Nanoelectronics

345

10

Drift-Diffusion and Multivalley Transport

A theory that makes an explicit connection between scattering-limited ohmic

mobility and quantum-emission-limited saturation velocity is presented.

The theory is applied to electrons in bulk silicon by taking a quantum equal

to the energy of an optical phonon. Because this quantum emission is indi-

cated, a modication in the mfp appears only in the high-eld regime. This

modication is shown to lead to electric-eld-induced degradation of the

diffusion coefcient. The theory presented agrees with the drift-diffusion

experimental data and empirical relations utilized in modeling devices. The

theory makes connections with an alternate description in terms of elec-

tron temperature under ac and dc conditions. As drift-diffusion processes

are central in the performance evaluation of submicron-scale devices where

high elds are necessarily present, these results contribute signicantly in

reshaping thinking processes in the high-eld regime. Transfer to higher

valleys in multivalley band structure is also discussed.

10.1 Primer

Considerable interest in the high-eld transport will continue to exist

because of scaled-down dimensions of devices that are now of submicron

scale in the quasi-free direction of the carrier ow. As we have seen, with

a high-electric eld present, the familiar linear velocity-eld characteristics

become sublinear and the velocity eventually saturates at an applied high-

electric eld. This nonlinear response of the carrier velocity to a high electric

eld has been extensively explored, both theoretically and experimentally

(for a review, see Reference 1). However, not much work has been reported

on diffusive transport in the presence of a high electric eld. An acceptable

procedure for studying such processes is that of the Monte Carlo simula-

tion with input parameters varied to get a desired output. A cumbersome

numerical procedure makes this approach impractical for device applica-

tions. Asimplied framework that extrapolates the well-established ohmic

behavior to the saturation regime is of more practical value to device design-

ers and researchers. Moreover, it provides an insight into processes that con-

trol the high-eld transport behavior.

346 Nanoelectronics

The Einstein relation with ratio of diffusion coefcient D

o

to ohmic mobil-

ity μ

o

for macro-devices has been around for many years. In the ohmic

domain, it can be described as

D

V

kT

q

o

o

t

B

µ

==

(10.1)

where V

t

= k

B

T/q is the thermal voltage at temperature T with a value at room

temperature (T = 300 K) 25.9 mV. In the high-eld domain, D(E), μ(E), and

electron temperatures T(E) in the published litrature are sporadically indi-

cated to depend upon the electric eld E. Considerable confusion exists while

evaluating each of these parameters in isolation. A unied expression for

each of these parameters allows us to dene hot-electron temperature and

also obtain its anlaytical expression.

10.2 Simplified Drift-Diffusion

In a free segment indicated by the free path, the electrons are traveling with

intrinsic velocity v

i

as discussed earlier. In a simplied model [1,2], this

intrinsic velocity for one dimension is the rms thermal velocity that strictly

applies to ND statistics. The kinetic energy related to the random motion of

anelectron,

() ,

12/

2

mv

n

∗

th

, is the thermal energy (1/2)k

B

T for one-degree of free-

dom in the direction of the carrier ow. The thermal velocity v

th

is given by

v

k

T

m

B

n

th

=

∗

(10.2)

where

m

n

∗

is the electron effective mass. For silicon at room temperature, this

thermal velocity is in the order of 10

7

cm/s when a carrier mobility effec-

tive mass

m

n

∗

= 0.26m

o

is considered. This random thermal velocity does not

contribute to any current (charge ow) in equilibrium as random velocity

vectors cancel each other out, resulting in zero ensemble (or time) average of

the velocity in a given direction.

When an electric eld, E, is applied, the random network drifts in a

direction that is opposite to that of an applied electric eld, in this case

–x-direction from the right to left. As an electron is accelerated by the elec-

tric eld, collisions impede their motion. An electron after suffering a col-

lision makes a fresh start of its drift velocity v(t) whose transient response,

as discussed in Chapter 4, is given by

vt

q

m

e

t

c

n

c

()=− −

−













∗

τ

τ

E

1

(10.3)