Order 1252715: Condensed Matter physics

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Lect-15-Semicond-2.pdf

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Semiconductors 2

Statistical Mechanics of Semiconductors

Recall from Lecture 3 – density of states for free electrons

per unit volume

Electrons in conduction band like

free electrons but with mass m*, can write:

Similarly the density of states for holes

near the top of the valence band are:

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The Fermi-Dirac distribution for a semiconductor

• For a metal, the Fermi energy is the highest occupied energy at 0 K. The chemical potential is temperature- dependent (but not much) and so the two are essentially the same.

• For a semiconductor, the definition of the Fermi energy is not so clear. We better use the chemical potential.

• Some (many) people also use the term “Fermi energy” for semiconductors but then it is temperature-dependent.

Earlier (Lecture 3) we wrote it as:

In this Section, we call it the Fermi-Dirac function to reflect relates to

electrons – recall it gives the probability that an available energy state E will

be occupied by an electron at absolute temperature T.

Statistical Mechanics of Semiconductors

For a given chemical potential, the total number of electrons in

the conduction band as a function of temperature is:

where

For

We have

And then,

Boltzman distribution

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Statistical Mechanics of Semiconductors

Want to solve integral: multiply Eq (1) by

(1)

Standard Equation for density of electrons

Statistical Mechanics of Semiconductors Similarly can get the number of holes in the valence band p as:

When substantially above the top of the valence

band we have:

and

Standard Equation for density of holes

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Number of electrons excited into conduction band must equal number

of holes left behind in valance band so

Law of Mass Action

Intrinsic Semiconductors

Forming product of density of electrons in conduction band, and holes in the

valence band we obtain important relation:

Depends only on band gap

Dividing the density of electrons in conduction band n(T), and holes in the valence

band p(T) we obtain:

Intrinsic Semiconductors

Taking the log of both sides of the below

and solving for gives:

That is, an expression for the chemical potential – at T=0 gives exactly in

the middle of the band-gap

Using the law of Mass Action above with n=p we obtain:

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Extrinsic/Doped Semiconductors

Law of Mass Action also holds for doping when we have

Concentrations n and p

From the law of mass action we have

Consider intrinsic case:

np =

Example

gap size (eV) n in m -3

at 150 K

n in m -3

at 300 K

InSb 0.18 2x10 22

6x10 23

Si 1.11 4x10 6

2x10 16

diamond 5.5 6x10 -68

1x10 -21

Using prefactor

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Dopants, n- and p-type

Majority and minority carriers

equal number of

electrons and holes

majority: electrons

minority: holes

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Band diagram, density of states, Fermi-Dirac distribution, and the carrier concentrations at thermal equilibrium

Intrinsic

semiconductor

n-type

semiconductor

p-type

semiconductor

Consider a Si sample maintained at T = 300K under

equilibrium conditions, doped with Boron to a

concentration 2×1016 cm-3 : Given the intrinsic

concentration n i = p i is 1x10 10 cm-3

• What are the electron and hole concentrations (n

and p) in this sample? Is it n-type or p-type?

Example

Suppose the sample is doped additionally with Phosphorus

to a concentration 6×1016 cm-3.

• Is the material now n-type or p-type?

What are the n and p concentrations now?