MODELING OF NANOSCALE ELECTRICAL JUNCTIONS AND ELECTRICAL CONTACTS B y Sneha Banerjee A DISSERTATION Submitted to Michigan State University i n partial fulfillment of the requirements for the degree of Electrical Engineering D octor of P hilosophy 2021 ABSTRACT MODELING OF NANOSCALE ELECTRICAL JUNCTIONS AND ELECTRICAL CONTACTS By Sneha Banerjee Nano - scale electrical contacts are essential for next generation electronics. Based on the materials of the contact members and the interfacial layers, these junctions can be of ohmic, Schottky or tunneling type. Nonuniform current distribution and current crowding across electrical contacts lead to nonuniform heat deposition, formation of local thermal hotspots, aggravation of electromigration, and in the worst scenario, lead to thermal runaway and breakdown of the device. C ontact resistance, on the other hand, severely restricts the current flow, an d affects the overall device properties. Devices based on thin film junctions, nanotubes or nanowires, and two - dimensional (2D) materials are especially sensitive to the current transport at electrical contacts, due to their reduced dimensions and increase d geometrical confinement for the current flow. The goal of this thesis is to develop theoretical models to understand, improve , and control current transport and to reduce contact resistance in nanoscale electrical contacts. First, we study the current de nsity - voltage ( characteristics of dissimilar metal - insulator - metal (MIM) nanoscale tunneling junctions using a self - consistent quantum model. Tunneling type contacts are ubiquitous as they can be formed when a thin insulator layer or gap exists betw een two contact ing members. Our model includes electron emissions from both the cathode and anode, and the effects of image charge potential, space charge and exchange correlation potential. The curves span three regimes: direct tunneling, field emis sion, and space - charge - limited regime. Unlike similar MIM junctions, the curves are polarity dependent. The forward and reverse bias curves and their crossover behaviors are examined in detail for various regimes, over a wide range of material properties. It is found that the asymmetry between the current density profiles increases with the work function difference between the electrodes, insulator layer thickness , and relative permittivity of the insulator. This asymmetry is profound i n the field emission regime and is insignificant in the direct tunneling, and space charge limited regimes. Next, we study the current distribution and contact resistance in ohmic, tunneling and two - dimensional (2D) material - based Schottky contacts. We modify the standard transmission line model (TLM) to include the effects of spatially varying specific contact resistivity along the contact length. Both Cartesian and circular (or annular ) contacts are analyzed. The local voltage - dependent along the contact length is calculated self - consistently by solving the lumped circuit TLM equations coupled with th e quantum tunneling model for MIM junctions, or the thermionic emission current injection model for 2D materials . We find that current distribution and contact resistance depend strongly on input voltage, contact dimension and geometry, and material proper ties . We also propose to reduce contact r esistance in 2D - m aterial - b ased e lectrical c ont acts by r oughness e ngineering of the contact interfaces. The results for ohmic contact are verified with finite element method (FEM) based simulations, and the 2D - material based calculations are validated with existing theory and experiments. We further extend this work and demonstrate a method to mitigate current crowding, by engineering the interface layer properties and geometry. We find that current steering and redistribution can be realized by strategically designing the specific contact resistivity along the contact length. We also find that introducing a nanometer scale thin insulating tunneling gap between highly conductive contact members can greatl y reduce current crowding while maintaining similar total contact resistance. iv To Maa and Baba: Lakshmi and Swapan Banerjee v ACKNOWLEDGEMENTS It would not have been possible to write this doctoral thesis without the help and support of the kind people around me. I would like to show my gratitude here. Above all, I would like to thank my parents for their constant support, patience, and encouragement . Their love, care, and uplifting words ha ve always motivated me to move forward . My grateful hugs to all my c ousins, uncles, and aunts who are undoubtedly the warmest people on earth . This thesis would not have been possible without the help, support, patience , advice, and guidance of my advisor, Professor Peng Zhang. During the four years of my graduate study at Michigan State University, Prof . Zhang ha s always been there for me whenever I needed help. I learned a lot from his mentorship. His advice has been invaluable to me on both academic and personal level s . I convey my sincerest gratitude to him. I would also like to thank my doctoral committee members Prof. Verboncoeur, Dr. Luginsland, Prof. Li, and Prof. Fan for their valuable suggestions and time . I thank the faculty and admi nistrative members of the Electrical Engineering department, for all the help and support they provided. I thank my colleagues (my academic family at MSU) Yi, Asif, Patrick, Janez, Yang, Yangyang and De q i. I could not have asked for a more supportive group . My special thanks to Yi and Asif for taking the courses with me, doing the homeworks and preparing for the exams together, and most importantly, for making my journey joyful. I also thank Xenofon and Tanvi from the Electrical Engineering department , for their help with homework s , Qualifier A preparation, and for being wonderful friends . Finally, I thank all my friends who have supported me, believed in me, fed me, reminded me to file tax returns, celebrated the small achievements with me, and listened to my complaints (about vi life). I thank M o yukh, Sampa, Ankita, Lahar, Srivalli, Gouree, Apoorva, Renu, Bharath, Abhijnan, Sudhakar, Abhiroop, Aditya, Affan, Sushant, Shayonee, Chirag, Pradyumna, Anson, Sunanda, Haritha, and Vivek. This work has been supported by the Air Force Office of Scientific Research (AFOSR) YIP Award No. FA9550 - 18 - 1 - 0061 . The Michigan Institute for Plasma Science and Engineering Fellowship ( Fellowship a re also acknowledged. vii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ................... ix LIST OF FIGURES ................................ ................................ ................................ .................. x CHAPTER 1 ................................ ................................ ................................ ............................. 1 INTRODUCTION ................................ ................................ ................................ .................... 1 1.1 Motivation and Background ................................ ................................ ................... 1 1.1.1 Quantum tunneling in metal - insulator - metal junctions ........................... 1 1.1.2 Current distribution and contact resistance ................................ .............. 4 1.2 Prior Works ................................ ................................ ................................ ............. 8 1.2.1 Quantum tunneling in metal - insulator - metal junctions ........................... 8 1.2.2 Contact resistance ................................ ................................ .................. 11 1.3 Organization of this thesis ................................ ................................ .................... 17 REFERENCES ................................ ................................ ................................ ....................... 20 CHAPTER 2 ................................ ................................ ................................ ........................... 30 QUANTUM TUNNELING IN METAL - INSULATOR - METAL NANOJUNCTIONS ....... 30 2.1 Introduction ................................ ................................ ................................ ........... 30 2.2 The Self - Consistent Model ................................ ................................ ................... 32 2.3 Results and Discussion ................................ ................................ ......................... 36 2.4 Concluding Remarks ................................ ................................ ............................. 43 REFERENCES ................................ ................................ ................................ ....................... 45 CHAPTER 3 ................................ ................................ ................................ ........................... 51 PLANAR TUNNELING ELECTRICAL CONTACTS ................................ ......................... 51 3.1 Introduction ................................ ................................ ................................ ........... 51 3.2 The 2D Transmission Line Model ................................ ................................ ........ 54 3.3 Results and Discussion ................................ ................................ ......................... 58 3.3.1 Constant specific contact resistivity along th e contact length ............... 58 3.3.2 Specific contact resistivity varies linearly along the contact length ...... 62 3.3.3 Tunneling contact resistance ................................ ................................ .. 64 3.4 Concluding Remarks ................................ ................................ ............................. 73 REFERENCES ................................ ................................ ................................ ....................... 76 CHAPTER 4 ................................ ................................ ................................ ........................... 82 CIRCULAR TUNNELING ELECTRICAL CONTACTS ................................ ..................... 82 4.1 Introduction ................................ ................................ ................................ ........... 82 4.2 The 2D - Circular Transmission Line Mo del ................................ .......................... 84 4.3 Results and Discussion ................................ ................................ ......................... 87 4.3.1 Constant specific contact resistivity along the contact length ............... 87 4.3.2 Tunneling dependent contact resistivity ................................ ................ 92 4.4 Concluding Remarks ................................ ................................ ............................. 99 REFERENCES ................................ ................................ ................................ ..................... 102 viii CHAPTER 5 ................................ ................................ ................................ ......................... 107 2D - MATERIAL - BASED SCHOTTKY CONTACTS ................................ ......................... 107 5.1 Introduction ................................ ................................ ................................ ......... 107 5.2 The Model ................................ ................................ ................................ ........... 109 5.3 Results and Discussion ................................ ................................ ....................... 111 5.4 Concluding Remarks ................................ ................................ ........................... 120 REFERENCES ................................ ................................ ................................ ..................... 121 CHAPTER 6 ................................ ................................ ................................ ......................... 125 INTERFACE ENGINEERING OF ELECTRICAL CONTACTS ................................ ....... 125 6.1 Introduction ................................ ................................ ................................ ......... 125 6.2 The Model ................................ ................................ ................................ ........... 127 6.3 Results and Discussion ................................ ................................ ....................... 130 6.4 Concluding Remarks ................................ ................................ ........................... 141 REFERENCES ................................ ................................ ................................ ..................... 142 CHAPTER 7 ................................ ................................ ................................ ......................... 148 CONCLUSION AND SUGGESTED FUTURE WORK ................................ ..................... 148 7.1 On Metal - Insulator - Metal Tunneling Junctions ................................ .................. 148 7.2 On Current Distribution and Contact Resistance ................................ ................ 149 REFERENCES ................................ ................................ ................................ ..................... 152 ix LIST OF TABLES Table 4.1 Material parameters for the Cu - insulator - Metal contacts. ................................ ............ 98 x LIST OF FIGUR ES Figure 1.1 High - speed transistors circuit made from carbon nanotubes. (a) Structure of a carbon nanotube transistor. CNTs, carbon nanotubes; SAM, self - assembled monolayer. (b) Optical image of nanotube transistors circuit fabricated on a flexible polyimide substrate, and (c) a scanning electron microscopy image of the nanotube thin film used in these transistors and circuits. [26] (d) A typical CNT - CNT contact present in the thin film. ................................ ................................ ...... 3 Figure 1.2 A schematic scanning tunneling microscope setup. When a tip is brought several angstroms away from a sample and a voltage is applied between them, a very small current flows between the last atom of th e tip and the sample. As the tip is scanned over the surface, image of the surface is recorded with atomic spatial resolution [34]. ................................ ................................ ................................ ........ 3 Figure 1.3 Current crowding at a metal - semiconductor contact. ................................ .................... 4 Figure 1.4 Atomic Force Microscopy (AFM) topography images of a device before (a) and after (b) device failure. Current crowding induced thermal hotspot in two - dimensional black - phosphorus field - effect transistors. [55] ................................ ....................... 5 Figure 1.5 Enlarged scanning electron microscope (SEM) images of flip chip solder joints showing pancake - type of void formation due to current crowding induced el ectromigration. [51] ................................ ................................ ............................. 6 Figure 1.6 A set of three scanning electron microscope (SEM) images of the initiation and growth of a whisker at the upper right corner a solder joint as a result of current crowding induced electromigration. [51] ................................ ................................ 7 Figure 1.7 - insulator - metal (MIM) junctions [68]. is the bias voltage, D is the insulator layer thickness, is the work function of the metal electrodes, is the electron affinity of the in sulator, is the Fermi level energy, and is the potential barrier. ............... 9 Figure 1.8 (a) Electrical contact and (b) its TLM. In layer is sandwiched between Regions I and II. is the specific contact resistivity. [84] ................................ ................................ ................................ ...... 12 Figure 1.9 Normalized potential under a contact versus as function of , and . [35] ................................ ................................ ...................... 13 Figure 1.10 Normalized contact resistance as a function of for various (Fig. 1.8a). [84] ................................ ................................ ................................ .............. 15 xi Figure 1.11 Modified transmission line model (TLM) including the non - linear characteristics of the contact. [102] ................................ ................................ ................................ .. 16 Figure 2.1 Dissimilar metal - insulator - metal (MIM) tunneling junction. The metal electrodes have equilibrium Fermi level and work function and (in these schematics we assume ). , where is electron affinity of the insulator. The insulator thin film thickness is . The applied voltage bias is . The current densities emitted from the electrode 1 and 2 into the gap are and , respectively. (a), (c) reverse bias ( is positively biased) condition; (b),(d) forward bias ( is negatively biased) condition. (a), (b) represent low and (c), (d) represent high bias voltage conditions. [1] .................. 32 Figure 2.2 ( a ) N ormalized (in terms of CL law) current density as a function of applied gap voltage , for two electrodes having work functions, and (Au) , separated by 1 nm vacuum gap ( = 1, ). Metal 2 is positively biased. The calculations are from SCM with both space charge and [28] , Fowler - Nordheim (FN) law (dash - dotted line) is from Ref. [51] [53] , calculated with the cathode work function = , and the quantum CL law (green dotted line) is from Ref. [32], [33] . (b) Current density in as a function of applied gap voltage , for = 1 nm and vacuum gap ( =1, ) . Solid and dashed lines in (b) represent RB and FB conditions respectively. Top to bottom, = 3.68 eV (Mg), 4.08 eV (Al), 5.1 eV (Au), 6.35 eV (Pt). The work function difference between the two metals is kept fixed, = = 1 eV. The inset in (b) represents the zoomed in view of the cross over behavior for the case of eV . [1] ................................ ................................ ........................... 37 Figure 2.3 The effects of work function difference on the characteristics of a dissimilar MIM junction with = 1 nm, vacuum gap ( =1, ). T op to bottom, = 4 eV, 3 eV, 2 eV, 1 eV, 0 eV, - 1 eV, - 3 eV. The work function of metal 2 is kept fixed, = 6.35eV (Pt). Solid and dashed lines represent RB and FB conditions respectively. [1] ................................ ................................ ............. 40 Figure 2.4 The effects of gap width ( ) o n the characteristics of a dissimilar MIM junction with vacuum gap (a) as a function of applied gap voltage . Top to bottom, = 0.5 nm, 1 nm, 1.5 nm, 2 nm, 3 nm. (b) as a function of gap width for different . Work function of the two electrodes are and . For the vacuum gap = 1 and . Solid and dashed lines represent RB and FB conditions respectively. ................................ ...................... 41 Figure 2.5 The effects of relative permittivity of the insulating thin film on the characteristics of a dissimilar MIM junction with = 1 nm. (a) a function of applied gap voltage . Top to bottom, and respectively . (b) as a function of for different . Work function of the two electrodes are and . Electron affinity of the insulator is . Solid and dashed lines represent RB and FB conditions respectively. ................................ . 42 xii Figure 3.1 A parallel, partially overlapped electric contact. The contacts are formed between (a) nanotube or nanowire 1 and 2, and (b) thin film 1 and 2; (c) side view of the contact; (d) its transmission line model. In (a), (b) and (c) a thin resistive interface layer (or a tunneling la yer of permittivity ) is sandwiched between the two contacting members. ................................ ................................ ............................. 54 Figure 3.2 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, ,for differ ent values of specific contact resistivity , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] ................................ ................................ .......................... 59 Figure 3.3 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , and (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, , for different values of , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] ................................ ................................ ................................ ....... 60 Figure 3.4 Normalized contact resistance of the parallel contact (Fig. 3.1). as a function of (a) no rmalized specific contact resistivity, and (b) normalized sheet resistance of contacting member 2, . Dashed lines are for Eq. (3.9), the limiting case of . The cross symbols are from COMSOL[46] 2D simulations. T he length and height of both upper and lower contacting members are assumed to be 20 nm and 10 nm respectively, and the thickness of the resistive interfacial layer is assumed to be 0.5 nm. The resistivities of the upper and lower contact members are in th e range of , and the resistivity of the interface layer is in the range of . [1] .................. 61 Figure 3.5 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , and (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, , for linear specific contact resistivity with different linear constant , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] .................... 63 Figure 3.6 Normalized contact resistance of the parallel contact (Fig. 3.1) with linear specific contact resistivity , for various value of . [1] 64 Figure 3.7 Similar material CNT - vacuum - CNT parallel tunneling contacts. (a) Voltage drop across the contact interface , and (b) tunneling current density across the contact interface for different contact len gth , with fixed V, and D = 0.5 nm; (c) and (d) for different , with fixed V and L = 50 nm; (e) and (f) for different applied voltage with fixed 0.55 nm, and 50 nm. All the material properti es are specified in the main xiii text. Solid lines are for self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical calculations from Eq. 3.7 with calculated using in Eq 3.10 and ohmic appro ximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] ................ 68 Figure 3.8 The total contact resistance of the CNT - vacuum - CNT parallel contact. Contact resistance is plotted as a function of (a) contact length, L , for different insulating layer thickness, D , (b) D , for different L , for a fixed applied voltage, V ; (c) and (d) applied voltage for different L and D respectively, in CNT - vacuum - CNT contacts. Solid lines are for self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical calculations from Eq. 3.8 with calculated using in Eq 3.10 and ohmic approximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] ................................ ................................ ................................ ..... 69 Figure 3.9 Dissimilar material CNT - insulator - metal parallel tunneling contacts. (a) Voltage drop across the interfacial insulating layer , and (b) tunneling current density , in CNT - insulator - Metal contacts, for fixed D = 0.5nm, L =50nm, V and different contacting metals (Ca, Al, Cu, Au). (c) , and (d) , in CNT - insulator - Al contacts, for different insulating layer permittivity , with fixed D = 0.5nm, L =50nm, V. Solid lines are for self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical calculations from Eq. 3.7 with calculated using in Eq 3.10 and ohmic approximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] ................................ ................................ ...................... 71 Figure 3.10 The total contact resistance of the CNT - i nsulator - metal parallel contact. Contact resistance is plotted as a function of (a) contact length L , (b) insulator layer thickness and (c) insulator layer permittivity , for CNT - insulator - metal contacts for different contacting metals (Ca, Al, Cu, Au). (d) Contact resistance as a function of work function of contacting member 2 ( ). The material properties and dimensions for (a) - (c) are specified in the text (the same as in Fig. 3.9). For (d), the resistivity of contacting member 2 is assumed to be . The results are from the self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10. [1] ................................ ................................ ............................ 73 Figure 4.1 Electrical contac t between two circular thin films: (a) top view; (b) side view; (c) its transmission line model. In (a), (b) and (c) a thin resistive interface layer (or a tunneling layer of permittivity ) of thickness D is sandwiched between the two contacting member s. The thicknesses of thin film 1 and 2 are and , respectively. [1] ................................ ................................ ................................ ..... 84 Figure 4.2 Normalized voltage drop across the contact interface , along the radial direction of an annular contact with uniform contact resistivity, for different values of (a) inner radius to outer radius ratio , with and , (b) specific contact resistivity , with and , xiv (c) sheet resistance ratio , with and . All the quantities are in their normalized forms defined in Sec. 4.2. ................................ .......................... 88 Figure 4.3 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of inner radius to outer radius ratio for different (a) normalized sheet resi stance of contacting member 2, , and (b) normalized specific contact resistivity . The dotted lines in (a) are calculated from Eq. (3.8) of Chapter 3, that is, for Cartesian parallel electrical contacts. Dashed lines are for Eq. (4.10), the li miting case of ................................ ................................ ................................ . 89 Figure 4.4 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of norm alized specific contact resistivity for different (a) normalized sheet resistance of contacting member 2, , and (b) inner radius to outer radius ratio . In (a), = 0.1, and in (b), =1. Dashed lines are for Eq. (4.10), the limiting case of The black dotted line in (b) The dotted lines in (a) are calculated from Eq. (3.8) of Chapter 3, that is, for Cartesian parallel electrical contacts. ................................ ................................ ................................ ................ 90 Figure 4.5 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of sheet resistance ratio for different (a) normalized specific contact resistivity and (b) inner radius to outer radius ratio . In (a), = 0.1, and in (b), = 1. Dashed lines are for Eq. (4.10), the limiting case of The black dotted line in (b) is calculated from Eq. (3.8) of Chapter 3, that is, for Cartesian parallel electrical contact s. ................................ ................................ ................................ . 90 Figure 4.6 Tunneling current density across the contact interface for different (a) input voltage , with fixed , , and ; (b) inner radius to outer radius ratio , with fixed , , and ; (c) outer radius , with fixed , , and ; (d) interfacial layer thickness , with fixed , , and . All the material properties are specified in the main text. Solid lines are for self - consistent numerical calculations using Eqs. (4.4), (4.5), and MIM quantum tunneling formulations (Chapter 2, Se c. 2.2) [5]. Dashed lines are for analytical calculations from Eq. (4.8) with being constant, calculated using in the MIM quantum model. ................................ ................................ ..................... 93 Figure 4.7 The total contact resistance across the Cu - vacuum - Cu contact interface as functions of input voltage for different (a) outer radius , with fixed an d ; (b) inner radius to outer radius ratio , with fixed and . as functions of for different (c) input voltage , with fixed and ; (d) interfacial layer thickness , with fix ed and . All of the material properties are specified in the main text. Solid lines are for self - consistent numerical calculations using Eqs. (4.4), (4.5), and MIM quantum tunneling formulations (Chapter 2, Sec. 2.2) [5], [6]. The black dotted line in (a) is calculated using Simmons tunneling current formula [5], [14], [15] and Eqs. (4.4), (4.5) for . Dashed lines are for xv analytical calculations from Eqs. (4.8) and (4.9) with calculated using in the 1D MIM tunneling model. ................................ ................................ .......... 96 Figure 4.8 Tunneling current density across the Cu - insulator - Metal contact interfaces for different (a) contacting metals with work functions for fixed , , , , and ; (b) insulator layer permittivity for fixed , , , , and . The to tal contact resistance across the Cu - insulator - Metal contact interfaces as functions of inner radius to outer radius ratio for different: (c) for fixed , , , and ; (d) , for fixed , , , and . All of the material properties are specified in the main text. ................................ ................................ ............... 99 Figure 5.1 (a) A typical parallel contact between Au and MoS 2 . (b) its transmission line model. The width (transverse dimension) of the two contact members is . ................. 109 Figure 5.2 (a) Voltage drop , (b) current density , and (c) specific contact resistivity across the contact interface for a monolayer (2D semiconductor) and gold (3D metal) contact for different carrier injection time with fixed applied voltage , and contact length 20 nm. Here, , , and . [1] ................................ ................................ ................................ ........................ 111 Figure 5.3 Contact resistance as a function of applied voltage for different , for (a) eV, (b) eV. as a function of temperature for different , for (c) eV, (d) eV. Here, the contact is between a monolayer and Au, with , and 50 nm. [1] ........................ 112 Figure 5.4 (a) Voltage drop , (b) current density , and (c) specific contact resistivity across the contact interface for /Ag con tact for different contact length with fixed applied voltage . Solid lines are for the self - consistent calculations of Model (A), and dashed lines are for Model (C). See text for details. Here, , , , and . [1] ................................ ... 113 Figure 5.5 Contact resistance for /Ag contact as a function of contact length for different applied bias (a) 0.1 V, (b) 0.3 V, (c) 0.60 V, and (d) 0.65 V. The two bias voltages in (c) and (d) are the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2030 and 2021, respectively. Solid lines are for the self - consistent calculations of Model (A), dotted lines are for Model (B), dashed lines are for Model (C), and dash - dotted lines are for Model (D). See text f or details. Here, , , , and . [1] ................................ ................................ ........... 115 xvi Figure 5.6 as a function of 2D carrier density for different temperatures for (a) Au contacts, and (b) In contacts. Crossed symbols are from experiments [10], [13], [14]; solid lines are from our self - consistent model from Eqs. 3.5, 3.6 and 5.2, Model (A); and dashed lines are from Eqs. (3.5), (3.6) with the Richardson - Dushman injection model, Model (C). In the calculation, we used , , for contacts [13], and for contacts [10]. The parameters [31], [10] , [31] [10] [13], and are used to fit the experimental results in Refs. [14], [10] , [13] respectively. For different cases, from top to bottom, different input voltages and are used in (a) and (b) respectively. [13] is assumed for all the cases. [1] ................................ ................................ .......... 116 Figure 5.7 as a function of temperature with (a) an increasing trend, and (b) a decreasing trend, for metal contacts with different . Crossed symbols are from exper iments[10], [13], [14]; solid lines are from our self - consistent Model (A), and dashed lines are extracted from model calculations in Ref. [13]. In the calculation, we used , and . The parameters [31] , and [13] are used for contacts respectively. In (a), for the three solid red lines, from top to bottom, , and all with In (b), and for the blue and purple solid lines, respetively. The value of [13], and [13] is assumed for all the cases. [1] ................................ ................................ .............. 117 Figure 5.8 (a) Roughness in Schottky barrier height , the resulting (b) current density , and (c) specific contact resistivity across t he contact interface for a monolayer MoS 2 - Au 2D/3D contact for different standard deviations (sd). (d) Contact resistance as a function of surface roughness (standard deviation/ ) for different mean values of . Here, applied voltage , and contact length . [1] ................................ ................................ ......................... 118 Figure 6.1 Electrical contact between contact member 1 and 2 for different electrode geo metry. (a) Electrical contact with uniform contact interface, (b) electrical contact with a spatially varying engineered interfacial layer, which is used to control the voltage and current distribution. ................................ ................................ ...................... 127 Figure 6.2 Electrical contact between two contacting members in (a) Cartesian, (b) circular geometry. (c), (d) its corresponding transmission line model. In (a) and (b), a thin interface layer (ohmic, Schottky, or tunneling type) is sandwiched between the two contact ing members. The thicknesses of thin film 1 and 2 are and , respectively. ................................ ................................ ................................ ........ 128 xvii Figure 6.3 Engineered ohmic contact in Cartesi an geometry (Fig. 6.2a) with specific contact resistivity . (a) Contact current density along the contact length for different values of ; (b) contact resistance as a function of for differ ent contact length . The input voltages is the required industry standards according to the International Roadmap of Devices and Systems (IRDS)[30] for year 2030. The thickness of both Cu contact members are , with a resistivity of [36], which gives sheet resistance . Contact length , and the width (transverse dimension) of the contact members . ............ 132 Figure 6.4 Ohmic contacts in circular geometry (Fig. 6.2b) with linearly varying specific contact resistivity. (a) Contact current density along the contact length for different values of linear constant . (b) Contact resistance as a function of for different sheet resistance ratio . Here, we use . In (a) . The input voltage is the required industry s tandards according to the International Roadmap of Devices and Systems (IRDS)[30] for year 2030. The contact member 1 is assumed to be copper (Cu) with sheet resistance [13], [36], outer radius of the upper contact member , and the inner radius of both the contact members . ................................ ................................ ........................... 133 Figure 6.5 Engineered Schottky contacts in Cartesian geometry. (a) Schottky barrier height , (b) the corresponding contact current density along the contact length for contacts for different values of , and (c) contact resistance as a function of for different input voltage . D ashed lines are for contacts with uniform [37]. The Fermi level [37], and carrier injection time . The bias voltage in (b) is the required industry standards according to the International Roadmap of Devices and Systems (IRDS)[30] for year 2021. Here, , , and . .................. 134 Figure 6.6 Tunneling type electrical contacts. (a) Contact current density , and (b) specific contact resistivity along the contact length for Cartesian tunneling contacts. Solid lines are for se lf - consistent numerical calculations using Eqs. (3.5) and (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29], for different values of gap distance D and work function of contact members . Sheet resistance of both the contact members is assumed to be . Dashed lines are calculated analytically with constant calculated using in the 1D MIM tunneling model. Black dotted lines are for an ohmic contact with , analyticall y calculated from the TLM equations. is the total contact resistance. ................................ ............. 136 Figure 6.7 Tunneling type electrical contacts. (a) Contact current density , and (b) specific contact resistivity along the contact length for Cartesian Cu - vacuum - Cu t unneling contacts. Solid lines are for self - consistent numerical calculations using Eqs. (3.5) and (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], xviii [29], for different values of gap distance D . Dashed lines are calculated analytically with c onstant calculated using in the 1D MIM tunneling model. Black dotted lines are for an ohmic contact with , analytically calculated from the TLM equations. ............................. 139 Figure 6.8 Contact resistance as a function of (a) input voltage , and (b) contact length for Cartesian Cu - vacuum - Cu tunneling contacts. Solid lines are self - consistent numerica l calculations using Eqs. (3.5), (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29], for different values of gap distance D. Dashed lines are calculated analytically with constant calculated using in the 1D MIM tunneling mo del. Black dotted lines are for an ohmic contact with , analytically calculated the TLM equations . Crossed points in (a) are for the four cases shown in Fig. 6.7. ................................ ........... 140 1 CHAPTER 1 INTRODUCTION Nanoscale electrical junctions are prevalent in a diverse set of electronic devices. They are naturally formed in transistors [1], [2] , scanning tunneling microscope s [3], [4] , thin film contacts, and two - dimensional (2D) material, nanowire, nanofiber, or nanorod based novel devices [5] [7] , as shown in Figures 1.1 and 1.2 below. Based on the materials of the contact members and th e interfacial layers, these junctions can be of ohmic, Schottky or tunneling type. Tunneling type contacts are especially common where the contacting members are separated by very thin insulating layers [8] [10] . The objective of this work, and the motivation behind, are discussed in detail in the following section. 1.1 Motivation and Background This thesis theoretically studies the current transport in nano - scale electrical junc tions and electrical contacts. First, we focus on the quantum tunneling induced electron transport in metal - insulator - metal junctions. Next, we study the effects and parametric dependence of current distribution and contact resistance for ohmic, tunneling, and Schottky type contacts. The objective is to better characterize electrical contacts and to optimize the current flow in such electrical junctions by improving controllability. 1.1.1 Quantum tunneling in metal - insulator - metal junctions Quantum tunnelin g phenomenon, which becomes important in nano scale junctions and circuits, imposes some serious challenges to the modern - day electronics. Due to the ever - increasing demands for physical scaling down, e lectrodes in the scale of 10 nm or sub - 10 nm are 2 commo n these days in silicon industry [1], [2], [11], [12] . The circuits in this range are so small that quantum mechanical effects become critical and cause device malfunction. On the other hand , the charge transport through tunnel junctions are utilized to develop novel devices that offer several advantages over the shortcomings of scaling . Next - generation transistors, such as, tunnel field - effect transistors (TFETs) [13], [14] , single electron transistors (SET s ) [15] [18] , and graphene - based field effect transistors, rely on quantum tunneling for their operation. TFETs, which outperform the traditional Si transistors at low by modulating the quantum tunneling current [14] . Recently, researchers were able to develop a highly sensitive detector for terahertz (THz) frequency utilizing the quantum tunneling effect in a graphene based TFET [13], [19] . Ultrasensitive detectors based on SETs are also attracting great attention [20] [22] . Nanoscale devices based on quantum tunneling principle s are expected to become increasingly important in future electronics industry . T o enable the practical use of such devices, a comprehensive study of the quantum current transport is n ecessary . Tunnel ing resistivity is one of the major obstacles for the development of low dimensional material - based devices. The performance of tr ansistors based on carbon nanotubes (CNTs), carbon nanofibers (CNFs), and graphene greatly depend on the tunneling current. In these transistors, parallel arrays of nanowires or dense networks of nanotubes (c.f. Fig. 1.1) are used for the channel material. The current transport is critically dependent upon the tunneling resistivity between these nanostructures [8], [23] [25] . An example is shown in Fig. 1.1 for illustration. Using carbon nanotube network channels, Tang and colleagues fabricated high - speed flexible CMOS ICs that offer sub - 10 ns stage delays [26] . They found on - state current of the transistors and thus the speed of the device, relied on electron tunneling between the nanotubes . To realize the excellent electrical 3 properties of novel low - dimensional materials on the circuit level and to develop flexible electronics based on them, contact engineering is crucial [27] . Tunneling electron emission through vacuum nanogap is also common in miniaturized vacuum and plasma electronic devices [28] [32] and plasmonic nanogaps [33] . Many new technologies combine the advantages of ballistic transport through vacuum with the scalability, reliability, and low cost of silicon technology [28] . Study of the tunneling induced charge transport across nanometer length scale is critical for the development of such technologies . F igure 1. 1 High - speed transistors circuit made from carbon nanotubes. (a) Structure of a carbon nanotube transistor. CNTs, carbon nanotubes; SAM, self - assembled monolayer. (b) Optical image of nanotube transistors circuit fabricated on a flexible polyimide substrate, and (c) a scanning electron microscopy image of the nanotube thin film used in these transistors and circuits. [26] (d) A typical CNT - CNT contact present in the thin film. F igure 1. 2 A schematic scanning tunneling microscope setup. When a tip is brought several angstroms away from a sample and a voltage is applied between them, a very small current flows between the last atom of the tip and the sample. As the tip is scanned over the s urface, image of the surface is recorded with atomic spatial resolution [34] . 4 F igure 1. 3 Current crowding at a metal - semiconductor contact. 1.1.2 Current d istr ibution and c ontact r esistance Current flow in an electrical circuit is usually non - homogeneous. Localized increase of current density or the current crowding effect [35] [38] is a serious and persistent problem in the electronics industry. Current crowds near a bend or a constriction, and it is especially strong at the vicinity of contact edges [39] [43] . An example of current crowding in a typical metal - semiconductor contact is shown in Fig. 1.3. Due to the resistivity mismatch between the contact members, the current transport is confined only near the front edge of the contact structure. Since Joule heating is proportional to the square of current densi ty [44] , current crowding leads to non - uniform heat generation in the contact area. On the other hand, the excessive amount of Joule heating deposited at the contact region b ecause of the large contact resistance is another critical concern of very - large - scale - integrated (VLSI) circuit engineers [45] [47] . There are various factors that can increase the total contact resistance, such as, fo rmation of oxide layers between the contact members, dielectric coating of the electrodes, presence of surface roughness , etc. The individual or combined effects of current crowding, non - uniform Joule heating, and contact resistance are responsible for abo ut 40% of all electrical/electronics failure, ranging from small scale modern consumer electronics, like, hand - held or wearable devices, personal computers etc. 5 to large scale space vehicles, particle accelerators, nuclear facilities, and military systems [28], [43], [48] [54] . F igure 1. 4 Atomic Force Micro scopy (AFM) topography images of a device before (a) and after (b) device failure. Current crowding induced thermal hotspot in two - dimensional black - phosphorus field - effect transistors. [55] In modern semiconductor industry, the contact problems have become more prominent nowadays with the growing demands for advanced computation, high speed, and high packing density. 2D materials, such as, molybdenum disulfide (MoS 2 ), black phosphorus, boron nitride, graphene have been demonstrated to be excellent channel materials for ultrathin field - effect transistors [2], [31], [55], [56] . However, the current crowding effect and the unusually high contact resistance at the 3D metal and the 2D semiconductor interfaces [55] [58] hinder the development of such electronics. Fig ure 1.4 shows current crowdin g induced break down in two - dimensional black - phosphorus field - effect transistors [55] . 6 To reduce cost and enhance performance, engineers are developing technologies that are shifting towards three - dimensional (3D) integrated circuits (ICs), where the dies are stacked on top of each other [51] . Both horizontal and vertical interconnects are used. In such densely packed structures, the power density is si gnificantly increased, albeit with limited choices of dissipation options. Heat dissipation is especially difficult for mobile units [51] . This leads to increase of the circuit temperature. Current crowding makes the situation worse by heating the junction nonuniformly. The thermal gradient at the contact area may result in thermal crossta lk and thermomigration. Interconnect junctions in 3D ICs, such as, flip chip joints and solder bumps, suffer from electromigration [51], [59] [63] which moves atoms based on the flow of current through a material. In high current density region, the generated heat br eaks atoms from the connections resulting in open - circuits, whi le the deposits or hillocks can grow and eventually close circuit connections resulting in short - circuits. Divergences in atomic flux, induced by current crowding, accelerates this process. These issues cause serious safety and reliability concerns [51], [63] . F igure 1. 5 Enlarged scanning electron microscope (SEM) images of flip chip solder joints showing pancake - type of void formation due to current crowding induced electromigration. [51] 7 F igure 1. 6 A set of three scanning electron microscope (SEM) images of the initiation and growth of a whisker at the upper right corner a s older joint as a result of current crowding induced electromigration. [51] Furthermor e, t he combined effect of electromigration and Joule heating can lead to catastrophic burn - out type failure [51] . Localized void formation at the copper (Cu) interconnects lead s to thinning of the conductor which increases the resistance and the corresponding Joule heating. Due to the high packing density and poor heat dissipation, the increased Joule heating increases the temperature, which in turn increases the electromigration rate. Electromigration, on the other hand, reduces the thickness of the electrode, further increasing the resistance and Joule heating. This positive feed - back causes thinning of Cu lines in the 3D integrated chips, builds up mechanical stress, and eventu ally, causes burn - out failure [51] pancake - void formation and - shown in Fig. 1.5 (Fig. 11 of Ref. [51] ) and Fig . 1.6 (Fig. 12 of Ref. [51] ), respectively. In addition to electromigration, strong c urrent crowding effect can also lead to localized overheating and formation of hotspots [55] . Several theoretical and experimental studies have found that the hotspots are usually formed at the contact edges. Figure 1.4 (Fig. 4 of Ref. [55] ) shows a failed device with a broken region at the inner contact edge after a prolonged operation at high voltage. The failure is often caused by the typically increases with temperature which causes more heating of the junction, which further 8 increases the junction temperature, in a positive feedback loop. If a circuit produces more heat than the heatsin k can dissipate, then thermal hotspots are formed leading to component breakdown, or in the worst - case scenario, a small explosion [51] . In high power microwave sources and pulsed power systems, poor electrical contact also prevents efficient power coupling to the load [52] , produces unwanted plasma [64] , and in the worst - case scenario, damages the electrodes and circuits. Therefore, a comprehensive and systematic modeling of the current transport and electrical contacts in nano scale electrical junctio ns is necessary to further advance modern electronics. 1.2 Prior Works 1.2.1 Quantum tunneling in metal - insulator - metal junctions The work on quantum tunneling started as early as 1926, when Schrodinger published his landmark equation for wave function i n quantum mechanics [65] . The same year, Wenzel, Kramers , and Brillouin developed a semiclassical method (WKB method) for finding approximations to the one - dimensional time independent Schrödinger equati on [66], [67] . This WKB method is widely used to calculate transmission and reflection coefficients through a smooth and slowly varying potential barrier. In 1933 Sommerfeld and Bethe published theoretica l study of tunneling in metal - insulator - metal (MIM) junctions for very low and high voltages using WKB approximation [68] . In 1935 Holm extended the theory to include intermediate voltages [69] . The challenge for this kind of study was to determine the actual shape of the potential barrier in the vacuum gap or insulator. 9 F igure 1. 7 - insulator - metal (MIM) junctions [68] . is the bias voltage, D is the insulator layer thickness, is the work function of the metal electrodes, is the electron affinity of the insulator, is the Fermi level energy, and is the potential barrier. Sommerfeld and Bethe first derived equations for the current density transmitted by a trapezoidal barrier (c.f. Fig. 1.7), where is the bias voltage, D is the insulator layer thickness, is the work function of the metal electrodes, is the electron affinity of the insulator, and is the Fermi level energy [68] . However, image charge potential rounds off the corners of a trapezoidal barrier and increases the flow of current between the electrodes. To obtain an analytic solution, Sommerfeld and Bethe approximated the barri er by a symmetric parabola. Later, Holm and Kirschstein, using the same method, improved upon the results of Sommerfeld and Bethe by using a symmetric parabola that was a closer fit to the potential barrier [69] . In 1963, Simmons modified the shape of the barrier and improved accuracy [9] . Tunneling ef fects between electrodes separated by thin insulating films have been studied extensively by Simmons [9], [70] in 1960s. His formulas have since been widely used for evaluating tunneling current in MIM junctions. From his study he concluded: a) tunneling current increases exponentially when gap distance decreases, b) for low voltages, MIM junctions can 10 be approximated as ohmic, c) tun neling current is polarity dependent for dissimilar electrodes. The tunneling current in Al - - Al structures have been experimentally studied and evaluated using [71] . Although widely used, formulas are derived by considering only the emission process from the electrodes, where the effects of image charge are consid ered, but the electron space charge potential and the electron exchange - correlation potential inside the insulator thin films are ignored . b) This model is reliable only in the low voltage regime for limited parameter space [10] . c) It fails to predict the maximum possible tunneling current in a metal - insulator - metal junction. There are several theoretical [72] [74] and experimental [75], [76] studies on space charge effects in a vacuum nanogap. Child - Langmuir (CL) law gives the space - charge limited current (SCLC) in a plane - parallel vacuum diode. This classical value for the limiting current can be exceeded by a large factor in nanoscale vacuu m gap because of tunneling. The new limit is referred as the quantum CL (QCL) law [72], [73] . The effects of exchange correlation potential in a vacuum nanogap have also been studied systematically [77] . In 2015, a general scaling law for the quantum tunneling current in nano - and sub - nanoscale MIM junctions has been developed by self - consistentl y solving the coupled Schrödinger and Poisson equations [10] [10] was for mulated for similar electrodes. It includes the effects of space charge and exchange - correlation potential, as well as current emission from both electrodes. The current - voltage (J - V) characteristics has three distinct regimes: a) the direct tunneling regi me, where it follows [9] , b) field emission regime, where it becomes close to Fowler Nordheim law [78] , and c) space charge limite d (SCL) regime, where it approaches quantum Child Langmuir law [72], [73] . 11 metal electrodes [79] . The rectification properties of such junctions have been studied for various materials and input parameters [79] . 1.2.2 Contact resistance Different theoretical models have been developed over time to characterize micro and nanoscale thin film - based contacts. In these thin - film contacts, the current flow lines bend sharply in the immediate vicinity of the contact edges [50] . Hall [80], [81] used conformal mapping techniqu e to investigate the two - dimensional (2D) thin film resistance for various patterns in Cartesian geometry. Denhoff [82] studied the spreading resistance of a round thin film contact by solving Laplace equation. Zhang [50] Cartesian thin - film contacts including the effects of dissimilar materials. By using Fourier series analysis, Zhang and Lau [83] in 2010, derived simple analytical scaling laws for the total resistance for arbitrary values of dimensions and resistivities. The models were then extended to horizontal [39], [84], [85] and vertical [38] type thin film contacts. Current crowding has been comprehensively studied by calculating the current flow patterns [28], [37], [85] . Kennedy and Murley [86] , in 1968, investigated the electrical properties of the diffused semiconductor resistor using a two - dimensional mathematical analysis. They calculated the constant voltage contours in the ohmic contact region of a diffused silicon resistor and found that only a small portion of the contact actively contributes to the electrical properties of the structure. The potential distribution is concentrated at the leading edge of the contact, resulting in extensive current crowding. They concluded that increasing the length of the ohmic c ontact has negligible influence on the current density at the metal - semiconductor interface. 12 F igure 1. 8 interface layer is sandwiched between Regions I and II. is the specific contact resistivity. [84] On the other hand, simple transmission line model (TLM) theory, in plenty of variant or extended forms, has been widely used for analyzing metal - semiconductor planar contacts. In 1969, Murrmann and Widmann [36] used a simple steady state TLM to characterize planar metal - semiconductor contacts. Although the formulation is simple, it can extract important contact characterization parameters, such as, contact resistance in ohm , semiconductor sheet resistance beneath the contact ) , and specific contact resistivity in , as shown in Fig. 1.8. is an important parameter in contact characterization , where is the contact current density. Berger [87] , in 1972, provided a thorough cha racterization of contact resistance and contact resistivity. The governing equations of the structure shown in Fig. 1.8 b are: (1.1) where and are the voltage drop and current flowing along the semiconductor at , is the width or the transverse dimension of the contact. The solution to E q. (1.1) is [35], [88] , 13 , (1.2) where is the contact length, the contact width, and the current flowing into the contact, and is the transfer length. F igure 1. 9 Normalized potential under a contact versus as function of , and . [35] The current transfer from semiconductor to metal takes place over , indicating strong current crowding. Equation 1.2 is plotted in Fig. 1.9 (Fig. 3.15 of Ref. [35] ). In 1980, Reeves extended the formulation for Cartesian contacts to circular contact structures [89] . In 1995, Reeves and Harrison [90] further extended the theory to alloyed ohmic contacts using a trilayer tr ansmission line model (TTLM). TTLM [90] [92] considers three layers (metal layer, alloyed semiconductor layer, and unalloyed semiconductor layer) and two interfaces between the three layers. Although TLM theory is one of the most commonly used models to characterize planar contacts, allowing important contact parameters ( ) to be extracted or calculated, this 14 simple model ha s some limitations: a) the sheet thickness of the contact material is assumed to be zero, forcing current flow to be one - dimensional, b) it is applicable only for planar ohmic contacts or contacts that can be approximated as ohmic, c) specific contact resi stivity is assumed to be constant along the contact length. Various modification of this theory ha ve been attempted to overcome some of the limitations. a) Overcoming zero sheet thickness limitation: For the sake of simplicity, Overmeyer [93] , in 1970, set and calculated the current density in the contact area as a function of semiconductor layer thickness and contac t length restriction was relaxed by Berger in his extended transmission line model (ETLM) with current still restricted to one dimensional flow [87] . Berger added a virtual specific contact resistivity of , which yields, , where is the resistivity and is the thickness of the semiconductor layer. This ETLM has been widely used in the literature [84], [94] . In 1986, Pimbley [95] - level transmission line model (DLTLM). His method introduces two - dimensional characteristics to the standard TLM by postulating two parallel lines (at one - quarter and three - quarters of the semiconductor thickness) to carry the semiconductor current instead of just one. DLTLM produces around 12% corrections to the TLM with source resistivity, thickness, and specific cont act resistivity typical of technologies. In 2014, Zhang and Lau [84] did a comprehensive comparative study (c.f. Fig. 1.10) of contact resistance usi ng exact field solution, TLM and ETLM for several geometric ratios and resistivity ratios. The solid, dashed, and dotted lines in Fig. 1.10 (Fig. 3 in Ref. [ 84] ) are for field solution, TLM and ETLM, respectively. The ratio determines the parameter regime where the TLM theory can be used with good accuracy [87] . They found that TLM can be used to evaluate contact 15 resistance if a nd ETLM can be used when and . Electrical junctions between thin films, nanotubes, and nanorods generally have since the height of the contact members are in nanometer. Hence, it is expected that these nanoscale electrical c - Recently, transmission line theory has been used to model and measure various low - dimensional material based contacts, such as, metal - CNT [96] , metal - nanofiber, Gold - [56], [97] , [98] , Indium - [56] , Nickel - [98] , and graphene - metal ohmic contacts [99] . F igure 1. 10 Normalized contact resistance as a function of for various (Fig. 1.8a). [84] b) Overcoming ohmic contact limitation, considering non - linear current transport: The figure of merit used for contacts characterization is the specific contact resistivity , defined as the ratio of the voltage drop to current density. It has a constant value for strictly linear contacts. If the junction is highly non - linear , the oh mic approximation fails to give accurate characterization of the contact. The dependence needs to be considered. In 1999, Onomura et 16 al. [100] found for Pt to p - GaN contacts changing nearly 3 orders of magnitude for current varying from to . Piotrzkowski et al. [101] , in 2011, presented analytical generalization of the TLM formulas which include functional dependence of , where and are the contact current density and contact voltage, respectively. In standard TLM measurement, , where is the width of the TLM test pattern, is the contact resistance, and is the resistivity of the semiconductor. Ref [101] , where is the differential resistance of the planar contact. In 2014, He et al. [ 102] proposed a numerical method to characterize non - linear metal - semiconductor contacts. They added a Schottky diode to the standard TLM in series with the original pure resistance to account for the nonlinearity, as shown in Fig. 1.11 (Fig. 1 of Ref. [102] ). Interfacial oxidation or sintering processes lead to . In their model, voltage , where is the voltage across the Schottky diode. F igure 1. 11 Modified transmission line model (TLM) including the non - linear characteristics of the contact. [102] c) Overcoming constant specific contact resistivity assumption along the contact length: The third and the most important limitation of the standard TLM theory is its constant assumption along the entire contact length. is calculated from the applied voltage and the corresponding current density for the two contacting members, treati ng the junction as one - 17 dimensional. This treatment is valid only for ohmic contacts with uniform interfacial layer properties. In practical contacts, depends on local voltage drop and local contact current density, and therefore, varies spatially. Phy sically, this spatial dependence of may be introduced by a variety of factors, such as, the inherent non - linearity of the current density - voltage ( ) profiles of tunneling and Schottky junctions, rough interfacial layer, nonuniform distribution of oxides, contaminants or impurities in the contact layer etc. The effects of this non - uniform along the contact length need to be considered for a more accurate characterization of the electrical properties in a contact structure. This important issue has not been addressed in the existing literature. Here , we propose a two - dimensional TLM theory by including the effects of spatially varying specific contact resistivity along the contact length [103] [106] . We use the two dimensional TLM [103], [104] for ohmic contacts, and TLM coupled with the thermionic injection model [105], [107] for Schottky contacts and the quantum self - consistent model [10], [79] for tun neling type contacts. We demonstrate a method to control current distribution, by engineering the interface layer properties and geometry. We find that current crowding can be mitigated by strategically designing the specific contact resistivity along the contact length. 1.3 Organization of this thesis Chapter 2 presents the study of quantum tunneling induced current transport in nano - and subnano - meter metal - insulator - metal (MIM) junctions with dissimilar metal electrodes . A self - consistent model is formulated to calculate the tunneling current density. The results are compared bias voltage ( J - V ) curves are examined in detail for various parameters in diffe rent voltage regimes. 18 Chapter 3 presents the study of current transport and contact resistance in nanoscale parallel electrical contacts in Cartesian geometry. Traditional lumped circuit transmission line model (TLM) is modified to include the effects of spatially varying specific contact resistivity . At first, simple analytical solutions for parallel contacts are derived for the special case of uniform . Next, self - consistent numerical solutions are obtained for contacts with linearly varying along the contact length. Finally, local voltage - dependent tunneling resistivity along the contact length is examined by solving the TLM equations coupled with the tunneling current self consistently. The current and voltage distribution and the overall contact resistance are analyzed in detail, for various input voltage, electrical contact dimension, and material properties. The results for ohmic contact are verified with finite element method (FEM) based simulations [103] . transmission line model [89], [108] (CTLM) is modified to include the effects of radially varying . First, ohmic contacts are analyzed and Bessel function based analytical solut ions are derived for the special case of uniform . Then, tunneling type contacts are analyzed and local voltage - dependent tunneling resistivity along the contact length is examined by solving the TLM equations coupled with the tunneling current self co nsistently. The current and voltage distributions in such contacts and their overall contact resistance are studied in detail, for various input voltages, contact dimensions, and material properties. Chapter 5 presents a theoretical study of the contact re sistance and the current flow distribution for electrical contacts between two - dimensional (2D) materials and three - dimensional (3D) metals. Self - consistent solutions are obtained by coupling the modified TLM developed in chapter 3 with the improved thermionic current injection model for 2D materials [109] . First, we study the current and voltage distributions in such contacts and their overall contact resistance for 19 various input voltages, temperatures, contact dimensions, and material properties . Then, we propose to reduce contact resistance in 2D - material - based electrical contacts by roughness engineering of the contact interfaces. Results are compared with existin g theoretical models and validated with experimental data [105] . Chapter 6 demonstrates a method to mitigate current crowding, by engineering the interface layer properties and geometry [106], [110] . Ohmic, Schottky and tunneling type contacts are studied based on the formulation derived in chapter 2, 3, 4 and 5. First, current steering and redistribution ar e realized by strategically designing the specific contact resistivity along the contact length. Then, a nanometer thin insulating tunneling gap is introduced between highly conductive contact members to reduce the severe current crowding effects whil e maintaining similar total contact resistance. 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It is presented here with the permission of the copyright holder. 2.1 Introduction Quantum tunneling [2], [3] is important to nanoelectronic circuit designs, tunneling electrical contacts [4] , scanning tunneling microscopes (STMs) [5], [6] , plasmonic resonators [7] [9] , carbon nanotubes [10] [14] , graphene [15], [16] and other two - dimensional (2D) materials based devices [17], [18] and novel vacuum nano - devices [19] [23] . Quantum tunneling effects impose serious challenges to the physical scaling down of traditional electronic circuits [24] . However, it enables the development of future tunneling field - effect transistors (TFETs), which are envisioned to [25] . Tunneling in electrical contacts can be utilized to mitigate current crowding and nonuniform heat deposition in the contact region [4], [26] . Tunneling phenomenon may also introduce new regimes in quantum plasmoni cs [27] . Hence, it is critical to accurately characterize the current density - voltage ( ) behaviors in nano - scale metal - insulator - metal (MIM) junctions, for a variety of material properties and junction dimensions. Tunneling effects between electrodes separated by thin insulating films have been studied extensively by Simmons [3], [28] [31] in 1960s. 31 image charge potential are considered, the electron space charge potential and the electron exchange - correlation potential inside the insul ator thin films, are ignored reliable only in low voltage regime for limited parameter space (insulator gap > 1 nm , barrier height > 3 eV) [27] . The effe cts of space charge in a vacuum nanogap have been studied [32] [34] extensively, with extensions to short pulse [35] . Recently, Zhang [27] proposed a self - con sistent model (SCM) to characterize quantum tunneling current in similar MIM junctions, considering current flowing from both the electrodes. It is found that the characteristics may be divided into three regimes: direct tunneling, field emission, and space - charge - limited regime [27] . However, the SCM for similar MIM junctions is not sufficient to characterize electron tunneling through MIM junctions formed between two electrodes with different work functions, where the characteristic is dependent upon the polarity of the bias vol tage [28] . The asymmetry of the polarity - dependent behavior is important to harmonic mixers, rectifiers, millimeter wave and infrared detectors [36] . Several efforts have been made to enhance this asymmetry in dissimilar MIM tunnel diodes [36] [38] . Moreover, dissimilar MIM junctions are naturally [5], [6] and in nanoscale electrical contacts [4], [39], [40] . In this Chapter, we extend the theory of Zhang [27] to dissimilar MIM junctions (Figure 2.1). Following Simmons [28] , we define the forward bias (FB) and reverse bias (RB) of the MIM junction, when the metal electrode with higher work function is negatively and positively biased, respectively. We provi de a detailed study of FB and RB asymmetry and its dependence on a wide range of input parameters (work functions of the electrodes, thickness and relative permittivity of the insulator), for different voltage regimes. The FB and RB characteristics are fou nd to cross over at high voltages in the field emission regime. The asymmetry between the current density profiles 32 increases with the work function difference of the electrodes, the thickness or permittivity of the insulator layer. 2.2 The Self - Consistent Model F igure 2. 1 Dissimilar metal - insulator - metal (MIM) tunneling junction. The metal electrodes have equilibrium Fermi level and work function and (in these schematics we assume ). , where is electron affinity of the i nsulator. The insulator thin film thickness is . The applied voltage bias is . The current densities emitted from the electrode 1 and 2 into the gap are and , respectively. (a), (c) reverse bias ( is positively biased) condition; (b),(d) forward bias ( is negatively biased) condition. (a), (b) represent low and (c), (d) represent high bias voltage conditions. [1] 33 Our self - consistent model (SCM) formulation is based on the formulation of similar MIM junctions [27] . The potential barrier formed between the two electrodes is, , (2. 1 ) where is the equilibrium Fermi level; = ; ; and are the work functions of metal electrode 1 and 2 respectively; is electron affinity of the insulator; = is the image charge potential energy including the effect of anode screening [34] , where e is the electron charge, is the permittivity of free space, is the relative permittivity of the insulator, and is the gap distance; the electric potential , where the two terms are the potential due to the external applied voltage and the potential due to the electron space charge, respectively; and = is the electron exchange - correlation potential calculated by the Kohn - Sham local density approximation (LDA) [41] , where is the local Seitz radius [ in terms of the Bohr radius nm , n ( x ) is the electron density, = 27.2 eV is the Hart ree energy, and is the exchange - correlation energy [41] [43] . , and are the exchange energy and the correlation energy respectively, for a uniform electron gas of density n , where , and , , , , , , and are constants obtained from [41] . The probability that an electron with longitudinal energy (normal to the surface) can penetrate the potential barrier is given by the WKBJ approximation [44] , , (2.2) 34 where and are the two roots of , is the electron rest mass. The tunneling current density from electrode 1 to the right, and from electrode 2 to the left, are respectively [3], [27], [34] , , (2.3a) , (2.3b) , (2.3c) , (2.3d) where is given in Eq. (2.2), is the total number of electrons inside electrode 1 (electrode 2) with longitudinal energy between and + impinging on the surface of electrode 1 (2) across a unit area per unit time, calculated by the free - electron theory of metal [45] , is the electron rest mass, is the reduced Planch constant, is the Boltzmann constant, and is the electrode temperature. Insi de the insulator, , we solve the coupled Schrödinger equation and the Poisson equation, for the electric potential and the exchange - correlation potential , , (2.4) , (2.5) where is the complex electron wave function, is the electron density, and is the electron emission energy (with respect to the Fermi energy ). We assume in t he calculation. 35 For a bias voltage , the boundary conditions are, , and . We also have the boundary conditions that both and are continuous at , and . Due to charge conservation, the net current density is constant for all x , where a prime denotes a derivative with respect to x , and . For convenience we use nondimensional quantities [27] , , , , , , , , where , is the Child - Langmuir law [46], [47] , , and is the Hartree energy. The wave function in the normalized form is , wh ere and are the nondimensional amplitude and phase respectively, both assumed real. Equations 2.4 and 2.5 are normalized to read, , (2.6) , (2.7) where is the net normalized current density. The boundary conditions to eqs. (2.6) and (2.7) are, , , , and . The normalized emission current density and are, , (2.8a) , (2.8b) 36 where , , and . By solving Eqs. (2.6) - (2.8) iteratively with the boundary conditions, we can self - consistently obtain the complete potential barrier profile , the curren t density emitted from both electrodes and , for any metal electrodes ( ), insulator layer ( , ), and bias voltage ( ). It is found the tunneling current emission is insensitive to the temperature and the Fermi level [27] . In our calculations, we assume room temperature 300 K and eV. It is worthwhile to note that, although the proposed model is developed for DC condition, it is applicable to the excitation of up to the Near Infrared frequency, since in typical metallic tunnel junctions, the tunneling events occur on a timescale much sh orter than the period of the driving fields [27], [48], [49] . In this formulation, we have assumed, 1) the electron transmission probability during the emission process can be approximated by the WKBJ solution, where the metal electrodes are based on the free electron gas model; 2) the surfaces of the electrodes are flat and the problem is one - dimensional; 3) the image potential can be approximated by the classical image charge methods; and 4) the two metallic electrodes are separated by a sufficiently thi n insulating film (in the nano - or subnano - meter scale), so that charge trapping in the insulator are ignored [30], [50] . 2.3 Results and Discussion Figure 2.2a shows the normalized current density as a function of applied gap voltage , for two electrodes having work functions, and (Au) , separated by 1 nm vacuum gap ( = 1, ). Metal 2 is given a positive bias (i.e. reverse bias, equivalent to Figs. 2.1a and 2.1c). The current densities are calculated from the SCM with both space charge 37 potential and exchange correlation potential included. The curves may be roughly divided into three regimes: direct tun neling regime ( ), field emission regime ( ), and space - charge - limited regime ( ), similar to the MIM with electrodes of the same material [27] . F igure 2. 2 ( a ) N ormalized (in terms of CL law) current density as a function of applied gap voltage , for two electrodes having work functions, and (Au) , separated by 1 nm vacuum gap ( = 1, ). Metal 2 is positively biased. The calculations are from SCM with both space charge and (dashed line) is from Ref [28] , Fowler - Nordheim (FN) law (dash - dotted line) is from Ref. [51] [53] , calculated w ith the cathode work function = , and the quantum CL law (green dotted line) is from Ref. [32], [33] . (b) Current density in as a function of applied gap voltage , for = 1 nm and vacuum gap ( =1, ) . Solid and dashed lines in (b) represent RB and FB conditions respectively. Top to bottom, = 3.68 eV (Mg), 4.08 eV (Al), 5.1 eV (Au), 6.35 eV (Pt). The work function difference between the two metals is kept fixed, = = 1 eV. The inset in (b) represents the zoomed in view of the cross over behavior for the case of eV . [1] In the direct tunneling regime, just like similar MIM junctions [27] , the tunneling current density from cathode and that from anode are comparable. The net curren t density can 38 be orders of magnitude lower than both and . Therefore, in this regime, both anode emission and cathode emission need to be considered for an accurate estimation of the tunneling current in the dissimilar MIM junction. In the direct tunneling regime, increases linearly with , which implies that the dissimilar electrode MIM junction behaves like an ohmic resister. The [28] . In the field emission regime, is much smaller compared to , because the effective barrier height at the cathode is reduced by the bias voltage. The net current densit y is approaching the Fowler - Nordheim (FN) law [51] [53] as V g increases. However, in the field emis formula gives a more accurate fit to the self - consistent SCM result, which is due to the inclusion . When the gap voltage reaches V , the effective barrier height is depressed by below the Fermi level of the cathode (i.e. equivalent to Fig. 2.1c). In the space - charge - limited (SCL) regime, when reaches 100V, the cathode current and therefore the net current approaches the qu antum CL law (QCL) [32], [33] , which gives the maximum current density that can be transported across a vacuum nano - gap for a given and , with quantum corrections. Figure 2.2b shows the net current density in as a function of applied gap voltage , in dissimilar MIM junctions separated by a 1 nm wide vacuum gap for a fixed = = 1 eV . Solid and dashed lines are for reverse biased (RB) (i.e. higher work function metal is positively biased) and forward biased (FB) (i.e. higher work function metal is negatively biased) current densities, respectively. The tunneling current density of a dissimilar MIM junction is very sensitive to its apparent barrier height. Figure 2.2b shows that, at low voltages , the characteristics are almost i dentical for the FB and RB conditions. In the region of 39 , the FB current exceeds slightly. At a higher voltage, the FB and RB characteristics cross over. The inset of Fig. 2.2b shows the zoomed in view of this cross over behavior for the case of eV. It is shown in Fig. 2.2b, as (and therefore ) increases, the FB and RB characteristics intersect at increased values of , which agrees with the results reported by Simmons, in 1960 [28] . The underlying reason for this crossover behavior is, in the high voltage region, the tilt of the potential barrier changes its direction for the RB condition (Figs. 2.1a an d 2.1c). For , the asymmetry between FB and RB characteristics becomes significant. In this region, for the same bias voltage, electrons see a lower effective potential barrier height in RB condition than that in the FB condition ( Fi g. 2.1 ). The asymmetry between FB and RB characteristics remains insensitive to the value of or , when the work function difference is kept fixed. When approaches 100 V, the net current density for both FB and RB conditions converges to the value of QCL, since the SCL current density depends only on and , but not on work function. The effect of the electron affinity of the insulating thin film on characteristics would be similar, that is, increasing would be equivalent to decreasing and , provided the relative permittivity of the insulator and the insulator thickness are unchanged. 40 F igure 2. 3 The effects of work function difference on the characteristics of a dissimilar MIM junction with = 1 nm, vacuum gap ( =1, ). Top to bottom, = 4 eV, 3 eV, 2 eV, 1 eV, 0 eV, - 1 eV, - 3 eV. The work fun ction of metal 2 is kept fixed, = 6.35eV (Pt). Solid and dashed lines represent RB and FB conditions respectively. [1] The effects of work function difference = on the characteristics of a MIM junction separated by a 1 nm wide vacuum gap, are shown in Fig. 2.3 . The work function of metal 2 is kept fixed, = 6.35 eV (Pt). Solid and dashed lines represent the RB and FB conditions, respectively. Unlike the previous case of fixed in Fig. 2.2b, in the field emission regime, th e asymmetry between FB and RB currents increases significantly as | | increases. Work function difference between the two metal electrodes in a dissimilar MIM junction influences the characteristics more profoundly than the individual work functions. The dotted line in Fig. 2.3 ( ) represents the similar MIM junction ( ) tunneling current density. The curves for lie above and below the reference, for and respectively. 41 F igure 2. 4 The effects of gap width ( ) on the characteristics of a dissimilar MIM junction with vacuum gap (a) as a function of applied gap voltage . Top to bottom, = 0.5 nm, 1 nm, 1.5 nm, 2 nm, 3 nm. (b) as a function of gap width for different . Work function of the two electrodes are and . For the vacuum gap = 1 and . Solid and dashed lines represent RB and FB conditions respectively. Figure 2.4 shows the effects of gap width (or insulator thickness) on the tunneling current density in dissimilar MIM junctions. In Fig. 2.4a, RB and FB tunneling current densities are plotted as functions of applied gap voltage for = 0.5 nm, 1 nm, 1.5 nm, 2 nm and 3 nm. In Fig. 2.4b, tunneling current densities are plotted as functions of for different externally applied bias voltages . For small gap width ( in Fig. 2.4a ), the asymmetry between FB and RB current densities tend to disappear. However, when is increased, the asymmetry increases significantly. The FB and RB characteristics tend to crossover at about the same voltage. However, this crossover voltage is not exactly the same for all (c.f. Fig. 2.4b, ), as previously reported by Simmons [28] . Figure 2.4b shows that the asymmetry between the FB and RB tunneling current densit ies appear only for high voltages. In low voltage regime ( for our MIM junction current calculations), for any given gap width ( ), the current 42 density profiles are almost identical for the two biases. The asymmetry increases with the app lied bias voltage and it tends to disappear as reaches into the SCL regime . Note that, when is large, the cathode emission current reaches the SCL current at a higher voltage. This explains the increase of asymmetry between FB and RB tunneling densities at high voltages (c.f. Fig. 2.4b, ) for large . F igure 2. 5 The effects of relative permittivity of the insulating thin film on the characteristics of a dissimilar MIM junction with = 1 nm. (a) a function of applied gap voltage . Top to bottom, and respectively . (b) as a function of for different . Work function of the two electrodes are and . Electron affinity of the insulator is . Solid and dashed lines represent RB and FB conditions respectively. Figure 2.5 shows the effects of insulator layer permittivity on the tunneling current density in dissimilar MIM junctions. In Fig. 2.5a, RB and FB tunneling current densities are plotted as functions of applied gap voltage for = 1, 2 and 6. In Fig. 2.5b, tunneling current densities are plotted as functions of for different externally applied bias voltages . The relative 43 permittivity of insulating layer greatly influences the image charge potential as well as the space charge potential (Eq. 2.7) , which in turn affect the current transport through the potential barrier. The asymmetry between FB and RB tunneling current densities increases with (Fig. 2.5a). However, for low voltages, there is no such asymmetry and the profiles are identical, since in this direct tunneling regime the MIM junction is ohmic (Figure 2.5b). The FB and RB characteristics crossover at higher voltages for increasing . It is important to note that, for low and int ermediate bias voltages, decreases with , but when reaches , this trend reverses because l arger reduces the effect of space charge (Eq. 2.7) . The asymmetry between FB and RB current densities tend to disappear as reaches the quantum CL limit (Fig. 2.5a and Fig. 2.5b, ). 2.4 Concluding Remarks Our self - consistent model characterizes the tunneling current in nano - and subnano - scale asymmetric (metal electrodes with dissimilar work functions) MIM junctions, taking into account the effects of both space charge and exchange - correlation potential. It provides accurate estimation of tunneling current density in different regimes over a wide range of input parameters. It is found imations of the tunneling current for only a limited parameter space in the direct tunneling regime. Their accuracy decreases when the effective barrier height decreases, where the self - consistent model would give a more accurate evaluation. We demonstrate d the influences of electrode work functions ( and ), insulator layer properties ( , ), insulator thickness ( ) and bias voltage ( ) on the FB and RB tunneling current density profiles. We found that the work function difference influences the asymmetry between forward and reverse bias characteristics more profoundly than their individual work 44 functions. This asymmetry increases with increasing insulat or layer thickness and relative permittivity. However, for very low (for our calculations, ) and very high voltages ( ), the tunneling current density profiles are almost similar for the two biased cases. It is worthwhile to note that, alt hough the proposed model is developed for DC condition, it is applicable to the excitation of up to the Near Infrared frequency, since in typical metallic tunnel junctions, the tunneling events occur on a timescale much shorter than the period of the driving fields [48] . 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Lett. , vo l. 88, no. 15, p. 154105, Apr. 2006, doi: 10.1063/1.2193776. 51 CHAPTER 3 PLANAR TUNNELING ELECTRICAL CONTACTS This chapter is based on the published journal paper Resistance Transmission Line Model for Nanoscale Paralle Sci Rep , vol. 9, no. 1448 4 , pp. 1 14, Oct. 2019, doi: 10.1038/s41598 - 019 - 50934 - 2, by S. Banerjee , J. Luginsland, and P. Zhang [1] . It has been reproduced here with the permission of the copyright holder. 3.1 Introduction Contact resistance and their electro - thermal effects have become one of the most critical concerns of very large scale integration (VLSI) circuit designers, because of the excessive amount of Joule heating being deposited at the contact region [2] [7] . The electrical contact properties have been extensively studied in m etal - semiconductor [8] [10] , metal - insulator - semiconductor and metal - insulator - metal [11] [14] junctions. The growing popularity of novel electronic circuits based on graphene, carbon nanotubes (CNTs) and other new materials has made contact engineering crucial. CNT based devices, in particular, experie nce significant challenges because of the inter - tube connections. On macroscopic level, the exceptional intrinsic electrical properties [15], [16] of CNTs become elusive [4], [15], [17] . C ontact resistances between CNTs profoundl y affect the electron transport and reduce the electrical conductivity of carbon nanofiber (CNF) [16] [18] , and greatly limit the performance of CNT thin film based Field Effect Transistors (FETs) [19] [22] . One can naturally expect these issues also arising from other novel two - dimensional materials (boro n nitride, molybedenum sulfide, black phosphorus, etc) as well as new nano - composites. While the work presented here is generalizable to these other material systems, here we choose carbon materials as examples. 52 53 54 3.2 The 2D Transmission Line Model F igure 3. 1 A parallel, partially overlapped electric contact. The contacts are formed between (a) nanotube or nanowire 1 and 2, and (b) thin film 1 and 2; (c) side view of the contact; (d) its transmission line model. In (a), (b) and (c ) a thin resistive interface layer (or a tunneling layer of permittivity ) is sandwiched between the two contacting members. Consider a parallel contact formed between two nanowires or nanotubes or between two conducting thin films or layers, as shown in Figs. 3.1(a) and 3.1(b), respectively. The distance between the two contact members is , and the contact length is . A thin resistive interface layer is sandwiched between them. Both contacts in Figs. 3.1(a) and 3.1(b) can be described by a two - dimensional (2D) model, as shown in Fig. 3.1(c). Note that the proposed formulation is generally 55 applicable to parallel Cartesian nanojunctions with different sh ape of the electrodes, for example, electrical contact between a nanowire and a thin film. In the 2D model, the effects of the transverse dimension (perpendicular to the paper) can be included in the effective sheet resistances and for conduc tor 1 and 2, respectively, such that there is no variation along the width in the transverse dimension. The spatial dependent specific interfacial resistivity (also termed specific contact resistivity) is , which is either predefined, or calcula ted from the local tunneling current in case of insulating tunneling layer [37] [39] . We use the DC equivalent lump circuit transmission line model (TLM) [12], [43] [45] , as shown in Fig. 3. 1(d), to model the 2D parallel contact in Fig. 3. 1(c). In the contact region PQNM in Figs. 3.1(c) and 3.1(d), using Kirchoff's laws for current and voltage, we get the following equations, , (3.1a) , (3. 1b ) , (3.1c) , (3.1d) where and represent the current flowing at through the lower contact member, MN and upper contact member, PQ respectively, and and the local voltage at along MN and PQ, respectively, and is the effective transverse dimension of the contacts. When , Equation ( 3. 1) becomes, , (3.2a) 56 , (3.2b) , (3.2c) , (3.2d) where and are the local current density and the local voltage drop across the contact interface at , respectively . Note that, from Eqs . (3.2a) and (3.2c), , where is the total current in the circuit, to be determined from the boundary conditions . The boundary conditions for Eq. (3.2) are, , (3.3a) , (3.3b) , (3.3c) , (3.3d) where, without loss of generality, we assume the voltage of the upper contact member at is 0, and the externally applied voltage at of the lower contact member is . Note that , and . From Eqs. (3.2) and (3.3), it is e asy to show , , , , where a prime denotes a derivative with respect to . For the contact model in Fig. 3.1(d), the contact resistance is defined as, . (3.4) 57 It is convenient to introduce non - dimensional quantities, , , , , , , , , , and , where we define , , and . In normalized forms, Eq. ( 3. 2) can be recast into the following second order differential equations, , (3.5a) , (3 .5b) , (3.5c) where , and . The corresponding boundary conditions to Eqs. 3. 5(a) - 3. 5(c) are respectively, , and , (3.6a) , , (3.6b) , (3.6c) where the unknown constant is the normalized total current in the circuit, and prime denotes a derivative with respect to . Note that integrating Eq. (3.5b) subject to Eq. (3.6b) gives , which means that the total current is conserved across the contact interface. Equations (3.5) and (3.6) are solved to give the voltage and current distribution along and across the contact interface as well as the total contact resistance, for a given electrical contact (Fig. 3.1) with spatially dependent interface speci fic contact resistivity . An example of the 58 procedure to solve Eqs. (3.5) and (3.6) numerically is as follows. For an initially guess on , Eq (3.5b) is solved using the shooting method, subject to Eq. (3.6b). Next, Eq (3.5a) is solved with the in itial values of and from Eq. (3.6a). It is then checked whether is equal to , as in Eq. (3.6a). The above - mentioned process repeats for different input until the condition is satisfied. Finally, Eq (3.5c) is solved to get (and ). In principle, Eqs. (3.5) and (3.6) can be solved numerically for arbitrary spatial dependence of specific contact resistivity . Here, we focus on a few special cases of practical im portance. We first consider the case of constant , where analytical solutions can be obtained (Sec. 3. 3 .1), which also serve to validate our numerical approach. We then consider the effects of spatially dependent on the parallel electrical con tacts. We focus on two situations: linearly varying specific contact resistivity along (Sec. 3. 3 .2), and thin tunneling junction with uniform thickness (Sec. 3. 3 .3), where analytical solutions to the TLM current and voltage equations are no longer avail able, and Eqs. (3.5) and (3.6) are solved numerically. 3. 3 Results and Discussion 3.3.1 Constant specific contact resistivity along the contact length For the special case of constant specific contact resistivity , the TLM equations, Eqs. (3.5) and (3.6), can be solved analytically to give, (3.7a) (3.7b) (3.7c) 59 (3.7d) (3.7e) and (3.8) where , and . F igure 3. 2 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, ,for different values of specific contact resistivity , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] Figure 3.2 shows the current and voltage di stributions along the contact length and across the contact interface for various specific contact resistivity , for a parallel contact formed between similar contact members, The voltage along both contact members and decrease with , as shown in Figs. 3.2(b) and 3.2(c), respectively. The current in contact 60 member 1 decreases with (Fig. 3.2(e)), whereas in contact member 2 increases with (Fig. 3.2(f)), with the total current being kept a constant along . The profiles of both normalized voltage drop and current density across the interface layer, are symmetric along the contact length, with the minimum at the center of the contact structure and the max imum at the contact edges, as shown in Figs. 3.2(a) and 3.2(d), respectively. The current crowding effects near the contact edges are well - known phenomena, as the current density is distributed to follow the least resistive path (i.e. minimum overall resis tance). It is important to note that as the specific contact resistivity decreases, the interface current density becomes more crowded towards the contact edges, as shown in Fig. 3.2(d). In other words, the less resistive the contact interface layer, the more severe of the current crowding effects, which is in agreement with previous studies using both TLM [44], [45] and field theory [12], [50], [51] . F igure 3. 3 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , and (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, , for different values of , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] 61 Figure 3.3 shows the current and voltage distributions along the contact length and across the contact interface for various parallel contacts formed between dissimilar materials, , with fixed specific contact resistivity . The vo ltage and the current along the two contact members show similar behaviors as those in Fig. 3.2. However, the voltage drop across the interface layer and the contact current density are no longer symmetric, as shown in Figs. 3.3 (a) and 3.3(d), respectively. When , the maximum of and occurs at ; when , the maximum of and occurs at . This current crowding effect can again be explained by the fact that current flows are self - arranged to take the least resistive path in the circuit by adjusting the current distribution according to the local resistance. F igure 3. 4 Normalized contact resistance of the parallel contact (Fig. 3.1). as a function of (a) normalized specific contact resistivity, and (b) normalized sheet resistance of contacting member 2, . Dashed lines are for Eq. (3.9), the limiting case of . The cross symbols are from COMSOL [46] 2D simulations. T he length and heigh t of both upper and lower contacting members are assumed to be 20 nm and 10 nm respectively, and the thickness of the resistive interfacial layer is assumed to be 0.5 nm. The resistivities of the upper and lower contact members are in the range of , and the resistivity of the interface layer is in the range of . [1] 62 The normalized contact resistance, calculated from Eq. (3.8) is plotted in Fig. 3.4 for various and . It is clear that increases with both and . In general, the contact resistance depends more strongly on the the specific contact resistivity of the interf acial layer than on the sheet resistance ratio of the contact members . For the special case of , Eq. (3.8) becomes, , (3.9) with , which is also plotted in Fig. 3.4. Note that Eq. (3.9) is identical to the expression typically used for metal - semiconductor contact. To verify the results obtained from our analytical solution, we performed numerical simulations using the COM SOL multiphysics software [46] , for various combinations of and on the geometry shown in Fig. 3.1. The finite - element - method (FEM) based COMSOL 2D simulation results are included in Fig. 3.4 (cross symbols), showing excellent agreement with our theory. The convergence iteration error was less than for each point. 3. 3 .2 Specific contact resistivity varies linearly along the contact length We assume the specific resistivity varies linearly along the contact length (Fig. 3.1) as . By solving Eqs . (3.5) and (3.6) numerically, we obtain the current and voltage distributions along the contact interface, as shown in Fig. 3.5. As increases, the overall contact interface becomes more resistive, therefore, the voltage drop across the interface layer increases (Fig. 3.5a), whereas the current density across the interface layer decreases in general (Fig. 3.5d). The maximum occurs at the contact edge with the highest specific resistivity 63 (i.e., at when , and at when ), while the maximum interface current occurs at the contact edge with the lowest (i.e., at when , and at when ). The effects of on the voltage and the current along the two contact memb ers are also shown in Figs. 3.5 (b), (c), (e) and (f), respectively. F igure 3. 5 (a) Voltage drop across the contact interface , voltage along (b) contact member 1 (MN), , and (c) contact member 2 (PQ), , (d) current density across the contact interface , current along (e) contact member 1, , and (f) contact member 2, , for linear specific contact resistivity with different lin ear constant , for . All the quantities are in their normalized forms defined in Sec. 3.2. [1] The normalized contact resistance, calculated from Eq. (3.4) for linear specific contact resistivity is plotted in Fig. 3.6. As increases, increases, since the contact interface becomes more resistive. As increases, the contact resistance depends more stro ngly on the linear constant . 64 F igure 3. 6 Normalized contact resistance of the parallel contact (Fig. 3.1) with linear specific contact resistivity , for various value of . [1] 3. 3 .3 Tunneling contact resistance Here, we assume the parallel contacts are formed through a tunneling interface layer between the two contact members. For simplicity, we have made the following assumptions: 1) the thickness of interfacial insulating film in the contact area is uniform, and 2) the insulating film is sufficiently thin (in the nano - or subnano - meter scale) so that charge trappings are ignored [52] [53] . For dissimilar contact members, the (normalized) current density at any location along the contact from con [38] , (3.10) where is the local voltage drop across the contact interface at , A , and . Definitions of , and 65 for forward bias (when lower work function contacting member is given positive bias ) are given below, , where , , , and and are the work functions of contacting member 1 and 2 respectively, is the e lectron affinity of the insulating layer, which is 0 for vacuum. For : , and for : , , where . On the other hand, th e definitions of , and for reverse bias (when higher work function contacting member is given positive bias) are: , for : , ; for : , and for : , . For the special case of the same material for contact members 1 and 2, in Eq. (3.10), where , , is the work function of contacting member 1 and 2, and, , for , for direct tunneling regime [11], [23] . More accurate results for the tunneling current may be calculated using quantum models developed in Chapter 2 by solving the coupled Schrodinger 66 equation and Poisson equation with the inclusion of space charge and exchange - correlation effects [11], [23] . We keep the normalization consistent with our previous calculations in Sec. 3. 2 . For a given parallel tunneling contact (Fig. 3.1), the inputs of our model are the applied voltage , sheet resistance ( and work function ( , ) of contacting members 1 and 2, permittivity ( ), thickness ( ), and electron affinity ( ) of the interfacial insulator layer, and the contact length L . Using Eq. (3.10), the specific contact resistivity is obtained from , which is inserted into the TLM equations, Eqs. (3.5) and (3.6), to give a self - consistent solut ion to the voltage and current profiles, as well as the contact resistance for the parallel tunneling contact. We consider CNT - vacuum - CNT parallel contact as an example. Both contact members are made of the same single - walled CNTs. Using the typical value of linear resistivity of single - walled CNT [54] [55] , and diameter (or the width ) of 3 nm, a n equivalent sheet resistance for both CNT contact members are estimated as , where the unit of the sheet resistance [12], [45] . The work function of CNTs is 4.5 eV [56] . The interfacial layer is assumed to be vacuum (relative permittivity = 1.0, and electron affinity ). The voltage drop across and the tunneling current density through the contact interface are shown in Fig. 3.7 for various contact length , vacuum gap distance , and applied voltage . The profiles of both and are symmetric about the cen ter of the contact, as expected for similar contact members (similar to Figs. 3.2a and 3.2d above). As the contact length increases, the local voltage drop across the contact interface decreases, so does the tunneling current density , as shown in Figs. 3.7a and 3.7b. However, the total current in the contact structure, increases with , 67 since the total contact resistance of the tunneling junction decreases as the contact length increases (cf. Fig. 3.8a below). As shown in Figs. 3.7c and 3.7d, when the gap distance increases, the voltage drop increases, but the current density decreases, which is because the tunneling junction becomes more resistive ( See Chapter 2) . Figures 3.7e and 3.7f shows both voltage drop and current density increase when the applied voltage increases. More importantly, both and exhibit a stronger spatial dependence as increases. This strong voltage dependence of electrical properties of the tunneling junction is in sharp contrast with those of ohmic contacts, where the profiles of and , and the total contact resistance is independent of the applied vol tage, and the current density scales linearly with the voltage drops, as discussed in Sec. 3. 3 .1 and Sec. 3. 3 .2 above. Also plotted in Fig. 3.7 are the analytical results from Eq. (3.7), by assuming constant tunneling contact resistivity across the contact length (i.e. the typically assumed one - dimensional tunneling junction [24] ), by (a), setting and using Eq. (3.10) (dashed lines) and (b), using ohmic approximations for the tunneling junction, in the limit o f (dotted lines) [37], [38] . In the latter case, the tunneling current density is a linear function of , , (3.11) where . and are th e same as for Eq (3.10). is calculated from the same expression for Eq (3.10) by setting . , for forward bias; , for reverse bias; and , for similar contacting members. , , , , have the same definition as in Eq. (3.10). 68 It is found that both assumptions of constant contact resistivity are not sufficiently reliable, especially when the tunneling thickness decreases or the applied voltage increases. As the tunneling junction resistance becomes nonlinear in these cases, it is necessary to use the coupled TLM equations, Eqs. (3.5) and (3.6), and the localized tunneling equation, Eq. (3.10), to provide mo re accurate predictions. F igure 3. 7 Similar material CNT - vacuum - CNT parallel tunneling contacts. (a) Voltage drop across the contact interface , and (b) tunneling current density across the contact interface for different contact length , with fixed V , and D = 0.5 nm; (c) and (d) for different , with fixed V and L = 50 nm; (e) and (f) for different applied voltage with fixed 0.55 nm, and 50 nm. All the material properties are specified in the main text. Solid lines are for self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical calculations from Eq. 3.7 with calculated using in Eq 3.10 and ohmic approximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] 69 F igure 3. 8 The total contact resistance of the CNT - vacuum - CNT parallel contact. Contact resistance is plotted as a function of (a) contact length, L , for different insulating layer thickness, D , (b) D , for different L , for a fixed appli ed voltage, V ; (c) and (d) applied voltage for different L and D respectively, in CNT - vacuum - CNT contacts. Solid lines are for self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical cal culations from Eq. 3.8 with calculated using in Eq 3.10 and ohmic approximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] The total contact resistance of the CNT - vacuum - CNT parallel contact is shown in Fig. 3.8, as functions of contact length , vacuum gap distance , and applied voltage . The total contact resistance increases very rapidly with increasing insulating layer thickness, D , and d ecreases with contact length, L . For the low applied voltage regime ( , is almost independent of , as shown in Figs. 3.8c and 3.8d. When the applied voltage , decreases sharply 70 with . This is because the junction is no lon ger ohmic and the tunneling resistivity decreases nonlinearly with the junction voltage, as a function of position along the contact length. Ohmic approximations (Eqs. 3.8, 3.11) fail to give accurate results in the latter case and it is necessary to use the self - consistent numerical model. As increases, the dependence of contact resistance on L becomes less significant. Similar profiles of contact resistance with were observed in other experimental and theoretical works [12], [25], [32] . The contact resistance lies between 5 k to 10 M for the cases shown in Fig. 3.8, which agrees with previously reported experimental and theoretical works [24], [27], [30] . The existing 1D models give an inaccurate estimation of the contact resistance because they do not consider the variation of tunneling current density along the contact length. Next, we extend our calculations for contacts of CNT with different metals calcium (Ca), aluminum (Al), copper (Cu) and gold (Au). The work functions of Ca, Al, Cu and Au are taken as 2.9, 4.08, 4.7 and 5.1 eV respectively [57] . The work functions and dimensions of the CNT are kept same as before. In addition, the dimensions of the CNT and contacting - metal - 2 are assumed to be same (wid th of 3 nm, thickness of 3 nm) for the simplicity of calculations. T he resistivity of Ca, Al, Cu and Au are known to be , , and respectively [57], [58] . 71 F igure 3. 9 Dissi milar material CNT - insulator - metal parallel tunneling contacts. (a) Voltage drop across the interfacial insulating layer , and (b) tunneling current density , in CNT - insulator - Metal contacts, for fixed D = 0.5nm, L =50nm, V and dif ferent contacting metals (Ca, Al, Cu, Au). (c) , and (d) , in CNT - insulator - Al contacts, for different insulating layer permittivity , with fixed D = 0.5nm, L =50nm, V. Solid lines are for self - consistent numerical calculations u sing Eqs. 3.5, 3.6, and 3.10, dashed and dotted lines are for analytical calculations from Eq. 3.7 with calculated using in Eq 3.10 and ohmic approximations for the tunneling junction, Eq. 3.11, in the limit of , respectively. [1] Figure 3.9 shows the effects of the work function of contacting member 2 ( ) and the permittivity of the thin insulating layer ( ), on the current and voltage characteristics in CNT - insulator - metal contacts. As the two contact members are different, the voltage drop and the tunneling current density are no longer symmetric along the contact length . Figure 3.9(a) and 3.9(b) show that the voltage drop increases and the tunneling current density decreases with 72 increasing . Figure 3.9(c) and 3.9(d) show that the voltage drop increases and the tunneling current density reduces significantly when the permittivity of the insulating layer increases from 1 to 3.9. Analytical solutions obtained by assuming constant tunneling resistivity along the contact length are also included, similar to the previous cases of Fig. 3.7. In general, for the chosen value of nm, t he ohmic approximations using Eq. 3.11 do not yield accurate results. The constant tunneling resistivity approximation using Eq. 3.10 by setting could be a good approximation for the self - consistent TLM model (Eqs. 3.5, 3.6, and 3.10), for tunneli ng layers with higher permittivity . Figure 3.10 shows the contact resistance (in ) for various contact metals and tunneling films for CNT - insulator - metal contacts. Contact resistance increases with insulating layer thickness , insulating layer per mittivity and work function of contacting member . It decreases with contact length , as in the similar contacts in Fig. 3.8. The potential barrier in the insulating layer increases with the increase of work function of the contact metal, resulti ng in lower tunneling current and higher contact resistance. 73 F igure 3. 10 The total contact resistance of the CNT - insulator - metal parallel contact. Contact resistance is plotted as a function of (a) contact length L , (b) insulator layer thickness and (c) insulator layer permittivity , for CNT - insulator - metal contacts for different contacting metals (Ca, Al, Cu, Au). (d) Contact resistance as a function of work function of contacting member 2 ( ). The material properties and dimensions for (a) - (c) are specified in the text (the same as in Fig. 3.9). For (d), the resistivity of contacting member 2 is assumed to be . The resul ts are from the self - consistent numerical calculations using Eqs. 3.5, 3.6, and 3.10. [1] 3. 4 Concluding Remarks In this chapter , we proposed a self - consistent model to characterize partially overlapped parallel contacts. Our model considers the spatial varia tion of contact resistivity along the contact structure. We solved the TLM equations for three cases: 1) constant specific co ntact resistivity, 2) linearly varying specific contact resistivity, and 3) spatial dependent specific contact resistivity along the contact length due to current tunneling . Our study provides a thorough understanding of 74 the contact tunneling resistance, c urrent and voltage distributions across nano and sub - nano scale MIM junctions in parallel electrical contacts . The effects of contact geometry (i.e. dimension of the contact, and distance between the contact electrodes), and material properties (i.e. work function, sheet resistance of the contact members, and permittivity of the insulating layer) on the spatial distributions of currents and voltages across these contacts , and the overall contact resistance are studied in detail . While predominately classica l in nature, the inclusion of tunneling current starts to address quantum effects in these small scale objects. It is found that in general the ohmic approximation of tunneling junctions (Eq. 3.11) is not reliable for predicting the contact resistance of parallel tunneling contacts. The one - dimensional (1D) tunneling junction models (Eq. 3.10 with constant voltage across the whole junction) are good approximations of the parallel contacts only when the thickness or the permittivity of the tunneling film is relatively large, or the applied voltage across the contact is relatively small. When the 1D models become unreliable for small or , or large , the self - consistent TLM equations coupled with the tunneling current (Eqs. 3.5, 3.6 and 3.10) need to be used to accurately characterize the parallel tunneling contacts. The parallel tunneling contact in this work may be considered as the basic building block to better understand the macroscopic electrical conductivity of CNT fibers, whi ch contains a very large number of such parallel contacts between individual CNTs . Furthermore, our study elucidates key parameters for parallel electrical contacts over a wide range of spatially dependent contact resistivity , which paves the way to strate gic ally design of contact structures with controlled current distribution profiles and contact resistance, by spatially varying the contact layer properties and geometry. 75 In this formulation, we have ignored the effects of space charge and exchange - correl ation inside the tunneling gap. We have also ignored possible charge trapping inside contact junctions. The model is assumed two - dimensional, where the effects of the transverse dimension are neglected. These issues will be the subjects of future studies. 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[53] - Charge Effects on Emission - Limited Current Journal of Applied Physics , vol. 38, pp. 832 840, Feb. 1967, doi: 10.106 3/1.1709421. [54] Materials , vol. 6, no. 6, pp. 2155 2181, May 2013, doi: 10.3390/ma6062155. [55] X. Ho et al. le - Nano Letters , vol. 10, no. 2, pp. 499 503, Feb. 2010, doi: 10.1021/nl903281v. [56] - walled and multiwalled carbon nanotubes: First - Phys. Rev. B , vol. 76, no. 23, p. 235413, Dec. 2007, doi: 10.1103/PhysRevB.76.235413. [57] D. R. Lide, G. Baysinger, S. Chemistry, L. I. Berger, R. N. Goldberg, and H. V. Kehiaian, CRC Handbook of Chemistry and Physics . 2003. [58] C. Kittel, Introduction to Solid State Physics , Seventh Edition. 1996. [59] P. Zhang, Y. Y. Lau, and R. M. Gilgenbac Appl. Phys. Lett. , vol. 97, no. 20, p. 204103, Nov. 2010, doi: 10.1063/1.3517497. 82 CHAPTER 4 CIRCULAR TUNNELING ELECTRICAL CONTACTS This chapter is based on the published journal paper crowding in tunneling type circular nano - J. Phys. D: Appl. Phys. , vol. 53, no. 35, p. 355301, Jun. 2020, doi: 10.1088/1361 - 6463/ab8fe0, by S. Banerjee, P. Y. Wong and P. Zhang [1] . It is presented here with the permission of the copyright holder. 4.1 Introduction 83 84 4.2 The 2D - Circular Transmission Line Model F igure 4. 1 Electrical contact between two circular thin films: (a) top view; (b) side view; (c) its transmission line model. In (a), (b) and (c) a thin resistive interface layer (or a tunneling layer of permittivity ) of thickness D is sandwiched between the two contacting members. The thicknesses of thin film 1 and 2 are and , respectively. [1] Consider a circular (ring) contact form ed between two conducting thin films or layers, as shown in Figs. 4.1(a) and 4.1(b). The outer radius of thin film 2 is and the inner radius of both the films is . A thin resistive interface layer of thickness D is sandwiched between them. Following Reeves [19] [21] , we modified the basic Cartesian geometry lumped circuit transmission line model (TLM) [22] [25] for circular structures, as shown in Fig. 4.1(c). The sheet resistance of the two conductors is and , respectively. The radially dependent specific interfacial resistivity (also termed specific contact resistivity) is , which is either predefined or calculated from the local tunneling current in the case of an insulating tunneling layer [14], [ 15], [26] . In the contact region in Fig. 4.1(c), when , give the following equations, 85 , (4.1a) , (4.1b) , (4.1c) , (4.1d) where and represent the currents flowing at along the radial direction of thin films 1 and 2, respectively, and and are the local voltages at along the radial direction of thin films 1 and 2, respectively. and are the local current density and the local voltage drop across the contact interface at , respectively . From Eqs. 4.1 (a) and 4.1(c), , where is the total current in the circuit to be determined from the following boundary conditions for Eq. (4.1), , , , , (4.2) where we assume the voltage of the upper contact member at is 0 and the external voltage is applied at to the lower contact member. Note that , , and . From Eqs. (4.1) and (4.2), we get , , , , where a prime denotes a derivative with respect to . For the contact model in Fig. 4.1(c), the contact resistance i s defined as, . (4.3) 86 For convenience, we introduce non - dimensional quantities, , , , , , , , , , and , where . In normalized forms, Eq. (4.1) can be written into the following coupled second order differential equations, , (4.4a) . (4.4b) Note that and . The corresponding boundary conditions to Eqs. (4.4) are, , , , (4.5a) , , , (4.5b) and the normalized total current, (4.5c) Equations (4.4) and (4.5) are solved to give the voltage distribution along and across the contact interface as well as the total contact resistance, for a given electrical contact (Fig. 4.1) with radially dependent interface specific contact resistivity , following a similar procedure as described in Chapter 3. Equations (4.4) and (4.5) can be solved numerically for arbitrary radial dependence of specific contact resistivity . Here, we focus on two special cases of practical importance. We firs t consider the case of constant , where analytical solutions can be obtained (Sec. 4.3.1). This serves to validate our numerical approach. We then consider the effects of 87 radially dependent on the tunneling type electrical contacts (4.3.2). The one - dimensional MIM quantum tunneling equations developed in Chapter 2 [5] are coupled with Eqs. (4.4), (4.5) and are solved self - consistently. 4.3 Results and Discussion 4.3.1 Constant specific contact resistivity along the contact length For the special case of constant specific contact resistivity , Eq. (4.4) can be rewritten as, . (4.6) The corresponding boundary conditions from Eqs. 4.5(a) and 4.5(b) are, , . (4.7) Using Eq. (4.7), the solution to Eq. (4.6) is, , (4.8) where and are the zeroth order modified Bessel functions of the first and second kind respectively and . The constants and are calculated from the boundary conditions as, , . The expressions of , , , and are, , , , and 88 , where and are the first order modified Bessel functions of the first and second kind respectively. The normalized contact resistance is, . (4.9) F igure 4. 2 Normalized voltage drop across the contact interface , along the radial direction of an annular contact with uniform contact resistivity, for different values of (a) inner radius to outer radius ratio , with and , (b) specific contact resistivity , with and , (c) sheet resistance ratio , with and . All the quantities are in their normalized forms defined in Sec. 4.2. Figure 4.2 shows the profiles of voltage drop along the radial contact length for a parallel annular thin film contact (Fig. 4.1) for different inner to outer radius ratio (Fig. 4.2(a)), specific contact resistivity (Fig. 4.2(b)), and sheet resistance ratio (Fig. 4.2(c)). Note t hat, since is constant along here, the profiles of contact current density = follow those of . The voltage drop across the contact interface increases with increasing inner radius to outer radius ratio and specific contact resistivity . For similar contacting members ( ) the maximum voltage drop occurs at the inner edge ( ) of the annular contact 89 (c.f. Figs. 4.2a and b). This is because the modified Bessel function of the second kind ( term in Eq. (4.8) ) increases sharply near the center. Physically this means the current in the contact interface is mostly crowded near . The current spreads out as it flows away from the center through the least resistive path. Figure 4.2(c) shows a similar trend in for dissimilar contacting members with . In fact, the current crowding at the inner edge ( increases wit h increasing . The voltage drop (and the contact current density) at the outer edge ( ) increases with decreasing . For , the majority of the contact current flows near . It is interesting to note from Fig. 4.2(a) that th e voltage drop profiles are highly asymmetric at the two edges ( , ) of the annular contact under study (Fig. 4.1) when is small, and the asymmetry reduces as increases. For the limiting case of (i.e. ) , CTLM reduces to th e planar limit (Chapter 3) [4] , where the maximum voltage drop occurs at both the edges ( ) and the minimum occurs at , making the profiles symmetric . F igure 4. 3 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of inner radius to outer radius ratio for different (a) normalized sheet resistance of contacting member 2, , and (b) normalized speci fic contact resistivity . The dotted lines in (a) are calculated from Eq. (3.8) of Chapter 3, that is, for Cartesian parallel electrical contacts. Dashed lines are for Eq. (4.10), the limiting case of 90 F igure 4. 4 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of normalized specific contact resistivity for different (a) normalized sheet resistance of contacting member 2, , and (b) inner radi us to outer radius ratio . In (a), = 0.1, and in (b), =1. Dashed lines are for Eq. (4.10), the limiting case of The black dotted line in (b) The dotted lines in (a) are calculated from Eq. (3.8) of Chapter 3, that is, for Cartesi an parallel electrical contacts. F igure 4. 5 Normalized contact resistance of the annular contact (Fig. 4.1) as a function of sheet resistance ratio for different (a) normalized specific contact resistivity and (b) inner radius to outer radius ratio . In (a), = 0.1, and in (b), = 1. Dashed lines are for Eq. (4.10), the limiting case of The black dotted line in (b) is calculated from Eq. (3.8) of Chapter 3, that is, for Cartesian parallel electrical contacts. 91 The normalized contact resistance is calculated from Eq. (4.9) and plotted in Figs. 4.3, 4.4, and 4.5 as functions of inner radius to outer radius ratio , normalized specific contact resistivity , and sheet resistance ratio of the two contacting members , respectively. Figures 4.3(a) and ( b) show that for , decreases with when is high or is low. Figures 4.4(b) and 4.5( b) also confirm this behavior. However, when is increased above , increases drastically with . Larger means shorter radial contact length (for a fixed or ), resulting in higher total contact resistance for the annular contact structure. In general, increases with the specific contact resistivity or the sheet resistance ratio . Profiles of total contact resistance for the case of are also plotted in Figs. 4.3, 4.4 and 4.5 as dashed lines. When , Eq. (4.9) becomes, , (4.10) with . Note that Eq. (4.10) is identical to the expression typically used for metal - semiconductor contact [24], [25] . The difference between solid lines (Eq. 4.9) and dashed lines (Eq. 4.10) decreases when or is large, as shown in Figs. 4.3(b) and 4.4(b). The dotted lines in Figs. 4.3(a), 4.4(b), and 4.5(b) are calculated from the contact resistance for parallel Cartesian contacts, that is, Eq. (3.8) of Chapter 3 [4] . Note that the spatial dimensions were normalize d by the contact length ( = in circular case) in Sec. 3.2 [4] , whereas they are normalized by the outer radius for the circular case here. To make the normalization consistent for direct comparison, we multiply the normalized specific contact resistivity by before inserting into Eq. (3.8), which is multiplied by ( ) to obtain the dotted lines in Figs. 4 .3(a), 4.4(b), and 4.5(b). For , the profiles of annular and Cartesian contact resistance match exceptionally well, as CTLM approaches the limit of Cartesian TLM. 92 4. 3 . 2 Tunneling dependent contact resistivity Next, we consider the case where the para llel annular contacts are formed through a tunneling interface layer between the two annular contact members. In this case, due to the nonlinear current density - voltage ( ) characteristic of metal - insulator - metal (MIM) tunnel junctions, specific contact resistivity varies radially. For simplicity, we have made the following assumptions: 1) the thickness of the interfacial insulating film in the contact area is uniform and 2) the insulating film is sufficiently thin (in the nano - or subnano - meter sca le) so that charge trappings are ignored (Chapter 2, Sec. 2.2). The local contact current density at any location from contact member 1 to contact member 2 is calculated based on the self - consistent 1D Schrödinger - Poisson solutions in the MIM junction developed in Chapter 2, Sec. 2.2 [5] . For given values of the work function of the two contact members , electron a ffinity , thickness , and relative permittivity of the insulator layer, the local contact current density can be calculated from this 1D quantum model for an input of the contact voltage drop at any location [5], [6] . The calculation of this - relation is coupled with CTLM, Eqs. (4.4), (4.5), and solved self - consistently. We keep the normalization consistent. Since solving the coupled quantum tunneling model and CTLM is time expensive, we calculate the one dimensional tunneling current density separately for the given MIM parameters ( , , , , ), over a wide range of bias voltages. The obtained curves are normalized (as in Sec. 4.2) and then fitted with polynomials. Those curve - fitted equations are used to find the specific contact resistivity , which is then inserted into the CTLM equations, Eqs. (4.4 ) and (4.5), to 93 give a self - consistent solution to the voltage and current profiles, as well as the contact resistance for the circular (annular) tunneling contact. F igure 4. 6 Tunneling current density across the contact inter face for different (a) input voltage , with fixed , , and ; (b) inner radius to outer radius ratio , with fixed , , and ; (c) outer radius , with fixed , , and ; (d) interfacial layer thickness , with fixed , , and . All the material properties are specified in the main text. Solid lines are for self - consistent numerical cal culations using Eqs. (4.4), (4.5), and MIM quantum tunneling formulations (Chapter 2, Sec. 2.2) [5] . Dashed lines are for analytical calculations from Eq. (4.8) with being constant, calculated using in the MIM quantum model. As an example, we consider Cu - vacuum - Cu circular thin film contacts. For our calculations, sub - nanometer scale interfacial layer thicknesses are assumed for the tunneling type of electrical contacts [12], [27], [28] . The dimensions of the contacting members are assumed to be in 94 nanoscale. We considered a wide range of electrode diameter, 10 160 nm, as electrodes of this range are common in transistors [29], [30] , nanowire, nanofiber, and nanorod based novel devices [31] [33] . Sheet resistance of both the contact members is estimated as [34] , where the unit of the sheet resistance The work function of Cu thin films is 4.56 eV [34] . The interfacial layer is assumed to be vacuum (relative permittivity = 1.0 and electron affinity eV). The tunneling current density through the contact interface is shown in Fig. 4.6 for various input voltages (Fig. 4.6(a)), inner radius to outer radius ratios (Fig. 4.6(b)), outer radii (Fig. 4.6(c)), and interfacial layer thicknesses (Fig. 4.6(d)). The profiles of are asymmetric and current crowding occurs mainly at the inner edge, as expected for similar contact members (similar to Fig. 4.2 above). As shown in Fig. 4.6(a), because of the strong nonlinearity in the characteristics of a tunneling junction, increases and exhibits a stronger radial dependence when the applied voltage increases. This strong voltage dependence of electrical properties of the tunneling junction is in sharp contrast with those of ohmic contacts (Sec. 4.3.1), where th e profiles of and the total contact resistance is independent of the applied voltage and the current density scales linearly with the voltage drop. Figure 4.6(b) shows that, as decreases, that is, the contact length increases, the tun neling current density decreases. The influence of outer radius for a fixed is shown in Fig. 4.6(c). The tunneling current density decreases when increases. However, the total current in the contact structure, , increases with because the total contact resistance of the tunneling junction decreases with (c.f. Fig. 4.7(a) below). In Figure 4.6(d), when the gap distance increases, the current density decreases quickly becau se the tunneling junction becomes more resistive [5], [6] . 95 Dashed lines in Fig. 4.6 are the analytical results calculated from Eq. (4.8) assuming constant tu nneling contact resistivity across the radial contact length, which is the typically assumed one - dimensional tunneling contact. This is calculated from the curve of the metal - insulator - metal tunneling junction by setting everywhere along the contact length. The constant contact resistivity assumptions are inadequate, especially when the tunneling thickness or inner radius to outer radius ratio decreases, or the applied voltage or outer radius increases. For these cases, one should solve the coupled TLM equations, Eqs. (4.4) and (4.5), and the localized MIM tunneling equation (Sec. 2.2) self - consistently to give more reliable predic tions. The total contact resistance of the Cu - vacuum - Cu circular thin film contact is shown in Fig. 4.7 as functions of applied voltage and inner to outer radius ratio . The total contact resistance decreases with , as shown in Fig. 4.7 (a). As increases, the dependence of contact resistance on becomes less significant. The dashed lines are for analytical solutions of the 1D tunneling model with constant calculated from Eqs. (4.8) and (4.9) as previously stated. The differe nce between the 1D model (Eq. (4.9)) and self - consistent numerical calculations (Eqs. (4.4), (4.5)) is significant when or is small, or when or is large. In these regimes where the 1D tunneling model with constant approximations fail to provide reliable predictions, it is necessary to use the self - consistent numerical model. 96 F igure 4. 7 The total contact resistance across the Cu - vacuum - Cu contact interface as functions of input voltage for different (a) outer radius , with fixed and ; (b) inner radius to outer radius ratio , with fixed and . as func tions of for different (c) input voltage , with fixed and ; (d) interfacial layer thickness , with fixed and . All of the material properties are specified in the main text. Solid lines are for self - consistent numerical calculations using Eqs. (4.4), (4.5), and MIM quantum tunneling formulations (Chapter 2, Sec. 2.2) [5], [6] . The black dotted line in (a) is calculated using Simmons tunneling current formula [5], [14], [15] and Eqs. (4.4), (4.5) for . Dashed lines are for ana lytical calculations from Eqs. (4.8 ) and (4.9) with calculated using in the 1D MIM tunneling model. [4], [14], [15] and Eqs. (4.4), (4.5) for . Clearly, the difference between the quantum based self - are widely used for these kinds of studies [4], [12] , are inadequate in sub - nm scale (see Fig. 3(a) of Ref. [6] ). For low applied voltages, in the direct tunneling regime, the MIM junction behaves 97 ohmically, thus varies slightly with , as shown in Figs. 4.7(a) and 4.7(b). When the applied voltage is increased into th e field emission regime (> 1 V), decreases sharply with . This is because the junction is no longer ohmic and the tunneling resistivity decreases rapidly and nonlinearly with the junction voltage, which is a function of position along the radi al contact length. As approaches 10 V, space charge effects become important [5], [6], [35], [36] , and saturates, increasing only slightly with . The effect of the inner radius to outer radius ratio of the upper contact member ( ) on the total contact resistance ( ) is shown in Figs. 4.7(b), (c), and (d), showing similar trends to Fig. 4.3(b) above for the case of constant contact resistivity . Figure 4.7(d) shows that reducing the insulator layer thickness even slightly can affect the contact resistance substantially. Next, we extend our calculations for Cu thin film contacts to different metals: magnesium (Mg), aluminum (Al), gold (Au), and platinum (Pt). The resistivity of a metal thin film is ulk resistivity . for the metals mentioned above are calculated from for and for [37], [38] , where is the electron mean free path and is the thickness of the thin film (Fig. 4.1). The work function [39] , bulk resistivity , and electron mean free path [37], [40], [41] for the metals are given in Table 4.1 . The film thickne ss is assumed to be for all the cases. 98 Table 4. 1 Material parameters for the Cu - insulator - Metal contacts. Metal Mg 3.68 [4 2 ] [4 3 ] [4 0 ] Al 4.08 [ 42 ] [4 4 ] [4 0 ] Au 5.1 [6] [ 37 ] Pt 6.35 [4 2 ] [4 4 ] [ 41 ] Figure 4.8 shows the effects of the work function of contacting member 2 ( ) and the permittivity of the thin insulating layer ( ) on the electrical characteristics of Cu - insulator - metal contacts. Figure 4.8(a) shows that the tunneling current density decreases with increasing . Figure 4.8(b) shows that reduces significantly when the permittivity of the insulating layer increases from 1 to 2.5. Figures 4.8(c) and (d) show the contact resistance (in ) for various contact metals and tunneling films for Cu - insulator - metal contacts as functions of . Contact resistance increases with insulating layer permittivity and work function of contacting member 2, . The potential barrier in the insulating layer increases with the increase of the work function of the contact metal, resulting in lower tunneling current and higher contact resistance. On the other hand, greatly influences the image charge potential and space charge potential [5], [6] in the contact interface. 99 F igure 4. 8 Tunneling current density across the Cu - insulator - Metal contact interfaces for different (a) contacting metals with work functions for fixed , , , , and ; (b) insulator layer permittivity for fixed , , , , and . The total contact resistance across the Cu - insulator - Metal contact interfaces as functions of inner radius to outer radius ratio for different: (c) for fixed , , , and ; (d) , for fixed , , , and . All of the material properties are specified in the main text. 4. 4 Concluding Remarks In this chapter, we presented a self - consistent tunneling model to characterize parallel electrical contacts between two annular thin films. Our model considers the radial variation of contact resistivity along the contact length. We solved the CTLM equations for constant specific contact resistivity and radially varying, tunneling dependent s pecific contact resistivity along the 100 contact length. Our study provides a thorough understanding of the contact tunneling resistance, current, and voltage distributions across nano and sub - nano scale MIM junctions in circular ring type electrical contacts using an inexpensive model from which many general conclusions may be drawn. The effects of contact geometry (i.e. inner and outer radius of the ring contact, distance between the contact electrodes) and material properties (i.e. work function, sheet resi stance of the contact members, and permittivity of the insulating layer) on the radial distributions of currents and voltages across these contacts and the overall contact resistance are studied in detail. The quantum tunneling model includes the effects o f image charge, space charge, and exchange correlation potential. It is found that the contact current density and voltage drop profiles are highly asymmetric at the two edges of the annular contact, even for similar contacting members. This is in sharp co ntrast to the current and voltage profiles of parallel Cartesian nanocontacts. However, the asymmetry reduces when inner radius to outer radius ratio increases; for , the profiles become almost symmetric. Our calculations for tunneling type contacts show that the contact resistance is voltage dependent, increases sharply with , and decreases with . If is increased above 0.9, the of the annular contact increases dramatically. It is found that the analytical solutions of one - dimensional (1D) tunneling junction models (constant voltage across the whole junction) are good approximations of the actual circular (annular) contacts only when the thickness or inner radius to outer r adius ratio is relatively large, or the applied voltage across the contact or outer radius is relatively small. Otherwise, the 1D tunneling model of constant contact resistivity becomes unreliable , and the self - consistent CTLM equations coupled with the spatially dependent tunneling current need to be used to accurately characterize the electrical contacts. 101 In existing CTLM [19], [20] , the interface contact resistivity is almost always assumed to be constant. Thus, the contact resistivity measured using the transmission line method (TLM) would consist of possible intrinsic errors when thin tunneling (e.g. oxide) layer at the contact interfaces is pre sent. In this case, our model would give more accurate evaluation of contact resistivity. The work presented here may be used to better understand the electrical conductivity of nanofiber and nanorod based thin - film devices, where such circular (or annular ) contacts naturally exist. Furthermore, our study reveals that, by varying the contact layer properties and geometry, one can strategically design the radially dependent contact resistivity in circular contacts to achieve desired current distribution. It is worth mentioning that while the work presented here is for tunneling type contacts, our modified CTLM equations with radially varying can also be used for other types of contacts such as ohmic and Schottky contacts. Although a TLM is less computat ionally expensive and easier to implement, it is a simplified approximation of practical 2D electrical contacts. Field solution methods need to be used in the future to accurately evaluate current crowding and fringing field effects, the impact of finite t hickness (or length) in the contact members, and the possible parallel component of current flows in the interface layer [4], [25], [45] [47] . The effects of reactive elements in the circuit, AC response, and imperfect insulator layer on the electrical properties of tunneling type contacts may also be studied in the future. Future studies may also consider the influence of properties of the materials forming the contact and the possible interaction of the semiconductor (or insulator) films under the contact region, such as Schottky barrier, band bending, charge redistribution, and material defects. 102 REFERENCES 103 REFERENCES [1] tunneling type circular nano - J. Phys. D: Appl. 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[46] - term inal thin film contact J. Phys. D: Appl. Phys. , vol. 46, no. 6, p. 065502, Feb. 2013, doi: 10.1088/0022 - 3727/46/6/065502. [47] film contact Appl. Phys. Lett. , vol. 97, no. 20, p. 204103, Nov. 2010, doi: 10.1063/1.3517497. 107 CHAPTER 5 2D - MATERIAL - BASED SCHOTTKY CONTACTS This chapter is based on the published journal paper - Dimensional - Material - vol. 13, no. 6, p. 064021, Jun. 2020, d oi: 10.1103/PhysRevApplied.13.064021, by S. Banerjee , L. Cao, Y. S. Ang, L. K. Ang, and P. Zhang [1] . It is presented he re with the permission of the copyright holder. 5.1 Introduction The undesirably large contact resistance between two - dimensional (2D) semiconductor and three - dimensional (3D) metallic electrodes represent one of the major obstacles towards the development of practical 2D electronic and optoelectronic devices [2] . The engineering of better electrical contacts has become a key research objective in recent years. Extensive efforts have been made to improve current flow through contacts and device performance in 2D material based devices [3] [8] . Recent experimental breakthroughs have demonstrated that the van der Waals metal contact to 2D semiconductor can significantly improve the quality of electrical con tact [9], [10] . This advancement opens up exciting avenues for the exploration how 2D/3D electrical contacts can be further improved. This motivates the need of a physical model that comprehensively includes both the material properties of 2D semicon ductors and the geometrical electrostatic effect in mixed - dimensional nanostructures, which remains rarely studied in the literature thus far. In this chapter, we present a consistent model for calculating the contact resistance, which is important for re alizing 2D - material - based electronics. The model is based on the self - consistent 108 spatial - dependent transmission line model (TLM) developed in Chapter 3, the correct charge injected model for 2D material based electrical contacts, and the effects of roughne ss [1] . The recently developed thermionic charge injection model in 2D materials [11] is coupled with the 2D transmission line model (TLM ) accounting for the varying specific contact resistivity along the contact length [12] . The profiles of current and voltage distribution along contact region and the total contact resistance are calculated for various input voltage, contact dimension, material properties, and temperature. It is found the one - dimensional (1D) models become less reliable when Schottky barrier height (SBH) becomes smaller or when the applied voltage becomes larger, where our self - consistent model is expected to provide an improved evaluation of the 2D - material - base d electrical contacts. Our self - consistent calculation results have been compared with that using the classic Richardson - Dushman (RD) thermionic law. We found that RD law significantly underestimates the contact resistance and overestimates the contact cur rent density for 2D - material - based contacts. We obtain excellent agreement by comparing our numerically calculated results with the reported experimental data [10], [13], [14] . We furt her incorporate the effects of interface roughness in the 3D/2D electrical contacts in our 2D TLM model. The interface roughness can be introduced (or engineered) by substrate doping [15] and is inherently present due to the inevitable presence of interfacial defects during the fabrication process. The impact of surface roughness on contact resistance for ohmic contacts have been studied previously [16] [19] . Previous experiment has also demonstrated that substrate roughness can improve the mobility of 2D transition metal dichalcogenide (TMD) by several orders of magnitude [20] . In our model, the contact interface roughness is modelled as fluctuating Schottky barrier heights (SBH) [21] along the electrical contacts. Using experimental device parameters of Au/MoS 2 electrical contact [13] , we show that the contact resistance at the 2D/3D Schottky contact 109 can be reduced by more than one order of magnitude. The key finding that roughness can improve the 2D/3D electri cal contact quality further highlights the technological importance of roughness engineering for improving the device performance of 2D electronics and optoelectronics. Our findings pave a theoretical foundation for the modeling of contact resistance in 2D /2D and 2D/3D electrical contact and establishes a new viable route towards the design of better electrical contacts to 2D materials using roughness engineering. 5.2 The Model F igure 5. 1 (a) A typical parallel contact between Au and MoS 2 . (b) its transmission line model. The width (transverse dimension) of the two contact members is . Due to the reduced dimensionality and the exotic electronic properties of 2D materials, the electron emission physics deviates significantly from traditional 3D materials [11], [22] [24] . The thermionic emission of cha rged carriers across a 2D - material - based metal/semiconductor Schottky contact is found to be universally governed by a simple current - temperature scaling law [11] , , where and are materials/device - d ependent parameters, and (and 3/2) for a vertical (and lateral) Schottky contact. For 2D transition metal dichalcogenide (TMDC), such as atomically - thin MoS 2 , the thermionic emission is governed by 110 , (5.1) where is the intrinsic SBH, the Fermi velocity m/s for MoS 2 , is the Fermi level, and ps is the carrier injection time determined experimentally [25] . Equation (5.1) deviates significantly from the classic Richardson - Dushman (RD) thermionic law for 3D materials, i.e. [26] . The Shockley diode equation can thus be modified as , (5.2) which is obtained based on the detailed balance principle [26] . Equation (5.2) represents the generalized 2D Shockley diode equation for 2D electronic systems. For comparison, the Shockley diode equation based on the 3D classic RD thermionic law is , where , [27] , is the mass of an electron. TLM equations developed in Chapter 3 (Eqs. 3.5 and 3.6 ) coupled with Eq. 5.2 are used to predict the profiles of current and voltage distributions, and the total contact resistance in nanoscale 2D/3D Schottky contacts (Fig. 5.1). The contact current density in Fig. 5.1 is calculated iteratively from the coupled TLM with Eq. 5.2. Sheet resistance of under the contact can be calculated semi - empirically [13] as, , where is the 2D carrier density with typical value of , is the electric charge, and is the temperature - dependent mobility. 111 5.3 Result s and Discussion F igure 5. 2 (a) Voltage drop , (b) current density , and (c) specific contact resistivity across the contact interface for a monolayer (2D semiconductor) and gold (3D metal) contact for different carrier inject ion time with fixed applied voltage , and contact length 20 nm. Here, , , and . [1] In Fig. 5.2, we show the self - consistent calculation of the voltage drop , injection current density , and the contact resistivity across the contact region for a fixed bias voltage of with different interface charge inject ion time, , for a MoS 2 /Au contact using the experimentally determined device parameter reported previously [13] . In Fig. 5.2(a), it is found that is nonuniform across the contact length. The variation of the increases with a decreasing charge injection time, which indicates a stronger current crowding effect in 2D/3D interface with h igh carrier injection efficiency, as shown in Fig. 5.2(b). This is consistent with previous studies that current crowding effect increases with more conductive contact interfaces [28], [29] . The spatial dependent contact resistivity is more evenly distributed across the contact region for contact with longer injection time [Fig. 5.2(c)]. 112 In Figs. 5.3(a) and 5.3(b), the contact resistance calculated from Eqs. (3.4) is shown as a function of the bias voltage for two different SBHs for 2D/3D contacts. The 2D/3D contac t exhibits a transition from Schottky contact characteristic, in which increases with decreasing temperature for , to a Ohmic contact characteristic, in which increases with increasing temperature for . Such transition is due to the offset of the SBH by the external bias voltage. In Figs. 5.3(c) and 5.3(d), the temperature dependence of further confirms the Schottky - to - ohmic transition at obs erved in Figs. 5.3(a) and 5.3(b), which is also consistent with the experiment s [13] . F igure 5. 3 Contact resistance as a function of applied voltage for different , for (a) eV, (b) eV. as a function of temperature for different , for (c) eV, (d) eV. Here, the contact is between a monolayer and Au, with , and 50 nm. [1] 113 F igure 5. 4 (a) Voltage drop , (b) current density , and (c) specific contact resistivity across the contact interface for /Ag contact for different contact length with fixed applied voltage . Solid lines are for the self - consistent calculations of Model (A), and dashed lin es are for Model (C). See text for details. Here, , , , and . [1] In Fig. 5.4 the , , and for MoS 2 /Ag contact are shown for varying contact length . The current crowding is strongly amplified in the case of long contact length, because the applied voltage is distributed over a longer resistive network, resulting in increased interface contact resistivity due to the voltage - dependent Schottky barriers. The influence of on is shown in Fig. 5.5 for different applied voltages. In Figs. 5.4 and 5.5, we calculate the results using four different approa ches: Model (A): self - consistent calculations using Eqs. (3.5),(3.6) and (5.2) (solid lines); Model (B): analytical solution (3.7), assuming constant , calculated using fixed in Eq. (5.2) (dotted lines); Model (C): calculations of Eq. (3.5),(3.6 ) with the 3D Richardson - Dushman injection model (dashed lines); and Model (D): analytical solution (3.7) assuming constant , calculated using fixed in the Richardson - Dushman injection model (dash - dotted lines). It is fo und that, for 2D - 3D contacts, the classic Richardson - Dushman 114 injection model significantly underestimates the contact resistance and overestimates the contact current density. We also found that, as increases, the analytical solutions [12] of the TLM with constant calculated using in the charge injection model, which is almost always used in the literature [10], [13], [14] , become less reliable; and our proposed self - consistent model may be used to obtain a more accurate evaluation of such contacts. This aspect is particularly important in the development of industrial - grade field - effect transistor based on 2D semiconductors . According to the International Roadmap of Devices and Systems (IRDS) [30] , the required industry standard bias voltage is 0.65 V and 0.60 V, respectively, for year 2021 and 2030. At these bias - voltage values, we found that the analytical model with both constant and Richardson thermionic injection model severely undere stimate the contact resistance by at least (Model (B)), (Model (C)), and (Model (D)) when compared to our self - consistent model combined with the 2D thermionic charge injection theory (Model (A)), over a wide range of contact length of 20 nm 100 nm. 115 F igure 5. 5 Contact resistance for /Ag contact as a function of contact length for different applied bias (a) 0.1 V, (b) 0.3 V, (c) 0.60 V, and (d) 0.65 V . The two bias voltages in (c) and (d) are the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2030 and 2021, respectively. Solid lines are for the self - consistent calculations of Model (A), dotted lines are for Mo del (B), dashed lines are for Model (C), and dash - dotted lines are for Model (D). See text for details. Here, , , , and . [1] Next, we compare our self - consistent model with the existing experimental works (c.f. Figs. 5.6 and 5.7), for various 2D carrier density , temperature, and metal interface. With suitable values of and , the results from our self - consistent model are in excellent agreement with the experimental data. Figure 5.6 shows that for a given temperature , the contact resistance decreases with , as it has been reported previously [10], [13], [14] . It is evident that, calculations from our self - consistent model provide much better fitting to the experimental data for 2D material - metal electrical contacts than models based on 3D R ichardson - Dushman injection law. 116 F igure 5. 6 as a function of 2D carrier density for different temperatures for (a) Au contacts, and (b) In contacts. Crossed symbols are from experiments [10], [13], [14] ; solid lines are from our self - consistent model from Eqs. 3.5, 3.6 and 5.2, Model (A); and dashed lines are from Eqs. (3.5), (3.6) with the Richardson - Dushman injection model, Model (C). In the calculation, we used , , for contacts [13] , and for contacts [10] . The parameters [31] , [10] , [31] [10] [13] , and are used to fit the experimental results in Refs. [14] , [10] , [13] respectively. For different cases, from top to bottom, different input voltages and are used in (a) and (b) respectively. [13] is assumed for all the cases. [1] 117 F igure 5. 7 as a function of temperature with (a) an increasing trend, and (b) a decreasing trend, for metal contacts with di fferent . Crossed symbols are from experiments [10], [13], [14] ; solid lines are from our self - consistent Model (A), and dashed lines are extracted from model calculations in Ref. [13] . In the calculation, we used , and . The parameters [31] , and [13] are used for contacts respectively. In (a), for the three solid red lines, from top to bottom, , and all with In (b), and for the blue and purple solid lines, respetively. The value of [13] , and [13] is assumed for all the cases. [1] Figure 5.7 shows the comparison of our self - consistent model with experiments on as a function of temperature for different , for metal contacts. For all the cases, our self - consistent calculation from Model (A) (solid lines) provides a much better fitting to the experimental data (symbols) compared to existing models [10], [13], [14] (dashed lines). The three red lines in Fig. 5.7(a), with input voltage from top to bottom, show that the increasing (ohmic characteristic) or decreasing (Schottky characteristic) trends of with temperature depends very sensitively on the input voltage to the contact, which is also evident in Fig. 5.3. This voltage dependence of the contact resistance has not been emphasized and is generally missing in the previous works [10], [13], [14] . Our model suggests that the input voltage 118 must be specifi ed in order to give a meaningful characterization of contact resistance for a given 2D material - metal contacts. F igure 5. 8 (a) Roughness in Schottky barrier height , the resulting (b) current density , and (c) specific contact resistivity across the contact interface for a monolayer MoS 2 - Au 2D/3D contact for different standard deviations (sd). (d) Contact resistance as a function of surface roug hness (standard deviation/ ) for different mean values of . Here, applied voltage , and contact length . [1] cal contact by including a SBH fluctuation term, i.e. , where is calculated by assuming that the SBH fluctuation follows a Gaussian distribution. The fluctuation of the SBH, injection current and contact resistivity profiles are shown in Figs. 5.8(a) - 5.8(c) respectively. As shown in Fig. 5.8(d), the SBH variation ha s a dramatic effect on the contact resistance. In general, is reduced significantly in the presence of roughness. Such reduction is particularly effective for 119 MoS 2 /Au contact with large SBH (e.g. 0.3 eV). Such roughness - induced contact resistance redu ction is reminiscent to the previously reported mobility enhancement of 2D TMDs due to the presence of crested rough substrate [20] . Thus, the finding in Fig. 5.8(d) s uggests that roughness not only improves the mobility but also decreases the contact resistance with 3D metals. The reduction of contact resistance with interface roughness is also achieved in 1D electrical contacts, i.e. both contact members with constant voltages applied uniformly across the contact region (not shown). The reduction in contact resistance due to fluctuation of interface resistance can be easily understood from a general circuit theory. Consider a rough resistive interface presented betwee n two conductors. Because of fluctuation of the conductivity along the interface, there will be local regions of highly conductive spots formed, whose resistance can be much smaller than that of a uniform interface. Such a resistive interface may be consid ered as a set of equivalent element resistors connected in parallel along the interface between the two contacting members, whose total interface resistance is , which is determined by the smallest resist or in the parallel connection, i.e. the equivalent resistor at the highly conductive spots. As a result, this leads to a reduced overall interface resistance compared to a uniform interface. ation in the conductivity along the interface (e.g. induced by doping, SBH, etc) and needs not be to physical roughness. Similar benefits of surface roughness are found to decrease the contact resistance in organic transistors [17] and to increase the mobility of charge carrier injection in organic and 2D transistors [17] , [20] . 120 5.4 Concluding Remarks A self - consistent transmission line model to quantify and model the current distribution and contact resistance in 2D materials based contacts is constructed and validated with the existing experimental works. It is found that interface roughness can signi ficantly reduce 2D/3D electrical contact resistance. Our findings provide a theoretical foundation for the modeling of 2D/2D and 3D/2D electrical contacts and further reveal a new route towards efficient 2D material electrical contact through roughness eng ineering. 121 REFERENCES 122 REFERENCES [1] Two - Dimensional - Material - Phys. Rev. Applied , vol. 13, no. 6, p. 064021, Jun. 2020, doi: 10.1103/PhysRevApplied.13.064021. 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Zhong et al. 2 Contacts with Scientific Reports , vol. 6, p. 21786, Mar. 2016, doi: 10.1038/srep21786. 125 C HAPT ER 6 INTERFACE ENGINEERING OF ELECTRICAL CONTACTS This chapter is based on the manuscript under review for a journal publication, S. Banerjee , J. Luginsland, and P. Zhang [1] . 6.1 Introduction Engineering electrical contacts to achieve desired interfa ce current transport is crucial for next generation electronics [2], [3] . Several efforts have been made to reduce the current crowding and improve the current transport in electrical contacts by making the proper ch oice for electrode thickness [4] , doping, electrode material and its geometry [5] [6], [7] , optimizing the current spreading layer [8] and the gate bias voltage [9] . The existing studies give no hint on the variation of current al ong the contact length and the importance of interface layer engineering to diminish the current crowding effects. The crowding is especially strong for contacts with low specific contact resistivity [10] [11], [12] . Increasing specific contact resistivity tends to reduce current crowding; however, it increases the total contact resista nce that may lead to increased Joule heating and degradation of the contact. Because of this tradeoff, it is particularly challenging to design electrical interfaces to reduce current crowding without decreasing the total current in the circuit. Our previous studies (Chapters 3, 4, and 5) [12] [14] showed that current and voltage distribution along the contact length greatly depend on the interfacial layer properties and geometry. In this chapter, we demonstrate how to precisely customize their profiles along the contact length by interface engineering. We characterize ohmic, Schottky [14], [15] and tunneling type [12], [16] [18] electrical contacts. Our goal is to maximize the control over electrical contact 126 operation and heat distribution by strategically varying the specific contact resistivity along the co ntact length. We use modified two - dimensional transmission line model (TLM) [12], [13] , where depends upon the local voltage drop and contact current density. The spatial variation of may be achieved by varying the doping, thickness, or shape of the contact layer, or by introducing impurities, such as, resistive co ntaminants, oxides, or foreign objects along the interface . Electrical properties of the engineered interfaces are investigated for various input voltage, contact dimension and geometry, and material properties. Solving the TLM equations self - consistently, we find spatial profiles of that can reduce current crowding, increase current transfer length, improve current transport, steer and redistribute current in the contact area. Most importantly, we find that the severe current crowding in highly conduc tive ohmic contacts can be eliminated by introducing a thin tunneling layer between the contact members. If the tunneling layer is sufficiently thin and the contact length is large, the change in the total contact resistance is found to be insignificant. The methods used here can be applied to characterize various contact geometries shown in Fig. 6.1. Controlled current and voltage distribution can be achieved via engineered spatially varying contact layer properties and geometry (Fig. 6.1) [3] . Note that, the transmission - line model, in general, underestimates the extent of the current crowding, which may be more accurately accounted for by the field solution approaches [9] [11] . However, such simplified models have been used successfully to capture the basic scalings and physics for the char acterization of mesoscale and nanoscale electrical contacts [12], [13], [19] [22] . In this chapter, we analyze nanoscale copper (Cu) thin film contacts and Gold - contacts as examples. The concepts, approaches, and results should be important to the design of any circuits where electrical contacts are of conc ern, such as semiconductor devices [20], [23] , integrated 127 circuits [24] , low - dimensional materials based electronics [2], [25] [27] , and all solid - state batteries [28] . F igure 6. 1 Electrical contact between contact member 1 and 2 for different electrode geometry. (a) Electrical contact with uniform contact interface, (b) electrical contact with a spatially varying engineered interfacial layer, which is used to control the voltage and current distribution. 6.2 The Model The formulation is based on the modi fied transmission line model for Cartesian (Chapter 3) [12] and circular (Chapter 4) [13] contact structures, coupled with the improved thermionic emission current injection model for 2D materials [14], [15] (for 2D/3D Schottky contacts), or the self - consistent quantum model for one - dime nsional MIM junctions (Chapter 2) [18], [29] (for tunneling type contacts). As shown in Fig. 6.2, the sheet resistance of the two contacting members is and , respectively. The spati ally dependent specific interfacial resistivity (also termed specific contact resistivity) is and for the Cartesian and circular contacts, respectively . The goal is to engineer a spatial profile of or in order to suppress current crowding. 128 While the modified TLMs have been presented in Chapters 3 and 4 before [12], [13] , the governing equations are given below for completeness. F igure 6. 2 Electrical contact between two contacting members in (a) Cartesian, (b) circular geometry. (c), (d) its corresponding transmission line model. In (a) and (b), a thin interface layer (ohmic, Schottky, or tunneling type ) is sandwich ed between the two contacting members. The thicknesses of thin film 1 and 2 are and , respectively. For Cartesian electrical contacts in Fig. 6.2a, its TLM in Fig. 6.2c gives [12] , , , , , (6.1) where represents the current flowing at through the lower or upper contact member respectively, and is the local voltage at along the lower or upper contact member, respectively, and is the effective transverse dimension of the contacts, and are the local current density and the local voltage drop across the contact interface at , respectively . Note that, from Eq. (1) , where is the total current in the circuit, to be determined from the boundary conditions, , , , , (6.2) 129 where we assume the voltage of the upper contact member at is 0, and the externally applied voltage at of the lower contact member is . Note that , and . For the contact model in Fig. 6.2(b), the contact resistance is defined as, . (6.3) For circular (ring) electrical contacts shown in Fig. 6.2b with its TLM in Fig. 6.2d, we have [13] , , , (6.4) where represents the current flowing at along the radial direction of thin films 1 and 2, respectively, and is the local voltage at along the radial direction of thin films 1 and 2, respectively. and are the local current density and the loc al voltage drop across the contact interface at , respectively . From Eq. 6.4, , where is the total current in the circuit to be determined from the following boundary conditions, , , , , (6.5) where we assume the voltage of the upper contact member at is 0 and the external voltage is applied at to the lower contact member, is t he outer radius of thin film 2 and is the inner radius of both the films. Note that , , and . For the contact model in Fig. 6.1(c), the contact resistance is defined as, . (6.6) For ohmic contacts, and can be prescribed. For 2D - semiconductor/3D - metal Schottky contacts, the local contact current density or is calculated from the 2D 130 thermionic emission model [14], [15] and for metal - insulator - metal (MIM) tunneling type contac ts, it is calculated from the one - dimensional MIM quantum tunneling model including space charge effects (Chapter 2) [18], [29] . and ar e then determined from these contact current densities by . The coupled equations are solved self - consistently, with more detailed descriptions in Chapters 3 and 4 [12] [13] . We first characterize both Cartesian and circular ohmic contacts with varying along contact length or radius, respectively. We find varying pa rabolically and linearly can effectively reduce the current crowding effects in planar and circular ohmic contacts, respectively. Next, we analyze the 2D - semiconductor/3D - metal contacts to increase the current transfer length by varying the Schottky barrier height (SBH) along the contact length . Finally, we introduce a thin (in sub - nanometer or nanometer) tunneling layer between the highly conductive contact members to reduce current crowding, without increasing the total contact resistance significantly. 6.3 Results a nd Discussion We analyze Cartesian ohmic contacts in Fig. 6.3 and circular ohmic contacts in Fig. 6.4. The input voltage is the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2030, which is given to contact member 1, at for the planar structure and at for the circular structure. Upper contact members at (Fig. 6.2a, 6.2 c ) and (Fig. 6.2 b , 6.2d) are grounded for the two structures under study. Thickness of both the contact members are assumed to be sam e, . The spatial or radial variation of can be realized by varying the doping or thickness or geometry of the contact layer, or by introducing impurities, such as, resistive contaminants, oxides, or foreign objects along the interface [12], [31] [35] . In Figure 6.3, we explore the reduction of the severe current crowding 131 (c.f. Fig. 6.3a, black dotted line) at the highly conductive planar (or Cartesian) Cu - Cu ohmic con tacts by varying the interfacial layer resistivity parabolically along the contact length. For our calculations, we assume with the minimum at half of the contact length , where is a constant. The sheet resistance of copper (Cu) is [13], [36] , where the unit of the sheet resistance [11], [20], [21] . Contact length , and the width (transverse dimension) of the contact members . Figure 6.3a shows that the profile of contact current density strongly depends on . The profiles of can be explained by simple current transport theory in a circuit, where electric current flows through the least resistive path. When is increased, the inhomogeneity of the contact current distribution dec reases. At around , the interfacial current becomes almost uniform along the contact length. The total contact resistance as a function of is plotted in Fig. 6.3b for different contact lengths. For all the contact lengths plotted here, increases only slightly with , e.g. for , is increased at most by 50% within the range of . Hence, evidently, it is possible to eliminate current crowding effects and achieve uniform contact current distribution without sacrificing the total current in the circuit. In practical circuit design and fabrication where it might be difficult to control the shape of a parabola, one can use a step variation by just making the edges of a contact interface (of planar, similar contact members) more resistive than the rest of the contact area. The approach used here to minimize the current crowding effects can be extended to contacts with different elec trode thickness, material, and geometry. 132 F igure 6. 3 Engineered ohmic contact in Cartesian geometry (Fig. 6.2a) with specific contact resistivity . (a) Contact current density along the contact length for different values of ; (b) contact resistance as a function of for different contact length . The input voltages is the required industry standards according to the Internati onal Roadmap of Devices and Systems (IRDS) [30] for year 2030. The thic kness of both Cu contact members are , with a resistivity of [36] , which gives sheet resistance . Contact length , and the width (transverse dimension) of the contact members . In Fig. 6.4, we investigate the current transport for circular ohmic contacts with linearly varying specific contact resistivity along the contact radius. Note that linearly varying specific contact resistivity is found to strongly modify the current densi ty profile for planar contacts [12] . Here, we assume radially varying , outer radius of the upper contact member (Fig. 6.2b) , and the inner radius of both the contact membe rs . is a linearization constant. The contact member 1 is assumed to be copper (Cu) with sheet resistance [13], [36] . 133 F igure 6. 4 Ohmic contacts in circular geometry (Fig. 6.2b) with linearly varying specific contact resistivity. (a) Contact current density along the contact length for different values of linear constant . (b) Contact resistance as a function of for different sheet resistance ratio . Here, we use . In (a) . The input voltage is the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2030. The contact member 1 is assumed to be copper (Cu) with sheet resistance [13], [36] , outer radius of the upper contact member , and the inner radius of both the contact members . As shown in Fig. 6.4a, linear variation of can reduce the current crowding effects for circular contacts. In particular, current crowding at the inner edge reduces significantly when is positive. Figure 6.4b shows that for circular contacts increases with rapidly for and remains almost constant when . Therefore, one can get a desired interfacial curre nt distribution profile without altering the overall contact resistance considerably. Hence, engineering the spatially varying interfacial contact resistivity can provide strategic thermal management of the integrated circuits and systems. 134 F igure 6. 5 Engineered Schottky contacts in Cartesian geometry. (a) Schottky barrier height , (b) the corresponding contact current density along the contact length for contacts for different values of , and (c) contact resistance as a function of for different input voltage . Dashed lines are for contacts with uniform [37] . The Fermi level [37] , and carrier injection time . The bias voltage in (b) is the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2021. Here, , , and . Current crowding is an unavoidable consequence of geometrical confinement and resistivity mismatch at the 2D - semiconductor/3D - metal Schottky junctions, where the current transport between the semiconductor and the metal contact is concentrated at the front edge of the contact [38] [9], [14], [26], [39] [41] . In Fig. 6.5, we study the engineering of such contacts by spatially varying the Schottky barrier height (SBH). We use the one - di mensional (1D) thermionic emission equation for 2D materials Eq. (5.2) [14], [15] , coupled with the TLM equations, Eqs. (3.5) and (3.6) [12] to analyze such 2D/3D contacts. For 2D transition metal dichalcogenide (TMDC), such as atomically - thin MoS 2 , the thermionic emission is governed by , where is the intrinsic Schottky barrier height (SBH), is the Fermi level, is the SBH, the Fermi velocity m/s for MoS 2 , and 135 ps is the carrier injection time determined experimentally [42] . The local contact current density at any position along the contact length is, . We assume that the SBH is a function of , eV , where is a constant, as shown in Fig. 6.5a. The injection current density at the contact interface for different values of is shown in Fig. 6.5b. It is found that the current crowding for uniform SBH (c. f. black dashed line in Fig. 6.5) can be reduced considerably by choosing the value of (e.g. Fig. 6.5b, ) . The bias voltage is the required industry standards accordin g to the International Roadmap of Devices and Systems (IRDS) [30] for year 2021. Figure 6.5c shows the contact resistance as a function of for different input voltage . Dashed lines are for contacts with uniform along the contact [37] . We see that the total contact resistance depends strongly on the parameter and the input voltage . The difference in contact resistance for engineered and uniform SBH is large for low bias voltages but becomes smaller for high bias voltages, for the chosen specific case here. Since the thermionic charge injection current for 2D materials sensitively d epends on both the bias voltage and temperature [14], [15] , [43] , the engineered SBH profile requires a more detailed characterization for practical implementation. 136 F igure 6. 6 Tunneling type electrical contacts. (a) Contact current density , and (b) specific contact resistivity along the contact length for Cartesian tunneling contacts. Solid lines are for self - consistent numerical calculations using Eqs. (3.5) a nd (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29] , for different values of gap distance D and work function of contact members . Sheet resistance of both the contact members is assumed to be . Dashed lines are calculated analytically with constant calculated using in the 1D MIM tunneling model. Black dotted lines are for an ohmic contact wi th , analytically calculated from the TLM equations. is the total contact resistance. Next, we investigate the reduction of current crowding for a highly conductive ( ) ohmic contacts by tunneling engineering. We introduce a thin insulating layer of uniform thickness along the contact length between the contact members. Current transport in the contact region is no longer ohmic and is governed by the quantum tunneling p henomenon [16] [18], [29] . We solve Eqs. (3.5) and (3.6) along with the metal - insulator - metal tunneling junction equation (Chapter 2) [ 18], [29] . The local contact current density at any location from contact member 1 to contact member 2 is calculated based on the coupled 1D Schrödinger - Poisson solutions in the MIM junction [18], [29] . Our quantu m model of the junction (Chapter 2) includes emissions from both cathode (contacting member 2) and anode (contacting member 1), the effects 137 of image charge potential [18] , space charge, and exchange correlation potentials [44] . For given values of the work function of the two contact members , electron affinity , thickness , and relative permittivity of the insulator layer, the local contact current density can be calculated from this 1D quantum model for an input of the contact voltage drop at any location [18], [29] . The calculation of this - relation is coupled with TLM, Eqs. (3.5), (3.6), and is solved self - consistently. We consider nanometer and sub - nanometer sc ale tunneling layers in Fig. 6.6 and Fig. 6.7, respectively. The current fabrication technology can manufacture nodes as small as 3 nm [45], [46] . The International Roadmap of Devices and Systems (IRDS) [30] predicts that 1.0 nm nodes may be implemented tentatively within few years, and the scale is expected to go down even further, in sub - nanometers. Figure 6.6 shows the contact current density , and the specific contact resistivity along the contact length for Cartesian contacts. For these calculations, the contact length is assumed to be 100 nm. Width and thickness of both the contact members are 10 nm. Solid lines are for self - consistent numerical calculations for the tunneling ty pe contacts, using Eqs. (3.5), (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29] , for different values of gap distance (insulator laye r thickness) D and work function of contact members . Dashed lines are for analytical calculations (See Eq. (3.8) of Chapter 3) of tunneling contacts with constant obtained using in the 1D MIM tunneling model. Sheet resistance of both the c ontact members is assumed to be . We solved two cases, i) for and , and ii) for and . The interfacial layer is assumed to be vacuum (relative permittivity = 1.0 and electron affinity eV). Black dotted lines are for an ohmic contact, calculated from Eq. (3.8) with specific contact resistivity , and sheet resistance ratio . We used 0.6 V as the input voltage, 138 which is the required industry standards according to the International Roadmap of Devices and Systems (IRDS) [30] for year 2030. It is clear that the interfacial current is much more evenly distributed for the contacts with a tunneling layer. The current crowding decreases significantly when the gap distance between two contact members is increased. The specific contact resistivity along the contact length, plotted in Figs. 6.6b, is about 2 orders of magnitude higher for tunneling contacts for the two cases considered. However, the total contact resistance, shown in the table in Fig. 6.6a, is still within the same order of the ohmic contact. This is because the total current in the circuit (i.e. area under the curves in Fig. 6.6a) does not decrease significantly. Similar calculations are done for Cu - vacuum - Cu contacts in Fig. 6.7 with a smaller gap distanc e (in sub - nanometer). The work function of Cu thin films is 4.56 eV [36] . For these calculations, the thickness of both Cu contact members are , with a resistivity of [36] , which gives sheet resistance . Contact length , and width , The interfacial layer is assumed to be vacuum (relative permittivity = 1.0 and electron affinity eV). 139 F igure 6. 7 Tunneling type electrical contacts. (a) Contact current density , and (b) specific contact resistivity along the contact length for Cartesian Cu - vacuum - Cu tunneling contacts. Solid lines are for self - consistent numerical calculations using Eqs. (3.5) and (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29] , for different values of gap distance D . Dashed lines are calculated analytically with constant calculated using in the 1D MIM tunneling model. Black dotted lines are for a n ohmic contact with , analytically calculated from the TLM equations. Figure 6.7 shows similar trends to those in Fig. 6.6. The current crowding decreases significantly when D increases. Although, (Fig. 6.7b) is orders of magnitude higher for tunneling contacts, the total contact resistance, plotted in Fig. 6.8a (crossed symbols) is still within the same order of the ohmic contact. Therefore, compared to a perfect ohmic contact with very small , tunneling type contacts with ultrathin insulator layer may help to achieve better contact current distribution and thermal management. Note that if the gap distance is increased for contacting members with high work function, then the junction will be come highly resistive and the total current transport will be reduced severely. 140 F igure 6. 8 Contact resistance as a function of (a) input voltage , and (b) contact length for Cartesian Cu - vacuum - Cu tunneling contacts. Sol id lines are self - consistent numerical calculations using Eqs. (3.5), (3.6), and MIM quantum tunneling formulations (Chapter 2) [18], [29] , for different values of gap distance D. Dashed lines are calculated analytically with constant calculated using in the 1D MIM tunneling model. Black dotted lines are for an ohmic contact with , analytically calculated the TLM equations. Crossed points in (a) are for the four cases shown in Fig. 6.7. Figure 6.8 shows the tunneling contact resistance as functions of input voltage and contact length . For low voltages, the difference betwee n the contact resistance for the ohmic contact and the corresponding tunneling contact is prominent. However, as voltage increases, the difference becomes smaller, which is caused by the saturation of the tunneling current in metal - insulator - metal due to s pace - charge effects [18], [29] . As shown in Fig. 6.8b, as contact length increases, the increase of total contact resistance due to the tunneling layer b ecomes smaller. Thus, our proposed method reducing current crowding with a tunneling layer would become more effective for longer electrical contacts. 141 6.4 Concluding Remarks In summary, we have proposed methods to effectively control current distribution and contact resistance in nanoscale electrical contacts. We have used the two dimensional TLM [12], [13] for ohmic contacts, and TLM coupled with the thermionic injection model [14], [15] for Schottky contacts and the quantum self - consistent mod el [18], [29] for tunneling type contacts. Our study shows that severe current crowding in highly conductive electrical contacts can be effectively reduc ed by spatially varying the contact layer properties and geometry, or by introducing a thin nanometer or sub - nanometer scale insulator layer between the contacting members. This theoretical study also provides insights for strategic current steering and re distribution at the contact interface, which can aid in better thermal management of the overall circuit. The local heating induced effects, such as thermal hotspots [47] and aggravation of electromigration [48] , can be miti gated by manipulating the specific contact resistivity along the contact length. It is worthwhile to note that the effects of the transverse dimension, possible charge trapping inside the contact layer, reactive elements and their effects on the time - depe ndent dynamics are ignored in the present study. Moreover, the transmission line model [12] cannot fully capture the current crowding and the fringing fields near the contact corners [9], [11], [21] . In future, field solution methods [11], [21], [49] may be used to have more accurate evaluation of these effects as well as the impact of finite thickness of the contact mem bers and the interfacial layer. 142 REFERENCES 143 REFERENCES [1] manuscript under review , 2021. [2] L. - Nat Electron , vol. 1, no. 3, pp. 158 159, Mar. 2018, doi: 10.1038/s41928 - 0 18 - 0045 - 9. [3] 10,755,975 B2, Aug. 25, 2020. [4] C. - K. Wang, T. - H. Chiang, Y. - Z. Chiou, and S. - Effect on Nitride - Based Light - Emitt ing Diodes Using Modulated P - Extension Electrode Jpn. J. Appl. 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This theoretical work also offers insights on the design and engineering of nanocontacts to reduce contact resist ance and improve current transport. 7.1 On Metal - Insulator - Metal Tunneling Junctions A self - consistent model has been developed to calculate tunneling current density in nano - and subnano - meter metal - insulator - metal (MIM) junctions with dissimilar metal e lectrodes [1] . [2] on similar MIM junction. Quantum mechanical analysis has been done to include the effects of exchange correlation potential and space charge potential, by solving the coupled Schrodinger equation and Poisson equation self consistently. The current in diss imilar MIM is found to be polarity dependent. The forward (lower work function metal is positively biased) and reverse (lower work function metal is negatively biased) characteristics cross over at higher voltages. The influence of the work function of the two metal electrodes, thickness and relative permittivity of the insulator layer on the reverse and forward bias J - V curves, have been examined in detail in various regimes from direct tunneling, field emission, to space charge limited regime. It is found that the work function difference between the two electrodes influences the asymmetry of J - V characteristics more profoundly than their individual work functions. The asymmetry increases with increasing insulator layer thickness and relative permittivity. 149 Future works on the metal - insulator - metal tunneling junctions may include effects of electrode geometry. The present model is developed for Cartesian structure. In future, similar formulation may be done for the cylindrical structure as well. Our model i s based on the assumption that the insulator layer is ideal, uniform and pristine. However, in real devices that may not be the case. There may be possible charge trapping inside the insulator film. In order to have a more accurate prediction of tunneling current density, the insulator layer imperfections need to be taken into account. In the future, frequency dependence of the tunneling junction may also be studied. Comparison of the theory and modelling with experiments may also be done. The application o f the proposed model is widespread. Although developed for DC condition, it is applicable to nanojunctions operating up to Near Infrared frequency, since the transit time for electron tunneling through a barrier of nm - scale thickness is typically less than 1 fs [3] [6] . Therefore, our model may be used to study the effects of tunneling resistance in several nanoscale electrical structures with time - varying excitations. For example, i n split - ring resonator arrays, one may include tunneling conductivity in the nanogap to investigate the tunneling induced changes in the frequency response and absorption behavior. Our quantum self - consistent model can also be applied to characterize the r ectification behavior of THz induced scanning tunneling microscopes (THz - STMs). The current profiles at STM tip - vacuum - sample junction can be studied for different THz peak fields, metal tips, sample metals and tip heights, and the basic scaling of the tim e dependent electron dynamics can be obtained. 7.2 On Current Distribution and Contact Resistance The standard transmission line model (TLM) has been modified to include the effects of spatially varying specific contact resistivity along the contact length. The current distribution 150 and contact resistance in ohmic, tunneling and two - dimensional (2D) material - based Schottky contacts have been studied for a large parameter space. Self - consistent solutions have been obtained from the tw o dimensional TLM [7], [8] for ohmic contacts, and TLM coupled with th e thermionic injection model [9], [10] for Schottky contacts and with the quantum self - consistent model [1], [2] for tunneling type contacts. Simple analytical solutions have been derived for the special case of uniform specific contact resistivity. Both Cartesian [7] and circular (or annular) [8] contacts are analyzed. It is found that the current and voltage distribution along the contacts and their overall contact resistance depend greatly on the input voltage, contact geometry, and material properties. For highly conductive ohmic contacts, the current distribution along the contact length is strongly nonhomogeneous [7], [8] . For tunneling contacts, the existing one - dimensional (1D) tunneling junction models become less reliable when the tunneling layer thickness becomes smaller or the applied voltage becomes larger. In these regimes, the proposed self - consistent model may prov ide a more accurate evaluation of the parallel tunneling contacts. On the other hand, a thorough study on the contact resistance of the novel 2D - material - based Schottky contacts showed that the junction characteristics transition from Schottky to ohmic reg ime when the input voltage (potential) is around the Schottky barrier height. We have also found that interface roughness can significantly reduce the electrical contact resistance for 2D material based contacts [9] . The results for ohmic contact are verified with finite element method (FEM) based simulations, and the 2D - material based calculations have been validated with existing theory and experiments. Next, we have proposed methods to effectively control current distribution and contact resistance in nanoscale electrical contacts by strategically designing the specific contact resistivity along the contact length [11], [12] . Our study shows that severe current crowding in highly conductive electrical contacts can be effectively reduced by spatially varying the contact layer 151 properties and geometry, or by introducing a thin nano meter or sub - nanometer scale insulator layer between the contacting members. Most importantly, we demonstrate that the current crowding effects in nanoscale electrical contacts can be mitigated while maintaining similar total contact resistance [11] . Future work on the current distribution and contact resistance may include the effec ts of various contact geometry, insulator layer non - uniformities and AC response. One may also investigate the role of capacitance and inductance in nano - scale contact structures. It would be interesting to study the time - dependent dynamics of such junctio ns, especially when a large contact resistance is coupled with the reactive elements. Our transmission line model is assumed to be two - dimensional, where the effects of transverse dimension are neglected. This issue might be included in future studies. Alt hough widely used, TLM is only a simplified approximation of the practical electrical contacts, where the current crowding and fringing fields near the contact corners cannot be fully accounted for. The impacts of the finite interfacial layer thickness and the possible parallel current components in the interface layer, are also ignored. To accurately quantify these effects, field solution methods need to be incorporated. In the future, one may also extend this work to a contact structure with multiple inte rfacial layers with anisotropic material properties. 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