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PAGE 1 1 CHARGE TRANSPORT IN GALLIUM NITRIDE NANOWIRES By RAMYA SHANKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Ramya Shankar PAGE 3 3 To my mother PAGE 4 4 ACKNOWLEDGMENTS I would like to take this opportunity to express my respect and gratitude to all the people who have cooperated to make this work possible I am indebted to my advisor, Dr Bosman. His guidance and encouragement have been invaluable during my graduate career. I w ould also like to thank my committee members Dr Ural and Dr Pearto n for their support. I am grateful to Dr. Ural and Dr Thompson for allowing the use of their laboratory equipment for this work. I would also like to thank Dr Biswas for his help in getting the magnetoresistance data. A special thank you goes out to Jaso n Johnson for fabricating the nanowires and for the SEM images used in this work. I would also like to thank Hemant Rao, Erdem Cicek and Ashkan Behnam for the many discussions and Al Ogden and Lin for their technical assistance with wire bonding. Lastly, I would like to thank my family for all their love and encouragement and the Almighty, for having shaped this piece of work through my hands. PAGE 5 5 TABLE OF CONTENTS ACKNOWLEDGMENTS ...............................................................................................................4 page LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 ABSTRACT ...................................................................................................................................11 CHAPTER 1 INTRODUCTION ..................................................................................................................13 Commercial Applications .......................................................................................................15 Properties of Nanowires .........................................................................................................15 Reported Properties and Applications of GaN Nanowires .....................................................16 2 FABRICATION .....................................................................................................................19 3 MEASUREMENTS ................................................................................................................23 4 SPACE CHARGE LIMITED TRANSPORT .........................................................................27 Introduction .............................................................................................................................27 Band Theory of a Solid ....................................................................................................27 Current Vo ltage Equations ..............................................................................................29 Traps ................................................................................................................................31 Limiting I V Characteristic .............................................................................................33 Determination of the Trap Distribution ...........................................................................38 Negative Differential Resistance ............................................................................................38 Parameter Extraction Using the Simple Model ......................................................................42 Trap Free Conduction .....................................................................................................43 Temperature Dependence of I V .....................................................................................43 Presence of a Single Shallow Trap ..................................................................................46 Correction Factors to Roses Equations .................................................................................46 Contact Effects ........................................................................................................................47 Application to Nanowires .......................................................................................................48 5 EFFECT OF POOR SCREENING ON SPACE CHARGE LIMITED TRANSPORT .........49 Correction for the Geometry of the Wire ...............................................................................52 Experimental Evidence for Poor Screening in Nanowires ..............................................52 Theoretical Calculation of Capacitance of the Nanowires ..............................................54 Formulating the SCL Current Equations .........................................................................55 PAGE 6 6 Calculation Using Correction for Aspect Ratio ......................................................................57 Summary .................................................................................................................................61 6 ANALYTICAL DETERMINATION OF MOBILITY ..........................................................64 7 MOBILITY OF GALLIUM NITRIDE NANOWIRES .........................................................71 8 DETERMINING MOBILITY FROM MAGNETORESISTANCE ......................................76 Theory .....................................................................................................................................76 Physical Magnetoresistance .............................................................................................77 Geomet rical Magnetoresistance ......................................................................................78 Mobility Equations ..........................................................................................................79 Geometrical and Physical Magnetoresistance ........................................................................80 Magnetoresistance in Nanowires ............................................................................................81 Measurement ...........................................................................................................................82 Calculation of Mobility ...........................................................................................................84 9 MONTE CARLO SIMULATION ..........................................................................................86 Analytical Computation of Mobility ......................................................................................89 Calculation of Hall Factor, Magnetoresistance Coefficients ..................................................93 Summary .................................................................................................................................97 10 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK ..........................................98 LIST OF REFERENCES .............................................................................................................100 BIOGRAPHICAL SKETCH .......................................................................................................104 PAGE 7 7 LIST OF TABLES Table page 11 Comparison of GaN with Si, GaAs, SiC ............................................................................14 21 Important impuri ty levels in Wurtzite GaN. ......................................................................22 31 Breakdown voltages (DC bias) of the nanowires for the first growth ...............................25 41 Slope and intercept of logJ 1/T plot for single traps and exponentially distributed traps ...................................................................................................................................38 42 Trap activation energy extracted from I V vs T curves. ....................................................46 51 Parameters extracted for two devices with deep traps .......................................................61 52 Summary of extracted device parameters using the SCL model .......................................63 61 Fitting parameters fo r GaN to find electron mobility ........................................................68 91 Material parameters for Monte Carlo simulation ...............................................................87 92 Hall factor and magnetoresistance coefficients for individual scattering mechanisms ....94 PAGE 8 8 LIST OF FIGURES Figure page 21 SEM images of GaN nanowires .........................................................................................20 22 AFM image of a GaN wire showing the diameter in nm (vertical scale). .........................21 31 Non linear I which the voltage is swept. ................................................................................................24 32 Linear I .........................................................................24 33 I V characteristics of Schottky devices that are .................................................................25 34 Current vs. voltage plotted for measurement of 2 different devices. The devices are rectifying for different polarities of vol tage. ......................................................................25 35 Current vs. voltage plot showing device breakdown. ........................................................26 36 I V measurements showing the effect of annealing. The solid line shows the I V characteristics before annealing and the dotted line shows the I V characteristic after annealing at 650C. .............................................................................................................26 41 Energy band diagrams ........................................................................................................28 42 Traps in an insulator. Trap Etn1 is a shallow trap and Etn2 is a deep trap ...........................32 43 The limiting triangle in log J log V space .........................................................................35 44 Current density vs. voltage plot in the loglog scale showing two sets of traps. ...............37 45 Current vs. voltage plot showing negative differential resistance .....................................40 46 Schematic diagram of the potential barrier at a repulsive trap. .........................................41 47 Plot of resistance vs. temperature showing increase in resistance with temperature. ........41 48 Plot of current vs. voltage ..................................................................................................44 49 Temperature dependence of I V at 50C 75C and 100C. ..................................................45 410 Plot of current density and the inverse of temperature on a linear log scale at 2.8Vand 2.9V. The fit to the log I vs 1/T curves are also shown. .....................................45 51 Injection into an ntype semiconductor..............................................................................52 52 Potential along InAs wires obtained by SCS measurements. ............................................53 PAGE 9 9 53 Electric field distribution in a wire ....................................................................................56 54 Plot of trap distribution vs quasi Fermi level. ....................................................................59 55 Current vs voltage plot of two nanowires of different radii. ..............................................60 56 Current vs. voltage characteristic of a 4 long device showing SCL behavior .................62 61 Current vs. voltage plots showing temperature dependence of current for ......................65 62 Temperature dependence of resistance ..............................................................................66 63 Fitting CaugheyThomas model to experimentally obtained mobility values. ..................68 64 Temperature dependence of mobility as a function of doping concentration. ...................69 65 Variation of mobility carrier concentration produc t with temperature. .............................69 71 Diameter dependence .........................................................................................................72 72 Diameter dependence of mobility ......................................................................................73 73 Variation of mobility with carrier concentration ..............................................................73 81 Hall geometry for magnetoresistance measurements. .......................................................80 82 Schematic of experimental set up to measure magnetoresistance, showing orientation of electric and magnetic fields. ..........................................................................................83 83 Temperature dependence of resistance with B field applied. ............................................83 84 Magnetoresistance vs temperature. ....................................................................................84 85 Low temperature mobility from magnetoresistance measurements ..................................85 86 Estimated high temperature mobility. ................................................................................85 91 Velocity electric field curve at different temperatures. .....................................................88 92 Doping dependence of the vE curve. ................................................................................88 93 Mobility variation with temperature. .................................................................................89 94 Velocity vs electric field curve obtained by Monte Carlo simulation of bulk GaN at 300K ...................................................................................................................................90 95 ...............................................................................92 96 Mobility variation with temperature. .................................................................................92 PAGE 10 10 97 Total mobility calculated from components shown in Fig 96. .........................................93 98 Hall factor variation with temperature. ..............................................................................95 99 Physical magnetoresistance coefficient plotted as a function of temperature. ..................96 910 Plot of geometrical magnetoresistance coefficient with temperature. ...............................96 PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CHARGE TRANSPORT IN GALLIUM NITRIDE NANOWIRES By Ramya Shankar August 2009 Chair: Gijs Bosman Major: Electric al and Computer Engineering In the last couple of decades gallium nitride devices have gone from the laboratory to commercially viable devices in the radio frequency domain and in solid state lighting. Nanowires of the same material should ideally be miniature versions of bulk devices, with the additional advantage of packing density. Carrier transport in nanowires is dominated by the type and geometry of the contact. Therefore, all device characteristics must be obtained in a circuit configuration that very nearly approximates that of the proposed application. Conventional methods of obtaining mobility, like the Hall Effect would not be very accurate because of the different contact geometry and circuit configuration involved. The present study focuses on asgrown nanowires with a mean diameter of 50 nm. The wires are too thick for energy quantization and thus are not one dimensional structures. But, the high aspect ratio of the nanowires modifies many of the electrostatic properties, main among them being poor screening. Space charge limited transport is investigated in these wires to obtain trap activation energy, trap density, equilibrium carrier concentration and electron mobility. The early onset of space charge limited transport is attributed t o poor screening in these wires. Electron mobility values obtain ed range from 100575 cm ^2 / V s ( 575 cm ^2 /V s for the first growth and 111 PAGE 12 12 cm^2 /V s for the second growth) Silicon, gallium vacancy and gallium on a nitrogen site have been identified as th e defects contributing to trap limited transport. The average carrier concentration is of the order of 10^17 /cm^3. Magnetoresistance has been employed to independently evaluate mobility and thus validate the results obtained by analyzing I V characteristics in the space charge limited regime. The mobility obtained by this experiment is 709 cm ^2 /V s (for the first growth) The difference between the mobility obtained by the two methods is 18.8%. PAGE 13 13 CHAPTER 1 INTRODUCTION The developed world today enjoys an unprecedented access to information and computational tools to process information, ther eby creating more information. The beginning of the digital age can be traced back to the invention of the silicon transistor in 1947, which heralded the beginning of the Integrated Circuits industry. The silicon MOSFET is the workhorse of the IC industry and engineers have for decades, worked toward making smaller transistors thereby increasing performance and reducing cost and power consumption. The exponential reduction in size has followed Moores law for decades, driving the quest for new materials, processing techniques and device structures 1. Despite the ubiquitous presence of silicon in logic circuits, there are some niche applications to which other materials are better suited. III V semiconductors are direct band gap materials and thus can be used in opto electronic applications. Gallium Arsenide is perhaps the most studied compound semiconductor, but the wide band gap III nitride materials have generated cons iderable interest since the 1960s for their potential to generate and detect green, blue and ultra violet light. Apart from using III Vs for optical switch ing, using GaAs for generating red light, IIInitrides for blue and green light, the entire color spe ctrum is covered, making white light illumination possible. Fluorescent and incandescent light sources can be replaced by these semiconductor light sources, reducing power consumption by close to 90%2. Gallium Nitride (GaN) crystallizes in both wurtzite and cubic phases with a direct band gap of about 3.5 eV. The wide band gap allows for operation at high temperatures because a GaN device would turn intrinsic at higher temperatur es as compared to semiconductors with smaller band gaps, like Si or Ge. Thus power devices can operate at higher temperatures with far less cooling than would be required for silicon or germanium devices. The critical electric field PAGE 14 14 for breakdown is propor tional to the square of the band gap. GaN has a high breakdown field greater than 4MV/cm. Table 1 1 lists some of the important properties of GaN as compared with that of Si GaAs and SiC A combined rating for high temperature and power is computed based on the band gap, breakdown field, saturation velocity and thermal conductivity. This combined figure of merit is much higher for GaN than for any of the other materials. Mechanical and thermal stability, possible passivation using Ga2O3 are other desirabl e properties of GaN2. Table 1 1. Comparison of GaN with Si, GaAs, SiC2 Property Si GaAs 4H SiC GaN Bandgap (eV) 1.12 1.42 3.25 3.4 Breakdown field (MV/cm) 0.25 0.4 3 4 Electron mobility (cm2/V s) 1350 6000 800 1300 Saturation velocity (107 cm/s) 1 2 2 3 Thermal conductivity (W/cmK) 1.5 0.5 4.9 1.3 Figure of merit 1 8 458 489 There were difficulties in finding a good growth substrate and effective doping because of which GaN development has lagged far behind Si and GaAs. Homoepitaxial growth and ptype doping of GaN are two problems that have been recently solved, leading to c ommercially viable GaNbased light emitting diodes (LEDs), lasers and UV detectors. Growing stable, ohmic contacts, especially to p type GaN, is a challenge that still needs to be addressed. A lot of progress has been made in GaN based devices driven by th e commercialization of photonic devices. GaN Schottky rectifiers have breakdown voltages above 4kV. GaN, in conjunction with AlGaN as a hetero structure has generated a lot of research interest due to high mobility of carriers in such transistors2. PAGE 15 15 Commercial A pplications GaN devices are no longer confined to research labs. With the launch of LED lighting sources that can be plugged to regular a.c power outlets by Seoul Semiconductors, the importance of GaN to complete the color spectrum and produce white light makes it a more lucrative investment. The GaN RF market was forecast to be commercially viable from 2008, with players like Cree, RF Micro Devices, Toshiba and Nitronex ta king a keen interest 3. Nitronex was ready to ship GaN WiMax transistors, with GaN grown on Si in the last quarter of 20054. Eudyna and NTT announced the deployment of the first GaN based WiMax network for testing in Tokyo in 2006. The GaN RF market is for ecast to reach $100M by 2010 5. The GaN LED market with an estimated 5 million units, mainly blue, green and white LEDs, is estimated to be $3.5B6. The late entry of GaN into the semiconductor device marke t has accelerated miniaturization of GaN devices. GaN is grown and used in bulk, thin film and heterostructure form. One dimensional (1D) nano structures are being investigated as the basic building blocks of nanoscale circuits. The large surface to vol ume ratio and the unique electronic properties of a 1D system could be exploited in a new class of devices. Some thermal, mechanical and electronic properties of a material change when grown as a nanostructure. Properties of N anowires The melting point of a solid grown as a nanostructure is far lower than in its bulk form. Therefore, the annealing temperature required to make defect free nanowires, would be much less than that needed for bulk devices. The low melting point is also advantageous when it comes to cutting and welding nanowires together to form a circuit. But, it would reduce tolerance to fluctuations in ambient temperature and limit high temperature applications. PAGE 16 16 Single crystalline 1D nanostructures have fewer defects per unit length and have m ore mechanical strength compared to their bulk counterparts. For polycrystalline materials, decreasing the size of the grain boundaries increases the mechanical strength up to a characteristic length, beyond which, the strength decreases again. The electri cal properties of a material change dramatically when the dimensions of the nanowire become comparable with the wavelength of electrons. The quantization of the density of states and change in behavior from conductors/ semi conductors to insulators are among the many effects of miniaturization. Unlike bulk circuits which are built top down, nanowire circuits have to be built bottom up. The small dimensions of the individual devices would allow for very high density packing on a chip, with improvement in mat erial synthesis. Well cleaved nanowires can be used as lasers without having to use mirrors to confine the phonons generated, with the nanowires acting as resonant cavities. Nanowires also exhibit non linear optical properties, making them useful as frequency converters and routing elements. The high surface to volume ratio of nanowires makes them more efficient sensors 7. Reported Properties and Applications of GaN N anowires Electrical and optical properties of GaN nanowires are being investigated by many groups. The growth direction of the nanowires and the shape of the cross section can be controlled by changing the substrate 8. Field emission devices with sufficient current d ensities to drive displays and for vacuum microelectronic devices have been demonstrated in GaN nanowires 9 10. Lasing has been demonstrated with the ends of the wire acting as the mirror 1112. GaN nanowires have also been used as gas sensors 13. Electrical properties of these wires seem to vary from growth to growth. The reported electron mobilities range from 30 1500 cm2/V s 1 4 16. The variation in mobility is due to the growth conditions and the diameter of the wires. The extent of the surface depletion region PAGE 17 17 depends on the size of the wire and changes the carrier concentration. Surface depletion in thin wires extends until the entire wire is depleted and only conducts when illuminated by UV light 16. Space charge limited current flow has also been observed in GaN nanowires 1617. The dependence of electrical characteristics on growth conditions necessitates a full electrical characterization of all devices for each growth. Though the literature reports space charge limited current in the se nanowires, the I Vs have not been used to extract device parameters (except 17). Mobility and carrier concentration as a function of temperature and the density and activation energy of traps are the parameters needed to completely characterize a device. An identification of traps and impurities present is the first step to developing better fabrication p rocedures. Nanowires have unique properties and applications due to their high surface area to volume ratio. The high aspect ratio also leads to electrostatic properties very different from bulk devices. Electrical transport in nanowires is dominated by th e properties of the contacts 18. This makes it necessary to obtain device characteristics from nanowires in a circuit configuration as close as possible to the one in which the use of the nanowires is proposed. Thus, if mobility of a wire is measured using a Hall set up, it is likely that the wire will not display the same mobility when operated as a FET. Transconductance is commonly used to compute mobility but the formula used to compute the capacitance of a wire assumes an infinitely long metallic wire. A metallic wire would completely screen an applied potential whereas most semiconductors are nondegenerately doped and have nonideal screening. The finite length of the nanowires implies that the fringing capacitance of the electrode s should also be considered. All these complexities make the accuracy of mobility values obtained from transconductance measurements questionable. PAGE 18 18 The focus of this work is to explain the different types of I V characteristics obtained from GaN wires and e xtract device parameters from the I V measurements where possible, considering the electrostatics unique to nanowires because of their high aspect ratio. Chapter 2 describes fabrication of nanowires and properties of GaN. Chapter 3 gives a brief overview of the types of I Vs obtained from the nanowires, which are explained in subsequent chapters. Chapter 4 deals with space charge limited transport and details a method to extract mobility, carrier concentration, trap density and activation energy from the I V measurements. Chapter 5 focuses on the effects of poor screening in nanowires on the onset of space charge limited transport. Chapter 6 describes a theoretical model to fit carrier concentration and mobility to resistance as a function of temperature. Chapter 7 presents a review of the reported values of mobility in GaN nanowires. Chapter 8 describes the theory and measurement of magnetor esistance. Chapter 9 deals with Monte Carlo simulation to estimate mobility and analytical determination of Hall factor and magnetoresistance coefficients. PAGE 19 19 CHAPTER 2 FABRICATION The GaN nanowires reported in this thesis were grown at P rofessor Urals N anotechnology lab using gold nanoparticles as catalyst. A gold layer about 15 thick was first e beam llium metal (99.9999% pure) was placed in a quartz boat within a tube furnace. The growth substrate was positioned 3 cm downstream of the quartz boat inside the furnace. Residual oxygen in the growth chamber was purged with Argon, which is streamed for 10 min at room temperature. To form the catalyst nanoparticles, the su bstrate was heated up to 850 C and annealed in Ar. High purity hydrogen and ammonia (both 99.999% pure) were streamed through the chamber for ~5 hours and the temperature was maintained at 850 C. The chamber was cooled and the wafer removed from the chamber when the temperature fell below 100 C to prevent oxidation of the wires13. The nanowires grown as outlined above have diameter s of around 100 nm. The as grown wires were scraped off to a silicon substrate. Contacts spaced 2, 4 m were deposited by lithography. Figure (21) shows as grown wires and a wire with contacts placed and Figure (2 2) shows some AFM images of the wires. Different metals were used to make contact to the wires. The first set of wires had a stack of four metals Pt/Ti/Al/Au, grown one on top of the other for contacts where Pt is the metal layer lying on the GaN nanowire. Many of these wires had a small Schottky barrier, though some showed pure ohmic behavior. Some of the devices also turned out r ectifying. Annealing the devices at 100 C for 5 minutes in an RTA produced more linear devices with ohmic contacts. There were no rectifying Schottky devices in the annealed sample. Using Ti/ Au, with Ti contacting the GaN surface, for contacts leads to Sch ottky devices, most of which were PAGE 20 20 rectifying. Thus the metals used to make contact to the nanowire play an important role in determining the properties of the device. A B Figure 21. SEM images of GaN nanowires A) As grown B) with contacts placed. PAGE 21 21 Figure 22. AFM image of a GaN wire showing the diameter in nm (vertical scale). Some of the wires grown have a circular crosssection and some have a triangular crosssection19. It can also be seen from Figure (2 2) that some wires are perfectly cylindrical with a constant diameter, while some wires are broad at one end and narrower at the other. The tapering effect in nanowires is attributed to the growth procedure the end close to the catalyst is believed to be widest, and the distribution of defects as investigated by photoluminescence also varies from one end to the other 20. As grown nanowires are believed to be ntype and the main contribution to defects arises out of N and Ga vacancies. Dislocations are almost non existent in nanowires unlike in bulk materials where grain boundary scattering plays a big role in determining mobility. PAGE 22 22 The activation energies of the commonly found traps in Wurtzite GaN are listed in Table (2 1). The growth chamber is likely to contain traces of Si and C and therefore these impurities are likely to be present in the nanowires. Ta ble 21. Important impurity levels in Wurtzite GaN. The energies are in eV 21 Impurity / Defect Ga site N site Si 0.012 0.02 N vacancy 0.03,0.1 C 0.11 0.14 Ga on N site 0.59 1.09 Ga vacancy 0.14 PAGE 23 23 CHAPTER 3 MEASUREMEN T S Gallium nitride wires from four different growths were studied, with the flow rate of ammonia varied in each growth. The GaN wires are just barely visible under an optical microscope, and every pair of contacts was thus checked to see which ones had a single wire bridging them. Room temperature DC I V characteristics were measured for every such nanowire. The GaN devices show both linear and non linear I V characteristics, with many having a Schottky barrier if not annealed. Linear characteristics dominate for devices of 4, 6, 10 micron lengths. The 2micron devices show more nonlinear beha vior, with some having distinct slope 1slope 2 transitions on a loglog plot of current vs. voltage which is characteristic of space charge limited transport. Figure 3 1 shows an I V characteristic obtained for one of the m devices, showing a sharp rise in current with one voltage sweep, and a slump in current with the other. The sharp rise in current is attributed to presence of traps and space charge limited conduction, and is discussed in detail in Chapter 4. The downward slump in current can als o be explained by the presence of traps, and the explanation for it is also found in Chapter 4. Figure 32 shows a linear I show a hysteresis. Figure 33 shows I V measurements of devices wit h a Schottky barrier, one of which conducts for both polarities of the applied voltage, and the other is showing a reverse saturation current. Going over all the IV plots, it is possible to find I V measurements for devices with a Schottky barrier that seem to be laterally reversed. One such pair is shown in Figure 34. Figure 33 shows I V measurements of devices with a Schottky barrier, one of which conducts for both polarities of the applied voltage, and the other is showing a reverse saturation PAGE 24 24 curren t. Going over all the I V plots, it is possible to find I V measurements for devices with a Schottky barrier that seem to be laterally reversed. One such pair is shown in Figure 34. Figure 31. Non linear I V characteristic m wire. The arrows i ndicate the direction along which the voltage is swept. Figure 32. Linear I V characteristic of a device. 4.00E 04 3.00E 04 2.00E 04 1.00E 04 9.00E 18 1.00E 04 2.00E 04 5 3 1 1 3 5I AmpereV Volt "5 to 5" 5 to 5 6.00E 04 5.00E 04 4.00E 04 3.00E 04 2.00E 04 1.00E 04 0.00E+00 1.00E 04 2.00E 04 3.00E 04 4.00E 04 5.00E 04 4 2 0 2 4I AmpereV Volt PAGE 25 25 A B Figure 33. I V characteristics of Schottky devices that are A) C onducting for both voltage polarities. B) R ectifying. A B Figure 34. Current vs. voltage plotted for measurement of 2 different devices. The devices are rectifying for different polarities of voltage. Table 3 1. Breakdown voltages ( DC bias) of the nanowires for the first growth Device Length (microns) Breakdow n voltage (V) 2 4 4 8 6 12 10 16 Both low and high temperature measurements were performed to determine the n product and trap occupancy as a function of temperature. 6.0E 04 5.0E 04 4.0E 04 3.0E 04 2.0E 04 1.0E 04 0.0E+00 1.0E 04 2.0E 04 3.0E 04 4.0E 04 4 2 0 2 4I AmpereV Volt 1.0E 05 0.0E+00 1.0E 05 2.0E 05 3.0E 05 4.0E 05 5.0E 05 6.0E 05 4 2 0 2 4I AmperesV Volt 1.8E 04 1.3E 04 8.0E 05 3.0E 05 2.0E 05 4 2 0 2 4I AmpersV Volt 5.0E 05 1.9E 18 5.0E 05 1.0E 04 1.5E 04 2.0E 04 2.5E 04 3.0E 04 4 1I AmperesV Volt PAGE 26 26 A B Figure 35. Current vs. voltage plot showing device breakdown. A detailed discussion of the results of I V T measurements is found in Chapter 4 for the Though a back gate is not fabricated, gate dependence of drain current in these devices was investigated by biasi ng a metal plate put under the sample. No change in current was observed even with gate bias as high as +/ 30V. The effect of annealing was studied by measuring the current voltage characteristics of some devices and then annealing them at different temperatures. The resulting change in IV is shown in Figure 36. Figure 36. I V measurements showing the effect of annealing. The solid line shows the I V characteristics before annealing and the dotted line shows the I V characteristic after annealing at 650C. 0.E+00 1.E 04 2.E 04 3.E 04 4.E 04 5.E 04 6.E 04 7.E 04 8.E 04 0 5 10I AmperesV Volts 0.E+00 5.E 06 1.E 05 2.E 05 2.E 05 0 5 10I AmperesV Volts 1.5E 03 1.0E 03 5.0E 04 0.0E+00 5.0E 04 1.0E 03 4.00 3.00 2.00 1.00 0.00 1.00 2.00 3.00 4.00I AmperesV Volts PAGE 27 27 CHAPTER 4 SPACE CHARGE LIMITED TRANSPORT Introduction A closer examination of the non linear I V characteristics reveals a common structure to the I V curves. The slope of the I V curves changes from 1 to 2 and then to an almost exponential rise. This characteristic is observed mostly in the shortest wires linear I V characteristics, with IV slope changing from linear to super linear are attributed to a phenomenon called Space charge limited current. A solid state analog to the Childs law of space charge limited currents in vacuum for thermionic emission was first predicted by Mott and Gurney in 1940 22.The initial advances to the theory explained the phenomenon for insulators, working with a very simple formalism Analyzing I V characteristics in the space charge limited regime yields device parameters like carrier concentration, mobility, trap density and trap activation energy 23 25. Space charge limited current has been reported before in GaN thin films 25 and nanowir es16 17. The onset of space charge limited currents occurs at much lower voltages in nanowires than in bulk devices. Calarco et al 16do not extract transport or device properties from space charge limited IV and Talin et al.17 estimate mobility from the SCL limited I V curves. Band Theory of a S olid It is easier to understand space charge limited transport by beginning with an account of the mechanism of conduction in a solid. A solid material has two allowed bands the valence band and the conduction band, separated by the forbidden band gap. The va lence band is completely filled by electrons at 0K. At higher temperatures, electrons absorb thermal energy and jump to the conduction band, leaving behind a void called a hole. This process, called PAGE 28 28 generation, is in equilibrium with the inverse process ca lled recombination by which electrons recombine with holes and jump back to the valence band, emitting energy. In metals, the conduction band is partially filled at 0K. Therefore, even at very low temperatures, the metal conducts electricity and there is no forbidden band gap. Insulators have a very wide band gap, making it impossible for electrons to jump from the valence band to the conduction band. Semiconductors demonstrate a behavior intermediate to that of metals and insulators. At low temperatures, the conduction band is empty and the material behaves like an insulator. As the temperature increases, some of the valence band electrons acquire energy to jump to the conduction band and the material begins to conduct. A simple representation of the band diagram is in Fig 3.1. Figure 41. Energy band diagram s of A) metals, B) semiconductors and C) insulators, showing the conduction and valence bands. The top of the valence band is said to have energy Ev and the bottom of the conduction band Ec. A theo retical energy level called the Fermi level is defined as the energy level below which all the states are filled at absolute zero. At higher temperatures, some states below the Fermi level are empty and some above are filled. The probability of an electron ic state being filled is given by the Fermi function. The energy level at midband gap is called the intrinsic Fermi level, and is the Fermi level for intrinsic semiconductors. For ntype semiconductors, the Fermi level is closer to the conduction band and for ptype semiconductors it is closer to the valence band. PAGE 29 29 The number of electrons in the conduction band is given as a function of the density of states NC, the Fermi level EF and the bottom of the conduction band EC as 0= exp (4 1) w here k is the Boltzmann constant and has a value 1.381023 J K1 and T is the temperature. Current Voltage E quations When a voltage is applied to a device, injecting electrons into it, the number of filled states increases and this might be thought of as a rise in the quasi Fermi level F, which is the Fermi level in the presence of injection and is related to the total carrier concentration n in the same way as the actual Fermi level EF is related to the equilibrium carrier concentration n0. At low levels of injection, the quasi Fermi level is very close to the actual Fermi level, and the number of free carriers can be approximated by n0. If a voltage is applied across the semiconductor then it conducts current and the electron current depe nds on the number of electrons in the conduction band and the applied voltage. Current density J is given by = (4 2) where is the mobile charge density and v is the drift velocity of electrons. In a semiconductor the mobile charge d ensity in equilibrium is given by the carrier concentration n0.Therefore, =  0  (4 3) Using the expression relating drift velocity and applied voltage = = (4 4) where is the mobility, V is the applied voltage and L is the length of the device. Combining the above equations, we have PAGE 30 30 = 0 (4 5) This is Ohms law and is valid for small injection levels. For small voltages, the injected carrier concentration ni is much smaller than the neutralized equilibrium carrier concentration n0. Here, the current is assumed to be due to the thermally generated electrons and does not take the injected electrons into consideration 26. If the space between the electrodes of the system has a capacitance per unit area C, then the injected charge per unit area stored by the system Q, with an applied voltage V takes the form = ( 46) Continuing as before, the mobile charge density becomes, = = = (4 7) where is the dielectric constant and L is the distance of separation between the electrodes. At higher injection levels, the injected electron concentration, ni, becomes much greater than n0. The increase in electron concentration is limited by the amount of charge that can be supported by the device at the applied voltage. Assuming the charge density is almost entirely due to ni, = ( 48) The exact mathematical expression, first derived by Mott and Gurney in 1940, has a scaling factor of 9/8. This is the solid state analog to Childs law, which is applicable in vacuum 26. This simple model of a pair of electrodes as a capacitor was proposed by Rose, and furthered by Lampert and Mark. The theory assumes an ohmic contact, which can pump in an infinite number of electrons into the insulator, a constant mobility, diffusion current much smaller than drift current and that the field between the contacts is uniform and can be approximated by 22 PAGE 31 31 = (4 9) There is a limit on the amount of current that can flow at an applied voltage and this limit is determined by the maximum charge that the device can support. This is called the space charge limit and conduction in this regime is called space charge limited conduction. Traps Thus far, only the presence of free carriers contributing to conduction has been considered. However, any defects or impurities present in the insulator will change the number of free carriers available for conduction. The traps can be localized o r distributed in energy, lie deep in the band gap, below the Fermi level (deep traps), or very close to the conduction band and above the Fermi level (shallow traps). Figure (4 2) shows a shallow and deep trap in an insulator23,26. Trapping and de trapping occurs more frequently at shallow traps as the energy needed for an electron to jump to the conduction band is smaller and affect conduction significantly. At deep traps, the trapping and de trapping requires more energy and occurs less f requently. The process of trapping and de trapping is assumed to be at quasi equilibrium which holds well in the presence and absence of applied voltage. When there is no applied voltage, the quasi equilibrium is explained by a thermal equilibrium betw een trapping and the thermal re emission. This assumption is true as long as the applied field does not substantially heat up the free carriers. The balance between the trapped and free carriers is thus altered only by the injected charge. The injected cha rge now has two components one that fills the traps and another that contributes to conduction. The charge is then given by = ( + ) = = ( 410) t is the trapped charge density. PAGE 32 32 Figure 42. Traps in an insulator. Trap Etn1 is a shallow trap and Etn2 is a deep trap23. At small voltages, the charge density is determined by the thermal generation and recombination processes and the presence of traps does not alter the linear dependence of current on applied voltage and an ohmic region is obtained. At higher levels of injection, the injected electrons start filling the traps rather than contributing to conduction. This can be thought of as a rise in the quasi Fermi level. The total carrier concentration is given by n = + 0 = exp (F kT ) (4 11) The occupancy of a trap level Et is is given by nt = ni i + n0 i = Nt1 + 1 g exp E t F kT (4 12) where Nt = trap density, g = degeneracy of the trap, ni,i = trapped injected charge and n0,i = trapped electron density under equilibrium. The number of trapped electrons thus depends on the temperature and the voltage applied. A significant increase in the number of free carri ers does not occur until the traps are filled, thus delaying the onset of space charge limited current. PAGE 33 33 A trap is said to be shallow if F, and thus F0, lies a few thermal voltages below Et. Assuming that no is smaller than ni and free charge dens ity is much smaller than the trapped charge density t, = = (4 13) which is a constant independent of injection and is given by The charge is thus = L = L (4 14) Thus the current is a factor = (4 15) Once the traps are filled, the current rises sharply and merges with that given by the space charge limit. This sharp rise is called the traps filled limit 23. Limiting I V Characteristic It follows from the above that for an applied voltage V, the current cannot be less than that given by Equation (45), because the applied voltage can only add to the available density of free carriers. The current cannot exceed that given by Equation (4 8), because (48) represents the upper limit of charge the system can support at the applied voltage V. The current can also not exceed the traps filled limit, since this rise with respect to voltage represen ts the case for which all the traps are filled. This leads to the definition of a triangle in the log I log V space within which the current flowing through a device is confined for all applied voltages. This is given in Figure (4 3). It is possible to de fine the corners of this triangle in terms of material parameters, and free and trapped carrier concentration. The lines of the triangle represent the different regimes of PAGE 34 34 conduction viz, the ohmic region in which the injected carrier concentration is far lower than the equilibrium concentration, the space charge limited region in which the injected carriers far exceed the equilibrium carriers, and the transition between the two regions, which may or may not be det ermined by the filling of traps. At the voltage at the intersection of the Ohms law curve and the space charge limited curve, the charge densities must be the same. This gives, = 0 = (4 16) = ( 417) This transition voltage can be used to determine the mobility and carrier concentration. T he t ime taken for an excess charge to be relaxed by redistribution is called the relaxation time and the time for an electron t o travel the length of a device is called transit time At the transition voltage, the relaxation time is equal to the transit time. Hence, = = =2 (4 18) The transition between Ohms law and the traps filled limit occurs at which using the same concept would be given by = (4 19) At the trap filling limit, the exact value for capacitance = 2 is taken rather than the geometrical capacitance, which differs by a factor 2. This difference arises because the electric field is taken as the ohmic value, = where Va is the applied voltage. The inequality < < 2 PAGE 35 35 where is the actual field across the device, is assumed true and this gives that the total charge in the device given by Equation (46) is bounded by the inequality 26 < <2 Since traps are filled at high voltages, the higher value of capacitan ce and field are taken to determine the trap filled limit transition voltage Figure 43. The limiting triangle in log J log V space 23 The transition from Ohms law to trapped square law occurs before V TFL and only a fraction of the total traps, nt are filled. The voltage at which this transition occurs is given by PAGE 36 36 V T = q ntL22 (4 20) For deep traps, the equilibrium occupancy of holes is given by = 0=1 + 1 g exp F kT (4 21) To fill the traps, the voltage needed is =q ,0L2 (4 22) The simple theory described above is taken from 24 and 23. This theory can be extended to any number of traps. The full I V showing both shallow and deep traps is shown in Figure (44). It is to be noted that in the case of deep traps, the traps start to fill while the conduction is still ohmic, and thus the departure from ohmic conduction, is a deep traps filled region, with slope greater than 2. If the device has 2 shallow traps, the trap lying deeper in the forbidden band gets filled first. This leads to shallow traps filing slope 2 region and shallow traps filled sharp rise in current for each set of traps, leading to a lot of structure in the IV. Each of the corner voltages can be used to compute the concentration of the traps. If deep traps are present, they start to fill while the device is still in the Ohmic region of conduction, therefore, there is no separate deep traps filling square law. This also implies that the device stays in the ohmic region longer than for the case of two shallow traps, and that shallow traps filling slope 2 region sets in earlier as can be seen from Figure (4 4) 25. If the device has Nt(e) of trap levels distributed exponentially within an energy band per unit volume, then the trapped conduction current takes the form, ~ ( ) + 12 + 1 (4 23) PAGE 37 37 where l is the ratio between a temperature Tt and the ambient temperature T. Trap density per unit energy range at an energy E below the conduction band is written as ( ) = ( ) exp ( ) (4 24) Figure 44. Current density vs. voltage plot in the log log scale showing two sets of traps. The dashed line represents JV for a deep and shallow trap, the continuous line for 2 shallow t raps. See text for explanation25. The temperature Tt represents the temperature at which annealing stops, and it can be assumed to be ~300K. Therefore, l is very likely to be close to 1. At l =1, Equation (423) changes form to the single shallow trap dominated curr ent, given by Equation (415)27. PAGE 38 38 Determination of the Trap Distribution The sharp increase in current is found for both single trap levels and traps exponentially distributed in energy and at l =1, current has the same dependence on voltage. It is thus necessary to determine the trap distribution before extracting any other parameter from the IV. Rewriting Equation (4 23) in terms of its temperature dependence, = 2 ( ) (4 25) Plotting the logarithm of current density against 1/T, the slope and intercept for each case will be different and should be as in Table 41. This measurement is necessary to determine the distribution of traps27. Table 4 1. Slope and intercept of logJ 1/T plot for single traps and exponentially distributed traps 27 Negative D ifferential R esistance Another distinctive structure in the I Vs of the shorter devices is hysteresis with one of the voltage sweep polarities giving a sharp rise in current and the other giving a slump in current. In the second case, increasing the applied voltage gives a smaller current, giving a negative differential resistance. There are also devices that show negative differential resistance without a slope 2 region in the I V. Some such I Vs are shown in Figure (410). GaN has a negative differential mobility at fields of 2.106 V/cm and the fields applied to these devices are two orders of magnitudes small er. The devices therefore operate in the constant mobility regime. Trap distribution Slope Intercept on log J axis Discrete 10 10 Exponential 10 2 ( ) 10 9 8 ( ) 2 3 PAGE 39 39 The negative differential resistance can be explained by the presence of repulsive traps. Repulsive traps present a potential barrier to the capture of electrons which can be visualized as in Figure (4 11). As the bias is increased, the energy of the electrons increases, enabling them to overcome the potential barrier which makes capture possible. This reduces the number of free carriers available and increases the resistance. A similar eff ect can be observed in some I V T curves as shown in F igure ( 412 ) As the temperature increases, the energy of electrons increases and the resistance reduces due to increased capture rate28. Figure ( 4 11 ) shows a schematic representation of a repulsive trap. The potential barrier presented by the trap to the electron is also expressible in terms of an effective cross section of the trap. At low temperatures or low electron energy, the potential barrier of the repulsive trap is too high for the electrons to cross and the capture cross section is around as ~1022 cm2. For attractive centers the carrier capture cross section is 1012 cm2 and for neutral centers, it is 1017 cm2. Change in electron energy would change the effective capture cross section of the trap and thus the capture rate 28. Repulsive traps are most effective when the field applied is such that the barrier to tr apping is overcome, in a temperature range where thermal generation does not increase significantly and impact ionization does not occur. For a capture rate for the electrons that have overcome a potential barrier the effective capture rate is given by 0= exp The condition for negative resistance to be observed is that > 1 (4 26) PAGE 40 40 A B Figure 45. Current vs. voltage plot showing negative differential resistance A) after a slope 2 region. B) without a slope 2 region 1.0E 07 1.0E 06 1.0E 05 1.0E 04 0.1 1I AmpereV Volt y1 y2 5 to 0 0 to 5 1E 07 1E 06 1E 05 0.0001 0.01 0.1 1 0.1 to 5 slope1 slope2 PAGE 41 41 Figure 46. Schematic diagram of the potential barrier at a repulsive trap 29. Figure 47. Plot of resistance vs. temperature showing increase in resistance with temperature. 1670 1680 1690 1700 1710 1720 1730 0 50 100 150 200 250 300 350 400 Resistance Temperature K PAGE 42 42 the applied field given as For p = 1/2, a field dependent temperature of electrons Te and B given by = 0 2 Equation (4 26) reduces to 28 > 2 .5 For GaN grown in nitroge nrich atmosphere, the dominant defects are Ga vacancies near the valence band which act as hole traps and N anti sites and N interstitial sites near the conduction band, which trap electrons. The N anti sites present a repulsive potential to the electrons and an observed capture cross section of 1022 cm2 is attributed to N antisites by29. Dislocations present in a d evice initially act as electron traps. The trapped electrons then present a repulsive potential to continued trapping of electrons30. However, GaN nanowires have a very small dislocation density 3132. Interstitial carbon can act as a double acceptor in n type GaN, with trap levels at 1.1 and 1.2 eV below the conduction band 33. Parameter Extraction Using the Simple M odel The model described above assumes a single trap en ergy level, and provides means to estimate carrier concentration, trap density and mobility provided the material constants are known. However, in a real device, the transitions between the regimes are not so sharp, neither are all the regimes observed as m devices which burn at ~4V continuous D.C or ~8V from a pulsed supply. Figure (48) shows an I V of one of the GaN wires, showing the transition between ohmic and super linear conduction. PAGE 43 43 Trap Free C onduction The simplest explanation and model woul d be to assume that the device transitions from ohmic to space charge limited conduction at the crossover voltage ( 1.7 V in this case, at a current of 10.9 A). The cross over voltage in this case should be given by Equation (417) with a correction fact or of (4/3) for accuracy. Using material constants from Table 21, and a wire diameter of 100nm, we have that the carrier concentration is n0 = 2.3 x 1014 cm3 and mobility = 4.4 x 105 cm2/V s. This no value is far lower than expected, even for an undoped wire, which can be taken to mean that there are traps, which this model does not take into account. Temperature D ependence of IV The first step is to determine the distribution of the traps, which is done by examining the temperature dependence of current density. Figure 49 shows the I V curves at 50C, 75C and 100C. The part of the I V curve in which, the current depends on the square of the voltage is plotted in Figure 410 logarithm of current vs 1/T. For traps exponentially distributed in energy, the slope of the current density depends on log10V. For a change in voltage of 0.1, the slope should change by 1. Since the slope is almost invariant, the traps must be at a discrete energy level. From the slope and intercept, we get a trap level of 0.008 eV and mobility 42637 cm2/V s. From the reported data on GaN nanowires, the mobility is far too high and the trap level too shallow. Trap levels were extracted for other wires using the temperature depende nce of current. The values extracted are in Table 4 2. PAGE 44 44 A B Figure 48. Plot of current vs. voltage A) in linear space B) in log log space. The square marks the transition from ohmic to square law conduction. Y1 is drawn to guide the eye along a line with slope 1 and y2 along a line with slope 2. 0 0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 0 1 2 3 4 5 6I AV Volt0.1 to 5 1E 07 1E 06 1E 05 0.0001 0.001 0.1 1I AV volt 0.1 to 5 y1 y2 PAGE 45 45 Figure 49. Temperature dependence of I V at 50C, 75C and 100C. Figure 410. Plot of current density and the inverse of temperature on a linear log scale at 2.8V and 2.9V. The fit to the log I vs 1/T curves are also shown. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 1 2 3 4I(Ampere)V(Volt) 25 50 75 100 y = 35.935x + 8.2906 y = 35.672x + 8.3175 7.8 7.82 7.84 7.86 7.88 7.9 7.92 7.94 7.96 7.98 0.005 0.01 0.015log J1/T (1/K) 2.8V 2.9V PAGE 46 46 Table 4 2. Trap activation energy extracted from I V vs T curves. Device Trap energy (eV) Possible e lement 1 0.08 Si (0.012 0.02) 2 0.16 0.02 Si 3 1.06 Gallium in Nitrogen site Presence of a S ingle S hallow T rap The corner voltage in this case is given by Equation (420), and thus the number of filled traps nt == 2.3 x 1014 cm3 = n0/ from Equation (4 13). Fitting the slope 2 part of the I V, given by Equation (415), we obtain a value for 4.4 x 105 cm2/V s. For a shallow trap to significantly affect the shape of the I V, values of carrier concentration lower and further increase mobility which is incorrect. The breakdown field in GaN is much higher than the fields used here, so the sharp rise in current cant be due to breakdown. To understand why the theory gives incorrect results, it is necessary to understand the factors that are introduced to correct for the phenomenological equation (48). Correction F actors to Roses E quations The theory proposed by Rose approximates the capacitance of the system and gives a phenomenological solution to the problem. However the current density equations are derived by solving the Poisson equations for the geometry with the correct boundary conditions. Neglecting diffusion current, current density can be written as = (4 26) The Poisson equation in a s i ngle dimension considering free and trapped charges is = + (4 27) PAGE 47 47 Roses theory was formulated for insulators. So, donor density is neglected in the Poisson eq uation and it is written as in Equation (4 27). a/a where Va is the applied voltage and a is the distance between the anode and the cathode Equation (48) is obtained, with the co the cathode, the free carrier density at the cathode should be infinite. The model assumes infinitely large contacts and a bulk material 26. To accurately model a nanowire, the equations derived by L ampert and Mark need corrections which take the geometry of the nanowire and the contacts into account. Contact E ffects Space charge theory as derived by Rose and Lampert assumes a 1 dimensional structure. If the length of the device is larger than its cross section, then two dimensional effects need to be considered. The enhancement of current density in the 2D model, J[2D], as compared to the 1D model, J[1D], is given by [2 ] [1 ] = 1 + (4 30) where F is the correction factor for the mean position of the injected electrons, given by Equation ( 4 31) and G depends on the geometrical pr operties of the emission area34. = ( ) 0 ( ) 0 (4 31) With ( ) + 1 = + 1 For shallow traps, with l =1, we have that F = 1/3. The correction factor for the geometry where a small circular patch of the cathode is injecting into the device, 35 = 1 + 4 (4 32) PAGE 48 48 Equation (4 32), derived in the limit R/L >> 1, should also approxim ate the case where R/L < 1 Application to Nanowires In the case of = 11. Therefore the enhancement is given by ( 1 + ) = 11/3 It is obvious that the correction factors derived above are insufficient to explain the observed I V characteristics in these nanowires. It is necessary to examine the space charge limit considering three important factors T he nanowire is a semiconductor, and hence its capacitance is different from the geometrical capacitance of an insulator between two conducting plates The nanowire has a very high aspect ratio, which leads to a redistribution of carrier concentration and poor electrostatic screening. The contacts to the nanowire are much bigger than the wire itself, thus fringe effects cannot be ignored 18. The above considerations are discussed in detail in the following chapter. PAGE 49 49 CHAPTER 5 EFFECT OF POOR SCREENING ON SPACE CHARGE LIMITED TRANSPORT Coulomb interactions in vacuum decay as 1/r and have a long range. In solids, this range is modified and shortened. This effect is called screening. Screening effects are also seen in electrolytes, but are not discussed here. Screenin g originates due to the presence of charges, both mobile and fixed, in the solid. In perfect insulators, screening is entirely due to the re alignment of atomic cores in the direction of the field. This creates a dipole moment, producing a field in the opposite direction to that applied. The resulting field reduces the applied field by a factor determined by the dielectric polarizability of the material. Therefore, in an insulator, the potential due to a charge q at a distance r is given by ( ) = 4 0 which is smaller than the potential in vacuum by a factor In metals, the contribution to screening by the atomic cores is much smaller than that of the electrons. The high concentration of mobile carriers effectively screens even very high fields, such that electric fields do not penetrate a metal. In semiconductors the density of mobile carriers is much less than in metals and thus screening is not as effective. The applied electric field penetrates the semiconductor upto a distance called the screening radius, given by = 0 02 1 2 (5 1) The screened potential is obtained by solving Poissons equation and takes the form, ( ) = 4 0 T he screening in semiconductors is far more effective than in insulators as the field decays exponentially and there is no potential beyond a few screening lengt hs. In bulk devices the PAGE 50 50 screening distance is usually much smaller than the device dimensions. For GaN, with a carrier concentration of 101 6 cm3 the screening distance is ~1000 In a bulk device, with planar current flow, a dipole sheet is created, beyond which there is no band bending and the potential is constant. The equilibrium carriers are free to move in a bulk device and since the field is confined within the device, only Poissons equation needs to be solved to compute the screened potential. When the dimensions of the system are reduced, making one of the dimensions comparable with the screening length or the wavelength of electrons, the decay of applied field slows down dramatically. In a low dimensional system, where the energy levels are quantized in either x, y or z, the movement of electrons is restricted to either a plane (2D) or a line (1D), wher eas the electric field lines are present over all space. This reduces the effectiveness of screening by the mobile carriers 36. If any of the geometric dimensions of the device is comparable to the screening distance, a similar reduc tion in screening results. This is the case in the nanowires under study. The aspect ratio R/L, of these systems varies between 0.02 and 0.05. The contacts are much larger being 100m x 300m The electric fields prevail in all the space between contacts, whereas the screening charges are only found within the wire. Therefore, to obtain the potential inside the wire, Poisson equation is solved inside the nanowire, and Laplace equation is solved outside it, and the potentials are matched at the interfaces. This gives a form of the contact potential that decays logarithmically. The charge density also decays logarithmically inside the wire. Electric fields penetrate much further in a nanowire than in a bulk device 18. The slow decay of charge density and contact potential implies that a contact depletion region formed in low dimensional structures is much wider, and is proportional to the applied PAGE 51 51 voltage rather than the square root of voltage as in the bulk case. Far from the contact, the elec tric potential is very low, but since the charge decay is hyperbolic, it remains non uniform for even longer distances. So, even for wires which are thick enough that the potential at the axis is well screened, contact potential decays very slowly. Poor sc reening is prevalent even when the radius of the wire is comparable to or slightly larger than the screening length 18. Space charge limited transport indicates the breakdown of screening as the injected non equilibrium charges, and thus t he applied potential are present throughout the length of the device. Current is space charge limited, when the injected carriers far exceed the equilibrium carriers. Considering an injecting contact, in which the semiconductor is under accumulation, the applied field appears across the semiconductor side. Considering a weakly doped ntype semiconductor, the conduction band bends downwards near the contact. There is thus an energy maximum, and a corresponding potential minimum along the length of the devic e, as shown in Figure 51. To the left of the energy maximum, the drift and diffusion currents are both large and nearly opposite. To the right, the field is much larger and the drift current dominates. If the potential drop in the space charge limited re gion is much larger than the thermal voltage, diffusion current can be neglected. Under space charge limited conduction, the electrons relax from the accumulation region into the bulk of the device. The field at the injecting contact due to the injected carriers opposes the field under which the electrons drift. Therefore, beyond the point of potential minimum, the device is assumed to be in near equilibrium. The field applied at the anode is screened beyond this point. As the injection level increases, the applied field also increases. The slope of the energy bandedges to the right of the potential minimum increases, trying to reduce (increase) the energy maximum (potential minimum). That is, screeni ng PAGE 52 52 becomes ineffective and the injected charge and field spread over the entire device. Therefore, the onset of space charge is determined by position of the potential minimum which is obtained in bulk devices by solving the Poisson equation in one dimens ion 36. Figure 51. Injection into an ntype semiconductor. Correction for the G eometry of the Wire For nanowires, it is important to c onsider the variation of the field and carrier concentration in two dimensions, r and z, assuming that the wire has rotational symmetry. In bulk devices, the width of the device is much larger than the length, and thus the field is considered uniform in the transverse direction. The electrons do not get pulled to the sides, and the problem reduces to a planar 1D flow of electrons 36. In nanowires, the length is much greater than the radius, and therefore, the same assumption cannot be ma de. Therefore, Poisson equation is solved in 2D to obtain the expression for space charge limited current and the corresponding critical voltage 1718,37. Experimental E vide nce for P oor S creening in N anowires A probe of either the potential profile within a nanowire, or the capacitance of a nanowire should show evidence of poor screening behavior Capacitance of a device is defined as charge PAGE 53 53 in the device over the applied vo ltage. If the potential is obtained along the length of a device, then its extent will determine the screening distance and its magnitude and dependence on distance from the contacts will determine the effectiveness of screening. The capacitance of nanowir es is so small, that conventional measurement methods are insufficient to probe nanowire capacity. Rece ntly, gate capacitance of InAs wires, having a diameter of 70115 nm, has been measured by scanning capacitance spectroscopy. The dC/dV behavior is analyzed and the potential along the length of the wire is obtained, as shown in Figure 52. The decay of potential is faster than exponential closest to the contact, then becomes exponential and has a region of very slow decay. This is a clear deviation from the behavior of bulk devices. The large drain contact screens the gate potential far longer than would be expected i n the case of bulk devices 37. This directly translates to the fa ct that the drain potential decays more slowly in the case of nanowires. Figure 52. Potential along InAs wires obtained by SCS measurements PAGE 54 54 Theoretical C alculation of C apacitance of the N anowires Another independent means of verifying the need for a correction factor is to calculate the capacitance of a nanowire as obtained from the I V measurements in either the trap limited or the space charge limited regimes. T his value can then be compared with the geometrical capacitance and capacitance derived taking the slow decay of contact potential into account = = = From the above equations we have that the total capacitance of the wire is given by = 2 2 Choosing one of the devices, showing SCL at 8V, with a current of 3.08 x 105 A, the capacitance is found to be 1.73 x 1016 F. Evaluating the geometrical capacitance with a length of 2 micron and a diameter of 160nm (as found from SEM measurements approx value ) we have 2.77 x 1019 F, which does not even come close to the capacitance measured from the I V dependence. The capacitance of a nanowire for fully depleted contacts is derived by solving Laplace and Poisson is given by18 = 2 exp eV 2 e n1 The above equation evaluates to 1.33 x 1017 F. This is still an order of magnitude smaller than the capacitance as found from the I V, but much closer than the geometrical value. PAGE 55 55 Formulating the SCL C urrent E quations In a nanowire, injected charges can be screened upto a radius R and a leng th L. The direction dependent screening in a nanowire introduces a scaling term in the distribution of charges and field, which shows up as the pre factor in the equation for current. Consider a bulk device, with an injected charge Q. This charge gets redistributed over a hemisphere of radius rs, due to electronelectron repulsion, and the radius increases with the applied voltage36. Space charge limited conduction begins when the drift time is equal to the relaxation time. That is, space charge just fills the device from contact to contact. This occurs at an applied voltage which is the critical voltage, at which rs = L, the length of the device. The total charge at which this occurs is given by = 2 3 3 ( 52) where is the injected charge per unit volume. In the case of a nanowire, the injected charge cannot move beyond the radius R, so the injected charge is redistributed along the wire, with peaks in the longitudinal direction. The charge thus reaches the anode much faster than in the bulk case. The non uniform redistribution of injected charge causes a change in the electric field profile both inside and outside the wire as shown in Figure (53). The charge that can totally fill a wire of radius R and length L is = 2 (5 3) From Equations (52) and (53), the injected charge depends on the geometry of the device. A scaling factor (R/L)2 can be deduced, by which the charge injected into a nanowire differs from the charge injected into bulk devices. PAGE 56 56 A B Figure 53. Electric field distribution in a wire A) showing contacts and fringing effect B) Field inside the wire 38 The same relationship is mathematically derived by solving Poisson equation in cylindrical co ordinates, Solving the equation in cyli ndrical co ordinates gives a scaling factor PAGE 57 57 (R/L)2, which gives more appropriate results. The current density for trap free space charge limited conduction becomes 17. = 223 (5 4) The transition voltage in this case is = 2 2 ( 55) This voltage is (R/L)2 times the bulk corner voltage, which implies that the onset of space cha rge limited current occurs sooner in nanowires than in bulk materials17. Extending the above theory, the shallow trap limited current, smaller than the SCL current by a factor = 2 23 (5 6) The corner voltage, VTFL, should also have a similar scaling factor, but it need not be (R/ L)2. This dependence can only be verified experimentally if wires of two different radii with the same trap activation energies are compared. One such pair of devices with deep traps around 0.1 eV were compared and the results prove that the voltage does scale as (R/L)2. These results will be presented in the subsequent sections. Calculation U sing C orrection for A spect R atio This section analyze s the same device analyzed with the bulk SCL equations assuming the same scaling factor for VTFL. Equations (430) and (431) are used for SCL limited current and int roduced for trap limited conduction to both the equations. The value of 0 increases by a factor 2 giving PAGE 58 58 0 = 3 .68 1017 3 The value of decreases by a factor 2, giving = 275 cm2/V s The trap density increases by a factor 2giving = 1 .77 1018 3 Using the trap energy estimated from the temperature dependence of current (8 meV), and a trap degeneracy of 2, from Equation (4 13) we have that Therefore, 0= 1 .75 1 017 3 = 575 cm2/V s From the temperature dependence of I V, using the nanowire model, mobility is 610 cm2/V s and this does not significantly change the values of either Nt or n0. Analysis of devices with deep traps: Two devices from the same growth with deep traps showing full SCLC characteristics are considered for this analysis. One of the wires has a diameter of 90 nm and the other 200nm. The devices survive voltages upto ~10V. Electron mobility is estimated from the space charge limited current using Equation (5 4). This value is used to compute the Fermi level at each applied voltage as given by = 0 .026 + 0 .026 The trap distribution can be calculated from a single SCL C curve if the Fermi level is known. Let h(E) describe the energetic distribution of traps, such that the number of occupied t raps is given by PAGE 59 59 nt = h (E) 1 + 1 g exp E t F kT Differentiating the above equation with respect to F/kT, we have ( ) = 00 1 where 0 is the smallest voltage applied and = ( ) ( ) Once the mobility is known, F and 0 can be calculated and thus h(F) can be calculated. A plot of h(F) vs F gives the approximate location of the trap. For deep traps, if the trap is lower than the Fermi level at the lowest voltage applied, then the result will only be approximate. However, the trap will not be more than a few thermal voltages away from the calculated trap activation energies. If the distribution of traps is spread over a range of energies, then h(F) will reflect that spread. The plot of h(F) vs F obtained is shown in Figure (5 4). The current voltage characteristics obtained are shown in Figure (5 5). Figure 54. Plot of trap distribution vs quasi Fermi level. 5.00E+19 0.00E+00 5.00E+19 1.00E+20 1.50E+20 2.00E+20 2.50E+20 0.0000 0.0500 0.1000 0.1500 F (eV)h(F) PAGE 60 60 Figure 55. Current vs voltage plot of two nanowires of different radii. Figure (5 5) shows that the onset of space charge limited transport is delayed in the thicker wire. The parameters extracted from these two devices are given in Table 5 1. The ratio of the voltages at which the transition to space charge limited conduction does show the predicted dependence on radius. It can be seen from equation ( 55 ) that the critical voltage for the onset of space charge limited current does not depend on the length of the device. But, a change in slope is more commonly observed at these voltages for the short 2 micron devices. This could be due to the fact tha t contact in homogeneities play a more important role in shorter devices and that the change in slope is due to trap limited conduction rather than space charge limited conduction. A change in slope of the I Vs is also observed in the 4 micron devices as shown in Figure (5 6). The change in slope occurs around a voltage of 3.5 V rather than around 2 volts. It can be seen from Figure (56) that the current increases sharply around 3 Volts. The exponential increase in current should be followed by the SCL r egime. However, if the device is trap free, the change from Ohmic behavior would occur at the start of the SCL regime. 2.00E 05 1.50E 18 2.00E 05 4.00E 05 6.00E 05 8.00E 05 1.00E 04 0.10 2.10 4.10 6.10 8.10I(A)V (V) r=100nm r=45nm PAGE 61 61 Table 5 1. Parameters extracted for two devices with deep traps Parameters #1 #2 Radius 45 nm 100 nm Length of the wire m m Trap filled voltage 2.3 V 5 V SCL conduction begins at 6.2 V 7.6 V Mobility ( with length of the wire) 494 cm 2 /V s 141 cm 2 /V s Mobility ( with distance between contacts) 334 cm 2 /V s 111 cm 2 /V s Equilibrium carrier concentration (from Fermi level with wire length ) 8 x 10 16 cm 3 3.6 x 10 16 cm 3 Equilibrium carrier concentration (from Fermi level, with spacing between contacts) 5.3 x 10 16 cm 3 2.3 x 10 16 cm 3 Equilibrium carrier concentration (from Resistance, with wire length ) 9 x 10 16 cm 3 3.61 x 10 16 cm 3 Equilibrium carrier concentration (from Resistance, with spacing between contacts) 6 x 10 16 cm 3 2.82 x 10 16 cm 3 Trap density 3.15 x 10 18 cm 3 5.42 x 10 17 cm 3 Trap activation energy 0.1 eV 0.1eV Summary This chapter presents the theory of space charge limited current flow. The current through any device is bounded within the limits represented by Ohms law, Mott Gurney law and the trap filling limit. It is possible to discern the presence of traps by ex amining the structure of the I V curves. The spatial distribution of traps is determined by measuring current vs. voltage at different temperatures. The high aspect ratio of nanowires result in poor screening and an early onset of space charge limited curr ent flow. The presence of repulsive traps leads to a negative differential resistance. PAGE 62 62 A B Figure 56. Current vs. voltage characteristic of a 4 long device showing SCL behavior A) showing slope 12 and B) showing slope 12 before breakdown. Carrier concentration, mobility and trap density can be extracted from the transition voltages and currents. The values obtained by employing the different models are presented in Table 52. 1.00E 06 1.00E 05 1.00E 04 1.00E 03 1.00E 02 0.10 1.00 S4R2C3 6 slope1 slope2 1E 06 1E 05 0.0001 0.001 0.01 0.1 1 10 slope1 slope2 S4R1C68 PAGE 63 63 Table 52. Summary of extracted device parameters using the SCL model Model 0 ( cm 3 ) (cm 2 /V s) (cm 3 ) Bulk device, no traps 2.3 x 10 14 4.4 x 10 5 Bulk device, presence of shallow traps 2.3 x 10 14 <1. 4.4 x 10 5 1.1 x10 15 Nanowire, with correction for high aspect ratio trap energy 8 meV and degeneracy 2. 1.75 x 10 17 575 610 1.177 x 10 18 Nanowire, with shallow trap at 0.06 eV 1.3 x 10 17 600 cm 2 /V s 2.51 x 10 17 Nanowire, with deep trap at 0.1eV 9 x 10 1 6 494 6.15 x 10 17 PAGE 64 64 CHAPTER 6 ANALYTICAL DETERMINATION OF MOBILITY The ohmic region of the I V curve can be used to extract mobility and carrier concentration using the temperature dependence of the current. The I V characteristics discussed in this chapter are thos e of devices which stay in the linear regime up to a voltage of 8V. Beyond this, the devices start to breakdown rather than transitioning to a space charge limited current. Figure (6 1) shows some linear I length device deviates from the o device also exhibits a trap dominated region in the I V at 128K. Figure ( 6 2) shows some plots of resistance against temperature. The change in resistance varies from almost nonexistence to double the values at room temperature. The huge shifts in resistance are attributed to trapping of electrons in shallow traps. The resistance of a semi conductor increases with decreasing temperature, as there is less thermal energy available for carrier generation. As the temperature is lowered, for an n type material the Fermi level moves toward the conduction band and the shallow traps start to fill. For devices dominated by shallow traps, lowering the temperature results in an increase in resistance over and above the thermal generation effect. The number of traps filled at a given voltage depends on the device dimensions and thus, the trap filling effect is more visible for short devices. The same phenomenon should occurs for the longer devices, if they have traps, but at higher voltages than measured in this work. The l inear part of an I V characteristic can be used to extract carrier concentration and mobility based on the temperature dependence of resistance. The product of mobility and carrier concentration can be determined as a function of temperature knowing the resistance and device PAGE 65 65 dimensions. The temperature dependence of mobility for a given carrier concentration can be estimated using th e Caughey Thomas model 39 as in 40. A B Figure 61. Current vs. voltage plots showing temperature dependence of current for A) 6 micron device B) 4 micron device. Labels on the right indicate the temperature at which the measurements were made.The 0T indicates magnetic field. 3.0E 05 8.0E 05 1.3E 04 1.8E 04 2.3E 04 2.8E 04 3.3E 04 3.8E 04 0.5 1.0 1.5 2.0 2.5 3.0Current (A)Voltage (V) 142 172 230 273 292 317 340 3E 05 2E 05 1E 05 9E 19 1E 05 2E 05 3E 05 0.7 0.5 0.3 0.1 0.1 0.3 0.5Current (A)Voltage (V) 100K 0T 4.2K 0T 150k 0t 250K 0T 200K 0T 300 K 0T PAGE 66 66 A B Figure 62. Temperature dependence of For a doping density N, the CaugheyThomas model is an empirical model which estimates mobility at a given carrier concentration as given by Equation ( 61). This allows for n product. ( ) = + 1 + ( 61) where the material. Mnatsakanov et al. 40 use observed GaN mobility, as shown in Figure ( 63) to compute the fitting parameters tabulated in Table 61. The temperature dep endency of the mobility is estimated depending on the main scattering 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 250Resistance ( )Temperature (K) 1050 1100 1150 1200 1250 1300 0 100 200 300 400Resistance ( )Temperature (K) PAGE 67 67 mechanism present at that temperature. At room temperature optical phonon scattering dominates and the contribution from impurity scattering is obtained by subtracting the phonon scatter ing component from the total mobility. A detailed discussion on scattering effects can be found in Chapter 9. The temperature dependence of mobility is written as ( ) = ( 0) ( ) 0 1 + ( ) 0 + ( 62) where ( ) = + evaluated at temperature T0 Table 61 The temperature dependence of mobility is shown in Figure ( 64). With data available on n product as a function of n products are shown in Figure ( 6 5) and the carrier concentrations determined are shown in Figure ( 6 6). The n products determined for the devices all range in the order of 1021 cm1/V s. Mobility at the temperatures is determined using Equation ( 6 2) to obtain carrier concentration which is of the order 1019 cm3. This carrier concentration is too high as the density of states in the conduction band is only 2.1 X 1018 cm3 and the wires are not intentionally doped. The fitting parameters used to in the CaugheyThomas model are for bulk Wurtzite GaN and the parameters might turn out quite different in nanowires. A study of variation in mobility in ntype GaN nanowires is necessary to determine if the fitting parameters used for bulk GaN can also be applied to the case of nanowires. PAGE 68 68 Figure 63. Fitting CaugheyThomas model to experimentally obtained mobility values 36. Table 61. Fitting parameters for GaN to find electron mobility 36 Parameters Values cm 2 /V s 55 cm 2 /V s 1000 cm 3 2 X10 17 1 2 0.7 PAGE 69 69 Figure 64. Tem perature dependence of mobility as a function of doping concentration. The doping concentrations are 13 x 1016 cm3 2101 7 cm3 31.5 x 101 7 cm3 42 x 101 7 cm3 53.5 x 101 7 cm3 6101 8 cm3 73 x 101 8 cm3 36 Figure 65. Variation of mobility carrier concentration product with temperature. 1E+21 1.1E+21 1.2E+21 1.3E+21 1.4E+21 1.5E+21 1.6E+21 1.7E+21 1.8E+21 1.9E+21 2E+21 0 50 100 150 200 250 300 350Carrier concentration Mobility productTemperature (K) PAGE 70 70 Summary : This chapter details the temperature dependence of current in the nanowires and the use of a the CaugheyThomas model to extract mobility and carrier concentration. Some of the nanowires show almost no change in resistance, in the measured temperature rang e from 128K to 340K, while some nanowires show a sudden change in resistance. These sudden changes in resistance are attributed to filling / emptying of traps. Some nanowires remain linear, but the resistance changes slightly with temperature. The mobility and carrier concentration products determined for the nanowires are in the range of 1021 cm1V1s1 and the corresponding carrier concentration and mobility from the CaugheyThomas model are of the order 1019 cm3 and 54 cm2V1s1. The results might be better if mobility values reported for GaN nanowires, are used in the model instead of those reported for bulk GaN. PAGE 71 71 CHAPTER 7 MOBILITY OF GALLIUM NITRIDE NANOWIRES Nanowires differ from bulk materials in many ways. The dimensions of the nanowires considered for this study are too big for quantization of energy levels, but the high surface to volume ratio, the absence of dislocations and the geometry of the device play a role in determining transport properties. As grown wires with diameters 20 300n m are reported in literature 1415. Growth conditions and type of contacts also change the electrical characteristics. Figure ( 7 1) shows t he diameter dependence of carrier concentration and mobility as reported for as grown GaN nanowires by 15. They use an alumina substrate, a gallium compound, ammonia and Ni catalyst at 800 C and 760 Torr to grow the wires and the mean diameter of the wires is 94 nm. Growing thinner wires using a silicon substrate led to an order magnitude smaller carrier concentrations and an order magnitude higher mobility. The carrier concentrations reported are in the order of 10191020 cm3 and mobility ranges between 2 and 20 cm2/V s. To illustrate the effect of growth conditions, the diameter dependence of mobility from another group 14 is presented in Figure (72). This set of wires was fabricated using vaporized gallium metal and ammonia at 850 900 C and the mean wire diameter is m uch higher than in the case of 15. The highest mobility reported is for wires of 200 nm diameter, at 319 cm2/V s and a carrier concentration of 1018 cm3. This is comparable t o that of bulk epitaxial films and lower mobility in thinner wires is attributed to increased surface scattering. The nanowire surface has a depletion layer of width w, in which the relaxation time is smaller than in the interior of the wire. If the surfa ce defect density does not vary from wire to wire, the thicker wires have a larger volume with a slower relaxation. Calculating an effective mobility taking the weighted relaxation times would give a higher value for the wires with a bigger diameter. Some of the thicker nanowires show the presence of PAGE 72 72 A B Figure 71. Diameter dependence of A) carrier concentration B) mobility.15 grain boundaries, which is ac companied by lower mobilities Motayed et al 14 thus identify two reasons for low mobility surface scattering and presence of grain boundaries. Huang et al. 41 report very high mobilities of 150 650 cm2/V s, much higher tha n thin films at the same carrier concentration of 1018 cm3 (100300 cm2/V of 10 nm and were grown by laser assisted catalytic growth. From Figures (7 1) (7 3) it is clear that there is a lot of variabi lity in transport parameters obtained for GaN nanowires. PAGE 73 73 Figure 72. D iameter dependence of mobility14. Figure 73. Variatio n of mobility with carrier concentration 41 The mobility in a ll these devices 1415,41 is measured using transconductance, with a FET structure. The capacitance of t he gate is given by PAGE 74 74 = 2 0 2 ( 71) Mobility is calculated using = 2 ( 72) Equation ( 7 1) is valid when the wire is entirely surrounded by oxide. This is not the case in the FET structures, so a corrected value of 2.2 is used for instead of 3.9. The model also assumes that the nanowire radius is much smaller than the thickness of the oxide. Another assumption is that the wire is an infinitely long metal and does not allow for depletion inside it. These factors make this method of calculating mobility unreliable 15. The effect of surface traps and surface depletion is also discussed in 16,42. The current through a nanowire depends on the cross section and the width of the surface depletion region. Below a critical diameter, dcrit, the nanowire is totally depleted and conduction occurs only with illumination from UV light. Holes are attracted to the surface by the electric field due to surface depletion, but electrons have to overcome a barrier to recombine with the holes. Below a critical diameter, surface Fermi level pinning defines the height of the recombination barrier. The depletion region in thin wires is small, and thus the recombination barrier is smaller. So, the current obtained is smaller because of the shorter lifetime of carriers. The dependence of current over the diameter is linear above the critical diameter, and exponential below. The critical diameter is written as = 16 2 ( 73) where is the barrier to surface recombination, is the density of states and is the doping density. The radius of the neutral zone in the wire is calculated using PAGE 75 75 = = 24 2 24 2 2 2 ( 74) The carrier concentration is 6.25 x 1017 cm3 for the undoped sample 3 8 and the surface barrier 0.55eV. In conclusion, the carrier concentration of undoped wires varies from 1017 to 1020 cm3 in literature, with a corresponding mobility variation from 2 to 650 cm2/V s. The ambiguity in the equations to determine mobility from transconductance necessitates an independent measurement of mobility Summary : There is a wide variation in the values of carrier concentration and mobility reported in the literature. The values calculated in Chapter 5, using the CaugheyThomas model, are in the same range reported the carrier concentration is 1019cm3 and mobility is 54 cm2/V s. PAGE 76 76 CHAPTER 8 DETERMINING MOBILITY FROM MAGNETORESISTANCE Magnetoresistance is the property of a material to change the value of its electrical resistance when an external ma gnetic field is applied to it. The phenomenon was discovered by Lord Kelvin in 1856 43. Magnetoresistance can be either physical or geometrical or both depending on the properties of the material and the geometry of the device being measured. The change in resistance of the device with an applied B field can be used to compute carrier mobility 44. If the magnetoresistance depends on the direction in w hich the B field is applied, then the material is said to show anisotropic magnetoresistance 45. Theory The force on an electron in an electr ic and magnetic field is given by 44 = ( + ) ( 81) Electrons accelerate due to this force gaining energy and a momentum m*v and subsequently lose energy and momentum to the lattice. In the relaxation time approximation, the net effect of the forces can be written as = ( + ) ( ) ( 82) where is the equilibrium velocity of the carriers. If an electric field is applied in the x direction and a magnetic field is applied in the z direction then an electric field is produced in the y direction giving electron velocity in x and y as = ( 83) = ( 84) PAGE 77 77 where = the cyclotron frequency. Using these equations and the charge distribution with applied electric and magnetic field, the current equations in x and y are = + ( 85) = + ( 86) w = 2 1 + 22 = ( 87) = 2 21 + 22 = ( 88) Physical M agnetoresistance In a Hall set up, current is allowed to flow only in the x direction. So a field is set up in y to compensate for the effect of the B field so that no current flows. Thus, = 0 and Hall co efficient is defined as = = 1 2+ 2 ( 89) However, there could be a change in resistance. If the relaxation time is independent of the energy of the electrons, applying a B field will not change the conductivity of the electrons. If relaxation time is energy dependent, and t here is no field applied, the Hall co efficient is = 1 2 2 = (8 10) where r is the Hall factor and varies between 1 and 2. At low B fields, 2 { 2 3 } ( 811) = 2 { 2 2 4 } ( 812) = { 1 222( 2 )} = 0{ 1 ( 2 ) 0 20 22} ( 813) PAGE 78 78 w here the physical magnetoresistance coefficient is given by = 3 22 1 ( 814) and the magnetoHall co efficient is = 4 3 23 1 (8 15) and 0, 0 are the Hall co efficient and conductivity with B = 0. The change in resistance with the applied B field, or the magnetoresistance, is 0 = 0 0 = 20 20 2 ( 816) For degenerate materials, with 1017 3 is constant. At low fields, there should be no magnetoresistance. At high B fields, = 1 ( 817) The conductivity a pproaches a constant at high fields, and the ratio of the high field conductivity to the zero field conductivity is 0 = 1 ( 818) Geometrical M agnetoresistance This effect is seen in wide devices where current and field in the y direction are zero because of the shorting effects of large contacts. Here, RH =0. If the relaxation time does not depend on energy, then = 01 + 22 ( 819) From Equation ( 819) we have that = 1 0 1 1 2 ( 820) If the relaxation time depends on the energy of the electrons, with no B field applied, PAGE 79 79 = ( 821) With a B field applied, the change in conductivity can be written as = 0( 1 22) ( 822) where the geometric magnetoresistance co efficient is given by = 3 3 ( 823) The change in resistivity, or the geometric magnetoresistance is giv en by 0 = 22 ( 824) Equation ( 8 16) gives the physical magnetoresistance, which is different from the geometrical magnetoresistance given by ( 824). Geometrical magnetoresistance arises only due to the geometry of the sample, but usually turns out to be a larger effect. For degenerate electrons, especially, = 1 but = 0 Mobility E quations The conductivity mobility, which goes into equations for current and the slope of the linear part of the v = ( 825) The mobility calculated from Halleffect experiments is given by = 2 2 = r ( 826) The mobility calculated from magnetoresistance experiments, for samples showing physical magnetoresistan ce is = 2 2 3 3 1 2 = 1 2 ( 827) The mobility calculated from magnetoresistance experiments, for samples showing geometrical magnetoresistance is PAGE 80 80 = 3 3 1 2 = 1 2 ( 828) The values of are very close to unity for most scattering processes and are calculated using Monte Carlo procedure and an analytical procedure as described by 44. A detailed discussion is presented in Chapter 9. Geometrical and P hysical M agnetoresistance Figure 81. Hall geometry for magnetoresistance measurements. (a) Hall geometry to measure physical magnetoresistance (b) Wide sample geometry to measure geomet rical magnetoresistance 46 The expressions for physical and geometrical magnetoresistance are discussed in the previous subsections. For long, thin samples, as shown in Figure ( 81 (a)), change in resistivity occurs if scattering is en ergydependent, conduction is anisotropic or due to more than one type of carriers. Physical magnetoresistance occurs because the Lorentz force compensates only for carriers with average velocity. All other carriers are either under or over compensated, c ausing increase in resistivity. PAGE 81 81 If the sample geometry is such that the Hall field is shorted out by the contacts, as shown in Figure ( 81 (b)), the Lorentz forces are no longer compensated and the carriers move at an angle to the applied electric field. S uch samples show a change in resistance with applied magnetic field, irrespective of whether the resistivity changes or not. This is geometrical mag netoresistance46. The effect of geometry on magnetoresistance is highest when the device is circular. That is, the ratio of the length to width ratio of the sample is very small. But, even in long devices, there is a geometrical contribution to magnetoresistance, though much lower 47. Magnetoresistance in N anowires Various reports on magnetoresistance in nanowires can be found in the literature. The main focus is on ferromagnetic nanowires Co balt doped ZnO wires 48, nickel an d cobalt wires45, M ndoped GaN wires49. A report on Bi nanowires can be found in50. Classical theory of magneto transport predicts a n increase in resistance with applied B field, but negative magnetoresistance is observed in the nanowires reported in literature Longitudinal magnetoresistance was measured in Bi nanowire arrays. The carrier paths become more confined when the B field i s applied parallel to the wire axis, reducing scattering at the wire boundary. Therefore, the resistivity reduces due to the increased apparent mean free path. When the magnetic field is applied perpendicular to the wire, scattering at the wire boundaries does not reduce as much as in the longitudinal case. At low transverse B field, the magnetoresistance varies as 20 where 0 is related to the carrier mobility. The higher the mobility the greater the magneto resistance observed. The nanowires show much lesser change in resistance as compared to bulk Bi. In the case of these wires, physical magnetoresistance is observed. Due to the confinement of electrons in a cylindrical potential PAGE 82 82 well, there exist both light and heavy electron bands. Applicat ion of B field moves the electrons to the band with lighter mass and a negative magnetoresistance is seen until temperatures where increased phonon scattering can compensate for the change in electron mass 50. Measurement chosen and the chip containing it was pasted onto a sample holder with epoxy. The contacts to the device were wire bonded to the sample holder. The I V characteristic of the wire was measured from 300K down to 50K with B = 0T, 2T and 5T, with the B field perpendicular to the plane of the sample. An American Magnetics 9 Tesla Superconducting Magnet was used to vary the temperature and B field. Liquid helium was used to coo l the sample down to low temperatures. The current voltage characteristic at 0T were measured first at all temperatures and the process repeated for the other B fields used. A similar measurement was made at 4.2K. A schematic of the measurement set up i s shown in Figure (82). The resistance of the device was obtained from a linear fit of the I V at 50K steps for each of the applied B fields. A plot of resistance with temperature for different B fields is shown in Figure ( 8 3). The 0T resistance of the device under test (DUT) drops abruptly at around 120K and increases again with almost the same slope as from 300 to 130K. This could be due to capture of electrons by repulsive traps at higher temperatures and emptying of the repul sive traps at low temperatures. A detailed discussion of repulsive traps can be found in Chapter 4. With the application of a B field, the resistance vs. temperature curve increases monotonically with reduction in temperature. The presence of the magnetic field reduces the probability of trapping 51and thus the resistance of the device with a B field applied is lower than the 0T resistance from 300K to 120K, but higher from 120K to 4.2 K. At 4.2K, where impurity PAGE 83 83 conduction dominates, magnetoresistance is negative again. A plot of magnetoresistance against temperature is shown in Figure (8 4). Figure 82. Schematic of experimental set up to measure magnetoresistance, showing orientation of electric and magnetic fields. Figure 83. Temperature dependence of resistance with B field applied. 18500 19000 19500 20000 20500 21000 21500 22000 22500 0 100 200 300 400Resistance Temperature K 0T 2T 5T B SQUID chip Sample holder PAGE 84 84 Figure 84. Magnetoresistance vs temperature. Calculation of M obility The product of mobility and carrier concentration is of the order 1021cm1V1s1 for this device. The high value of this product indicates a high density of electrons. Any change in PMR GMR 1 for degenerate electrons. The high band gap of GaN the high energy sepa ration between valleys in the conduction band and the isotropy of the gamma valley rule out multiple carrier conduction, change in effective mass and anisotropic conduction. These further point to geometric magnetoresistance. Using only the low temperature ( below 120K) resistance values, the mobility obtained is plotted in figure 85 using E quation 8 24. Extrapolating the low temperature resistance values with no B field applied to higher temperatures, an estimate of mobility at high temperatures can be o btained. This is shown in Figure ( 8 6). At 300K, the mobility is 960 cm2/V s and the peak mobility of 1300 cm2/V s is at a temperature of 110K. With the geometrical magnetoresistance factor as computed in Chapter 9, the mobility at 300K is 709 cm2/V s. 12 10 8 6 4 2 0 2 4 0 100 200 300 400Magnetoresistance %Temperature (K) 2T 5T PAGE 85 85 Figure 85. Low temperature mobility from magnetoresistance measurements Figure 86. Estimated high temperature mobility 500 550 600 650 700 750 800 850 900 950 0 20 40 60 80 100 120 140Mobility cm2/V sTemperature (K) 705 710 715 720 725 730 735 740 745 750 0 50 100 150 200 250 300 350Mobility (cm2/V s)Temperature (K) PAGE 86 86 CHAPTER 9 MONTE CARLO SIMULATI ON The Monte Carlo method is used to calculate mobility by simulating transport considering acceleration due to an applied field and loss of momentum and energy due to scattering. The velocity of carriers for a range of applied fields at a different temperatu res can be calculated by this method. The velocity of an electron in a semiconductor is limited by lattice scattering. Polar optical phonons, acoustic phonons, inter valley optical phonons, piezoelectric scattering and impurity scattering are considered f or GaN to estimate the total scattering rate and compute mobility 49. A list of material parameters necessary for the calculations is in Table 91 A discussion on the doping and temperature dependence of mobility can be found i n 52 Figures ( 9 1) and ( 92) showing the same are reproduced here for convenience. Low temperature low field drift velocity show maximum change with doping. This is bec ause impurity scattering dominates at these temperatures, and doping has a direct effect on the ionized impurity concentration. At high temperatures, and high fields polar optical phonons dominate and the effect of doping is almost insignificant. Increasing the electron concentration, while holding the background doping concentration steady, significantly enhances mobility because of the increased screening due to electrons. The angular distribution arising due to polar, piezoelectric and impurity scatterin g become more peaked along the direction of the electrons and this causes an enhancement in mobility. This effect is best noticed in a 2 DEG. A plot of electron mobility in a bulk device and in a 2DEG is shown in Figure ( 9 3). The experimental Hall mobility is higher than the conductivity mobility by a factor <2. This is easily seen from Figure 11. This arises due to the fact that the Hall factor is ignored. Hall mobility is related to the conductivity mobility as PAGE 87 87 Table 91. Material parameters for M onte C arlo simulation 52 Parameter Value Band gap 3.5 eV Mass density 6.1 g/cm 3 Longitudinal sound velocity 6.6X10 5 cm/s Relative Static dielectric constant 8.9 Relative high frequency dielectric constant 5.35 Optical phonon equivalent temperature 1078 K Intervalley phonon equivalent temperature 1078 K Piezoelectric constant 0.375 Acoustic deformation potential 8.3 eV Intervalley deformation potential 10 9 eV/cm Number of equivalent valleys L M 1 6 1 Effective mass L M 0.2 0.4 0.6 Non parabolicity L M 0.183 0.065 0.029 Intervalley separation L M 2 eV 2.1 eV = ( 91) where, r, the Hall factor has a value between 1 and 2. A detailed description of the relatio nship between conductivity mobility, hall mobility and magnetoresistance mobility can be found in Chapter 8. To get accurate mobility from Hall experiments, a computation of the Hall factor is necessary. Similarly, to get conductivity mobility from magneto resistance measurements, magnetoresistance coefficients need to be computed. PAGE 88 88 Figure 91. Velocity electric field curve at different temperatures. Dotted line 77 K, line 150K, dashed line 300K, circles 500K, plusses 1000K 52. Figure 92. Doping dependence of the vE curve. Dotted line, Nd = n = 1015 cm3,without ionized impurity scattering, line Nd = n = 1015 cm3, dashed line Nd=n = 1016 cm3, circles Nd = n= 1017 cm3, plusses Nd=n=1018cm3 52 PAGE 89 89 Figure 93. Mobility variation with temperature. Plusses Hall data for 2DEG, line Monte Carlo for 2 DEG, circles Hall data for bulk GaN, dotted line Monte Carlo for bulk GaN 52. With the parameters listed in Table ( 91) the velocity electric field was first calculated considering only polar optical, acoustic and deformation potential scattering to check accuracy of the parameters and if the scattering mechanisms considered sufficientl y represent scattering in bulk GaN devices at room temperature. The velocity field curve obtained is shown in Figure 94. The low field mobility is 978 cm2/V s. This is a fairly good agreement, considering that piezoelectric scattering was not considered in the Monte Carlo simulation. Analytical C omputation of M obility The various scattering mechanisms have an energy dependence which also translates to a temperature dependence, so that at a given temperature, it is possible to determine the dominant scatt ering mechanism and thus the dominant mobility contribution. PAGE 90 90 Figure 94. Velocity vs electric field curve obtained by Monte Carlo simulation of bulk GaN at 300K When mobilities due to different mechanisms are present at the same temperature, which is o ften the case, the total mobility is given by Matth ie ssens rule 1 = 1 1 +1 2 +1 3 + ( 92) This implies, 1 = 1 1 1 +1 2 2 +1 3 3 + ( 93) Therefore, = 1 1 ( 94) Using a similar argument, the magnetoresistance coefficients can also be calculated. This equation can be used to estimate the Hall scattering factors knowing the component mobilities. The individual mobility components are estimated after 44. Mobility components are calculated using the relaxation time approximation. Recalling Equation ( 825), mobility is written as PAGE 91 91 = where is the averaged relaxation time. Each mobility component can be determined from the averaged relaxation time of the corresponding scattering mechanism. This approximation holds true for impurity, piezoelectric and deformation potential scattering but is not true for polar optical phonon scattering, because it is an inelastic process. Therefore, an expression obtained from a variational approach is us ed to compute relaxation time, and thus mobility for polar optical phonon scattering. This study is limited to low electric fields, and thus to low electron energies. Since the inter At low energies there is no intervalley scattering, so optical phonon, acoustic (deformation potential and piezoelectric) and impurity scattering are considered. The scattering rates are calculated using the parameters of Table 1 using the standard expr essions 53. For ionized impurity scattering an impurity concentration of 1017 cm3 is assumed. The scattering rates obtained are shown in Figure ( 95). The various mobility components are estimated from the scattering rates and a plot of the mobility versus temperature is shown in Figure ( 9 6). Mobility at a temperature is limited by the smallest mobility component at that temperature. Mobilities due to polar optical scattering, acoustic and piezoelectric scattering dominate in the high temperature range and impurity scattering dominates at low temperatures. Acoustic and optical phonon scattering increase with increase in temperature. Inter valley scattering is insignificant until the average electron energy exceeds the gammaL valley energy separation of 1.9 eV. Impurity scattering and piezo electric scattering have an inverse dependence on temperature. Fig ( 9 7) shows the total mobility obtained from the component mobilities obtained using Matth ie ssens rule from Equation ( 92). PAGE 92 92 The mobility at 300K is found to be ~700 cm2/V s, which is comparable to that obtained using the Monte Carlo procedure. Figure 90K Open circles impurity scattering, dottedline polar optical phonon emission, + polar optical phonon absorption, x piezoelectric scattering, line acoustic deformation potential scattering. Figure 96. Mobility variation with temperature. The various components are due to the following scattering mechanisms line impurity, dotted line optical phonons, dash dot acoustic phonons, dashed piezo electric. PAGE 93 93 Figure 97. Total mobility ca lculated from comp onents shown in Fig 96. Calculation of Hall Factor, M agnetoresistance C oefficients The relaxation time is the reciprocal of the scattering rate. It can be written in the form m = ( 95) To average over all energy we use the formula = 3 / 2exp 3 / 2exp ( 96) This can be simplified to a function ( ) = 1 ( 97) which evaluates for integers as ( ) = ( 1 ) ( 98) For all p, ( ) = ( 1 ) ( 1 ) ( 99) Therefore, we have that = ( ) ( 5 2 ) ( 5 2 ) ( 910) PAGE 94 94 2 = 22 2 ( 5 2 2 ) ( 5 2 ) ( 911) The Hall factor r is 2 2 = 5 2 (5 2 2 ) (5 2 )2 ( 912) The physical magnetoresistance coefficient is = 5 2 3 5 2 5 2 2 2 1 ( 913) The geometrical magnetoresistance coefficient is given by = 5 2 3 5 2 2 5 2 3 ( 914) Equations ( 9 12) ( 9 14) can be calculated analytically using the value for s from the energies. Table ( 92) lists the calculated parameters for r, and 44. Table 9 2. Hall factor and magnetoresistance coefficients for individual scattering mechanisms Mechanism s Hall factor (analytical) Hall factor (from scattering table) Acoustic Def. Pot 1.18 1.1203 0.117 1.4020 Piezoelectric 1/2 1.1 1.09 0.0856 1.3123 Impurity 3/2 1.93 1.73 0.4889 4.4758 Polar optical phonons See text See text 1.19 0.0119 1.4504 For Polar optical phonons, at low temperatures the relaxation time is independent of energy and at higher temperatures, it varies roughly as 1 / 2. From the above table, it is obvious that when conduction is limited by impurity scattering, the Hall factor is high enough that there is a significant discrepancy between the Hall concentration, hall mobility and carrier concentration PAGE 95 95 and conductivity mobility. The Hall factor calculated from the scattering table is probably lower than the analytical term becaus e the scattering rate is evaluated at the Brooks Herring limit in the Born approximation and that depends on the carrier concentration chosen. At low carrier concentrations ( low temperatures/ high trap concentration), the numerical value approaches the ana lytical value. Fig ( 9 8) shows a plot of the Hall factor with temperature. Figure 99 shows a plot of the physical magnetoresistance with temperature, Figure 910 shows geometrical magnetoresistance plotted against temperature. From figure (9 8), it can be seen that the Hall factor remains almost invariant with temperature. The magnetoresistance coefficients for impurity scattering are much higher than for other types of scattering. Therefore, at low temperatures, the magnetoresistance mobility needs a bi gger correction than at higher temperatures. However, for degenerate electrons, geometrical magnetoresistance coefficient is 1 and physical magnetoresistance coefficient is 0 44. Figure 98. Hall factor variation with temperature. PAGE 96 96 Figure 99. Physical magnetoresistance co efficient plotted as a function of temperature. Figure 910. Plot of geometrical magnetoresistance co efficient with temperature. PAGE 97 97 Summary This chapter describes the dependence of electron mobility on electric field as determined by Monte Carlo simulations. Values obtained by considering optical, acoustic and inter valley scattering are compared with those reported in literature. The Hall factor and magnetoresistance coefficients are calculated for various scattering mechanisms. The magnetoresistance coefficients obtained here are used in Chapter 8, to determine mobility. PAGE 98 98 CHAPTER 10 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK A framework to understand charge transport in GaN nanowires under the influence of electric and magnetic fields has been put together. Extraction of device parameters from I V data obtained for devices from four different growth runs simulation and analytical methods has been presented. A detailed discussion on the effects of different types of traps on the I V has been presented, along with simple techniques for extracting carrier concentration, mobility and trap density from such I V. Mobility has been independently determined from magnetoresistance data. The highest measured mobility of 710 cm2/V s was obtained from magnetoresistance measurements and the carrier concentration is around 1017 cm3. Trap activation energies were e stimated from space charge limited transport and the possible presence of gallium vacancies, silicon, gallium in nitrogen sites is indicated. A Monte Carlo simulation was performed to extract values for Hall factor and magnetoresistance coefficients. At 3 00K, the Hall factor is calculated to be 1.125, physical magnetoresistance coefficient 0.035 and geometric magnetoresistance coefficient 1.35. In this study, Ti/Pt/Al/Au contacts have yielded the best results. But, the formation of an ohmic contact is by no means a given, and annealing does not necessarily improve the I V characteristics. Ti/Au contacts yield Schottky devices and the mechanism of forming ohmic contacts could be further studied. The Poissons equation solved for the nanowire assumes a cyli ndrical cross section. Some GaN wires have a triangular cross section and this has been previously reported in the literature. More work is needed to analyze the effects of field crowding at the sharp corners. Surface pinning of the Fermi level has not been investigated or considered in this work, and the electric PAGE 99 99 field distribution could be studied by imposing this boundary condition. For thin wires, in the surface depletion region, the relaxation time is smaller than in the bulk of the device. Since the onset of space charge is attributed to slow relaxation, if the depletion region is very thick, the onset of space ch arge might be delayed. 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Apart from her thesis on GaN nanowires, she has worked on automating recognition of citrus fruits and leaves with Dr Lee of UFs Agricultural Engineering Department. Her research interests are device characterization and modeling. She ob tain ed her Masters degree from UF in August 2009. 