Lecture 1 Nanoscale MOSFETs. Course Structure Basic Concepts (1-38)

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1 Lecture 1 Nanoscale MOSFETs Course Structure Basic Concepts (1-38) 1

2 Course Layout 7.5 ECTS 7 Lectures 7 Excercises Written exam (t.b.d) Textbook: Nanoscale Transistors: Device Physics, Modeling and Simulation by Lundstrom/Guo. Avaliable electronically. We will cover everything except the last chapter(1-181).

3 Motivation I Modern transistors have thin 1D or D bodies 8 nm SOI 3

4 Motivation II Modern transistors have very short gate lengths L g < l diffusive transport gives a poor description of the device performance HEMTs: nm Lg. 14 nm Si FINFET: 0 nm Lg. We need to understand the behaviour of close to ballistic FETs 4

5 Course Topics Contemporary description of MOSFETs Basic semiconductor concepts (1) Basic MOSFET / CMOS physics () D Ballistic FETs (3-4) Scattering (5) 1D Nanowire and CNT FETs (6) Molecular FETs Single Electron Transistors (7) 1D Tunneling Field Effect Transistors. Ferroelectric FETs. (7). 5

6 Band Structure, Group Velocity Direct band gap semiconductors Isotropic G-valley Parabolic bands for small E E k = E C + ħ m k x + k y + k z Indirect band gap semiconductors (Si, Ge) Multiple anisotropic X/L-valleys Parabolic bands for small E E k = E C + ħ k x + k y m t + k z m l The group velocity for a conduction band electron v x = 1 de ħ dk X 6

7 Nonparabolicity InAs E >> E c introduces nonparabolicity E k 1 + αe k = ħ m k x + k y + k z (-band kp) a: ev-1 (InAs) 0.5 ev-1 (Si) TB: Atomistic bandstructure modeling EMA: Effective mass kp: -band k dot p 7

ħ n π m W E k = ε n + ħ k m x ψ r = φ y, z φ y, z sin nπz W z 1 L eik

8 D / 1D structures The channel of a modern FET is D today - 1D tomorrow (?) ψ r = φ z 1A eik xx+ik y y φ z = W sin k nz = sin nπz/w W y z ε n = ħ n π m W E k = ε n + ħ k m x ψ r = φ y, z φ y, z sin nπz W z 1 L eik xx sin nπy W y y z x ε n,m = ħ π m n W z + m E k = ε n,m + ħ k x 8 W y m

9 Fermi-Dirac Integrals F j η F = 1 Γ j + 1 න 0 ξ j 1 + e ξ η F dξ df j dη F = F j 1 Gamma function Γ 0.5 = π F 0 (η F ) = log 1 + e η F η F = E F E n kt In the non-degenerate limit: F j η F e η F η F 0 Γ 1 = 1 Γ 1.5 = Γ = 1 π Γ.5 = 3 π 4 In the degenerate limit: F j η F η j+1 F / Γ(j + ) η F 0 n positive integer Γ n = n 1! Notes on FD/integrals: Γ p + 1 = pγ(p) 9

10 Fermi-Dirac Integrals η F 3/ F1 n/n xd e η F F In the non-degenerate limit: F j η F e η F In the degenerate limit: F j η F η F j+1 / Γ(j + ) n/n xd η F = E F E n kt F Γ = 3 π 4 Γ 0 + = 1 Γ = π η F η F 1/ F 0 F 1

11 Carrier Statistics: 3D,D,1D Density of states: D D = m πħ D 1D = m πħ 3 D 3D = m N 3D = π ħ 3 E E C 1 E E C Effective density of states: N D = m kt πħ N 1D = πm kt h m kt π ħ 3 v x = 1 de ħ dk X We will derive these later on. n 0 k F = f 0 k, k F = න f 0 E, E F D E de F j η F = k 1 η F = (E F E C )/kt ξ j dξ Γ j + 1 න 1 + e ξ η F 0 0 n 0 = N 3D F 1/ η F n s = N D F 0 η F n L = N 1D F 1/ η F 3D D 1D 11

12 Semi-classical transport f(x, p, t) t f + v x x qe f x = መC(x, p, t)f p x Boltzmann Transport Equation f: distribution function. C: collision operation Zeroth Moment: continuity equation First Moment: drift diffusion nd Moment: energy balance J nx = qnμ n E x + 3 μ n μ n = qτ m m dw dx W 3 nkt Kinetic energy near equilibrium High field, short devices: τ m (E) Scattering increases with energy W > 3 nkt Mean energy larger Drift-diffusion is a poor approximation for nm-scale transistors! 1

13 Ballistic Transport Directed moments E FS f 0 v x x qe f 0 x = 0 p x E 1 Contact in equlibrium f=f 0 qv DS E FD E 1 f=f 0 in the active region but with two Fermi levels! Contact in equlibrium f=f 0 E n s + = 1 A f 0 E F,d, E k k x,k y >0 E f,s E J + = 1 f,d A E(0) k g k k k x,k y >0 qv x f 0 E k E F,d Ld π d න k g k dk We will work out these expressions in detail for the 1D and D FETS in comming lectures 13

14 Quantum Transport 1D The Schrödinger Equation with open boundary conditions: E ψ x = 1e ik 1x + re ik 1x ψ x = te ik x x<0 x>l ψ(x, k) Local Density of states / electron density: LDOS 1 x, E 1 π dk 1 de ψ x, k 1 x n 1 x, E = f 0 E E F LDOS 1 (x, E) Current: I D E = q h T 1 E f 0 E E F f 0 E E F qv D T 1 E = 1 ψ 0, E 1 14

15 Quantum Transport Numerical Solutions a d dx f x n f x n 1 + f x n f x n+1 a EI H Σ 1 Σ ψ = iγ 1 N N matrix Self-Energies models the open boundary contacts: Σ 1 i, j = t 0 e ik 1a δ 1,i δ 1,j Σ i, j = t 0 e ik 1a δ N,i δ N,j N N matrices with 1 element 0 ħ t 0 = m a Source Injection: γ 1 = i Σ 1 1,1 Σ 1 (1,1) = ħ v k a N 1 vector with (1,1) element 0 ψ = igγ G = EI H Σ 1 Σ 1 Retarded Green s function ψ xn (E) 15

16 Quantum Transport NEFG G r = EI H Σ 1 Σ 1 If H is a real space representation: A 1 E = GΓ 1 G Γ 1 = i(σ 1 Σ 1 ) Spectral Density LDOS Tr(A 1 ) equals to the local density of states G 1 n (E) = f 0 E F1 E A 1 π T 1 (E) = trace(γ 1 GΓ G ) Correlation Function Transmission I D E = q h T 1 E f 0 E E F f 0 E E F qv D Tr(G 1n ) equals to the position dependent electron concentraion Can treat scattering and many-body effects. G = EI H Σ 1 Σ Σ s 1 Self-Energy due to scattering 16

17 Excercises Download and work on the excercise sets 1 every week. 17

Lecture 8: Ballistic FET I-V

Lecture 8: Ballistic FET I-V 1 Lecture 1: Ballistic FETs Jena: 61-70 Diffusive Field Effect Transistor Source Gate L g >> l Drain Source V GS Gate Drain I D Mean free path much shorter than channel length