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Prof. Ming-Jer ChenProf. Ming-Jer Chen
Department of Electronics EngineeringDepartment of Electronics Engineering
National Chiao-Tung UniversityNational Chiao-Tung University
October 2, 2014October 2, 2014
DEE4521 Semiconductor Device PhysicsDEE4521 Semiconductor Device Physics
Lecture Lecture 3a3a::
Transport: Drift and DiffusionTransport: Drift and Diffusion
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This lecture accompanies pp. 111–131 on
drift and diffusion, as well as pp. 159-175 on
non-uniform doping, of textbook.
Textbook pages involved
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Hole drift current density JHole drift current density Jp,x,drift p,x,drift = qp<v= qp<vxx> = qp> = qpppxx
vx0 +-
f(vx)
Maxwellian velocity distribution (equilibrium) for holes (<vx> = 0)
Shifted Maxwellian velocity distribution for holes (electric field x: must be small)
x
<vx>: drift (NOT thermal) velocityp: hole mobility
Similarly, for electronsJn,x,drift = qnnx
Note: Polarity
Holes
Drift
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Electron diffusion current density JElectron diffusion current density Jn,x,diffusion n,x,diffusion = qD= qDnndn/dxdn/dx
porn
x
Gradient of carrier density
Dp: Hole diffusion coefficient
Dn: Electron diffusion coefficientvery hot
very cold
Diffusion
Hot to Cold: Diffusion
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Why D and its unit?
-3d -2d -1d 0 1d 2d 3d x
N = 0
N = 1
N = 2
This is a random walk problem.
Variance [x2] =Nd2
1
1/21/2
1/41/4 1/4
1/4
f(x)
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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 = qnn
Hole conductivity p = qpp
Total conductivity = n + p
= V/LI/A = J
Applied V must be small.
I-V in a biased semiconductor
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4-2
Non-uniformly doped semiconductoris 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).
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4-8
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Built-in Field in Non-uniform Semiconductors
4-9
You must be able to distinguish between built-in electric field and applied electric field. (hint: Superposition principle)
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3-10
The experimentally measured dependence of the drift velocity on the applied field.
Figure 3.9
drift =
Focus on low field region(<103 V/cm)
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3-9
Mobility as a function of temperature. At low temperatures, impurity scattering dominates, but at hightemperatures, lattice vibrations dominate.
Figure 3.8
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(a) An electron approaching an ionized donor is deflected toward it, but a hole is deflected away from thedonor. (b) Electrons deflect away from the negatively charge ionized acceptors but holes deflect toward them.
Figure 3.5
3-6
Coulomb (or Impurity) Scattering
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Displacement of atomic planes under the influence of a pressure wave. For a longitudinal wave (a), thedisplacement is in the direction of motion. For a transverse wave (b), the displacement is transverse to thedirection of motion. For a three-dimensional crystal, for each longitudinal wave there are two transversewaves. The dashed lines represent the equilibrium positions, and the solid lines indicate the deflected positionsat a given time.
Figure S1B.6
S1B-6
Lattice Vibrations
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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
3-5
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n = qfe/m*ce
Electron Mobility
Electron Mean Free Timeor Electron Average Scattering Time
Electron Conductivity Effective Mass
Time constants are relevant in device physics.So, the ability to experimentally extract those is essential.
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How to derive n = qfe/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 timel: free path, scattering lengthTwo random variables: and ln: total number of collision events
v
Slope = a = q/m
a
t
v = (a + a +…….+ a)/n
vdrift = <v> = na<v>/n =a<v>…L = a<2>n/2 follows exponential distributionn = L/vdrift<>