Greens Function Methid

Superlattices and Microstructures, Vol. 28, No. 4, 2000
doi:10.1006/spmi.2000.0920
Available online at http://www.idealibrary.com on

Nanoscale device modeling: the Green’s function method
S UPRIYO DATTA†
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285,
U.S.A.
(Received 24 July 2000)
The non-equilibrium Green’s function (NEGF) formalismprovides a sound conceptual basis for the devlopment of atomic-level quantum mechanical simulators that will be needed
for nanoscale devices of the future. However, this formalism is based on concepts that are
unfamiliar to most device physicists and chemists and as such remains relatively obscure.
In this paper we try to achieve two objectives: (1) explain the central concepts that define
the‘language’ of quantum transport, and (2) illustrate the NEGF formalism with simple examples that interested readers can easily duplicate on their PCs. These examples all involve
a short n++ –n+ –n++ resistor whose physics is easily understood. However, the basic formulation is quite general and can even be applied to something as different as a nanotube or
a molecular wire, once a suitableHamiltonian has been identified. These examples also underscore the importance of performing self-consistent calculations that include the Poisson
equation. The I –V characteristics of nanoscale structures is determined by an interesting
interplay between twentieth century physics (quantum transport) and nineteenth century
physics (electrostatics) and there is a tendency to emphasize one or the otherdepending on
one’s background. However, it is important to do justice to both aspects in order to derive
real insights.
c 2000 Academic Press
Key words: non-equilibrium Green’s function, Keldysh or Kadanoff–Baym formalism,
self-energy, nanoscale device modeling, quantum transport

1. Introduction
MOS transistors with channel lengths as small as 10 nm are now being actively studied boththeoretically
and experimentally [1]. At the same time recent demonstrations of molecular switching make molecular
electronic devices seem a little closer to reality [2]. It is clear that quantitative simulation tools for this new
generation of devices will require atomic-level quantum mechanical models. The non-equilibrium Green
function (NEGF) formalism (sometimes referred to as the Keldysh or theKadanoff–Baym formalism) provides a sound conceptual basis for the devlopment of this new class of simulators. 1D quantum devices like
tunneling and resonant tunneling diodes have been modeled quantitatively using NEMO [3] which is based
on the NEGF formalism. Although the transport issues in MOS transistors or molecular electronics are completely different, the NEGF formalism should provide asuitable conceptual framework for their analysis as
well. However, this formalism is based on concepts that are unfamiliar to most device physicists and chemists
† E-mail: d—tt—dpurdueFedu

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Superlattices and Microstructures, Vol. 28, No. 4, 2000

A

B

C
O

U←n

C
O

N
T
A
C

µ

[H + U]
Device"Poisson" Equation

N
T

µ

A

Equilibrium
Statistical Mechanics

C

T,

T,

1

2

H,U,µ→ n

Fig. 1. A, A device in equilibrium; B, self-consistent procedure for the analysis of electronic devices in equilibrium. ‘Poisson’ is written
within quotes as a reminder that it may need to be supplemented with a exchange-correlation potential.

and as such remains relatively obscuredespite the obvious value of a fundamentally sound approach on which
practical simulation tools for nanoscale devices can be based. In this paper we try to achieve two objectives:
(1) explain the central concepts that define the ‘language’ of quantum transport, and (2) illustrate the NEGF
formalism with simple examples that interested readers can easily duplicate on their PCs. The numerical...