Setup for Low Temperature Measurement The TO-8 package containing the device wa s placed in the sample chamber of a cryostat. The TO-package was mounted in a sample holder and placed on a copper finger to be cooled with liquid N2. A layer of Iridium is placed between the TO-8 package and the sample holder to aid the conduction of heat The connections to the pins of the TO-8 package are made through feed-throughs in the windows of the chamber using cryogenic wires and then the chamber is closed. The chamber is evacuated first using a mechanical

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45 pump and followed by a rotary turbine pump in order to pr ovide thermal isolation. A vacuum level of ~5mTorr was used for our experiments. To cool the system, N2 was placed in a pressurized container. N2 liquid guided by the pressure travels through a heat exchanger that is ther mally connected to the copper finger containing the sample. The N2 flow rate is controlled by pressure applied to the N2 container. The main temperature control is ach ieved by the flow rate and the heat transfer control in the heat exchanger. The lowest temperature achievable using this system is limited by the choice of N2 as the coolant, which is approximately 77K. To achieve a finer temperature control a feedback heating system consisting of a temperature sensor and a variable gain amplif ier is used. The amplifier delivers a current to a resistive heater proportional to the diffe rence of the temperatur e set by the user and the current temperature reading from the sensor Using the amplifier gain for fine control a stable temperature level to 0.1K within a few degrees of the desired temperature can be achieved easily. Overview of the Observed Noise Spectra The voltage noise spectral density Sv was measured for frequencies between 10Hz and 100kHz and converted in to current noise spectral density Si using [39] 2R S Sv i (4-4) where R is the resistance of the device unde r test (DUT). A typical plot of Si is given in Fig. 4-4. For both CNT and SNW devices the noise observed was a combination of excess noise and thermal noise. The current spec tral density of the thermal noise is given by [37] R T K SB thermal4 (4-5)

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46 10-24 10-23 10-22 10-21 10-20 1 10 100 1000 10000 100000Frequency (Hz) Thermal Noise Total Noise 1/f Noise Lorentzian Current Noise Spectral Density (A2/Hz) Figure 4-4. A typical plot of the measured current noise spec tral density showing thermal noise and the 1/f noise and the Lorent zian components of the excess noise. The effect of RC roll off is visible at around 70 kHz

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47 CNTs reportedly exhibit shot noise but the full shot noise magnitude for the maximum bias level used in the study 13.9A, is 4.510-24 A2/Hz, below observed excess noise levels. Moreover the actual shot noise level is expected to be suppressed, placing it outside the range of observation [18]. Th e thermal noise level was higher (6.624 10-23 A2/Hz for 1 k resistor at 300K). The observe d excess noise was mostly 1/f type noise [40]. The current noise spectral density for 1/f type of noise can be described by the following expression [39]; f I A Sf / 1 (4-6) where I is the dc current level, f is the frequency, A, a and are constants expressing the relative noise magnitude, current and freque ncy exponent, respectively. The values of A and are estimated by fitting a line to the experi mental data as shown in Fig. 4-4. The parameter was found to be approximately equal to 1; hence this noise component is called 1/f noise. The value of the current exponent is equal to 2 for a device in linear mode as was the case for both CNT and SNW devices used in this study. An illustration of this relationship is shown in Fig. 4-5 for a CNT de vice. Note that at the lowest current level the 1/f noise is below the thermal noise floor at 1kHz, hence the plot for 1kHz deviates from slope 2. From this point on, = 2 is assumed and the following expression for 1/f noise will be used; f I A Sf 2 / 1 (4-7)

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48 Si = 9 10-7 I2.03 10-25 10-24 10-23 10-22 10-21 10-20 10-9 10-8 10-7Current (A) 10 Hz 100 Hz 1 kHz Fitted (10 Hz)Si (A 2 /Hz) Figure 4-5. Plot depicting the relationship between the 1/f noise and device current. The magnitude of 1/f noise is shown at di fferent frequency poi nts. The frequency spectra of the 1/f noise for the three cu rrent levels are shown in the inset.

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49 For the CNT devices in some cases Lorentzian spectra were observed. These Lorentzian components arise wh en random number fluctuatio ns are caused by processes with a single characteristic time and activation energy. A Lorentzian component can be characterized as [13], 2 21 ) 0 ( ) ( L LS f S (4-8) where SLis the plateau value of the Lorentzian is the angular frequency and is a characteristic time. No Lorentzian com ponent was observed in the SNW devices. The result and analysis of the 1/ f and Lorentzian components will be discussed in details in subsequent chapters.

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50 CHAPTER 5 NOISE IN SILICON NANOWIRES Introduction The result of the resistance measurement for these devices was presented in chapter 3. The measured resistivity and the 1/f noise coefficient A presented in Table 5-1 show significant variations from device to device. However the plot of A vs. (Fig. 5-1) shows that the devices from both wafers that have th e lowest effective resistivity also generally show the lowest noise. Based on discussions in chapter 3, th ese devices can be identified as the devices with low contact resistance. Th e fact that the low-noise devices also have low contact resistance suggests that the source of the noise is the contact. To check this possibility further a noise model was developed and will be presented next. Noise Model The proposed noise model includes bulk c ontact components just like the bulk and contact components of resistance. The circuit diagram of the model is shown in figure 52. From this circuit, the measured open-ci rcuited noise voltage across the terminals is given by c cn b bn nR i R i v (5-1) where ibn and Rb are the noise current source and the resistance respectively for the bulk region, and icn and Rc are the noise current source and the resistance respectively for the contact region of a wire.

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51 10-11 10-10 10-9 10-8 10-7 10-2 10-11 Resistivity ( -m) 1/f noise coefficient Wafer 1 Wafer 2 Figure 5-1. 1/f noise coefficient A vs. effective resistivity The uncertainties in A and due to the measurement uncertainties are shown.

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52 TABLE 5-1 Relative noise magnitude Device Number of wires A a 2 6.7110-9 b 2 1.2010-9c 1 2.9210-11Wafer 1 d 1 8.4510-9 a 2 1.2210-9 b 3 3.3710-10c 1 1.6110-10d 5 4.0410-11e 2 1.6910-10f 1 2.7610-10g 2 3.5110-11h 2 4.2110-11Wafer 2 i 2 4.7310-10

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53 ibn Bulk Contacticn vn R c R b Figure 5-2. Circuit repres entation of the noise model

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54 From (5-1) the total 1/f voltage noise spectral density Sv in terms of the individual current noise spectral densities is given by 2 2 c ic b ib vR S R S S (5-2) where Sib and Sic are the current noise densities of the bulk and contact noise sources, respectively. From (5-2), with the total resistance, R = Rb + Rc, (5-3) the total current noise spectral density can be written as, 2 2 R R S R R S Sc ic b ib i. (5-4) Using (4-7), the expression for 1/f noise with 1 2 2 2 2 2 R R I f A R R I f A I f A Sc dc c b dc b dc i (5-5) where Ab and Ac are the 1/f noise coefficients fo r the bulk and the contact region, respectively. Equation (5-5) can be simplified to 2 2 R R A R R A Ac c b b. (5-6) Most of the devices studied contained multip le nanowires. For those cases, the total noise of the device is the sum of the noise contribution from all the nanowires in the device. From (4-7) N i i i t t tI f A I f A S1 2 2 (5-7) N i t i i tI I A A1 2 2 (5-8)

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55 where At and It are the combined noise coefficient and current for all the nanowires in the device under study and, Ai and Ii are the noise coefficient and current for the i-th nanowire. Using (5-6) (5-8) N i t i i ci ci i bi bi tI I R R A R R A A1 2 2 2. (5-9) From (5-9), the bulk and contact noise components can be separated. The bulk noise is given by N i t i i bi bi bI I R R A A1 2 2 (5-10) and the noise component from the contact is given by, N i t i i ci ci cI I R R A A1 2 2, (5-11) with c b tA A A (5-12) If either Ab or Ac is known, the other one can be calculated from (5-12). The noise component of the bulk can be accurately determined from the devices that have negligible contact resistance. To understand this consider (5-9). For negligible contact resistance, i.e. Rb >> Rc, we have bR R and R >> Rc. Then from (5-9) bA A (5-13) Also, from (5-10) with b iR R N i t i bi bI I A A1 2. (5-14) From the well known Hooge model for bulk 1/f noise [39],

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56 i Hb biV p A (5-15) where Hb is the Hooge parameter, p is the density of carriers and Vi is the volume of the i-th wire. Using the Hooge model in (5-14), N i t i i Hb bI I V p A1 21 (5-16) N i t i i b HbI I V p A1 21. (5-17) The Hooge parameters were calculated from the devices with the lowest resistivity and noise. As these devices have th e lowest contact resistance, the Hb calculated using (5-17) gives the best estimate of the bul k Hooge parameter. The calculated Hooge parameters are 1.1 10-5 and 7.5 10-6 for wafer 1 and wafer 2, respectively. In general the value of the Hooge parameter is a good indicator of the pr ocess quality, a nd the values obtained for the Si nanowires are comparable to Hooge para meters for modern low noise silicon bulk devices [40]. Usi ng these calculated Hooge parame ters the bulk and contact noise A values for the other devices were calcu lated using (5-10), (5-11) and (5-16). However, unlike bulk noise, there is no known model fo r contact noise, so the contact noise magnitude per wire ( Aci) cannot be calculated dir ectly from (5-11). To calculate the contact noise it is necessary to assume a functional dependence between the noise magnitude and some physical parameter such as the radius or length. One can expect the contact noise to be some function of radius but independent of length. Hence, the following model for the contact noise was adopted, m i cir A (5-18)

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57 where ri is the radius of the nanowire. The exponent m determines how the noise is related to the physical parameter of the corresponding nanowire. For example, for m = 0, 1 and 2, the noise is independe nt, proportional to the radius and proportional to the crosssectional area respectively. The values for m tested for a fit were –3, -2, -1, 0, 1, 2 and 3. The best fit to the data was obtained for m = -2, in other words th e best-fit model suggests the relative noise is invers ely proportional to the cross se ctional area of the nanowire i.e. 21i cir A. (5-19) The proportionality constants for wafers 1 and 2 are 4.710-18 cm2 and 4.610-19 cm2, respectively. It was shown in chapter 3 th at the contact resistance is also inversely proportional to the contact area. Comparing (3-13) and (5-19); ci ciR A (5-20) This model suggests that the contact noise is proportional to the contact resistance, which is reasonable, considering that both th e noise and the resistance indicate the quality of the contact. The calculated contact noise from measurements and the modeled noise are shown in Fig. 5-4. The plot shows good agreement between the model and the measured noise and also shows that the agr eement is worse if the magnitudes of the bulk and contact noise components become comp arable. This is expected because the calculation involves subtracti ng two statistical quantities; consequently, when the magnitudes of the two noise compone nts are comparable, the error in Ac calculated using (5-12) would become higher.

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58 10-13 10-12 10-11 10-10 10-9 10-8 10-7 Wafer1 a Wafer1 d Wafer2 a Wafer2 b Wafer2 c Wafer2 d Wafer2 f Wafer2 i Ac Ab Ac (Fitted) Figure 5-4. Contact and bul k noise components calculated from measured data. The contact noise component calculated fr om the model is also shown. The calculated uncertainties due to measurement uncertainties are indicated.

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59 Summary and Conclusion The resistance model presented enables the calculation of the bulk and contact components of the resistance. The contact re sistance is believed to originate from the impinging end of the nanowire where the na nowire connects to the uncatalyzed silicon layer. To estimate the relative magnit udes of the bulkand contact-resistance components, consider a typical nanowi re with a length and radius of 8m and 50nm, respectively. For a doping level of 11018/cm3, the bulk resistance is 424 k. The contact resistance calculated from the model is 215 k, and is a significant portion of the total resistance, which, may be reduced by improve d processing. The bulk 1/f noise coefficient for this nanowire for a Hooge parameter of 110-5 is 510-10. The 1/f noise coefficient for the contact noise from the model is 110-8, for a proportionality constant of 110-19. Hence the contact noise is the dominant noise mechanism in this nanowire. The likely mechanism for noise in the case of our device s is carrier trappingdetrapping in defects producing the well-known 1/f-like number fluc tuation noise spectra [37]. The impinging end of the wire, where contact to the silic on electrode is made through possibly pinholes in the native oxide, is expected to be defect rich and thus the dominant source of contact noise whereas the base end of the nanowire is connected epitaxially to the silicon sidewall creating a defect lean lower noise contact configurat ion. Furthermore, because of the higher resistance on the impinging side, an y fluctuations in this contact will couple out more to the device contacts. This analysis shows that the Hooge para meter for SNWs, the figure of merit for bulk noise performance, is within the range of Hooge parameters for modern low noise bulk devices [40]. Because the contact was id entified as the dominant source of noise,

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60 further noise reduction can envisioned by op timizing the contact. Reducing the contact resistance can potentially reduce contact noi se because they originate from a common mechanism, as indicated by (5-20); moreover less contact noise will couple out into the remainder of the circuitry as the contact resi stance becomes a smaller fraction of the total resistance.

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61 CHAPTER 6 SEQUENTIAL ABLATION OF CARBON NANOTUBES During the CNT growth process typically multiple CNTs grow and bridge the gap between the metallic contacts. Due to the random nature of the CNT growth of the total number of these CNTs two thirds are expected to be semiconducting and the rest metallic tubes [41]. In order to st udy a single CNT, a single CNT needs to be identified. Commonly that is done using an Atomic Force Microscope (AFM). Once a CNT is located and marked, the contacts can be placed and electrical measurements made. Alternatively, CNTs are grown on a wafer cont aining a matrix of contacts. It is then necessary to find a single CNT connecting two contacts with the help of AFM for this case as well. Both of these processes are very time consuming and limit the number of CNTs that can be studied. In our experime nts due to the large size of the contacts, multiple metallic and semiconducting CNTs are contacted simultaneously and a pulsed voltage bias tube ablation method was implem ented to delineate the noise and charge transport properties of the metallic and semiconductor tubes [42-43]. Although destructive, this method is suitable for ex tracting and comparing the noise properties of the metallic and semiconducting CNTs because it allows the measurement of multiple CNTs in a short period of time and due to th e close proximity of the measured CNTs the process variation is kept to a minimum. Setup for the Sequential Ablation Technique The samples were placed at room temperat ure in a pulsed bias circuit that allowed for approximately 10s duration voltage bias pulses appl ied at the drain terminal at a

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62 repetition rate of 80Hz as depicted in Fig. 6-1. A positive, continues gate bias of +14V was applied so that only the metallic tube s contribute to conduction. Upon slowly increasing the level of the pulsed bias volta ge at the drain and monitoring the device current on an analog oscilloscope, the onset of Joule heating in a metallic tube could be observed from small fluctuations in the curr ent. Upon further increase of the pulsed drain bias, a tube was ablated and the pulse d bias was switched quickly to the 1k dissipative resistor preventing damage to additional t ubes. The ablation occu rred approximate at +17V for a pulse width of 6sec and a repetition rate of 80 Hz. Current-voltage and noise measurements were carried out before and after the ablation of each tube using a HP4351 semiconductor parameter analyzer and a HP 3561A low frequency spectrum analyzer operating between 10Hz and 100KHz. As mentione d in chapter 4, the noise observed is a combination of low frequency 1/f-like excess noise and thermal noise. Shot noise levels were below the detection range of our experiment. Result and Analysis The conductances of the metallic and the semiconducting CNTs were extracted from the measured current-voltage characte ristics. At sufficiently high gate voltage Vg, in this case +14V or higher, the semiconducti ng CNTs are turned off, as a result the measured conductance, Gm represents the total parallel conductance of the metallic CNTs. The semiconducting CNTs are turned ON at Vg = -14V. So at this gate voltage the measured conductance Gt represents the conductance of all the CNTs including the metallic and semiconducting CNTs. Now th e total parallel conductance of the semiconducting CNTs, Gs is obtained using, m t sG G G (6-1)

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63 50 o o o Gate Bias Pulse Generato r Analog Oscilloscope C N T 1 k 100 Figure 6-1. Pulsed bias expe rimental set-up for sequential ablation of metallic CNTs.

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64 The low bias conductance of metallic and semiconducting CNTs in sample A measured after an ablation even t, are plotted as a function of the number of tube ablations in Fig. 6-2. The plot clearly shows the c onductance change of the metallic CNT matrix due to ablation, while the total conductan ce of the semiconducting tube matrix remains unchanged as expected. As metallic tubes are ablated successively, the total number of CNTs remaining becomes less and as a re sult the total conduct ance drops, which is confirmed by the measured data shown in Fig. 6-2, except after the first ablation attempt. The increase in conductance after the first ablation attempt appears to be the result of an annealing effect, which will be discussed later in the next section. From the difference in measured conductance before and after an abla tion the conductance of the ablated CNT is obtained. The calculated conductances of the ablated CNTs are presented in Fig. 6-3. From the data the average conductance of a single metallic tube is calculated to be 2.710-5S or 37k. No metallic or semiconducting CNTs were found to be connecting after the 7th ablation attempt; the most probable cause is static damage during the measurements. However, the total number of metallic CNTs in the sample can be estimated from extrapolating the fitted line in Fig. 6-2. It wa s estimated that the original sample contained 8 metallic tubes. Under ra ndom CNT growth conditions typically 1/3 of the tubes show metallic properties while 2/3 becomes a p-type semiconductor for the Pd contact system used in this experiment [41]. Consequently sample A is expected to have contained 16 semiconductor tubes for a total of 24 tubes originally. By measuring the linear conductance of the samples at Vg = -14V (semiconductor and metallic tubes both conducting) and at Vg = +14V (metallic tubes only) the conductance of the semiconductor tubes at Vg = -14V can now be estimated.

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65 0 5.0 10 -51.0 10 -41.5 10 -42.0 10 -42.5 10 -40 12345 6Number of Ablation Conductance (S) Metallic CNT Semiconducting CNT Figure 6-2. The measured low bias conduc tances of metallic and semiconducting CNTs in sample A, plotted as a function of the number of CNT ablation attempts performed on the sample

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66 For sample A, this resulted in 1.7 M per semiconductor tube. The spread in data displayed in Fig. 6-3 is in line with values reported in the literat ure and may be due to small variations in the Pd metal contact-CNT interfaces from sample to sample [44]. From (4-7) the measured 1/f noise of carbon nanotubes can be expressed as f I A f St t t 2) ( (6-2) where the excess noise factor At models the relative magnitude of the excess noise and is the frequency exponent. The value of At /Rt, where Rt is the total ohmic resistance of the sample, has been shown to be close to 10-11 -1 for single CNTs [7]. Assuming that the parallel CNTs in our samples produce noise independently, the total 1/f noise can be formulated as a combination of noise contri butions from the metallic and semiconducting CNTs; S j sj sj M i mi mi t t tf I A f I A f I A f S1 2 1 2 2) ( (6-3) where the subscript m and s denotes the metallic and semiconducting noise and current contributions, and, i and j subscripts represent the indivi dual metallic and semiconducting CNTs respectively. M and S are the total number of metallic and semiconducting CNTs in the sample. Simplifying (6-3), S j t sj sj M i t mi mi tI I A I I A A1 2 1 2 (6-4) Writing in terms of the total and individual metallic and semiconducting CNT conductances, S j t sj sj M i t mi mi tG G A G G A A1 2 1 2 (6-5)

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67 1.0 10-5 1.5 10-5 2.0 10-5 2.5 10-5 3.0 10-5 3.5 10-5 4.0 10-5 2 3 4 5 6 Number of Ablation Conductance (S) Figure 6-3. Calculated conductances of th e ablated metallic CNTs for sample A.

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68 where Gmi and Gsj subscripts represent the conductan ce of the individual metallic and semiconducting CNTs respectively. And Gt represents the tota l conductance of the device. From (6-5) S j sj sj M i mi mi t tG A G A G A1 2 1 2 2. (6-6) From (6-6) it is evident that the noise measurements at Vg = +14V represents the 1/f noise contributions for the metallic CNTs. B ecause at this gate bias, the semiconductor tubes are cut-off, GS 0, and presumably contribute negligible noise levels. This assumption was validated by the result of the analysis as it will be shown later in this section. From (6-6) it also follows that at each successive ablation of a CNT the measured 2 t tG A will decrease. The change in 2 t tG A represents the lost noise contribution (and conductance) from the ablated CNT. This successive drop in the measured2t tG A value is illustrated in Fig. 6-4 for Vg = +14V where the second term on the right hand side of (6-6) has a negligible contribution. The conducta nce of the ablated CNTs is known from the current-voltage measurements, so the 1/f noi se coefficient of the ablated CNT can be obtained from (6-6). For sample A, the calculated values for Am and Am/Rm are presented in Fig. 6-5. The average value of Am for sample A was found to be 210-5 resulting in Am/Rm = 4.910-10S. Note that the measurement data after the very first ablation was not used to calculate the average because of the observed annealing effect to be discussed in detail later. Analyzing th e noise data measured at Vg = -14V which includes semiconductor and metallic tube noise contributions resulted in As = 3.710-3 or As/Rs = 2.210-9S.

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69 At/Rt 1 10-10 1 10-9 1 10-8 0 1 2 3 4 5 6 Number of Ablation1 10-141 10-131 10-12At Gt 2 At/Rt At Gt 2 Figure 6-4. Measured At/Rt and AtGt 2 at Vg = +14V for sample A, plotted as a function of the number of CNT ablation attempts performed on the sample.

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70 1 10-6 1 10-5 1 10-4 2 3 4 5 6 Number of AblationAmi 1 10-101 10-91 10-8Ami/Rmi Ami Ami/Rmi Figure 6-5. Calculated A and A/R for the ablated metallic CNTs for sample A.

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71 Measurements were performed on sample B on the same wafer and on sample C on a different wafer. For both of the samples B and C, complete ablation of the metallic CNTs was achieved, so only semiconducti ng CNTs remained at the end of the experiment. A summary of the data is presented in Table 6-1 and the A vs. R plot for all the data points collected is presented in Fi g. 6-6. The values of the frequency noise exponent range between 0.8 and 1.2 for the samples studied. The measured A/R values almost all exceed the average 910-11 -1 value reported by Snow et al. [8]. From this we conclude that our samples are relatively noisy which may be attributed to the sputtering deposition of the Pd contacts. Also it is clear that in contrast to conventional metal and semic onductor systems where the excess noise of metals can be typically ignored, the excess noi se of metallic tubes needs to be accounted for. Also note that the extracted A values validate our initial assumptions made in the analysis of the data. Details of the Observed Annealing Effect As mentioned before, an annealing effect was observed in our experiment. The first time a sample was subjected to the pulsed bi as, the conductance of the sample increased while the noise level improved. The conduc tance improvement at the first ablation attempt is obvious from Fig. 6-2. The effect of the 1/f noise level improvement can be seen in Fig. 6-4. Note that 2t tG A decreases after the first ablation attempt even though the conductance Gt increases, which suggests a reducti on in the 1/f noise coefficient At. Furthermore the difference between the measur ed noise level before and after the first metallic CNT ablation was found to be roughly 2 to 4 times higher than the rest. This again suggests an improvement of nois e during the first ablation attempt.

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72 10-6 10-5 10-4 10-3 10-2 10-1 104 105106107R ( )A Sample A Sample B Sample C Figure 6-6. Calculated A vs. R values for the ablated CNTs The solid and open symbols represent the metallic and semiconducting CNTs respectively. The data points estimated from the remaining semiconducting CNTs after all the metallic CNTs have been ablated are indicated with a circle.

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73 So it appears that during the application of power to a virg in CNT sample an annealing effect is occurring due to Joule heating. This also raises the question whether the results for the individual CNTs presented are valid as the ablation process it self may change the property of the CNT being measured. To answer this question we turn our attention to the At/Rt plot shown in Fig. 6-4. The parameter At/Rt can be considered as a figure of merit for the comparison of relative noise magnitudes of CNT sample s [7]. The noise reduction at the first ablation attempt due to annealing is clearly visible from th e large change in the At/Rt plot. However after the very first ablation attempt the change in the At/Rt values lies within the statistical spread of our data and would not affect the result appreciably. Furthermore, the calculated parameters fo r the individual ablated CNTs conductances (see Fig. 6-3) and noise magnitudes (see Fig. 6-5) also do not show an appreciable trend over successive ablations. So we conclude th at the ablation process does not change the properties of the CNTs appreciably after the initial ablation. Summary and Conclusions The pulsed bias experiments were succe ssful in demonstrating the sequential ablation of single metallic carbon nanotubes. The conductance and relative excess noise factors of semiconductor and metallic tubes were determined showing that both produce 1/f like excess noise. However the relative 1/f noise level in the metallic tubes as expressed by the parameter A is two orders of magnit ude lower, and the ratio A/R is between a factor 3 to three orders of magnitude lower than in the semiconducting CNTs. There is a large spread in the resistance of both metallic and semiconducting CNTs in all three samples measured (see Table 6-1). However both A and the factor A/R, used as a figure of merit for noise comparison is cons istently higher for semiconducting CNTs in all cases. In two instances during ablation attempts on sample B and C, it was observed

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74 that semiconducting CNTs were ablated along with metallic CNTs. The most likely reason for this is static damage caused by connecting the devices to the measurement equipment as mentioned earlier. This uni ntended ablation of semiconducting CNTs provided an opportunity to measure the resi stance and the noise contribution of the ablated CNT directly, rather than calculati ng the resistance and noi se contribution from the remaining semiconducting CNTs after al l the metallic CNTs were ablated. The resistance and the noise c ontribution for these semiconducting CNTs were found to be within the same range as the semiconducting CNTs remained after metallic CNTs were ablated. Sample C was from a different wafer than sample A and B. Sample C was annealed in Ar at 220C for 10 minutes, while sample A and B were not annealed. The resistance of the metallic CNTs in sample C is higher compared to the samples A and B. However it is possible that this higher resi stance is not caused by annealing but simply results from process variations from wafer to wafer. Furthermore, the resistance of the semiconducting CNTs for sample C is with in the range of resistances of the semiconducting CNTs in sample A and B. The 1/f noise level in sample C does not show an appreciable change from the other two samples. Hence the thermal annealing process used did not have an effect on the noise properties of CNTs. The fact that the annealing caused by the J oule heating due to current flow in the device improves the conductance and lowers the 1/f noise leve l points to the contact as the dominant source of 1/f noise. Because the CNT operates as a one-dimensional conductor, most of the power is dissipated in the contact region [45], and an improvement of the quality of the contact between the CNT and Pd cau sed by the Joule heating is

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75 evident from the lowering of the contact resi stance. Similarly, the improvement in the noise level can be attributed to annealing out defects in the contact area. An annealed contact is likely to have a lower number def ect related traps, hence results in a lower noise level.

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76 Table 6-1 Noise and conductance data of samples A, B and C. Device Data Sample A Sample B Sample C Number of Metallic Tubes 8 3 2 Estimated Number of Semiconducting CNTs 16 6 4 Estimated Total Number of CNTs 24 9 6 Rm Average 37k 103k 4.6M Std. Dev. 12k 44k Rs Average 1.7M 189k 498k Std. Dev. 180k 398k Am Average 2.010-5 1.910-5 1.110-4 Std. Dev. 1.910-5 1.310-6 Am/Rm Average 4.910-10S 2.010-10S 2.310-11S Std. Dev. 3.110-10S 7.410-11S As Average 3.710-3 3.810-3 2.710-2 Std. Dev. 5.510-3 1.310-2 As/Rs Average 2.210-9S 1.410-8S 1.210-7S Std. Dev. 1.110-8S 1.310-7S

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77 CHAPTER 7 LORENTZIAN TYPE OF NOISE IN CARBON NANOTUBES Introduction to the Lorentzian Noise Component An overview of the noise properties of CNT devices was presented in chapter 4. In Fig. 4-4 a measured plot of th e current noise spectral density (Si) was given. A closer examination of this plot reveals the pr esence of Lorentzian noise components superimposed on the 1/f like noise. The pr esence of a Lorentzian component becomes clear when the fitted 1/f like noise is subtracted from the measured excess noise as shown in Fig. 7-1. Lorentzian components arise when rando m number fluctuations are caused by processes with a single characteristic time and activation energy. A Lorentzian component can be characterized as [13] 2 21 ) 0 ( ) ( L LS f S (7-1) where SL(0) is the plateau value of the Lorentzian, is the angular frequency and is a characteristic time. The plateau values and th e characteristic times can be obtained from fitting the curve given by equation (7-1) to th e data. The fitted curve is shown as a dotted line in Fig. (7-1). The Lorentzian(s) that showed up at a pa rticular temperature shifts to down vertically and to a higher frequency as shown in Fig.7-2. This movement of a particular Lorentzian over temperature can be tracked until it moves out of the frequency range of measurement.

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78 10-18 Frequency (Hz) 10 10-22 10-24 102103104105 Excess Noise ( A2/Hz ) 10-20 Lorentzian (A2/Hz)10-2410-2210-2010-1810-16 (a) (b)Figure 7-1. Extraction of Lore ntzian spectra from excess noise; (a) a typical excess noise plot (top) with fitted 1/f noise line, and (b) bottom plot shows the Lorentzian spectra obtained after subtracting 1/f noise from the excess noise.

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79 log(f) log (Si) Lorentzian at T1 Lorentzian at T2 T1 < T2 1/f noise Lorentzian shift with increasing T Total noise Figure 7-2. A diagram describing the shift of Lorentzian spectra with temperature.

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80 Results and Analysis The noise measurement was repeated over a temperature range of 77K to 300K on two devices (A and B). Several Lorentzians were observed and tracked across this temperature range and their corre sponding characteristic times () were extracted. The calculated values for at different temperatur e are shown in Fig. 7-3 for devices A and B, respectively. Note that the hor izontal axis is 1000/T, where T is temperature in Kelvin. From the plot, it is evident that the characteristic times increase with decreasing temperature and are clearly thermally activated i.e., ) / exp(0T k EB (7-2) where E is the activation energy and kB is the Boltzmann constant. This equation is well known to explain the characteristic times of thermally activated generationrecombination noise component s in semiconductors with E = ET -EF, where ET is the trap or defect energy level to which the carrier is activated and EF is the Fermi level [13]. The activation energies calculated from the plot sh own in Fig. 7-3 are lis ted in table 7-1. The position of EF in the semiconducting carbon nanotubes is determined by the workfunction difference of the Pd /CNT contact system placing EF in these devices near the valence band edge, independent of temperature [44]. As a result the values listed in table 7-1 can be interpreted as hole activation en ergies for carrier de trapping from different defect or trap centers. Figure 7-3 also shows that at sufficiently low temperatur es the characteristic times tend to reach a constant value. This trend is expected and originates from the mechanism of the carrier trapping and de trapping process itself.

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81 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 2 4 6810 12 141000/K (Sec) I-A I-B II-A III-A IVA II-B IV-B III-B Figure 7-3. Lorentzian characteristic times for device A and B shown in solid and dashed lines respectively. For sample A, the ac tivation energies of the Lorentzians are 0.31, 0.27, 0.21 and 0.08 eV for Lo rentzians I-A, II-A, III-A and IV-A, respectively. For sample B, the activ ation energies are 0.42, 0.51, 0.2 and 0.29 eV for Lorentzians I-B, II-B III-B and IV-B, respectively.

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82 The measured characteristic time is the reciprocal sum of the carrier capture time c, and the emission time e i.e., c e 1 1 1 (7-3) e is a strong function of temperature because the energy available to the trapped carrier is a function of temperature. At a higher temper ature, it is easier fo r the trapped carrier to acquire the energy required to come out of the trap and as a result, the time a carrier spends in the trap on average decreases, hence e is lower at higher temperatures. Conversely, c is relatively independent of temper ature [39] because the trapping process is a random process dependent on the capture cross-section of the trap and trap availability. At higher temperature, e <>c, so c dominates and becomes independent of temperature. From data it was observed that the plateau values of the Lorentzians SL(0), have a linear relationship with that can be described as 0LS. (7-4) For device A, the value for was found to be 3.110-17. For device B, the noise measurement at each temperature was perf ormed at the gate voltage levels of Vg = 0 and +14V. In Fig. 7-4, a plot of the SL(0) vs. measured at different temperatures is presented. From the plot it is apparent that the linear relationship described in (7-4) is also valid here. For this case the estimated values for are 3.810-17 and 4.710-17 for Vg = 0 and +14V respectively. This linear re lationship suggests that the variance of the number of carriers N2 is a constant [13].

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83 Table 7-1 Activation energies calculated using equation (7-2). Activation energies in eV Device A 0.08 0.21 0.27 0.31 Device B 0.2 0.29 0.42 0.51

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84 10-23 10-22 10-21 10-20 10-19 10-18 10-17 10-16 106 105 104 103 102 101 (sec)SL(0) (sec) Plateau Value Vg = 0V Fitted data Vg = 0V Plateau Value Vg = 14V Fitted data Vg = 14V Figure 7-4. A plot of the Lore ntzian Plateau value vs. charact eristic time for device B at Vg = 0 and 14V. The linearly fitted data is also shown.

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85 (sec) SL(0) (sec) 10-6 10-5 10-4 10-3 10-2 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 Vg (V) 10-22 10-21 10-20 10-19 10-18 10-17 Lorentzian 1: Characteristic Time Lorentzian 2: Characteristic Time Lorentzian1: Plateau Value Lorentzian2: Plateau Value Figure 7-5. A plot of the Lorent zian Plateau value and characteri stic time at different gate voltages for device A.

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86 Effect of Gate Voltage Figure 7-4 and the results discussed above also suggest that th e gate voltage does not change the Lorentzians a ppreciably. To verify this observation, sample B was measured at different gate voltages at r oom temperature (approximately 300K). The results of these measurements shown in Fig. 7-5, do not show a clear dependence on gate voltage; thus confirming our previous observa tion. This observation is important and may help pin point the physical location of the noise generating trap center. First note that the Lorentzians are visible from Vg = -14V to +14V. In a conventional device Lorentzians are associated with semiconducting materials, since in a metallic system, because of the presence of a large number of carriers, an individual trap center does not produce a noticeable effect on the transport characteristics. However, in this case the Lorentzian is clearly associated with a metallic CNT as at Vg = +14V, where a semiconducting CNT would not conduct and he nce an associated Lorentzian will not show up. The calculated activation ener gies for the noise component s are within the range of energies of physical processes possibly pr esent in the carbon nanotubes studied. For example, the fluctuations caused by carrier transitions between one-dimensional subbands will have activation energies equal to the energy difference between sub-bands. The sub-band energy differences in carbon nanotubes can be close to the discrete activation energies presented in table 7-1. Also, it may be po ssible that the noise originates from local defect centers due to ad sorption of a chemical species. It has been reported that oxygen chemisorption on the su rface of a semiconducting carbon nanotube lowers the energy band gap. For example on a (8,0) SWNT the energy gap decreased

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87 from 0.56eV to 0.23eV due to oxidation [46]. Local defects of this nature may produce noise with activation energies in the range reported in this work. However, the gate voltage insensitivity sugge sts that the defects associated with the Lorentzians are located at or very close to the contact region of the CNT device since the band bending due to the gate voltage is minimum near the contact region of a semiconducting CNTs. As shown in Fig. 3-4 th e energy band bends due to the application of gate voltage and as a result the quantity E (= ET -EF) described in equation (7-2) will change. The change in E is not uniform across the device. It has a maximum at the center of the device and a minimum near the contac ts. For a large change in gate voltage one would expect a sign ificant change in E and a noticeable change in via equation (7-2). In the absence of any such effect on we postulate that the defects that generate the observed Lorentzians are not lo cated at the bulk region but are located at or near the contact region of the CNT device.

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88 CHAPTER 8 ENERGY BAND SIMULATION OF CARBON NANOTUBES Introduction From the gate bias insensitivity of the measur ed Lorentzian noise reported in chapter 7, the physical location of the noise generating trap centers was identified to be most likely the metal-CNT contact region. Similarly, the relationship between the observed 1/f noise and the gate bias level may help to identify th e physical location of the associated traps as well. Although the fluctuation process associated with the 1/f noise does not have a single activation energy like Lorentzian noise, but instead interacts with a distribution of activation energies, it will be shown in the ne xt chapter that the plot of the calculated distribution of activation ener gies shows a certain energy shift between two different levels of gate bias. Because the band-bendi ng of a CNT due to the applied gate bias varies over the length of the de vice, it may be possible to re late the observed shift of the activation energy distribution to the band-bending due to the gate bias change and thus pin point the physical location of the associated trap centers In order to do this, the profile of the CNT energy band at a specified gate bias, is necessary. Simulation can provide an estimate of the band-bending due to the work-function di fference between the CNT and the metal contact and also due to th e application of the gate voltage. Over the last few years a number of simulation st rategies were proposed and validated by experimental results [47-51].

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89 E D(E) D ( E ) F ( E si g n ( E )) E E E x = x1 EF F(E) F ( E si g n ( E-Ei )) q si g n ( E ) D ( E ) F ( E si g n ( E )) (b) E D(E-Ei) F(E) D(E-Ei) F(E sign(E-Ei)) E E E x = x2 EF F(E sign(E-Ei)) Ei q sign(E-Ei) D(E-Ei) F(E sign(E-Ei)) (c) (a) E Metal Contact W EC EV Fermi Level Ei(x) x CNT x = x2x = x1 q (x) Intrinsic Fermi Level Figure 8-1. Explanation of the charge calcu lation using equation (8-1). (a) Symbol definition and step by step graphical repr esentation of factors appearing in (81) for two cases; (b) the intrinsic and the equilibrium Fermi levels are the same and (c) when the intrinsic Fermi level is higher than the equilibrium level.

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90 Self Consistent Solution of Char ge Density and Poisson’s Equation The two basic steps are (1) the calculation of charge density a nd (2) the calculation of potential. First the procedure for the charge density calculation will be described. Calculation of Charge Density The one dimensional charge density (x) is obtained from integrating the product of local density of states (DOS) and Fermi func tion over energy. The process is explained in Fig. 8-1 and can be expresse d mathematically by [48], dE E sign x E E F E sign E D q xi) ( ) ( ) ( ) ( ) (. (8-1) The unit of (x) is in coulomb/m. Here the axial direction of the CNT is assumed to be the x direction. The symbols used are described bellow, -q is the electron charge in coulomb. E is the variable denoting energy measured from the equilibrium Fermi level in J. D(E) is the one dimensional density of state of CNT in (J-m)-1. Ei(x) is the so called charge neutrality level measured from the Fermi level. In electrical engineering term inology it can be understood as the intrinsic Fermi level referenced from the equilibrium Fermi level [52]. sign(E) is the sign function defined as -1, for E < 0, sign(E) = 0, for E = 0, (8-2) = 1, for E > 0. F(E) is the Fermi function defined as ) / exp( 1 1 ) ( T k E E FB (8-3)

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91 where, kB is the Boltzmann constant and T is temperature in K. The density of states (DOS) is given by [47-48,53] n n nE E E E D 2 2) ( 4 ) ( (J-m)-1 (8-4) where the symbols are defined as follows, is the band parameter given by [29], 2 30 a J (8-5) where a = 0.24 nm, is the graphite lattice constant and 0 = 2.6 eV, is the nearest neighbor C-C tight binding overlap energy [29,48]. n is the wave number given by [48,53] where, R is the CNT radius. In the presented work R = 0.5 nm is used. (E) is the step function defined as The DOS plots for semiconducting and metallic CN Ts using (8-4) are presented in Fig. 82 Note that the half band-gap eV R 37 0 3 and the CNT radius R are used as the unit for energy and length respectively to give the reader an intuitive feel for the magnitudes involved. (E) = (8-7) 1, for E 0, 0, for for E < 0. R n 3 / 1 for semiconducting CNT R n, for metallic CNT n = (8-6)

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92 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20 25 30 DOS (eV-1 R-1) E ( ) (a) -10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20 25 30 35 40 45 E ( )DOS (eV-1 R-1) (b) Figure 8-2. Density of state plots calculated using (8-4); (a) for a semiconducting CNT (b) for a metallic CNT. The unit of the energy axis is the half band-gap,

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93 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 101 10 0 10 1 10 2 Ei (eV) Net carrier density (R-1) Metallic with higher sub-bands Semiconducting with higher sub-bands Metallic with lowest sub-bands Semiconducting with lowest sub-band Hole Electron Figure 8-3. A plot of the calculated density of carriers per unit length for metallic and semiconducting CNTs at T = 300K. Note th at the CNT radius is used as the unit of length. For the metallic CNT, at low band-bending (| Ei| < 1eV) a constant DOS approximation yields suffici ently accurate result. Similarly, for a semiconducting CNT, the calculation with only the lowest sub-band is sufficiently accurate up to | Ei| < 0.7eV.

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94 The one-dimensional carrier density plot calc ulated using (8-1) is presented in Fig. 8-3. Note, that the presence of the band-gap is clear for the semiconducting CNTs. Also it is evident from the plot that for a small band-bending, i.e. for a small Ei it is sufficient to take into account only the low est sub-bands. The DOS for th e metallic CNT considering only the lowest sub-bands with the low temp erature approximation becomes a constant and is given by ME D 4 ) ( (J-m)-1 (8-8) In this case, from (8-1) the charge density (x) becomes linearly related to Ei and is given by ) ( ) ( x E q xi M coulomb/m. (8-9) This simplifies the calculation for metallic CN Ts because this lead s to a closed form solution for the charge density and the corresponding band-bending, as will be shown later. This closed form solution matches very closely with the iterative solution scheme for T = 300K and with the higher sub-bands in cluded, hence the closed form solution for the metallic CNTs was used. The calculation fo r the semiconducting CNTs is not straight forward even with the lowest sub-ba nd and requires an iterative solution. Calculation of Potential The potential on the surface of the CNT is governed by Poisson’s equation. The total potential (x) includes the contribution from the potential due to th e charge on the CNT ()(x) and the potential due to the gate voltage (g)(x) i.e. ) ( ) ( ) ( x x xg V. (8-10)

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95 ()(x) and (g)(x) are related to the charge density and the gate potentia l respectively via the following Fourier space relationships [47-48]; k k kU (8-11) And, k k g kM (8-12) where the symbols are defined as follows; k () is the Fourier transform of ()(x) k (g) is the Fourier transform of (g)(x) k is the Fourier transform of (x) k is the Fourier transform of the gate potential profile, Vg(x) = Vg(0). Note Vg is the gate voltage in volts and (E) is the step function defined in (8-7). Uk is the cylindrical Poisson’s kernel re lating the charge density to the potential given by [47] ) ( ) ( ) ( ) ( ) ( 20 0 2 0 0 0 s s kkR I kR K kR I kR K kR I U (8-13) where is the dielectric consta nt of the oxide layer in Gaussian units given by 04 r. 0 is the free space permittivity, and r = 3.9 is the relative dielectric constant of SiO2. I0 and K0 are the modified Bessel function of the first kind. R and Rs are the CNT radius and the ga te oxide thickness respectivel y. In our calculation 0.5 nm and 600 nm is used for R and Rs respectively. Mk relates the gate potential to the pote ntial on the CNT surface, given by [47], ) ( ) (0 0s kkR I kR I M (8-14) k is the wave number given by

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96 L n k (8-15) where L is the device length and n is an integer. For the devices used in the study; L = 1 m. Note that the Poisson’s kern el described above is for cy lindrical geometry but the device structure used in our experiments was planar. Consid ering the large gate oxide thickness and the order of accuracy sought, this approximation was considered sufficient. Self Consistent Solution Method The intrinsic Fermi level and the potential ar e related due to the conservation of the total electron energy, i.e. W y q x Ei ) ( ) ( (8-16) where W is related to the work-function diffe rence between the CNT and the contact metal. This relationship is depicted graphica lly in Fig. 8-1(a). Note that in our work W is assumed to be equal to the work-function difference between the CNT and the metal, which is only true for an ideal contact. We believe that this assumption of an ideal contact is sufficient as we are only interested in an order of magnit ude estimate. Also the work-function of the CNT is assumed to be 4.66 eV, i.e. equal to the graphite workfunction [54-55], and the work-functi on of Pd is 5.1 eV [34]. To enforce the boundary condition (0) = 0 ; we use the basic idea from Odintsov et al. [47], where, anti-symmetric sources were used. In this method instead of W Wsign(x) and instead of the gate voltage Vg, Vgsign(x) is used and the region of space –L x L is solved for. Although this method forces the correct boundary condition, it requires solving a larger space (2 device length). To avoid this limitation we have used a

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97 different approach. Note that the use of th e anti-symmetric sources has the effect of forcing the cosine terms in a Fourier series to zero. Hence, we use the following sine series expansion of the Fourier series and solve for only the space 0 x L ix i i kkx x N) sin( ) ( 2 (8-17) where xi is the discretized position vector and N is the total number of grid point. The inverse Fourier tr ansform is given by k i k ikx x ) sin( ) ( (8-18) The simulation results presented by Odintsov et al. [47] were accurately reproduced by the simplified approach described above. Th e matrix size can be further reduced by observing that the charge, pot ential and energy profiles ar e symmetric across the device at zero drain bias, which means that all the even terms of the Four ier expansion will be zero and thus only the calculation of the odd terms is sufficient. First, the closed form solution for the metallic CNT considering only the lowest sub-band will be described. Taking the Fourie r transform of (8-16) and using (8-10) g k k k kq W q E (8-19) where Ek and Wk are the Fourier transforms of W(x) and Ei(x) respectively. Using (8-11) and (8-12) k k k k k kqM W qU E (8-20) Using (8-9) we obtain the closed form solution for Ek; k M k k k kU q qM W E21 (8-21)

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98 With all the quantities in the right hand side of the expression known, Ei(x) can be calculated by evaluating (8-21) and then taking the Fourier transform. For the iterative solution we start from (8-16). Using (8-10) and re-arranging the terms, x q x q W x Eg i ) (. (8-22) Using (8-11) and (8-12) x F U F q M F q W x Ek k k i 1 1) ( (8-23) Since (x) is a function of Ei(x) (8-23) can be written as the familiar vector equation, a form suitable for iteration, i iE E f (8-24) where Ei is the discretized vector form of Ei(x) Now (8-24) can be solved iteratively. The flow chart of the solution scheme used is presented in Fig. 8-4. Results and Discussions The calculated band-bending for a metallic CNT at a gate voltage of 0 and +14V are shown in Fig. 8-5. Note that the noise measurements were performed at very low drain-source bias condition; he nce, in the simulation the dr ain-source voltage is ignored i.e. Vds = 0V. Fig. 8-5 and 8-6 both show that the band-bending reaches its maximum level at the middle of the devi ce as expected. The gate coupli ng is relatively weak as the maximum band-bending is only ~0.7eV for Vg = +14V. The reason for this weak coupling is the large gate oxi de thickness (600 nm) of the device. The plots show two distinct regions which agrees with the obs ervation made by other researchers [47-51]. Near the contact the change in Ei is abrupt, and then from x 10 nm the changes occur at a much slower pace.

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99 Calculate potential using Using (8-11) to (8-12) If En Tolerance STOP Y N Calculate Fourier Transform: q q Calculate Charge using (8-1) Input: Initial Guess for Ei n START Calculate error: Ei n = Ei n+1 Ei n Calculate new guess value: Ei n = Ei n Ei n < 1, step size for convergence Calculate Ei n+1 using Using (8-16) Figure 8-4. Flow-chart for the iterative solution scheme.

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100 These two regions arise because in the middl e of the device the band-bending is mostly gate controlled and near the contact the effect of the gate is diminished and the effect due to the charge accumulation near the contact dominates. Also note that the use of the closed form expression to calculate the bandbending is validated by the results obtained, the use of the lowest sub-bands are su fficient enough for the small band-bending produced by the highest Vg = +14V. The carrier density and the band-be nding for a semiconducting CNT at Vg = -14V was simulated using the iterative scheme show n in Fig. 8-4. The calculated band profile is presented in Fig. 8-6, and the resulting 1D charge density is also shown. Note that as expected, the charge density is non zero only wh en the equilibrium Ferm i level is close to the band-edges. Also note that the band-be nding near the contact for the semiconducting CNT is much more pronounced than the metallic case. This is to be expected as due to the presence of a band-gap more bending is ne cessary to obtain the equivalent amount of charge. In the next chapter the application of the energy band calculation to identify the physical location of the 1/f noise sources will be presented.

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101 0 200 400 600 800 1000 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Position (nm)Ei (eV) Vg = 0V Vg = +14V (a) (b) 80 0 20 40 60 100 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Position (nm)Ei(eV) Vg= 0V Vg= +14V 80 0 20 40 60 100 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Position (nm)Ei(eV) Vg= 0V Vg= +14V Figure 8-5. The plot of Ei vs. position at Vg = 0 and +14 V for a metallic CNT; (a) over full device length, L = 1 m, and (b) close up near the left contact.

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102 Position (nm) -2 -1 0 1 0 100 200 300 400 500 600 700 800 900 1000 -2 -1 0 1 Carrier density, (nm-1)Energy(eV) Ei ECEV EiEC EV Position (nm) -2 -1 0 1 0 100 200 300 400 500 600 700 800 900 1000 -2 -1 0 1 Carrier density, (nm-1)Energy(eV) -2 -1 0 1 0 100 200 300 400 500 600 700 800 900 1000 -2 -1 0 1 Carrier density, (nm-1)Energy(eV) Ei ECEV EiEC EV Figure 8-6. Band-bending at Vg = +14 V for a semiconducting CNT. Ec, Ev and Ei are the conduction band edge, valence band edge and the intrinsic Fermi level respectively. The carrier density is also shown.

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103 CHAPTER 9 THERMALLY ACTIVATED 1/F NOISE IN CARBON NANOTUBES Introduction A brief introduction to the 1/ f noise measurement was presented in chapter 2. In the beginning of this chapter the results and analysis of the 1/f noise measurement over temperature will be presented. Later an es timate of the physical location of the traps centers associated with the measured noise based on the energy band simulation discussed earlier will be presented. The Lorentzian noise components of the two devices (devices A and B) to be discussed here were presented earlier. In th is chapter the focus is on the 1/f noise these devices produce. These devices were current biased at 13.9 A during noise measurements. This bias point was well within the linear range of device operation at all temperatures used in our experiments. For device A, the resistance changed from 0.98 k at 77K to 1.04 k at 300K and for device B, from 0.83 k at 77K to 0.91 k at 300K. This rather weak dependence of device resist ance on temperature seems to indicate that a strong electron-phonon coupling is absent; lending support to a ba llistic charge transport model with tunneling metal-to-nanot ube contacts as proposed by Javey et al. [44]. The observed 1/f noise was calculated by subtracting the Lorentzian and the thermal noise components from the total noise as explained in Chapter 4. The 1/f noise can be expressed by Eq. (4-7) repe ated here for convenience [39] f I A f Sf 2 / 1) ( (9-1)

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104 where I is the dc current magnitude, f is the frequency, and A and are constants. The values of A and are estimated by fitting a line to the e xperimental data as shown in Fig. 4-4. The parameter was found to be close, but not exactly equal, to 1.0, as expected for 1/f-like noise. This is an indication that number fluctuation as opposed to mobility fluctuation noise dominates the measured sp ectra since Hooge type mobility fluctuation noise strictly requires a = 1.0 value for Hooge’s equation to be valid [37]. The parameter A represents the relative magnitude of the 1/f noise. As mentioned earlier, for CNTs, the parameter A / R was shown to be close to a constant of 10-11 S and can serve as a figure of merit for the comparison of 1/f noise between different samples [7,8]. The values of A/R over all temperatures of our experiment range from 1.3 10-11 to 7.6 10-11 S, which is within the range of values reported. The experimental values of A and over temperature are shown in Fig. 9-1. Calculation of the Distribution of Activation Energies Unlike the Lorentzian noise, 1/f noise resu lts when the fluctuation process does not have a single activation ener gy, but instead interacts with a distribution of activation energies. The plot of A vs. T presented in Fig. 9-2 shows p eaks and valleys, that can be attributed to the non-uniform di stribution in energy of the act ivation energies associated with the fluctuation processes. The distribution of th e activation energies D(E) can be obtained from the 1/f noise magnitude and frequency dependence using [56] ) ( ) (/ 1T S T k E Df B (9-2) where is the angular frequency.

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105 10A 0.8 1.0 1.2 50100150200250300350Temperature (K) -810-7 10A 0.8 1.0 1.2 50100150200250300350Temperature (K) -810-7 Figure 9-1. Measured 1/f Noise coeffi cient over temperature for sample A.

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106 The energy E is calculated from ) / exp(0T k EB (9-3) with 0 = 10-14 sec. Note that E is rather insensitive to the value used for 0. The calculated D(E) vs. E plot is shown in Fig. 9-2. The distribution of activati on energies also shows distinct peaks and valleys as expected from the A vs. T plot. The plot for sample B shows a broad peak around 0.45 eV, which indicates that for this sample there are a higher number of defects in this energy range. Note that activ ation energies for the Lorent zian components for sample B includes 0.42 and 0.51 eV which roughly corresponds to the peak shown by the D(E) vs. E plot. For sample A, the activation energies calculated from the Lorentzian components are lower than in sample B and r oughly correspond to the peak in the D(E) vs. E plot as well. Please refer to chapter 7 for the activation energies of the Lorentzian components. The frequency exponent can also be calculated from the magnitude of the 1/f noise using [56] 1 ) ln( ) ( ln ) ln( 1 1 ) (/ 1T T S Tf (9-4) The calculated and measured values of are presented in Fig. 9-3. The agreement between the calcul ated and measured values of is very good considering the fact that the values are calculated from the slopes of the A vs. T plot which has only a limited number of measured points. This validates the results for the energy distribution calculation

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107 E (eV) E (eV) 0 1 2 3 4 5 6 0.10.20.30.4 0.50.6 0.7D(E) (arb. Units) Sample A Sample B 0 1 2 3 4 5 6 0.10.20.30.4 0.50.6 0.7 Sample A Sample B Figure 9-2. Distributions of activation energies for sample A and B.

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108 0.6 0.8 1.0 1.2 1.4 50 100 150 200 250 300 350T (K) (Measured) (calculated) 0.6 0.8 1.0 1.2 1.4 50 100 150 200 250 300 350T (K) (Measured) (calculated) Figure 9-3. Measured and cal culated frequency exponent vs. T for sample A

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109 Note that the agreement be tween the plots is worse at around 280K, the reason for this is again the limited number of points from which the plot is generated from. Both samples showed similar range of activation en ergies (Fig. 9-2) and the calculated energy range is within the range of charge trans port and generation-recombination processes in CNTs. Physical Location of the Trap Centers The interpretation of the obs erved excess noise is further complicated by the fact that carbon CNTs operate as mesoscopic quant um devices, where traditional methods of noise measurements and interpretation may not readily apply. For example, the common 4-point probe measurement method to elimin ate contact noise from the measurements will not be conclusive for a CNT device. The source and drain contact reservoirs are an integral part of a nanotube de vice and the addition of two more contacts may completely change the operation of the device itself [44]. A possible source of noise suggested in the literature [7-8], is the interface between the supporting oxide layer and the nanotubes. To check this possibi lity further, the measurement was repeated on a sample (C) containing 54 nanotubes grown under identical conditions as samples A and B, ex cept that the oxide be neath the nanotubes was etched away so that the nanotubes were suspe nded in air and were not in contact with the oxide layer. No change in the magnitude of the 1/f noise was observed. The value of the factor A/R for the suspended sample was 4.7 10-11S, of the same order as the nonsuspended sample. This rules out the oxide interface as a dominant source of excess noise.

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110 0 1 2 3 4 5 6 7 8 9 0.10.20.30.40.50.60.7E (eV)D(E) (arb. units) Vg = 0V Vg = 14V Vg = 14V shifted by 0.04eV 0 1 2 3 4 5 6 7 8 9 0.10.20.30.40.50.60.7E (eV)D(E) (arb. units) Vg = 0V Vg = 14V Vg = 14V shifted by 0.04eV Direction of shift Figure 9-4. D(E) vs. T plot calculated for Vg = 0 and +14V. The trace for Vg = +14V shifted down by 0.04eV is also shown

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111 In Fig. 9-4, the calculated D(E) vs. E plot is for shown for two different gate voltages, Vg = 0V and +14V. Note that both plot s show similar features. Since at Vg = +14V, the semiconducting CNTs do not conduct, it can be concluded that the energy distribution shown in the plot is associated with the metallic CNTs. A closer inspection of the plot also reveals that the trace of D(E) for Vg = +14V if shifted down by about ~0.04eV, matches with the trace of D(E) for Vg = 0V. This indicates that the both traces, i.e. for Vg = 0V and +14V, corresponds the same metallic CNT(s) and are associated with the same trap centers. In this cas e, the shift in the energy distribution D(E) can be explained by the band-bending due to the applic ation of the gate voltage. Now, from the simulation results for the metallic CNTs presented in chapter 8, an estimate of the bandbending can be obtained. The plot of the charge neutrality level, Ei (or the intrinsic Fermi level) for the gate bias levels of Vg = 0V and +14V is shown in Fig. 8-5. Ei the difference of Ei between these two gate bias le vels is shown in Fig. 9-5. Ei is defined as, V V E V V E Eg i g i i14 0 (9-5) The plot of Ei shown in Fig. 9-5 reveals that the 0.04eV band-bending occurs at a distance of only ~17nm from the contact. So it can be concluded that the defects associated with the measured 1/f noise level are physically located near the contact and thus the bulk origination of the observed noise can be ru led out. The contact region is expected to be defect rich, furthermore, due to the 1D nature of CNT, there is a limited availability of electr on states in the CNT. As a resu lt the carrier transport through the Schottky barrier at the contact will be heavily modulated by th e presence of a trap centers in this region. This is may explain the high level of noise observed in CNT systems.

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112 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Position (nm)Ei (eV) (17nm, 0.04eV) Figure 9-5. The difference of Ei due to the change in Vg = 0V to +14V vs. position. The physical location at which the band-bending is equal to 0.04eV is also marked.

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113 Since, a positive Vg would shift the band down (pl ease see Fig. 8-5), we can conclude from the positive shift of D(E) for a positive Vg that the traps associated have lower energy than the Fermi level. Note that the defect density distribution becomes high at very low values of E This means that the most of th e defects are concentrated at an energy level close to the Fermi le vel. Although the maximum of the D(E) plot is outside of the range of the measurement range, from the plot we can estimate that the difference between the two peaks to be around ~ 0.4eV (the first peak is the maximum close to the origin and the second one bei ng the broad peak around 0.4eV) This difference is once again within the ra nge of the physical proce sses in the CNT system (f or example, the half band-gap, = 0.37eV, please see Fig. 8-2). It may be possible that these two peaks correspond to the energy levels formed due to contact formation or local defect centers due to adsorption of a chemical species. Fo r example, it has been reported that oxygen chemisorption on the surface of a (8,0) SWNT the energy gap decreased from 0.56eV to 0.23eV. However, our observation of annea ling effect during CNT ablation, linked the quality of the contact to the 1/f noise level. He nce, we identify the defects associated with the formation of the contacts to be the most likely source of the observed noise phenomenon.

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114 CHAPTER 10 CONCLUSIONS Summary of Results and Conclusions In the previous chapters the results and analysis of the excess noise measurements on silicon nanowires (SNW) and carbon nanotube s (CNT) were presented. In the SNW case, the excess noise was observed to have a 1/f like spectrum and the Hooge parameter, H was found to be of the order of ~10-5, which is in line with modern state of art silicon processes and shows the high qua lity of the devices measured In the case of the CNTs, Lorentzian spectra were observed superimpo sed on the 1/f noise. Recently published data estimates the H for CNTs to be ~2 10-3 [16-17]. Although this estimate is smaller than the previous estimate of 0.2 [7], it is still tw o orders of magnitude larger than the SNWs. The SNW technology is based on modern Si pr ocesses, which is considered a mature technology. The Hooge parameter in this case can be thought of as an indicator of the level of maturity of a particular technology. The Hooge parameter for mainstream silicon technology evolved from a magn itude of the order of ~10-2 in the 1980’s to the value of ~10-5 today. This noise reduction was achie ved by improvements in the material processing techniques that resu lted in defect lean materials and by well controlled and improved device fabrication processes. In comparison, the CNT technology is a novel one. It is very likely that as the CNT t echnology progresses towards maturity from its current nascent state, the noi se figure of merit will c ontinue to improve. However a question can be raised about th e validity of the Hooge parameter as an effective figure of merit for the noise comparison in 1D systems like CNTs. First,

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115 consider the fact that the resi stance in a CNT is mostly at the contact, the bulk resistance is generally comparatively small. Since, the ratio A/R for a CNT is close to a constant [7], two different length devices with identical A and R and bias condition would yield different values for H, because the total number of carriers might be different. The Hooge parameter should be thought of as a meas ure for the “per carrie r” noise level. So, the physical significance of H is diminished unless all the carriers are equally affected by the noise generating trap centers. Hence, for a fair comparison let us compare the 1/f noise level in CNTs to the contact noise of the SNWs. The quantity A/R for the contact region of the SNW ( i.e. Ac/Rc) from our noise model (presented in chapte r 5) is also a constant, and from (5-20) the values for wafer 1 and 2 are 2.8 10-13S and 2.7 10-14S respectively. Comparing with the published value of A/R = 10-11S for CNTs, it is apparent that the SNW has at least two orders of magnitude better noise perfor mance than a typical CNT device. Which may be attributed to better qualit y contacts with a lower defect density at the contact for a SNW. As mentioned earlier the H for the bulk noise component, is in the range of Hooge parameters for modern low noise bulk devices [40]. Our research identified the contact region as the origin of th e dominant source of noise in CNT devices and also in SNWs. No te that a defect rich contact not only increases the noise level but al so likely adds to the total resistance as well. This may explain the observed invariance of the A/R ratio for both CNT and SNWs. At this time the CNT technology is still nascent and the contact mechanism between the metal and the CNT is not well characterized. Because the co ntact was identified as the dominant source of noise, further noise reduction can envisi oned by optimizing the contact. Contact

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116 optimization may have a dramatic effect on noise because not only may it reduce the noise generating mechanism, it also may help reduce the contact resistance as well. As a result less contact noise will couple out into the remainder of the circuitry as the contact resistance becomes a smaller fract ion of the total resistance. Future Work Since the contact region is identified as th e most critical area for a 1D nanowire in terms of noise, a logical next step would be to focus future research on the contact mechanism in these devices with the inten tion of finding ways to minimize the noise level. For the Si nanowires, th e next research effort should be on the quality improvement of the impinging end of the nanowire [57] sin ce it was shown to be the likely source of the observed noise. The presence of the na tive oxide at the im pinging end seems to critically affect both the de vice resistance and the noise performance, so a research project to minimize its effect may produce dramatic performance improvements. For CNTs, more research is necessary to understand the deta ils of the contact mechanism and the nature of the trap forma tion at the contact. The activation energies presented in this work may provide additio nal clues to determine if these activation energies can be associated w ith a certain physical processes. Other possible avenues, not covered in this study should al so be explored. For example the gate bias dependence of the 1/f noise in a semiconducting CNT ma y provide important clues about noise producing mechanism as the observed noise char acteristics can potentially be linked to the device characteristics operating as a FET. A detailed modeling effort, similar to what was done on SNWs in this work (chapter 5), may shed more light on the noise phenomena. The large spread in the publishe d data on CNTs seems to indicate that the CNT devices are very much process depende nt. So in order to understand the noise

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117 mechanism and to make improvements upon the current noise level it is necessary to continue research on differ ent type metal-CNT contact s and different processing techniques. The positive effect of annealing was demonstrated in our ablation experiment where A/R the figure of merit for 1/f noise show ed clear improvements due to annealing (chapter 6). So, research on similar post-pro cessing techniques appe ars to be worthy of further investigation.

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APPENDIX A EQUIVALENT NOISE SOURCES OF BR OOKDEAL-5004 LNA

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119 Spectral density of the equivalent voltage source10-1910-1810-1710-1610-1510-14110100100010000100000Frequency (Hz)Sv(V2/Hz) Spectral density of the equivalent voltage source10-1910-1810-1710-1610-1510-14110100100010000100000Frequency (Hz)Sv(V2/Hz) Spectral density of the equivalent current source10-2710-2610 -251 10100100010000100000Frequency (Hz)Si(A2/Hz) Spectral density of the equivalent current source10-2710-2610 -251 10100100010000100000Frequency (Hz)Si(A2/Hz)

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APPENDIX B FLOW CHART OF THE INSTRUMENT CONTROL PROGRAM

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121 Y Y Initialize SA Select Min. and Max. freq. span Select additional input attenuation START Set span Set number of average Auto range Add input attenuation Start measurement STOP All spans complete ? Save data Flag Overload ? Y N N Averaging complete ?

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122 LIST OF REFERENCES 1. Y. Taur, D. A. Buchanan, W. Chen, D. J. Frank, K. E. Ismail, S. H. Lo, G. A. SaiHalasz, R. G. Viswanathan, H. J. C. Wann, S. J. Wind a nd H. S. Wong, “CMOS scaling into the nanometer regime,” Proc of IEEE, 85 (4), pp. 486-504, April 1997. 2. S. Iijima, “Helical microtubeles of graphitic carbon,” Nature, 354 (6348), pp. 56-58 November 1991. 3. S. J. Tans, M. H. Devoret, H. J. Dai, A. Th ess, R. E. Smalley, L. J. Geerligs and C. Dekker, “Individual single-wall carbon na notubes as quantum wires,” Nature, 386 (6624), pp. 474-477, April 1997. 4. H. W. Kroto, J. R. Heath, S. C. Obrien R. F. Curl and R. E. Smalley, “C-60 – Buckminsterfullerene,” Nature, 318 (6042), pp. 162-163, November, 1985. 5. S. Frank, P. Poncharal, Z. L. Wang, W. A. de Heer, “Carbon nanotube quantum resistors,” Science, 280 (5370) pp. 1744-1746, June 1998. 6. A. B. Dalton, S. Collins, E. Munoz, J. M. R azal, V. H. Ebron, J. P. Ferraris, J. N. Coleman, B. G. Kim and R. H. Bau ghman, “Super-tough carbon-nanotube fibres These extraordinary composite fibres can be woven into electronic textiles,” Nature, 423 (6941), pp. 703-703, June 2003. 7. P. G. Collins, M. S. Fuhrer, and A. Zettl e, “1/f noise in carbon nanotubes,” App. Phys. Lett., 76(7), pp. 894-896, February 2000. 8. E. S. Snow, J. P. Novak, M. D. Lay and F. K. Perkins, “1/f noise in single-walled carbon nanotube devices,” App. Phys. Lett., 85 (18), pp. 4172-4174, November 2004. 9. D. J. Frank and C. J. Lobb, “Highly effi cient algorithm for percolative transport studies in two dimensions,” Phys. Re v. B, 37(1), pp. 302-307, January 1988. 10. S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller and P. Avouris, “Carbon nanotubes as Schottky barrier tran sistors,” Phys. Rev Lett., 89 (10), no. 106801, September, 2002. 11. D. Kingrey and P. G. Collins, “Noise in carbon nanotube electronics,” Int. Sym. On Fluctuation and Noise, Austin, Proc of SPIE, vol. 5846, pp. 92-100, 2005.

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123 12. R. Tarkiainen, L. Roschier, M. Ahlskog, M. Paalanen and P. Hakonen, “Lowfrequency current noise and resistance uctuations in multiwalled carbon nanotubes,” Physica E, 28, pp. 57–65, April 2005. 13. A. D. Van Rheenen, G. Bosman and R. J. J. Zijlstra, “Low-frequency noise measurements as a tool to analyze deep-l evel impurities in se miconductor-devices,” Solid-State Elec. 30 (3), pp. 259-265, March 1987. 14. A. L. McWhorter, “Surface traps and 1/f noise in Germanium,” Phys. Rev., 98 (4), pp. 1191-1192, January 1955. 15. F. N. Hooge, “1/f noise is no surface effect ,” Phys. Lett. A, Physics Letters A, 29 (3), pp. 139-140, April 1969. 16. M. Ishigami, J. H. Chen, E. D. Williams, D. Tobias, Y. F. Chen and M. S. Fuhrer, “Hooge's constant for carbon na notube field effect transist ors,” App. Phys. Lett., 88 (20), 203116, May 2006. 17. Y. M Lin, J. Appenzeller, J. Knoch, Z. H. Chen, P. Avouris, “Low-frequency current fluctuations in individual se miconducting single-wall carbon nanotubes,” Nano Letters, 6 (5), pp. 930-936, May 2006. 18. P.-E. Roche, M. Kociak, M. Ferrier, S. Guron, A. Kasumov, B. Reulet and H. Bouchiat, “Shot noise in car bon nanotubes,” Int. Sym. On Fluctuation and Noise, Santa Fe, Proc. SPIE, 5115, pp. 104–115, 2003. 19. V. L. Gurevich and A. M. Rudin, “Shot noise in the presence of phonon-assisted transport through quasiballistic nanowire s,” Phys. Rev. B, 53 (15), pp. 1007810085, April 1996. 20. A. G. Scherbakov, E. N. Bogachek and U Landman, “Noise in three-dimensional nanowires,” Phys. Rev. B, 57 (11), pp. 6654-6661, March 1998. 21. H. W. C. Postma, I. Kozinsky, A. Husa in and M. L. Roukes, “Dynamic range of nanotubeand nanowire-based electromechanical systems,” App. Phys. Lett., 86 (22), 223105, May 2005. 22. T. Li, J. N. Wang and Y. M. Zhang, “El ectrical transport in doped one-dimensional nanostructures,” J. of Nanoscience and Nanotechnology, 5 (9), pp. 1435-1447, September 2005. 23. J. Kong, A. M. Cassell and H. J. Dai, “Chemical vapor deposition of methane for single-walled carbon nanotubes,” Chem. Phys. Lett., 292 (4-6), pp. 567-574. August 1998. 24. S. Iijima and T. Ichihashi, “Single-shell carbon nanotubes of 1-nm diameter,” Nature, 363 (6430), pp. 603-605, June 1993.

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124 25. A. Thess, R. Lee, P. Nikolaev, H. J. Dai, P. Petit, J. Robert, C. H. Xu, Y. H. Lee, S. G. Kim, A. G. Rinzler, D. T. Colbert, G. E. Scuseria, D. Tomanek, J. E. Fischer and R. E. Smalley, “Crystalline ropes of metallic carbon nanotubes,” Science 273 (5274), pp. 483-487, July 1996. 26. Y. Li, W. Kim, Y. Zhang, M. Rolandi, D. Wang and H. Dai, “Growth of singlewalled carbon nanotubes from discrete cataly tic nanoparticles of various sizes,” J. Phys. Chem. B., 105 (46), pp. 11424-11431, August 2001. 27. J. A. Sippel-Oakley, “Charge-induced act uation in carbon nanotube s and resistance changes in carbon nanotube networks,” Ph .D. dissertation, Dept. of Physics, University of Florida, 2005. 28. M. Saif Islam, S. Sharma, T. I. Kamins and R. Stanley Williams, “A novel interconnection technique for manufacturing nanowire devices,” Appl. Phys. A, 80, pp. 1133–1140, 2005. 29. R. Satio, G. Dresselhaus and M. S. Dre sselhaus, Physical Pr operties of Carbon Nanotubes, 1st ed., Imperial College Press, London, 1998. 30. M. S. Dresselhaus, R. Saito and A. Jorio, “Semiconducting Carbon Nanotubes,” 27th Int. Conf. on the Physics of Semiconduc tors, Flagstaff, Proc. of ICPS-27, pp. 25-31, 2004. 31. S. Datta, Quantum Transport: From Atom to Transistor, Cambridge Univ. Press, Cambridge, 2005. 32. J. Appenzeller, J. Knoch, V. Derycke, R. Martel, S. Wind and P. Avouris, “Fieldmodulated carrier transport in carbon nanotube transist ors,” 89(12), Phys. Rev. Lett., 126801, September 2002. 33. M. Anantram, L. Delzeit, A. Cassell, J. Han and M. Meyyappan, “Nanotubes in nanoelectronics: transport, growth and modeling,” Physica E, 11 (2-3), pp. 118125, October 2001. 34. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., John Wiley & Sons, New York, 1981. 35. Y. Cui, X. F. Duan, J. T. Hu and C. M. Lieber, “Doping and elec trical transport in silicon nanowires,” J. of Phys Chem. B, 104 (22), pp. 5213-5216 2000. 36. S. Sharma, T. I. Kamins, M. S. Islam, R. S. Williams and A. F. Marshall, “Structural characteristic s and connection mechanism of gold-catalyzed bridging silicon nanowires,” J. of Crys tal Growth, 280, pp. 562-568, 2005. 37. A. Van der Ziel, Noise in Solid State Devices and Circuits, John Wiley and Sons, New York, 1986.

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125 38. C. D. Motchenbacher and J. A. Connelly Low-noise electronic system design, John Wiley and Sons, New York, 1993. 39. F. N. Hooge, T. G. M. Kleinpenning a nd L. K. J. Vandamme, “Experimental studies on 1/f noise,” Rep. on Prog. in Phys., 44 (5), 479-532, 1981. 40. F. N. Hooge, “1/f noise sources,” IEEE Trans. on Electron Dev., 41 (11), pp. 19261935, 1994. 41. M. S. Dresselhaus, G. Dresselhaus, a nd P. Avouris, Carbon Nanotubes: Synthesis, Structure, Properties, and Application, 1st ed., Springer-Verlag, Berlin, 2001. 42. P. G. Collins, M. Hersam, M. Arnold, R. Martel, and Ph. Avouris, “Current saturation and electrical breakdown in multiwalled carbon nanotubes,” Phys. Rev. Lett. 86 (14), 3128, 2001. 43. R. V. Seidel, A. P. Graham, B. Rajasekhara n, E. Unger, M. Liebau, G. S. Duesberg, F. Kreupl, and W. Hoenlein, “Bias depe ndence and electrical breakdown of small diameter single-walled carbon nanotubes,” J. of App. Phys. 96 (11), pp. 6694-6698, 2004. 44. A. Javey, J. G. Guo, Q. Wang, M. Lundstrom, and H. Dai, “Ballistic carbon nanotube field-effect transistors,” Nature, 424, pp. 654-657 2003. 45. S. Dutta, Quantum Transport: Atom to Tr ansistor, 2nd ed., Cambridge University Press, Cambridge, 2005. 46. V. Barone, J. Heyed, and G. E. Scuser ia, Chem. Phys. Lett., “Effect of oxygen chemisorption on the energy band gap of a chiral semiconducting single-walled carbon nanotube,” 389, pp. 289-292, 2004. 47. A. A. Odintsov and Y. Tokura, “Contact phenomena and Mott transition in carbon nanotubes,’ J. of Low Temp. Phys., 118, pp. 509-518, 2000. 48. T. Nakanishi, A. Bachtold and C. Dekker, “Transport through th e interface between a semiconducting carbon nanotube and a me tal electrode,” Phys. Rev. B, 66, 073307, 2002. 49. D. L. John, L. C. Castro, H. Clifford and D. L. Pulfrey, “Elect rostatics of coaxial Schottky-barrier nanotube fi eld-effect transistors,” I EEE Trans. Nanotech., 2(3), September 2003. 50. J. Guo, J. Wang, E. Polizzi, S. Dutta and M. Lundstrom, “Electrostatics of nanowires transistors,” IEEE Tran s. Nanotech., 2(4), December 2003. 51. L.C. Castro, D.L. John and D.L. Pulfrey, “Towards a compact model for Schottkybarrier nanotube FETs,” Conf. on Optoel ectronic and Microelectronic Materials and Devices, Sydney, Proc. I EEE COMMAD-02, pp. 303-306, 2002.

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126 52. B. G. Streetman and S. Banarjee, Solid State Electronic Devices, 5th ed., Prentice Hall, London, 2000. 53. J. W. Mintmire and C. T. White, “U niversal density of states for carbon nanotubes,” Phys. Rev. Lett., 81( 12), pp. 2506-2509, September 1998. 54. F. Leonard and J. Tersoff, “Role of Fermi level pinning in nanotube Schottky diodes,” Phys. Rev. Lett., 84(20), pp. 4693-4696, May 2000. 55. B. Shan, “First principles study of work functions of single wall carbon nanotubes,” Phys. Rev. Lett., 94, 236602, 2005. 56. P. Dutta and P. M. Horn, “Low-frequency fl uctuations in solids – 1/f noise,” Rev. of Mod Phys., 53 (3), pp. 497-516, 1981. 57. M. S. Islam, S. Sharma, T. I. Kamins a nd W. R. Stanley. “Ultr ahigh-density silicon nanobridges formed between two vertical silicon surfaces,” Nanotechonology, 15, L5-L8, 2004.

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127 BIOGRAPHICAL SKETCH Shahed Reza obtained his BS in electri cal engineering from the Bangladesh University of Engineering and Technol ogy (BUET) in 1994 and then worked for Bangladesh Atomic Energy Research Establishm ent. He received his MS in electrical engineering from the University of Centra l Florida in 1998. He worked as a design engineer at Piezo Technology Inc. (PTI) from 1997 to 2000. At PTI, he designed precision crystal oscillators and LC filters. From 2000 to 2003 he designed analog microwave circuits at Agilent Technologies. As a Ph.D. candidate in the Department of Electrical and Computer Engineering at the University of Florida, Gainesville, he conducted research on the noise properties of carbon nanotubes and silicon nanowires. He is also currently working as a research engineer at Invivo Di agnostic Imaging. At Invivo, he is conducting rese arch on image guided noise tomography and the modeling and simulation of RF probes used in MR Imaging.


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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
    List of Tables
        Page viii
    List of Figures
        Page ix
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Device fabrication and mounting
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Device operation and DC characteristics
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
    Noise measurements
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Noise in silicon nanowires
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
    Sequential ablation of carbon nanotubes
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
    Lorentzian type of noise in carbon nanotubes
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
    Energy band simulation of carbon nanotubes
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
    Thermally activated 1/F noise in carbon nanotubes
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
    Conclusions
        Page 114
        Page 115
        Page 116
        Page 117
    Appendices
        Page 118
        Page 119
        Page 120
        Page 121
    References
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
    Biographical sketch
        Page 127
Full Text











EXCESS NOISE IN ONE DIMENSIONAL QUANTUM NANOWIRES


By

SHARED REZA















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006





























Copyright 2006

by

SHARED REZA






















Dedicated to

My mother

Nurunahar Tahera

And my father

ATM Naderuzzaman Talukdar















ACKNOWLEDGMENTS

I would like to thank my wife, Misty Sabina, for her support through all my years

of graduate school and research. Without her limitless patience and constant

encouragement, this dissertation would have been impossible. Also there is our little

bundle of joy, Ronan Reza, whose smile at the end of a long day at work gave me the

energy to continue my struggle well past midnight and start afresh the next day.

My parents have been the key people to influence my life and I am indeed grateful

to them for all that they have done to help make me what I am today. Over all these

years, in every difficult period of time they were the source of support and comfort. I

consider every achievement in my life theirs as well.

My work would have been impossible without the guidance of Dr. Gijs Bosman. I

was introduced to this topic of noise through one of his lectures back in 1998 and have

been interested in ever since. It is from him that I learned the numerous tricks of the

trade, learned literally how to extract information from noise. My heartfelt thanks go to

him for his constant patience and trust in me and for all of his advice about my research

and professional career. Thanks go to Quyen T. Huynh, for her help in performing the

ablation experiments.

I would like to thank Dr. Andrew G. Rinzler and his student Dr. Jennifer Sippel-

Oakley for giving us the carbon nanotubes devices that I have used in my experiments. I

sincerely appreciate their patience at the beginning of the collaboration when I was just

learning how to make the connections to the devices without damaging them.









Also, I appreciate Dr. M. Saif Islam at UC Davis and his colleagues Theodore I.

Kamins and R. Stanley Williams at HP labs for providing us with the silicon nanowire

devices. Special thanks go to Dr. Saif Islam for the fruitful discussions during the

analysis phase, which led to the successful modeling of the data.

I have to thank Dr. Charles Saylor, Dr. Randy Duensing, Sathya Vijayakumar,

and Dr. Mark Limkeman at Invivo Diagnostic Imaging, for their constant support and

thought provoking discussions on multiple aspects of my research.

I am grateful to and Dr. Jing Guo and Dr. Ant Ural for allowing me to attend their

lectures on carbon nanotubes and countless helpful discussions and advice. And finally I

would like to thank my committee members Dr. Scott E. Thompson and Dr. Kirk J.

Ziegler for making the time to serve on my supervisory committee with their busy

schedule.















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .................................................... ....... .. .............. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

ABSTRACT .............. ..................... .......... .............. xii

CHAPTER

1 IN T R O D U C T IO N ............................................................................. .............. ...

2 DEVICE FABRICATION AND MOUNTING ................................................. 11

Fabrication of C arbon N anotubes ...................................... ................... ...............11
Fabrication of Silicon N anow ires .................................................................... .... .... 12
M counting of D devices .............................................................................. .... .18

3 DEVICE OPERATION AND DC CHARACTERISTICS .......................................21

D C Characteristics of Carbon N anotubes ..................................................................21
Electronic Structure and Properties of Carbon Nanotube .................................21
DC Characteristics and Measurements............... ..................................25
DC Characteristics of Silicon Nanowires .......................................................30
Resistance M easurem ents............................................................... ........ ....30
Resistance M odel ............................................... .. ..... ................. 33

4 N OISE M EA SUREM EN TS ............................................................ .............39

N oise M easurem ent System ............................................... .............. ............39
Noise Characterization of the LNA ................................ ......................... ........ 39
Instrument Control and Other Measurement Issues .................................................41
Setup for Low Temperature Measurement.............................................................44
Overview of the Observed Noise Spectra ......................................... ...............45

5 NOISE IN SILICON NANOWIRES............................................................50

Introduction ......... .... ..............................................................................50
N oise M odel.......................................................................................... 50
Sum m ary and C onclu sion ..................................................................................... 59









6 SEQUENTIAL ABLATION OF CARBON NANOTUBES...................................61

Setup for the Sequential Ablation Technique..........................................................61
R esu lt an d A n aly sis ................ .. .......................................... .... ...... ..............62
Details of the Observed Annealing Effect..... .......... ....................................... 71
Sum m ary and C onclu sions .............................................................. .....................73

7 LORENTZIAN TYPE OF NOISE IN CARBON NANOTUBES ...........................77

Introduction to the Lorentzian Noise Component ........................................ ...........77
R results and A naly sis............ ............................................................ ..... .... ..... 80
E effect of G ate V oltage ............... .................................................... ............... 86

8 ENERGY BAND SIMULATION OF CARBON NANOTUBES .............................88

In tro d u ctio n ............... .. ............ ..... ......... ............ ........ ...... ................ 8 8
Self Consistent Solution of Charge Density and Poisson's Equation.......................90
Calculation of Charge D ensity ........................................ ........................ 90
Calculation of Potential ............ ..... .............. ............... 94
Self Consistent Solution Method...... ................. ...............96
R results and D iscu ssions........ ................................................................ .. .... ........ 98

9 THERMALLY ACTIVATED 1/F NOISE IN CARBON NANOTUBES.............103

Intro du action ................... .......... ....... ......... ..... ............... ................ 10 3
Calculation of the Distribution of Activation Energies ................. ................104
Physical Location of the Trap Centers................................................................... 109

10 CONCLUSIONS ................... ............................ ....... .................. 114

Summary of Results and Conclusions .......................................... ...............114
F u tu re W o rk ...................................... .............................................. 1 16

APPENDIX

A EQUIVALENT NOISE SOURCES OF BROOKDEAL-5004 LNA.......................118

B FLOW CHART OF THE INSTRUMENT CONTROL PROGRAM ......................120

LIST OF REFEREN CES ........................................................... .. ............... 122

BIOGRAPHICAL SKETCH ............................................................. ............... 127
















LIST OF TABLES

Table page

2-1 Measured nanowire dimensions from SEM.............................................................17

3 -1 D ev ice re sistan ce .................................... ....................................................... .. 3 5

5-1 R elativ e noise m agnitu de ................................................................. .....................52

6-1 Noise and conductance data of samples A, B and C................................................76

7-1 Activation energies calculated using equation (7-2) ................................................83



































viii
















LIST OF FIGURES


Figure page

1-1 Carbon nanotube and the planer graphene sheet....................................... ........... 2

1-2 Scanning electron microscope (SEM) image of silicon nanowires connecting
tw o e le ctro d e s ...................................... ............................... ................ .. 9

2-1 Cross-section of the device showing different layers ...........................................13

2-3 SEM image of suspended carbon nanotubes ..............................15

2-4 Illustration of fabrication steps for silicon nanowires............................................16

2-5 Mounting of the device on a TO-8 package. ..................................... .............19

3-1 Nanotube formation from graphene sheet. .......................................................22

3-2 Formation of Zigzag and armchair nanotube from graphene..............................23

3-3 E-k diagram of graphene in the first Brillouin zone using the 7t-band nearest-
neighbor tight-binding m odel .......... .................... ........................ .... ........... 24

3-5 Measured drain current vs. voltage characteristics at different gate bias.................28

3-6 Measured dynamic conductance and the calculated components vs. gate bias........31

3-7 Measured dynamic conductance vs. gate voltage bias at 77K and 300K.................32

3-8 Calculated vs. measured resistance, including contact resistance............................38

4-1 N oise m easurem ent setup .............................................................. .....................40

4-2 Setup for the noise characterization of the low noise amplifier............... ..............42

4-3 Voltage noise spectral density at the input side of Brookdeal-5004 LNA for
different source resistance. .............................................. .............................. 43

4-4 A typical plot of the measured current noise spectral density..............................46

4-5 Plot depicting the relationship between the 1/f noise and device current ...............48









5-1 1/f noise coefficient A vs. effective resistivity p. ................................ ................51

5-2 Circuit representation of the noise model ..... ......... ....................................... 53

5-4 Contact and bulk noise components calculated from measured data....................58

6-1 Pulsed bias experimental set-up for sequential ablation of metallic CNTs .............63

6-2 The measured low bias conductances of metallic and semiconducting CNTs in
sa m p le A ......................................................................... 6 5

6-3 Calculated conductances of the ablated metallic CNTs for sample A ...................67

6-4 Measured At/R and At.Gt2 at Vg = +14V for sample A, plotted as a function of
the number of CNT ablation attempts performed on the sample ...........................69

6-5 Calculated A and A/R for the ablated metallic CNTs for sample A .......................70

6-6 Calculated A vs. R values for the ablated CNTs..................... ............................ 72

7-2 A diagram describing the shift of Lorentzian spectra with temperature .................79

7-3 Lorentzian characteristic times for device A and B ........................................... 81

7-4 A plot of the Lorentzian Plateau value vs. characteristic time for device B at Vg
= 0 a n d 14 V .............................................................................................................8 4

7-5 A plot of the Lorentzian Plateau value and characteristic time at different gate
voltages for device A ................................... ... ... ...... ...............85

8-1 Explanation of the charge calculation using equation (8-1)............................... 89

8-2 Density of state plots calculated using (8-4). ................................ .................92

8-3 A plot of the calculated density of carriers per unit length for metallic and
sem conducting CN Ts at T = 300K .................................... ........................ 93

8-4 Flow-chart for the iterative solution scheme.............. .............................................99

8-5 The plot ofE, vs. position at Vg= 0 and +14 V for a metallic CNT. ...................101

8-6 Band-bending at Vg = +14 V for a semiconducting CNT. .....................................102

9-1 Measured 1/f Noise coefficient over temperature for sample A ...........................105

9-2 Distributions of activation energies for sample A and B .......... ................107

9-3 Measured and calculated frequency exponent pvs. Tfor sample A...................108









9-4 D(E) vs. Tplot calculated for Vg = 0 and +14V. ................ ..... ..................110

9-5 The difference of E, due to the change in Vg = OV to +14V vs. position. ............. 112















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EXCESS NOISE IN ONE DIMENSIONAL QUANTUM NANOWIRES
By

Shahed Reza

December 2006

Chair: Gijs Bosman
Major Department: Electrical and Computer Engineering.

Silicon nanowires and carbon nanotubes are two promising novel devices in the

nanotechnology area. A study of the current-voltage and the low frequency excess noise

properties of these devices is presented.

From the silicon nanowire current-voltage and noise characteristics measured at

room temperature in the linear regime of device operation, the bulk and contact resistance

contributions are extracted and modeled. The excess noise observed at low frequencies is

interpreted in terms of bulk and contact noise contributions, with the former comparable,

in terms of Hooge parameter values, to the low noise levels observed in high quality

silicon devices. The contact noise is significant in some devices and is attributed to the

impinging end of the bridging nanowires.

The charge transport and noise properties of three terminal, gated devices

containing multiple single-wall metallic and semiconducting carbon nanotubes were

measured at room temperature. A method to separate contributions from the metallic and

semiconducting carbon nanotubes by sequential ablation using high voltage pulsed bias is









presented. The relative low frequency excess noise of the metallic tubes was observed to

be two orders of magnitude lower than that of the semiconductor tubes.

The low frequency noise of single-walled carbon nanotubes is studied over the 77K

to 300K temperature range. Lorentzian shaped spectra along with 1/f noise spectra have

been observed. From the Lorentzian noise components, a range of thermal activation

energies from 0.08 to 0.51 eV for the associated fluctuation mechanisms is obtained.

From the 1/f noise spectra, a distribution of activation energies of fluctuation processes

ranging from 0.2 to 0.7eV is derived. These findings indicate that the observed noise

spectra are caused by number fluctuations. Using simulation results and the observed gate

dependence of the noise producing activation energy distribution, the physical origin of

the observed noise phenomena was shown to be the contact region.














CHAPTER 1
INTRODUCTION

In the last decades of the 20th century there has been a remarkable progress in

electronics. The technology that has been the vehicle of this revolution is the

complementary metal oxide silicon field effect transistor (CMOSFET or CMOS). Since

the introduction of this technology the basic device remained mostly unchanged, but the

device size gradually became smaller. This miniaturization enabled semiconductor

industries to pack more devices in a chip, thus increasing the functionality, speed and cost

reduction. Over the last three decades this miniaturization trend continued in a very

consistent manner, roughly every three years quadrupling the packing density, obeying

Moore's law. There are strong indications that the limit of the miniaturization is rapidly

approaching. The current state of the art MOS device has a feature size of 20nm and the

physical limit for scaling is expected to be -10nm. Below this feature size the error in the

lithography and the uncertainty of the doping profile will be a great obstacle for

manufacturing high performance devices [1]. Unless a new technology is discovered, the

semiconductor industry faces the danger of becoming stagnant. This necessitates the

research for device structures that may take the place of the current CMOS technology.

Several new devices have been proposed. Among them, ferromagnetic field effect

transistors, single electron transistors, molecular transistors, silicon nanowires and carbon

nanotubes are currently in the forefront of research.








































Figure 1-1. Carbon nanotube and the planer graphene sheet (Image courtesy of Dr. Ant
Ural, Dept. of ECE, University of Florida).









Carbon nanotubes (CNT) were discovered by Iijima in 1991 [2]. The arrangement

of atoms in a CNT is the same as the atomic arrangement in graphite. A CNT can be

described as a graphene sheet rolled on to itself and forming a hollow tube [3] (see Fig. 1-

1). The ends of the nanotube can either be open or capped by the so-called Fullerenes, a

half sphere structure [4]. CNTs can be either single walled or multi-walled. The diameter

of a single wall CNT generally is between 0.7 to 10nm and its length can vary from nm to

|tms depending on the growth conditions. Because of the small diameter CNTs operate as

one-dimensional quantum structures. Furthermore, the CNTs can be grown and the

critical feature size, the diameter, is determined by the growth process and thus, unlike in

MOS technology, does not depend on lithography. This is a significant advantage of

CNTs over the current CMOS technology.

The carbon nanotube is a unique device. First, the CNT can be either metallic or

semiconductor, so it can potentially be used both as the device and the interconnects.

Furthermore, the band-gap of the semiconducting tube depends on the diameter. This can

potentially eliminate the need for complex band-gap engineering and there are many

possible ways this property can be utilized. CNT is a very good conductor of heat and

current; the current density of CNTs can be orders of magnitude higher than copper [5].

The carbon nanotube is a very stable structure having a tensile strength of about 60 times

greater than steel [6]. Despite these very attractive properties, there remain several

significant obstacles before the CNT technology can be considered for mainstream

electronics.

The fabrication method of CNT is still far from perfect. During the CNT growth

process a mixture of CNTs of different length, diameter and orientation is formed. The









process to selectively grow a CNT with a certain feature size at a precise location is still

unknown. Another problem, first reported by Collins et al. [7], is high excess noise. One

would expect lower excess noise levels since the atoms in a graphene structure are well

ordered and properly terminated. On the contrary the 1/f like noise in CNTs was observed

to be extremely high. Compared to a carbon film resistor, which is generally considered

too noisy for many applications, CNT is several orders of magnitude more noisy. Unless

a way to reduce this noise is found, the applicability of CNTs in mainstream electronics

will be severely limited.

Collins et al. [7] observed that both single-wall and multi-wall CNTs are equally

noisy. The observed excess noise obeys the well-known expression for 1/f type noise,

A.I2
S,,/(f )= (1-1)
f,8

where Idc is the dc current level, fis the frequency, and A and Pare constants. The

exponent /was found to be approximately equal to one as expected for 1/f type noise.

The parameter A represents the relative magnitude of the 1/f noise.

Additionally, Collins el al. established the following empirical relationship between

the 1/f noise coefficient A and the resistance of the device R,

A/R O 1011S (1-2)

Recently, Snow et al. [8] reported an additional dependence in (1-2) for a two

dimensional network devices or mats,

9x10
A/R = S (1-3)


where L is the device sample length in |tm and which may be longer than the length of

the constituent CNTs. A reduction in 1/f noise with device length is expected for an









electronic system where 1/f noise varies inversely with the number of carriers, but the

fact that A/R is inversely proportional to 1/L1.3 cannot be explained from the number of

carriers; normally one would expect a 1/L2 relationship. Snow et al. also reported on 1/f

noise gate voltage dependence. The noise magnitude is minimum at OV gate bias and

maximum at the gate voltage where the device inverts from p-type to n-type mode of

operation. Furthermore, it was reported that at positive gate voltage the resistivity

fluctuation Sp is related to the resistivity p by

S Oc p3 6 (1-4)

This type of power law dependence is characteristic of percolating systems. And in fact

the two-dimensional mats used by Snow et al. resemble a percolation system [9] where

the device consists of many interconnected CNTs. Unlike a conventional one-

dimensional device, where a carrier flows from one contact to the other ballistically

through one CNT, in these devices the carriers flow through the multiple intersecting

network of CNTs. This can result in a length dependent transport properties and it is the

likely cause of the 1/L13 dependence shown in (1-3).

In an attempt to exclude possible noise from the contact, Collins et al. [7]

conducted a four-point probe measurement, but did not observe a change in the measured

noise between a two- and four-point probe experiments. Hence, they concluded that the

source of the noise is not the contacts. However, now it is understood that a four-probe

technique cannot be used for carbon nanotubes, because the contacts are an integral part

of the mesoscopic CNT device and the addition of another set of contacts completely

changes the device itself [10]. Hence, the result of the four-probe measurement does not

rule out contact as a potential source of 1/f noise. Recently Kingrey et al. [11] reported









the effect of annealing and passivation studies. It was observed that after annealing, the

noise magnitude of a CNT at room temperature decreased but it was still substantial and

varied widely across the temperature range of 80 to 450K. In an attempt to prevent

absorption of species back on to the CNT after heating, two different types of passivation

layers, SiO2 and Polymethylmethacrylate, were tried. In both cases no significant change

in noise characteristics was observed. In the same study the presence of Lorentzian

spectra along with 1/f noise was reported.

The presence of Lorentzian spectra is also reported by Tarkiainen et al. [12]. The

observed Lorentzian spectra can be expressed by the following expression [13],

SL (0)
SL (f) S= (1-5)
1+0 )2 '

where SL(O) is the plateau value of the Lorentzian, ca is the angular frequency and r is a

characteristic time. This type of noise spectra is associated with a two energy level

system. The characteristic time of the observed spectra was found to be a function of the

bias voltage. Most importantly, the noise characteristics were observed to be dependent

on the direction in which the bias was applied, which may correspond to the location of

the trap centers.

The exact mechanism of the 1/f noise phenomena in general is still unclear.

However, two widely accepted models for this phenomenon exist. The first one proposed

by McWhorter [14] was successful in explaining the noise in MOSFETs. This model

assumes that the noise is caused by carrier number fluctuations. The second model,

proposed by Hooge [15], postulates that the 1/f noise phenomena originates in the bulk

because of mobility fluctuation of carriers. Hooge's model requires the exponent f in (1-

1) to be exactly 1; however McWhorter's model does not have this restriction.









Commonly, the Hooge's parameter aH is used as a figure of merit to compare the 1/f

noise levels between devices and technologies even if the source is of McWhorter type.

This figure of merit aH is related to the 1/f noise coefficient A by

aH = AN (1-6)

where N is the number of carriers associated with the fluctuation process.

Ishigami et al. [16] in their recent publication addressed the issue of which

mechanism is responsible for the observed 1/f noise. They established that the 1/f noise

parameter A is inversely proportional to the number of carriers N. Since

N = CL Vg Vh /e (where cg, L, Vg and Vth are gate capacitance, device length, gate

voltage and gate threshold voltage respectively), it follows that


Sc Vg Vh (1-7)


Based on (1-7) Ishigami et al. concluded that the noise is of the mobility fluctuation

origin. Their argument is based on a bulk MOSFET analogy where equation (1-7) is true

only if the noise has a mobility fluctuation origin. For a number fluctuation origin one

would expect -oc h 12 relationship. Although this statement is shown to be true for


MOSFETs, it is not clear whether it would be applicable for a CNT device, because,

unlike a MOSFET where the device operation is controlled by the modulation of the

channel conductance using gate voltage, for a CNT device the control is achieved by

modulating the Schottky barriers formed at the contact region. In addition, CNT devices

typically have a very significant contact resistance which dominates the current voltage

characteristic, and this further complicates the analogy with a simple MOSFET 1/f noise

model.









Ishigami et al. estimated the value of al to be (9.3 + 0.4)x10-3, which is much

lower than the previously reported value of- 0.2 by Collins et al. [7]. This is also

supported by a recent publication by Lin et al. [17]. Their estimate for al is 2x10-3. Lin

el al. also established that A is inversely proportional to the device length, which is


consistent with the oc N behavior. The findings of Lin et al. further suggested that 1/f
A

noise in CNTs is not substantially affected by the acoustic phonon scattering or ionized

impurity scattering. To understand the effect of contact and bulk Lin et al. added a third

gate to modulate the bulk region of the device. The 1/f noise level in this modified device

also showed similar dependence on device resistance as a conventional device. In this

1
case the measured relative noise magnitude supported the oc N model established for a
A

conventional device.

In summary, the CNT exhibits thermal, shot and low frequency excess noise. The

thermal and shot noise have been characterized and modeled satisfactorily [18]. The 1/f

noise magnitude in a CNT is several orders of magnitude higher than in the conventional

silicon devices and the origin and mechanism of this noise component are still unclear.

Now the silicon (Si) nanowire devices used in our study will be introduced. Silicon

nanowire (SNW) is another leading candidate in the nanotechnology research arena.

Similar to CNTs, these devices have a large length over diameter ratio. SNWs are grown;

thus, like a CNT, the need for lithography is eliminated. The major advantage of SNW

over CNT technology is the constituent material. Si is the material of choice for the

mainstream semiconductor industry. As a result, integrating SNW in currently available

processes would be easier than the CNT technology.












































Figure 1-2. Scanning electron microscope (
two electrodes.


4) image of silicon nanowires connecting









Currently available SNWs typically have a larger radius than the CNTs. The typical

SNW radius is on the order of 50nm, and the radius of a typical CNTs is on the order of

-lnm. Although there has been theoretical work done on the noise of nanowires [19-22],

no experimental work has been reported on the noise in SNW. The result of our study on

Si nanowires presented in this manuscript will help fill the void.

In this dissertation, the result of a study of the low frequency excess noise of CNT

and Si nanowires is presented. Chapter 2 describes the fabrication methods and sample

geometry used in this study. In chapter 3 the details of device operation, DC

measurements and in chapter 4 the details of noise measurement techniques are

discussed. In chapter 5 the results and analysis of the noise measurements on SNWs and

the possible source of the observed noise are discussed. In chapter 6 a sequential ablation

technique to determine the characteristics of individual CNTs is presented. The analysis

of the thermally activated Lorentzian component of noise is discussed in chapter 7. In

chapter 8 the details of CNT simulation technique are discussed. Finally, the analysis of

the thermally activated 1/f noise in CNTs and the conclusions resulting from the

dissertation work are presented in chapter 9 and chapter 10, respectively.














CHAPTER 2
DEVICE FABRICATION AND MOUNTING

Fabrication of Carbon Nanotubes

The growth method for the devices used in this study is the chemical vapor

deposition (CVD) growth method [23]. There are several other methods for growing

carbon nanotubes; among them the arc discharge method, [24] and laser ablation method

are popular [25]. In the CVD growth method, a substrate with a catalyst (typically Fe, Co

or Ni) is placed in a furnace and a hydrocarbon and hydrogen gas flow are added. The

hydrocarbon gas acts as the source of carbon. At a high temperature (between 5000C and

10000C) the hydrocarbon, usually methane, is catalytically decomposed and CNTs are

formed. The diameter of the grown CNT is approximately equal to the diameter of the

catalyst particles used [26]. The CNT can grow outwards from the catalyst particle while

the catalyst particle is attached to the substrate or the nanotube grows in between the

catalyst particle and the substrate, with the catalyst particle traveling on the tip of the

nanotube. A localized growth of nanotube can be obtained by controlling the placement

of the catalyst.

In the case of the devices used in this study, the nanotubes were grown on a silicon

substrate with a top oxide layer, typically 600nm thick (Fig. 2-1). The catalyst, 10mg/ml

Fe(NO3)3-9H20 particles, was dispersed over the entire surface of the substrate so the

nanotube growth occurred over the entire surface. The gas flow rate for both hydrogen

and methane was 200sccm [27]. Optical lithography was used to pattern Cr/Pd (sputtered

on to the wafer, 5nm/45nm thick respectively) 500[tm long electrodes spaced 1 tm apart









(the Pd likely overcoats the edge of the Cr layer so that electrical contact to the nanotubes

is via the Pd; see Fig. 2-2).

Fabrication of Silicon Nanowires

Silicon nanobridges were grown between electrically isolated electrodes formed

from the top silicon layer of (110)-oriented silicon-on-insulator (SOI) substrates [28].

Approximately Inm Au was deposited on the (11 1)-oriented sides of the electrodes and

annealed in a H2 ambient at 6700C to form nanoscale Au-Si alloy catalyst islands. The

structure was then exposed to a mixture of 15sccm SiH4, 60sccm HC1, and 30sccm B2H6

(100ppm in H2) in a H2 ambient at 6800C and a total pressure of 1.3kPa for 30min to

grow nanowires bridging between electrodes with a separation of 10m or less. Note that

B2H6 is added to provide p-type dopants in the form of boron. It was found that the

doping concentration can be controlled by increasing the partial pressure of B2H6. Since

the dopant is incorporated during the growth process as opposed to adding it later using

ion implantation, it is expected that the lattice structure of the device will be relatively

defect free. As will be shown later, the noise measurement results support this idea.

Before nanowire growth, reactive ion etching was used to remove Au catalyst from all

areas of the substrate other than the sidewalls. This helped to suppress the uncatalyzed

growth of Si between electrodes, ensuring good electrical isolation. Highlights of the

fabrication process for the bridging nanowires are illustrated in Fig. 2-4. The dimensions

of the Si nanowires used in our experiments were measured using a scanning electron

microscope (SEM) and are presented in Table 2-1.



















Pd/Cr

Source


SiO2


Carbon Nanotube

. \


Drain


600nm
+


Si

Gate
contact

Figure 2-1. Cross-section of the device showing different layers.































-- 500Am -1


Figure 2-2. Top view of the device showing the nanotubes and the


layout.




















1.5 gpm




Electrode
^


Figure 2-3. SEM image of suspended carbon nanotubes.
























Si Electrodes


Angled catalyst
o' deposition


Si electrode


Figure 2-4. Illustration of fabrication steps for silicon nanowires; (a) Etching to form
electrodes on a SOI substrate (b) Angled deposition of Au catalyst particles
and (c) Nanowire growth in [111] direction. SEM image of multiple
nanowires bridging across the gap between the Si electrodes shown in (d)
[28].


























TABLE 2-1 Measured nanowire dimensions from SEM


Wire 1
Length Radius
(cm) (cm)
7.5x10-4 6.8x10-6
4. x10-4 5.8x10-6
8.3x10-4 8.0x10-6
7.0x10-4 6.8x10-6


Wire 2
Length Radius
(cm) (cm)
6.6x10-4 6.2x10-6
4.7x10-4 4.8x10-6


Wire 3
Length Radius
(cm) (cm)


Wire 4 Wire 5
Length Radius Length Radius
(cm) (cm) (cm) (cm)


a 3.0x10-4 6.7x10-6 3.3x10-4 7.5x10-6
b 3.4x10-4 3.8x10-6 3.6x10-4 1.1x10-5 3.0x10-4 5.4x10-6
c 6.0x10-4 7.5x10-6
c d 6.3x10-4 4.2x10-6 6.5x10-4 5.0x10-6 6.4x10-4 8.3x10-6 6.0x10-4 6.7x10-6 9.0x10-4 4.7x10-6
e 8.4x10-4 4.6x10-6 1.1x10-3 6.0x10-6
Sf 1.0x10-3 5.0x10-6
g 1.Ox10-3 6.3x10-6 8.5x10-4 4.2x10-6
h 1.0x10-3 4.8x10-6 1.0x10-3 4.3x10-6
i 1.5x10-3 4.7x10-6 1.1x10-3 5.6x10-6


Device









Mounting of Devices

The silicon substrate containing the devices was mounted on a standard TO-8

package (Fig. 2-5). First a 50 mil alumina substrate with top and bottom gold-plated

surfaces was attached to the TO-8 package using silver loaded conductive epoxy. Then

the silicon substrate containing the devices was attached on the alumina substrate using

the conductive epoxy. This way the silicon substrate is isolated from the body of the

package and thus can be used as the back gate. Next the electrodes and the back gate

were bonded to the pins of the TO-8 package using a wedge bonder.

Some of the CNT devices were fabricated on Pd only electrodes. For these devices

it was not possible to bond to the electrode using the available bonders. Because of the

softness of Pd, the Au bond wire would not attach to Pd. Hence, conductive epoxy was

used to connect the bond wire to the Pd electrode. First a bond was made on the pin of the

TO-8 package, and then the bond wire was cut at an appropriate length so that the wire

reached the surface of the electrode. A minute amount of epoxy was then dispensed on

the electrode using a fine needle and then the bond wire was carefully pushed into this

drop of epoxy using a fine set of tweezers. The epoxy was then cured by baking in an

oven at 1200C for 8 to 12 hours.

During our first attempt to measure the CNT devices it was discovered that these

devices are extremely sensitive to static electricity. For a noise measurement the pins of

the TO-8 package need to be soldered to the measuring equipment. But the soldering

process and occasionally the mere handling was enough to destroy these highly sensitive

devices.
















Access to back gate


Wire bond


A Sample

Alumina Substrate


Conductive Epoxy

I/A
/ I]


TO-8 Package

Pin
Figure 2-5. Mounting of the device on a TO-8 package.


LI






20


To eliminate this problem all of the pins of the TO-8 package were soldered to the body

of the package using copper wires before attaching the substrates to the package. In this

way both ends of a nanotube in the sample always remain shorted together thus

preventing build-up of static charge that may damage the device. After the pins were

soldered to the measurement system the wire connecting the pins to the body of the

package was removed and measurements were made. The SNW devices did not show a

static sensitivity hence, did not require the procedure described above.














CHAPTER 3
DEVICE OPERATION AND DC CHARACTERISTICS

DC Characteristics of Carbon Nanotubes

Electronic Structure and Properties of Carbon Nanotube

A graphene sheet is a two dimensional structure of carbon atoms arranged in a

honeycomb like formation, one carbon atom at each vertex of the hexagon. A CNT has

the same basic structure of graphene but instead of a planer structure, it has a hollow

tubular shape. The physical, chemical and electrical properties of a CNT depend on how

the CNT was formed from the basic graphene structure.

Consider the graphene lattice presented in Fig. 3-1, a CNT can be formed by cutting

along the dotted lines joining them together forming a cylindrical shape. The vector

represented by AB is called the chiral vector C. The length of the chiral vector defines

the circumference of the CNT. The chiral vector can be expressed in terms of the lattice

vectors al and a2, mathematically,

Ch = na 2 +ma2 (3-1)

where n and m are integers and different combinations of n and m yield CNTs of different

chirality (the length and the angle of the chiral vector). The chirality uniquely defines a

particular type of CNT (n,m) [29], for example zigzag (n,0) and armchair (n,n) CNTs.

The formation of these two types of CNTs are explained in Fig. 3-2.

Typically, the length of a CNT is on the order of |tm and the diameter is on the order of

nm. Because of the small diameter the CNT operates as a one-dimensional device.




















Circumference vector: C = n a1 + m a2

(a)


0 (chiral angle)

Bb-


C (b)


(4,2)


Figure 3-1. Nanotube formation from graphene sheet; (a) definition of the unit vectors
and (b) the directions of the chiral vector for a (4,2) CNT is shown. (Image
courtesy of Dr. Ant Ural, Dept. of ECE, University of Florida).



































(4,0)
\




(n O) zigzag


(3,3)


(n, n) armchair


Figure 3-2. Formation of Zigzag and armchair nanotube from graphene. (Image courtesy
of Dr. Ant Ural, Dept. of ECE, University of Florida)


I
























15.0 '
^- 100 -
50
S0.0

K -lo. -
K


conduction


_- valence

F MK
(b)


Figure 3-3. E-k diagram of graphene in the first Brillouin zone using the 7t-band nearest-
neighbor tight-binding model; (a) in 3-dimension, and (b) 2D representation
showing the conduction and valence bands (Image courtesy of Dr. Ant Ural,
Dept. of ECE, University of Florida).









The energy dispersion relation (E-k plot) of a CNT can be obtained from the E-k plot of

graphene shown in Fig. 3-3. The E-k plot for a CNT can only be subsets of the two

dimensional plot for graphene, because of the quantization in the circumferential

direction [30]. These subsets or the energy band are given by slicing the E-k surface for

graphene with vertical periodic parallel planes with a constant spacing. The spacing of

these planes is determined by the chirality of the CNT [18]. If, for a CNT, one of these

planes intersects the K points, where the conduction band intersects the valence band,

then for that particular CNT, the band gap is zero, i.e. the CNT is metallic. This can only

happen if,

-n m = multiple of 3. (3-2)

If the planes do not intersect the K points, the CNT is semiconducting. For example, the

armchair (n,n) CNT is always metallic, but the zigzag (n,O) CNT is metallic if n is a

multiple of 3, semiconducting otherwise. For the same reason, of a randomly grown

collection of a total of n CNTs, n/3 are expected to be metallic and the rest

semiconducting. The energy band-gap of a semiconducting CNT is related to the

diameter of the CNT by [18]


E = ac- (3-3)
d

where t is the nearest neighbor tight binding overlap energy, ac-c is the nearest neighbor

atomic distance and dt is the diameter of the tube.

DC Characteristics and Measurements

As mentioned before, the conduction and valence band of a metallic CNT intersects

and as a result the resistance of a metallic CNT is independent of the gate bias. The

resistance of a metallic CNT can be written as









Rmeta ic R quantum +Rcontact + Rbulk (3-4)

Rquantum is the lower limit of resistance for a quantum resistance, equals to 12.9 kM per

sub-band [31]. Rcontat and Rbulk are the contact and bulk resistances due to non-idealities

present in the device. So, the lower limit for the resistance of metallic CNTs is 12.9 kM.

The characteristics of a semiconducting CNT are more complicated as the current

conduction is a function of the gate bias.

The operation of a semiconducting CNT also depends on the nature of the source

and the drain metal contact pads. Typically metals with high work function such as Ti

and Pd are used for these contacts. In this case at a low drain-source bias (Vds) the Fermi

level of the contacts lines up close to the valence band. As a result the potential barrier

for the holes becomes small but the barrier for the electrons becomes large and hole

conduction dominates via tunneling. For this reason a typical CNT FET device (CNFET)

resembles the characteristics of a conventional p-channel MOSFET. If a metal with a

smaller work function is used, then the Fermi level aligns somewhere between the

conduction band (Ec) and valence band (Ev), depending on the work function of the

metal. Fig. 3-3 shows a case where the Fermi level is aligned at the middle of the band

gap. In this case at a gate voltage close to OV, both electron and holes experience a

potential barrier so conduction is negligible (see Fig. 3-4(a)). With increasingly negative

gate bias, the width of the barrier for the holes decreases and increasing number of holes

can tunnel through (see Fig. 3-4(b)). At a sufficiently high negative bias the barrier is

completely removed and the current through the devices becomes independent of gate

bias.
































Drain E .- Drain

... qVd qVd
Source ,-"-- qVds Source / qVds
Ev

(a) (b)


Figure 3-4. Qualitative response of the nanotube conduction and valence band at (a) a
gate voltage below threshold voltage so that the CNT is off and (b) at a gate voltage
above threshold voltage so that the CNT is on [32]






























3x10-4 v (Volt)

2x10-4 -15V
-10V
1x10-4 --5V
SOV
5V
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
10V
1x10-4 15V
2x10_ 20V
-2x10-4
25V
30V
-3x 10-4 30V

-4x 10-4

Vd, (V)
Figure 3-5. Measured drain current vs. voltage characteristics at different gate bias









Because of the symmetry of the Fermi level alignment, electron transport can be achieved

with a sufficiently high positive gate, which makes this particular CNT an ambipolar

device.

The Id, vs. Vds plot at different gate bias for a typical device used in our study is

presented in Fig. 3-5. The plot is almost linear, but with higher Vds, sub-linear

characteristics were observed in most devices due to phonon dispersion, in few cases

super-linear characteristics were observed as well [33]. From the devices studied, each

contained a randomly grown matrix of metallic and semiconducting CNTs between the

drain and source Pd metal contacts. Fig. 3-5 indicates p-type channel operation and it

shows an on-state at Vg = OV gate bias as expected based on the earlier discussion. The

device becomes more conductive at increasing negative gate bias and saturates at Vg = -

10V gate bias. With increasing positive gate bias the conduction decreases because the

barrier for hole tunneling becomes increasingly wider. This device does not show

ambipolar transport because the gate bias required to lower the barrier enough for

electron transport cannot be achieved. A high gate bias is required because the Fermi

level is pinned close to the valence band edge and the gate oxide layer in these devices

was thick.

The metallic CNTs do not respond to the gate bias, so at a high positive gate bias,

when the semiconducting CNTs are off the measured current It is the total current

through the metallic CNTs Im. With Im known, at any gate bias the current through the

semiconducting CNTs Is, can be calculated from

I, = I I (3-5)









Alternatively, the total conductance of the semiconducting CNTs can be obtained by

subtracting the total conductance of the metallic CNTs (measured when the

semiconducting CNTs are off) from the measured device conductance. Plots of the total

dynamic (ac) conductance Gac and the metallic and semiconducting components are

presented in Fig. 3-6.

The DC characteristics were found to be a weak function of temperature for the

temperature range of our experiment 77K to 300K. A plot of the ac conductances at 77K

and 300K is shown in Fig. 3-7.

DC Characteristics of Silicon Nanowires

Resistance Measurements

The SNW devices used in this study have a much larger radius than the CNT

devices described earlier, around -50 nm. As a result, unlike CNTs, these devices do not

operate in the quantum domain. The resistance of these devices can be characterized by

equation (3-4) with Rquantum, 0. Since, the calculation of the bulk component of the

resistance is relatively straightforward, it opens up an opportunity to model the total

resistance in terms of bulk and contact components. The details of method employed for

this modeling is presented next.

The I-V characteristics of the SNW devices were measured using an HP4145B

semiconductor parameter analyzer. The devices were found to be very linear over the

voltage range of +5V.


































U
3x10-4 -

2x10-4

1x10-4-

0
-15 -10 -5 0 5 10 15 20 25 30
Vg (V)


Figure 3-6. Measured dynamic conductance and the calculated components vs. gate bias.































5.5x10-


5.0x10-4


0 4.5x10-4


4.0x10-4

3.5x10-4

3.0x10-4
-15 -10 -5 0 5 10 15
Vg (V)
Figure 3-7. Measured dynamic conductance vs. gate voltage bias at 77K and 300K.









The device resistance Rm calculated from the slope of the I-V plot (see Table 3-1), is

the parallel combination of the resistances of the bridging wires (R,) in the device, i.e.,

1
Rm =-- (3-6)
Z1-
zi=l,Nj

where Nis the number of wires in the device.

The total resistance of an individual wire is the sum of bulk resistance Rb, and contact

resistance, R,, i.e.,

R, = Rb + R, (3-7)

The bulk resistance of a wire is related to the bulk resistivity pb by,


Rb, = Pb. (3-8)


where r, and 1, are the radius and length of wire i, respectively. The effective resistivity p

of a device is calculated from the measured resistances and the dimensions of the wires

measured from SEM images using

2
p =Rm *. (3-9)
i=l,N /

From (3-7) (3-9),

1
P = pb Rm- (3-10)
=1,NR, Rc,

So the effective resistivity calculated using (3-10) is equal to the bulk resistivity pb only

in the absence of contact resistance (Rc, & 0) and greater otherwise.

Resistance Model

Bulk resistivity was calculated from the devices with the lowest resistivity and

noise. As these devices have the lowest contact resistance, the resistivity calculated using









(3-10) and neglecting contact resistance gives the best estimate of the bulk resistivity.

The carrier density p is related to the bulk resistivity by the following expression;

p=- (3-11)


where /p is the hole mobility and -q is the electron charge.

The corresponding carrier densities calculated using the resistivity versus impurity-

concentration relationship for bulk Si at 300K [34], are 5x 10s cm-3 and 1.3 x10 cm-3 for

wafers 1 and 2, respectively. Cui et al. [35] reported that the carrier mobility in highly

doped silicon nanowires is comparable to that observed in bulk silicon. Consequently, the

above listed values are assumed good estimates of the nanowire carrier densities. The

bulk resistances of all other devices were calculated using these carrier concentrations.

Note that dopant fluctuation is ignored in our analysis. This is a reasonable assumption

given the large number of dopant atoms per wire. Also the physical diameters of the

nanowire are used in the calculation. Due to the presence of surface charge, the surface

region of the nanowire is expected to be depleted, and as a result the effective diameter

becomes less than the physical diameter. The depletion width depends on the surface

charge density and the number of traps filled. Our initial estimate of surface charge

density is 2x1012 q/cm-2. The calculation for the worst-case scenario shows that the

resistance model presented here and the noise model (to be presented in chapter 5)

remains valid. Since the surface charge density is not well characterized at this time, the

analysis using the physical diameters of the nanowires is presented.

























TABLE 3-1 Device resistance
Device Number Rm (Q)
of wires
a 2 1.76x105
b 2 3.11x105
c 1 1.38x105
d 1 2.47x105

a 2 5.55x104
b 3 4.85x104
c 1 1.89x105
" d 5 7.47x104
e 2 5.08x105
f 1 6.30x105
g 2 1.76x105
h 2 2.64x105
i 2 3.93x105









From the devices containing only one nanowire, the contact resistance was

obtained from

Rc = Rm Rb. (3-12)

It was observed that the contact resistance is inversely proportional to the cross-

sectional area with a proportionality constant Kc of 1.69x10-5 Q-cm2, i.e.,


R = K. (3-13)


This model was applied to all other devices to calculate the contact resistance. The

minimum and maximum contact resistances obtained from (3-13) for individual

nanowires were 48.9 kM and 383 kM, respectively. The total resistance was calculated by

combining the calculated contact and the bulk resistances. A plot of the measured

resistance and the resistance calculated using the above model is shown in Fig. 3-8. The

plot shows good agreement between the measured and the calculated resistance for all

devices except for two devices.

Our model suggests a common mechanism for the contact resistance in all devices,

most likely resulting from the interface between the impinging end of the nanowire and

the sidewall. The base end of the nanowire is connected epitaxially to the silicon

sidewall, and thus the contact resistance on this side should be negligible. On the

impinging side however, the nanowire makes contact to the silicon electrode through the

pinholes of the native oxide [36], so the contact resistance on this side is expected to be

dominant. It is possible for the actual contact area to be different from the wire cross-

section, because the nanowire has to burrow through a native oxide layer. However, the

good fit of the model indicates that for all but two of the nanowires, the impinging






37


contacts are very uniform. A closer SEM examination of these two devices showed a

nanowire with a contact area much smaller than the cross-section of the nanowire, which

may explain why these two are different from the other devices. These two devices were

not used for subsequent noise modeling.



















8

C 7 Wafer 1
6o A Wafer 2
S Slope 1 Line
5


3


c2



o 0 I I I I
0 1 2 3 4 5 6 7
Measured Resistance (105 n)
Figure 3-8. Calculated vs. measured resistance, including contact resistance. The
uncertainties in the calculated resistance due to the measurement uncertainties
are shown. The uncertainties for the measured resistance are too small to be
displayed in the plot. The devices having only one nanowire are marked with
a circle.














CHAPTER 4
NOISE MEASUREMENTS

Noise Measurement System

A typical noise measurement system is shown in Fig. 4-1. The bias circuit shown is

required to measure excess noise. The bias circuit needs to have lower noise than the

device under test (DUT). Often the noise generated by the DUT is lower than the

detectable range of the spectrum analyzer, for this reason the low noise amplifier (LNA)

is used to amplify and bring the noise signal within the dynamic range of the spectrum

analyzer. Brookdeal-5004, a commercially available LNA was used in this study.

HP3561A, a low frequency spectrum analyzer (SA) was used to acquire the time domain

data and calculate the voltage noise spectral density [37]. The computer workstation was

used as an instrument controller for the SA and for importing the data for further

processing.

Noise Characterization of the LNA

The noise characteristics of the LNA is of the highest importance as it is at the front

of the signal chain [38]. For this reason the LNA was characterized for noise performance

in the beginning of the study. The noise of the LNA can be adequately described in terms

of a voltage and current noise source at the input node as shown in Fig. 4-2. If a resistor

R, is connected to the input, the voltage noise spectral density measured at the output of

the LNA is given by [38]

S, = A f [4kRTR + SnR + S,, (4-1)






































L-HP. P *.3
HP 3561A



- i


Instrument Control
and
Data Import


Workstation
Figure 4-1. Noise measurement setup.









where Aef = A, R,/(R + R,), R,, S,, and S,, are the effective gain, input resistance,

equivalent voltage and current noise sources of the LNA respectively. The current noise

spectral density for the resistance R, is given by [37]

S,, = 4kBT/ R. (4-2)

From (4-1) referring to the input side of the LNA;

Svo/A2f = 4kBTRj + S,,R + S,. (4-3)

According to (4-3) a log-log plot of Svo/Aff vs. R, should show three distinct regions (i)

Svo/A,f = S,, a constant for S,, > 4kTR, > S,,,R (ii) a slope -1 region for

S, B 4kBTR, and S,,R2 <_ 4k TR, and (iii) a slope -2 region for S,,R2 > 4kBTR, > S,.

One such plot is presented in Fig. 4-3. The plot indicates the different regions clearly

however in region (iii) the effect of RC roll-off at higher frequency is visible. If the

device resistance is within region (ii) a meaningful noise measurement can be made. If it

is in region (i), the equivalent voltage noise of the LNA dominates and if in region (iii)

the current noise of the LNA dominates. Regions (i) and (iii) are only visible at certain

frequencies, this indicates that the equivalent noise sources are thus frequency dependent.

The extracted spectral densities S,, and S,, using (4-3) are presented in appendix A.

Instrument Control and Other Measurement Issues

As mentioned in the previous section the LNA is used for amplification, as the

noise signal is typically too small for the SA to detect directly. But a situation can arise

when the noise signal is large enough to saturate the LNA and that will result in an

incorrect noise reading.




























vn, Svn


Ri Av


Figure 4-2. Setup for the noise characterization of the low noise amplifier.







43



















10-11

10-12
(iii)
re" ^10-13

> 10-14

15 -4- 10Hz
c'a -- t50 Hz
S 10-16 (i) 100 Hz
3S 500 Hz
C 0-17 r -- 1 kHz
10 5 kHz
101 --- 10kHz
-- 4KTR Line
10 -19 I I 1 I II I I IIII I I II I I III, I I II I II I I II
101 102 103 104 105 106 107 108
Resistance (Q)
Figure 4-3: Voltage noise spectral density at the input side of Brookdeal-5004 LNA for
different source resistance.









To prevent this situation, an analog oscilloscope was connected in parallel to the

SA to monitor the output of the LNA for clipping during noise measurement. Also, even

if the LNA does not saturate, a random high noise spike can overload the SA input during

a noise measurement and again cause an incorrect reading. An instrument control routine

was written in the HPVEE graphical programming language to continuously check for

an overload condition during measurement and stop the measurement if overload is

detected. Additionally an option to select the attenuation level for the internal attenuator

of the SA is added to the code that allowed precise control of the noise magnitude

presented to the SA input. The SA collects a set of noise data for a certain time span, and

then calculates the spectral noise density in the frequency domain. The calculated spectral

noise density for the first 10% of the bandwidth is considered not accurate due to finite

averaging time. So a control routine is written to start with the specified minimum

frequency span and repeat the measurement for every decade of frequency range up to the

maximum specified frequency and discarding the first 10% of the data for each decade. A

flow chart of the instrument control program is presented in appendix B.

Setup for Low Temperature Measurement

The TO-8 package containing the device was placed in the sample chamber of a

cryostat. The TO-package was mounted in a sample holder and placed on a copper finger

to be cooled with liquid N2. A layer of Iridium is placed between the TO-8 package and

the sample holder to aid the conduction of heat. The connections to the pins of the TO-8

package are made through feed-throughs in the windows of the chamber using cryogenic

wires and then the chamber is closed. The chamber is evacuated first using a mechanical









pump and followed by a rotary turbine pump in order to provide thermal isolation. A

vacuum level of -5mTorr was used for our experiments.

To cool the system, N2 was placed in a pressurized container. N2 liquid guided by

the pressure travels through a heat exchanger that is thermally connected to the copper

finger containing the sample. The N2 flow rate is controlled by pressure applied to the N2

container. The main temperature control is achieved by the flow rate and the heat transfer

control in the heat exchanger. The lowest temperature achievable using this system is

limited by the choice of N2 as the coolant, which is approximately 77K.

To achieve a finer temperature control a feedback heating system consisting of a

temperature sensor and a variable gain amplifier is used. The amplifier delivers a current

to a resistive heater proportional to the difference of the temperature set by the user and

the current temperature reading from the sensor. Using the amplifier gain for fine control

a stable temperature level to +0.1K within a few degrees of the desired temperature can

be achieved easily.

Overview of the Observed Noise Spectra

The voltage noise spectral density S, was measured for frequencies between 10Hz

and 100kHz and converted in to current noise spectral density S, using [39]

S, = S,/R2 (4-4)

where R is the resistance of the device under test (DUT). A typical plot of S, is given in

Fig. 4-4. For both CNT and SNW devices the noise observed was a combination of

excess noise and thermal noise. The current spectral density of the thermal noise is given

by [37]


Sthermal = 4KBT/R


(4-5)







46

















10-20



10-21
2- -Thermal Noise
o -Total Noise
1/f Noise
10-22
) 1
o
Sn Lorentzianntt

S10-23
C


n 24
1 10 100 1000 10000 100000
Frequency (Hz)

Figure 4-4. A typical plot of the measured current noise spectral density showing thermal
noise and the 1/f noise and the Lorentzian components of the excess noise.
The effect of RC roll off is visible at around 70 kHz









CNTs reportedly exhibit shot noise but the full shot noise magnitude for the

maximum bias level used in the study 13.9[tA, is 4.5x10-24 A2/Hz, below observed excess

noise levels. Moreover the actual shot noise level is expected to be suppressed, placing it

outside the range of observation [18]. The thermal noise level was higher (6.624 x10-23

A2/Hz for 1 kQ resistor at 300K). The observed excess noise was mostly 1/ft type noise

[40]. The current noise spectral density for 1/fI type of noise can be described by the

following expression [39];

A.I"
Slf -p (4-6)
f8

where I is the dc current level,f is the frequency, A, a and 8 are constants expressing the

relative noise magnitude, current and frequency exponent, respectively. The values of A

and fare estimated by fitting a line to the experimental data as shown in Fig. 4-4. The

parameter fwas found to be approximately equal to 1; hence this noise component is

called 1/f noise.

The value of the current exponent a is equal to 2 for a device in linear mode as was

the case for both CNT and SNW devices used in this study. An illustration of this

relationship is shown in Fig. 4-5 for a CNT device. Note that at the lowest current level

the 1/f noise is below the thermal noise floor at 1kHz, hence the plot for 1kHz deviates

from slope 2. From this point on, a= 2 is assumed and the following expression for 1/f

noise will be used;

A1I2
Silf = (4-7)
f,8







48



















10-20





10-21





10-23


-10 Hz
10-24 I 100 Hz
-1 kHz
-Fitted (10 Hz)

10-25
10-9 10-8 107
Current (A)
Figure 4-5. Plot depicting the relationship between the 1/f noise and device current. The
magnitude of 1/f noise is shown at different frequency points. The frequency
spectra of the 1/f noise for the three current levels are shown in the inset.









For the CNT devices in some cases Lorentzian spectra were observed. These

Lorentzian components arise when random number fluctuations are caused by processes

with a single characteristic time and activation energy. A Lorentzian component can be

characterized as [13],

S, (O)
SL(f) =SL( (4-8)
1f --2Z 2

where SL(O)is the plateau value of the Lorentzian, o) is the angular frequency and r is a

characteristic time. No Lorentzian component was observed in the SNW devices. The

result and analysis of the 1/f and Lorentzian components will be discussed in details in

subsequent chapters.














CHAPTER 5
NOISE IN SILICON NANOWIRES

Introduction

The result of the resistance measurement for these devices was presented in chapter

3. The measured resistivity and the 1/f noise coefficient A presented in Table 5-1 show

significant variations from device to device. However the plot of A vs. p (Fig. 5-1) shows

that the devices from both wafers that have the lowest effective resistivity also generally

show the lowest noise. Based on discussions in chapter 3, these devices can be identified

as the devices with low contact resistance. The fact that the low-noise devices also have

low contact resistance suggests that the source of the noise is the contact. To check this

possibility further a noise model was developed and will be presented next.

Noise Model

The proposed noise model includes bulk contact components just like the bulk and

contact components of resistance. The circuit diagram of the model is shown in figure 5-

2. From this circuit, the measured open-circuited noise voltage across the terminals is

given by

vn = ibn Rb + i,, R (5-1)

where ibn and Rb are the noise current source and the resistance respectively for the bulk

region, and icn and Rc are the noise current source and the resistance respectively for the

contact region of a wire.






51
















10-7
4- Wafer 1
.D A Wafer 2
S10 8




o
u 10





10-10
10-"11 . .l . .I .
10-2 10-1 1
Resistivity (f-m)
Figure 5-1. 1/f noise coefficient A vs. effective resistivity p. The uncertainties in A and p
due to the measurement uncertainties are shown.























TABLE 5-1 Relative noise magnitude
Device Number of A
Device A
wires
a 2 6.71x10-9
S b 2 1.20x10-9
Sc 1 2.92x10-11
d 1 8.45x10-9

a 2 1.22x10-9
b 3 3.37x10-10
c 1 1.61x10-10
d 5 4.04x10-11
e 2 1.69x10-10
f 1 2.76x10-10
g 2 3.51x10-11
h 2 4.21x10-11
i 2 4.73x10-10








































Figure 5-2. Circuit representation of the noise model









From (5-1) the total 1/f voltage noise spectral density S, in terms of the individual

current noise spectral densities is given by

S, = Sb R2 + S, R (5-2)

where Sb and S, are the current noise densities of the bulk and contact noise sources,

respectively.

From (5-2), with the total resistance,

R =Rb +Rc, (5-3)

the total current noise spectral density can be written as,


S, = Sb b + S,- (5-4)


Using (4-7), the expression for 1/f noise with P 1,


S, A A A2 = A .b. .. (5-5)
f dc f dc R) f dc (5)
f f R f R

where Ab and Ac are the 1/f noise coefficients for the bulk and the contact region,

respectively. Equation (5-5) can be simplified to


A = Ab + AC.- (5-6)


Most of the devices studied contained multiple nanowires. For those cases, the total

noise of the device is the sum of the noise contribution from all the nanowires in the

device. From (4-7)


St = .2 = I2 (5-7)
S11 f


SA, = 4A, (5-8)
I=1 it









where At and It are the combined noise coefficient and current for all the nanowires in the

device under study and, A, and I, are the noise coefficient and current for the i-th

nanowire. Using (5-6) (5-8)


At Ab,. +Ac, I (5-9)


From (5-9), the bulk and contact noise components can be separated. The bulk

noise is given by


A = Ab Rb \-2 (5-10)


and the noise component from the contact is given by,


Ac = A. R (5-11)


with A,= Ab + A. (5-12)

If either Ab or Ac is known, the other one can be calculated from (5-12).

The noise component of the bulk can be accurately determined from the devices

that have negligible contact resistance. To understand this consider (5-9). For negligible

contact resistance, i.e. Rb >> Rc, we have R Rb and R >> R,. Then from (5-9)

A Ab. (5-13)

Also, from (5-10) with R,J Rb,


Ab = [AbI--l (5-14)
I=\ IVt


From the well known Hooge model for bulk 1/f noise [39],










Ab, cHb (5-15)
P-V,

where aHb is the Hooge parameter, p is the density of carriers and V, is the volume of the

i-th wire. Using the Hooge model in (5-14),


Ab =aHb (5-16)



1 )2 (5-17)
1 p-V, \I,

The Hooge parameters were calculated from the devices with the lowest resistivity

and noise. As these devices have the lowest contact resistance, the aHb calculated using

(5-17) gives the best estimate of the bulk Hooge parameter. The calculated Hooge

parameters are 1.1 x 105 and 7.5 x 106 for wafer 1 and wafer 2, respectively. In general the

value of the Hooge parameter is a good indicator of the process quality, and the values

obtained for the Si nanowires are comparable to Hooge parameters for modern low noise

silicon bulk devices [40]. Using these calculated Hooge parameters the bulk and contact

noise A values for the other devices were calculated using (5-10), (5-11) and (5-16).

However, unlike bulk noise, there is no known model for contact noise, so the

contact noise magnitude per wire (Ac,) cannot be calculated directly from (5-11). To

calculate the contact noise it is necessary to assume a functional dependence between the

noise magnitude and some physical parameter such as the radius or length. One can

expect the contact noise to be some function of radius but independent of length. Hence,

the following model for the contact noise was adopted,

A,,c r," (5-18)









where r, is the radius of the nanowire. The exponent m determines how the noise is

related to the physical parameter of the corresponding nanowire. For example, for m = 0,

1 and 2, the noise is independent, proportional to the radius and proportional to the cross-

sectional area respectively. The values for m tested for a fit were -3, -2, -1, 0, 1, 2 and 3.

The best fit to the data was obtained for m = -2, in other words the best-fit model suggests

the relative noise is inversely proportional to the cross sectional area of the nanowire i.e.,


Ac, oc 2. (5-19)


The proportionality constants for wafers 1 and 2 are 4.7x10-1 cm2 and 4.6x1019 cm2,

respectively. It was shown in chapter 3 that the contact resistance is also inversely

proportional to the contact area. Comparing (3-13) and (5-19);

Ac, oc Rc (5-20)

This model suggests that the contact noise is proportional to the contact resistance,

which is reasonable, considering that both the noise and the resistance indicate the quality

of the contact. The calculated contact noise from measurements and the modeled noise

are shown in Fig. 5-4. The plot shows good agreement between the model and the

measured noise and also shows that the agreement is worse if the magnitudes of the bulk

and contact noise components become comparable. This is expected because the

calculation involves subtracting two statistical quantities; consequently, when the

magnitudes of the two noise components are comparable, the error in Ac calculated using

(5-12) would become higher.






58














10-7
10-8 Ac Ab -..Ac (Fitted)
10-8 | -


10-9
109


10-10


10-11


10-12


10-13
(0 "0 () .Q "O 4, -




Figure 5-4. Contact and bulk noise components calculated from measured data. The
contact noise component calculated from the model is also shown. The
calculated uncertainties due to measurement uncertainties are indicated.









Summary and Conclusion

The resistance model presented enables the calculation of the bulk and contact

components of the resistance. The contact resistance is believed to originate from the

impinging end of the nanowire where the nanowire connects to the uncatalyzed silicon

layer. To estimate the relative magnitudes of the bulk- and contact-resistance

components, consider a typical nanowire with a length and radius of 8tlm and 50nm,

respectively. For a doping level of Ixl101/cm3, the bulk resistance is 424 kM. The contact

resistance calculated from the model is 215 kM, and is a significant portion of the total

resistance, which, may be reduced by improved processing. The bulk 1/f noise coefficient

for this nanowire for a Hooge parameter of 1x10-5 is 5x10-10. The 1/f noise coefficient for

the contact noise from the model is l 10-8, for a proportionality constant of 1x1019.

Hence the contact noise is the dominant noise mechanism in this nanowire. The likely

mechanism for noise in the case of our devices is carrier trapping-detrapping in defects

producing the well-known 1/f-like number fluctuation noise spectra [37]. The impinging

end of the wire, where contact to the silicon electrode is made through possibly pinholes

in the native oxide, is expected to be defect rich and thus the dominant source of contact

noise whereas the base end of the nanowire is connected epitaxially to the silicon

sidewall creating a defect lean, lower noise contact configuration. Furthermore, because

of the higher resistance on the impinging side, any fluctuations in this contact will couple

out more to the device contacts.

This analysis shows that the Hooge parameter for SNWs, the figure of merit for

bulk noise performance, is within the range of Hooge parameters for modern low noise

bulk devices [40]. Because the contact was identified as the dominant source of noise,






60


further noise reduction can envisioned by optimizing the contact. Reducing the contact

resistance can potentially reduce contact noise because they originate from a common

mechanism, as indicated by (5-20); moreover less contact noise will couple out into the

remainder of the circuitry as the contact resistance becomes a smaller fraction of the total

resistance.














CHAPTER 6
SEQUENTIAL ABLATION OF CARBON NANOTUBES

During the CNT growth process typically multiple CNTs grow and bridge the gap

between the metallic contacts. Due to the random nature of the CNT growth of the total

number of these CNTs two thirds are expected to be semiconducting and the rest metallic

tubes [41]. In order to study a single CNT, a single CNT needs to be identified.

Commonly that is done using an Atomic Force Microscope (AFM). Once a CNT is

located and marked, the contacts can be placed and electrical measurements made.

Alternatively, CNTs are grown on a wafer containing a matrix of contacts. It is then

necessary to find a single CNT connecting two contacts with the help of AFM for this

case as well. Both of these processes are very time consuming and limit the number of

CNTs that can be studied. In our experiments due to the large size of the contacts,

multiple metallic and semiconducting CNTs are contacted simultaneously and a pulsed

voltage bias tube ablation method was implemented to delineate the noise and charge

transport properties of the metallic and semiconductor tubes [42-43]. Although

destructive, this method is suitable for extracting and comparing the noise properties of

the metallic and semiconducting CNTs because it allows the measurement of multiple

CNTs in a short period of time and due to the close proximity of the measured CNTs the

process variation is kept to a minimum.

Setup for the Sequential Ablation Technique

The samples were placed at room temperature in a pulsed bias circuit that allowed

for approximately 10ts duration voltage bias pulses applied at the drain terminal at a









repetition rate of 80Hz as depicted in Fig. 6-1. A positive, continues gate bias of +14V

was applied so that only the metallic tubes contribute to conduction. Upon slowly

increasing the level of the pulsed bias voltage at the drain and monitoring the device

current on an analog oscilloscope, the onset of Joule heating in a metallic tube could be

observed from small fluctuations in the current. Upon further increase of the pulsed drain

bias, a tube was ablated and the pulsed bias was switched quickly to the IkQ dissipative

resistor preventing damage to additional tubes. The ablation occurred approximate at

+17V for a pulse width of 6[tsec and a repetition rate of 80Hz. Current-voltage and noise

measurements were carried out before and after the ablation of each tube using a HP4351

semiconductor parameter analyzer and a HP3561A low frequency spectrum analyzer

operating between 10Hz and 100KHz. As mentioned in chapter 4, the noise observed is a

combination of low frequency 1/f-like excess noise and thermal noise. Shot noise levels

were below the detection range of our experiment.

Result and Analysis

The conductances of the metallic and the semiconducting CNTs were extracted

from the measured current-voltage characteristics. At sufficiently high gate voltage Vg, in

this case +14V or higher, the semiconducting CNTs are turned off, as a result the

measured conductance, Gm represents the total parallel conductance of the metallic CNTs.

The semiconducting CNTs are turned ON at Vg = -14V. So at this gate voltage the

measured conductance Gt represents the conductance of all the CNTs including the

metallic and semiconducting CNTs. Now the total parallel conductance of the

semiconducting CNTs, Gs is obtained using,

G, = G, G (6-1)
























Pulse (ji
Generator









Analog Oscilloscope


1 kn


50 n


Gate
Bias


Figure 6-1. Pulsed bias experimental set-up for sequential ablation of metallic CNTs.









The low bias conductance of metallic and semiconducting CNTs in sample A

measured after an ablation event, are plotted as a function of the number of tube ablations

in Fig. 6-2. The plot clearly shows the conductance change of the metallic CNT matrix

due to ablation, while the total conductance of the semiconducting tube matrix remains

unchanged as expected. As metallic tubes are ablated successively, the total number of

CNTs remaining becomes less and as a result the total conductance drops, which is

confirmed by the measured data shown in Fig. 6-2, except after the first ablation attempt.

The increase in conductance after the first ablation attempt appears to be the result of an

annealing effect, which will be discussed later in the next section. From the difference in

measured conductance before and after an ablation the conductance of the ablated CNT is

obtained. The calculated conductances of the ablated CNTs are presented in Fig. 6-3.

From the data the average conductance of a single metallic tube is calculated to be

2.7x0l5S or 37kQ. No metallic or semiconducting CNTs were found to be connecting

after the 7th ablation attempt; the most probable cause is static damage during the

measurements. However, the total number of metallic CNTs in the sample can be

estimated from extrapolating the fitted line in Fig. 6-2. It was estimated that the original

sample contained 8 metallic tubes. Under random CNT growth conditions typically 1/3 of

the tubes show metallic properties while 2/3 becomes a p-type semiconductor for the Pd

contact system used in this experiment [41]. Consequently sample A is expected to have

contained 16 semiconductor tubes for a total of 24 tubes originally.

By measuring the linear conductance of the samples at Vg= -14V (semiconductor

and metallic tubes both conducting) and at Vg = +14V (metallic tubes only) the

conductance of the semiconductor tubes at Vg = -14V can now be estimated.






















2.0x10-4


8 1.5x10-4
"U
1.0x10 -4

S-4- Metallic CNT
5.0x10-5 Semiconducting CNT

0 &----- -- -W -- -I- -
0 1 2 3 4 5 6
Number of Ablation
Figure 6-2. The measured low bias conductances of metallic and semiconducting CNTs
in sample A, plotted as a function of the number of CNT ablation attempts
performed on the sample









For sample A, this resulted in 1.7 MQ per semiconductor tube. The spread in data

displayed in Fig. 6-3 is in line with values reported in the literature and may be due to

small variations in the Pd metal contact-CNT interfaces from sample to sample [44].

From (4-7) the measured 1/f noise of carbon nanotubes can be expressed as


S, (f) (6-2)


where the excess noise factor At models the relative magnitude of the excess noise and /

is the frequency exponent. The value of A/Rt, where Rt is the total ohmic resistance of

the sample, has been shown to be close to 10-11 Q-1 for single CNTs [7]. Assuming that

the parallel CNTs in our samples produce noise independently, the total 1/f noise can be

formulated as a combination of noise contributions from the metallic and semiconducting

CNTs;

S A(f 2 1 mt Am s A -
S(f) = = + (6-3)


where the subscript m and s denotes the metallic and semiconducting noise and current

contributions, and, i andj subscripts represent the individual metallic and semiconducting

CNTs respectively. M and S are the total number of metallic and semiconducting CNTs

in the sample. Simplifying (6-3),

M (j>2S (J 2
At,= A. m + A. (6-4)
,=I \,, Ij J

Writing in terms of the total and individual metallic and semiconducting CNT

conductances,

M 2 (6-5
A, = A-m, + A (6-5)
-1 I Gt J-1 I t
























4.0x10-5

3.5x10-5

S3.0x10-5

2.5x10-5

0 2.0x10-5

1.5x10-5

1.0x10-5 I I I
2 3 4 5 6
Number of Ablation

Figure 6-3. Calculated conductances of the ablated metallic CNTs for sample A.









where Gm, and Gsj subscripts represent the conductance of the individual metallic and

semiconducting CNTs respectively. And Gt represents the total conductance of the

device. From (6-5)

M S
2G 2+ + GA (6-6)
A, tG,2 = A,, .G, +- A, G (6-6)
1=1 j=1

From (6-6) it is evident that the noise measurements at Vg = +14V represents the 1/f

noise contributions for the metallic CNTs. Because at this gate bias, the semiconductor

tubes are cut-off, Gs & 0, and presumably contribute negligible noise levels. This

assumption was validated by the result of the analysis as it will be shown later in this

section.

From (6-6) it also follows that at each successive ablation of a CNT the measured

A, Gt2 will decrease. The change in A, G,2 represents the lost noise contribution (and

conductance) from the ablated CNT. This successive drop in the measured A, G,2 value is

illustrated in Fig. 6-4 for Vg = +14V where the second term on the right hand side of (6-6)

has a negligible contribution. The conductance of the ablated CNTs is known from the

current-voltage measurements, so the 1/f noise coefficient of the ablated CNT can be

obtained from (6-6). For sample A, the calculated values for Am and Am/Rm are presented

in Fig. 6-5. The average value of Am for sample A was found to be 2x105 resulting in

(Am/Rm)= 4.9x10-10S. Note that the measurement data after the very first ablation was not

used to calculate the average because of the observed annealing effect, to be discussed in

detail later. Analyzing the noise data measured at Vg = -14V which includes

semiconductor and metallic tube noise contributions resulted in (A,)= 3.7x 103 or (As/Rs)

= 2.2x10-9S.

































\ I


I I I I I I
0 1 2 3 4 5 6


1x10-12







1x10-13 <







1x10-14


Number of Ablation
Figure 6-4. Measured At/R and At.Gt2 at Vg = +14V for sample A, plotted as a function of
the number of CNT ablation attempts performed on the sample.


1x10-8







, 1x10-9







1x10-10


--- At/Rt

-- At.Gt2

































l AmlmmV -




4 A'mi/Rmi

I I I I \ 1x10-10
2 3 4 5 6
Number of Ablation
Calculated A and A/R for the ablated metallic CNTs for sample A.


1x10-4







j 1x10-5







1x10-6


Figure 6-5.









Measurements were performed on sample B on the same wafer and on sample C on

a different wafer. For both of the samples B and C, complete ablation of the metallic

CNTs was achieved, so only semiconducting CNTs remained at the end of the

experiment. A summary of the data is presented in Table 6-1 and the A vs. R plot for all

the data points collected is presented in Fig. 6-6. The values of the frequency noise

exponent /range between 0.8 and 1.2 for the samples studied.

The measured A/R values almost all exceed the average 9x 1011 Q-1 value reported

by Snow et al. [8]. From this we conclude that our samples are relatively noisy which

may be attributed to the sputtering deposition of the Pd contacts. Also it is clear that in

contrast to conventional metal and semiconductor systems where the excess noise of

metals can be typically ignored, the excess noise of metallic tubes needs to be accounted

for. Also note that the extracted A values validate our initial assumptions made in the

analysis of the data.

Details of the Observed Annealing Effect

As mentioned before, an annealing effect was observed in our experiment. The first

time a sample was subjected to the pulsed bias, the conductance of the sample increased

while the noise level improved. The conductance improvement at the first ablation

attempt is obvious from Fig. 6-2. The effect of the 1/f noise level improvement can be

seen in Fig. 6-4. Note that A, G,2 decreases after the first ablation attempt even though

the conductance Gt increases, which suggests a reduction in the 1/f noise coefficient At.

Furthermore the difference between the measured noise level before and after the first

metallic CNT ablation was found to be roughly 2 to 4 times higher than the rest. This

again suggests an improvement of noise during the first ablation attempt.






72














10-1
S D O Sample A
10-2 A Sample B
O Sample C

10 -3


10-4


10-5 r


10-6
104 105 106 107
R (Q)
Figure 6-6. Calculated A vs. R values for the ablated CNTs. The solid and open symbols
represent the metallic and semiconducting CNTs respectively. The data points
estimated from the remaining semiconducting CNTs after all the metallic
CNTs have been ablated are indicated with a circle.









So it appears that during the application of power to a virgin CNT sample an annealing

effect is occurring due to Joule heating. This also raises the question whether the results

for the individual CNTs presented are valid as the ablation process itself may change the

property of the CNT being measured. To answer this question we turn our attention to the

AtRt plot shown in Fig. 6-4. The parameter A/Rt can be considered as a figure of merit

for the comparison of relative noise magnitudes of CNT samples [7]. The noise reduction

at the first ablation attempt due to annealing is clearly visible from the large change in the

A/IR plot. However after the very first ablation attempt the change in the A/Rt values lies

within the statistical spread of our data and would not affect the result appreciably.

Furthermore, the calculated parameters for the individual ablated CNT's conductances

(see Fig. 6-3) and noise magnitudes (see Fig. 6-5) also do not show an appreciable trend

over successive ablations. So we conclude that the ablation process does not change the

properties of the CNTs appreciably after the initial ablation.

Summary and Conclusions

The pulsed bias experiments were successful in demonstrating the sequential

ablation of single metallic carbon nanotubes. The conductance and relative excess noise

factors of semiconductor and metallic tubes were determined showing that both produce

1/f like excess noise. However the relative 1/f noise level in the metallic tubes as

expressed by the parameter A is two orders of magnitude lower, and the ratio A/R is

between a factor 3 to three orders of magnitude lower than in the semiconducting CNTs.

There is a large spread in the resistance of both metallic and semiconducting CNTs in all

three samples measured (see Table 6-1). However both A and the factor A/R, used as a

figure of merit for noise comparison is consistently higher for semiconducting CNTs in

all cases. In two instances during ablation attempts on sample B and C, it was observed









that semiconducting CNTs were ablated along with metallic CNTs. The most likely

reason for this is static damage caused by connecting the devices to the measurement

equipment as mentioned earlier. This unintended ablation of semiconducting CNTs

provided an opportunity to measure the resistance and the noise contribution of the

ablated CNT directly, rather than calculating the resistance and noise contribution from

the remaining semiconducting CNTs after all the metallic CNTs were ablated. The

resistance and the noise contribution for these semiconducting CNTs were found to be

within the same range as the semiconducting CNTs remained after metallic CNTs were

ablated.

Sample C was from a different wafer than sample A and B. Sample C was

annealed in Ar at 2200C for 10 minutes, while sample A and B were not annealed. The

resistance of the metallic CNTs in sample C is higher compared to the samples A and B.

However it is possible that this higher resistance is not caused by annealing but simply

results from process variations from wafer to wafer. Furthermore, the resistance of the

semiconducting CNTs for sample C is within the range of resistances of the

semiconducting CNTs in sample A and B. The 1/f noise level in sample C does not show

an appreciable change from the other two samples. Hence the thermal annealing process

used did not have an effect on the noise properties of CNTs.

The fact that the annealing caused by the Joule heating due to current flow in the

device improves the conductance and lowers the 1/f noise level points to the contact as

the dominant source of 1/f noise. Because the CNT operates as a one-dimensional

conductor, most of the power is dissipated in the contact region [45], and an improvement

of the quality of the contact between the CNT and Pd caused by the Joule heating is






75


evident from the lowering of the contact resistance. Similarly, the improvement in the

noise level can be attributed to annealing out defects in the contact area. An annealed

contact is likely to have a lower number defect related traps, hence results in a lower

noise level.























Table 6-1 Noise and conductance data of samples A, B and C.
Device Data Sample A
Number of Metallic Tubes 8
Estimated Number of Semiconducting CNTs 16
Estimated Total Number of CNTs 24
(Rm)


(R,)


KAm)


Average
Std. Dev.

Average
Std. Dev.


37kQ
12kQ

1.7MQ


2.0x105
1.9x10-5


Average
Std. Dev.
Am/Rm)
Average
Std. Dev.
(A,)
Average
Std. Dev.
(AR,)
Average
Std. Dev.


4.9x10-10S
3.1x10-10S

3.7x10-3


2.2x10-9S


Sample B
3
6
9

103kQ
44kQ

189kQ
180kQ

1.9x10-5
1.3x10-6


2.0x10-10S
7.4x10-11S

3.8x10-3
5.5x10-3

1.4x10 SS
l.lxlO10S


Sample C
2
4
6

4.6MQ


498kQ
398kQ

1.1x10-4


2.3x1011S


2.7x10-2
1.3x10-2

1.2x107S
1.3x107S














CHAPTER 7
LORENTZIAN TYPE OF NOISE IN CARBON NANOTUBES

Introduction to the Lorentzian Noise Component

An overview of the noise properties of CNT devices was presented in chapter 4. In

Fig. 4-4 a measured plot of the current noise spectral density (S,) was given. A closer

examination of this plot reveals the presence of Lorentzian noise components

superimposed on the 1/f like noise. The presence of a Lorentzian component becomes

clear when the fitted 1/f like noise is subtracted from the measured excess noise as shown

in Fig. 7-1.

Lorentzian components arise when random number fluctuations are caused by

processes with a single characteristic time and activation energy. A Lorentzian

component can be characterized as [13]

S, (O)
SL(f)= (7-1)
l+ 2Z2

where SL(O) is the plateau value of the Lorentzian, ac is the angular frequency and r is a

characteristic time. The plateau values and the characteristic times can be obtained from

fitting the curve given by equation (7-1) to the data. The fitted curve is shown as a dotted

line in Fig. (7-1).

The Lorentzian(s) that showed up at a particular temperature shifts to down

vertically and to a higher frequency as shown in Fig.7-2. This movement of a particular

Lorentzian over temperature can be tracked until it moves out of the frequency range of

measurement.






78














10-16

0_18



10

N1
10)20 10-20 "

T








10-24 1024
10 102 103 104 105
Frequency (Hz)

Figure 7-1. Extraction of Lorentzian spectra from excess noise; (a) a typical excess noise
plot (top) with fitted 1/f noise line, and (b) bottom plot shows the Lorentzian spectra
obtained after subtracting 1/f noise from the excess noise.





















Total noise


Lorentzian shift with
increasing T


Lorentzian at T1


Lorentzian at T2
Lorentzian at T2


1/f noise


log(f)
Figure 7-2. A diagram describing the shift of Lorentzian spectra with temperature.









Results and Analysis

The noise measurement was repeated over a temperature range of 77K to 300K on two

devices (A and B). Several Lorentzians were observed and tracked across this

temperature range and their corresponding characteristic times (r) were extracted. The

calculated values for rat different temperature are shown in Fig. 7-3 for devices A and B,

respectively. Note that the horizontal axis is 1000/T, where T is temperature in Kelvin.

From the plot, it is evident that the characteristic times increase with decreasing

temperature and are clearly thermally activated i.e.,

r = rO exp(E / kT), (7-2)

where E is the activation energy and kB is the Boltzmann constant. This equation is well

known to explain the characteristic times of thermally activated generation-

recombination noise components in semiconductors with E = Er-EF, where ET is the trap

or defect energy level to which the carrier is activated and EF is the Fermi level [13]. The

activation energies calculated from the plot shown in Fig. 7-3 are listed in table 7-1. The

position of EF in the semiconducting carbon nanotubes is determined by the work-

function difference of the Pd/CNT contact system placing EF in these devices near the

valence band edge, independent of temperature [44]. As a result the values listed in table

7-1 can be interpreted as hole activation energies for carrier detrapping from different

defect or trap centers.

Figure 7-3 also shows that at sufficiently low temperatures the characteristic times

tend to reach a constant value. This trend is expected and originates from the mechanism

of the carrier trapping and detrapping process itself.






81











10-1
/ I-A III-A
10-2 I-B I

10-3 /
/ / -A

10-4 /


10-5 / IV-A

*/ IV-B /
10-6 /
11-B

10-7 I I I
2 4 6 8 10 12 14
1000/K
Figure 7-3. Lorentzian characteristic times for device A and B shown in solid and dashed
lines respectively. For sample A, the activation energies of the Lorentzians are
0.31, 0.27, 0.21 and 0.08 eV for Lorentzians I-A, II-A, III-A and IV-A,
respectively. For sample B, the activation energies are 0.42, 0.51, 0.2 and
0.29 eV for Lorentzians I-B, II-B, III-B and IV-B, respectively.









The measured characteristic time r is the reciprocal sum of the carrier capture time

zT, and the emission time re i.e.,

1 1 1
-+- (7-3)
T Te Tc

re is a strong function of temperature because the energy available to the trapped carrier

is a function of temperature. At a higher temperature, it is easier for the trapped carrier to

acquire the energy required to come out of the trap and as a result, the time a carrier

spends in the trap on average decreases, hence Te is lower at higher temperatures.

Conversely, zT is relatively independent of temperature [39] because the trapping process

is a random process dependent on the capture cross-section of the trap and trap

availability. At higher temperature, re << z, hence re dominates the observed value of T.

At lower temperature e >> c, so zr dominates and becomes independent of temperature.

From data it was observed that the plateau values of the Lorentzians SL(O), have a

linear relationship with that can be described as

SL(O)=X T. (7-4)

For device A, the value for x was found to be 3.1x10-7. For device B, the noise

measurement at each temperature was performed at the gate voltage levels of Vg = 0 and

+14V. In Fig. 7-4, a plot of the SL(0) vs. r measured at different temperatures is

presented. From the plot it is apparent that the linear relationship described in (7-4) is

also valid here. For this case the estimated values for x are 3.8x10-17 and 4.7x10-17 for Vg

= 0 and +14V respectively. This linear relationship suggests that the variance of the

number of carriers (AN2) is a constant [13].





























Table 7-1 Activation energies calculated using equation (7-2).
Activation energies in eV
Device A 0.08 0.21 0.27 0.31
Device B 0.2 0.29 0.42 0.51






84












10-16


10-17

10-18

10
S10-19 E

ci 10 n -20

.102 Plateau Value Vg = OV
1 0 -2 1 [
Fitted data Vg = OV
22 Plateau Value Vg = 14V
10-22 4 Q[rtnA
Fitted data Vg = 14V

10-23
10-5 10- 10-4 -410 10- 10-1
r (sec)
Figure 7-4. A plot of the Lorentzian Plateau value vs. characteristic time for device B at
Vg = 0 and 14V. The linearly fitted data is also shown.





















10-2 10-17




10-

1 19
-- Lorentzian 1: Characteristic Time 10
0 -- Lorentzian 2: Characteristic Time
10-4 / ,
u 10 -D- Lorentzianl: Plateau Value
S-A-- Lorentzian2: Plateau Value 100 C)



10 10-21



10-6 10-22
-14-12-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

vg (V)
Figure 7-5. A plot of the Lorentzian Plateau value and characteristic time at different gate
voltages for device A.









Effect of Gate Voltage

Figure 7-4 and the results discussed above also suggest that the gate voltage does

not change the Lorentzians appreciably. To verify this observation, sample B was

measured at different gate voltages at room temperature (approximately 300K). The

results of these measurements shown in Fig. 7-5, do not show a clear dependence on gate

voltage; thus confirming our previous observation. This observation is important and may

help pin point the physical location of the noise generating trap center.

First note that the Lorentzians are visible from Vg = -14V to +14V. In a

conventional device Lorentzians are associated with semiconducting materials, since in a

metallic system, because of the presence of a large number of carriers, an individual trap

center does not produce a noticeable effect on the transport characteristics. However, in

this case the Lorentzian is clearly associated with a metallic CNT as at Vg = +14V, where

a semiconducting CNT would not conduct and hence an associated Lorentzian will not

show up.

The calculated activation energies for the noise components are within the range of

energies of physical processes possibly present in the carbon nanotubes studied. For

example, the fluctuations caused by carrier transitions between one-dimensional sub-

bands will have activation energies equal to the energy difference between sub-bands.

The sub-band energy differences in carbon nanotubes can be close to the discrete

activation energies presented in table 7-1. Also, it may be possible that the noise

originates from local defect centers due to adsorption of a chemical species. It has been

reported that oxygen chemisorption on the surface of a semiconducting carbon nanotube

lowers the energy band gap. For example on a (8,0) SWNT the energy gap decreased









from 0.56eV to 0.23eV due to oxidation [46]. Local defects of this nature may produce

noise with activation energies in the range reported in this work.

However, the gate voltage insensitivity suggests that the defects associated with the

Lorentzians are located at or very close to the contact region of the CNT device since the

band bending due to the gate voltage is minimum near the contact region of a

semiconducting CNTs. As shown in Fig. 3-4 the energy band bends due to the application

of gate voltage and as a result the quantity E (= Er-EF) described in equation (7-2) will

change. The change in E is not uniform across the device. It has a maximum at the center

of the device and a minimum near the contacts. For a large change in gate voltage one

would expect a significant change in E and a noticeable change in rvia equation (7-2). In

the absence of any such effect on r, we postulate that the defects that generate the

observed Lorentzians are not located at the bulk region but are located at or near the

contact region of the CNT device.