————————————————————————————————————————– NUMERICAL QUANTUM MODELING OF FIELD-EFFECT-TRANSISTOR WITH SUB-10NM THIN FILM SEMICONDUCTOR LAYER AS ACTIVE CHANNEL: PHYSICAL LIMITS AND ENGINEERING CHALLENGES ————————————————————————————————————————– A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY TONY LOW AIK SENG B ENG (HONS.), NUS DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE ’The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.’ P A M Dirac ’ it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds sway.’ Richard P Feynman ’The important thing is not to stop questioning Curiosity has its own reason for existing One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality It is enough if one tries merely to comprehend a little of this mystery every day Never lose a holy curiosity.’ Albert Einstein Acknowledgements This thesis is the fruit of four years of PhD work whereby I have been accompanied, supported and influenced by many people It is my pleasure to take this opportunity to express my gratitude to all of them First and foremost, I would like to thank my direct supervisor Prof Li Ming-Fu I have been in his research group since my third year undergraduate studies due to the University Undergraduate Research Opportunity Program During these years, I have known him as a very dedicated researcher and most importantly a kind person His enthusiasm in research and very rigorous and critical thinking has made a deep impression on me I owe him lots of gratitude for having shown me this way of research and inspiring me onto my PhD pursuit In addition, his continual support of my graduate study at National University of Singapore is pertinent for me finally able to complete this work smoothly I am really glad that I have come to get know Professor Li in my life I would also like to expresse my sincere gratitude to my thesis co-advisor Professor Kwong Dim Lee for his taking the effort to monitor my work despite his extremely busy schedule at Silicon Nano Device Lab And Institute of Microelectronics He has a deep appreciation of current technological problems Thus his opinon and suggestions are valuable for my research During my initial stage of graduate study, I am really fortunate to have the help and guidance from my ex-colleague Dr Hou Yong Tian We had many fruitful discussions In particular, his meticulous approach to solving research problems has influenced me tremendously In the later stage of my research, I have the great pleasure to work with Dr Zhu Zhen Gang who has impressed me with his patience and persistence while we were working on our Pseudopotential program I also enjoyed the numerous discussions we had on quantum mechanical problems Both of them had also provided me with brotherly advises and tips for my graduate studies and career that helped me a lot in staying at the right track Thanks to Davood Ansari for introducing me to Finite Element Method (FEM) He had generously taught me valuable tricks in solving problems using FEM I also enjoyed the numerous discussions we had He often create sparks of wit and intelligence that triggers me to rethink the problem in a new light I am also grateful for helpful researchers like Prof M V Fischetti, Prof D K Ferry, Prof D Esseni and Prof S Takagi for their patience in attending to my queries while I was working on mobility problems I owe much gratitude to the computational electronics team at Purdue University, especially Prof Mark Lundstrom, Dr Anisure Rahman and Dr Ramesh Venugopal for providing me with assistance on their program NanoMOS (which is freely available on www.nanohub.org) I would also like to thank Dr Yeo Yee Chia and Prof Samudra who had monitored my work and took effort in reading many of my manuscripts and diligently gave me many invaluable advices and insights on my work I would also like to thank Dr Bai Ping for his help and support in the use of computational resources at Institute of High Performance Computing There are also many others at Silicon Nano Device Laboratory and Chartered Semiconductor who have influenced me over the course of my study and I would like to thank them all for their contributions, however indirect, to this thesis Especially Ang Kah Wee, Chui King Jien, Loh Wei Yip, Shen Chen, Tan Kian Ming, Wang Xing Peng, Wu Nan, Xu Bing and Yu Hong Yu I also like to thank the excellent teachers at Electrical Engineering and Physics department who have imparted invaluable knowledge useful for my work In particular, I very much enjoyed the series of courses on quantum mechanics by Prof Berthold-Georg Englert, which have armed me with the indispensible fundamentals that helped me tremendously in my graduate studies I have also benefited much from the series of courses on Semiconductor Technology conducted by Dr Lap Chan from Chartered Semiconductor I thank my family for understanding me Where would I be without my family? My mother, Annie Sim, who sincerely raised me up and place in me the seed of intellectual pursuit since I was a child My siblings, Linus Low and Kim Low for their patience and understanding I’m am extremely grateful to have the zealous support from my extended family, especially my grandparents, Aunt Cho Hua and Uncle Cho Phong I am very blessed to have the love and friendship of Vera Kim Last but not least, this thesis is dedicated to the spirit of my late father who will always have an important place in my heart This work is supported by Scholarships from Singapore Millenium Foundation (SMF) and Chartered Semiconductor Manufacturing (CSM) In particular, I would like to thank the SMF secretariat John De Roza and Dr Lap Chan (CSM) for their help and support And also a graduate felloship from IEEE Electron Device Society 2005 Abstract The framework of this thesis can be divided into three main segments; (1) Electronic Subband Structure and Device Electrostatics (2) Homogeneous Transport and Low-Field Mobility (3) Quantum Ballistic Transport and Device Limits, all conducted using the Ultra-Thin Body (UTB) devices with a sub-10nm thin film In segment (1), we begins with an assessment of Si and Ge thin film semiconductors’ electrostatics properties in the framework of effective mass approximation We explained how one can perform a unitary transformation to obtain the required effective masses under all common surface orientations The studies of valence bandstructure for various orientations are conducted using the Kohn Luttinger Hamiltonian We also addressed the experimental observation of enhanced threshold voltgae shifts due to surface roughness and how this will impact workfunction designs in these devices Following this, we disucss the empirical pseudopotential method and the methodology to calculate the bandstructure of semiconductor thin films Finally, we performed an ab initio calculation of Si and Ge bandstructure under all common surface orientations We highlighted the important features of our atomistic calculations and the cases where the effective mass approximation will fail In segment (2), we discuss the result of our numerical calculation of electronic transport in the dissipative regime We began from Boltzmann equation and derive the important expressions for calculation of momentum relaxation time for various scattering processes, i.e phonon, surface roughness, Coulomb, for the case of electron For hole, we only consider the more important surface roughness scattering processes in thin film semiconductor devices We assessed the mobilities in all common surface orientations In segment (3), we discuss the result of our numerical calculation of two-dimensional quantum transport in the framework of Non-Equilibrium Green Function (NEGF) Various methods of numerical approach are developed within the effective mass approximation, namely the mode-space approach and real-space approach under the finite differencing schemne and finite element analysis We began with the more numerically viable mode-space Non-Equilibrium Green Function (NEGF) approach We dsicussed the formulation of this method in detail We then studied the performance limits of Ge Double-Gated MOSFETs considering common crystal orientations for surface and transport However, more realistic devices simulations entails at least a 2D description of real-space in order to capture the access geometry effects Thus we discuss two methods to address this issue; the Scattering Matrix and Real-Space NEGF approach Lastly, we conducted a Finite Element Analysis of the quantum transport problem in a 2D waveguide under the NEGF framework We sought to address the issues of how surface roughness configuration on the two SiO2 /Si surfaces of double-gated device will affect the transport properties Contents List of Figures 14 List of Tables 26 Introduction 31 1.1 Overview Of Device Scaling 31 1.2 Objectives of This Work 31 1.3 Overview of Thesis 32 Electronic Subband Structure and Device Electrostatics 2.1 Motivation 2.2 34 34 An Analysis of Subband Structure and Electrostatics of Two-Dimensional Electron Gas in Thin Film Silicon and Germanium Semiconductor Using Effective Mass Theory 35 2.2.1 Electron Quantization Under Different Orientations 35 2.2.2 Hole Quantization Under Different Orientations 36 2.2.3 Model for Enhanced VT H shifts 39 2.2.4 Additional Secondary Effects for Enhanced VT H shifts 42 2.2.5 Electron Quantization on L Valley Occupations 44 2.2.6 Body Thinkness Scaling and Charge Overdrive 44 2.2.7 Impact of Tbody and surface orientation on hole quantization effect 46 2.2.8 Energy dispersion and anisotropy 47 2.2.9 Enhanced VT H for Various Bandstructure 50 2.2.10 Impact on Metal Gate Workfunction Requirement 58 2.2.11 Impact on Threshold Voltage Variation σV th 59 2.3 Empirical Pseudopotential Method For Efficient Thin Film Semiconductor Bandstructure Calculation 66 2.3.1 Concept of Pseudopotential 66 2.3.2 Computational Theory For Pseudopotential Method 67 2.3.3 The Matrix Form For Pseudopotential Method 68 2.3.4 Empirical Atomic Pseudo-Potential 69 2.3.5 Bandstructure Calculation of Silicon and Germanium Thin Film using Empirical Pseudopotential Methods 2.3.6 Method of Surface Passivation 72 2.3.7 2.4 70 Comparison of Empirical Pseudopotential Mthod With ab inito Method 72 Pseudopotential Calculation of Silicon and Germanium Bandstructure Including Exchange Correlation Effects (ab initio Calculation) For UTB MOSFETs Applications 76 2.4.1 Brief Theory Outline of the ab initio Method 77 2.4.2 Important Features of Thin Film Electronic Structures 84 2.4.3 Energy Anisotropy and Impact on Transport Property 85 Homogeneous Transport and Low-Field Mobility 3.1 Motivation 3.2 91 91 Discussion on Theory and Methodology for Calculation of Electron and Hole Mobilities in Si and Ge Thin Film Semiconductors 3.2.1 3.3 92 Fundamentals of Scattering Processes in the Linear Response Regime 92 Momentum Relaxation Time Expression for Two Special Cases 94 3.3.1 Relaxation Time For Electron-Phonon Scattering Process 100 3.3.2 Relaxation Time For Electron-Coulomb Scattering Process 104 3.3.3 Relaxation Time For Electron-Surface Roughness Scattering Process 112 3.3.4 3.4 Calculation of Surface Roughness Limited Hole Mobility 113 Electron Mobility in Germanium and Strained Silicon Channel Ultra-Thin Body Metal Oxide Semiconductor Field Effect Transistors 3.4.1 Calculated mobility in Si UTB MOSFETs 117 3.4.2 Body thickness to power of six dependency 117 3.4.3 Strained Silicon for Mobility Enhancement 118 3.4.4 3.5 116 Germanium UTB MOSFETs 119 Surface Roughness Limited Hole Mobility in Germanium and Silicon channel in Ultra-Thin Body Metal Oxide Semiconductor Field Effect Transistors 120 3.5.1 Optimum channel orientation 120 3.5.2 Optimum surface orientation 121 Quantum Ballistic Transport and Device Limits 131 4.1 Motivation 131 4.2 The Landauer Formalism and Concepts For Mesoscopic Transport 132 4.3 Theory of Quantum Transport Simulation Using Mode-Space Non-Equilibrium Green Function In a Finite Diffference Schemne 134 4.3.1 The System Hamiltonian 134 4.3.2 The Density Matrix 134 4.3.3 Density Matrix in Terms of Green Function 136 4.3.4 Open Boundary Condition and Self-Energy 137 4.3.5 Coupling Function 138 4.3.6 Computing Device Observables: Calculating Charge Density 139 4.3.7 Computing Device Observables: Calculating Current 140 4.3.8 Hamiltonian In Discrete Lattice Representation 141 10 Hence, we have the following matrix elements for i ∈ ΓO ∈ (ΓS ∪ ΓD ), j = Nz ; / ˆ < xi , zNz |P |xi , zNz > = ˆ < xi , zNz |P |xi−1 , zNz > = ˆ < xi , zNz |P |xi+1 , zNz > = ˆ < xi , zNz |P |xi , zNz −1 > = − ∆x ∆z ∆x2 ∆x2 ∆z − and the vector a is again just simply ai,Nz = −ρ(xi , zNz )/ r for i ∈ ΓO ∈ (ΓS ∪ ΓD ) / Consider ΓGt , where r = (xi , zj ) ∈ ΓGt and j = The potential V (x, z) satisfy the Dirichlet condition, V (x, z) = VGt , which is equivalent to the equality Vi,0 = VGt for i ∈ ΓGt Vi+1,1 − 2Vi,1 + Vi−1,1 Vi,2 − 2Vi,1 + Vi,0 ρi,1 + =− 2 ∆x ∆z r Vi+1,1 − 2Vi,1 + Vi−1,1 Vi,2 − 2Vi,1 ρi,1 VGt + =− − ∆x2 ∆z ∆z r (345) (346) (347) Hence, we have the following matrix elements for i ∈ ΓGt and j = 1; ˆ < xi , z1 |P |xi , z1 > = ˆ < xi , z1 |P |xi−1 , z1 > = ˆ < xi , z1 |P |xi+1 , z1 > = ˆ < xi , z1 |P |xi , z2 > = 2 − ∆x ∆z ∆x2 ∆x2 ∆z − and the vector a has to now be modified to include the boundary condition and is given as ai,1 = −ρ(xi , z1 )/ r − VGt /∆z for i ∈ ΓGt and j = Similarly, consider ΓGb , where r = (xi , zj ) ∈ ΓGb and j = Nz The potential V (x, z) satisfy the Dirichlet condition, V (x, z) = VGb , which is equivalent to the equality Vi,Nz +1 = VGb for i ∈ ΓGb Vi+1,Nz − 2Vi,Nz + Vi−1,Nz Vi,Nz +1 − 2Vi,Nz + Vi,Nz −1 ρi,Nz + =− 2 ∆x ∆z r Vi+1,Nz − 2Vi,Nz + Vi−1,Nz −2Vi,Nz + Vi,Nz −1 ρi,Nz VGb + =− − ∆x2 ∆z ∆z r (348) (349) (350) 223 Hence, we have the following matrix elements for i ∈ ΓGt and j = Nz ; 2 − ∆x2 ∆z ∆x2 ∆x2 ∆z ˆ < xi , zNz |P |xi , zNz > = − ˆ < xi , zNz |P |xi−1 , zNz > = ˆ < xi , zNz |P |xi+1 , zNz > = ˆ < xi , zNz |P |xi , zNz −1 > = and the vector a has to now be modified to include the boundary condition and is given as ai,Nz = −ρ(xi , zNz )/ r − VGb /∆z for i ∈ ΓGb and j = Nz Now we are only left with the nodes at the four corners First off, consider r = (xi , zj ) ∈ (ΓO ∩ ΓS ) = (x1 , z1 ), (x1 , zNz ) The potential V (x, z) satisfy the Neumann condition, ∂x V (x, z) = ∂z V (x, z) = 0, which is equivalent to the equality V1,1 = V0,1 = V1,0 and V1,Nz = V0,Nz = V1,Nz +1 respectively Consider first r = (x1 , z1 ); V2,1 − 2V1,1 + V0,1 V1,2 − 2V1,1 + V1,0 ρ1,1 + =− 2 ∆x ∆z r V2,1 − V1,1 V1,2 − V1,1 ρ1,1 + =− ∆x2 ∆z r (351) (352) Hence, we have the following matrix elements for i = and j = 1; ˆ < x1 , z1 |P |x1 , z1 > = ˆ < x1 , z1 |P |x2 , z1 > = ˆ < x1 , z1 |P |x1 , z2 > = and the vector a is given as a1,1 = −ρ(x1 , z1 )/ 1 − ∆x ∆z ∆x2 ∆z − r (353) for i = and j = Next, consider r = (x1 , zNz ); V2,Nz − 2V1,Nz + V0,Nz V1,Nz +1 − 2V1,Nz + V1,Nz −1 ρ1,Nz + =− ∆x2 ∆z r V2,Nz − V1,Nz V1,Nz +1 − V1,Nz ρ1,Nz + =− ∆x2 ∆z r 224 (354) (355) Hence, we have the following matrix elements for i = and j = Nz ; ˆ < x1 , zNz |P |x1 , zNz > = ˆ < x1 , zNz |P |x2 , zNz > = ˆ < x1 , zNz |P |x1 , zNz +1 > = and the vector a is given as a1,Nz = −ρ(x1 , zNz )/ 225 r 1 − ∆x2 ∆z ∆x2 ∆z − for i = and j = Nz (356) F An Iterative Schemne to Solving Poisson Equation in a Self-Consistent Manner We shall describe this schemne for the one-dimensional case, which is applicable to the context of finding a solution in the two-dimensional semi-infinite leads, of which the solution is not a function of the longitudinal direction ∂ ∂ (x) V (x) = −ρ(x) ∂x ∂x where (x) is the material’s electrical permittivity (in C N −1 m−2 ) ρ(x) is the charge density (in Cm−3 ) V has the dimension JC −1 One can easily verify that the dimension is consistent For our context of the semi-infinite lead, we take (x) to be constant and the Dirichlet boundary condition V (x) = Using a constant mesh size of ∆x with N nodes, then we can discretized the above differential equation as follows (t ≡ ∆x−2 );  ··· 0  −2   −2 · · · 0    −2 · · · 0   t    0 · · · −2    0 · · · −2   0 ··· −2    V1     V2     V3       V   N −2     VN −1  VN  ρ1       ρ2       ρ3      = −       ρ   N −2       ρN −1   ρN ⇒ DV                  = −ρ (357) (358) In our numerical algorithm, we begin with an initial description of the potential, V (x) = V (x) And based on this potential, one can obtain the set of eigen-functions ψi (x) and eigen-energies Ei governed by Schroedinger equation The charge density is then solved using the following relation; |ψi (x)|2 ρ(x) = q0 |ψi (x)|2 i D2D Ei i = q0 ∞ 1 + exp( E−µ ) kT dE gi md,i µ − Ei kT log + exp( ) π kT (359) where gi is the subband degeneracy, md,i is the subband density of states mass and µ is the Fermi level So we can yield the corresponding inital charge density ρ0 (x) 20 20 So what can µ has to be adjusted so that the charge neutrality condition is satisfied Therefore, once ρ(x) has been 226 we next to arrive a next better approximation to V (x)? First, define the following; F n = D V n + ρn When self-consistency is reached, the set of V n (x) and ρn (x) (solved based on Eq 359) will yield F = A Newton iteration schemne can be written as; F n+1 = F n + J(V n+1 − V n ) V n+1 = V n − J −1 F n (360) where we have impose the new set of V (x) and ρ(x) to be constraint by F n+1 = J is the Jacobian with matrix elements given by; [J]ij = ∂fi ∂ρi = [D]ij + ∂Vj ∂Vj The problem about this Jacobian is that it does not have a well-defined analytical form for ∂ρi /∂Vj (due to the complicated form of ρ(x) function) To circumvent this problem, we are going to approximate the form of ρ(x) using the familiar charge density relation in bulk semiconductor for electron inversion layer (where we uses q for its charge and qV (x) for its potential energy); ∞ dE µ (x)−E + exp( q kT ) 1 2mde 3/2 ( ) E − qV (x) dE = q µ (x)−E qV (x) 2π + exp( q kT ) ∞ 2mde 3/2 √ = q ( ) E dE µq (x)−E −qV (x) 2π + exp( ) kT ∞ 2mde kT 3/2 √ ( ) E dE = q µ (x)−qV (x) 2π + exp( q kT − E) ρ(x) = q D3D (E) qV (x) ∞ = 2q 2πmde kT h2 = qDef f 0.5 ( 3/2 √ π ∞√ E 1+ µ (x)−qV (x) exp( q kT µq (x) − qV (x) ) kT where mde is the density of states mass for 3D case states j (u) dE − E) (361) Def f is the effective density of is the Fermi-Dirac integral of order j, defined as [Blakemore82](a review found self-consistently, we check for the system total charge µ is to be adjusted accordingly and this process continues until the charge neutrality condition is satisfied 227 paper)[Halen85]; j (u) = Γ(j + 1) ∞ Ej dE + exp(E − u) ≡ d du And Γ(j) is the mathe√ matical Gamma function [W eissteingamma] Some commonly use result are Γ( ) = π, √ 1√ Γ( ) = π and Γ( ) = π Then our Jacobian is well defined analytically as; 2 which have the useful property that [J]ij = j (u) q Def f ∂fi = [D]ij − ∂Vj kT j (u) −0.5 ( = (362) j−1 (u) µq (xi ) − qV (xi ) ) kT Hence, with the Jacobian defined, one can employ the Newton iterative procedures Now, we describe the iterative schemne We began with initial set of V (x) and ρ0 (x) With ρ0 (x), one needs to find the corresponding µ0 (x) for the Jacobian; q µq (x) = kT −1 0.5 ρ(x) qDef f + qV (x) Then using 360, one can find the next better approximation to V (x), V (x) The schemne is re-iterated until the next update to V (x) is negligible to satisfactory criterion set prior to simulation Here, we solved for the case of positive charge q0 The solution for that of electron is then easily obtainable from the solution of the positive charge 228 G Computing the Finite Element Matrix for the General Case of Mass Tensor We see that the FEM matrix involves evaluation of the following mass tensor term; [T ]hi = = M −1 (r) r αh (r) · r αi (r)dΩ ˆ ˆ r∈Ωi      −1 m−1 ˆ ˆ m xz   ∂x αh (r)   ∂x αi (r)   xx dΩ · −1 r∈Ωi mzx m−1 ∂z αi (r) ˆ ∂z αh (r) ˆ zz (363) m−1 ∂x αh (r)∂x αi (r) + m−1 ∂z αh (r)∂z αi (r)dΩ ˆ ˆ ˆ ˆ xx zz = r∈Ωi m−1 ∂x αh (r)∂z αi (r) + m−1 ∂z αh (r)∂x αi (r)dΩ ˆ ˆ ˆ ˆ xz zx + (364) r∈Ωi where αi (r) is the node-wise shape functions for node i M −1 is the 2D mass tensor We ˆ express these shape functions into element-wise, for e.g.; Ph Pi r∈Ωi eh δ(eh , ei ) a b ∂x αh (r)∂x αi (r)dΩ = ˆ ˆ a=1 b=1 r∈Ωi ei ∂x αha (r)∂x αi b (r)dΩ eh One can evaluate ∂x αha (r) using an explicit formula, where the subscript can be related to the element local nodes Where the shape function at node i has the following form within e the triangular element defined by the triangle vertex i, j and k; αi (r) = 1/(2A)[(xj yk − xk yj ) + (yj − yk )x + (xk − xj )y] (where A is the area of the element defined by node i, j and k) Then, [yj − yk ] 2A e ∂y αi (r) = [xk − xj ] 2A e ∂x αi (r) = (365) We also enforce the requirement that the node index i, j and k are arranged in an anticlockwise manner 229 H FEM Matrix Involving Integral of Three Shape Functions Consider the evaluation of the following integral involving three shape functions; [A]hi ≡ αh (r)αi (r)V (r)dΩ r∈Ωi = αh (r)αi (r)αj (r)dΩ Vj r∈Ωi j Ph Pi Pj eh δ(eh , ei , ej ) a b c = a=1 b=1 c=1 r∈Ωi j ei j e αha (r)αi b (r)αj c (r)dΩ Vj (366) The above integral appears in the formulation of Schroedinger equation in FEM framework We began by making the approximation V (r) ≈ j Vj αj (r) And eventually, the above integral can be easily evaluated when the integral of the three shape functions are known We proceed with determining the explicit form for this integral Firstoff, one would find (x 3,y 3) A A (x 1,y 1) A d (x 2,y 2) h Figure 83: Illustration of the derivation of Local Coordinates from Cartesian coordinates it extremnely convenient to use local coordinates instead of cartesian coordinates for the evaluation of this integral The local coordinates are dimensionless with values ranging from to By definition, ξi at any point within the triangle is the ratio of the perpendicular distance from the point to the side opposite to vertex i to the length of the altitude drawn from vertex i Thus, from the Fig 83, one can write; ξi = Ai di = hi A 230 (367) where A = A1 + A2 + A3 One can derive their linear transformation relation;  x   x1 x2 x3        y  =  y1    1 y2  ξ1      y3   ξ2    ξ3 (368) One can also derive the differential form dxdy = 2Adξ1 dξ2 Recalling that the element-wise shape functions αi can be expressed explicitly as; ˜ αi = ˜ [(xj yk − xk yj ) + (yj − yk )x + (xk − xj )y] 2A (369) and based on this, one can obtain the useful identity αi = ξi in the local coordinates repre˜ sentation Thus, one can rewrite the integral in this new representation; Thij ≡ 1−ξ2 ξh ξi ξj dξ1 dξ2 αh (x, y)αi (x, y)αj (x, y)dΩ = 2A ˜ ˜ ˜ r∈Ωi (370) We can easily work out the Thij for all possible h, i and j and the results is simply; Thij =          A 10 h=i=j A 60 h=i=j A 30 otherwise 231 (371) I Matrix Elements For Green Function Using NodeWise Shape Functions as Non-Orthogonal Bases In this section, we would formulate the FEM-BEM equation into its corresponding matrix equation using the node-wise shape functions αi as our basis functions Since, we are dealing ˆ with a set of non-orthogonal basis, it will be advantageous to employ tensor analysis Let us term this set of αi functions the covariant basis and write them as: {|ˆ µ >} Note ˆ α that these functions are real in our context We have the covariant metric gµυ ≡ Sµυ =< αµ |ˆ υ > S basically accounts for the overlap between two covariant basis function We have ˆ α a matching dual function that can be readily derived from the covariant functions whose members have the property of being biorthogonal (we will illustrate this property very soon) to the covariant functions These are the contravariant basis functions defined as follows; |ˆ µ >= α υ |ˆ υ > (S −1 )υµ One can then show that the contravariant functions are indeed α biorthogonal to the covariant functions: < αµ |ˆ υ >= ˆ α space representation, this biorthogonality is expressed as λ (S −1 µλ ) µ < αλ |ˆ υ >= δυ ( in real ˆ α αµ (r)ˆ υ (r)dr One can also easily ˆ α show that the overlap matrix (contravariant metric) of the contravariant functions is in fact the inverse of S; g µυ ≡< αµ |ˆ υ >= ˆ α λσ (S −1 µλ ) < αλ |ˆ σ > (S −1 )συ = (S −1 )µυ With these ˆ α results, one can then relate the covariant and contravariant functions in terms of their metric as follows; |ˆ µ = α |ˆ υ g υµ α (372) |ˆ υ gµυ α (373) υ |ˆ µ = α υ Using the biorthogonality property of covariant and contravariant functions, one can write ˆ the projection operator as follows; I = |ˆ µ >< αµ | Or entirely in terms of just the α ˆ µ ˆ covariant or contravariant basis functions; I = µυ |ˆ µ > g µυ < αυ | = α ˆ µυ |ˆ µ > gµυ < αυ | α ˆ (See Appendix E for more discusssion on this) Hence, we can express the Green function as follows; G(r, r ) = ˆ r| G r = r| αi ˆ αi G |ˆ j ˆ ˆ α αj r ˆ ij αi (r)ˆ j (r )Gi ˆ α •j = ij 232 (374) The placeholder (•) means that the first index is contravariant and the second index is covariant (See Appendix E for more usage of this) αj (r ) is simply the contravariant function ˆ αj expressed in real space ˆ We shall now make use of our new expression of G(r, r ) and substitute into our FEM-BEM equation Following the mathematical workings from Eq 375 to Eq 379, we reduce to solving the following matrix equation G = (A + B)−1 S Hence, our next immediate task is to compute the A (related to FEM) and B (related to BEM) matrix elements 233 234 + ij + ij + ij αh (r ) = ˆ + ij ij r∈Ωi r∈Ωi k 2 M −1 (r) rek ∈∂Ωik G(rek , r ) F (r) 2 M −1 (r) 2 · rek ∈∂Ωik 2 M −1 (r) rek ∈∂Ωik αi (rek ) ˆ r∈∂Ωik αh (r) ˆ 2 M −1 (r) rek ∈∂Ωik · 2 · · r r j (r )Gi dΩ •j rek Gek (rek , r)d∂Ωek Gi αj (r ) •j ˆ rek Gek (rek , r)d∂Ωek Gi αj (r ) •j ˆ · nd∂Ωi Gi αj (r ) ˆ •j ˆ · nd∂Ωi Gi αj (r ) ˆ •j ˆ · nd∂Ωi ˆ · nd∂Ωi ˆ rek Gek (rek , r)d∂Ωek ˆ r αi (r)dΩ M −1 (rek ) 2 r ˆ ˆ r αi (r)α ˆ r αi (r)dΩ M −1 (rek ) 2 rek Gek (rek , r)d∂Ωek )dΩ M −1 (rek ) ˆ r αh (r) αi (rek ) ˆ αh (r) (−V (r) + E + iη) αi (r) − M −1 (r) ˆ ˆ r∈∂Ωik F (r) r F (r) 2 r F (r) r r G(r, r M −1 (rek ) 2 αi (rek )ˆ j (r )Gi ˆ α •j F (r) (−V (r) + E + iη) αi (r) − M −1 (r) ˆ r∈∂Ωik k F (r) r F (r) F (r) (−V (r) + E + iη) αi (r)ˆ j (r )Gi − M −1 (r) ˆ α •j r∈∂Ωik r∈Ωi ijk k r∈Ωi F (r) (−V (r) + E + iη) G(r, r ) − M −1 (r) Let F (r) = αh (r), we have; ˆ = = F (r ) = (376) (375) 235 = + ij + ij ij ij k r∈Ωi k r∈Ωi 2 M −1 (r) rek ∈∂Ωik r∈∂Ωik αh (r) ˆ 2 M −1 (r) rek ∈∂Ωik αi (rek ) ˆ 2 2 · · r Gi •j rek Gek (rek , r)d∂Ωek Gi g jg •j · nd∂Ωi Gi g jg ˆ •j · nd∂Ωi Gi ˆ •j α αj (r )ˆ g (r )dr ˆ rek Gek (rek , r)d∂Ωek ˆ r αi (r)dΩ M −1 (rek ) 2 r ˆ r αi (r)dΩ M −1 (rek ) ˆ r αh (r) 2 ˆ r αh (r) αi (rek ) ˆ αh (r) (−V (r) + E + iη) αi (r) − M −1 (r) ˆ ˆ r∈∂Ωik αh (r) ˆ αh (r) (−V (r) + E + iη) αi (r) − M −1 (r) ˆ ˆ k r∈∂Ωik αh (r) ˆ G is then simply equals to (A + B)−1 S [B]hi ≡ 2 [A]hi ≡ M −1 (r) r∈Ωi rek ∈∂Ωik αi (rek ) ˆ 2 M −1 (rek ) r ˆ r αh (r) 2 · rek Gek (rek , r)d∂Ωek αh (r) (−V (r) + E + iη) αi (r) − M −1 (r) ˆ ˆ · nd∂Ωi ˆ ˆ r αi (r)dΩ (379) (378) (377) αj (r )ˆ g (r )dr ˆ α We can write them in compact form as I = AGS −1 + BGS −1 where [G]ij ≡ Gi , S is the overlap matrix defined earlier and; •j δgh = α αh (r )ˆ g (r )dr ˆ Multiply both sides by αg (r ) and integrate over r ; ˆ J Matrix Elements Contribution Due to Barrier Potential in Empirical Pseudopotential Methods In this section, we discuss how we can add in the effect of a barrier potential The vacuum layers in our supercell is in fact a barrier layer, which in our context, is the dielectric layer Hence, one should include a barrier potential V0 in this region, where V0 is relatively large compared to the energy range considered For a l layer supercell with m layer of atoms (one layer correspond to a thickness of lattice constant a0 ), the barrier potential VB can be expressed as VB (z) = V0 U (z − ma0 ) where U (z) is the Heaviside step function P W, G + k VB (z) P W, G + k = V0 = = = V0 V V0 V V0 Ω = V0 dr exp(i(G − G) · r)U (z − ma0 ) V dr exp(i(G − G) · (r + n1 a1 + n2 a2 + n3 a3 ))U (z − ma0 ) n1 n2 n3 Ω n1 n2 n3 Ω dr exp(i(G − G) · r )U (z − ma0 ) dr exp(i(G − G) · r )U (z − ma0 ) Ω a1 a1 dx exp(i∆Gx x ) = V0 δGx ,Gx δGy ,Gy a3 a2 a2 dy exp(i∆Gy y ) a3 a3 dz exp(i∆Gz z )U (z − ma0 ) a3 dz exp(i∆Gz z )U (z − ma0 ) la0 = V0 δGx ,Gx δGy ,Gy dz exp(i∆Gz z ) la0 ma0   V0 δGx ,Gx δGy ,Gy l−m , ∆Gz = l = exp(i∆Gz ma0 )  V δ , ∆Gz = 0 Gx ,Gx δGy ,Gy − i∆Gz la0 (380) where r = r + n1 a1 + n2 a2 + n3 a3 , r ∈ Ω, Ω is the domain within a chosen unit cell ∆Gx,y,z = Gx,y,z − Gx,y,z 236 K General Mobility Tensor Form Making use of the reduced Boltzmann equation and the relaxation time approximation, we have as in Eq 58; f (k) = f (k) − τsp (k)e ∂f (k) F ·v ∂E (381) F here is defined as the electric field The perturbation from the equilibrium produces a current expressed in its tensor form; Jij = −e 2π f (k) − f (k) v · ei dk (382) where i and j denotes the particular axis for two dimensional transport (ei and ej are their unit vector respectively) The current J in direction i due to a field applied in j (F is applied in j direction), where v = Jij 2π = e2 = = kE τsp (k) 2π e2 4π = ∂i Eei + ∂j Eej ) ∂f (k) F · v v · ei dk ∂E ∞ τsp (E) e2 4π 2 kT E0 2π ∂f (E ∂k F · v v · ei k(E, θ) dEdθ ∂E ∂E ∞ τsp (E)f (E) − f (E) (383) Fj ∂j E∂i E (384) k(E, θ) E0 ∂k ∂E dEdθ (385) and we have the specific result for Jii ; Jii = 4π 2 kT ∞ 2π e2 τsp (E)f (E) − f (E) Fi (∂i E)2 k(E, θ) E0 ∂k ∂E dEdθ (386) The mobility µii is therefore given by; µii = 4π e2 kT n inv 2π ∞ τsp (E)f (E) − f (E) E0 237 Fi (∂i E)2 k(E, θ) ∂k ∂E dEdθ (387) ... Ballistic Transport and Device Limits, all conducted using the Ultra -Thin Body (UTB) devices with a sub- 10nm thin film In segment (1), we begins with an assessment of Si and Ge thin film semiconductors’... 2.2 34 34 An Analysis of Subband Structure and Electrostatics of Two-Dimensional Electron Gas in Thin Film Silicon and Germanium Semiconductor Using Effective Mass Theory ... based on quantum phenomena 1.3 Overview of Thesis Chapter discusses the result of our numerical calculation of the electronic structure of thin film Si and Ge semiconductor One begins with an assessment