L35 Intro to Quantum Transport - nanoHUB.orgL35_Intro_to_Quantum_Transport.pdf · 5 model SOI...

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 non-equilibrium 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.

2 Lundstrom ECE-656 F11

(Thanks to Xufeng Wang for help in preparing this lecture.)

for more information…

1) View an online short course on nanoscale MOSFETs (especially Lecture 6) at http://nanohub.org/resources/5306

2) Consult the NEGF Resource page at http://nanohub.org/topics/Negf

3 Lundstrom ECE-656 F11

Lundstrom ECE-656 F11 4

outline

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

1)   Introduction 2)  Semiclassical ballistic transport 3)  Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)  Discussion 6)  Summary


5

model SOI device

sketch of IBM structure simulation

domain

•  LG = 40 nm and larger •  TCH = 8.6 nm Si •  TINS = 1.1 nm SiON •  VDD = 1 V

A. Majumdar et al., IEEE TED, 56, pp 2270, 2009 (The ETSOI Devices studied here were provided by IBM Research)

LG = 40 nm

(Measured at Purdue Univ. by Himadri Pal.)

Lundstrom ECE-656 F11

6

semiclassical vs. quantum

d k( )

dt= −q

E

r t( ) = r t0( ) + υ ′t( )t0

t

∫ d ′t

υ t( ) = 1

dEdk

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

7

quantum confinement

E =p2

2m*

E =p2

2m* =32kBT

λB =

3m*kBT 10 nm (Si)

p = k = 2πλ

ψ ~ e± ikx

TSi

gate

gate

S D


8

quantum confinement

z

ener

gy --

>

0 TSi

EC = 0

EF

ψ = 0 ψ = 0

−2

2m*

d 2ψdx2

+ EC (x)ψ = Eψ

Hψ = Eψ

ε1

ε2Eigenvalue problem

εn =2n2π 2

2m*TSi2 n = 1,2,3...


9

quantum effects on MOSFETs

1) increases VT

2) decreases the gate cap

3) affects transport along the channel…

Quantum mechanics:

(D. Esseni et al. IEDM 2000 and TED 2001)


10

the MOSFET: an open quantum system

channel drain source

re!ik

1x

teik2x

EC (x)! "qV (x)SOURCE

DRAIN

1eik1x

xL0



outline


1)  Introduction 2)   Semiclassical ballistic transport 3)  Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)  Discussion 6)  Summary


the BTE

12

∂f∂t

+∇r f •υ − q

E •∇ p f =

dfdt coll

f 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 (EF1 and EF2).

f r ,k( ) = 1

1+ e(E−EF )/kBT= 11+ e(EC

r( )+Ek( )−EF )/kBT

?

semiclassical transport

13

EF2

EF1

x

Ener

gy

x0

EC (x)

k

E(k) unchanged from bulk Si with a constant potential.

E

Bottom of E(k) moves up and down with the spatially varying EC(x)


filling states in ballistic transport

14

E

k EF2

EF1

x

Ener

gy

x0

EC (x)

ETOP


15

solving the ballistic BTE

E

k EF2

EF1

x

Ener

gy

x0

n(x0 ) = ∫ LDOS1(E, x0 ) f0 EF1( ) + LDOS2 (E,x0 ) f0 EF 2( )⎡⎣ ⎤⎦dE

�

ε(x)

ETOP

VGS=VDS = 0.6 V

J-H Rhew, Z. Ren, and M.S. Lundstrom, Solid-State Electron. 46, 1800, 2002

16

LDOS(E, x)

17

n(E, x) Electrons injected from source Electrons injected from drain

All injected electrons

18

I(E, x)



outline


1)  Introduction 2)  Semiclassical ballistic transport 3)   Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)  Discussion 6)  Summary


objectives

20

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


21

solving the Schrödinger equation

−2

2m*

d 2ψdx2

+ EC (x)ψ = EψE

nerg

y

EC (x)

x1 3 2 4 a

finite differences �

ψ1

�

ψ2

�

ψ3

�

ψN

N (N-1)

H[ ]ψ = Eψ

ε1

ε2

ε3 Schred nanoHUB.org

ψ 1 = 0 ψ N = 0

22

Schred results: wide Q well

Wide quantum well dense energy levels and surface inversion

EF


23

open quantum systems: source injection

device contact 2 contact 1

SOURCE

DRAIN

xL0

teik2 xre−k1x1eik1x

no E(k) well-defined

E(k) well-defined

E(k)

EC x( )∝−qV x( )

ψ 1 ≠ 0

ψ N ≠ 0


24

solving the wave equation

{ψ} = G[ ]{S}formal solution:

G(E)[ ] = E I[ ]− H[ ]− Σ1[ ]− Σ2[ ]( )−1

(N x N retarded Green’s function)

E[I ]− [H ]− [Σ1]− [Σ2 ]( ){ψ} = {S}(not an eigenvalue problem - energy is continuous)

[H ]{ψ} = E[I ]{ψ}→ E[I ]− [H ]( ){ψ} = 0

Σ1 , Σ2 “self energies”

25

finding n(x) from ψ(x) the device is attached to a bulk contact …..

L

n1(x) = ψ k1(x)

k1 >0∑ 2

f1 E( )

absorbing contact

device contact

k of injected electron

computed wave function within device

Fermi function

of contact

�

E k1( )

x

�

ψk1 = 1Leik1x

f1 E( )

26

finding n(x) from ψ(x)

n1(xi ) =

1L

ψ k1(xi )

k1 >0∑ 2

f1 E( )

n1(xi ) =1πdk1dE

ψ k1(xi )

2⎡⎣⎢

⎤⎦⎥f1 E( )dE

0

∞

∫ = LDOS1 xi ,E( ) f1 E( )dE0

∞

∫

g1D E( ) = 2πdk1dE

Repeat for contact 2 and add the results…..

n(xi ) = LDOS1 xi ,E( ) f1 E( )dE0

∞

∫ + LDOS2 xi ,E( ) f2 E( )dE0

∞

∫

just like the semi-classical ballistic case!


�

EC (x)

position

ener

gy

1)  Guess EC (x)

2)  For each energy:

3)  Determine n (x):

4)  solve Poisson for EC (x)

E[I ]− [H ]− [Σ]( ){ψ} = {S}

�

n(xi) = n1(xi) + n2(xi)

independent energy channels (ballistic)

5) Determine ID I E( ) = 2q

hT (E) f1 − f2( )

ID = I E( )∫ dE 27

recap

28

LDOS (x, E) LDOS from source LDOS from drain

Total LDOS

29

n(x, E) Electrons injected from drain Electrons injected from source


30

I(x, E)



outline


1)  Introduction 2)  Semiclassical ballistic transport 3)  Quantum ballistic transport 4)   Carrier scattering in quantum transport 5)  Discussion 6)  Summary


32


�

Σ1

�

Σ2

state at energy, E position, x

n(xi ) = LDOS1 xi ,E( ) f1 E( )dE0

∞

∫ + LDOS2 xi ,E( ) f2 E( )dE0

∞

∫


33


[Gn (E)] = GΣinG† Σin (E) = Γ1(E) f1(E)+ Γ2 (E) f2 (E)

“in-scattering” function connection to source

population of source


34

scattering

�

Σ1

�

Σ2

state at energy, E position, x in- scattering

out- scattering

G(E)[ ] = E[I ]− [H ]− [Σ1]− [Σ2 ]− [ΣS ][ ]−1

[Gn (x,E)] = G Σ1in + Σ2

in + ΣSin( )G+


35

in-scattering function

EGn x,E( )⎡⎣ ⎤⎦ = G[ ] Σ1

in⎡⎣ ⎤⎦ G+⎡⎣ ⎤⎦

“in-scattering” from contact 1

Σ1in⎡⎣ ⎤⎦ = Γ1[ ] f1

strength of connection to source



36

in-scattering function (phonons)

EGn x,E( )⎡⎣ ⎤⎦ = G[ ] ΣS

in⎡⎣ ⎤⎦ G+⎡⎣ ⎤⎦

in-scattering from another state

ΣSin⎡⎣ ⎤⎦ ~ D[ ] Gn[ ]

strength of connection to phonons



phonon in-scattering

37

E

E + ω

E − ω

ΣSin E( ) ≈ D0 Nω +1( )Gn E + ω( ) + D0NωG

n E − ω( )(absorption) (emission)

phonon emission

phonon absorption

Note: ΣSin⎡⎣ ⎤⎦ Gn⎡⎣ ⎤⎦depends on

solution procedure

38

1) Solve:

G(E, x)[ ] = E[I ]− [H ]− [Σ1]− [Σ2 ]− [ΣS ][ ]−1

[Gn (x,E)] = G[ ] Σ1in⎡⎣ ⎤⎦ G[ ]+ + G[ ] Σ2

in⎡⎣ ⎤⎦ G[ ]+

+ G[ ] ΣSin⎡⎣ ⎤⎦ G[ ]+

2) Compute:

depends on [Gn] solve by iteration!

3) Solve Poisson’s equation


current

39

I(E) ≠ T E( ) f1 − f2( )

I(E) = Trace Σin[ ] A[ ]( )− Trace ΓS[ ] Gn⎡⎣ ⎤⎦( )


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

I ds (u

A/um

)

Vds (V)

Ids vs. Vds, Vg = 0.55V, Vback = 0V

40

IV comparison

ETSOI (measured)

ETSOI (semiclassical - ballistic)

ETSOI (quantum ballistic) ETSOI (quantum with phonon / SR scattering)


41

internal quantities

−30 −20 −10 0 10 20 300

5

10 x 107

Elec

tron

velo

city

(cm

/s)

Transport direction (nm)

Electron velocity

−20 0 20

−1

0

Firs

t con

duct

ion

band

(eV)

ETSOI (semiclassical - ballistic) ETSOI (quantum ballistic) ETSOI (quantum with SRS and phonon scattering)


42

LDOS (x, E) LDOS from source LDOS from drain

Total LDOS

43

n(x, E) Electrons injected from drain


Electrons injected from source

44

I(x, E)

LEFT going current RIGHT going current


outline


1)  Introduction 2)  Semiclassical ballistic transport 3)  Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)   Discussion 6)  Summary


46

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.


47

why is quantum transport important?

from M. Luisier, ETH Zurich / Purdue

4) 3)

2) 1)


outline


1)  Introduction 2)  Semiclassical ballistic transport 3)  Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)  Discussion 6)   Summary


summary

49

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.



questions

1)  Introduction 2)  Semiclassical ballistic transport 3)  Quantum ballistic transport 4)  Carrier scattering in quantum transport 5)  Discussion 6)  Summary

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