Chapter 6
Lecture Notes
Semiconductors in Nonequilibrium Conditions
Excess electrons in the conduction band and excess holes in the valence band
may exist in addition to the thermal-equilibrium concentrations if an external
excitation is applied to the semiconductor.
The creation of excess electrons and holes means that the semiconductor is no
longer in thermal equilibrium.
Carrier generation and recombination
Generation ⇒ process whereby electrons and holes are created
Recombination ⇒ process whereby electrons and holes are coincided and lost.
A sudden change in temperature or optical excitation can generate excess
electrons and holes creating a nonequilibrium condition.
Semiconductors in Nonequilibrium Conditions
Injection ⇒ A process of introducing excess carriers in
semiconductors.
Generation and recombination are two types:
(i) Direct band-to-band generation and recombination and
(ii) the recombination through allowed energy states within the bandgap,
referred to as traps or recombination centers.
Direct band-to-band generation and recombination
Thermal equilibrium:
The random generation-recombination of electrons-holes occur continuously
due to the thermal excitation.
In direct band-to-band generation-recombination, the electrons and holes are
created-annihilated in pairs:
Gn 0 = G p 0
Rn 0 = R p 0
At thermal equilibrium, the
concentrations of electrons and
holes are independent of time;
therefore, the generation and
recombination rates are equal, so we
have,
G n 0 = G p 0 = Rn 0 = R p 0
Excess carrier generation and recombination
Symbol
n0, p0
Definition
Thermal equilibrium electron and hole concentrations
(independent of time)
n, p
Total electron and hole concentrations (may be functions of
time and/or position).
δn = n- n0 Excess electron and hole concentrations (may be functions
δp = p- p0 of time and/or position).
Gn0, Gp0
Thermal electron and hole generation rates (cm-3s-1)
Rn0, Rp0
Thermal equilibrium electron and hole recombination rates
(cm-3s-1)
g n′ , g ′p
Rn′ , R ′p
τn0,τp0
Excess electron and hole generation rates (cm-3s-1)
Excess electron and hole recombination rates (cm-3s-1)
Excess minority carrier electron and hole lifetimes (s)
Nonequilibrium Excess Carriers
When excess electrons and holes are created,
n(t ) = n0 + δn(t ),
p(t ) = p 0 + δp(t )
np ≠ ni2
and
The excess electrons and holes are created in pairs,
g n′ = g ′p
δ n (t ) = δ p (t )
and
The excess electrons and holes recombine in pairs,
Rn′ = R ′p
The excess carrier will decay over time and the decay rate depends on the
concentration of excess carrier.
dn(t )
∝
dt
[n
2
i
]
− n(t ) p (t )
or,
[
]
dn(t )
= α r ni2 − n(t ) p (t )
dt
αr is the constant of proportionality for recombination.
Nonequilibrium Excess Carriers
[
]
dn(t )
= α r ni2 − n(t ) p (t )
dt
n(t ) = n0 + δn(t ),
p(t ) = p0 + δp(t )
and
δn(t ) = δp(t )
dδn(t )
= −α r δn(t )[(n0 + p 0 ) + δn(t )]
dt
( p0 + n0 ) >> δn(t )
For low-level injection condition:
For p-type material,
Thus,
p 0 >> n0
dδn(t )
= −α r δn(t ) p 0
dt
The solution is,
δn(t ) = δn(0)e −α
r
p0 t
= δn(0)e −t / τ n 0
where τn0 is the excess minority carrier lifetime for electron and τn0 = 1/(αrp0)
dδn(t )
= −α r δn(t )[(n0 + p 0 ) + δn(t )]
dt
Rn′ = −
τ0 =
dδn(t )
δn
= α rδn(t )[(n0 + p0 ) + δn(t )] =
τ0
dt
1
α r ( p0 + n0 + δn )
For high-level injection condition:
Examples
( p0 + n0 ) ⟨ δn(t )
The recombination rate for electrons in p-type material,
Rn′ = −
dδn(t )
δn(t )
= α r δn(t ) p 0 =
τ n0
dt
The excess electrons and holes recombine in pairs, (in p-type material)
Rn′ = R ′p =
δn(t )
τ n0
The recombination rate is a positive quantity.
And, for n-type
Rn′ = R′p =
δp(t )
τ p0
where τp0 is the minority carrier lifetime of holes.
The excess recombination rate is determined by the excess
minority carrier lifetime.
Continuity equations:
The continuity equation describes the behavior of excess carriers with time and
in space in the presence of electric fields and density gradients.
Δx
Fp(x)
Fp(x + Δx)
∼
F =F+ represents hole
particle flux (# / cm2 –s)
Area, A
x
x + Δx
The net increase in hole concentration per unit time,
∂p
∂t
= lim Δx → 0
x → x + Δx
Fp+ ( x ) − Fp+ ( x + Δx )
Δx
+ g p − Rp
Continuity equations
+
∂F p
∂p
+ g p − Rp
=−
∂t
∂x
The hole flux,
Fp+ = J p / e
The unit of hole flux is holes/cm2-s
∂p
1 ∂J p
=−
+ g p − Rp
∂t
e ∂x
Similarly for electrons,
∂n 1 ∂J n
=
+ g n − Rn
∂t e ∂x
These are the continuity equations for holes and electrons respectively
Recall: In one dimension, the electron and hole current
densities due to the drift and diffusion are given by:
∂p
J p = eμ p pE − eD p
∂x
J n = eμ n nE + eDn
∂n
∂x
Substitute these in the continuity equations:
∂ ( pE )
∂2 p
∂p
+ g p − Rp
= Dp 2 − μ p
∂x
∂x
∂t
Or,
∂n
∂ 2n
∂ (nE )
+ g n − Rn
= Dn 2 + μ n
∂x
∂t
∂x
∂n
∂ 2n
∂E ⎞
⎛ ∂n
= Dn 2 + μ n ⎜ E + n
⎟ + g n − Rn
∂t
∂x
∂x ⎠
⎝ ∂x
∂E ⎞
∂p
∂2 p
⎛ ∂p
= Dp 2 − μ p ⎜ E + p
⎟ + g p − Rp
∂t
∂x
∂x ⎠
⎝ ∂x
n = n0 + δn
and
p = p 0 + δp
The thermal-equilibrium concentrations, n0 and p0, are not functions of time. For
the special case of homogeneous semiconductor, n0 and p0 are also
independent of the space coordinates. So the continuity may then be written in
the form of:
∂δn
∂ 2δn
∂E ⎞
⎛ ∂δn
= Dn
+
+
E
n
μ
⎟ + g n − Rn
n⎜
2
∂t
∂x
∂x ⎠
⎝ ∂x
∂δp
∂ 2δ p
∂E ⎞
⎛ ∂δp
= Dp
−
+
E
p
μ
⎜
⎟ + g p − Rn
p
2
∂t
∂x
∂x ⎠
⎝ ∂x
Time –dependent
diffusion equations
Ambipolar transport equation
Under applied external electric field, the excess electrons and holes created by
any mean say, light illumination, tend to drift in the opposite directions.
Separation of these charged particles induces an internal electric field opposite to
the applied external electric field.
This induced internal electric field attract the electrons and holes move toward
each other, holding the pulses of excess electrons and holes together. So, the
electric field appears in the time-dependent diffusion equations is composed of the
external and the induced internal electric fields.
Electrons and holes drift or diffuse together with a single effective mobility or
diffusion coefficient. This phenomena is called ambipolar transport.
e (δ p − δ n )
εs
=
∂ E int
∂x
;
As the electrons and holes
are diffusing and drifting
together:
gn = g p = g
and
Rn = R p = R
E int ⟨⟨ E app
The effect of internal field
∂ 2δn
∂E ⎞
∂δn
⎛ ∂δn
n
E
μ
+
= Dn
+
⎟+ g −R
⎜
n
x
x
∂
∂
∂t
∂x 2
⎠
⎝
∂δn
∂ 2δn
∂E ⎞
⎛ ∂δn
μ
E
+
p
= Dp
−
⎜
⎟+ g −R
p
∂
x
∂
x
∂t
∂x 2
⎝
⎠
Eliminating
∂E
∂x
∂δn
∂ 2δn
∂δn
′
′
=D
+μE
+g−R
2
∂t
∂x
∂x
D′ =
Dn D p (n + p )
Dn n + D p p
μ′ =
μ n μ p ( p − n)
μnn + μ p p
D′ is the ambipolar diffusion coefficient
μ′ is called the ambipolar mobility
Assumption:
Charge neutrality (quasineutrality) δn ≈δp
Ambipolar transport equation
For p-type semiconductors
For n-type semiconductors
D′ ≈ Dn and μ′ ≈μn
D′ ≈ Dp and μ′ ≈ - μp
For p-type semiconductors
∂δn
∂ 2δn
∂δn
δn
= Dn
+
+
−
μ
g
E
n
∂t
∂x 2
∂x
τ n0
And for n-type semiconductors
δp
∂δp
∂ 2δp
∂δp
= Dp
− μpE
+g−
2
τ p0
∂t
∂x
∂x
Applications of ambipolar transport equation
(1) Steady state :
∂ (δn ) ∂ (δp )
=
=0
∂t
∂t
(2) Uniform distributi on of excess Carriers : D p
∂ (δn )
∂ (δp )
=E
=0
∂x
∂x
(4) No excess carrier generation : g ' = 0
∂ 2 (δn )
∂x 2
= 0, Dn
(3) Zero Electric Field : E
(5) No excess carrier recombinat ion : R′ =
δn δp
=
=0
τ n0 τ p0
(6) Infinite carrier lifetimes , τ p 0 = τ n 0 = ∞, R′ =
δn
δp
=
=0
τ n0 τ p 0
∂ 2 (δn )
∂x 2
= 0,
∂δn ∂δp
=0
=
∂x
∂x
Indirect recombination:
In real semiconductors, there are some crystal defects and these defects
create discrete electronic energy states within the forbidden energy band.
Recombination through the defect (trap) states is called indirect recombination
The carrier lifetime due to the recombination through the defect energy state
is determined by the Shockley-Read-Hall theory of recombination.
Shockley-Read-Hall recombination:
Shockley-Read-Hall theory of recombination assumes that a single trap center
exists at an energy Et within the bandgap.
There are four basic processes
The trap center here
is acceptor-like.
It is negatively
charged when it
contains an electron
and is neutral when
it does not contain
an electron.
Capture rate ∝ (free carrier concentration )×(concentration of empty defects states)
Emission rate ∝ (trapped carrier concentration )×(concentration of empty
conduction or valence band states)
Emission rate ∝ (concentration of trapped carriers)
The electron capture rate (process 1),
The electron emission rate (process 2),
Rcn = C n n[1 − f F (Et )]N t
Ren = E n N t f E (Et )
The relationship between the capture Cn and emission coefficients En can be
determined by the principle of detailed balance.
Principle of Detailed Balance: Under equilibrium conditions each fundamental
process and its inverse must self-balance independent of any other process that
may be occurring inside the material.
Under thermal equilibrium, Rcn = Ren, which gives a relation between Cn and En.
In nonequilibrium, the net rate at which electrons are captured from the
conduction band is given by,
Rn′ = Rcn − Ren
Under steady state condition, there is no change of trap carrier concentration.
Thus
Where,
Rn′ = R ′p =
(
C n C p N t np − ni2
)
C n (n + n ′) + C p ( p + p ′)
⎡ − (E c − E t ) ⎤
and
n ′ = N c exp ⎢
⎥
kT
⎦
⎣
⎡ − (E t − E v ) ⎤
p ′ = N v exp ⎢
⎥
kT
⎦
⎣
The trap level energy is near the midgap so that n′ and p′ are not too different from
the intrinsic carrier concentration ni.
n′ = p′ = ni
At thermal equilibrium,
np = n0 p0 = ni2
so that
Rn′ = R ′p = 0
R′ =
(
CnC p N t np − ni2
)
(np − n )
2
i
≈
Cn (n + n′) + C p ( p + p′) τ p 0 (n + ni ) + τ n 0 ( p + ni )
(
n0 + δn )( p0 + δn ) − ni2
=
τ p 0 (n0 + δn + ni ) + τ n0 ( p0 + δn + ni )
(
n0 + p0 )δn + (δn )2
R′ =
τ p 0 (n0 + δn + ni ) + τ n 0 ( p0 + δn + ni )
Limits of extrinsic doping and low injection:
For p-type semiconductors
Rn′ =
δn
τ n0
For n-type semiconductors
R′p =
δp
τ p0
For high level injection,
R ′p = Rn′ =
δp
τ p 0 + τ n0
τ n0
1
=
Cn N t
τ p0
1
=
C p Nt
Example A step illumination is applied uniformly to an n-type
semiconductor at time t = 0 and switched off at time t = toff (toff >> τp0).
No applied bias.
(1)Find the expression of the hole concentration and
(2) Sketch hole concentration vs time for 0 < t < ∞ .
(3) Also find the expression of the conductivity for 0 < t < ∞ .
Light
g and δp(t)
Illumination
po+δp(∞)
0
δp(t') = δp(0)exp(-t'/τp0)
τp0g'
po
0
δp
∂δp
∂ 2δp
∂δp
′
= Dp
− μpE
+g −
2
τ p0
∂t
∂x
∂x
toff
Time, t
t'
Example 6.2:
A semiconductor has the following properties:
The semiconductor is a homogeneous, p-type material in thermal equilibrium
for t ≤ 0. At t = 0, an external source is turned on which produces excess
carriers uniformly at rate of g ' = 10 20 cm −3 s −1 . At, t = 2 × 10 −6 s the external
source is turned off.
(1) Derive the expression for the excess-electron concentration as a function
of time for 0 ≤ t ≤ ∞.
(2) Determine the value of excess electron concentration at (i) t = 0,
(ii) t = 2 × 10 −6 s , (iii) t = 3 × 10 −6 s and (iv) t = ∞.
(3) Plot the excess electron concentration as a function of time.
.
a
.
b
D n = 25cm 2 / s
τ n0 = 10 −6 s
D p = 10cm 2 / s
τ p 0 = 10 − 7 s
Example : A continuous illumination is applied to an n-type semiconductor at
time x = 0 and no carrier generation for x > 0. That means g′ = 0 for x > 0.
Determine the steady state excess concentrations for holes for x > 0 and the
hole diffusion current.
Numerical: Consider p type Silicon at T = 300 K. Assume that τn0 = 5 × 10-7 s,
Dn = 25 cm2/s and δn(0) = 1015 cm-3. Determine excess hole concentration profile.
n-type semiconductor
∂δp
∂ δp
∂δp
δp
′
E
= Dp
−
μ
g
+
−
p
∂t
∂x
τ p0
∂x 2
2
0 = Dp
∂ δp δp
−
2
∂x
τ p0
2
δp ( x) = Ae
x
Lp
+ Be
x=0
x
Excess concentration
−x
Lp
δ p(0) δ n(0)
Diffusion
δ p( x ),δn(x)
x
δp ( x) = Ae
x
Lp
+ Be
−x
Lp
Dielectric relaxation time constant:
If an amount of net charge is injected suddenly in a semiconductor, the free
charge carriers of opposite sign try to balance the injected charge and establish
charge neutrality. How fast the charge neutrality can be achieved is determined
by the dielectric relaxation time constant, τd. This phenomenon is called dielectric
relaxation. The excess net charge decays exponentially,
∇ .E
=
ρ
,
ε
J
= σ E ,
dρ σ
+ ( )ρ = 0
ε
dt
ρ (t ) = ρ (0) e −t / τ
d
where
∇ .J
= −
∂ ρ
∂ t
ε
τd =
σ
Example 6.6
(a) Find the dielectric relaxation time constant if the resistivity of the sample is
1014 Ω-cm and relative permittivity is 6.3.
(b) Find the dielectric relaxation time constant of an n-type Silicon with Nd = 1016 cm-3.
Surface Effects
∂ 2δp
δp
0 = Dp
+ g′ −
2
∂x
τ p0
δp ( x) = g 'τ p + Ae
x
Lp
+ Be
−x
Lp
τ
δp B
=
δp S
τ
pB
pS
∂ 2δp
δp
0 = Dp
+ g′ −
2
∂x
τ p0
As x → ∞
x
Lp
−x
Lp
Ae → ∞; ; Be → 0 ∴B ≠ 0, but
δp ( x → ∞ ) = g 'τ p
; δp ( x → 0) = g 'τ p + B
⇒ A=0
Time and Spatial Dependence of Excess Carrier Concentration
Example : A finite number of carriers is generated instantaneously at time t = 0
and x = 0, but g′ = 0 for t > 0. Assume an n-type semiconductor with a constant
applied electric field E0, which is applied in the +x direction. Determine the excess
carrier concentration as a function of x and t.
The ambipolar equation for t > 0 and x > 0,
∂δp δp
∂ 2δp
∂δp
−
−
μ
E
= Dp
p 0
2
∂x τ p 0
∂t
∂x
The solution of the above equation,
δp ( x , t ) =
e
−t τ p 0
(4πD p t )1/ 2
(
⎡ − x − μ p E0t
exp ⎢
4D pt
⎢⎣
)2 ⎤⎥
⎥⎦
Quasi-Fermi level:
If excess carriers are created in a semiconductor, the Fermi energy is strictly no
longer defined. Under nonequilibrium condition, we can assign individual Fermi
level for electrons and holes. These are called quasi-Fermi levels and the total
electron or hole concentration can be determined using these quasi-Fermi
levels. We can write,
⎛ E Fn − E Fi ⎞
n0 + δn = ni exp⎜
⎟
kT
⎝
⎠
⎛ E Fi − E Fp
p0 + δp = ni exp⎜⎜
kT
⎝
⎞
⎟⎟
⎠
Example Given: n-type semiconductor at 300K. n0 = 1015 cm-3, p0 = 105 cm-3, ni =
1010 cm-3, δn = δp=1013 cm-3. Calculate (a) the Fermi energy level and (b) the quasi
Fermi energy levels.