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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:
Kinetic energy:
+E
K .E.  E  EC
K.E.
EC
Ev
Electric field:
P.E .   qV  EC  E ref
1
V   EC  E ref
q

P.E.
+V
Potential Energy:

Eref
dV 1 dEC
ε  V , or in 1 D ε  

dx q dx
EEE 531: Semiconductor Device Theory
I
Energy-wavevector relation for free electrons:
 Definition:
2
p
E
, m0  free electron mass
2m0
 de Broglie hypothesis:
h h
p 
k  k
 2
2k 2
E
2m0
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:
 2 2

  V (r )k (r )  Ek k (r )

 2m

EEE 531: Semiconductor Device Theory
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Bloch Theorem:
If the potential energy V(r) is periodic, then the solutions of
the SWE are of the form:
k (r )  exp(ik  r )un (k, r )
where un(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
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Periodic potential
Bloch function
Cell periodic Part
Plane wave component
EEE 531: Semiconductor Device Theory
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Reciprocal Space:
A 1D periodic function:
f ( x)  f ( x  l ); l  na
can be expanded in a Fourier series:
f ( x)   An e
i 2nx / a
  Ag e
n
g
igx
2n
g
a
The Fourier components are defined on a discrete set of
periodically arranged points (analogy: frequencies) in a
reciprocal space to coordinate space.
3D Generalization:
un k, r   
G
n
iGr
f G k e ;
G  hb1  kb 2  lb3
Ga Where hkl are integers. G=Reciprocal lattice vector
EEE 531: Semiconductor Device Theory
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First Brillouin Zone (in reciprocal space):
• 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)
First Brillouin Zone for Zincdirection, X is on face in
Blende and Diamond real space
(100) direction
FCC lattices
EEE 531: Semiconductor Device Theory
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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
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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
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• 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 (T )  E g (0)  T 2 /(T  )
Si
Eg (eV)
1.12
Ge
GaAs
0.66
1.42
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:
2E
 2
*
mij  ki k j
1
1
Si
EEE 531: Semiconductor Device Theory
I
 1
 *
 m xx
1 
 0
*
m


 0

0
1
m*yy
0

0 


0 

1 
* 
m zz 
• 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
Ge
Si
E  EC i
2
2
2 

k
k
  kx
y
z 



2  m*xi m*yi m*zi 


2
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
Constant energy
surfaces
Valence
bands
2



  2
2 4
2 2 2
2 2
2 2 1 2
E (k )  
 Ak  B k  C k x k y  k y k z  k z k x

2m 

mo
mo
mo
*
*
*
mhh 
, mlh 
, mso 
2
2
2
2
A
A B  C /6
A B  C /6
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
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Introductory comments -Counting states:
• Let us consider a one-dimensional chain of atoms:
a
a
a
L =Na  length of the chain
• According to Bloch theorem, the solutions of the 1D SWE
for periodic potential are of the form:
 k ( x)  uk ( x)eikx , uk (x  a)  uk (x)
• The application of periodic boundary conditions, leads to:
 k (0)   k ( L)  uk (0)  uk ( L)eikL  1  ei 2n  eikL
2n
, n  0,1,2,  ( N  1)
 the allowed k-values are: k n 
L
EEE 531: Semiconductor Device Theory
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Note on the boundary conditions:
• If one employs vanishing boundary conditions, it would
give as solutions standing waves (sinx or cosx functions),
which do not describe current carrying states.
• Periodic boundary conditions lead to traveling-wave (eikx)
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:
2
k  k n  k n 1 
L
Length in the reciprocal space
associated with one state (2 if
we account for the spin)
EEE 531: Semiconductor Device Theory
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• In 3D, the unit volume in the reciprocal space associated
3
with one state is 2 L  (not accounting for spin).
Calculation of the DOS function:
• Consider a sphere in k-space with volume:
4 k 3
3
• The total number of states we can accommodate in this
volume is:
4k 3 / 3 L3 3
N (k ) 
 2k
3
2 L  6
• The # of states in a shell of radius k and thickness dk is, by
similar arguments, equal to:
3
L
2
4

k
dk  M (k )dk  dN (k )
3
(2)
EEE 531: Semiconductor Device Theory
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M (k )  # of states per unit length dk
• Use the fact that the number of states is conserved, i.e.
M (k )dk  g ' ( E )dE
where
g ' ( E )  M (k )dk / dE  L3 g ( E )
Spin degeneracy
1 2 dk
g (E)  2  2 k
 # of states per unit volume per
dE
2
unit energy interval dE around E
• For parabolic energy bands, for which E=2k2/2m*
* 3 / 2
1  2m
g ( E )  2  2 
2   
EEE 531: Semiconductor Device Theory
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E
DOS effective masses:
• For single valley and parabolic bands, the DOS function in
3D equals to:
3/ 2
*
1  2m 
g C ( E )  2  2 
E  EC , for electrons in the
2   
conduction band
* 3 / 2
1  2m
gV ( E )  2  2 
2   
EV  E , for holes in the
valence band
E
gC(E)
EC
EV
gV(E)
EEE 531: Semiconductor Device Theory
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• For many-valley semiconductors with anisotropic effective
mass, using Herring-Vogt transformation:
mi* *
*
ki 
k , i  x, y, z, md  density of states effective
* i
md
mass
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):
*
mdn
Z
2/3


2 1/ 3
ml mt
 1.08m0
Z(# of equivalent valleys)=6, ml=0.98m0, mt=0.19m0
*
m
GaAs (electrons): dn  0.067m0 <= isotropic mass
EEE 531: Semiconductor Device Theory
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• For holes, which occupy the light-hole (lh) and heavy-hole
(hh) bands, the effective DOS mass equals to:
32
32 23
*
mdp
 (mhh
 mlh
)
*
Si (holes): mlh  0.16m0 , mhh  0.53m0 , mdp
 0.59m0
*
 0.64m0
GaAs (holes): mlh  0.074m0 , mhh  0.62m0 , mdp
Side note:
• For two-dimensional (2D) and one-dimensional (1D)
systems, one has:
g 2 D ( E )  E 0  const.
g1D ( E )  E 1 / 2
EEE 531: Semiconductor Device Theory
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Conductivity effective mass:
• Consider a many-valley semiconductor, such as Si:
3
1
2
1
3
2
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 contributions from each valley separately, i.e.
3
J  2J1  2J 2  2J 3  2  J i
i 1
EEE 531: Semiconductor Device Theory
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• The contribution from an individual valley is given by:
1 0 0
2  mxi

n 2  1 
nq 

J i   i E  q  *  E 
 0 m1 0  E
yi
6
6 
 m i

1
0 0 m 


zi 
Conductivity tensor
Effective mass tensor
• Thus, the total current density equals to:
1  2
mt
2  ml
nq 
J2
 0
6 
 0
0
1
ml
 m2
t
0


2 1
0  E  nq  * E
mc

2
m
t
0
1
ml
1 1 1
2
     The conductivity effective mass is
*
mc 3  ml mt 
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:
vdn=-mnF (for electrons) and vdp=mpF (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:
*
mn
dv dn
* v dn
  qF  mn
dt
m
• Low frequency limit:
*
 qF  mn
v dn
q m
q m
 0  v dn   * F  m n F  m n  *
m
mn
mn
EEE 531: Semiconductor Device Theory
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• 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:
E  E0 steadystate
dE
 qv  F 
   E  E 0  q E v  F
dt
E
• Description of the momentum relaxation time m and energy
relaxation time E:
t=0
t= m
t= E
(m=10-14-10-12 s) (E=10-13-10-11 s)
EEE 531: Semiconductor Device Theory
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• Drift velocity for GaAs and Si:
Intervalley transfer
10
Drift velocity [cm/s]
7
T = 300 K
1
10
6
0.1
5
10
-1
10
10
0
10
1
10
2
Electric field [kV/cm]
GaAs
Slope dvd/dF=m
EEE 531: Semiconductor Device Theory
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Silicon
0.01
10
3
Average energy [eV]
10
• Small devices => non-stationary transport
velocity overshoot=> faster devices (smaller transit time)
Drift velocity [cm/s]
Velocity overshoot effect
3x10
7
2.5x10
7
2x10
7
1.5x10
7
1x10
7
5x10
6
E=1kV/cm
E=4kV/cm
E=10kV/cm
E=20kV/cm
E=40kV/cm
E=100kV/cm
T = 300 K
Silicon
0
0
0.5
1
1.5
time [ps]
EEE 531: Semiconductor Device Theory
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2
2.5
3
Carrier Mobility:
1
1
1
1
1
1
1






 m ii  ni  ac  npo  po  pe
Ionized
impurities neutral
Si, GaAs impurities
(low T)
Si, GaAs
Mathiessen’s rule:
Acoustic
phonons
Si, GaAs
polar
optical phonons
GaAs
Non-polar
optical phonons
Si
Piezoelectric
(low-T)
GaAs
m l  T 3 / 2
1 1
1
1
1
1
1






 mii  T 3 / 2 N I1
m mii m ni m ac m npo m po m pe
m ni  N 01
EEE 531: Semiconductor Device Theory
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Carrier Mobility (Cont’d):
Electron mobility
1400
1200
2
Mobility [cm /V-s]
1600
1000
800
600
400
200
0 15
10
Monte Carlo
Experimental data
Simulation results
Simulation results
(mesh force only)
10
16
10
17
10
-3
Doping [cm ]
EEE 531: Semiconductor Device Theory
I
18
10
19
Drift velocity in Si:
(A) Electrons:
vn 
mn F
1  mn F / vs  
2 1/ 2
mon
m n ( N I , T )  m min 
1  N I N cn 
 0.57 
2
2

T
cm
cm
8  2.33 




m min  88
,
m

7
.
4

10
T

on
 Vs 
 Vs 
 300 




 0.146
2.4
T 
17  T 
3

  0.88
, N cn  1.26  10 

 cm
 300 
 300 
Saturation velocity:
2.4  107
cm 

vs 
T / 600  s 
 
1  0.8e
EEE 531: Semiconductor Device Theory
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(B) Holes:
vp 
m pF
1  m p F / vs
m p ( N I , T )  m min
mon

1  N I N cp 

 0.57 

2

cm
, m op  1.36  108 T  2.33 

m min

 Vs 



 0.146
2.4
T 
17  T 
3

  0.88
, N cp  2.35  10 

 cm
 300 
 300 
T 

 54

 300 
2
 cm
 Vs

EEE 531: Semiconductor Device Theory
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Hall measurements:
• Resistivity measurements
• carrier concentration characterization
• low-field mobility (Hall mobility)
y
Bz
t
z
x
VH
vx
Eyp
Eyn
w
vx
L
Va
Ix
EEE 531: Semiconductor Device Theory
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• The second law of motion for an electron moving in a
electric and magnetic field, at low frequencies is of the form:
 * vx
mn
  qFx  qv y B

 m
* v
 qF - q[ v  B]  mn
0
vy
m
*
mn
  qFy  qv x B  0  o.c.
  m
• One also has:
J  qnv  J x  qnvx  qnmeff Fx
• Hall coefficient:
Fy
rn
1
RH 
 
J xB
qn
qn
Determine n
Sign=>carrier type
where rn is the Hall scattering factor:
rn 
EEE 531: Semiconductor Device Theory
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

2
2
mH

1
m eff
• The effective carrier mobility is obtained in the following
manner:
1. Calculate the conductivity of the sample:
Jx
Ia L
x 

Fx Va wt
2. Evaluate the Hall mobility:
m H   x RH
3. Based on the knowledge of the Hall scattering factor,
determine the effective mobility using:
m eff  m H / rn
EEE 531: Semiconductor Device Theory
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Diffusion process:
p(x)
J p   qD p p
n(x)
J n  qDn n
-
+
• Dn, Dp  Diffusion constants for electrons and holes
• Total current equals the sum of the drift and diffusion
components:
J n  qnm n F  qDn n
J p  qpm p F  qD p p
EEE 531: Semiconductor Device Theory
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Einstein relations (derivation):
Assumptions:
• equilibrium conditions
• non-degenerate semiconductor
n
J n  qnm n F  qDn
0
x
 E F  Ec 
 E F  Ei 
  n  ni exp

n  N c exp
 k BT 
 k BT 
 E F  Ei  n
n  E F Ei  1
q
 


ni exp

nF

x  x
x  k BT
k BT
 k BT  x

 q 
 Dn / m n  k BT / q  VT
nF  0  
qnm n F  qDn  
 D p / m p  k BT / q  VT
 k BT 
EEE 531: Semiconductor Device Theory
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Generation-Recombination Mechanisms:
Photons and phonons (review):
• Photons  quantum of energy in an electromagnetic wave
E  hf  1  4  eV
p  h / ,   c / f  0.4mm ( E  1eV )

large energy, small momentum
• Phonons  quantum of energy in an elastic wave
E  hf  0.02  0.06 eV
p  h / ,   v s / f  1.8nm
( E  0.02eV , v s  8.5  103 m / s )

small energy, large momentum
EEE 531: Semiconductor Device Theory
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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 highfield and breakdown phenomena
EEE 531: Semiconductor Device Theory
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Classification:
One step
(Direct)
Two Energy-level
particle consideration
•
•
•
•
• Shockley-Read-Hall (SRH)
generation-recombination
• Surface generationrecombination
Two-step
(indirect)
Impact
ionization
Three
particle
Auger
Photogeneration
Radiative recombination
Direct thermal generation
Direct thermal recombination
Pure generation process
•
•
•
•
EEE 531: Semiconductor Device Theory
I
Electron emission
Hole emission
Electron capture
Hole capture
(1) Direct processes
Diagramatic description:
Ec
Ec
Light
E=hf
Light
Ev
Photogeneration
Important for:
• narrow-gap semiconductors
• direct band-gap SCs used
for fabricating LEDs for
optical communications
heat
Ev
Radiative
recombination
Ec
heat
Ec
Ev
x
Ev
Direct thermal Direct thermal
generation
recombination
Not the usual means by which
the carriers are generated or
recombine
EEE 531: Semiconductor Device Theory
I
• Photogeneration band-diagramatic description:
E
E
Virtual
states
Phonon emission
Ec
Phonon absorption
Eg
Eg
EV
Direct band-gap SCs
k
Indirect band-gap SCs
k
Momentum and energy conservation:
p f  pi  ps
p f  pi
E f  Ei  E ph
final
initial
E f  Ei  Es  E ph
photon
final
EEE 531: Semiconductor Device Theory
I
initial phonon
photon
Near the absorption edge, the absorption coefficient can be
expressed as:

  hf  E g

g
Light
intensity
Distance
hf = photon energy
1/
Eg = bandgap
light-penetration depth
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
EEE 531: Semiconductor Device Theory
I
• Photogeneration-radiative recombination  mathematical
description
- Both types of carriers are involved in the process
r  Bpn, g  Bp0 n0  Bni2


R  r  g  B pn  ni2  p  p0  p, n  n0  n
R  B  n  p0  n0  n 
 B(GaAs)  1.3  0.3  1010 cm 3 / s

15
3
 B( Si)  2  10 cm / s
- Limiting cases:
n
1

R Bp0
n
1


R Bn
(a) Low-level injection: n, n0  p0   rad 
(b) High-level injection: n  n0 , p0   rad
EEE 531: Semiconductor Device Theory
I
(2) Auger processes:
Diagramatic description:
Electron
capture
Ec
Ec
Ec
Ec
Ev
Ev
Ev
Ev
Hole
capture
Electron
emission
Hole
emission
Recombination process
Generation process
(carriers near the band edges involved)
(energetic carriers involved)
• Auger generation takes place in regions with high concentration 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
r  Cn pn 2  C p p 2 n, g  Cn p0 n02  C p p02 n0




R  r  g  Cn pn 2  p0 n02  C p p 2 n  p02 n0

p  p0  p, n  n0  n
a  p 2  2n p  n 2 p  n  n 
n
1
0
0 0
0
0
 Auger 


R aC p  bCn
 b  n02  2n0 p0  n 2n0  p0  n 
- Limiting cases (p-type sample):
(a) Low-level injection:
(b) High-level injection:
  C p ( 2n
1
2
 Auger  n C p  Cn 
 Auger 
EEE 531: Semiconductor Device Theory
I
C p p02
n 0
0
 n )

1
- Auger Coefficients:
(Silvaco)
T [K]
Cn [cm6/s]
Cp [cm6/s]
77
2.3x10-31
7.8x10-32
300
2.8x10-31
9.9x10-32
400
2.8x10-31
1.2x10-31
(3) Impact ionization:
Diagramatic description  identical to Auger generation

1
Gimpact  n J n   p J p
q

Ionization rates => generated electron holepairs 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:
3.5x10
7
3x10
7
2.5x10
7
2x10
7
1.5x10
7
1x10
7
5x10
6
An [cm-1]
Encrit [V/cm]
n
Si
1x106
1.66x106
1
GaAs
3x105
6.85x105
1.6
2x106
2x106
1
1.55x107
1.56x105
1
Ge
Impact ionization
1
(a)
0.35
0.18
0.25
0.15
0
Material
Average energy [eV]
Velocity [cm/s]
  crit n 
En  


 n  An exp 
  E  


  crit  p 
Ep  


 p  Ap exp 
  E  
 
 
0.3
0.4
0.5
0.6
0.7
Distance along the channel mm]
0.35
(b)
0.25
0.8
0.18
0.6
0.15
0.4
0.2
0
0.3
0.4
0.5
0.6
0.7
Distance along the channel [ mm]
VG=3.3 V, VD =3.3, 2.5, 1.8 and 1.5 V
EEE 531: Semiconductor Device Theory
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(4) Shockley-Read-Hall Mechanism:
Diagramatic description:
Electron
capture
Hole
capture
Ec
Electron
emission
ET
Hole
emission
Ev
Recombination
Ec
ET
Ev
cn
pT
nT
en
nT
Ec
c
pT
ET
e
p
p
Ev
Generation
Mathematical model:
dn
 en nT  cn npT
dt
dp
 e p pT  c p pnT
dt
Two types of carriers
involved in the process
nT  N T f T

 nT  pT  N T
pT  N T 1  f T 
EEE 531: Semiconductor Device Theory
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- Thermal equilibrium conditions:
dn / dt  0, dp/dt  0

 cn npT  en nT
 en  cn n1
e p  c pn  e  c p
p
T
p 1
 p T
 p
n1 and p1 are the electron and hole densities when EF=ET
- Steady-state conditions:
 Rn  Rnc  Rne  cn npT  en nT  cn ( npT  n1nT )
 R  R  R  c pn  e p  c ( pn  p p )
pc
pe
p
T
p T
p
T
1 T
 p

cn n  c p p1
np  ni2
fT 
 R
1
1
cn ( n  n1 )  c p ( p  p1 )
( n  n1 ) 
( p  p1 )
c p NT
cn N T
EEE 531: Semiconductor Device Theory
I
- Define carrier lifetimes:
1
1
1
1
p 

, n 

c p N T  p vth N T
cn N T n vth N T
- Empirical expressions for electron and hole lifetimes:
n 
0n
1
N A  ND
, p 
N nref
1
0p
N A  ND
N ref
p
n0 [s]
Nnref [cm-3]
p0 [s]
Npref [cm-3]
Source
5x10-5
5x1016
5x10-5
5x1016
D'Avanzo
3.94x10-4
7.1x1015
3.94x10-4
7.1x1015
Dhanasekaran
EEE 531: Semiconductor Device Theory
I
- Limiting cases:
n  p (n0  n  n1 )   n ( p0  n  p1 )
 SRH 

R
n0  p0  n
n
 n
(a) Low level injection (p-type sample):  SRH 
R
n
 n   p
(b) High-level injection:  SRH 
R
- Generation process (pn  0):
 ni2
ni
R
 G  G 
 p n1   n p1
g

g=generation rate =>  g   p e   n e
EEE 531: Semiconductor Device Theory
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
ET  Ei
, 
k BT
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
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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 equations:
B
E  
t
  H  J cond
D  
B  0
D

t
EEE 531: Semiconductor Device Theory
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(2) Current-Density Equations:
J n  qnm n E  qDn n
J p  qpm p E  qD p p
• For non-degenerate SC’s, the carrier diffusion constants and
the mobilities are related through the Einstain’s relations:
Dn / mn  k BT / q, D p / m p  k BT / q
• The above equations are valid for low fields. Under high
field conditions, the terms mnE and mpE 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:

t
    H     J cond    D  0

  Jn  J p



 0,   q p  n  N D  N A 
t
 n  1   J  G  R
n
n
n
 t q
 
p
1
     J p  Gp  Rp
q
 t
• Low-level injection (SRH lifetime dominated by the minority carrier lifetime):
Rn 
n p  n p0
n
pn  p n 0
, Rp 
p
EEE 531: Semiconductor Device Theory
I
(4) Poisson’s equation:
• Derived from the Maxwell’s equations (electrostatics case):
  E  0  E  

  D    E        

2
2
Quasi-Fermi levels:
• In non-equilibrium conditions, one needs to define separate
Fermi levels for n and p:
 E Fn  Ei 

n  ni exp 
 k BT   J n  nm n E Fn
J p  pm p E Fp
 Ei  E Fp 

p  ni exp 
 k BT 
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 pn(t) for t>0 .
hf
n-type sample
x=0
EEE 531: Semiconductor Device Theory
I
x
Solution:
pn
• Boundary conditions: E  0,
0
x
• Minority hole continuity equation:
pn
pn  pn 0
pn  p n 0
1
    J p  Gp 
G
t
q
p
p
• General form of the solution: pn (t )  Ae
• Boundary conditions:
pn (t  0)  pn 0  G p
pn (  )  pn 0
pn (t )  pn0   pGe
t /  p
pn 0   p G
t /  p
B
pn (t )
Light turned off
pn 0
EEE 531: Semiconductor Device Theory
I
t
(2) Steady-state injection from one side:
Consider a sample under constant illumination by a light source. Calculate pn(x).
hf
n-type sample
x=0
x
Solution
• Minority carrier continuity equation:
pn
1
1 J p pn  pn 0
    J p  Gp  Rp  

t
q
q x
p
pn
d 2 pn pn  pn 0
 Dp

2
t
p
dx
EEE 531: Semiconductor Device Theory
I
• Steady-state situation: pn  0
Dp
d 2 pn
dx2
pn  pn0

p
t
d 2 pn pn
0
 2  0, L p   p D p
2
dx
Lp
pn ( x )  Ae
 x / Lp
 Be
x / Lp
• Boundary conditions for a long sample:
Diffusion length
pn ()  0  B  0
pn (0)  pn (0)  pn 0  A
pn (x )
• Final solution:
 x / Lp
pn ( x )  pn 0   pn (0)  pn 0 e
qD p
 x / Lp
 pn (0)  pn0 e
J p ( x) 
Lp
EEE 531: Semiconductor Device Theory
I
pn (0)
pn 0
Lp
x
(3) Surface recombination:
Consider a sample under constant illumination by a light source. There is a finite surface recombination at x=0. Calculate
pn(x).
Gp
Surface recombination:
dpn
Dp
 S r  pn (0)  pn 0 
x=0
dx x 0
n-type sample
x=L
Solution
• Minority carrier continuity equation at steady-state:
Dp
d 2 pn
dx2
Gp
pn  pn0
d 2 pn pn

 Gp  0 
 2 
2
p
Dp
dx
Lp
EEE 531: Semiconductor Device Theory
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x
• General form of the solution:
pn ( x )  Ae
 x / Lp
 Be
x / Lp
C
B=0 (asymptotic condition)
• Boundary conditions for a long sample:
pn ( )  C   p G p
pn (0)  A  C  A   p G p
• Use boundary condition at the surface to determine pn(0):
dpn
Dp
dx
x 0
• Final solution:
 Sr  pn (0)  pn0   pn (0) 
 p D pG p
D p  Sr L p

Sr  p
x / Lp 
pn ( x)   pG p 1 
e

 L p  Sr  p

EEE 531: Semiconductor Device Theory
I
• Graphical representation of the solution:
pn (x )
 pG p
Sr  0
Sr increasing
Sr  
x
EEE 531: Semiconductor Device Theory
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Additional terms:
Dielectric relaxation time:
Assume a parallel resistor-capacitor model:
A
L
R  L / A, C  k s 0 A / L, d  RC  k s 0
Dielectric relaxation time is the time in which
any charge imbalance neutralizes itself.
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.
k BT k s 0
LD 
 Dn  d
2 N
q
D
EEE 531: Semiconductor Device Theory
I