3. ANALYSIS OF DISTRIBUTION FUNCTIONS
3.1. Objective of the test
Analysis of distribution functions and properties of electrons in
metals and semiconductors.
3.2. Theory and the main formulae
Statistical methods are used for the investigation of systems
consisting of large number of particles. Electrons in a metal are
investigated by quantum statistics.
The density of available energy levels for conduction electrons
(the number of available states per unit volume and unit of energy
centred at W) is given by
g W  
4 2V 3 / 2
m W  Wc 1 / 2 ,
3
h
(3.1)
where Wc – is the energy corresponding to the bottom of the
conduction band, h – Plank’s constant, V – volume, m – electron
mass.
The probability that an allowed energy level W is occupied by
electrons is described by the Fermi-Dirac function:
1
.
(3.2)
exp W  WF  kT  1
Here WF is the Fermi energy, k – Boltzmann’s constant, T –
fF 
absolute temperature.
The energy distribution of electrons in a metal is given by
4V 2m3 2
W
N W   2 g W  f F W  

. (3.3)
3
exp W  WF  kT   1
h
If the inequality exp W  WF  kT  1 is satisfied, a system of
electrons is non-degenerate and the classical Maxwell-Boltzmann
statistics can be used. Then
11
nh3
,
2(2mkT ) 3/ 2
e(WF Wc ) / kT 
(3.4)
where n is the density of free electrons.
So, distribution of free electrons in a semiconductor is described
by
f (Wk ) 
2
( kT )  3 / 2 Wk e Wk / kT .

Here Wk  W  Wc is the kinetic energy.
(3.5)
Integrating the distribution function we can find the part of the
particles with energies that exceed W  :
N

N

 f (W ) d W
k
W
k
.
(3.6)

Here N  is the total number of particles, N  – the number of
particles with Wk  W  .
If W *  0 , then N  N   1 .
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1.
2.
3.
4.
5.
6.
7.
8.
3.3. Preparing for the test:
Using lecture-notes and referenced literature [1, p. 38–52],
examine principles of statistical physics, distribution functions
and properties of electrons in metals and semiconductors.
Prepare to answer the questions:
What statistics can by applied to electrons in a metal?
What statistics is applied to a non-degenerate system of microparticles?
What statistics is usually applied to electrons in a semiconductor?
Explain the meaning of a distribution function.
What are the meanings of products N (W ) d W and f (W ) d W ?
Write the expression and sketch the graph of the Fermi-Dirac
function.
Describe the significance of the Fermi-Dirac function.
Explain the meaning of the Fermi energy.
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9. Write and explain the non-degeneration condition.
10. Can we use the classical statistics for electrons in a solid?
11. How does the mean energy of a particle in a non-degenerate
system change when temperature increases?
3.4. In laboratory:
1. Answer the test question.
2. According to specified data simulate behaviour of free electrons
in metals and semiconductors:
a) calculate the Fermi-Dirac distribution function at various
temperatures;
b) find energy distributions of electrons in a metal at various
temperatures;
c) find energy distributions of electrons in a non-degenerate
semiconductor at given temperatures;
d) find variation of the Fermi energy with temperature at
constant density of electron in a semiconductor;
e) calculate what part of electrons in a semiconductor has kinetic
energy exceeding given energy value W  , find dependence of
this part on W  and temperature T .
3. After necessary calculations plot graphs and examine the results.
4. Prepare the report.
1.
2.
3.
4.
5.
3.5. Contents of the report
Objectives.
Initial data.
Results of calculations.
Graphs.
Conclusions.
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