Carrier Transport
Carrier Diffusion
Carrier diffusion
All random movement has the tendency of
‘diffusion’
Particle transport from higher density part to lower
density part until balanced either in homogeneous
density or balanced by other induced effect
Diffusion of charged particle induces current
Diffusion current density: (note the sign and why?)
J n − dif = eDn∇n
J p − dif = −eD p ∇p
µn (cm2/V-s)
Dn (cm2/s)
µp (cm2/V-s)
Dp(cm2/s)
Si
1350
35
480
12.4
GaAs
8500
220
400
10.4
Ge
3900
101
1900
49.2
Total Current Density
Combining drift and diffusion current.
J = enµ n E + epµ p E + eDn∇n − eD p ∇p
Diffusion current: Example
The electron concentration in silicon is given by n(x)=1e15exp(x/L) cm-3 (x>0) where L=1e-4 cm. The electron diffusion
coefficient is Dn=25 cm2/s. Determine the electron diffusion
current density at (a) x=0, (b) x=1e-4 cm, and (c) x-> infinity.
J n − dif = eDn∇n
15
-3
dn
1015 − x L
2 10 cm
e
e − x L = −1.6 × 25 e − x L A/cm 2
J x = eDn
= −eDn
= −e × 25 cm /s − 4
dx
L
10 cm
x = 0 ⇒ J x = −1.6 × 25 A/cm 2 = −40 A/cm 2
x = L ⇒ J x = −1.6 × 25 × exp(−1) A/cm 2 = −14.7 A/cm 2
x → ∞ ⇒ J x → 0 A/cm 2
Graded Impurity Distribution
 E − E Fi 
n0 = ni exp  F

 kT 
 E − E Fi 
p0 = ni exp − F
kT 

n0 p0 = ni2
If carrier concentration is a function of position, the difference
between Fermi level EF and the intrinsic Fermi level EFi must be a
function of position.
But at equilibrium, Fermi level must be the same for all positions.
Therefore, the intrinsic Fermi level and therefore the bandedges
has to change with the position.
This implies an induced electric field
Induced Electric Field
1
1
1
E = ∇E Fi = ∇Ec = ∇Ev
e
e
e
The induced electric field is proportional to the gradient of the
function dependent intrinsic Fermi level or equivilently function
dependent bandedges
Similarly, applied electric field also ‘bends’ bandedges and
intrinsic Fermi level.
Induced Electric Field and Carrier density
1
E = ∇E Fi
e
for n-type extrinsic semiconductors
 E − EFi 
n0 = ni exp  F
= Nd

 kT 
E=−
kT 1
kT 1
∇n0 = −
∇N d
e n0
e Nd
Induced Electric Field and Doping: Example
Assume that the donor impurity concentration in a semiconductor
is given by
 x
15
-3
N d ( x ) = 10 exp −  (cm )
 L
for x>0 and where L=1e-4 cm. Determine the electric field induced
in the material due to this impurity concentration.
kT 1
E=−
∇N d
e Nd
kT 1 dN d
1
Ex = −
= 0.0259 × = 259 V/cm
e N d dx
L
• note the exponential doping profile gives rise to a constant induced field.
Induced Electric Field and Doping: Example
Determine the induced electric field in a semiconductor in thermal
equilibrium given a linear variation in doping concentration.
Assume that the donor concentration in an n-type semiconductor
at T=300K is given by,
16
19
-3
N d = 10 − 10 x (cm )
where x is given in cm and ranges between 0<x<1 um
kT 1
E=−
∇N d
e Nd
kT 1 dN d
1019
= 0.0259 × 16
Ex = −
V/cm
19
e N d dx
10 − 10 x
at x=0, for example
E x = 25.9 V/cm
The Einstein Relation
Assume there is no electrical connections, the induced electrical
field will induce a drift current that exactly balance the diffusion
current for each type of carriers. This gives rise to the Einstein
relation
0 = J n = enµ n E + eDn∇n
but the induced electric field is
therefore
Dn
µn
=
similarly for holes
kT
e
Dp
kT
=
µp
e
E=−
kT 1
∇n
e n
Excess Carriers and Excess Carrier Life Time
At thermal equilibrium
n0 p0 = ni2
when in nonequilibrium state, excess carriers are (typically)
excited in pairs
p = p0 + δp
n = n0 + δn
recombination can be approxiamted as
(
dn
= α r ni2 − np
dt
)
to the first order, or for low-level injection
dδn
= −α r (n0 + p0 )δn
dt
δp = δn
Excess Carrier Life Time
dδn
δn
= −α r (n0 + p0 )δn = −
dt
τ ex
τex is called the excess carrier life time
for n-type material n0>>p0 and
for p-type mateiral p0>>n0 and
1
τ ex ≈
α r n0
1
τ ex ≈
α r p0
Generation-Recombination Processes
Band-to-Band Generation and Recombination
recombination-generation centers: defects, surface states, etc.
Auger Recombination
HW4: Chapter 4
4.2, 7, 17, 24, 28,35,38,44,54