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Lecture 8
Semiconductor Physics VINonequilibrium Excess Carriers in
Semiconductors
Nonequilibrium conditions.
Excess electrons in the conduction band and excess holes in the valence band
Ambipolar transport(双极输运) : Excess electrons and excess holes diffuse, drift, and recombine with the same effective diffusion coefficient, drift mobility, and lifetime.
Ambipolar transport equation
Quasi-Fermi energy (准费米能级)for electrons and quasi-Fermi energy for holes
2
CARRIER GENERATION
AND RECOMBINATION
The Semiconductor in Equilibrium
3
0 0n pG G
The electrons and holes recombine in pairs, so the
recombination rates of electrons and holes are equal.0 0n pR R
The concentrations of electrons and holes are
independent of time; therefore, the generation and
recombination rates are equal:
The electrons and holes are created in pairs, so the
thermal-generation rates of electrons and holes are equal.
0 0 0 0n p n pG G R R
Under nonequilibrium conditions
When high-energy photons are incident on a
semiconductor, electrons in the valence band may be
excited into the conduction band.
=> Electron-hole pairs are generated.
=> The additional electrons and holes created are
called excess electrons and excess holes.
The generation rate of excess electrons and holes are
equal ' '
n pg g
4
Fig Creation of excess electron and hole densities by photons.
Excess electron and hole concentrations
2
0 0 inp n p n
5
0n n n
0p p p
As in the case of thermal equilibrium, an electron in
the conduction band may "fall down" into the valence
band, leading to the process of excess electron-hole
recombination.
The recombination rate of excess electrons and holes
are equal ' '
n pR R
Fig Recombination of excess carriers reestablishing thermal equilibrium6
The net rate of change in the electron concentration
The carrier concentration is dependent on time.
Since n0 and p0 are independent of time and ( ) ( )n pt t
If we consider a p-type material (p0 >> n0) under
low-level injection ( )0( )n t p
0
( )( )
n
r n
d tp t
dt
7
2( )[ ( ) ( )]r i
dn tn n t p t
dt
2
0 0
( ( ))[ ( ( ))( ( ))]r i
d n tn n n t p p t
dt
0 0( )[( ) ( ))]r n t n p n t
called the excess minority carrier lifetime
The excess carrier concentration is an exponential decay
from the initial excess concentration
The recombination rate
For the direct band-to-band recombination, the excess majority
carrier holes recombine at the same rate, so that for the p-type
material
8
0 0/( ) (0) (0)r np t t
n t n e n e
1
0 0( )n r p a constant for the low-level injection
'
0
0
( ( )) ( )( )n r
n
d n t n tR p n t
dt
' '
0
( )n p
n
n tR R
In the case of an n-type material (n0 >> p0) under low-
level injection0( )n t n
The excess minority carrier lifetime
9
' '
0
( )n p
p
p tR R
1
0 0( )p rn
In summary, ' ' ( )n pR R f n t
' ' ( )n pG G f n t
' ' time, spacen pG G f
CHARACTERISTICS OF EXCESS CARRIERS
The excess carriers behave with time and in space in
the presence of electric fields and density gradients
is of equal importance.
The excess electrons and holes diffuse and drift with
the same effective diffusion coefficient and with the
same effective mobility. This phenomenon is called
ambipolar transport.
10
• Continuity Equations
Fig Differential volume showing
x component of the hole-particle flux
A one-dimensional hole particle
flux is entering the differential element
at x and is leaving the element at x + dx.
The net increase in the
number of holes per unit time
due to x-component hole flux
the increase in the
number of holes per
unit time due to the
generation of holes
the decrease in the
number of holes per
unit time due to the
recombination of holes.
11
The net increase in the number of
holes per unit time in the differential
volume element is:
includes the thermal equilibrium carrier lifetime and the excess carrier lifetime.pt
Pp
pt
Fp pdxdydz dxdydz g dxdydz dxdydz
t x
12
The net increase in the hole concentration per unit time is
the continuity equation for holes
Similarly, the one-dimensional continuity equation for electrons is
p
p
pt
Fp pg
t x
nn
nt
Fn ng
t x
13
•Time-Dependent Diffusion Equations
The hole and electron current densities are
The hole and electron flux are
Substitute them into the continuity equation
where
2
2
( )p p p
pt
p pE p pD g
t x x
2
2
( )+ n n n
nt
n nE n nD g
t x x
p
p p p
J pF pE D
e x
nn n n
J nF nE D
e x
p p p
pJ e pE eD
x
n n n
nJ e nE eD
x
pE p EE p
x x x
14
The time-dependent diffusion equations for holes and electrons are
The thermal equilibrium concentrations, no and po are not functions
of time. For the special case of a homogeneous semiconductor, no
and po are also independent of the space coordinates.
Therefore, the time-dependent diffusion equations can be
Which describe the space and time behavior of the excess carriers
2
2( )p p p
pt
p p E p pD E p g
x x x t
2
2( )n n n
nt
n n E n nD E n g
x x x t
2
2
( ) ( ) ( )( )n n n
nt
n n E n nD E n g
x x x t
2
2
( ) ( ) ( )( )p p p
pt
p p E p pD E p g
x x x t
AMBIPOLAR TRANSPORT
15
Fig The creation of an internal electric field
as excess electrons and holes tend to separate
• With an applied electric field,
the excess holes and electrons
are created.
• This separation will
induce an internal electric
field between the two sets
of particles.
The negatively charged electrons and positively charged holes
then will drift or diffuse together with a single effective mobility
or diffusion coefficient. - called ambipolar transport.
intappE E E
16
To relate the excess electron and hole concentrations to the
internal electric field, Poisson's equation is introduced
where is the permittivity of the semiconductor materials
Since
then charge neutrality condition n p
The diffusion equations can be written as
intint
( )
s
Ee p nE
x
n pg g g n p
nt pt
n pR R R
2
2
( ) ( ) ( )( )p p
p p E pD E p g R
x x x t
2
2
( ) ( ) ( )( )n n
n n E nD E n g R
x x x t
int, E 0If p n
17
Combining the two diffusion equations yields
The ambipolar transport equation is derived as
where the ambipolar diffusion coefficient
and the ambipolar mobility is
2
2
( ) ( ) ( )' '
n n nD E g R
x x t
'n p p n
n p
nD pDD
n p
( )'
n p
n p
p n
n p
2
2
( ) ( )( ) ( )( )n p p n n p
n nnD pD p n E
x x
( )( )( ) ( )n p n p
nn p g R n p
t
18
The Einstein relation relates the mobility and diffusion coefficient
The ambipolar diffusion coefficient may be written in the form
Since both n and p contain the excess-carrier concentration,
which are dependent on time and space, the coefficient in
the ambipolar transport equation are not constants.
The ambipolar transport equation is a nonlinear differential equation.
pn
n p
e
D D kT
( )'
n p
n p
D D n pD
D n D p
2
2
( ) ( ) ( )' '
n n nD E g R
x x t
19
•Limits of Extrinsic Doping and Low Injection
In a p-type semiconductor with low-level
injection
In a n-type semiconductor with low-level
injection
The ambiploar
parameters
reduce to a
minority
carrier value,
which are
constants.
The equivalent ambipolar particle is negatively charged.
0 0
0 0
[( ) ( )]'
( ) ( )
n p
n p
D D n n p nD
D n n D p n
' n ' nD D
' pD D
( )'
n p
n p
p n
n p
' p
0 0p n0n p
0 0 0 and np n n
20
Consider the generation and recombination terms in
the ambipolar transport equation.
For electrons
For holes,
The generation rate for excess electrons must equal the
generation rate for excess holes.
The minority carrier lifetime is essentially a constant for low
injection.
'
0 0( ) ( ')n n n n n ng R g R G g R R
' ' 'n n n
n
ng R g R g
' ' 'p p p
p
pg R g R g
'
0 0( ) ( ')p p p p p pg R g R G g R R
0 0n nG R
0 0p pG R
' ' 'n pg g g
21
The ambipolar transport equation can be written in
terms of minority carrier parameters.
For an n-type semiconductor under low injection,
•The condition of charge neutrality:
The behavior of excess majority carriers is determined by the
minority carrier parameters.
2
2
0
( ) ( ) ( )'n n
n
n n n nD E g
x x t
2
2
0
( ) ( ) ( )'p p
p
p p p pD E g
x x t
For a p-type semiconductor under low injection,
p n
22
Table Common ambipolar transport equation simplification
23
Consider an infinitely large, homogeneous n-type semiconductor with zero
applied electric field. Assume that at time t = 0, a uniform concentration of
excess carriers exists in the crystal, but assume that g' = 0 for t> 0. If we assume
that the concentration of excess carriers is much smaller than the thermal-
equilibrium electron concentration, then the low injection condition applies.
Calculate the excess carrier concentration as a function of time for t > 0.
Example1
For the n-type semiconductor, we need to consider the
ambipolar transport equation for the minority carrier holes
Solution
We are assuming uniform concentration of excess holes so that
2
2
0
( ) ( ) ( )'p p
p
p p p pD E g
x x t
2 2( ) / ( ) / 0p x p x
24
So the bipolar transport equation reduces to
The solution is
where is the uniform concentration of excess
carriers that exists at time t = 0.
(0)p
From the charge-neutrality condition, we have that
So the excess electron concentration is given by
0
( )p p
p
d
dt
0/( ) (0) pt
p pt e
n p
0/( ) (0) pt
n pt e
For t>0, we are also assuming that g’=0
25
Consider an infinitely large, homogeneous n-type semiconductor
with a zero applied electric field. Assume that, for t < 0, the
semiconductor is in thermal equilibrium and that, for t>0, a
uniform generation rate exists in the crystal. Calculate the excess
carrier concentration as a function of time assuming the
condition of low injection.
Example2
The condition of a uniform generation rate and a homogeneous
semiconductor again implies that
Solution
The equation, for this case, reduces to
The solution to this differential equation is
0
( )'
p
p d pg
dt
2 2( ) / ( ) / 0p x p x
0/
0( ) ' (1 )pt
pp t g e
26
Consider a p-type semiconductor that is homogeneous and infinite
in extent. Assume a zero applied electric field. For a one-
dimensional crystal, assume that excess carriers are being generated
at x=0 only. The excess carriers being generated at x = 0 will begin
diffusing in both the +x and -x directions. Calculate the steady-state
excess carrier concentration as a function of x.
Example3
Solution
Since E=0, g’=0 for , and for steady state 0n t
Dividing by the diffusion coefficient
The parameter Ln, has the unit of length and is called the minority
carrier electron diffusion length.
2
2
0
( )0n
n
d n nD
dx
2 2
2 2 2
0
( ) ( )0
n n n
d n n d n n
dx D dx L
0x
27
The general solution to the equation is
The minority carrier electron concentration will then
decay toward zero at both x = and x =
These boundary conditions mean that B = 0 for x > 0
and A = 0 for x < 0.The solution is
The steady-state excess electron concentration decays
exponentially with distance away from the source at x = 0.
/ /( ) n nt L x L
n x Ae Be
/( ) (0) nx L
n x n e 0x
/( ) (0) nx L
n x n e 0x
28
Fig Steady-state electron and hole concentrations for the
case when excess electrons and holes are generated at x= 0.
29
Assume that a finite number of electron-hole pairs is generated
instantaneously at time t = 0 and at x = 0. But assume g' = 0 for t > 0.
Assume we have an n-type semiconductor with a constant applied
electric field equal to E0. which is applied in the +x direction.
Calculate the excess carrier concentration as a function of x and t.
Example4
Solution
The one-dimensional ambipolar transport equation is
The solution to this partial differential equation is of the form
using Laplace transform techniques
2
2
0
( ) ( ) ( )p p
p
p p p pD E
x x t
0/( , ) '( , ) pt
p x t p x t e
2
0
1/2
( )1'( , ) exp[ ]
(4 ) 4
p
p p
x E tp x t
D t D t
30
The total solution is
Excess-hole concentration versus distance
at various times for zero applied electric field
Excess-hole concentration versus
distance at various times for a
constant applied electric field.
0 2/
0
1/2
( )( , ) exp[ ]
(4 ) 4
pt
p
p p
x E tep x t
D t D t
31
•Dielectric Relaxation Time Constant
Initially, a concentration of
excess holes is not balanced by a
concentration of excess electrons.
How is charge neutrality
achieved and how fast?Fig The injection of a concentration of
holes into a small region at the surface
of an n-type semiconductor.
Poisson's equation is
The current equation, Ohm's law, is
The continuity equation, neglecting the effects of generation
and recombination, is
is the net charge density and the initial value is
J E
E
Jt
( )e p
32
Taking the divergence of Ohm's law and using Poisson's equation,
Substituting it into the continuity equation
Its solution is
where
called the dielectric relaxation time constant(介电弛豫时间).
It means the time to charge neutrality.
d
t dt
d( / )( ) (0)
tt e
0d
dt
d
J E
33
Assume an n-type semiconductor with a donor impurity
concentration of Nd = 1016 cm-3. Calculate the dielectric relaxation
time constant.
Example
Solution
The conductivity is found as
The permittivity of silicon is
The dielectric relaxation time constant is then
19 16 1(1.6 10 )(1200)(10 ) 1.92( )n de N cm
14
0 (11.7)(8.85 10 ) /r F cm
1413(11.7)(8.85 10 )
5.39 101.92
d s
QUASI-FERMI ENERGY LEVELS
34
The thermal-equilibrium electron and hole concentrations
are functions of the Femi energy level.
If excess carriers are created in a semiconductor, we are no
longer in thermal equilibrium and the Fermi energy is
strictly no longer defined.
However, a quasi-Fermi level can be defined for nonequilibrium.
0 exp( )F Fii
E En n
kT
0 exp( )Fi F
i
E Ep n
kT
0 exp( )Fn Fii
E En n n
kT
0 exp( )Fi Fp
i
E Ep p n
kT
35
Example
Solution
Consider an n-type semiconductor at T = 300 K with carrier
concentrations of
In nonequilibrium, assume that the excess carrier concentrations
are
Calculate the quasi-Fermi energy levels.
The Fermi level for thermal equilibrium is
The quasi-Fermi level for
electrons in nonequilibrium is
The quasi-Fermi level for holes
in nonequilibrium is
0
i
( ) 0.2982F Fi
nE E kTIn eV
n
0( ) 0.2984Fn Fi
i
n nE E kTIn eV
n
0
i
( ) 0.179Fi Fp
p pE E kTIn eV
n
13 310n p cm
15 3 10 3 5 3
0 010 , 10 , 10in cm n cm and p cm
36
Thermal-equilibrium
energy-band diagram
Quasi-Fermi levels for
electrons and holes
Note: Fn FE E
Summary
The excess carrier generation rate and recombination rate.
Ambipolar transport: Excess electrons and holes do not move
independently of each other, but move together.
The ambipolar transport equation: the excess electrons and
holes diffuse and drift together with the characteristics of the
minority carrier under low injection and extrinsic doping.
Excess carrier behavior is a function of time and a function of
space.
The quasi-Fermi level for electrons and the quasi-Fermi level
for holes were defined to characterize the total electron and
hole concentrations in a semiconductor in nonequilibrium.
37
An n-type gallium arsenide semiconductor is doped
with Nd = 1016 cm-3 and Na = 0. The minority carrier
lifetime is
Calculate the steady-state increase in conductivity and
the steady-state excess carrier recombination rate if a
uniform generation rate, g' = 2 x 1021 cm-3s-1, is
incident on the semiconductor
HOMEWORK7
7
0 2 10p s