Efficient Simulation of Nanoelectronic Devices: Contact Block Reduction Method Overview, Slides of Computer Science

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Efficient Simulation

of Nanoelectronic Devices with

Contact Block Reduction method

What we will discuss

• A brief overview of general approaches to

quantum transport simulations in nano-devices

• Major computational methods for the quasi-ballistic

quantum transport in nano-devices

• Contact block reduction (CBR) method
  • #1 point: reduction to contact block(s)
  • #2 point: use of an incomplete set of eigenstates (generalized von

Neumann boundary conditions)

• #3 point: mode space reduction
• Numerical efficiency estimation

NEGF theory and applications

• Microscopic theory for quantum transport that may

include all interactions

• Known as Keldysh formalism

(sometimes also referred to as

“generalized Kadanoff-Baym approach”)

• Uses second quantization language
• Equivalent to Green’s functions formalism in the case

of the ballistic (coherent) transport

The lowest order perturbation terms in NEGF

include

• Electron-electron interaction through the Hartree-

Fock approximation (correlation & exchange

integrals)

• Does not affect coherence (“ballisticity”) of the

transport [Datta]

• Electron-phonon interaction through a self-

consistent solution for the self-energy term

• No rigorous self-consistent scheme has been

implemented so far even in 1D!

‘True NEGF’ problem

• The computational costs:

From the numerical point of view this

approach is almost hopeless for realistic 2D-

3D devices…

 

loop E

N N O N

 

 

 

Wigner functions

• Transport is rigorously quantum-mechanical
• Similarities to the quasi-classical transport theory

(Boltzmann equation)

• Scattering can be taken into account in a convenient

(standard to MC) way

• Integral equations can be solved using EMC

technique

So far, the method is rather slow (days to obtain a

converged solution). Known simulations are

restricted to quasi 1D systems and RTDs.

Landauer approach & ballistic transport

• Applicability

ballistic or quasi-ballistic quantum

transport

• Main assumption

applied voltage drops at the

interfaces with the device (contact

resistance) or inside the device: no

voltage drop in the leads

• Model

leads have to be “infinitely” more

conducting than the device, and have

known distribution functions and

potentials

Device

( ) ( ) ( ) [ ]

'

' '

2 e

J T E f E f E dE

h

l l

l l l l

= -

ò

Transfer matrix and QTBM

• Transfer matrix methods
  • Standard (1969)
  • Usuki method (1995),

Ferry’s “recursive scattering matrix” (J.Appl.Phys, 2004)

• Boundary conditions are given by

the Quantum Transmitting Boundary Method (QTBM)

• Frensley (1990)
• Lent and Kirkner (1992)
• Ting (1995)
• Laux & Fischetti method (PRB,2004)

Transfer Matrix

1D 2D

Tight-binding k ∙ p Single-band

3D

Single-band,

transport is quasi-1D!

Recursive Green’s functions

Widely used due to the popularity of NEMO 1D

Works with NEGF

Efficient (in 1D and 2D)

Flexible: can be applied to different geometries

RGF

1D

Tight-binding k ∙ p

NEGF with

phonon scattering:

NEMO 1D

2D

Single-band,

multi-band;

MOSFETs,

DGFETs, etc.

3D

Small structures,

single-band;

nano-wires.

Scales as

Unfortunately works only for two contacts!!

Cannot be applied to calculate (at least self-

consistently) the gate leakage current, gate

charge, and, generally, multi-terminal devices.

Recursive Green’s functions

 

2

E x y z

N N O N N

   

 

System’s Hamiltonian

• “Big” (infinite) system’s Hamiltonian:

0 0

0 0

0 0

everything

L L

L

H W

H

H W

W W H

æ ö

÷

ç

÷

ç

÷

ç

÷

ç

÷

ç

÷

=

ç ÷

÷

ç

÷

ç

÷

ç

÷

ç

÷

ç

÷

ç ÷

è ø

O M

K

H

l

is the Hamiltonian of lead l ( l = 1.. L )

W

l

is the coupling of the device to lead l

0

H is the Hamiltonian of the device

Important notations

• D denote the (internal) device region
• C denote contact (boundary) region

D

Device

“contacts”

C

The boundary conditions

D

C

“open” boundary conditions

(for the retarded Green’s function)

“closed” boundary conditions

(for the “decoupled” G

0

)

1D example:

L L

ik x k x

y Ie re

= +

( )

2

2

L

ik x

m x

e

• D

D

= -

h

S

0

C

y =

0

n

C

¶ y

=

¶

Dirichlet

von Neumann

0

n

C

C

a b

y

y

¶

• =

¶

Robbins

We want to find such boundary conditions for G

0

, which would “mimic”

the open boundary conditions, but still form a Hermitian linear eigenvalue

problem…

Then, the elements of GR can be obtained using even an incomplete

set of such eigenstates of the closed system…

Incomplete set of eigenstates and von Neumann

boundary conditions

• Using the von Neumann boundary conditions we can

use incomplete set of eigenstates.

• Typically it is enough to find only <1% for 3D or

5-7% for 2D of all eigenstates to obtain quite accurate

results!

• The explanation is simple:
  • As one can easily check that von Neumann boundary conditions

are zero-th order approximation to the open boundary

conditions (close to the band edge or for a very small real-space

grid step).