Nanoelectronics

163

4

Nonequilibrium Carrier Statistics

andTransport

An ensemble of carriers in equilibrium follows Fermi–Dirac statistics. In

equilibrium, randomly oriented velocity vectors or associated mean free

paths (mfps) in vector addition give a zero vector sum. The application of

an electric eld tends to align these tiny electric diploes in the direction of

an electric eld. In this chapter, nonequilibrium Arora distribution function

(NEADF) is discussed [1,2]. NEADF is an anisotropic distribution function

that is an outgrowth of the isotropic Fermi–Dirac distribution function. The

velocity-eld proles briey discussed in Chapter 3 are microscopically ana-

lyzed as randomly oriented velocity vectors streamline in the direction of an

electric eld making the velocity unidirectional in an extremely high electric

eld. This is the source of ultimate saturation of velocity vectors.

4.1 Tilted Band Diagram in an Electric Field

In equilibrium, the bands are at and velocity vectors are randomly oriented.

The bands tilt with drop in potential energy over the length of the sample by

−qV as the electric eld E = V/L exists due to the voltage source of potential

difference V, as shown in Figure 4.1. The electric eld is given by the gradi-

ent of the potential V(x) = U(x)/q, where U(x) = qV(x) is the potential energy

(PE). As the reference level of the potential is arbitrary, it can be taken to be

V(0) = 0 at the left edge of the resistor of length L in Figure 4.1a. In that case,

U(x) = E

c

(x) is the potential energy prole. The electric eld E is then given by

any of the following forms as each of E

c

(x), E

v

(x), E

i

(x), and E

vac

(x) differs from

the other only by a constant:

E =− =− =− =− =−

11111

q

dU x

dx q

dE

dx q

dE

dx q

dE

dx q

dE

dx

cviv

ac

()

(4.1)

The conduction band edge (as well as the Fermi energy) is the function of

the distance along the length of the device

EEqx Eq

V

L

xE qV

x

L

ccxc

xc

x

()x =−=−

=−

== =00 0

E

(4.2)

164 Nanoelectronics

The potential increases linearly V(x) = Ex as transition takes place from

x = 0 with V(0) = 0 to x = L with V(L) = V with reference voltage level V = 0

pegged on the left edge (x = 0). However, the electron potential energy (PE)

EqVx Eqx

cxocxo

−=−() E

decreases as x is increased with reference level

E

cxo

at

x = 0. PE at x = L is

EEqV

cxLcxo

=−

, as shown in Figure 4.1b.

4.2 Velocity Response to an Electric Field

The carrier motion in equilibrium is stochastic with net (average) veloc-

ity equal to zero, as shown in Figure 4.2. The magnitude of each randomly

E

E

Collisions

Conduction

band

Valence

band

Forbidden

band

(a) (b)

E

–

+

qV

E

K

dE

C

E

V

x

x

–

dx

F =

E

C

= PE(x) or U(x

)

dU(x)

–

dx

=

FIGURE 4.1

The tilted band diagram in an electric eld when the potential source exists across the sample.

(a) Physical sample with voltage applied across its ends with electric eld going from right to

left. (b) The titled energy band diagram over the length of the sample with dip equal to qV as

one moves from left to right.

e

e

e

eh

h

h

h

Electron

s

Holes

Atoms

h

FIGURE 4.2

Example of the path of an electron in a conductor.