# EEE 531: Semiconductor Device Theory I

EEE 531: Semiconductor Device Theory I

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

Energy bands

Effective masses

EEE 531: Semiconductor Device Theory I

Energy bands

Basic convention:

E
C

E
v

E
ref

K.E.

P.E.

+E

+V

Kinetic energy:

Potential Energy:

Electric field:

EEE 531: Semiconductor Device Theory I

Energy
-

Definition:

de Broglie hypothesis:

Energy
-
wavevector relation for electrons in a crystal:

The dispersion relation in a crystal (E
-
k relation) is obtained
by solving the Schrödinger wave equation:

EEE 531: Semiconductor Device Theory I

Bloch Theorem:

If the potential energy
V
(
r
) is periodic, then the solutions of
the SWE are of the form:

where
u
n
(
k
,
r
) is periodic in
r

with the periodicity of the direct
lattice and
n

is the band index.

Methods used to calculate the energy band structure:

Tight
-
binding method

Orthogonal plane
-
wave method

Pseudopotential method

k

p

method

Density functional technique (DFT)

EEE 531: Semiconductor Device Theory I

Periodic potential

Bloch function

Cell periodic Part

Plane wave component

EEE 531: Semiconductor Device Theory I

Reciprocal Space:

A 1D
periodic

function:

can be expanded in a Fourier series:

The Fourier components are defined on a discrete set of
periodically arranged points (analogy: frequencies) in a
reciprocal space to coordinate space.

3D Generalization:

Where
hkl

are integers.
G
=Reciprocal lattice vector

EEE 531: Semiconductor Device Theory I

First Brillouin Zone (in reciprocal space):

First Brillouin Zone for Zinc
-

Blende and Diamond real space

FCC lattices

The periodic set of allowed
points corresponding to the
Fourier (reciprocal) space

associated with the real
(space) lattice form a
periodic lattice

The
Wigner
-
Seitz

unit cell
corresponding to the
reciprocal lattice is the
First
Brillouin Zone

is zone center, L is on
zone face in (111)
direction, X is on face in
(100) direction

EEE 531: Semiconductor Device Theory I

Examples of energy band structures:

Si

GaAs

Based on the energy band structure, semiconductors can be
classified into:

Indirect band
-
gap semiconductors (Si, Ge)

Direct band gap semiconductors (GaAs)

EEE 531: Semiconductor Device Theory I

Model Energy Bands in III
-
V and IV Semiconductors:

Conduction Band
-

3 Valley Model (

, L, X minima).
Lowest minima: X (Si), L (Ge),

(GaAs, most III
-
Vs)

Valence Band
-

Light hole, heavy hole, spin
-
split off band

EEE 531: Semiconductor Device Theory I

The energy band
-
gaps
decrease

with
increasing temperature
.
The variation of the energy band
-
gaps with temperature can
be expressed with a universal function:

E
g

(eV) 1.12 0.66 1.42

Si

Ge

GaAs

EEE 531: Semiconductor Device Theory I

Effective Masses

Curvature of the band determines the effective mass of the
carriers in a crystal, which is different from the free electron
mass.

Smaller curvature

heavier mass

Larger curvature

lighter mass

For
parabolic bands
, the components of the effective mass
tensor are calculated according to:

Si

EEE 531: Semiconductor Device Theory I

From the knowledge of the energy band structure, one can
construct the plot for the allowed k
-
values associated with a
given energy =>
constant energy surfaces

Si

Ge

Note:

The electron effective mass in GaAs is isotropic, which
leads to spherically symmetric constant energy surfaces.

EEE 531: Semiconductor Device Theory I

Due to the
p
-
like symmetry

and mixing of the V.B. states, the
constant energy surfaces are warped spheres:

The
hh
-
band is most warped

The
lh
-

and
so
-
band are more spherical

Valence

bands

Constant energy

surfaces

EEE 531: Semiconductor Device Theory I

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

Counting states

Density of states function

Density of states effective mass

Conductivity effective mass

EEE 531: Semiconductor Device Theory I

Let us consider a one
-
dimensional chain of atoms:

According to
Bloch theorem
, the solutions of the 1D SWE
for periodic potential are of the form:

The application of
periodic boundary conditions

the allowed
k
-
values are:

L

=
Na

length of the chain

a

a

a

-
Counting states:

EEE 531: Semiconductor Device Theory I

Note on the boundary conditions:

If one employs vanishing boundary conditions, it would
give as solutions
standing waves
(sin
x

or cos
x
functions),
which
do not describe

current carrying states
.

Periodic boundary conditions lead to
traveling
-
wave

(e
ikx
)
solutions, which represent
current carrying states
.

Counting of the states:

Each atom in the 1D chain contributes one state (two if we
account for the spin: spin
-
up and spin
-
down states).

The difference between two adjacent allowed
k

values is:

Length in the reciprocal space
associated with one state (2 if
we account for the spin)

EEE 531: Semiconductor Device Theory I

In 3D, the unit volume in the reciprocal space associated
with one state is (not accounting for spin).

Calculation of the DOS function:

Consider a sphere in
k
-
space with volume:

The total number of states we can accommodate in this
volume is:

The # of states in a shell of radius
k

and thickness d
k

is, by
similar arguments, equal to:

EEE 531: Semiconductor Device Theory I

Use the fact that the number of states is conserved, i.e.

where

For parabolic energy bands, for which
E=

2
k
2
/2m
*

# of states per unit length
dk

# of states per unit volume per

unit energy interval
dE

around
E

Spin degeneracy

EEE 531: Semiconductor Device Theory I

DOS effective masses:

For
single valley
and
parabolic bands
, the DOS function in
3D equals to:

for electrons in the

conduction band

for holes in the

valence band

E
C

E
V

g
C
(E)

g
V
(E)

E

EEE 531: Semiconductor Device Theory I

For
many
-
valley semiconductors

with
anisotropic

effective
mass, using Herring
-
Vogt transformation:

the expression for the density of states function reduces to
the one for the single valley case, except for the fact that
one has to use the
density of states effective mass
:

Si (electrons):

Z
(# of equivalent valleys)=6,
m
l
=0.98
m
0
,
m
t
=0.19
m
0

GaAs (electrons):
<=

isotropic mass

density of states effective
mass

EEE 531: Semiconductor Device Theory I

For
holes
, which occupy the light
-
hole (
lh
) and heavy
-
hole
(
hh
) bands, the effective DOS mass equals to:

Si (holes):

GaAs (holes):

Side note:

For
two
-
dimensional

(2D) and
one
-
dimensional

(1D)
systems, one has:

EEE 531: Semiconductor Device Theory I

1

1

2

2

3

3

Conductivity effective mass:

Consider a many
-
valley semiconductor, such as
Si
:

Under the assumption that the

valleys are
equally populated
,

the electron density in each

valley equals
n/6
.

The total current density equals the sum of the contributi
-
ons from each valley separately, i.e.

EEE 531: Semiconductor Device Theory I

The contribution from an individual valley is given by:

Thus, the total current density equals to:

Conductivity tensor

Effective mass tensor

The conductivity effective mass is
used for mobility calculations!

EEE 531: Semiconductor Device Theory I

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

Drift (mobility, drift velocity, Hall effect)

Diffusion

Generation
-
recombination mechanisms

EEE 531: Semiconductor Device Theory I

Drift process:

Under
low
-
field conditions
, the carrier drift velocity is
proportional to the electric field:

v
dn
=
-
m
n
F

(for electrons) and
v
dp
=
m
p
F

(for holes)

These expressions can be obtained from the second law of
motion. For example, for an electron moving in an electric
field, one has:

Low frequency limit:

EEE 531: Semiconductor Device Theory I

The linear dependence of
v

on
F

does not hold at
high fields
when electrons gain considerable energy from the electric
field, in which case one has:

Description of the
momentum relaxation time

m

and
energy
relaxation time

E
:

t=0

t=

m

(

m
=10
-
14
-
10
-
12

s)

t=

E

(

E
=10
-
13
-
10
-
11

s)

EEE 531: Semiconductor Device Theory I

Drift velocity for GaAs and Si:

GaAs

Silicon

Slope d
v
d
/d
F
=
m

Intervalley transfer

EEE 531: Semiconductor Device Theory I

Small devices

=> non
-
stationary transport

velocity overshoot=> faster devices (smaller transit time)

Silicon

Velocity overshoot effect

EEE 531: Semiconductor Device Theory I

Carrier Mobility:

Ionized

impurities

Si, GaAs

neutral

impurities

(low T)

Si, GaAs

Acoustic

phonons

Si, GaAs

Non
-
polar

optical phonons

Si

polar

optical phonons

GaAs

Piezoelectric

(low
-
T)

GaAs

Mathiessen’s rule:

EEE 531: Semiconductor Device Theory I

Carrier Mobility (Cont’d):

Electron mobility

EEE 531: Semiconductor Device Theory I

Drift velocity in Si:

Saturation velocity:

(A) Electrons:

EEE 531: Semiconductor Device Theory I

(B) Holes:

EEE 531: Semiconductor Device Theory I

Hall measurements:

Resistivity measurements

carrier concentration characterization

low
-
field mobility (Hall mobility)

EEE 531: Semiconductor Device Theory I

The second law of motion for an electron moving in a
electric and magnetic field, at low frequencies is of the form:

One also has:

Hall coefficient:

where
r
n

is the Hall scattering factor:

Determine
n

Sign=>carrier type

EEE 531: Semiconductor Device Theory I

The
effective carrier mobility

is obtained in the following
manner:

1.

Calculate the conductivity of the sample:

2.

Evaluate the Hall mobility:

3.

Based on the knowledge of the Hall scattering factor,

determine the effective mobility using:

EEE 531: Semiconductor Device Theory I

Diffusion process:

+

p(
x
)

-

n(
x
)

D
n
,
D
p

Diffusion

constants for electrons and holes

Total

current

equals the sum of the drift and diffusion
components:

EEE 531: Semiconductor Device Theory I

Einstein relations (derivation):

Assumptions:

equilibrium conditions

non
-
degenerate semiconductor

EEE 531: Semiconductor Device Theory I

Generation
-
Recombination Mechanisms:

Photons and phonons (review):

Photons

quantum of energy in an electromagnetic wave

Phonons

quantum of energy in an elastic wave

EEE 531: Semiconductor Device Theory I

Generation
-
Recombination mechanisms:

Notation:

g

generation rate

r

recombination rate

R=r
-
g

net recombination rate

Importance:

BJTs

R plays a crucial role in the operation of the

device

Unipolar devices

(MOSFET’s, MESFETs, Schottky
diodes

No influence except when investigating high
-
field and breakdown phenomena

EEE 531: Semiconductor Device Theory I

Classification:

Two

particle

One step

(Direct)

Two
-
step

(indirect)

Energy
-
level

consideration

Photogeneration

Direct thermal generation

Direct thermal recombination

Shockley
-
-
Hall (SRH)
generation
-
recombination

Surface generation
-
recombination

Three

particle

Impact

ionization

Auger

Electron emission

Hole emission

Electron capture

Hole capture

Pure generation process

EEE 531: Semiconductor Device Theory I

(1) Direct processes

Diagramatic description:

E
c

E
v

Light

E
=
hf

E
c

E
v

Light

E
c

E
v

heat

E
c

E
v

heat

x

Photo
-

generation

recombination

Direct thermal

generation

Direct thermal

recombination

Not the usual means by which

the carriers are generated or

recombine

Important for:

narrow
-
gap semiconductors

direct band
-
gap SCs used
for fabricating LEDs for
optical communications

EEE 531: Semiconductor Device Theory I

Photogeneration

band
-
diagramatic description:

Momentum and energy conservation:

E

k

E
g

Phonon emission

Phonon absorption

Indirect band
-
gap SCs

Virtual

states

final

initial

photon

final

initial

photon

E
c

E
V

E

k

Direct band
-
gap SCs

E
g

phonon

EEE 531: Semiconductor Device Theory I

Near the absorption edge, the absorption coefficient can be
expressed as:

hf

= photon energy

E
g

= bandgap

g

= constant

g
=1/2 and 1/3 for allowed direct

transitions and forbidden direct transitions

g
=2 for indirect transitions where phonons

are involved

Light

intensity

Distance

1/

light
-
penetration depth

EEE 531: Semiconductor Device Theory I

Photogeneration
-

mathematical
description

-

Both types of carriers are involved in the process

-

Limiting cases:

(a) Low
-
level injection:

(b) High
-
level injection:

EEE 531: Semiconductor Device Theory I

(2) Auger processes:

Diagramatic description:

E
c

E
v

E
c

E
v

E
c

E
v

E
c

E
v

Electron

capture

Hole

capture

Electron

emission

Hole

emission

Recombination process

(carriers near the band edges involved)

Generation process

(energetic carriers involved)

Auger generation

takes place in regions with high concent
-
ration of mobile carriers with
negligible current flow

Impact ionization

requires
non
-
negligible current flow

EEE 531: Semiconductor Device Theory I

Auger process

mathematical description

-

Three carriers are involved in the process

-

Limiting cases (
p
-
type sample):

(a) Low
-
level injection:

(b) High
-
level injection:

EEE 531: Semiconductor Device Theory I

-

Auger Coefficients:

(Silvaco)

(3) Impact ionization:

Diagramatic description

identical to Auger generation

Ionization rates

=>
generated electron hole
-

pairs per unit length of travel per carrier

EEE 531: Semiconductor Device Theory I

-

Ionization rates dependence upon the electric field

component parallel to the current flow:

0

0.2

0.4

0.6

0.8

1

0.3

0.4

0.5

0.6

0.7

Average energy [eV]

Distance along the channel [

m

m]

0.15

0.18

0.25

0.35

(b)

Impact ionization

V
G
=3.3 V, V
D

=3.3, 2.5, 1.8 and 1.5 V

EEE 531: Semiconductor Device Theory I

(4) Shockley
-
-
Hall Mechanism:

Diagramatic description:

Mathematical model:

E
c

E
v

Electron

emission

Hole

emission

E
c

E
v

Electron

capture

Hole

capture

E
T

E
T

Recombination

Generation

E
c

E
v

E
T

p
T

c
n

e
n

n
T

c
p

e
p

Two types of carriers

involved in the process

n
T

p
T

EEE 531: Semiconductor Device Theory I

-

Thermal equilibrium conditions:

-

-
state conditions:

n
1

and
p
1

are the electron and hole densities when
E
F
=
E
T

EEE 531: Semiconductor Device Theory I

-

-

Empirical expressions for electron and hole lifetimes:

EEE 531: Semiconductor Device Theory I

-

Limiting cases:

(a) Low level injection (
p
-
type sample):

(b) High
-
level injection:

-

Generation process (p

n

0):

g
=
generation rate

=>

EEE 531: Semiconductor Device Theory I

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

Description of basic equations for
semiconductor device operation

Concept of quasi
-
Fermi levels

Sample solution problems

Dielectric relaxation time and Debye length

EEE 531: Semiconductor Device Theory I

Basic equations for SC device operation:

Maxwell’s equations

Current density equations

Continuity equations

Poisson’s equation

(1) Maxwell’s equations:

Any carrier transport model must satisfy the Maxwell’s equa
-
tions:

EEE 531: Semiconductor Device Theory I

(2) Current
-
Density Equations:

For non
-
degenerate SC’s, the carrier diffusion constants and
the mobilities are related through the Einstain’s relations:

The above equations are valid for low fields. Under high
field conditions, the terms
m
n
E

and
m
p
E

must be replaced
with the saturation velocity.

Additional terms appear in the presence of a magnetic field.

EEE 531: Semiconductor Device Theory I

(3) Continuity equations:

Derived from Maxwell’s equations:

Low
-
level injection (SRH lifetime dominated by the
minori
-
):

EEE 531: Semiconductor Device Theory I

(4) Poisson’s equation:

Derived from the Maxwell’s equations (electrostatics case):

Quasi
-
Fermi levels:

In non
-
equilibrium conditions, one needs to define separate
Fermi levels for
n

and
p
:

EEE 531: Semiconductor Device Theory I

Sample problems:

Decay of the photo
-
excited carriers

-
state injection from one side

Surface
-
recombination

(1) Decay of photo
-
excited carriers:

Consider a sample illuminated with light source until
t

0
. The
generation rate equals to
G
. At
t
=0 the light source is turned
off. Calculate
p
n
(t) for
t
>0 .

n
-
type sample

x

x
=0

hf

EEE 531: Semiconductor Device Theory I

Solution:

Boundary conditions:

Minority hole continuity equation:

General form of the solution:

Boundary conditions:

Light turned off

EEE 531: Semiconductor Device Theory I

-
state injection from one side:

Consider a sample under constant illumination by a light sour
-
ce. Calculate
p
n
(
x
).

Solution

Minority carrier continuity equation:

n
-
type sample

x

x
=0

hf

EEE 531: Semiconductor Device Theory I

-
state situation:

Boundary conditions for a long sample:

Final solution:

L
p

Diffusion length

EEE 531: Semiconductor Device Theory I

(3) Surface recombination:

Consider a sample under constant illumination by a light sour
-
ce. There is a finite surface recombination at
x
=0. Calculate
p
n
(
x
).

Solution

Minority carrier continuity equation at steady
-
state:

n
-
type sample

x

x
=0

x
=
L

Surface recombination:

G
p

EEE 531: Semiconductor Device Theory I

General form of the solution:

Boundary conditions for a long sample:

Use boundary condition at the surface to determine

p
n
(0):

Final solution:

B
=0 (asymptotic condition)

EEE 531: Semiconductor Device Theory I

Graphical representation of the solution:

S
r

increasing

x

EEE 531: Semiconductor Device Theory I

Dielectric relaxation time:

Debye length:

Debye length is the spatial counterpart of the dielectric relaxation time.

It is a measure of the smallest length over which charge can neutralize
-
state conditions.

L

Assume a parallel resistor
-
capacitor model:

Dielectric relaxation time is the time in which
any charge imbalance neutralizes itself.

A