Nanoelectronics

117

3

Carrier Statistics

In Chapter 2, extensive discussion on quantum wells was held and how

analog-like continuous energy and momentum states get digitized was dis-

cussed. The analog-like continuous spectrum thus can be viewed as a collec-

tion of digitized quantum states with step-size miniscule compared to the

thermal energy as we go from one energy level to the next higher one. In this

chapter, the focus is on the lling of quantum states. The probability that

a quantum state is occupied is described by the Fermi–Dirac distribution

function. The carrier concentration is a multiplication of the effective density

of quantum states multiplied by the effective probability.

3.1 Fermi–Dirac Distribution Function

The probability f(E) that a quantum state of energy E is occupied is given

by [1,2]

fE

e

EE kT

FB

()

()/

=

+

−

1

1

(3.1)

where E

F

is the Fermi energy with the probability of occupation 1/2 at E = E

F

(see Appendix 3A for more discussion). This probability function is shown

in Figure 3.1. The placement of the Fermi energy with respect to the conduc-

tion band edge describes the degeneracy of the system. The degeneracy of a

carrier sample depends on the carrier concentration with respect to the DOS.

Asample is nondegenerate (ND) if the carrier concentration n

d

< N

cd

, where

d = 1, 2, 3 is the dimensionality of the nanoscale sample and N

cd

is the effec-

tive density of states (EDOS) for which an expression will be found later in

the chapter. On the other hand, when n

d

⨠ N

cd

, a sample is strongly degen-

erate. The three-dimensional (3D) conduction band edge E

co

is lifted by the

ground-state energy ε

o

in a quantum well, so the lifted conduction band edge

E

c

= E

co

+ ε

o

. The minimum energy of occupation is E

co

or E

c

in a quantum

well as no quantum states exist in the forbidden bandgap. In the Maxwell–

Boltzmann (ND) approximation, the 1 in the denominator of Equation 3.1 is

neglected as the Fermi energy E

F

is below E

co

as shown in Figure 3.1. In this

case, the reduced Fermi energy η = (E

F

− E

co

)/k

B

T is negative. As a general

rule of thumb, the “1” in the denominator of Equation 3.1 is negligible if

118 Nanoelectronics

η < −3. Figure 3.1 shows that the conduction band E

F

< E

cND

and hence η < 0

for ND statistics. In the limit η < −3, the distribution follows the dashed line

of Figure 3.1, its tail tted with the exponential distribution given by

fE

e

e

EE kT

EE

kT

FB

FB

()

()/

()

≈=

−

−−

1

/

(3.2)

On the other hand, in the strongly degenerate regime, where the Fermi

energy is above the conduction band edge, the probability of occupa-

tion is 1 up to the Fermi energy level and zero above it. An extreme case

exists at T = 0 K when (E − E

F

)/k

B

T → ∞ for E > E

F

. This limit makes f(E) = 0.

As (E − E

F

)/ k

B

T → −∞, the probability of occupation is f(E) = 1. Figure 3.1

shows the conduction band E

F

> E

cD

and hence η > 0 for a degenerate statis-

tics. As shown by the solid line in Figure 3.1, the probability is 1 for every state

being occupied below the Fermi level and is 0 above it. Step-like distribution

at 0 K near the Fermi energy point smears out at a nite temperatureT. The

degenerate approximation is always valid in case the Fermi energy is well

above the conduction band edge.

3.2 Bulk (3D) Carrier Distribution

The carrier concentration is a function of probability f(E) and the DOS

between E and E + dE. The differential carrier concentration per unit volume

dn

3

between E and E + dE is given by

Fermi–Dirac

function at T = 0 K

Maxwell–Boltzmann

approximation

EE

cND

E

cD

1/2

f(E)

E

F

FIGURE 3.1

Fermi–Dirac distribution function as a function of energy. In the ND limit, the Fermi energy

is below the conduction band edge making the tail of the distribution (E > E

cND

) Maxwellian.

In the degenerate regime (E

cD

> E > E

F

), every state is occupied up to the Fermi level and every

state is empty above it at 0 K.