Lecture 35: Introduction to Quantum Transport in Devices

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ECE-656: Fall 2011 Lecture 35: Introduction to Quantum Transport in Devices Mark Lundstrom Purdue University West Lafayette, IN USA 1 11/21/11 objectives 1) Provide an introduction to the most

1 ECE-656: Fall 2011 Lecture 35: Introduction to Quantum Transport in Devices Mark Lundstrom Purdue University West Lafayette, IN USA 1 11/21/11 objectives 1) Provide an introduction to the most commonly-used approach for simulating quantum transport (the nonequilibrium Green s function (NEGF) approach) and discuss the interpretation of NEGF simulations. 2) In the process, discuss how quantum effects influence the performance of nanoscale MOSFETs. (Thanks to Xufeng Wang for help in preparing this lecture.) 2

2 for more information 1) View an online short course on nanoscale MOSFETs (especially Lecture 6) at nanohub.org/resources/5306 2) Consult the NEGF Resource page at nanohub.org/topics/negf 3 outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 4

3 model SOI device sketch of IBM structure simulation domain L G = 40 nm L G = 40 nm and larger T CH = 8.6 nm Si T INS = 1.1 nm SiON V DD = 1 V A. Majumdar et al., IEEE TED, 56, pp 2270, 2009 (The ETSOI Devices studied here were provided by IBM Research) (Measured at Purdue Univ. by Himadri Pal.) 5 semiclassical vs. quantum d ( k ) = qe dt r ( t) = r ( t 0 ) + υ ( t ) t d t t 0 υ ( t) = 1 de dk k = k ( t ) Must treat electrons as waves when the potential energy (bottom of the conduction band) varies rapidly on the scale of the electron s wavelength. ΔpΔx Uncertainty principle 6

4 quantum confinement gate ψ ~ e ±ikx p = k = 2π λ S T Si D E = p2 2m * gate E = λ B = p2 2m = 3 * 2 k BT 10 nm 3m * k B T (Si) 7 quantum confinement 2 d 2 ψ 2m * dx + E 2 C (x)ψ = Eψ Hψ = Eψ Eigenvalue problem ε n = 2 n 2 π 2 2m * T Si 2 n = 1,2, 3... energy --> E F ε 1 E C = 0 ψ = 0 ψ = 0 ε 2 0 T Si z 8

5 quantum effects on MOSFETs Quantum mechanics: 1) increases V T 2) decreases the gate cap 3) affects transport along the channel (D. Esseni et al. IEDM 2000 and TED 2001) 9 the MOSFET: an open quantum system source 1e ik 1 x re!ik 1 x SOURCE channel E C (x)! "qv (x) drain DRAIN te ik2x 0 L x 10

6 outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License f the BTE f t + f r υ qe p f = df dt coll ( r, k,t ) a number between zero and 1 r f υ q E p f = 0 equilibrium or ballistic Boundary conditions: Deep source and drain are assumed to be in thermodynamic equilibrium with well-defined but separate Fermi levels (E F1 and E F2 ). f ( r, k ) = 1 1+ e = 1 (E E F )/k B T 1+ e (E C r ( )+E( k ) E F )/k B T? 12

7 semiclassical transport E E(k) unchanged from bulk Si with a constant potential. E F1 Energy k E F2 E C (x) x 0 x Bottom of E(k) moves up and down with the spatially varying E C (x) 13 filling states in ballistic transport E E F1 E TOP Energy k EF2 E C (x) x 0 x 14

8 solving the ballistic BTE E V GS =V DS = 0.6 V E F1 E TOP Energy x 0 n(x 0 ) = LDOS 1 (E,x 0 ) f 0 k E F2 ε(x) x ( E F1 ) + LDOS 2 (E, x 0 ) f 0 E F 2 ( ) de J-H Rhew, Z. Ren, and M.S. Lundstrom, Solid-State Electron. 46, 1800, LDOS(E, x) 16

9 n(e, x) Electrons injected from source Electrons injected from drain All injected electrons 17 I(E, x) 18

10 outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License objectives To: Illustrate the NEGF approach to quantum transport in nanoscale MOSFETs Not To: Derive the NEGF equations Discuss implementation and numerical issues Discuss nanoscale MOSFET device physics 20

11 solving the Schrödinger equation 2 d 2 ψ 2m * dx + E 2 C (x)ψ = Eψ [ H ]ψ = Eψ Energy ε 3 ε 2 ε 1 Schred nanohub.org E C (x) ψ 1 = 0 ψ 1 ψ 2 ψ 3 ψ ψ = 0 N N a (N-1) N x finite differences 21 Schred results: wide Q well E F Wide quantum well dense energy levels and surface inversion 22

12 open quantum systems: source injection contact 1 1e ik 1x re k 1x device contact 2 te ik 2 x SOURCE ψ 1 0 E C ( x) qv ( x) DRAIN well-defined E(k) 0 no E(k) ψ N 0 L x well-defined E(k) 23 formal solution: solving the wave equation [H ]{ψ } = E[I]{ψ } ( E[I] [H ]){ψ } = 0 ( E[I] [H ] [Σ 1 ] [Σ 2 ]){ψ } = {S} (not an eigenvalue problem - energy is continuous) {ψ } = [ G]{S} [ G(E) ] = E I Σ 1, Σ 2 self energies ( [ ] [ H ] [ Σ 1 ] [ Σ 2 ]) 1 (N x N retarded Green s function) 24

13 finding n(x) from ψ(x) the device is attached to a bulk contact.. contact device f 1 ( E) ψ k1 = 1 L eik 1x E( k 1 ) L x n 1 (x) = k 1 >0 ψ k1 (x) k of injected electron computed wave function within device 2 f1 ( E) absorbing contact Fermi function of contact 25 finding n(x) from ψ(x) n 1 (x i ) = 1 L k 1 >0 ψ k1 (x i ) 2 f1 ( E) 1 dk n 1 (x i ) = 1 π de ψ k 1 (x i ) 2 f ( E)dE 1 = LDOS 1 x i, E 0 0 ( ) f 1 E ( )de g 1D ( E) = 2 dk 1 π de just like the semi-classical ballistic case! Repeat for contact 2 and add the results.. ( ) f 1 E n(x i ) = LDOS 1 x i,e ( )de + LDOS 2 x i,e 0 0 ( ) f 2 E ( )de 26

14 recap energy E C (x) 1) Guess E C (x) 2) For each energy: ( E[I] [H ] [Σ]){ψ } = {S} 3) Determine n (x): n(x i ) = n 1 (x i ) + n 2 (x i ) 4) solve Poisson for E C (x) position independent energy channels (ballistic) 5) Determine I D I ( E) = 2q ( ) h T (E) f f 1 2 I D = I ( E) de 27 LDOS (x, E) LDOS from source LDOS from drain Total LDOS 28

15 n(x, E) Electrons injected from source Electrons injected from drain All injected electrons 29 I(x, E) 30

16 outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License filling states in ballistic transport Σ 1 Σ 2 state at energy, E position, x ( ) f 1 E n(x i ) = LDOS 1 x i, E ( )de + LDOS 2 x i,e 0 0 ( ) f 2 E ( )de 32

17 filling states in ballistic transport [G n (E)] = GΣ in G Σ in (E) = Γ 1 (E) f 1 (E) + Γ 2 (E) f 2 (E) in-scattering function connection to source population of source 33 scattering in- scattering state at energy, E position, x Σ 1 Σ 2 out- scattering [ G(E) ] = [ E[I] [H ] [Σ 1 ] [Σ 2 ] [Σ S ]] 1 [G n (x,e)] = G Σ in 1 + Σ in in ( 2 + Σ S )G + 34

18 in-scattering function in-scattering from contact 1 G n [ ] Σ 1 in E ( x,e) = G G + in Σ 1 = [ Γ 1 ] f 1 strength of connection to source population of source 35 in-scattering function (phonons) in-scattering from another state G n [ ] Σ S in E ( x,e) = G G + in Σ S ~ [ D] [ G n ] strength of connection to phonons population of source 36

19 phonon in-scattering E + ω phonon emission phonon absorption E E ω Σ S in ( E) D 0 ( N ω +1)G n E + ω ( ) + D 0 N ω G n ( E ω ) (emission) (absorption) Note: in Σ S depends on G n 37 solution procedure 1) Solve: [ G(E, x) ] = [ E[I] [H ] [Σ 1 ] [Σ 2 ] [Σ S ]] 1 2) Compute: [G n (x,e)] = G in [ ] Σ 1 [ G] + in + [ G] Σ 2 in + [ G] Σ S [ G] + [ G] + 3) Solve Poisson s equation depends on [G n ] solve by iteration! 38

current I(E) T ( E) ( f 1 f 2 ) I(E) = Trace ([ Σ in ][ A] ) Trace Γ S ([ ] G ) n 39 4000 IV comparison I ds vs. V ds, V g = 0.

20 current I(E) T ( E) ( f 1 f 2 ) I(E) = Trace ([ Σ in ][ A] ) Trace Γ S ([ ] G ) n IV comparison I ds vs. V ds, V g = 0.55V, V back = 0V ETSOI (semiclassical - ballistic) I ds (ua/um) ETSOI (quantum ballistic) ETSOI (quantum with phonon / SR scattering) ETSOI (measured) V ds (V) 40

internal quantities Electron velocity (cm/s) 10 x Electron velocity 107 0 5 0 30 20 10 0 10 20 30 Transport direction (nm) 1 First conduction band (ev) ETSOI

21 internal quantities Electron velocity (cm/s) 10 x Electron velocity Transport direction (nm) 1 First conduction band (ev) ETSOI (semiclassical - ballistic) ETSOI (quantum ballistic) ETSOI (quantum with SRS and phonon scattering) 41 LDOS (x, E) LDOS from source LDOS from drain Total LDOS 42

22 n(x, E) Electrons injected from source Electrons injected from drain All injected electrons 43 I(x, E) RIGHT going current LEFT going current 44

23 outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License quantum vs. semi-classical transport 1) Boltzmann Transport Equation ( ) f r,k In equilibrium, this is the Fermi function. 6D, 3 in position and 3 in momentum space 2) Non-equilibrium Green s function formalism ( ) G r, r,e 7D because E is an independent variable. Energy channels are coupled for dissipative scattering. 46

24 why is quantum transport important? 4) 3) 2) 1) 47 from M. Luisier, ETH Zurich / Purdue outline 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary (Thanks to Xufeng Wang for help in preparing this lecture.) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 48

4) NEGF provides a rigorous prescription for including scattering. 5) NEGF is limited by a single particle, mean-field assumption.

25 summary 1) For ballistic transport, the NEGF approach is identical to solving the Schrödinger equation with open boundary conditions. 2) The local density of states divides into parts fillable by each contact. 3) Conceptually, scattering processes are like contacts. 4) NEGF provides a rigorous prescription for including scattering. 5) NEGF is limited by a single particle, mean-field assumption. 6) A basic familiarity with quantum transport should be part of every device engineer s training. 49 questions 1) Introduction 2) Semiclassical ballistic transport 3) Quantum ballistic transport 4) Carrier scattering in quantum transport 5) Discussion 6) Summary 50

Simple Theory of the Ballistic Nanotransistor

Simple Theory of the Ballistic Nanotransistor Mark Lundstrom Purdue University Network for Computational Nanoechnology outline I) Traditional MOS theory II) A bottom-up approach III) The ballistic nanotransistor