Download 2014.10.1

DEE4521
Semiconductor Device Physics
Lecture 3a:
Transport: Drift and Diffusion
Prof. Ming-Jer Chen
Department of Electronics Engineering
National Chiao-Tung University
October 2, 2014
1
Textbook pages involved
This lecture accompanies pp. 111–131 on
drift and diffusion, as well as pp. 159-175 on
non-uniform doping, of textbook.
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Drift
Hole drift current density Jp,x,drift = qp<vx> = qppx
Maxwellian velocity distribution (equilibrium) for holes (<vx> = 0)
Shifted Maxwellian velocity distribution
f(vx) for holes (electric field  : must be small)
x
Holes
-
<vx>: drift (NOT thermal)
velocity
p: hole mobility
0
x
vx +
Similarly, for electrons
Jn,x,drift = qnnx
Note: Polarity
3
Diffusion
Electron diffusion current density Jn,x,diffusion = qDndn/dx
p
or
n
very hot
Dn: Electron diffusion coefficient
Hot to Cold: Diffusion
Gradient of carrier density
very cold
x
Dp: Hole diffusion coefficient
4
Why D and its unit?
f(x)
N=0
1
This is a random walk problem.
-3d -2d -1d 0 1d 2d 3d
x
N=1
1/2
1/2
N=2
1/4
Variance [x2] =Nd2
1/4
1/4
1/4
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I-V in a biased semiconductor
 = V/L
I/A = J
Applied V must be small.
For electrons in a band, Jn = Jn, drift + Jn,diffusion
For holes in another band, Jp = Jp,drift + Jp,diffusion
Total J = Jn + Jp
= (n + p) + diffusion components
=  + diffusion components
Electron conductivity n = qnn
Hole conductivity p = qpp
Total conductivity  = n + p
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Non-uniformly doped semiconductor
is a Good Vehicle,
1. to derive Einstein’s relationship.
2. to prove that in equilibrium case, Fermi level remains
constant, through any direction in all spaces (real
space, energy space).
7
4-2
8
4-8
Built-in Field in Non-uniform Semiconductors
You must be able to distinguish between built-in electric field and
applied electric field. (hint: Superposition principle)
9
4-9
The experimentally measured dependence of the drift velocity on the applied field.
Figure 3.9
Focus on low field region
(<103 V/cm)
drift =  
10
3-10
Mobility as a function of temperature. At low temperatures, impurity scattering dominates, but at high
temperatures, lattice vibrations dominate.
Figure 3.8
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3-9
(a) An electron approaching an ionized donor is deflected toward it, but a hole is deflected away from the
donor. (b) Electrons deflect away from the negatively charge ionized acceptors but holes deflect toward them.
Figure 3.5
Coulomb (or Impurity) Scattering
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3-6
Displacement of atomic planes under the influence of a pressure wave. For a longitudinal wave (a), the
displacement is in the direction of motion. For a transverse wave (b), the displacement is transverse to the
direction of motion. For a three-dimensional crystal, for each longitudinal wave there are two transverse
waves. The dashed lines represent the equilibrium positions, and the solid lines indicate the deflected positions
at a given time.
Figure S1B.6
Lattice Vibrations
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S1B-6
Room temperature majority and minority carrier mobility as functions of doping in p-type and n-type silicon.
Solid lines: minority carriers; dashed lines: majority carriers.
Figure 3.4
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3-5
Electron Mobility
n = qfe/m*ce
Electron Conductivity Effective Mass
Electron Mean Free Time
or Electron Average Scattering Time
Time constants are relevant in device physics.
So, the ability to experimentally extract those is essential.
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How to derive n = qfe/m*ce?
This is a collision (scattering) event. This event is a Poisson event.
Given applied  in a conductor with a length L





l
l
l
l
l
: time to scatter, free time
l: free path, scattering length
Two random variables:  and l
n: total number of collision events
v
v = (a + a +…….+ a)/n
a
t
Slope = a = q/m
vdrift = <v> = na<v>/n =a<v>
…
L = a<2>n/2
 follows exponential distribution
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n = L/vdrift<>