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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

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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

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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

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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<>