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

wavevector relation for free electrons:
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
, leads to:
the allowed
k

values are:
L
=
Na
length of the chain
a
a
a
Introductory comments

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
•
Radiative recombination
•
Direct thermal generation
•
Direct thermal recombination
•
Shockley

Read

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
Radiative
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

radiative recombination
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

Read

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:

Steady

state conditions:
n
1
and
p
1
are the electron and hole densities when
E
F
=
E
T
EEE 531: Semiconductor Device Theory I

Define carrier lifetimes:

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

ty carrier lifetime
):
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
•
Steady

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
(2) Steady

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
•
Steady

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
Additional terms:
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
itself under steady

state conditions.
L
Assume a parallel resistor

capacitor model:
Dielectric relaxation time is the time in which
any charge imbalance neutralizes itself.
A
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