Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0047043/00001
## Material Information- Title:
- Device Simulation of Functionalized 2D Material
- Creator:
- Seol, Gyungseon
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2014
- Language:
- english
- Physical Description:
- 1 online resource (4 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Electrical and Computer Engineering
- Committee Chair:
- GUO,JING
- Committee Co-Chair:
- BOSMAN,GIJSBERTUS
- Committee Members:
- YOON,YONG KYU
ZIEGLER,KIRK JEREMY - Graduation Date:
- 8/9/2014
## Subjects- Subjects / Keywords:
- Atoms ( jstor )
Binding energy ( jstor ) Charge density ( jstor ) Doping ( jstor ) Electric current ( jstor ) Electrons ( jstor ) Energy gaps ( jstor ) Graphene ( jstor ) Orbitals ( jstor ) Transistors ( jstor ) Electrical and Computer Engineering -- Dissertations, Academic -- UF dft -- graphene -- molybdenum-disulfide -- negf -- transition-metal-dichalcogenides -- tungsten-diselenide - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Electrical and Computer Engineering thesis, Ph.D.
## Notes- Abstract:
- This work presents simulation work on electrical properties of functionalized 2D materials and the devices that utilize them as channel. The proposal is divided into 4 chapters. Chapter 1 reviews the basic electronic property of graphene and graphene nanoribbons. The basic concepts dealt here will help understanding the works introduced from chapter 2. In chapter 2 and 3, we introduce studies on graphene derivate 2D material. In that domains of h-BN can be grown in graphene using chemical vapor deposition (CVD) process, there may be ways to utilize insulating properties of h-BN to provide confinements to the graphene system. Chapter 2 contains study on electrical properties of armchair nanoribbon (AGNR) confined by h-BN by conducting density functional theory (DFT) calculation. In the following chapter, we cover studies on chemical modifications such as hydrogenation or fluorination on graphene, where the change of initial sp2 hybridization of graphene to sp3 generates bandgap. By adapting selective chemical modifications, bandgap can be generated in 2D graphene while conserving the continuous two-dimensional atomistic layer. We adopt a ballistic transistor model, where the band structure were calculated using the ab-intio simulations to assess the performance of graphene nanoroad and nanomesh transistors. In chapter 4, we explore potassium doping of transition metal dichalcogenides (TMDs) such as MoS2 and WSe2. Also DFT calculations were conducted to understand the characteristics of the doping and how it is different compared to the graphene case. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2014.
- Local:
- Adviser: GUO,JING.
- Local:
- Co-adviser: BOSMAN,GIJSBERTUS.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-02-28
- Statement of Responsibility:
- by Gyungseon Seol.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 2/28/2015
- Resource Identifier:
- 969977012 ( OCLC )
- Classification:
- LD1780 2014 ( lcc )
## UFDC Membership |

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PAGE 1 D EVICE SIMULATION OF FUNCTIONALIZED 2D MATERIAL By GYUNGSEON SEOL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 PAGE 2 Â© 2014 Gyungseon Seol PAGE 3 To my family PAGE 4 4 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor, Prof. Jing Guo, for his excellent guidance, caring and patience. Studying as a student of Prof. Guo has been a great privilege that not much Ph.D . student would be lucky enough to have . It is a great pleasure to thank my colleagues. I thank Dr. Kai Tak Lam and Dr. Yang Lu, for all the discussions. It had been great joy to work with Dr. Wenchao Chen, Dr. Qun Gao, Leitao Liu, Xi Cao, Zhipeng Dong and Runlai Wan. Finally I want to thank my wife and my daughter. T his work was supported by ONR, ARL , and NSF. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREV IATIONS ................................ ................................ ........................... 11 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 1.1. O verview ................................ ................................ ................................ .......... 14 1.2. Graphene B ased E lectronics ................................ ................................ ........... 15 1.2.1. Tight Binding Approach of Graphene Electronic Properties ................... 16 1.2.2. Electronic Properties of Armchair Graphene Nanoribbon ....................... 17 1.3. Outline ................................ ................................ ................................ ............. 18 2 BANDGAP OPENING IN BORON NITRIDE CONFINED ARMCHAIR GRAPHENE NANORIBBON ................................ ................................ ................... 22 2.1. Overview ................................ ................................ ................................ .......... 22 2.2. Simulation M ethod ................................ ................................ ........................... 23 2.3. R e s ults ................................ ................................ ................................ ............. 24 2.4. Summary ................................ ................................ ................................ ......... 28 3 PERFORMANCE PROJECTION OF GRAPHENE NANOMESH AND NANOROAD TRANSISTORS ................................ ................................ ................. 33 3.1. Overview ................................ ................................ ................................ .......... 33 3.2. Approach ................................ ................................ ................................ ......... 35 3.3. Results ................................ ................................ ................................ ............. 36 3.3.1. DFT R esults : E k D ispersion R elation ................................ ................... 36 3.3.2. DFT R esults : Binding E nergy C alculation U sing Basis S et S uperposition E rror (BSSE) ................................ ................................ .......... 38 3.3.3. Device P erformance ................................ ................................ ............... 39 3.4. Summary ................................ ................................ ................................ ......... 41 4 n DOPING OF TRANSITION METAL DICHALCOGENIDES BY POTASSIUM ...... 48 4.1. Overview ................................ ................................ ................................ .......... 48 4.2. Method ................................ ................................ ................................ ............. 49 PAGE 6 6 4.3. Results ................................ ................................ ................................ ............. 51 4.4. Summary ................................ ................................ ................................ ......... 52 5 STRAIN INDUCED INDIRECT TO DIRECT BANDGAP TRANSITION IN BI LAYER WSE 2 ................................ ................................ ................................ ......... 57 5.1. Overview ................................ ................................ ................................ .......... 57 5.2. Method ................................ ................................ ................................ ............. 59 5.3. Results ................................ ................................ ................................ ............. 61 5.4. Summary ................................ ................................ ................................ ......... 65 6 ELECTROSTATIC SCREENING PROPERTIES OF MULTILAYER MOS 2 ............ 75 6.1. Overview ................................ ................................ ................................ .......... 75 6.2. Method ................................ ................................ ................................ ............. 76 6.3. Results ................................ ................................ ................................ ............. 78 6.4. Summary ................................ ................................ ................................ ......... 83 7 TWO DIMENSIONAL HETEROJUNCTION DIODE AND TRANSISTOR ............... 88 7.1. Overview ................................ ................................ ................................ .......... 88 7.1. Graphene MoS2 Junction Diode ................................ ................................ ...... 90 7.1.1. One Dimensional Simulation for Vertical Junction ................................ .. 90 7.1.2. Classical Transport Calculation ................................ .............................. 92 7.1.2. Two Dimensional Simulation to Treat Lateral Variations ........................ 93 7.2. MoS 2 Graphene MoS 2 Bipolar Junction Transistor ................................ .......... 95 7.2.1. 1D Capacitance Model with NEGF Tr ansport Calculation ...................... 95 7.2.2. Results of the MGM BJT ................................ ................................ ........ 97 Summary ................................ ................................ ................................ .............. 100 8 CONCLUSION ................................ ................................ ................................ ...... 108 APPENDIX A BANDGAP CALCULATION OF AGNR AND AGNR WITH EDGE PERTURBED BY THE IONIC POTENTIAL ................................ ................................ ................. 111 B BASIS SET SUPERPOSITION ERROR ................................ ............................... 119 C G 0 W 0 CALCULATION APPROACH ................................ ................................ ...... 122 D VASP INPUT SCRIPT FOR GGA, HSE AND G 0 W 0 SIMULATION ...................... 123 E ANALYTICAL DERIVATION OF DEBYE LENGTH ................................ .............. 127 F EVALUATION OF 2D THOMAS FERMI MODEL ................................ ................. 130 G C N ................................ .................. 132 PAGE 7 7 H EXTRACTION OF INTERLAYER COUPLING FACTOR AND RELATION TO NEGF ................................ ................................ ................................ .................... 134 LIST OF REFERENCES ................................ ................................ ............................. 137 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 146 PAGE 8 8 LIST OF TABLES Table page 2 1 Tight binding parameters for A C x (BN) y s ................................ ............................ 29 3 1 Summary of the MOSFET performance ................................ ............................. 42 4 1 Summary of bonding properties of MoS 2 and graphene ................................ ..... 53 5 1 DFT calculated structural parameters ................................ ................................ . 67 5 2 DFT calculated BL WSe 2 bandgap ................................ ................................ ..... 68 7 1 Scattering potential matrix element value ................................ ......................... 101 D 1 GGA step 1 and 2 ................................ ................................ ............................. 123 D 2 GGA step 3 ................................ ................................ ................................ ....... 124 D 3 GGA step 4 ................................ ................................ ................................ ....... 124 D 4 HSE Self consistent calculation ................................ ................................ ........ 125 D 5 HSE Band calculation ................................ ................................ ....................... 125 D 6 HSE Band calculation INCAR file. ................................ ................................ .... 126 G 1 Classical and NEGF transport calculation comparison ................................ ..... 133 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Schematic of 2D graphene and corresponding lattic vectors. . ............................ 20 1 2 Schematic of an AGNR and a ZGNR. ................................ ................................ 21 2 1 Schematic of an A C x (BN) y s. calculation. ................................ ........................... 29 2 2 Bandgap properties and C to C distance as a function of AGNR and A C x (BN) y s width. ................................ ................................ ................................ ... 30 2 3 A Pseudocharge density plot of A C x (BN) y and H terminated AGNR. ................ 31 2 4 DFT and TB model bandgap calculation comparison . ................................ ........ 32 3 1 Schematic of the device structure and the top of barrier ballistic transistor model. ................................ ................................ ................................ ................. 43 3 2 Schematic of the graphene Nanoroad and the bandgap properties. . ................. 44 3 3 Schematic of the graphene nanomesh and the bandgap properties. ................. 45 3 4 C H and C F binding energy of A X NR structures with N a =3 or 4. ..................... 46 3 5 The ballistic performance limits of Graphene nanoroad FETs compare to Si.. ... 46 3 6 The ballistic performance limits of Graphene nanomesh FETs compare to Si and GaAs. ................................ ................................ ................................ .......... 47 3 7 A HNRs and A FNRs device performance comparison . ................................ ..... 47 4 1 DFT calculated structures of K doped graphene and MoS 2 . ............................... 54 4 2 K to graphene, MoS 2 and WSe 2 binding energy comparison. ............................ 55 4 3 Charge density plot of K doped graphene and K doped MoS 2 . .......................... 55 4 4 Charge transfer calculated using Bader analysis ................................ ................ 56 5 1 Schematic of bi layer WSe 2 and the mechanical strain setup. ............................ 69 5 2 DFT calculated E k relations of bi layer WSe 2 , comparing the XC functions and the effect of including and excluding spin orbital coupling (SOC). ............... 70 5 3 Comparison of the bandgap features of BL WSe 2 . ................................ ............. 71 5 4 Comparison of PL measurement with HSE DFT calculation . ............................. 72 PAGE 10 10 5 5 Bandgap features of ML WSe 2 . ................................ ................................ .......... 73 5 6 ML WSe 2 atomic and orbital contribution on the E k relation. ............................. 74 5 7 T he partial charge calculation of at each band points of interest. A) K C V . The energy range specified in sub figure represents the integration interval. ................................ ................................ ............................. 74 6 1 Schematic of multi layer MoS 2 , and 1D capacitance model. .............................. 84 6 2 Normalized charge density profile in 10 layer of MoS 2 FET with varying gate charge density . ................................ ................................ ................................ ... 85 6 3 Effective screening length in 10 layers of MoS 2 FET with varying gate charge density . . ................................ ................................ ................................ .............. 85 6 4 Effective screening length between the first (bottom) two layers( i=1, eff ) compared with the Debye length ( D i ) at different temperatures. ........................ 86 6 5 Charge and potential profile along the MoS 2 stack. ................................ ............ 87 7 1 Simulated structure of graphene MoS 2 junction. ................................ .............. 101 7 2 MoS 2 Graphene junction properties. ................................ ................................ 102 7 3 Band alignment results of 2D Poisson calculation. ................................ ........... 103 7 4 Isolated diode model and its results. ................................ ................................ 104 7 5 Schematic of the 3 terminal MGM BJT and NEGF current calculation results. . 105 7 6 Current gain and intrinsic delay of the MGM BJT.. ................................ ........... 106 A 1 Graphene nanoribbon schematic. ................................ ................................ ..... 111 A 2 Schematic of AGNR and perturbation modeling. ................................ .............. 115 B 1 BSSE calculation method. ................................ ................................ ................ 121 C 1 Bilayer WSe 2 HSE DFT and G 0 W 0 calculation. ................................ ................ 122 PAGE 11 11 LIST OF ABBREVIATIONS 1D 1 dimensional 2D 2 dimensional 3D 3 dimensional ABNNR Armchair boron nitride nanoribbon AGNR Armchair graphene nanoribbon BNNR Boron nitride nanoribbon BSSE Basis set superposition error CMOS Complimentary metal oxide semiconductor CP Counter poise CVD Chemical vapor deposition DFT Density functional theory DIBL Drain induced barrier lowering DIBT Drain induced barrier thinning DOS Density of states FET Field effect transistor GNR Graphene nanoribbon h BN hexagonal boron nitride ITRS International technology roadmap for semiconductors I V Current voltage MOSFET Metal semiconductor field effect transistor NEGF Non equilibrium Green ' s funct ion TB Tight binding TMD Transition metal dichalcogenides PAGE 12 12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy D EVICE SIMULATION OF FUNCTIONALIZED 2D MATERIAL By Gyungseon Seol August 2014 Chair: Jing Guo Major: Electrical and Computer Engineering In the recent years, since the first advent of graphene in 2004, two dimensional (2D) semiconducting materials, such as h BN and transition metal dichalcogenides (TMDs) have gained of a great interest. To minimize the short channel effect at extreme scaling of future sub 5 nm gate length field effect transistors (FETs), large bandgap semiconductors with ultrathin body are ess ential. In this context adopting 2D semiconductors as channel will serve a great advantage. Graphene , due to its excellent charge transport properties, ranks itself as one of the most promising future device material. TMDs are also of great interest due to sufficient bandgap they provide, which is desired in achieving large on off current ratio . C ombinations of these materials are possible, leading to various device structures to be studied. T his dissertation can be categorized into to topic. The first is t he Ab initio calculations to study the material properties of the 2D semiconductor. We present g raphene derivatives which bandgap can be engineered without reducing its dimensionality, which is preferred in achieving higher current density when applied as channel material of FET s . Doping properties of TMDs (MoS 2 and WSe 2 ) and effect of strain on bilayer WSe 2 are investigated . These studies served as a fundamental in PAGE 13 13 understanding the device operation mechanism and performance prediction, which is the second topic of this work. Performance limit of FETs using grapehene derivative materials is examined and show that they out perform Si based FETs. Next, the electrostatic screening properties of MoS 2 is presented, which we predict the screening length can be as short as half the interlayer distance of the 2D crystalline to infinite screening length depending on the temperature and applied gate voltage. Following is the e lectrical properties of Graphene MoS 2 heterojunction . To assess the mechanism of the junction we perfrom 1D and 2D Possion calculation followed by transport calculation using classical method and non bipolar junction transistor using MoS 2 Graphene MoS 2 hetero structure is studied. PAGE 14 14 CHAPTER 1 INTRODUCTION 1.1. O verview In the past few decades, scaling down the device feature size has been the major driving force behind sustaining the growth of semiconductor industry. So far by increasing the device density, it has successfully fulfilled its task in delivering more functionality. However, we are now in the era of 22nm node, and the International Technology Roadmap for Semiconductors (ITRS) 2011 suggests that in 2022, the gate length of a metal oxide f ield effect transistor (MOSFET) will reach sub 10nm regime [1]. Further reduction of feature size lies as a question mark in terms of physical realization and enhancement in the device performance. With the channel length in such extreme scale, the gate lo ses its controllability, due to the drain induced electrostatic effect that lowers the barrier (DIBL) [2]. At the same time the barrier itself gets thinner, drain induced barrier thinning (DIBT), which results in increase of source to drain (S/D) tunneling [3]. Both of these effects make it difficult to turn the device off, which lead to poor power efficiency that is undesired in mobile applications. To overcome the limitations, various new classes of materials as well as device structures have been explore d [4 6]. One approach is to make the channel thin. As the gate size shrinks, to maintain the electrostatic control, the channel needs to be scaled down accordingly. So , the question is, what if the channel is atomi c thin. Since the first demonstration of g raphene [7], 2D layered materials have been investigated in depth. Although graphene exhibits superior carrier mobility, the material innately suffers from zero bandgap. Thus, employing a pristine graphene as a channel material in the field effect transist or (FET), result in a low I ON /I OFF ratio. This imposes a PAGE 15 15 major obstacle for using graphene in future digital electronic device applications. To address this problem, few ways have been propose d to indu ce bandgap in a graphene system, such as , graphene nano ribbons (GNRs), graphene nanomesh and graphene antidot [8~11]. Also there exists other layer ed structure material family with bandgap , such as, hexagonal boron nitride ( h BN), t ransition m etal d ichalcogenides (TMDCs) [12,13]. In this work, we theoretically investigate the material properties of such 2D functionalized materials that can provide , or bear a sufficient bandgap and elucidate the physics behind. Also we conduct device simulations that employ the materials as channel to project the performance limit and compare to that of the Si MOSFET . 1.2. Graphene B ased E lectronics Graphene is a monolayer (ML) of carbon tightly packed into a dense 2D honeycomb lattice as visualized in Figure 1 1 A . Previously, it was studied as a building block of carbon allotropes such as, graphite and carbon nanotubes (CNTs). It was believed until recently, that the 2D material is unstable and cannot exist in free state. However, in 2004, Novoselov et al . proved this belief to be false, demonstrating by micromechanical cleavage technique [7]. Further investigation was carried out and in 2007, the subtle optical effect created on top of a chosen SiO 2 substrate allowed its observation even with an ordinary optical m icroscope [14]. Carbon is the sixth element in the periodic table and is smallest atom in column IV. A carbon atom has six electrons in 1s 2 , 2s 2 and 2p 2 orbital. Two electrons are in the 1s 2 orbital and are strongly bound as core electrons. When the carbo ns form a crystalline, the 4 valence electron occupying 2s 2 and 2p 2 orbital s give rise to 2s , 2p x , 2p y and 2p z orbital. Since 2s and 2p orbital are close in energy, these orbital mix and form sp 1 , sp 2 and sp 3 hybridization. In the case of graphene, sp 2 hybridization, and forms a PAGE 16 16 trigonal planar structure , where each carbon atoms are connected by s with bonding distance of 1.42Ã…. This o nd is responsible for the mechanical strength of the carbon allotropes and this filled band lie as deep valence band. The remaining p z orbital is perpendicular to the x y plane with one electron occupying it. This half filled band results in strong tight binding character and thus functions as main contributor of graphenes' fascinating electronic propert ies . Also t his is the reason that the tight binding (TB) description of this system which includes just one p z orbital per carbon atom and accounting for only the nearest neighbor interaction provides an accurate graphene band structure at the energy range near the F ermi point [ 1 5]. 1.2.1. Tight Binding Approach of Graphene Electronic Properties Band structure of a graphene can be calculated by a tight binding (TB) model in a p z orbital basis set per carbon atom by taking account of its nearest neighbors only . The Hamiltonian matrix can be express as [16], . (1 1) The is tight binding parameter between carbon atoms . and are the basis vectors of graphene as shown in Figure 1 1 B . By calculating the eigenvalues of the Hamiltonian matrix, the band structure of graphene can be obtained as in equation (1 2). , (1 2) This can be further simplified by Taylor expansion to, , (1 2) PAGE 17 17 where is the Fermi velocity. The highly linear dispersion relation differs from conventional semiconductors, where the energy spectrum can be approximated in parabolic manner. Although the graphene exhibits superior carrier velocity and near ballistic transport characteristics, it is innately a zero bandgap. This is an unappealing feature for nano electronic applications. 1.2.2. Electronic Properties of A rmchair Graphene Nanoribbon One way to generate a bandgap in a graphene system is to reduce its dimension to a quasi 1D nanostructure, such as CNT and GNR. Depending on the orientation, a GNR can be categorized into two group, z igzag graphene nanoribbon (ZGNR) and a rmchair graphene nanoribbon (A GNR), as depicted in Figure 1 2 A and Figure 1 2 B, respectively. Although ZGNR s ha ve distinctive electronic and magnetic features due to their localized edge states at each end, h ave zero bandgap [ 18 ]. Therefore we will only focus on AGNRs. An AGNR is also metallic or semiconducting as a function of the width and can be categorized into 3 groups, namely, 3p , 3p+1 and 3p+2 , where p is an integer [19 21]. Within the TB model the bandgap of AGNRs can be calculated as , , , and 0, for the each 3 groups mentioned, respectively. The detail s of the derivation using TB model are in A ppendix A.1. Deeper investigations have revealed that the edge bonds terminated by H atoms ( h ydrogen passivation) generate shortening of the C C bonds at the edge of AGNRs. The perturbation by the edge bond relaxation give s rise to a small bandgap even in the 3p+2 family [1 8 ]. PAGE 18 18 1.3. Outline This work is organized as the following. Chapter 2 talks about 2D continuous sheet of hybrid material, h BN and Graphene. Creating domains of h BN in graphene has been reported. By precise control of the shape of the domain, one can engineer the electronic properties of the graphene/ h BN hybrid material. We have condu cted study on GNRs in between BN nanoribbons (BNNRs). In this structure, the confinement of the wave functions can be given to the AGNRs without rapid bond termination, thus generate bandgap in a continuous 2D structure. Similar approach has been studied in Chapter 3. Instead of the h BN, functionalized graphene was used to generate the confinement. By hydrogenation of fluoridation, the sp 2 hybridization of the graphene changes to sp 3 hybridization which is responsible for generation of bandgap. By selective chemical functionalization, we can create nanoroads or nanomeshes and control the bandgap in a 2D graphene derivate system. Simulations were conducted on FETs utilizing the material s as channel to estimate and evaluate the performance limit. While in the previous chapters focus on generating bandgap in the graphene system and evaluating them, the scope of coming chapters are on TMDs, which are 2D crystalline materials which innately bear bandgap. In Chapter 4, potassium (K) doping of MoS 2 and WSe 2 was studied and compared with graphene via A b initio simulations . We find that he binding energy of the K to the is larger than to the graphene , which indicates that the K acts as more stable dopant in the TMD system In Chapter 5 , we report the effect of uniaxial strain on bi layer WSe 2 via first principle calculations. We find indirect to direct bandgap transition in this system, and PAGE 19 19 confirm this feature with measurement results. We also find that the transition originates from the dz 2 orbital of the W atom. While Chapter 4 and 5 deals with the material features, the following Chapter 6 and 7 are oriented on the application device of the TMDs. Electrostatic screening behavior of multi layer of MoS 2 is examined in Chapter 6. In Chapter 7, we discuss the properties of the graphene MoS 2 heterojunction which resembles the metal semiconductor junction. Also operating mechanism of metal base transistor like MoS 2 graphene MoS 2 heterostructure transistor is examined Last, we summarize and conclude the work in the conclusion chapter, Chapter 8. PAGE 20 20 Figure 1 1 . Schematic of 2D graphene and corresponding lattic vectors . A ) Visualization of 2D graphene. B ) Schematic of honeycomb lattice. The red dashed rhombus represents the unitcell of graphene, which consist of A and B sublattice. PAGE 21 21 Figure 1 2 . Schematic of an AGNR and a ZGNR. A ) N a AGNR, where N a represents the AGNR width, 9 in this case. B ) N z ZGNR, where N z =6 represents the ZGNR width . PAGE 22 22 CH APTER 2 BANDGAP OPENING IN BORON NITRIDE CONFINED ARMCHAIR GRAPHENE NANORIBBON G raphene nanoribbons (GNRs) have seized strong interest . Recent studies show that domains of graphene in monolayer hexagonal boron nitride (h BN) can be synthesized. Using the first principle calculations we have studied the electronic properties of armchair GNRs (AGNRs) confined by BN nanoribbons (BNNRs). Whi le, H terminated AGNRs have a close to zero bandgap with the width index of 3p+2 , AGNRs confined by BNNRs exhibit a considerable bandgap. The bandgap opening is primarily due to perturbation to the on site potentials of atoms at AGNR edges. A tight binding (TB) model is parameterized to confirm this mechanism and enable future device studies. 2.1. Overview Since the first demonstration of graphene, the ballistic transport property of the material has attracted interest for nano electronic applications [7] . Graphene, having a honeycomb structure results in a zero bandgap . Stripped into a few nanometer wide graphene nanoribbon (GNR) , bandgap can be tuned to a certain extent , by the confinement of electronic wave function [17, 19, 23, 24] . Single layer hexagon al boron nitride ( h BN), also a honeycomb lattice structure, can be formed into boron nitride nanoribbons (BNNRs). Due to large ionicit y of B and N atoms, BNNRs exhibit qualitatively different properties from those of GNRs, insulating and magnetic behavior s [25 27] . C, B and N are all in the same period of the periodic table, thus graphene and h BN have similar lattice constant. This makes combining BN and graphene attractive and has been extensively studied [28 30] . PAGE 23 23 While H terminated GNRs requires breaki ng of bonds, BNNR confined GNRs forms a continuous 2D atomistic layer [29] . Th is provide s a natural way of realizing a densely packed parallel array of semiconducting GNRs, which is necessary in providing large on currents for transistor applications. Focu sing on A rmchair GNRs (AGNRs) confined by Armchair BNNRs (ABNNRs) , we observed considerable bandgap opening in 3p+2 categories of AGNRs and alteration in bandgap relations between the three AGNR families. We claim that these differen ce from H terminated A G NRs , originate from the charge redistribution at the edges of AGNRs confined by BNNRs, and we exclude edge bond relaxation effect. 2.2. Simulation M ethod W e conducted ab initio density functional theory (DFT) calculations with SIESTA codes on sets of H terminated AGNRs and AGNRs bounded by ABNNRs for comparison [31] . From now on we will denote an AGNR confined by ABNNR as A C x (BN) y , where the x and y represents the width of AGNR and BNNR portion of the system, respectively. Simulations were run using the double employing the generalized gradient approximation (GGA) method. The Perdew Burke Ernzerhof (PBE) exchange correlation functional is adopted and the Troullier Martins scheme is used for the norm conserving pseudopotentials. A grid cutoff of 210 Ry was used and the Brillouin zone sampling is done by the Monkhost pack mesh of k points (16Ã—4Ã—1). Figure 2 1 A shows a random structure of an A C ncc (BN) nbn 1+nbn2 used throughout the work. The width of BNNRs on each side of GNR, n bn1 and n bn2 , are set in a manner that the sum, n cc +n bn1 +n bn2 is even, i.e. for an odd n cc , n bn1 =9 and n bn2 =10 , and for even n cc , n bn1 =n bn2 =10 , so that the unit cell replicate along the y direction. Total length of the BNNR in the unit cell is chosen as so that it is wide enough to act as an PAGE 24 24 insulator. The DFT calculation results of A C 14 (BN) 20 and A C 14 (BN) 32 were compared. Bandgap of the two cases are, 0.4016 eV and 0.4193 eV, respectively, resulting in an around 4.5% difference. Thus, larger value of n bn1 and n bn2 would result in similar dispersion relations, meaning that n bn1 +n bn2 =19 and 20 functions well for our purpose. 2.3 . Re s ults Figure 2 2 A shows the DFT calculated bandgap of the A C x (BN) y s and H terminated AGNRs. They exhibit qualitatively different electrostatic features between the two set. The width of AGNRs is defined by the number of dimer lines ( N ) and are categorized into 3 groups, N=3p, N=3p+1, and N=3p+2 ( p is an integer). The bandgap of ideal GNRs are inversely proportional to the width, with all N=3p+2 group remaining a zero bandgap. However, in H terminated AGNRs, the edges are terminated in a rather abrupt manner. Thus for the atomic bonds to be relaxed, the C C bonds parallel to the dimer line are shorter at the two edges than the rest. The perturbation from this generates a small bandgap opening in 3p+2 category of H terminated AGNRs, which can be observe in Figure 2 2 A [19] . For A C x (BN) y s, due to the similarity in the lattice constant of GNRs and BNNRs, we may expect different. Figure 2 2 B compares the C C bonds length parallel to the dimer of the H terminated 14 AGNR and GNR portion of the A C 14 (BN) 20 . Average C C bond length (lattice co nstant) of A C x (BN) y and AGNR is 1.425 Ã… (4.292 Ã…) and 1.413Ã… (4.262 Ã…), respectiv ely, resulting in only a 0.856% (0.694 %) difference. For the bonds at the edge, H terminated 14 AGNR have 3.4 % difference at while of A C 14 (BN) 20 exhibit only 1.4 %. Thus, the effect of edge bond relaxation can be eliminated as a source of bandgap opening in A C x (BN) y s. Another point observed in Figure 2 2 A is the relation of the width to the bandgap. In H terminated AGNRs, the bandgap in the three groups follow the hierarchy of PAGE 25 25 E 3p+2 PAGE 26 26 The E o 3p and E o 3p+1 are the bandgap for ideal AGNR which are given by, t CC 2] and t CC [2 , respectively [19] . The analytical calculation results are included in Fig ure . 2 2 A , marked as gray crosses. The cases for 3p+2 are not included. Due to the initial zero bandgap of the 3p+2 systems, perturbation calculation does not hold for this category. However shows perturbation indeed affects the bandgap hierarchy. For better understanding the mechanism, a Tight Binding ( TB ) model was conducted. Though DFT calculation provides accurate description of the system, it is computationally expensive. The TB model is a computationally cost effective method, which can facilitate further device studies. The Hamiltonian of this mo del can be expressed as equation ( 2 2) and the values of the par ameters used are listed in T able 2 1. ( 2 2) Indices i and j denote the site, c i + and c i are creation, annihilation operators. Also, , PAGE 27 27 . The 1 st and 2 nd nearest tight binding parameters were introduced for better fitting. Also, potential changes in the confined GNR portion due to the adjunct B and N atoms were considered , as in the previous analytic calculations. Because the TB model is constructed within the unit cell of the system, we assumed that the potential would effectively influence nearby atoms only in lateral direction, as indicated in Fig ure . 2 1 C . The fourth term in the RHS of equation ( 2 2) represents the changes in potential profile at the atomistic sites. Where the value of C,i , is 0 for ABNNR part and denotes the decaying potential from the B or N atoms affecting the nearby C atoms, which can be expressed as equation ( 2 3) in the AGNR part. ( 2 3) ( 2 4) ( 2 5) In equation ( 2 4) and ( 2 5), the potential is defined in exponential manner, where, P B and P N are the streng th of the potential and is the decay length [25] . d B Cn ( d N Cn ) denotes distance from the B (N) at the edge to the n th C atom. Fig ure 2 4 A plots the bandgap results of ab initio calculation an d TB model, which fits within 6 % range. Fig ure 2 4 B depicts few of the resulting dispersion relations in the region of interest. Thus the TB model employing potential changes in the AGNR portion of A C x (BN) y s describes the system well. Fig ure 2 4 C shows the extended TB calculation. Although PAGE 28 28 the bandgap is inve rsely proportional to the width for both cases, the minimal bandgap of A C x (BN) y s is considerably larger than that of H terminated GNR s . Both TB calculation and analytic calculation match. 2.4. Summary W e have investigated electronic properties of A C x (BN) y s through DFT calculation. The edge bond relaxation that is evident in H terminated AGNRs has a significantly small effect in A C x (BN) y s due to the similar lattice constant of two materials and thus can act as stable passivation method. The source of considerably bandgap opening for A C x (BN) y s with width index of 3p+2 , and distinctive width to bandgap relations from that of H terminated AGNRs is evaluated by the perturbation calculation and verifies that it originates from the change in the potenti al energy of the C atoms in the AGNR edges that are adjacent to B and N. A TB model of the system was constructed as a simple and effective method to predict the electronic properties. PAGE 29 29 Table 2 1 . Tight binding parameters for A C x (BN) y s Parameters (eV) CC BN t CC t BN t BC t NC t BC,2 0.015 1.95 2.5 2.9 2.0 2.5 0 t CC,2 t BB t NN t NC,2 P B P N 0 0 0.15 0.59 0.75 0.68 2.1 Figure 2 1. Schematic of an A C x (BN) y s . A ) Simulated structure of A C x (BN) y . x Axis is the transport direction. B ) Schematic of AGNR portion of A C x (BN) y unitcell. Inverted open triangle (closed triangle) represents C atom perturbed by adjacent B (N), for analytical calculation. C ) Schematic of A C x (BN) y unit cell, B and N, affec ting the potential profile of at C atomic sites within the unit cell. PAGE 30 30 Figure 2 2 . Bandgap properties and C to C distance as a function of AGNR and A C x (BN) y s width. A ) Bandgap ( E g ) vs. width plot of the DFT calculation of AGNRs (the solid line with closed square) and A C x (BN) y s (the solid line with open triangle), and analytical calculations of A C x (BN) y (gray cross ). B ) C to C distance of bonds that are parallel to the dimer line vs. the atomic indices. Data extracted from the DFT calcul ation of H terminated 14 AGNR (the solid line with open square) and A C 14 (BN) 20 (the solid line with closed triangle). Average of the H terminated 14 AGNR is represented as dot and A C 14 (BN) 20 as dash. Top panel is in absolute scale, distance unit in Ã…. Bo ttom panel is in relative scale regards to the average. PAGE 31 31 Figure 2 3 . A Pseudocharge density plot of A C x (BN) y and H terminated AGNR. A ) A C x (BN) y . B ) H terminated AGNR. Units are in e / Ã… 3 . PAGE 32 32 Fig ure 2 4 . DFT and TB model bandgap calculation comparison. A ) Bandgap ( E g ) of A C x (BN) y s as a function of width. The DFT calculation (solid line with closed square) and the TB model results (dashed line with open triangle) are compared. On the left y axis is the bandgap and on the right y axis is the relative err or of the two calculations. Relative error is defined as, err(%)=(E g,ab intio E g,TB )/E g,ab intio . B ) E k band plots of A C 8 (BN) 20 , A C 9 (BN) 19 , A C 13 (BN) 19 ,and A C 14 (BN) 20 comparing the DFT calculations (solid line) and TB results (dashed line). C ) TB calculated Bandgap PAGE 33 33 CHAPTER 3 PERFORMANCE PROJECTION OF GRAPHENE NANOMESH AND NANOROAD TRANSISTORS We examine the performance limits of the field effect transistors (FETs) with chemically modified graphene as the channel materials. G raphene nanoroad ( XNR) and graphene nanomesh (XNM) can be created t hrough selective chemical modification of X adsorb a te (either H or F) on graphene, which generates a bandgap while conserving the continuous two dimensional atomistic layer. We adopt a ballistic transistor m odel, where the band structure were calculated using the A b intio simulations to assess the performance of graphene nanoroad and nanomesh transistors. It is shown that array of graphene nanoroads , defined by hydrogenation or fluorination of atomically narr ow dimmer lines in a 2D graphene , are most ideal for transistor channel material in terms of delivering a large on current, which significantly outperforms Si metal oxide semiconductor (MOS) FETs. Alternatively, comparable performance to silicon can be ach ieved by careful designed graphene nanomesh through patterned hydrogenation or fluorination. As for the chemical modification, both hydrogenation and fluorination leads to similar transistor performance, with fluorination more preferred in terms of chemica l energetics. 3.1. Overview Graphene, a material with high mobility, has been extensively explored for electronics applications [32 , 33 ] . However, due to its honeycombs structure, the material innately suffers from zero bandgap. Thus, employing pristine graphene as a channel material in the field effect transistor ( FET ), result in a low I ON /I OFF ratio. This imposes a major obstacle for using graphene in future digital electronic device applications . To address this problem, few ways have been pro posed to induce bandgap in graphene. PAGE 34 34 For example, in graphene nanoribbons (GNRs) the confinement of the wave functions tunes the electronic properties, resulting in a bandgap [ 34 , 35 ]. Similarly, the graphene nanomesh or antidot, a two dimensional (2D) grap hene with nanoscale array of holes, has also been proposed [ 9 11 ]. However, all these structures have abrupt C C bond terminations at the edges, and hence the atomistic control is crucial for reliable performance. At current status, the fabrication of such precisely defined edge is still unreliable. A bandgap can also be induced by introducing chemical modification, such as hydrogenation and fluorination, to the graphene. These modifications alter the initial sp 2 hybridization to sp 3 , which induces a bandga p of a few eV in graphane or fluorinated graphene [ 36 39 ]. The combination of the two mentioned methods has been proposed and investigated by Singh et al [ 40 ]. Graphene nanoroad ( X NR, X denotes the type of adsorbate, i.e. H or F) is a narrow graphene stri p, similar to the GNR, sandwiched in between two hybridized graphene s. The hybridized graphenes confine the electron within the X NR, giving rise to a finite bandgap. Arrays of X NRs provide a natural way of realizing a densely packed parallel semiconductor with potentially uniform edge orientation. Applying the same idea to antidot systems, we propose a structure with vacancy replaced by selective chemical modification of graphene . We denote this as graphene nanomesh ( X NM). Note that, both X NR and X NM are formed by selective chemical modifications, without breaking the covalent C C bonds. Thus, these structures have continuous carbon 2D atomistic layer, in which the excessive edge defects can be prevented. In chapter , we evaluate the performance of FETs using the X NRs and X NMs as the channel materials. Our results indicate that array of X NRs, which behave as densely PAGE 35 35 packed current paths defined by hydrogenation or fluorination functionalization, can outperform Si metal oxide semiconductor (MOS) FETs. Bo th the hydrogenation and the fluorination structure exhibited similar performance, even with atomistic narrow separation. On the other hand, X NM exhibits similar performance to that of Si. 3.2. Approach To compare the performance of the FETs employing X NRs and X NMs as the channel, we used a single gated FET structure with high 2 ( r =25) insulator thickness of 3 nm . The resulting gate insulator capacitance is C ins = 7.38Ã—10 2 F/m 2 . The applied power supply voltage is V DD =0.5 V. The schematic of the struc ture is in Fig ure 3 1 A of MOSFET model was used with self consistent electrostatics that is treated by a capacitance model [ 38 ]. This semi classical model is describe d in Fig ure 3 1 B . In this model, the positive velocity states ( +k states) are filled by the source Fermi level ( E FS ) and the negative velocity states ( k states) are filled by the drain Fermi level ( E FD ). The current, I DS is determined by the difference in these two carrier distribution. The E k relation of the material is calculated from ab initio density functional theory (DFT). Si ha s been used as the MOSFET performance reference. For the E k relations of Si, we consid er the parabolic effective mass model with (100) as the channel direction. The X NR and X NM channels are as depicted in Figure 3 2 A, 3 2 B and Figure 3 3 A, 3 3 B respectively and are denoted as A HNRs, A FNRs, and HNMs as defined in the captions in the figur es accordingly. We conduct DFT calculations with SIESTA codes on a set of A HNRs, A FNRs, and HNMs to extract the E k relation of the materials [ 31 ]. For the simulations we used the double employing the generalized gradient appr oximation (GGA) method. The Perdew Burke PAGE 36 36 Ernzerhof (PBE) exchange correlation functional is adopted and the Troullier Martins scheme is used for the norm conserving pseudopotentials. A grid cutoff of 300 Ry is used and the Brillouin zone sampling is done by the Monkhost pack mesh of k points (31Ã—11Ã—1) for A X NRs and (31Ã—31Ã—1) for X NMs. 3.3. Results 3.3.1. DFT R esults : E k D ispersion R elation We first examine the E k relation of the X NR arrays. Figure 3 2 C shows 2D E k dispersion relations of the A HNR{ 9 , 3 } obtained from the DFT calculation . T he Brillouin zone is indicated by the red rectangle in the center. The A HNR s ha ve a small effective mass a long the transport direction ( m t ) and a large effective mass in the width direction ( m l ), resulting in a high l y anisotropic E k relations . This high ly anisotropic E k relation , would maximize both the band structure limited velocity in the transport direction and the 2D density of states (DOS), simultaneously, ideal for improving transistor performance. Figure 3 2 D compares the E k dispersion relations of the A HNR s { 9 ,N a }, w ith N a of 1 , 3 and 5. The results are plot t ed along Y X band lines in the Brillouin zone . Notice that the x axis is not in scale but normalized to the Y X directions. As shown in Figure 3 2 D , the slope (which represents the group velocity, v g ) of the lowest conduction band in the Y direction is negligibly small, compared to that of the X direction . Even for N a of 1, isolation in transverse direction is achieved . The bandgap of few other samples of A HNRs with varying N g and N a are plotted in Figure 3 2 E . The N a is set to 1 or 2, 3 or 4, and 5 or 6, depending on the width of the chemically modified portion, such that the total unitcell is an even dimer line structure, i.e. N a +N g is even. This enables the unit cell to be replica t ed along the y direction for DFT calculations . The plot indicates that for the N g = 3p and 3p +1 ( p is an integer) PAGE 37 37 groups , the bandgap values remain considerably high even with one atomistic line separation . This confirms that the atomistic chemical modification provides sufficient confinement to the system , resulting in A HNRs functioning similar to armchair GNRs (AGNRs). The inset of Figure 3 2 E show s the effective mass along the transport direction m t as a function of N g . The m t of A HNR{ 9,N a } and A HNR{ 12,N a } is smaller than that of A HNR{ 10 ,N a } , as well as the transverse effective mass of Si, which is 0.19 m 0 . A small effective mass along the transport direction is preferred for a larger carrier injection v elocity and faster transistor intrinsic speed. Next we examine the band structures of X NMs. T he structure of a X NM is illustrated in Figure 3 3 A . The X NM consists of periodic arrangement of chemical ly modified pattern. We consider only the modified pattern that result in hexagonal pattern with six C and H atoms. We define this structure by the distance between the neighboring chemically modified hexagons, which is refer ed to as nanoneck. Fu r thermore, only the even indexed structures are taken into account, since the bandgap of the odd indexed structures is zero [ 11 ]. T he unitcell of the X NM is shown in Figure 3 3 B . The symmetry of the unitcell provides clear definition of nanoneck and also efficiency in DFT calculation. Fig ure 3 2 C, 3 2 D shows the band str uct u re s of HNM{ 1,2 } and HNM{ 1,6 }, obtained from the DFT calculations. In the 2D gaphene, the Dirac cone lies in the K and lattice causes the Brillouin zone fold ing , int . The two lowest conduction bands shares the same minim point . These bands sepa rate with PAGE 38 38 increasing | k| , similar to the light hole band and heavy hole band in Si. We denote the two conduction band as the light electron (LE) and the heavy electron (HE) band. The extracted effective mass of both the LE and HE bands are shown in the inset of Figure 3 3 E . The narrower nanoneck induces stronger quantum confinement to the system and as a result, the bandgap decr eases with increasing W . In general, t he H N Ms introduce sufficient bandgap in the graphene system . However, due to its nearly isotropic E k relation with a larger effective mass , HNM are less advantages compared to A HNRs in terms of carrier transport . 3.3 .2. DFT R esults : Binding E nergy C alculation U sing Basis S et S uperposition E rror (BSSE) In the earlier work, schemes of hydrogenation and fluorination on carbon system have been studied, and it was suggested that fluorination is energetically more preferre d and easier to be achieved [ 39 ]. We confirm the argument by calculating the binding energy of the A HNRs and A FNRs by the DFT calculation. SIESTA, a DFT calculation code used in this work utilizes localized pseudo atomic orbital (PAO) basis. However, it is well known that in calculating binding energy of two systems, the PAO basis set may generate error. The unequal basis set between interacting bonded system and non interacting separate system gives over estimation. This is known as basis set superposit ion error (BSSE), and need to be compensated for. Detail of the method is in appendix B . Figure 3 4 compares the C H and C F binding energy of the A HNRs and A FNRs as a function of width of the AGNR portion (N g ). The N a is set a 3(4) for odd (even) N g to set the width of the whole unitcell as even for simulation purpose. Notice that C F binding energy is ~1.2 eV larger than C H. Thus, we conclude that the PAGE 39 39 fluorination is advantageous over hydrogenation in terms of fabrication and structural stability, without sacrificing the performance as we will demonstrate in the C hapter 3.3.3 . 3.3.3. Device P erformance Next we study the ballistic performance of the MOS FET devices using X NMs and X NRs as the channel. We apply the top of barrier model to the E k disper sion relation s calculated from DFT. The performance of the MOSFETs using the A HNRs (HNMs), as is co mpared to Si , is shown in Figure 3 5 ( Figure 3 6 ). Figure 3 5 A shows the I V characteristic of the A HNR FETs. For a fair comparison of the on current ( I ON ), we adjust the gate workfuntion such that the off current density ( I OFF ) is set to 100nA/Âµm for all the MOSFETs. The A HNR FETs with an index of { 9,3 } or { 12,4 } offer the largest I ON . Especially , A HNR{ 12,4 } MOSFET show 52.4% larger I ON compared to the Si MOSFET. Since we have adopted a ballistic transistor model to evaluate the performance of the channel materials, the current is calculated as product of v avg and number of mobile carriers that are at the top of the barrier. A s shown in Figure 3 5 B , at a fixed gate bias of V G =0.5 V, v avg of A HNR{ 12,4 } MOS FET is 82.9% higher than Si MOSFET, resulting in a larger I ON . The larger v avg in A HNR{ 9,3 } and A HNR{ 12,4 } is due to the large m l , as mentioned previously. The solid line s with closed polygon in the Figure 3 5 B are thermal velocity of the carriers expressed as equation (3 1). (3 1) The, k B is the Boltzmann constant and T=300K . The analytic values and the numerical results match well in the lo w V G range at the non degenerate limit. The I ON is PAGE 40 40 the product of the charge density and the average velocity. It has been reported before that a smaller effective mass in an isotropic E k does not necessarily leads to a larger on current, because increase of the carrier injection velocity can be offset by decrease of the density of states (DOS)[ 42 ]. A small effective mass along the transport direction (for a larger carrier injection velocity) and a large effective mass in the transverse direction (for a hi gh DOS) are most ideal, which require a highly anisotropic 2D E k . We also compared the intrinsic speed of the FETs, using the intrinsic delay, . This serves as a fair comparison metric in that it takes account of both ON state and OFF state of the device, and also the power supply voltage. The details of the method to compute the intrinsic delay are delineated in ref . 43 . The calculated results are shown in Figure 3 5 C , which indicate that, at a common on off ratio of 10 3 , A HNR{ 12,4 } is two times faster than Si. This is caused by the larger average carrier injection velocity in A HNR{ 12,4 }. Figure 3 6 shows performance limits of HNMs compared to that of Si. The modeled HNM{ 1,2 } and HNM{ 1,4 }, whose bandgaps are larger than 0.7 eV exhibit comparable performance to Si, but are less preferable than A HNRs with N g =3p+1 category. The main limiting factor in the HNMs, is the large effective mass in the transport direction, which lowers the carrier injection velocity. The extracted effect ive mass for HNM{ 1,4 } is 0.15 m 0 for LE band and 0.17 m 0 for the HE band. Even though the effective mass in HNM can be further decrea sed by increasing the nanoneck ( Figure 3 3 E ) , this would also decrease the bandgap, thus limit the achievable maximum on off current ratio. PAGE 41 41 Finally we further investigate the performance of HNRs as shown in Figure 3 7 . First, we study the effect of varying separation ( N a ) of HNRs, on the device performance. Referring to Figure 3 7 A , at V G of 0.5 V, comparing A HNR{ 9,1 } and A HNR { 9,3 }, the current show a difference of less than 5% and v avg show a difference of 12.5 %. Thus we can verify that even one atomistic dimer line separation g e nerated by chemical modification can offer enough isolation and provide comparable performance to those of large N a . Second, we compare the effect of hydrogenation and fluorination of HNRs , by comparing the performance of A HNRs and A FNRs. Referring to Fi gure 3 7 A, and 3 7 B , A HNR{ 9,3 } and A F NR{ 9,3 } show close performance resemblance, only 2 % difference in I ON , and 3% in v avg . The summary of the performance of each device is compared with Si in Table 3 1 . 3.4. Summary The electronic properties of selectiv ely chemical modified X NRs and X NMs were evaluated using A b initio calculations. For X NRs, a considerable band gap comparable to A GNR of the same confinement width can be achieved e ven with a single atomistic line of hydrogenation or fluorination in a 2D graphene sheet . Hydrogenation and fluorination leads to similar transistor performance, with fluorination more preferred in terms of chemical energetic. The ballistic performance limits of devices using X NRs and X NMs as channel were investigated using a to p barrier method. Due to high anisotropic feature of the E k relations, the performance of XNR transistors is better than that of XNM transistors. A carefully defined pattern of the selective chemical modification to the graphene system is promising in ach ieving high on off ratio and speed, which are required in digital electronic device applications . PAGE 42 42 Table 3 1 . S ummary of the MOSFET performance Channel material E g [eV] I ON [ 10 3 ÂµA/Âµm] v avg [10 5 m/s] [fs] Si 1.12 2 . 46 2 . 1 6 94.7 A HNR{9,3} 0.68 3.74 (52.0%) 3.29 (52.3%) 56.5 ( 40.3%) A HNR{10,4} 0.98 2 .61 (6.10 %) 1.91 ( 11.6%) 10 2.2 (8.0%) A HNR{12,4} 0.60 3.75 (52.4%) 3.95 (82.9%) 47.9 ( 49.4%) HNM{1,2} 1.41 2.08 ( 15.4%) 1.37 ( 36.6%) 148.5 (56.8%) HNM{1,4} 0.75 2.68 (8.94%) 2.20 (1.85%) 91.9 ( 2.94%) A HNR{9,1} 0.46 3.90 3.70 A FNR{9,1} 0.44 3.45 3.29 A FNR{9,3} 0.73 3.66 3.18 A HNR{10,2} 1.01 2.74 1.80 A FNR{10,2} 1.03 2.28 1.54 A FNR{10,4} 1.00 2.48 1.80 The value s in the bracket s represent the percentage difference with regard to the value of Si. The percent difference is calculated as, ( x x Si ) /x Si , where x is the value of each material and x Si is that of Si. A t the bias point of , V DD =0.5 V with V G =0.5 V cubic Hermite interpolation within the values in Figure 3 5 C and Figure 3 6 C at I ON / OFF =10 3 . PAGE 43 43 Figure 3 1. Schematic of the device structure and the top of barrier ballistic transi s tor model. A ) Schematic of device structure. Single top gated MOSFET with high 2 ( r =25) insulator with thickness of 3 nm . C ins = 7.38Ã—10 2 F/m 2 , V DD =0.5 V B ) S chematic of s tor model. T he +k states are filled by the source Fermi level ( E FS ) and the k states are filled by the drain Ferm i level ( E FD ). PAGE 44 44 Figure 3 2 . Schematic of the g raphene Nanoroad and the bandgap properties. A ) Graphene Nanoroad array . B ) Atomistic structure of the Armchair Nanoroad. The unit cell of the 2D structure which is denoted as A X NR{ N g ,N a }. N g is the number of dimer lines of the graphene portion and N a is the number of dimer lines of chemically modified graphene. The N g =9 , and N a =3 for this case. X denotes the adsorbate (i.e. H or F). C ) The 2 D band structure of A HNR{ 9,3 }. The red dash rectan gle is the Brillouin zone. D ) E k relation of A HNR{ 9,N a } along the Y X band lines in the Brillouin zone with varying N a . Notice that the x axis is not in scale but normalized for Y X direction. E ) The bandgap as a function of the N g for a few d ifferent N a values . N a is 1 or 2 (solid line), 3 or 4(dashed line), and 5 or 6 (dotted line), depending on the width of the chemically modified portion. The inset shows the effective mass in the transport direction ( m t ) as a function of N g . PAGE 45 45 Figure 3 3 . Schematic of the g raphene nanomesh and the bandgap properties. A ) Atomistic structure of the nanomesh, which is denoted as X NM{ R,W }. R , fixed to 1 lattice constant throughout the work, is the radius of the selectively chemical modified graphene portion. W is the width of the graphene nanoneck between two neighboring selectively chemical modified graphene. X denotes the adsorbate i.e. H. The structure shown here is HNM{ 1,4 }. B ) The unit cell of the 2D nanomesh superlattice structure. Band structure of C ) HNM { 1,2 }, and D ) HNM{ 1,6 }. The red dash hexagon is the Brillouin zone. E ) The bandgap as a function of W . The inset shows the electron effective masses of the structure. LE (HE) stands for light electron (heavy electron) band effective mass. PAGE 46 46 Figure 3 4 . C H and C F binding energy of A X NR structures with N a =3 or 4. The results are from DFT calculations. The figure indicates that C F bonding is more energetically favorable than C H bonds. Figure 3 5. The ballistic performance limits of Graphene nanor oad FETs compare to Si. A common off current density of 100nA/Âµm is defined by adjusting the gate workfunction for all transistors for a fair comparison of the I ON . A ) The I DS vs. V G characteristics are c ompared to that of Si MOSFETs. B ) Average carrier in jection velocity ( v avg ) as a function of the gate voltage for graphene nanoroad FETs. The solid line s with closed polygons are thermal velocity. C ) Intrinsic transistor delay vs. on off current ratio where the channel length was assumed as 22nm. The detailed method is described in [14] . PAGE 47 47 Figure 3 6 . The ballistic performance limits of Graphene nanomesh FETs compare to Si and GaAs. A ) The I DS vs. V G characteristics. B ) Average carrier injection velocity ( v avg ) as a function of the gate voltage f or graphene nanomesh FETs. The solid lines with closed polygons are the thermal velocities. C ) Intrinsic transistor delay vs. on off current ratio where the channel length was assumed as 22nm Figure 3 7 . A HNRs and A FNRs device performance comparison A ) Comparison of I D vs. V G charact eristics of A HNRs and A FNRs. B ) Average carrier injection velocity ( v avg ) as a function of the V G for A HNRs and A FNRs. PAGE 48 48 CHAPTER 4 n DOPING OF TRANSITION METAL DICHALCOGENIDES BY POTASSIUM Potassium (K) has a small electron affinity and thus is acts as a strong electron donor, however, its high reactivity impairs it stability as a dopant in graphene. T ransition metal dichalcogenides (TMDs) are a new material of interest due to its atomistic thickness and nominal bandgap. A b initio simulations were performed to understand K doping of MoS 2 and WSe 2 in comparison with graphene . The results indicate that K dopant in MoS 2 or WSe 2 induce larger charge transfer compared to K dopant in graphene. Also the binding energy of the K to the MoS 2 and WSe 2 is larger than K to graphene indicating that the K acts as more stable dopant in the TMD system. 4.1. Overview The first demonstration of graphene [ 7 ] unveiled a class of 2D layered material to the palette of future nano device s . Such material includes h BN, topological insulators, and transition metal dichalcogenides (TMDs) . Graphene has been of a great interest due to its fascinating properties such as mobility in the order of 10 5 cm 2 /V.s, thermal conduct ivity up to 3000 W/K.m [48,49] . However, due to the absence of band gap, the graphene based field effect transistors (FETs) have high off current (I OFF ) and thus is not suitable for current logic applications. On the contrary the h BN, has large direct ban dgap of 5.9 eV which makes it appropriate for optoelectronic applications [50,51], but rather too high for low power device applications. The TMDs, e.g., MoS 2 and WSe 2 are also layered 2D material with crystal structure built up of X M X monolayers (M=Mo , W; X= S, Se ) . Each layer is interacted through van der Waals force. Previous research reports that TMDs have a nominal bandgap of 1.1 2 eV [ 52,53 ] , which is comparable to Si, and thus it is a promising candidate for future nano electronics . PAGE 49 49 Recent study of monolayer and few layered MoS 2 , w hich has a bandgap of 1.8 eV, has shown the potential use high performance n type FETs (n FETs) [13] . Another study has shown WSe 2 , which has a bandgap of ~1.2 eV, used as p type FETs with doped source and drain [55]. Als o theoretical studies show promising results on adapting TMDs as a channel material [56,57]. Motivated upon these studies, we focus on understanding the doping of monolayer (ML) of MoS 2 and WSe 2 , in particular, n type doping with potassium (K). K has small electron affinity and thus is acts as a strong electron donor. K doping in graphene has already been studied in depth and the high reactivity of the dopant make it less practical and t his is illustrated by the need of cryogenic temperatures and ultra high vacuum conditions to stably adsorb potassium on graphene surfaces [58]. Our question is would K serve as decent n type dopant in TMD system and if it does, why. In this chapter we compare the doping nature of graphene , monolayer (ML) MoS 2 and ML WSe 2 with potassium as a dopant . Ab initio DFT simulations were performed to understand the charge transfer and doping mechanism. Our results indicate that the potassium has stronger interaction with the TMDs compare to graphene. In both the K doped MoS 2 and the K doped Wse 2 , the K to the neighboring chalcogen atom bond distance w as shorter th a n the K C bond distance of K graphene. We can infer that the K doping in TMD system to be more stable than the case for graphene. 4.2. Method We have conducted two sets of ab initio DFT calculation by varying the density of K dopant atom in graphene, ML MoS 2 and ML WSe 2 system. The unitcell of graphene consists of two C atoms, namely A, B sublattice. For the TMDs the unitcell consists of PAGE 50 50 one transition metal atom and two chalco gen atoms. For simul ation purpose we group the 2Ã—2 unitcell of graphene with one K dopant (one side doping of 1/4 K atom per unitcell) and refer it as 0.25K/UC graphene and 3Ã—3 unitcell of graphene with one K dopant as (1/9 K atom per unitcell) and refer i t as 0.11K/UC graphene and appl ied the same notation for the TMDs. Figure 4 1 A shows the 2D array of 0.25K/UC of graphene, where the supercell is depicted within the red dotted rhombus. Figure 4 1 C shows that of the TMDs. In Figure 4 1 E , the supercell of 0 .25K/UC TMDs is depicted for clarification in the definition of the supercell. Figure 4 1 F is the schematic for the 0.11K/UC TMDs supercell. The simulation was conducted by density functional theory (DFT) with Vienna ab initio simulation package (VASP) cod es [59]. Compare to SIESTA codes [31], the VASP simulator uses atom independent basis set to solve the Kohn Sham equation. Meaning that, the VASP calculation does not need compensation for the BSSE that occurs in atom centered basis set. Thus, to extract t he binding energy of the dopant to the 2D materials 2 sets of DFT calculation is needed. For example binding energy of K to graphene, one calculation for K doped graphene and one for K with graphene but distance is far enough that they will not interact wi th each other. The extracted binding energy would be total energy of the first calculation sub tract the value of the second. The d ouble gradient approximation (GGA) method. The Perdew Burke Er nzerhof (PBE) is used for the exchange correlation potential. The cutoff energy for the wave function expansion is set to 500 eV. Bader analysis [60 62] was followed by the DFT calculation to understand the charge contribution from the potassium dopant. PAGE 51 51 4 .3. Results For both the graphene and the TMDs, the K dopant is locat ed above the centers of the hexagons lattice. Figure 4 1 A and 4 1 B ( Figure 4 1 C and 4 1 D) depicts the top and side view of K doped graphene (ML TMD) , respectively, with one side doping and a density of 1/4 K atom per unit cell ( 0.25K /UC) . Despite of S being a much larger atom than C, the distance between the K dopant and the S plane in MoS 2 (2.60 Ã…) is shorter than that between K and C plane in graphen e (2.89 Ã…). The bond length of K S is 3.06 Ã… in MoS 2 , compared to the K C bond length of 3.24 Ã… in graphene. The shorter bond length indicates a stronger binding of K to MoS 2 , which is consistent with a larger binding energy in K doped MoS 2 and more stable doping as shown in Fig ure 4 2 A . The results are also summarized in Table 4 1 . The simulation results indicate that the bond length decreases and the binding energy increases as the doping density decreases from 0.25 K /UC to 0.11 K /UC , but the qualitative di fference between K doped MoS 2 and K doped graphene remains unchanged. Until now we have discussed in terms of doping density per unitcell, which may be misleading in that the size of the unitcell are rather different. Lattice constant of graphene is 2.46 Ã… and MoS 2 is 3.16 Ã… , 1.28 times larger, and 1.64 time larger in terms of area. For a fair comparison , Figure 4 2 B plots the binding energy as function of doping density. The binding energy of the K to MoS 2 is still larger than K to graphene. Figure 4 3 plots the charge density distribution of K doped graphene and K doped MoS 2 , where the cutting plane is shown in Figure 4 1 E . For K doped MoS 2 system, we can confirm the larger interaction between the S atom to K compare to C K interaction in K doped graphe ne. We also performed similar simulations of K WSe 2. The distance between the K dopant and the Se plane is 2.76 Ã… and the K Se bond length is 3.36 Ã…. Because WSe 2 have larger atoms than MoS 2 , the bond length is PAGE 52 52 larger and the binding energy is smaller, but the K WSe 2 binding energy is still larger than that of a K graphene. To examine charge transfer between the K dopant and the ML 2D materials, we performed Bader analysis of charge transfer as shown in Fig ure . 4 4 , with a K concentration of one atom per f our MoS 2 unit cells ( 0.25K/UC ). After placing one K atom in the center of the super cell, there is 0.52e of charger transferred from the K atom to MoS 2 , with the top atoms on the top S layer sharing the charge, while in ML WSe 2 , 0.50 of the K charge is tra nsferred, slightly lower than that of K MoS 2 4.4. Summary In this chapter , we have studied the potassium n doping of ML TMD and compared with the graphene through DFT calculations. The result s indicate larger charge transfer of K to TMD system then gra phene when doped. Also the larger binding energy indicate s that K doping in TMD system is more stable than graphene case. Confirming that TMDs can be doped, we may expect development of layered semiconductor MOS devices. PAGE 53 53 Table 4 1 . Summary of bonding properties of MoS 2 and graphene MoS 2 Graphene E b (eV) d K S (Ã…) Interlayer dist (Ã…) E b (eV) d K C (Ã…) Interlayer dist (Ã…) 0.25K/UC 0.612 3.19 2.60 0.233 3.22 2.89 0.11K/UC 1.248 3.0 7 2.43 0.532 3.11 2.77 0.06K/UC 0.813 2.97 2.6 PAGE 54 54 Figure 4 1. DFT calculated structure s of K doped graphene and MoS 2 . A )Top view of the 1 K atom in 2Ã—2 unitcell array of graphene (denoted as 0.25K/UC graphene, doping density of 4.77Ã—10 14 /cm 2 ). The red dot rhombus indicates the supercell. B ) Side view of the 0.25 K/UC graphene. The DFT calculation results show C plane to K plane distance of 2.89 Ã…. C ) Top view of the 1 K atom in 2Ã—2 unitcell array of ML TMDs (denoted as 0.25K/UC ML TMDs, for ML MoS2 the doping density is translated to 2.89Ã—10 14 /cm 2 ). The difference in doping density between graphene and ML MoS 2 is due to the size of lattice constant of 2.46 Ã… for graphene and 3.16 Ã… for MoS 2 . D ) Side view of the 0.25K/UC TMDs. For ML MoS 2 , the S plane to K plane distance of 2.60 Ã…. E ) 0.25K/UC supercell. (f) 0.11K/U C supercell. PAGE 55 55 Figure 4 2 . K to graphene, MoS 2 and WSe 2 binding energy comparison. A ) Binding energy comparison between simulated materials and the size of the supercell. B ) Binding energy as a function of doping density. Figure 4 3. Charge density plot of K doped graphene and K doped MoS 2 . A ) K doped grpahene, B ) K doped MoS 2 . The cutting plane is for A 1 E . PAGE 56 56 Fig ure 4 4. Charge transfer calculated using Bader analysis [15 17] A ) K doped MoS 2 0.25K/UC and B ) K doped WSe 2 0.25K/UC. PAGE 57 57 CHAPTER 5 STRAIN INDUCED INDIRECT TO DIRECT BANDGAP TRANSITION IN BI LAYER WSE 2 I n this work we report the i nfluence of the uniaxial strain on the electronic properties of bi layer WSe 2 via first principle calculations. With the uniaxial tensile strain of ~0.6%, we find the indirect bandgap of the bi layer WSe 2 transit to direct bandgap and confirm this with the measurement. We confirm that this originates from the dz 2 orbital of the W atom. 5 .1. Overview Since the first demonstration of g raphene [ 7 ], 2D materials, such as h BN, and transition metal dichalcogenides (TMDs, MoS2, WSe2, etc.) have seized a great interest. To minimize the short channel effect at extreme scaling of future sub 5 nm gate length field effect transistors (FETs), lar ge bandgap semiconductors with ultrathin body are essential [ 6 ]. In this context adopting 2D semiconductors as channel will serve a great advantage. Amongst the 2D materials, unlike the semi metallic graphene, TMDs in general have bandgap of 1~2 eV [ 67 ], p lacing them as a promising candidate for future electronic and optoelectronic applications . Transition metal dichalcogenides MX 2 (M : transition metal, X : chalcogen) are members of the layered materials, whose crystal structure is built up of X M X single layers stacked together by Van der Waals (vdW) force . Just as in graphene, mono layer (ML) and few layers of TMDs can be achie ved by using mec hanical exfoliation technique [69 ]. WSe 2 has gained interest as a p type device channel as opposed to n type MoS 2 and exhibit s high carrier mobility and electrostatic modulation of conductance similar as to MoS 2 [ 55 ]. Theoretical studies ha ve shown that a s a ML, MoS 2 and WSe 2 amongst with other TMDs, are direct bandgap material at point ( K C to K V ) of the B rillouin zone , and as a bulk, become PAGE 58 58 indirect bandgap V ) [ 70 ]. The C V V represent the four local CBM and VBM points of the B rillouin zone (refer to Figure 5 2 C) . Th is indirect bandgap nature of multilayer TDMs limits their application in optoelectronic devices. One way to engineer the band structure of a material is to apply strain. In TMDs, lattice constant and interlayer distance which is the vdW gap, can be modified by applying strain. This results in modification of elec tronic band structure, especially in the energy regions of interest: the CBM and VBM. A dramatic transition from semiconducting to metallic with application of strain has been reported in ML and bi layer (BL) MoS 2 by the DFT calculations [ 71 ]. In applicati on involving light harvesting or detection , thicker films with direct optical bandgap are favorable. Thus, for an indirect bandgap BL or even for few layer TMD, assuming that the energy difference of the indirect and direct bandgap is small enough , applyin g strain may lead to indirect to direct bandgap crossover . Previous study shows that the direct and indirect bandgap differ ence (equivalent to K C of BL MoS 2 is relatively large , value of ~300 meV [ 54 ]. Also no transition to direct bandgap was observed up to 2.2% uniaxial tensile strain [ 72 ]. Compared to MoS 2 , WSe 2 BL has smaller K C 40 meV . Thus, it may be possible to achieve direct bandgap in BL WSe 2 under certain strain. In this work we prese nt density functional theory (DFT) calculations on a BL WSe 2 to understand the effects of the strain on the band structure. BL WSe 2 exhibits V at zero strain. As the strain is applied, we find that the K C energy is indeed lowere V and V energy are relatively less perturbed. At PAGE 59 59 uniaxial tensile strain of ~0.6 %, BL WSe 2 transits to K C K V direct bandgap which is confirmed with the photoluminescence (PL) measurements. By further examining the band structure of ML WSe 2 , we find the cause of the K C lowering in the modification of W d z 2 orbital. 5.2. Method In this study DFT calculations were performed using the projector augmented wave (PAW) pseudo potential through the Vienna Ab initio simulation package (VASP) [ 59 , 73 ]. Structure relaxation was first performed to determine the relaxed atomistic structure in the presence of uniaxial strain, and the band structure in the presence of strain was subsequently calculated. For the atomic structure relaxation, generalized gradi ent approximation (GGA) method was used with Perdew Burke Ernzerhof (PBE) exchange correlation (XC) functional [ 74 ]. The weak interaction of vdW force was taken account via DFT D2 method [ 75 ]. The Brillouin zone was sampled by the 9Ã—9Ã—1 Monkhorst Pack (MH) scheme. As for the band calculation, GGA and local density approximation (LDA) , which are semi local XC functional s , typically underestimate experimental band gap [ 76,77 ]. Heyd Scuseria Ernzerhof ( HSE ) gives a larger bandgap than the LDA and GGA simulations, but a smaller bandgap than the GW calculations [ 78 ] . It has also been indicated that the optical gap is related to the GW bandgap subtracting the exciton binding energy. To verify, GW quasi particle (QP) calculation, non self consi stent G 0 W 0 was also performed (details can be found in the A ppendix C ) [ 79 ]. The band structure by the G 0 W 0 calculations has an approximately k independent shift of the conduction C V ), wit hout PAGE 60 60 changing the direct or indirect bandgap feature (see F igure C . 1), similar to what has been shown before for ML TMD materials with or without strain [ 80 ]. The GW bandgap is larger than HSE bandgaps. An exciton binding energy of about 1 eV has been repo rted for ML TMD materials, but with uncertainty. The value varies by about a factor of two if different k point grids are used in simulation [ 80 ], and it decreases by about a factor of two from monolayer to bilayer [ 81 ]. It is also dependent on the surrounding electrostatic environment, which is the capping PMMA and the substrate material whose dielectric properties are not exactly clear. The uncertainty of the exciton binding energy makes a comparison to experiment difficult. On the other hand, it i s found that the PL peak energy agrees with the bandgap of the HSE calculation with spin orbit coupling (SOC) taken into account . This is a coincidence, which could be due to cancellation of the blue shift by GW correction and red shift by exciton binding. Nevertheless, instead of introduce fitting parameters for uncertainty of the exciton binding energy, the band structures from the HSE calculations with SOC are presented next, since we are mainly interested in the indirect to direct bandgap transitions wi th the applied strain . T hus, for band structure calculation, HSE XC potential with SOC was used with increased k point sampling of 15Ã—15Ã—1 . Also as reference to the HSE results, GGA band structure calculation s w ere conducted (VASP input script s in generati ng HSE and G 0 W 0 results are included in A ppendix D) . Figure 5 1 A is the top view BL WSe 2 . T he arrows indicate the orientation of the applied uniaxial tensile strain. Figure 5 1 B shows the side view of the calculated structure, which the layers are in 2H st acking order. Throughout the work, Poisson's ratio of 0.19 was used to the transverse direction of the applied strain [ 67 ]. PAGE 61 61 As for the measurement results , F igure 5 1 C illustrates the two point bending method used to apply uniaxial tensile strain to the WSe 2 flakes . First, WSe 2 is exfoliated onto a 260 nm thick Si/SiO 2 substrate. Each flake is transferred to a 1.5 mm thick polyethylene terephthalate glycol modified (PETG) substrate via poly methyl methacrylate (PMMA) as the transfer medium [ 82 ]. The flexi ble property of PETG makes it possible to bend , thus apply strain , while the capping PMMA serves as a clamp. Applied strain ( ) is calculated as equation in Fig ure 5 1 A , where the bent PETG is approximated by a circular arc . Details of the preparation of t he sample and the measurement techniques are in ref. 83 . 5.3. Results Under the presence of uniaxial strain , structure relaxation was first performed to determine the relaxed atomistic structure for the ML and BL WSe 2 . The calculation results are listed in Table 5 1. For both ML and BL WSe 2 , increase in strain results in increase of lattice constant and intra layer W Se ( d W Se ) distance, while intra layer Se Se distance ( d Se Se ) and lattice constant in z (out of plane) direction ( c ) decrease. As exp ect ed , the applied tensile strain results in material becoming wider and thinner. Figure 5 2 shows the DFT calculated E k relat ions of the b i layer WSe 2 , comparing the XC functions used and the effect of including and exclud ing spin orbital coupling (SOC) interactions . The solid line denotes the case with no applied strain. Dot line denotes the case for the 1% uniaxial tensile strain. The extracted bandgap values are in Table 5 2. First to note is that regardless of the calculation method, K C is lowered while the change in K V , and V is small . For HSE with SOC (Figure 5 K V indirect bandgap does transit to K C K V direct bandgap with 1% strain . Second, cases that take account for the SO C (Figure 5 2A and 5 2 C) show large r split in the first and PAGE 62 62 second VBM at K point, compared to the non SOC case s (Figure 5 2B and 5 2 D) . Figure 5 2E is a re plot of SOC and non SOC calculation using HSE XC functional, without any strain. The band split in and 0.13 eV for non SOC. Because of this band split, the bandgap profile changes. To K V indirect bandgap and non SOC V indirect bandgap . This results in larger bandgap for non SOC calculation . Before going into the SOC effects in BL WSe 2 , understanding of SOC effect in ML WSe 2 is required . Figure 5 5 A shows the band plot of ML WSe 2 with SOC taken into account. It is well known that t he broken inversion symmetry in ML WSe 2 results in large spin splitting of VBM with opposite spin moments at two K and valley and are in time reversal to each other [ 84,85 ] . Our calculation results indicate ~0.6 4 eV . Difference of the split size from the past stu dies (0.46 eV) comes from the difference in XC function used [ 84 ~ 86 ]. This can be confirmed by the split size in Figure 5 2A, for BL calculation with GGA XC function shows split of 0.48 eV. Even though the value is for the BL, the split size is known to be similar with the ML [ 86 ]. The strong SOC originates from W d orbitals. The K C is dominated by the W d z 2 orbital (which we will confirm later on) , SOC is inactive and thus, K C holds its degenera cy [ 87 ] . In the case of 2H stacked BL TMDs , the two layers are 180 degree in plane rotated with each other. For one layer the K and valley is switched while the spin is unchanged. Thus the 2 split bands are spin degenerate [ 86 ]. For V of the ML WSe 2 , the time reversal symmetry which keeps the spin degenera te. But in spin degenerate to bands. Last to note is that the bandgap value from measurement of ~ 1.52 eV fit closer to HSE results (1.585 eV) th a n GGA (1.188 eV ) as previously PAGE 63 63 stated [83 ]. Fig ure 5 3A and 5 3 B show the four direct and indirect K C K V , K V , K C V , V bandgaps as function of applied strain. The calculation results from this point on take account of SOC. The GGA results in Figure 5 K V to K C V , indirect to indirect bandgap transition with extremely small window at near 1% strain of K C K V direct bandgap. HSE results (Figure 5 3B) is in contrast to GGA, where at strain of ~0.6 % BL WSe 2 transits to a direct bandgap. The difference in the two result s is due to the closer K V and V energy in GGA calculation than HSE calculation. For both GGA and HSE, it can also be seen that the E g,KC , decreases more rapidly th a n E g,KC K V , which indicate s that, even for HSE prediction, with much larger strain the B L WSe 2 will eventually transit to indirect K C V ba n dgap. Another to note is from 0% strain to the K V indirect bandgap increase. In this strain region, V is l thus the bandgap increases. K C is also lowered with applied strain and at near 0.6%, K C becomes the CBM, thus direct bandgap. As the K C continues to be lowered, the bandgap decreases. Figure 5 4 is compar ison of the PL measurements which are extracted fro m ref 83 , and the DFT calculations. Fig ure 5 4 A shows the shift of PL peak as the strain is applied. The PL peak is in direct relation with the bandgap of the material, and the HSE DFT results show close match . As DFT results predict, the PL peak increases (bandgap increase) until strain of ~0.5% and decrease after on. Figure 5 4 B shows PL intensity at the peak which features exponential increase. More can be interpreted from the PL amplification plot in Figure 5 4 C . The PL amplification can be calculated a s in equation ( 5 1), where the Boltzmann approximation is used in calculating carrier concentrations . Also we assume that C - PAGE 64 64 K V ) and all the radiative recombination is from the direct transi tion (K C K V ) only. This is reasonable given that a radiative recombination event for the indirect bandgap requires a phonon for momentum conservation, which make s the probability of the event negligible . ( 5 1) The and are the K C K V direct ba n d gap with strain and no strain, respectively, which are from the DFT calculations. k B is the Boltzmann constant and T is the temperature. The CB valley shifts linearly with strain, and thus the PL amplification changes exponentially, which can be observed in Figure 5 4C . The comparison between the two show s that the HSE DFT predicts the actual case very well and also confirm s that the K C point indeed is lowered as the strain is applied. As the BL Wse 2 is t wo single layers held together by vdW force, closer examination of ML WSe 2 can reveal the origin of the K C point shift. Figure 5 5A plots the E k dispersion relation of the ML WSe 2 , with no str ain and 2% uniaxial tensile strain. Same as the BL case the K C is lowered, and thus the initial direct bandgap changes from 1.63 eV to 1.51 eV. We can also see the increase of V , however, in ML WSe 2 , the V is much lower than K V , and thus does not affect the bandgap transition in the range of strain of interest. In F igure 5 5 B , we can see the monotonic decrease of the direct bandgap. We further investigate the e lectronic band structure with relative contribution from atomic orbitals . It has been reported in previously that in TMDs, d orbital in the metal PAGE 65 65 atom contributes largely to the lowest conduction band and highest valence band [ 88 , 89 ]. Further DFT calculations on MoS 2 have revealed that the at the two eys, the K V is composed by the Mo d x 2 y 2 and d xy orbitals with some mixing from the S p x and p y orbitals, while the K C is dominated by Mo d z 2 orbitals [88 ]. We may expect the similarity in WSe 2 . Figure 5 6 A and 5 6 B verifies the se previous works , where the atomic contribution of W atom and Se atom are plotted with the size of the circles representing the contributi ng rate. Figure 5 6 C~G shows the d orbital contribution from W atom , namely d xy , d yz , d z 2 , d xz and d x 2 y 2 respectively. As shown in Figure 5 6 D and F , the d yz , and d xz play negligible role in the CBM and VBM. The d xy and d x 2 y 2 (Figure 5 6 C and 5 6 G) cont r i butes to nd K V evenly. Most importantly W atom d z 2 (Figure 5 6 E) orbital is solely responsible for K C . Another to note is that V is affected by W atom d z 2 orbital too, with small contribution from Se atom. Thus, the large increase in V for ML to BL WSe 2 , is also caused by the modification of W d z 2 orbital. Figure 5 7 are the partial charge calculation of at each band points of int erest (K C V ), to visualize the orbital character s of single states. The partial charge is defined as partial charge density as respect to by the volume of unit cell. In calculating the partial charge density, the weighting of the states is not in Fermi Dirac distribution. Rather, the occupancies of the states are set as 1.0 within the integration energy range (indicated in the figure) and as 0.0 otherwise. Figure 5 7A and 5 7 C clearly show that the K C and V are dominated by the W d z 2 orbital which is in contrast to 5 7 B which confirms the in plane d xy , d x 2 y 2 contribution at K V . 5.4. Summary BL WSe 2 exhibits large PL amplification when uniaxial tensile strain is applied . We find the cause of this to be lowering of the local conduction band minima at the point PAGE 66 66 of the Brillouin zone. Origin of this behavior is caused by the modification of d z 2 orbital in the W atom with the applied strain . At strain of ~0.6% indirect bandgap BL WSe 2 transits to direct bandgap of 1.6 eV (from PL measurement) which is confir med with DFT calculations and PL measurements. We also find that to achieve accurate electronic band structure, HSE exchange correlation function with account for spin orbital coupling to be reasonable choice, which can also be used to predict the effect o f the strain. PAGE 67 67 T able 5 1. DFT calculated structural parameters (lattice constant, a 0 ; intra layer W Se bond lengths, d W Se ; intra layer Se Se bond lengths, d Se Se ; intra layer Se W Se bond angle, lattice constant in z (inter plane) direction, c ) and energy band gap for mono layer MoS2 as a function of the applied strain; negative values of the band gap indicate overlapping of conduction and valence bands Applied strain a 0 (Ã…) d W Se (Ã…) d Se Se (Ã…) c (Ã…) Mono layer WSe 2 0 % 3.308 2.539 3.347 82.46 N/A 0.5 % 3.319 2.543 3.342 82.16 N/A 1.0 % 3.331 2.547 3.337 81.86 N/A 1.5 % 3.343 2.551 3.332 81.56 N/A 2.0 % 3.354 2.555 3.327 81.26 N/A Bi layer WSe 2 0 % 3.308 2.539 Â§ 3.345 82.42 12.98 2.538 0.5 % 3.319 2.543 Â§ 3.340 82.12 12.96 2.542 1.0% 3.331 2.547 Â§ 3.335 81.82 12.95 2.545 1.5% 3.343 2.551 Â§ 3.330 81.52 12.94 2.549 2.0 % 3.354 2.555 Â§ 3.325 81.22 12.93 2.553 Â§ Intra layer W Se distance for Se atom exterior of the two WSe 2 layers. layer W Se distance for Se atom interior of two WSe 2 layers. PAGE 68 68 T able 5 2 . DFT calculated BL WSe 2 bandgap Strain XC : GGA XC : HSE E g [eV] E g [eV] SOC 0 % 1.188 V 1.585 V 1 % 1.210 V 1.575 C V Non SOC 0 % 1.316 V 1.826 V 1 % 1.305 V 1.784 C V PAGE 69 69 Figure 5 1. Schematic of b i layer WSe 2 and the m echanical strain setup . A ) Schematic of b i layer WSe 2 (top view), where the arrows indicat e the orienta t ion of the applied uniaxial tensile strain. B ) Side view of the b i layer WSe 2 . The details of geometric parameters are in Table 5 1. C ) Mechanical setup to apply strain. Arrows indicate the direction of the applied force. PAGE 70 70 Figure 5 2. DFT calculated E k relations of b i layer WSe 2 , comparing the XC functions and the effect of including and excluding spin orbital coupling (SOC). The solid line denotes the case with no applied strain. Dot line denotes the case f or 1% uniaxial tensile strain. A ) GGA with SOC. B ) GGA w/o SOC. C ) HSE with SOC. D ) HSE w/o SOC. Despite the method used, with applied strain, E ) Comparison of HSE DFT calculation with and without SOC for no strain applied case. PAGE 71 71 Figure 5 3. Comparison of the bandgap features of BL WSe 2 . A ) GGA, B ) HSE PAGE 72 72 Figure 5 4. Comparison of PL measurement with HSE DFT calculation A ) PL peak energy and bandgap from DFT as function of strain. B ) PL intensity at peak point vs. strain. C ) PL ampli fi cation comparison be tween measurement and DFT calcu lation. PAGE 73 73 Figure 5 5. B andgap features of ML WSe 2 . A ) HSE DFT E k cal culation of ML WSe 2 . B ) B andgap features as function of applied strain. PAGE 74 74 Figure 5 6. ML WSe 2 a tomic and orbital contribution on the E k relation. A ) W atom . B ) Se atom. C ~ G are orbita l contribution from the W atom d xy , d yz , d z 2 , d xz and d x 2 y 2 respectively. The size of the circles is proportional to the contribution. Figure 5 7. T he partial charge calculation of at each band points of interest. A ) K C , B and C V . The energy range specified in sub figure represents the integration interval. PAGE 75 75 CHAPTER 6 ELECTROSTATIC SCREENING PROPERTIES OF MULTILAYER MOS 2 The n onlinear e lectrostatic screening behavior of a multi layer MoS 2 is studied, through numerical calculations composed of one dimensional capacitance model and equilibrium charge statistics. We find that the electrostatic screening properties depend highly on the appli ed gate voltage and the temperature . Layers of MoS 2 can be electrostatically non screening in the insulating limit due to finite bandgap of the two dimensional material. In the opposite limit, the screening length can be as short as half of the interlayer distance of MoS 2 layer , indicating strong screening. 6 .1. Overview Since the first demo nstration of graphene in 2004 [7 ], the two dimensional ( 2D ) material has been of a great interest for the future electronic devices [ 92 ]. However, the absence of a band gap in the graphene results in high leakage current, i.e. low on off current ratio (I ON /I OFF ) when applied as a channel material of a field effect transistor (FET) [ 32 ]. While there have been efforts to induc e a bandgap to the semi metallic material, such as patterning to nano ribbon [ 23,24,34 ], application of electric field across few layers of graphene [ 93 ] , and chemical derivatives [ 36 ] and etc. , it still remains as a challenge which force to search for alternative 2D materials . h BN and transition metal dichalcogenides (TMD s , such as MoS 2 , MoSe 2 , WS 2 , WSe 2 and etc. ) are a few of such. TMDs are semiconducting layered materials, composed of vertically stacked X M X layers (M=Mo,W and X=S,Se,Te) that interact through a weak van der Waals (vdW) force. Each layer consists of a hexagonal plane of M atoms sandwiched between the two X atom layers. As a monolayer (ML) , TMDs exhibit bandgap of 1~2 eV [ 67 ]. A ML MoS 2 for example exhibits direct bandgap of 1.8 eV where, Radisavljevic et al .,[ 1 3 ] PAGE 76 76 have demonstrated a ML MoS 2 based FET with a room temperature electron mobility in the similar range of graphene nanoribbons and a high I ON / I O FF . Fang et al . , [ 94 ] also demonstrated n doped MoS 2 and WSe 2 FETs using potassium as a dopant . Other device topologies that adapt TMDs have been introduced. Vertical quantum tunneling FET is one of them, which use vertically stacked TMDs together with graphene layers [ 65,95,96 ]. In this sense, understanding of the electrostatic screen ing behavior of these 2D materials is c rucial. Kuroda et al .[ 97 ], studied electrostatic screening features of multi layer graphene, where the charge density profile in each layers show non exponential decaying behavior. N umerical analysis showed th at the screening length of the graphene can be less than 1 layer thick to as large as few layers, depending on screened charge density, intrinsic doping and temperature. In this theoretical work, we use a 1 dimensional ( 1D ) capacitance model self consistently iterated with the equilibrium charge calculation , to determine the electrostatic screening behavior of few layers of MoS 2 . We find that the electrostatic screening is highly nonlinear due to the 2D density of states (DOS) properties of the MoS 2 . Depending on the gate bias and the temperature, the screening length can be less than 1 layer thickness of the MoS 2 and extend to infinite, meaning the insulating limit. Also, we derive analytical expression for the Debye length which is essentially equivalent to the screening length. Last, we compare the results with Thomas Fermi model and assess its validity. 6.2. Method The studied structure is in the scheme of a FET, consisting N layers of MoS 2 on top of a SiO 2 substrate with a bottom gate as in the Fig ure 6 1A . The equilibrium condition is considered , thus the Fermi energy is same through the layers. T he PAGE 77 77 interlayer coupling b etween the stacked layers is neglected, assuming that the layers are stacked in an incommensurate manner. Fig ure 6 1 B describes t he 1D capacitance model to calculate the electrochemical potential of the MoS 2 layers. The numerical calculation procedure is following that of [ 95 ] for the vertical stacking of 2D materials. The Q i in the figure represents the charge density in the i th layer and is a function of vacuum ener gy level ( E vac ) as in eq uation ( 6 1) for N layer MoS 2 . , . ( 6 1) is equivalent to the gate oxide capacitance ( C ox ) . T he dielectric constant ( ) of 4 is used to represent the SiO 2 gate oxide with thickness ( t ox ) of 280 nm . is the vacuum permittivity. F or , where is the interlayer dielectric constant of MoS 2 , and d 0 = 6 Ã… is the interlayer distance between the layers. is crucial in determining the charge in each layer. As the in TMDs vary with number of layer, the interpolated values from the work of Kumar et al ., are used [ 98 ] . For most part of the work, N =10 and corresponding value of 6.84 is used . The boundaries are, C 0 = C N+1 =0, E vac, 1 =E vac,N+1 =0 , and eq ( 6 1) can be expressed as . ( 6 2) The v acuum energy level at the gate is, , where is the bottom gate work function. The equilibrium charge density is calculated through the PAGE 78 78 Hamiltonian. The e xpress ion for the i th layer of MoS 2 is , , where, ( ) is the electron (hole) charge density. . ( 6 3) We define and . k B is the Boltzmann constant and T is the temperature. are the Fermi energy levels of the i th layer MoS 2 and are zero under equilibrium state. (5.1 eV) is the electron affinity energy of the MoS 2 . The conduction band energy of i th layer is and the valence band energy is . A bandgap value ( ) of 1.80 eV and a Fermi velocity ( ) of 5.33Ã—10 5 m/s is used. The corresponding effective mass is 0.52 m 0 for the tight binding parameter of t 0 =1.1eV from the Dirac equation [ 84 ]. is the j th order Fermi Dirac integral. The applied gate voltage ( V G ) induces 2D charge density of , which depends highly upon the geometry of the gate insulator. Thus, for general quantification, we will use quantity of gate charge density of Q 0 instead o f V G . Through the work, we consider the un doped MoS 2 layers only. 6.3. Results Figure 6 2 A and 6 2 B shows the charge density profile normalized by Q 0 , for 10 layer s of MoS 2 at 1 0K and 300 K respectively. The charge density profile features not a simple exponential decay . The screening strength depends highly on the magnitude of Q 0 : with the larger Q 0 , stronger the electrostatic screening. Also at higher (away from PAGE 79 79 the gate) layers , the slope of the Q i /Q 0 decreases, which indicates weakening of the electrostatic screening. As shown in Figure 6 2C, the conduction band ( E C ) is further away from the Fermi level ( E F ) in the higher layers due to the less induced electrons. Thus, at small Q 0 (for example Q 0 =10 11 cm 2 in F ig ure 6 2B), the screening length can increase significantly . Temperature is also critical factor in modulating the screening behavior. At T=300K, broadening of the Fermi Dirac tail is larger than T=10 K. To induce similar charge, the E C for T=300K is further from the E F t hen T=10K case (Figure 6 2C). Thus at larger temperature, the charge profile along the layers is more even , than lower temperature, which implies weaker screening (i.e., longer screening length). The e lectric field profile along the z axis in Fig ure 6 2D c onfirm s this temperature dependency, where the profile at 300 K is more in an exponential manner compared to the more radical profile at 10 K. The electric field at i th layer is calculated as equation ( 6 4). ( 6 4) As in the previous work of screening effects in the graphene system [ 97 ] , we will use the equation ( 6 5 ) to quantify the screening strength and denote it as an effective screening length ( ), which is determined from the charge den sity of a two neighboring i th and ( i+1 ) th MoS 2 layers. , . ( 6 5 ) Figure 6 3 A and 6 3 B shows the calculated for T= 1 0 K and 30 0 K, respectively. As pointed out previously, the strength of the electrostatic screening PAGE 80 80 decreases (screening length increase) at higher layers, which is due to reduced induced charge density. Also, the higher temperature results in a l arger . Depending on the Q 0 , the vary from less than an interlayer distance ( d 0 ) of a MoS 2 to as large as insulator limit (in given Q 0 range, ~100 d 0 at the bottom layer) . A Debye length ( ) is a critical length which the mobile charge carriers screens the external electric field, which is essentially equivalent to the . We will use t he Debye length at each layer to quantify the screening length and compare with the . This can be linearized charge density approximation. The l inearized charge calculation is derived from eq uation ( 6 3 ) as, . ( 6 6 ) The , and , . As we will be dealing with n type case only, the hole charge density, is neglected. By self consistently calculating equation ( 6 6 ) with the 1D capacitance model [equation ( 6 2)], we can find the poten tial . The analytic expression of is in equation ( 6 7 ). ( 6 7 ) PAGE 81 81 The details of the derivation are in the Appendix E . W ith the limit ing condition of , which corresponds to the high gate bias with high temperature condition, the Debye length can be further simplified to . This indicates that the minimum screening length to be as short as half the MoS 2 interlayer distance. Figure 6 4 shows the comparison of the two methods used to quantify the screening length (i.e., and ). Since the strength of the screening is largest at the bottom layer, we will examine only the screeni ng length for i=1 . The plots verify that in the case of low Q 0 , the screening length can extend to infinity. This differentiates from the graphene case, where the screening length saturates to a certain value [ 97 ]. The origin of the difference is in the bandgap. Because of the zero bandgap property of the graphene, together with the thermal broadening of the Fermi Dirac charge distribution , the charge is induced even when Dirac point is at the Fermi energy . In turn , the induced charges partially screen th e electrostatics , thus, the screening length saturates to a temperature dependent finite value. However, this is not the case for MoS 2 . The bandgap permits the layers to be non screening , or have an infinite screening length . The Figure 6 4 plot shows that the two models used to model the screening length match well. Especially at higher temperature case, the range of Q 0 where two plots agree is larger. This is due to the linear approximation made earlier in calculating the charge. The linearization limits accuracy only in the range of . With larger temperature this condition can be met with larger range of , which is in direct relation with Q 0 . Although this study deals with multi layer of PAGE 82 82 MoS 2 , the method used here can be adopted to any stack of 2D material with finite bandgap. Previous studies have successfully modeled screening behavior of graphene via nonlinear Thomas Fermi (TF) model [ 100 ]. Following the analytical derivation from ref. 20 , we evaluate the validity of TF model by comparing with the 1D capacitance model. The details of the derivation can be found in the reference or summar y in A ppendix F . The charge density in the z (vertical) position of the thickness system in the continuum limit is expressed as, , where . Figure 6 5A, compares the charge profile of 10 layer MoS 2 stacks with varying Q 0 using the two models. The two model s only agree in small range of layer index i (or z ), at large Q 0 . The limitation of the analytical TF model is due to the missing bandgap information. The derivation (see A ppendix F ) relies on the 2D constant DOS, and the resulting show exponential behavior . However, in real , there exists a bandgap in MoS 2 system where DOS becomes zero. This is evident in the conduction band ( E C ) plot in Figure 6 5B. At Q 0 = 10 11 cm 2 , the Fermi level of all the layers of MoS 2 are positioned with in t he bandgap, thus, only the thermal broadened charges exists. This results in large discrepancy between the two model s in charge density calculation. Likewise, even for Q 0 =10 1 3 cm 2 case, at i ~3, the E C becomes larger than E F . From this point the charge density of the 1D capacitance model saturates while the TF model show continuous decrease. PAGE 83 83 6.4. Summary In this chapter , we have investigated the electrostatic screening behavior of multi layer MoS 2 , using numerical methods accounting for applied gate voltage (or gate charge) and temperature. The screening length of the layers depends highly on the both variables. The major difference between multi layer graphene is that, the graphene has saturating screening length which is due to its semi metallic fea ture . On the other hand, MoS 2 , due to the sufficient large bandgap of 1.8 eV, the screening length can extend to infinity. We also derive analytical expression for the Debye length which suggests that the lower limit of the screening length to be around ha lf of the interlayer distance of MoS 2 layers. PAGE 84 84 Figure 6 1 Schematic of multi layer MoS 2 , and 1D capacitance model . A ) Schematic of multi layer MoS 2 , in a form of field effect transistor (FET) with a back gate. B ) Corresponding 1D capacitance model of A . PAGE 85 85 Figure 6 2. Normalized charge density profile in 10 layer of MoS 2 FET with varying gate charge density, Q 0 at A ) T=10 K and B ) T=300 K. C ) Corresponding conduction band ( E C ) profile and D ) the electric field at each i th layer with the Q 0 = 10 12 cm 2 . Figure 6 3. Effective screening length in 10 layer s of MoS 2 FET with varying gate charge density, Q 0 at A ) T=10 K and B ) T=300 K. PAGE 86 86 Figure 6 4. Effective screening length between the first (bottom) two layers( i=1, eff ) compared with the Debye length ( D i ) at different temperatures. PAGE 87 87 Figure 6 5. Charge and potential profile along the MoS 2 stack. A) Charge profile comparison along the layers of MoS 2 stack (N=10) . C omparing the results of the 1D capacitance model (solid line) and the TF model (dashed lin e). B) Conduction band profile of 10 layer ed MoS 2 calculated from the 1D capacitance model. PAGE 88 88 CHAPTER 7 TWO DIMENSIONAL HETEROJUNCTION DIODE AND TRANSISTOR A vertical transistor based on a stack of atomically thin two dimensional (2D) semiconductors is theoretically examined. The structure under study resembles that of metal based transistor in that the metallic graphene is used as a base and semiconducting MoS 2 is used as an emitter and a collector. To understand the device operating mechanism, we first investigate the properties of graphene MoS 2 junction. We find that the characteristic of this junction depend not only on the intrinsic junction itself, but also highly on the lateral extension region, thus detailed 2D Poisson calculation is required for the study. As for the three terminal metal base transistor like structure, the excessive in plane transport feature. This is desired in conventional lateral transport device t opologies. However, for the structure under study which utilizes the interlayer carrier transport, the large leakage current in the base is undesired. Rather, in this type of structure, base material with sufficient bandgpap is preferred. 7.1. Overview Si nce the first demonstration of graphene in 2004 [ 7 ], the 2D material has been of great interest for the future electronic devices [ 92 ]. However, graphene as a channel for field effect transistor (FET) where the current path is along the in plane direction of the 2D material, absence of the bandgap leads to a low current on off ratio (I ON /I OFF ). This is a major obstacle in utilizing graphene as a c hannel material in current digital logic applications [ 32 ]. There have been considerable efforts to induce a ban dgap, however resulted in trade off with the mobility [ 102,23,36 ]. As a contrast to the in plane transport of the graphene, out plane transport properties can be exploited as PAGE 89 89 demonstrated by Britnell et al , using vertical stacking of 2D semiconductor mater ials to create a quantum tunneling FET [ 65,95 ]. In the similar sense, graphene base hot electron transistor (GB HET) was proposed and explored, which show high current on off ratio of 10 4 ~10 5 [ 103~105 ]. The hot electron transistors (HETs) were introduced i n . The main expectation of such device was high speed applications due to their short base transit time [ 106,107 ] . The GB HETs exploits exploit this attribute to an extreme in that in that the width of the base can be as short as single atomic thickness and the semi metallic property of the graphene which result in relatively low base series resistance. While the reported structure of the GB HETs exhibits tunneling barrier between emitt er and base, we would like to focus on a more simple structure that consists of a 2D semiconductor, mono layer (ML) MoS 2 , as a collector and an emitter and the graphene as a base. As graphene have a semi metallic E k relation, this structure resembles the metal base transistor (MBT) [ 108 ]. By adopting fabrication technique as mechanical transfer process [ 64 ], various types of 2D material can be stacked and may expect such device to be of low cost. In the following chapters we will first, observe the operat ion principle of the metal semiconductor (MS) junction like graphene MoS2 (GM) junction. The measured I V curve of this 2 terminal structure show exponential behavior as expected in a MS junction. However, the reverse bias shows non saturating linear incre ase of current. This is due to the geometry of the structure, where at the edge of the overlapping GM junction, an extrinsic lateral tunneling path is generated. We confirm this theory with 2D Poisson calculations coupled with the equilibrium charge calcul ation. We also extract PAGE 90 90 the graphene to MoS 2 interlayer electron escape rate for transport prediction. In the second part of this work, we focus on the MBT like structure, which is composed of two GM junctions. Here, we consider only the intrinsic junctions to elucidate the device mechanism. We will denote this as MoS 2 graphene MoS 2 bipolar junction transistor ( MGM BJT). We find that the large base current hinders the device from achieving high current gain. Engineering of interlayer escape rate is required and base material with finite bandgap is preferred. 7.1. Graphene MoS2 J unction D iode 7.1.1. One D imensional S imulation for V ertical J unction Figure 7 1A is a schematic of the GM junction , where the bottom MoS 2 layer is grounded and the bias ( V Gra ) is applied to the top graphene layer. To examine the intrinsic junction property, one dimensional (1D) model is sufficient as in schematic in Figure 7 1 B, which can be electrostatically modeled by a 1D capacitance model as depicted in Figure 7 1 C. By sol ving the 1D capacitance model coupled with the carrier statistics, potential profile can be determined. The general form to calculate the charge at stacked system is presented in equation ( 7 1). , . ( 7 1) For this case, the total number of layer is 2, where i=1 ( i=2 ) represents the MoS 2 (graphene) layer. , is equivalent to gate oxide capacitance ( C ox ) . For the SiO 2 gate oxide , dielectric constant ( ) of 4 is used with thickness ( t ox ) of 280 nm . is the vacuum permittivity. represents the dielectric constant between MoS 2 and graphene ( ) and is set as 1 with interlayer distance d 0 = 6 Ã… . The boundaries conditions are, C 0 = C 3 =0, E vac, 1 =E vac,3 =0 , and equation ( 7 1) can be expressed as [ 95 ], PAGE 91 91 ( 7 2). The vacuum energy level at the gate is, , where is the bottom gate work function. This 1D electrostatic equation is self consistently solved with equilibrium semi classical carrier statistics from method. The charge density of i th layer ( i= 1 is MoS 2 and i= 2 is Graphene) is expressed as, , where, . ( 7 3) We define , and , . k B is the Boltzmann constant and T is the temperature. and are quasi Fermi energy levels of MoS 2 and graphene. When a bias of V Gra is applied, they are assumed to be separated by . The bandgap =1.80 eV and =0 eV for MoS 2 and graphene respectively. The MoS 2 c onduction band energy is and valence band energy is . T he D irac energy of graphene is . (5.1 eV) and (4.5 eV) are the electron affinity energy of MoS 2 and graphene work function , respectively . The Fermi velocity of MoS 2 is =5.33Ã—10 5 m/s and graphene is =1Ã—10 6 m/s. The corresponding effective mass for MoS 2 is 0.52 m 0 with tight binding parameter t 0 =1.1eV from the Dirac equation [ 8 4 ] . The PAGE 92 92 is the j th order Fermi Dirac integral. By self consistent iteration of equation ( 7 2) and ( 7 3), the potential the two 2D layers can be found which in turn is used to calculate the current. 7.1.2. Classical T ransport C alculation Within the classical representation, the MoS 2 and graphene interlayer tunneling current can be expressed by the density of state (DOS) of the two materials as, . ( 7 4) The energy dependent DOS of MoS 2 , (for and ) and (elsewhere), is used. The graphene DOS is and , . The ML MoS 2 is grounded and the bias is applied to the graphene layer thus, and . Lastly, h is the plank constant. C is an interlayer tunneling parameter whose value proportionally determines the interlayer transport rate and can be extracted as a fitting factor to the experimen tal I V characteristics. The I V of 3.3Âµm 2 device is plotted in Figure 7 2A which is measured at 220 K with varying back gate bias ( V G ). In the forward bias region, the curve exhibits exponential behavior, however, in the reverse bias show linear current characteristic which is not typical a typically expected in a MS or a p n junction. The reason for this is in the lateral extension of the junction and will be dealt in the next sub chapter. Figure 7 2B is the conductance plot of diode at V Gra = 0. The interlayer tunneling parameter for PAGE 93 93 classical transport is extracted as a fitting parameter between the classical transport model and the measurement. In the numerical model, the MoS 2 layer is n type doped to dop i ng density of N D,MoS2 = 1Ã— 10 16 /m 2 and graphene layer p doped to N A,Gra = 1.1Ã— 10 16 /m 2 . From the non equilibrium greens function (NEGF) formalism, is extracted [ 109 ]. The relation between and is shown in the A ppendix G , which by the nature of the method differs by a factor of . Both the classical transport model and NEGF calculations are in a close match. I n Figure 7 2C, the MoS 2 graphene interlayer electron transport time ( ) is extracted from equation ( 7 5), as 3.2 nsec at V G =0 and increases up to 14 nsec at V G =15 V. ( 7 5) The increase of as V G increase is due to V G dependent E C of MoS 2 and E Dirac of graphene. As can be noted from the Figure 2 D and E, the energy distance between the two values decreases as the V G increases, thus, leading to smaller DOS of graphene at the corresponding E C of the MoS 2 layer . 7.1.2. Two Dimensional Simulation to T reat L ateral V ariations Previous 1D capacitance model describes only the electrostatic properties of the intrinsic junction. To capture the barrier height variation along the horizontal direction, we solve the two dimensional (2D) Poisson equation [equat ion ( 7 6)] self consistently with the quasi equilibrium carrier statics [equation ( 7 3)] via a 2D finite difference method . The schematic of the diode is in Figure 7 1A. ( 7 6) PAGE 94 94 The r and 0 are the relative dielectric constan t and the vacuum permittivity, respectively. V is the pot e ntial . n and p are the electron density and hole density , respectively. N A is p type doping concentration in graphene and N D is n type doping concentration in MoS 2 . The results are in Figure 7 3A and 7 3 B, for the forward ( V MoS2 = 0.3 V) and reverse bias ( V MoS2 =0.3 V), respectively. With this 2D picture, the I V characteristics in Figure 7 2A can be explained. At forward bias, the electron flows from the E C of the MoS 2 to the graphene layer without any barrier. In the reverse bias region, the E C E F,Gra bec omes the Schottky barrier height (SBH), and thus blocks transport and only the thermionic emission should contribute to the transport. However, due to the lowering of E C at the lateral extension region of the MoS 2 ( 0~1Âµm region in the Figure 7 3B ) , the potential is also effected at the edge of the junction area where a lateral tunneling path is created. This results in linear reverse current. To confirm above mentioned argument, we use a parallel isolated diode model as depicted in Figure 7 4A. The potential profile is obtained through the 2D Poisson calculation, and the I V characteristics can be calculated by extending equation ( 7 4) to consider the potential barrier variation in the lateral direction. The total current is calculated as, , where x i , W i , and I i are the length, width and current density of the D i diode. Because the barrier height is no longer uniform and the current density is a function of the lateral position, a summation along the lateral direction is performed, which can be physically interpreted as multiple Schottky barrier diodes with differ ent barrier heights in parallel. The modeling approach only accounts for the intrinsic vertical junctions and PAGE 95 95 neglects the parasitic resistance in the horizontal direction, such as metal MoS 2 and metal graphene contact resistance and the lateral diffusion resistance of MoS 2 . The parasitic resistance value is expected to be gate voltage dependent especially near the threshold voltage value. The calculated current of the intrinsic junction 1D model current (solid line) is compared with the accounted for extri nsic parallel diode (dotted line) in Figure 7 4B and 7 4 C, as function of applied V G . The reverse bias current in 1D model is the thermionic emission due to the thermal broadening of the Fermi Dirac distribution of charges. With the 2D effect, reverse bias dependency of the leakage current does not saturate due to the fore mentioned lateral tunneling effect. 7.2. MoS 2 Graphene MoS 2 Bipolar Junction Transistor 7.2.1. 1D Capacitance M odel with NEGF T ransport C alculation With appropriate modifications, equation ( 7 1~2) can be used to for electrostatic calculation of three terminal MGM BJT. Likewise equation ( 7 3) can be used to calculate the charge density in each layer: with i=1 as collector MoS 2 layer, i=2 as graphene base layer and i=3 as MoS 2 emitter layer. By self consistent calculation, band profile can be calculated and used for transport prediction. The current equation of current equation ( 7 4) is valid only for 2 terminal devices. To study the transport of MGM BJT structure, we adopt the NEGF formalism [ 109 ]. The schematic of the structure is shown on Figure 7 5 A , where due to the nature of 1D model, the base contact is treated as a virtual contact, which the graphene is extended to the metal contact. We have confirmed (Figure 7 2B) that for a two terminal GM vertical Schottky contact, equation ( 7 4) agrees well with the NEGF method (details in A ppendix G ). The Hamiltonian of the orbital tight binding model given as, PAGE 96 96 , ( 7 7) where , , intra layer nearest neighbor (NN) hopping parameter, t 0 =2.7 eV and NN C to C atomic distance of a=1.42 Ã… . In the NEGF formalism, electron transport across the layered device at given energy E is, , ( 7 8 ) m,n is emitter ( E ), base ( B ) and collector ( C ) and is the retarded greens function. A is the total area of the Brillouin zone used for the numerical calculation. The two MoS 2 layers , emitter and collector , are dealt as phenomenological contacts to the graphene base and expressed in the NEGF formalism as energy dependent self energy E and C . E,C in the definition of the self energy ( E,C ) is related to the inter layer transport coupling between graphene MoS 2 layers, same as of current for GM diode. Self energy ( B ) describes the phenomenological extension of graphene base and the contact, and is expressed as, . The broadening matrix is . PAGE 97 97 And lastly , the current density between the two nodes is calculated using transmission from equation ( 7 8) as, . The is the Fermi Dirac distribution function at temperature T . The chemical potential are, . The applied voltage at each port, emitter, collector and base ( V E , V C , V B , respectively ) alters the H(K) in equation (7 7 ) and thus varies the current. The current density at each node is summation of each current density elements defined as, . As implied in the derivation, the properties of the contact self energy defines the characteristic of the MGM BJT, thus careful engineering of E,C and B is required. 7.2.2. Results of the MGM BJT As fore mentioned the structure under study resembles that of the metal base transistor (MBT) [ 108 ]. The short interlayer transit time is essential for high performance, where the parameter is responsible in this model. Another to note is that d ue to the high Fermi velocity of graphene ( V F =3Ã—10 6 m/s) [ 110 ], the excessive base current will be the major drawback. T o achieve large current gain ( =I C /I B ), suppressing the base current is critical. As the main purpose of this study is to illustrate the operation mechanism of the device, we will examine the device behavior as a function of . Assuming the lateral extension of the graphene base is ~1 Âµm, and in that the Fermi velocity of graphene is 3 Ã—10 6 m/s , the intrinsic delay time is in the PAGE 98 98 order of 0.33 psec. The v alue results in the intrinsic delay time in the similar order, and thus will be used through the investigati on. The corresponding scattering potential matrix element value is listed in Table 7 1 . Figure 7 5B shows the current density component ( I CE ,I CB ,I BE ) plot as function of V CB with MGM BJT in common base configuration at V EB = 0.4 V, and . The value of is in the similar range as have been extracted earlier in GM junction section of the study. The corresponding I C plot is in Figure 7 5D where, as V CB increase the two current plots with varying V EB cross and at V CB >0 , with larger (more negative) V EB , the current is smaller. Figure 7 5B, where the I BE is larger than I CE . In this ca se, we cannot expect gain (current gain or transconductance gain ) from this device . As o pposed to this case, with increased graphene MoS 2 interlayer transport parameter to ( Figure 7 5C ), we can find that the I CE exceeds I BE and as result, Figure 7 5E shows I C larger with larger V EB in all V CB range as in a typical BJT. Figure 7 5F compares the I C and I B of the two results. While I B is orders larger than I C for , current gai n can be observed for intrinsic delay ( int ) and g raphene intralayer transport time ( B ) as function of . increase while int show exponential decrease. The int was calculated as the below equation. ( 7 10 ) Likewise, the PAGE 99 99 . The is smaller than at lower , however, at , become similar, and at this point the and further increase of results in faster and larger . Figure 7 6B is to identify the effect of on the current components. The bias condition is V CB =0.2V which can be inferred from Figure 7 5E that is past the saturation point, and V EB = 0.2 V. Both I CB and I BE are directly related to and , thus show similar trend but shifted , w hile I CE show monotonic increase. Although this study is focused on the GM junction and has utilized graphene as a base of a MGM BJT, other combination is possible. Here, the effect of introducing a bandgap material in the base is investigated. Figure 7 7 shows the numerical calculation results on BJT structures with artificial 2D material in the base. The adopted 2D material has DOS a s a graphene but with a bandgap of 0.1 eV and 0.8 eV. Figure 7 7A, depicts the energy dependent DOS of the graphene and graphene with bandgap. Figure 7 B shows the band profile at V CB =0 V and V EB = 0.4 V. From this figure we can expect the dramatic reductio n of I CB and I BE with larger bandgap, which translates to large decrease in I B . Figure 7 7 C ~ E compares the t ransmission of the two cases as function of energy between, C E, B E and C B respectively. The large bandgap clearly blocks the transmission, especially Figure 7 7 E show large reduction for C B transmission. Thus, large decrease in both the I C and I B is expected and is confirm by Figure 7 7 F. Although the overall current level has been reduced , the introduction of bandgap benefits the current gain as plotted in Figure 7 7G. PAGE 100 100 Summary Graphene MoS 2 junction was studied using numerical methods. Comparing with measurement and 2D Poission calculation, we have found the cause of linear reverse I V ch aracteristic in the lateral tunneling junction. Also, interlayer transport parameter has been extracted to predict the performance of 3 terminal MGM BJT. The major drawback of this MBT like device is the large base current. Careful engineering of and ratio, which is physically related to the interlayer transport property is required to achieve certain current gain. We also find that the base material with bandgap results in reduction of overall current density, however is preferred in achieving large current gain. PAGE 101 101 Table 7 1. Scattering potential matrix element value E,C / B Â§ 1.0Ã—10 3 1.0Ã—10 2 1.0Ã—10 1 2.0Ã—10 1 1 2 3 E,C [eV 2 nm 2 ] 8.0Ã—10 8 8.0Ã—10 7 8.0Ã—10 6 1.6Ã—10 5 8.0Ã—10 5 1.6Ã—10 4 2.4Ã—10 4 M B0 2 [eV 2 nm 2 ] 0.0825 0.825 8.25 16.5 82.5 165 247 Â§ is set to a constant value of Figure 7 1. Simulated structure of graphene MoS 2 junction. A) 2 Dimensional (2D) and B) 1 dimensional (1D) schematic of the graphene MoS 2 junction. C) 1D capacitance model. PAGE 102 102 Figure 7 2. MoS 2 Graphene junction properties. A ) Measured I V plot of MoS 2 Graphene diode with varying gate bias (V G ). B ) Extracted conductance from measured I V compared with numerical results. C ) Graphene MoS 2 interlayer electron transport time ( ). D ) Conduction band energy of bottom MoS 2 (E C,MoS2 ) and Dirac point of top graphene (E Dirac ) as function of V G and E ) difference between the two value. PAGE 103 103 Figure 7 3. Band alignment results of 2D P oisson calculation. A ) Forward bias and, B) reverse bias. PAGE 104 104 Figure 7 4. Isolated diode model and its results. A) Isolated diode model to describe the leakage current at reverse bias. B) I V curve in logarithm scale . C) I V curve in linear scale . The solid line represents r esult from 1D model. Dashed line represents results from isolated diode model. PAGE 105 105 Figure 7 5. Schematic of the 3 terminal MGM BJT and NEGF current calculation results. A) Schematic of the 3 terminal MGM BJT . Calculated current components with B ) and C) . D ) I C as function of V CB with and E ) I C as function of V CB with . F ) Comparison of I C and I B in logarithmic scale. PAGE 106 106 Figure 7 6. Current gain and intrinsic delay of the MGM BJT. A ) E xtracted current gain ( ) , intrinisic delay ( int ) and g raphene intralayer transport time ( B ) as function of . B ) E xtracted current components as function of . PAGE 107 107 Figure 7 7. Predicted performance of MGM BJT with bandgap base. A ) A rtificial badgap induced graphene DOS with bandgap of 0.1eV and 0.8 eV . B) Band diagram of the MGM BJT. ( C~E ) Transmission as function of energy between, C E, B E and C B respectively. F ) I C and I B as function of V CB and G ) corresponding c urrent gain ( ). PAGE 108 108 CHAPTER 8 CONCLUSION The first demonstration of graphene [7] unveiled series of 2D layered material for th e future nano devices. Such material includes h BN, topological insulators, and transition metal dichalcogenides (TMDs). Graphene has been of a great interest due to its fascinating properties such as mobility in the order of 10 5 cm 2 /V s, thermal conductiv ity up to 3000 W/K m [48,49]. However, due to the absence of ban d gap, the graphene based FETs have high off current and thus is not suitable for current logic applications. In this work, we have proposed few ways to induce the bandgap in the graphene by f unctionalization. Also we have looked into TMDs, which are 2D materials that naturally bear bandgap. One purpose of this work is to understand the material properties, such as band structure which convey the essential character of the material, doping feat ure, and electrostatic screening behaviors. Another is to understand operation mechanism and project the performance of devices that use these 2D materials. In the first topic, which is Chapter 2 of this dissertation, we have investigated electronic proper ties of A C x (BN) y s through DFT calculation. An earlier study has shown that the domains of graphene can be grown in the 2D h BN sheet [29]. Motivated from this, we proposed a structure, A C x (BN) y , which is analogous to the AGNRs. The confinement is provide d from the h BN to induce bandgap in the graphene system. The bandgap properties of A C x (BN) y resembles that of the AGNR. However, considerably bandgap opening for A C x (BN) y s with width index of 3p+2 is observed. Also the relation of the width to the bandg ap in A C x (BN) y s, follows the hierarchy as E 3p+2 PAGE 109 109 perturbation from the large ionic potential difference between B and N. A TB model of the system was constructed as a simple and effective method to confirm out theory and predict the electronic properties. As an extension of the first work on A C x (BN) y , we have looked into chemical modified graphene, i.e. , graphane, to induce confinement on the graphene system. The band structure of partially hydrogenated graphene, graphene nanoroad and graphene nanomesh were studied and showed that a considerable bandgap can be achieved in a continuous 2D layer. We also f ound through top barrier ballistic MOSFET model that the highly non isotropic 2D E k of graphene nanoroad is preferred in achieving higher device performance. Moving the focus from generating bandgap in graphene to 2D material with bandgap, we look into T MDs. In Chapter 4 of this dissertation, we have studied the potassium (K) doping features in MoS 2 and WSe 2 , and compared it with the graphene, through DFT calculations. K has a small electron affinity and thus is acts as a strong electron donor. While the K doping in graphene has been reported to be unstable [58], our study shows that K MoS 2 and K WSe 2 bonding to be stronger, thus more reliable. By adopting mechanical transfer process [64], various types of 2D material can be stacked, which leads to study of stacked heterojunctions. As prior to understanding the junction physics, we studied the electrostatic screening behavior of multilayer TMDs (we have work on MoS 2 , however, same method can be applied to other TMDs) following the method of a graphene stac k [97]. By using numerical methods, we found the electrostatic screening to be highly dependent on applied gate voltage and temperature. The major difference between multi layer graphene is that, the graphene PAGE 110 110 has saturating screening length which is due to its semi metallic feature . On the other hand, MoS 2 , due to the sufficient large bandgap of 1.8 eV, the screening length can extend to infinity. We also derive analytical expression for the Debye length which suggests that the lower limit of the screening length to be around half of the interlayer distance of MoS 2 layers. As for the last topic, we have investigated the properties of graphene MoS2 vertical junction. As graphene is semi metallic material, this junction resembles the metal semiconductor juncti on. However, due to their 2D nature and also geometry of the junction, which extends to form a contact, leads to lateral tunneling junction at the reverse bias. This result in a linear I V characteristic, that cannot be found in the traditional MS junction . Also, interlayer transport parameter has been extracted to predict the performance of the 3 terminal MGM BJT. The major drawback of this MBT like device is the large base current. Careful engineering of and ratio, which is physically related to the interlayer transport property is required to achieve certain current gain. We also find that the base material with bandgap is preferred for large current gain. These works can be extended to explore other 2 D materials. One of such material may be black phosphorus. This has recently gained interest in that it has highly non isotropic direct bandgap 2D band structure [114,115]. The peculiar crystalline structure enables the band be tailored with applied strain which the origin is yet unclear. In depth study via DFT is needed to understand the physics. Also the anisotropic band structure will lead to interesting study of the transport properties. PAGE 111 111 APPENDIX A BANDGAP CALCULATION OF AGNR AND AGNR WITH EDGE PERTU RBED BY THE IONIC POTENTIAL A.1. Derivation o f Bandgap f or A GNRs AGNRs are semiconductors with bandgap which decrease as the width increase and in the larger width limit will function as graphene. However, the variation of the bandgap follows a distinctive behavior as function of the width and can be categorized into three groups depending on the width, e.g., n=3p, 3p+1 and 3p+2. Here, we will derive the bandgap characteristic of each family member and carry on the derivation of the perturbed AGNR system in appendix A.2, where the perturbation is due to the neighboring B and N atoms that are dealt in chapter 2 of the text. Figure A 1 . Graphene nanoribbon schematic. A ) Schematic of AGNR with width(n) of N a =9 in this case. B ) Topologically equivalent structur e of N a AGNR C ) Topologically equivalent structure of N a AGNR in the case of k=0. All the sublattices are linked with tight binding parameter t 0 . The AGNR, as depicted in Figure A 1 A can be viewed topologically same as Figure A 1 B which can be further si mplified, in the condition that k =0, to Figure A 1 C PAGE 112 112 with the TB of t 0 between each sublattices [17]. The condition k =0, is valid one in that our interest is in calculating the bandgap only. Using this concept, and following the derivation from ref.49, th e tight binding Hamiltonian reduces to a two leg ladder lattice system. The Hamiltonian can be written as equation (A 1). (A 1) where t 0 corresponds to either the left or the right ladder. The eigenstates and the eigenenergies of H 0 in the q th band , are (A 2) Equation (A.2) describes only the n of the 2n eigenstates. To complete the solution we consider symm etric and anti symmetric wave function where, (A 3) First, to calculate the eigenstate of the q th band, PAGE 113 113 which results in , . (A 3) And to solve for the Hamitonian, which leads to , . Thus, as for the general solution of the eigenenergies of the q th subband is, and , (A 4) where, . We can further derive the solution of the eigenenergies into the 3 categories of the AGNR family. A.1.1. Case 1. n=3p , with . For this to have minimum value, . Thus, at q = p , PAGE 114 114 So, the bandgap of the AGNR with width of n=3p is, (A 5) A.1.2. Case 1. n=3p +1 Similarly, , with and for this to have minimum value, . Thus, at q = p+1 , , (A 6) A.1.2. Case 3 . n=3p +2 , with and for this to have minimum value, and thus, at q=p+1, , PAGE 115 115 A.2. Bandgap AGNR Perturbed b y t he Ionic Potential o f B a nd N a t t he AGNR Edge Figure A 2 . Schematic of AGNR and perturbation modeling. A ) Schematic of AGNR perturbed at the four edge C atoms by the neighboring B and N of the AGNR of the for A C x (BN) y system in chapter 2. B ) Simplified ladder representation. C ) Case for the even and odd AGNR width. To calculate the bandgap of the AGNR confine d by B NNR ( A C x (BN) y ) through perturbation calculation, we need to account for the perturbed Hamiltonian. The is the Hamiltonian of the AGNR itself, which has been derived in appendix A.1, and is the first order perturbation. We set the perturbation Hamiltonian as, . PAGE 116 116 The off diagonal terms are zero matrix in that we assume nothing has affected the C C interaction of the original AGNRs (we simpl ify the problem to just perturbation in the ionic onsite energy of the C atoms at the edge as represented in Figure A 2 A and B ). n=odd n=even The 1 st order perturbed eigenenergies can be written as, (A 7) First, solving the denominator, , where Thus, PAGE 117 117 Moving on to the nominator, . For both even and odd cases, , Thus, A.2.1. Case 1. n=3p With q = p , , and thus, PAGE 118 118 Resulting bandgap calculation for n=3p is, (A 8) A.2.2. Case 1. n=3p +1 With q = p+1, as in the AGNR case the anti symetric solution results in the minimum value of the eigenergy, and thus solving for that results in. The resulting bandgap calculation for n=3p is, PAGE 119 119 APPENDIX B BASIS SET SUPERPOSITION ERROR B .1. Overview DFT is a quantum mechanical method to describe the electronic structure and dynamics of a many body system. SIESTA code [26] and VASP code [xx] is used throughout this work. Both use pseudo potential method to reduce the basis set size, therefore enhance the efficiency, however, SIESTA uses localized combination of atomic orbital (LCAO) basis set and VASP uses plane wave basis set. When calculating for the binding energy of two interacting system, the atomic orbital basis set results in over estimation of the binding energy. This is due to the unequal basis set between the interacting bonded system and non inter acting separate system. This is known as b asis set superposition error (BSSE) and should be accounted for. One of the ways to do so is conducting counter [51 interacting separate system, same as the interacting bonded system. B .2. Counterpoise Correction In the uncorrected interaction energy is, , where is the coord inate of the geometry of the whole bonded structure, is the total energy of the calculated with the full basis set AB at that geometry. and are the total energy of the non interacting parts, each calculated from the basis set A and B respectively. On the other hand c ounterpoise corrected interaction energy is , PAGE 120 120 , where and is the total ene rgies of part A and B, calculated within the basis set of AB at geometry , respectively . This can be understood as increasing the basis set size of each element A and B to the whole AB structure. Thus BSSE can be calculated as, . B .3. Calculating the C H B inding E nergy of A HNRs U sing Counterpoise Correction By using the CP corrected method we can extract the H or the F binding energy to the graphene system. Figure A 3 A shows the atomic structure of the A HNR. From Figure A 3 B , the side view of the unitcell , we can see that the H atoms are puliing the binding C atoms out of the graphene plane which indicates that the basis set of the A HNRs will differ from that of jus t the H atoms and the graphene. To account for the by the DFT calculation, we calculate the molecular dynamically relaxed geometry (or the atomic coordinates) and the total energy ( ) of the A HNR structure. From the obtained atomic coordinates, we run the second un relaxed DFT calculation with all the C atoms set to ghost atoms as in Figure A 3 D and obtain the total energy, . Third conduct un relaxed DFT calculation of the graphene with the H atoms as ghost atoms as in Figure A 3 (e) and get . The resulting binding energy can be calculated as, , PAGE 121 121 where n H is the numbe r of H atoms in the unitcell of the A HNR and is the binding energy of a H atom. Figure B 1 . BSSE calculation method. A ) Atomic structure of unitcell of A HNR, top view. B ) Side view of the unitcell of A HNR. Notice that C atoms with H bonded is pulled out of the graphene plane indicating that the atomic basis has been modified from that of graphene. C ) Schematic of A HNR. D ) Schematic of H atoms with the basis set extended t o the same basis set of the A HNRs, where the dotted atoms represent the ghost atoms of the graphene. E ) Schematic of graphene with the basis set extended to the same basis set of the A HNRs, where the dotted atoms represent the ghost atoms of the H. PAGE 122 12 2 APP ENDIX C G 0 W 0 CALCULATION APPROACH In the following GW quasi particle (QP) calculation, non self consistent G 0 W 0 was used to reduce the computational cost [ 79 ]. Same atomistic relaxed structure was used as in the main text. HSE exchange correlation function al was used to obtain wave functions for the GW calculation. Although this step is not essential, this method has been suggested to improve agreement with experiments [ 90 ]. The Brillouin zone sampled with a 6 Ã— 6 Ã— 1 k point mesh. The band structure was Wa nnier interpolated using the WANNIER90 program [ 91]. From the F igure CA , ~0.65 eV shift of CBM as regard to the va lence band can be observed. In F igure CB , splitting the valence bands due to spin orbital coupling is shown. Figure C 1 . Bilayer WSe 2 HSE DFT and G 0 W 0 calculation . A ) Comparison of Wannier interpolated HSE DFT and G0W0 calculation. SOC is not considered and no strain is applied. B ) Comparison of spin orbital coupling (SOC) effect in HSE DFT calculation. Band splitting in the K point of t he valence band can be observed. PAGE 123 123 APPENDIX D VASP INPUT SCRIPT FOR GGA, HSE AND G 0 W 0 SIMULATION D.1. GGA In Cpater 5, structural relaxation was performed using GGA XC functional. In the VASP calculation, for fast and accurate results, first, (1) ISIF was s et as 7 in the input script (INCAR). This maintains the overall shape of the geometry from the initial guess during the relaxation process but modified the cell volume. Once this calculation is converged, using the updated geometry (saved as CONTCAR, updat e the POSCAR to CONTCAR), another relaxation calculation was performed, with (2) ISIF=2. This procedure relaxes the ions without changing the cell shape or the volume. The resulting structure will be relatively precise, however, to further improve accuracy , these two steps can be iterated. With the resulting relaxed structure, (3) self consistent calculation is followed with higher k mesh grid. Then last step is the (4) bandline calculation. Below is the INCAR script for the process. T able D 1. GGA step 1 and 2 INCAR file Comments KPOINT file System = BL_WSe2 PREC = NORMAL SYMPREC = 1E 8 ENCUT = 400 ISTART = 0 ICHARG = 2 NELMIN = 6 EDIFF = 1E 5 NSW = 100 IBRION = 2 ADDGRID = .TRUE. ISIF = 7 ISMEAR = 0 SIGMA = 0.05 GGA = PE LVDW = .TRUE. VDW_C6 = 12.64 48.894 VDW_R0 = 1.771 1.8413 ISTART=0 for step (1), 1 for step (2). For iterating of these steps use 1. ICHARG=2 for step (1), 1 for step (2). For iterating of these steps use 1. ISIF=7 for step (1) and 2 for step (2) Last three lines are to include the effects of the vdW force vias DFT D2 m ethod k points 0 Gamma 9 9 1 0 0 0 PAGE 124 124 T able D 2 . GGA step 3 INCAR file Comments KPOINT file PREC = ACCURATE LSORBIT = .TRUE. LORBIT = 11 SYMPREC = 1E 8 ENCUT = 400 ISTART = 1 ICHARG = 1 NELMIN = 6 EDIFF = 1E 7 NSW = 0 ADDGRID = .TRUE. ISMEAR = 0 SIGMA = 0.05 GGA = PE LVDW = .TRUE. VDW_C6 = 12.64 48.894 VDW_R0 = 1.771 1.8413 LSORBIT=.TRUE. switches on spin orbit coupling LORBIT = 11 : quick method for the determination of the spd and site projected wave function character Increase the k mesh points for improved accuracy KPOINTS k points 0 Gamma 15 15 1 0 0 0 T able D 3 . GGA step 4 INCAR file Comments KPOINT file PREC = ACCURATE LSORBIT = .TRUE. LORBIT = 11 SYMPREC = 1E 8 ENCUT = 400 ISTART = 1 ICHARG = 11 NSW = 0 IBRION = 1 LWAVE = .FALSE. LCHARG=.FALSE. ADDGRID = .TRUE. ISMEAR = 0 SIGMA = 0.05 GGA = PE LVDW = .TRUE. VDW_C6 = 12.64 48.894 VDW_R0 = 1.771 1.8413 ICHARG=11: To obtain the eigenvalues (for band structure plots) or the DOS for a given charge density read from CHGCAR. k points 50 Line mode rec 0.66666667 0.33333333 0.00000000 ! K 0.00000000 0.00000000 0.00000000 ! Gamma 0.00000000 0.00000000 0.00000000 ! Gamma 0.50000000 0.00000000 0.00000000 ! M 0.50000000 0.00000000 0.00000000 ! M 0.66666667 0.33333333 0.00000000 ! K D.2. HSE VASP input script (INCAR and KPOINT S) are shown below. The script contains self consistent calculation, which require relaxed atomistic structure, and the band calculation. Note that the INCAR file is basically same. However, for band calculation, the k points in KPOINTS file must be specif ied with weighting of 0. Also to generate LSORBIT = .TRUE. results in output file of PROCAR that contains the information. PAGE 125 125 T able D 4 . HSE Self consistent calculation INCAR file KPOINT file System = HSE_BL_WSe2 PREC = ACCURATE LSORBIT = .TRUE. SYMPREC = 1E 8 ISTART = 1 LHFCALC = .TRUE. HFSCREEN = 0.2 ALGO = A TIME = 0.4 PRECFOCK = Normal NELMIN = 5 EDIFF = 1E 8 IBRION = 1 IMIX = 1 ISMEAR = 0 SIGMA = 0.02 k points 0 Gamma 15 15 1 0 0 0 T able D 5 . HSE Band calculation INCAR file KPOINT file System = HSE_WSe2_noStrain PREC = ACCURATE LSORBIT = .TRUE. ISTART = 1 LHFCALC = .TRUE. HFSCREEN = 0.2 ALGO = N TIME = 0.4 PRECFOCK = Normal NELMIN = 4 EDIFF = 1E 8 IBRION = 1 IMIX = 1 ISMEAR = 0 SIGMA = 0.02 LWAVE = .F. LCHARG = .F. Automatically generated mesh 229.00000 Reciprocal lattice 0.00000 0.00000 0.00000 1 The k points are generated 0.11111 0.00000 0.00000 1 in the IBZPT file in the 0.22222 0.00000 0.00000 1 previous SCF calculation 0.33333 0.00000 0.00000 1 0.44444 0.00000 0.00000 1 K points added for bandline the weight is set to 0 0.66667 0.33333 0.00000 0 K point 0.65306 0.32653 0.00000 0 0.63946 0.31973 0.00000 0 0.01360 0.0068 0.00000 0 0.01020 0.0000 0.00000 0 0.48980 0.00000 0.00000 0 0.50000 0.00000 0.00000 0 M point PAGE 126 126 D.3. G 0 W 0 The G 0 W 0 calculation is performed over HSE wave functions. Thus require WAVECAR file obtained from HSE self consistent calculation. For G 0 W 0 , the input of the KPOINTS file is the Monkhorst Pack (MH) scheme . The bandline calculation as in HSE or other XC is not adaptable for GW approach. The band is Wannier interpolated from the k mesh using WANNIER90 [ 91 ]. For honeycomb lattice structure, to sample all high calculation cost purpose we use 6x6x1. T able D 6 . HSE Band calculation INCAR file. Step 1 Step 2 Step 3 System = WSe2 PREC = ACCURATE ISTART = 1 ICHARG = 1 # charge: 1 file EDIFF = 1E 5 IBRION = 1 ISMEAR = 0 SIGMA = 0.02 LHFCALC = .T. HFSCREEN= 0.2 ALGO = A TIME = 0.4 LWANNIER90_RUN=.T. LWRITE_MMN_AMN=.T. System = WSe2 PREC = ACCURATE ISTART = 1 ICHARG = 1 EDIFF = 1E 5 IBRION = 1 ISMEAR = 0 SIGMA = 0.02 LHFCALC = .T. HFSCREEN= 0.2 TIME = 0.4 ALGO = Exact NBANDS = 300 LOPTICS = .T. NEDOS = 2000 System = WSe2_BL PREC = ACCURATE ISTART = 1 ICHARG = 1 IBRION = 1 ISMEAR = 0 SIGMA = 0.02 LHFCALC = .T. HFSCREEN= 0.2 TIME = 0.4 NBANDS = 304 ALGO = GW0 LSPECTRAL = .T. NOMEGA = 50 LWANNIER90_RUN=.T. LWRITE_MMN_AMN=.T. PAGE 127 127 APPENDIX E ANALYTICAL DERIV ATION OF DEBYE LENGTH In a 1D system, with the varying space charge density of ,the potential can be calculated by solving the Poisson equation, . In the n type semiconductor this rewritten as , (E 1) , where n is the electron density and q is the elementary charge. Here, the gradient of n to the V is the DOS . F or the n type semiconductors , the DOS w ithin a small potential range near the E C can be approximated as, , which results in . T h us equation (E 1) can be simplified to, , and this second order differential equation can be solved by comparing with, which results in the Debye length ( ) of, . (E 2) If we apply Boltzmann's carrier distribution to approximate the charge density as, PAGE 128 128 , , the equation ( E 2 ) result in , which is a well known equation for bulk semiconductors [ 99 ]. states (DOS) properties of MoS 2 . We re derive the Debye length as to quantify the screening length through charge density calculated from linearized method. Rewriting equation ( 6 3) from the main text, t he electron charge density at each i th layer is, , where , , and . The DOS at each layer is , ( E 3 ) and thus, linearized electron charge density can be expressed as, ( E 4 ) . PAGE 129 129 Through self consistent iteration with 1D capacitance model [eq uation ( 6 2) in the main text ] and equation ( E 4 ) , the potential and electron charge density ( ) can be found. Inserting to equation (E 2), Debye length can be found as, . ( E 5 ) PAGE 130 130 APPENDIX F EVALUATION OF 2D THOMAS FERMI MODEL Here we provide the derivation of the two dimensional ( 2D ) Thomas Fermi (TF) model to evaluate the screening characteristics on multi layer MoS 2 . The details can be found in ref 101 . Also, derivations for graphene multilayer can be found in ref 100 . We use parabolic conduction band , located at the K points of the Brillouin zone, with valley and spin degeneracy ( , ) of 2 and in plane effective mass ( ) of 0.52 m 0 . The kinetic energy (per unit area) of each layer of MoS 2 can be expressed as, , where . In the continuum limit, the grand potential for this system is , . ( F 1) is the chemical potential, is the total thickness of the MoS 2 layers. is the interlayer distance between the MoS 2 layer s with a value of 6Ã… as in the main text. The screening properties are obtained by minimizing with respect to the charge density . This leads to a differential equation of, , (F 2) where, . The corresponding boundary conditions are as follows. PAGE 131 131 ( F 3 ) Equation F 2 can be rewritten in an integral form as, , . (F 4) Solving this results in an analytic solution of , which is a function of vertical position at given thickness . is equivalent to the in the main text [eq uation ( 6 3)], except that former is in a continuous limit and later is in a discrete form. ( F 5) PAGE 132 132 APPENDIX G C N In the main text, to calculate the interlayer current, classical approach is used. This method utilizes the DOS of the graphene and MoS 2 layers and an interlayer transport parameter C . The NEGF formalism can also be used for the same purpose. To do so, interlayer transport parameter (we denote as N ) needs to be redefined to suit the formalism. As in the main text C hapter 7 .2. 1 , the Hamiltonian of the device is treated orbital tight binding model, , where , , intra layer nearest neighbor (NN) hopping parameter, t 0 =2.7 eV and NN C to C atomic distance of a=1.42 Ã… . Transmission across the layered device at given energy E is, , is the retarded greens function. A is the total area of the Brillouin zone used for the numerical calculation. The contact self energy needs to be defined in a way that the physical fitting parameter can be introduced. The T able G 1 compares the classical method to the NEGF formalism for convenience of understanding. The term is physically equivalent to the transmission function derive form the NEGF. As the MoS 2 layer is treated as a contact to the graphene channel (Hamiltonian ma trix), interlayer transport parameter is PAGE 133 133 buried inside the self energy (or the broadening matrix). Within the NEGF formalism, the self energy can be expressed in terms of DOS as below. Here, represents t he device the in the main text is , and connection to the classical parameter is as, . . Table G 1. Classical and NEGF transport calculation comparison Classical Approach NEGF approach PAGE 134 134 APPENDIX H EXTRACTION OF INTERLAYER COUPLING FACTOR AND RELATION TO NEGF A transport model for determining the interlayer current for tunneling transistor adopting 2D semiconductors have been proposed [ 111 ]. The procedure was followed to determine the fore mentioned interlayer tunneling parameter C and N of this work . The interlayer current between two 2D semiconductors (top and bottom represented as subscript T and B, respectively) can be computed by the Landauer Buttiker formula [ 112 ] , . ( H 1 ) The ( ) is the valley (spin degeneracy), is the Fermi Dirac distribution function and is the interlayer transmission between the wave state with wave vectors of and . e is the elementary charge. Under assumption of weak interlayer coupling, which has been consistent within this work, can be express in terms of DOS (or the spectral function) and the scattering potential matrix element, as, . The spectral function is, , where is the E k relation of the top and bottom layer. Thus, returning to equation ( H 1 ) results in the interlayer current as [ 111 ], . ( H 2 ) PAGE 135 135 Although here, the scattering potential matrix element , has been expressed as a function of wave vectors, with assumption that and are small compared to the s ize of the Brillouin zone, thus, is also negligible, and the wave function decay exponentially in the interlayer region with decay constant of , the scattering potential matrix can be simplified as, . ( H 3 ) is the area of the 2D semiconductor overlap (or the device area). The is the interlayer distance, equivalent to in the main text. The decay constant , can be expressed in tunneling as, , where is the barrier height (300 meV in the work) and is the parallel momentum of graphene with lattice constant =2.46 Ã… [ 113 ]. The is the power spectrum of the long range random fluctuation which can be further approximated within a unitcell as , where is the correlation length set as 10nm. Thus equation ( H 3 ) can be expressed as wave function independent constant of, . The extracted scattering matrix element can be applied to NEGF calculation via energy dependent broadening matrix. In the NEGF formalism for a 2 te rminal device, the transmission can be expressed as , PAGE 136 136 with . In this work the Hamiltonian of the device is treated with mono layer of graphene orbital tight binding model. The MoS 2 layer is dealt as phenomenological contact, the broadending matrix can be expressed as, with the extracted value of as in the main text, the resulting . 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Castro Neto Strain Induce d Gap Modification in Black Phosphorus PRL 112, 176801 , 2014. PAGE 146 146 BIOGRAPHICAL SKETCH Gyungseon Seol was born in Seoul, Republic of Korea, in 1980. He received his B.S . degree in Department of Electrical and Computer Engineering in 2005 from Hanyang University, Korea. From 2005 to 2007, he worked on his M.S . degree in Seoul National University, Korea . His scope of study at th e period was on optimization and characterization of high electron mobility transistors (HEMTs) for millimeter wave applications. To be specific, Ohmic contact of the HEMT device and its thermal reliability. From 2008, he is pursu ing Ph. D . degree in Department of Electrical and Computer Engineering at University of Florida. His current research topi cs focus on theory, modeling and simulation of nano scale electronic devices. PAGE 1 Bandgap opening in boron nitride confined armchair graphene nanoribbon Gyungseon Seol and Jing Guo Citation: Appl. Phys. Lett. 98, 143107 (2011); doi: 10.1063/1.3571282 View online: http://dx.doi.org/10.1063/1.3571282 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v98/i14 Published by the American Institute of Physics. Related Articles Origin of charge separation in III-nitride nanowires under strain Appl. Phys. Lett. 99, 262103 (2011) ZnO/Sn:In2O3 and ZnO/CdTe band offsets for extremely thin absorber photovoltaics Appl. Phys. Lett. 99, 263504 (2011) Cooperative transition of electronic states of antisite As defects in Be-doped low-temperature-grown GaAs layers J. Appl. Phys. 110, 123716 (2011) Measurement of valence band structure in boron-zinc-oxide films by making use of ion beams Appl. Phys. Lett. 99, 261502 (2011) Contactless electroreflectance study of E0 and E0+SO transitions in In0.53Ga0.47BixAs1x alloys Appl. Phys. Lett. 99, 251906 (2011) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 03 Jan 2012 to 128.227.55.129. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions PAGE 2 BandgapopeninginboronnitrideconÃžnedarmchairgraphenenanoribbonGyungseonSeola andJingGuoDepartmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville,Florida32611,USA Received2December2010;accepted5March2011;publishedonline8April2011 Graphenenanoribbons GNRs haveseizedstronginterest.Recentstudiesshowthatdomainsof grapheneinmonolayerhexagonalboronnitride h -BN canbesynthesized.UsingtheÃžrstprinciple calculationswehavestudiedtheelectronicpropertiesofarmchairGNRs AGNRs conÃžnedbyBN nanoribbons BNNRs .While,H-terminatedAGNRshaveaclosetozerobandgapwiththewidth indexof3 p +2,AGNRsconÃžnedbyBNNRsexhibitaconsiderablebandgap.Thebandgapopening isprimarilyduetoperturbationtotheon-sitepotentialsofatomsatAGNRedges.Atightbinding modelisparameterizedtoconÃžrmthismechanismandenablefuturedevicestudies.Â© 2011 AmericanInstituteofPhysics . doi: 10.1063/1.3571282 SincetheÃžrstdemonstrationofgraphene,theballistic transportpropertyofthematerialhasattractedinterestfor nanoelectronicapplications.1Graphene,havingahoneycomb structureresultsinazerobandgap.Strippedintoafew nanometer-widegraphenenanoribbon GNR ,bandgapcan betunedtoacertainextent,bytheconÃžnementofelectronic wavefunction.2 Ã 5Single-layerhexagonalboronnitride h -BN ,alsoahoneycomblatticestructure,canbeformed intoboronnitrideNRs BNNRs .Duetolargeionicitiesof BandNatoms,BNNRsexhibitqualitativelydifferentpropertiesfromthoseofGNRs,insulatingandmagnetic behaviors.6 Ã 8C,B,andNareallinthesameperiodofthe periodictable,resultinggrapheneand h -BNtohavesimilar latticeconstant.ThismakescombiningBNandgraphene attractiveandhasbeenextensivelystudied.9 Ã 11ComparedtoH-terminatedGNRswhichrequiresbreakingofbonds,BNNR-conÃžnedGNRsformsacontinuous two-dimensionalatomisticlayerwhichdoesnotrequire breakingofbonds.10Thematerialalsoprovideanaturalway ofrealizingadenselypackedparallelarrayofsemiconductingGNRs,whichareconsiderednecessarytoprovidelarge enoughon-currentsfortransistorapplications.Focusingour attentiontoarmchairGNRs AGNRs conÃžnedbyarmchair BNNRs ABNNRs ,weobservedconsiderablebandgap openingin3 p +2categoriesofAGNRsandalterationin bandgaprelationsbetweenthethreeAGNRfamilies.We claimthroughtheworkthatthesedifferentbehaviorsfrom H-terminatedAGNRsoriginatefromthechargeredistributionattheedgesofAGNRsconÃžnedbyBNNRs,andwe excludeedgebondrelaxationeffect. Weconducted abinitio density-functionaltheory DFT calculationswithSIESTAcodesonsetsofH-terminatedAGNRsandAGNRsboundedbyABNNRsforcomparison.12FromnowonwewilldenoteanAGNRconÃžnedbyABNNR asA-Cx BN y,wherethe x and y representsthewidthof AGNRandBNNRportionofthesystem,respectively.Simulationswererunusingthedouble-polarizedbasissetemployingthegeneralizedgradientapproximationmethod.The PerdewÃBurkeÃErnzerhofexchange-correlationfunctionalis adoptedandtheTroullierÃMartinsschemeisusedforthe norm-conservingpseudopotentials.Agridcutoffof210Ry wasusedandtheBrillouinzonesamplingisdonebythe Monkhostpackmeshof k -points 16 4 1 .Figure 1 a showsarandomstructureofanA-Cncc BN nbn 1+ nbn 2used throughoutthework.Thetransportisinthe x direction.The widthofBNNRsoneachsideofGNR, nbn 1and nbn 2,areset inamannerthatthesum, ncc+ nbn 1+ nbn 2iseven,i.e.,foran odd ncc, nbn 1=9,and nbn 2=10,andforeven ncc, nbn 1= nbn 2=10,sothattheunitcellreplicatealongthe y -direction.Total lengthoftheBNNRintheunitcellischosenassothatitis longenoughtoactasaninsulator.TheDFTcalculationresultsofA-C14 BN 20andA-C14 BN 32werecompared. Bandgapofthetwocasesare,0.4016eVand0.4193eV, respectively,resultinginanaround4.5%difference.Thus, largervalueof nbn 1and nbn 2wouldresultinsimilardispersionrelations,meaningthat nbn 1+ nbn 2=19and20functions wellforourpurpose. Figure 2 a showstheDFT-calculatedbandgapofthe A-Cx BN y'sandH-terminatedAGNRs.Theyexhibitqualitativelydifferentelectrostaticfeaturesbetweenthetwoset. First,wewillverifytheedgerelaxation.ThewidthofAGNRsisdeÃžnedbythenumberofdimerlines N andare categorizedintothreegroups, N =3 p , N =3 p +1,and N =3 p +2 p isaninteger .ThebandgapofidealGNRsareinverselyproportionaltothewidth,withall N =3 p +2group remainingazerobandgap.However,inH-terminatedAGNRs,theedgesareterminatedinaratherabruptmanner. a Electronicmail:sworainbow@tec.uÃŸ.edu. FIG.1. Coloronline a SimulatedstructureofA-Cx BN y. x -Axisisthe transportdirection. b SchematicofAGNRportionofA-Cx BN yunitcell. Invertedopentriangle closedtriangle representsCatomperturbedbyadjacentB N ,foranalyticalcalculation. c SchematicofA-Cx BN yunitcell, BandN,affectingthepotentialproÃžleofCatatomicsiteswithintheunit cell.APPLIEDPHYSICSLETTERS 98 ,143107 2011 0003-6951/2011/98 14 /143107/3/$30.00Â©2011AmericanInstituteofPhysics 98 ,143107-1 Downloaded 03 Jan 2012 to 128.227.55.129. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions PAGE 3 Thusfortheatomicbondstoberelaxed,theCÃCbonds paralleltothedimerlineareshorteratthetwoedgesthanthe rest.Theperturbationfromthisgeneratesasmallbandgap openingin3 p +2categoryofH-terminatedAGNRs,which canbeobserveinFig. 2 a .3ForA-Cx BN y's,duetothe similarityinthelatticeconstantofGNRsandBNNRs,we mayexpectdifferent.Figure 2 b comparestheCÃCbonds lengthparalleltothedimeroftheH-terminated14-AGNR andGNRportionoftheA-C14 BN 20.AverageCÃCbond length latticeconstant ofA-Cx BN yandAGNRis1.425Â 4.292Â and1.413Â 4.262Â ,respectively,resultingin onlya0.856% 0.694% difference.Forthebondsatthe edge,H-terminated14-AGNRhave3.4%differenceatwhile ofA-C14 BN 20exhibitonly1.4%.Thus,theeffectofedge bondrelaxationcanbeeliminatedasasourceofbandgap openinginA-Cx BN y's. AnotherpointobservedinFig. 2 a istherelationofthe widthtothebandgap.InH-terminatedAGNRs,thebandgap inthethreegroupsfollowthehierarchyof E3 p +2 E3 p E3 p +1,where Exrepresentsthebandgapofthecategorized group.However,A-Cx BN y's,followsthehierarchyas E3 p +2 E3 p +1 E3 p.TheCatomsclosetothejunctionwill encounterperturbationofon-siteenergywhichweexpect alsobethecauseforbandgapopeningfor3 p +2group. EvidenceoftheperturbationcanbeobservedinFig. 3 , whichshowsthepseudochargedensityofanA-Cx BN yand aH-terminatedAGNR.ForanH-terminatedAGNR,the symmetryofthechargedensityinpreservedwithonlya smallperturbationattheH-terminatededge.AnA-Cx BN yontheotherhand,thechargesredistributeandpopulate highlyatnitrogen,whichiscausedbythelargeionicpotentialdifferencebetweenBandN,whereBhasthehighestand NhasthelowestamongC,B,andN. Foranalyticalpurpose,wecanconstructaneffective HamiltonianmatrixonlyfocusingontheGNRpartofthe A-Cx BN ystructureasintheschematicofFig. 1 b .The differenceconsideredforthiseffectiveHamiltonianmatrix fromHamiltonianmatrixofAGNR,istheperturbedpotentialenergiesontheCatomsadjacenttotheBandN.Here, forsimplicitywewillassumethatthelargeabsoluteon-site energyofBandNwillonlyaffectthenearestCatoms,thus totalfouratoms.Wewillrefertheperturbedpotentialenergiesas Ea Eb asperturbedbyB N .Thentheperturbation wascalculatedonequivalenttwo-laddersystemat k =0.3The bandgapforthe3 p and3 p +1categoryoftheA-Cx BN yis asinEq. 1 . E3 p= E3 p 0+2 Ea+ Eb N +1sin22 p +1 3 p +1+sin23 p 2 p +1 3 p +1, E3 p +1= E3 p +1 02 Ea+ Eb N +1sin2p +1 3 p +2+sin23 p p +1 3 p +2. 1 The E3 p 0and E3 p +1 0arethebandgapforidealAGNRwhich aregivenby, tCC 4cos p/ 3 p +1 2 and tCC 2 4cos p/ 3 p +2 ,respectively.3TheanalyticalcalculationresultsareincludedinFig. 2 a ,markedasgraycrosses. Thecasesfor3 p +2arenotincluded.Duetotheinitialzero bandgapofthe3 p +2systems,perturbationcalculationdoes notholdforthiscategory.Howevershowsperturbationindeedaffectsthebandgaphierarchy. Forbetterunderstandingthemechanism,atightbinding TB modelwasconductedbasedonthe abinitio results. ThoughDFTcalculationprovidesaccuratedescriptionofthe system,itiscomputationallyexpensive.TheTBmodelis alsoacomputationallycosteffectivemethod,whichcanfacilitatefurtherdevicestudiesbasedonthismaterial.The HamiltonianofthismodelcanbeexpressedasEq. 2 and thevaluesoftheparametersusedarelistedinTable I . H =iici +ci i , j ti ci +cj+ cj +ci i , j ti 0 ci +cj+ cj +ci +iEC, i. 2 TABLEI.Tightbindingmodelparameters. A-GNRx/ BNNRyParameters eV CCBNtCCtBNtBCtNCtBC ,20.0151.952.52.92.02.50 tCC ,2tBBtNNtNC ,2PBPN Ã… 000.150.590.750.682.1 FIG.2. Coloronline a Bandgap Eg vswidthplotoftheDFTcalculationofAGNRs thesolidlinewithopensquare andA-Cx BN ys thesolid linewithclosedtriangle ,andanalyticalcalculationsofA-Cx BN y gray cross . b CtoCdistanceofbondsthatareparalleltothedimerlinevs.the atomicindices.DataextractedfromtheDFTcalculationofH-terminated 14-AGNR thesolidlinewithopensquare andA-C14 BN 20 thesolidline withopentriangle .AverageoftheH-terminated14-AGNRisrepresented asdotandA-C14 BN 20asdash.Toppanelisinabsolutescale,distanceunit in Ã… .Bottompanelisinrelativescaleregardstotheaverage. FIG.3. Coloronline Apseudochargedensityplotof a A-Cx BN yand b H-terminatedAGNR.Unitsarein e / Ã…3. 143107-2G.SeolandJ.Guo Appl.Phys.Lett. 98 ,143107 2011 Downloaded 03 Jan 2012 to 128.227.55.129. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions PAGE 4 Indices i and j denotethesite, ci +and ciarecreation,annihilationoperators.Also, i=BN:Siteenergyofboron BN:Siteenergyofnitrogen CC:SiteenergyofcarbonatsublatticeA CC:SiteenergyofcarbonatsublatticeB, ti=tBN:BtoNbondingparameter tCC:CtoCbondingparameter tBC:BtoCbondingparameter tNC:NtoCbondingparameter, ti 0=tBC,2:2ndnearestBtoNbondingparameter tCC,2:2ndnearestCtoCbondingparameter tBB:2ndnearestBtoBbondingparameter tNN:2ndnearestNtoNbondingparameter tNC,2:2ndnearestNtoCbondingparameter.The1stand2ndnearesttightbindingparameterswereintroducedforbetterÃžtting.Also,potentialchangesintheconÃžnedGNRportionduetotheadjunctBandNatomswere takenintoaccount,asinthepreviousanalyticcalculations. BecausetheTBmodelisconstructedwithintheunitcellof thesystem,weassumedthatthepotentialwouldeffectively inÃŸuencenearbyatomsonlyinlateraldirection,asindicated inFig. 1 c .ThefourthtermintheRHSofEq. 2 represents thechangesinpotentialproÃžleattheatomisticsites.Where thevalueof EC, I,is0forABNNRpartanddenotesthedecayingpotentialfromtheBorNatomsaffectingthenearby Catoms,whichcanbeexpressedasEq. 3 intheAGNR part. EC, n= n+ VB n + VN n , 3 VB n = PBBNe dBÃC n/ , 4 VN n = PN BN e dNÃC n/ . 5 InEqs. 4 and 5 ,thepotentialisdeÃžnedinexponential manner,where PBand PNthestrengthofthepotentialand isthedecaylength.5dBÃC n dNÃC n denotesdistancefromthe B N attheedgetothe n thCatom.Figure 4 a plotsand comparesthebandgapresultsof abinitio calculationandTB model,whichÃžtswithin6%range.Figure 4 b depictsfew oftheresultingdispersionrelationsintheregionofinterest. ThustheTBmodelemployingpotentialchangesinthe AGNRportionofA-Cx BN y'sdescribesthesystemwell. Figure 4 c showstheextendedsimulationresultsbasedon TBcalculation.Althoughasthewidthincrease,thebandgap ofbothA-Cx BN y'sandH-terminatedAGNRsdecrease,but theminimalbandgapofA-Cx BN y'sisconsiderablylarger thanthatofH-terminatedGNRs.BothTBcalculationand analyticcalculationmatch. Insummary,wehaveinvestigatedelectronicproperties ofA-Cx BN y'sthroughDFTcalculation.TheedgebondrelaxationthatisevidentinH-terminatedAGNRshasasigniÃžcantlysmallereffectinA-Cx BN y'sduetothesimilarlattice constantoftwomaterialsandthuscanactasstablepassivationmethod.Thesourceofconsiderablybandgapopening forA-Cx BN y'swithwidthindexof3 p +2,anddistinctive widthtobandgaprelationsfromthatofH-terminatedAGNRsisevaluatedbytheperturbationcalculationandveriÃžes thatitoriginatesfromthechangeinthepotentialenergyof theCatomsintheAGNRedgesthatareadjacenttoBandN. ATBmodelofthesystemwasconstructedasasimpleand effectivemethodtopredicttheelectronicproperties. ThisworkwassupportedbyONR,ARL,andNSF.1K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang,Y.Zhang,S.V. Dubonos,I.V.Grigorieva,andA.A.Firsov, Science 306 ,666 2004 .2M.Ezawa, Phys.Rev.B 73 ,045432 2006 .3Y.-W.Son,M.L.Cohen,andS.G.Louie, Phys.Rev.Lett. 97 ,216803 2006 .4M.Y.Han,B. zyilmaz,Y.Zhang,andP.Kim, Phys.Rev.Lett. 98 , 206805 2007 .5Z.Chen,Y.-M.Lin,M.J.Rooks,andP.Avouris, PhysicaE Amsterdam 40 ,228 2007 .6F.Zheng,K.Sasaki,R.Saito,W.Duan,andB.-L.Gu, J.Phys.Soc.Jpn. 78 ,074713 2009 .7F.Zheng,G.Zhou,Z.Liu,J.Wu,W.Duan,B.-L.Gu,andS.B.Zhang, Phys.Rev.B 78 ,205415 2008 .8Z.ZhangandW.Guo, Phys.Rev.B 77 ,075403 2008 .9Y.Ding,Y.Wang,andJ.Ni, Appl.Phys.Lett. 95 ,123105 2009 .10L.Ci,L.Song,C.Jin,D.Jariwala,D.Wu,Y.Li,A.Srivastava,Z.F. Wang,K.Storr,L.Balicas,F.Liu,andP.M.Ajayan, NatureMater. 9 ,430 2010 .11Z.Wang,H.Hu,andH.Zeng, Appl.Phys.Lett. 96 ,243110 2010 .12J.M.Soler,E.Artacho,J.D.Gale,A.GarcÂ’a,J.Junquera,P.Ordejo,and D.SÂ‡nchez-Portal, J.Phys.:Condens.Matter 14 ,2745 2002 . FIG.4. Coloronline a Bandgap Eg ofA-Cx BN y'sasafunctionof width.TheDFTcalculation solidlinewithclosedsquare andthe TBmodelresults dashedlinewithopentriangle arecompared.On theleft y -axisisthebandgapandontheright y -axisistherelativeerrorof thetwocalculations.RelativeerrorisdeÃžnedas, err % = Eg , ab intio Eg , TB / Eg , ab intio. b E k bandplotsofA-C8 BN 20,A-C9 BN 19, A-C13 BN 19,andA-C14 BN 20comparingtheDFTcalculations solidline andTBresults dashedline . c TBcalculatedBandgap Eg asafunction ofwidth,from5to50.A-Cx BN ys solidblackline ,H-terminatedAGNR solidgrayline ,andanalyticalcalculationfromEq. 1 blackopen rhombus . 143107-3G.SeolandJ.Guo Appl.Phys.Lett. 98 ,143107 2011 Downloaded 03 Jan 2012 to 128.227.55.129. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions |