Nanoelectronics

23

2

Atoms, Bands, and Quantum Wells

Charge carriers (electrons, holes, or ions) in multimicron mesoscopic sys-

tems behave like particles. Their energies have analog (or classical) charac-

ter meaning that their energy variables are continuous. However, when one

or more of the three Cartesian directions are constrained to a nanometer

scale, the carriers exhibit a wave (or quantum) character. In this setup, the

energies are digitally (or quantum) separated by forbidden regions. Analog

characteristics can be built from digital energies by expanding a nanoscopic

system to macro- or microscopic scales in which case the quantum jumps

are miniscule. This transformation leads to the concept of density of states

(DOS), which is the number of quantum states per unit volume (or area or

length) per unit energy.

2.1 Birth of a Quantum Era

The origin of the quantum era is traced to the work of Louis de Broglie in the

early part of the twentieth century. He postulated that just like a photon is

a packet (quantum) guided by electromagnetic waves, electrons are guided

through the existence of quantum waves. This wave–particle duality is rep-

resented by the relation

λ

D

h

p

=

(2.1)

where wave property, represented by the de Broglie wavelength λ

D

, is con-

nected to the particle property represented by crystal momentum of a carrier

p = m*v. Here, m* is the effective mass and v is the carrier velocity. m*includes

the effect of the force eld of the crystal lattice on a carrier and hence dif-

fers from the free electron mass m

o

. m* = 0.067m

o

for gallium arsenide (GaAs).

m*is actually a tensor as it varies from one direction to the other in the crys-

tal lattice. However, for most engineering approximations, it is taken to be a

scalar that is averaged depending on the application.

Equation 2.1 has a close resemblance to that of a photon (quantum of light)

with energy E = hf = mc

2

and momentum p = mc, where f is the frequency of

24 Nanoelectronics

a photon and m is its relativistic mass. The Newtonian laws do not apply

to photons for which the theory of relativity denes the energy–mass con-

nection (E = hf = mc

2

). The mass as known in Newtonian mechanics has an

entirely different meaning when electron speed is near that of light. The

mass increases with the speed of the particle. As speed approaches the speed

of light c = λ/T = fλ, the mass tends to rise to innity unless the rest mass

of the photon is negligible. This zero mass or zero momentum of a photon

enters into the description as selection rule for electrons making a transition

must necessarily have zero change in momentum for photon to be emitted or

absorbed. Otherwise, the probability of emission or absorption of a photon is

greatly reduced. The wavelength of a photon is similarly related to that of the

massless (rest mass m

o

= 0, not moving mass m) photon as

λ= == =

c

f

hc

hf

hc

mc

h

p

2

(2.2)

In a crystal with an ensemble of carriers (electrons or holes), the average

energy is thermal energy k

B

T. In fact, it depends on the degrees of freedom.

For one-dimensional (1D), two-dimensional (2D), and three-dimensional

(3D) motion, it is (1/2)k

B

T, k

B

T, and (3/2)k

B

T, respectively, or (d/2)k

B

T, where

d = 3 (bulk), 2 (layer), or 1 (line). The thermal de Broglie wavelength λ

Dth

cor-

responding to the thermal momentum

pm

vm

EmkT

th th B

== =

*2 2

**

for

an energy E = k

B

T is given by

λ

Dth

th

B

h

mv

h

mkT

==

*

*2

(2.3)

with thermal velocity equal to

v

kT

m

th

B

=

2

*

(2.4)

Equation 2.2 for photons and Equation 2.3 for electrons are easily adaptable

for numerical computation if E = hf for a photon or E = p

2

/2m* for an electron

is expressed in eV:

λ= =

⋅

hc

EE() ()eV

eV nm

eV

(photon)

1240

(2.5)

λ

D

o

mmE

=

⋅123

12

.()

(* )( )

/

nm eV

/eV

(electron)

(2.6)