ELEKTRONIKOS PAGRINDAI
1
2008
3. ELECTRONS IN METALS AND SEMICONDUCTORS
Large numbers of free electrons can exist in a metal or semiconductor.
Statistical methods are the most useful for the investigation of systems that
consist of a large number of particles.
The type of statistics developed to describe such distributions depends on the
type of particles and also on the possible interactions between them.
Classical statistics is applied to the systems of classical particles. Classical
physics assumes that the particles are distinguishable (easily recognised), can
be named numbering them and that each state can hold an unlimited number
of particles. These assumptions yield the Maxwell-Boltzmann distribution
function that will be denoted by fB(W).
Systems of microparticles are investigated by quantum statistics.
There are two groups of microparticles.
Photons and phonons are indistinguishable and have spin quantum numbers
that are integers: 1, 2, 3,… An unlimited number of photons and phonons can
be accommodated in an allowed state. Such particles have been investigated
by Bose and Einstein. Now they are called bosons. The Bose-Einstein
distribution function is used to describe distribution of bosons.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
2
2008
Statistical methods. Classical and quantum statistics
Electrons, protons and some other microparticles have only spin quantum
numbers that are fractional numbers (-1/2, 1/2, 3/2,…). These microparticles are
also indistinguishable, as bosons, but they obey Pauli’s exclusion principle. This
results in the Fermi-Dirac distribution function denoted by fF(W). Such
particles are called fermions.
We will be interested in the properties of electrons in solids. So first of all we will
examine fermions. Fermions reveal their characteristic features meeting each
other and occupying allowed states.
… Characteristic features of fermions reveal when number of particles NΣ is large
and near to the number of allowed states G.
If NΣ<<G, fermions do not meet occupying allowed states. Then the system of
fermions is non-degenerate. We will see that classical statistics may be
applied to the non-degenerate system of fermions.
Statistical
physics
VGTU EF ESK
Quantum
statistics
BoseEinstein
FermiDirac
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NΣ/G<<1
MaxwellBoltzmann
ELEKTRONIKOS PAGRINDAI
3
2008
Distribution functions
Distribution functions are used to characterise the systems of particles.
There are two types of distribution functions.
Let us consider a system containing NΣ particles.
The number of particles dN having energy in the range from W to W+dW is
proportional to NΣ and dW:
d N = N Σ f (W ) d W
The proportionality coefficient f(W) is distribution function.
Ratio dN/NΣ=f(W)dW means probability dP that particle energy is in the range
from W to W+dW .
f (W ) =
d N / NΣ d P
=
dW
dW
… Ratio dP/dW is probability density.
So distribution function means probability density.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
4
2008
Distribution functions
d N / NΣ d P
f (W ) =
=
dW
dW
At dW = 1, we have that dP = f(W). So distribution function means the
probability of a particle to have energy in a unit range given at energy value
W.
Besides f(W), the other distribution function is used:
N(W)=NΣf(W).
Because N(W)dW = dN, the product N(W)dW means the number of particles dN
with energy from W to W+dW .
At dW = 1, we have that dN = N(W). So distribution function N(W) describes the
number of particles per unit of energy centred at energy value W.
∞
∞
0
0
∫ f (W ) d W = 1
∫ N (W ) d W = N Σ
At given distribution function, we can find parameters characterizing the system
(mean energy, most probable momentum, mean-square-root velocity, etc.)
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
5
2008
Energy distribution of electrons in metals
Objectives:
Analysis of properties of fermions and distribution of electrons in metals
(derivation and analysis of the energy distribution function for electrons in
metals)
Steps:
1. Analysis of density of states of free electrons.
2. Analysis of probabilities of allowed states ocupation.
3. Derivation and analysis of the energy distribution function for electrons in
metals.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
6
2008
Density of states
The state of a classical particle is defined by the set of three co-ordinates and
three components of momentum.
… A point of the six-dimensional space with coordinates x, y, z , px, py, pz
corresponds to the classical particle.
According to the uncertainty principle and Heisenberg’s inequalities:
∆ x∆ y∆ z ∆ p x ∆ p y ∆ p z ≥ h 3
... A cell of the six-dimensional space corresponds to a microparticle. Its
volume is h3.
… The density of the allowed states of microparticles in the six-dimensional space
is limited.
Let us consider that a microparticle is free and can move in the volume V. Then
∆ p x ∆p y ∆ p z ≥ h / V
3
VGTU EF ESK
… The cell of volume h3/V of the three-dimensional
space with co-ordinates px, py, pz corresponds to the
free microparticle.
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ELEKTRONIKOS PAGRINDAI
7
2008
Density of states
... Lets find the number of states u in the momentum
range from p to p+dp.
… A spherical shell corresponds to the momentum
range from p to p+dp. Its volume is
dV p =
Then
4π
( p + d p )3 − 4π p 3 ≅ 4πp 2 d p
3
3
dVp
4 πV
u= 3
= 3 p2 d p
h /V
h
p = 2mWk
u=
dp=
m
d Wk
2Wk
u=
4 2 πV
4 2 πV 3 / 2
m
W − Wc d W = g (W ) d W
3
h
Density of states:
g (W ) =
4 2 πV
VGTU EF ESK
h3
m 3 / 2 W − Wc
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h3
m 3 / 2 W k d Wk
ELEKTRONIKOS PAGRINDAI
8
2008
Fermi-Dirac function
Any allowed energy level may be occupied by two electrons having different spin
quantum numbers. On the other hand, not all available energy states are filled.
Usually electrons cannot gain sufficient energy to occupy very high levels.
The probability that allowed energy level is occupied by electrons is
given by the Fermi-Dirac function.
f F (W ) =
1
exp[(W − WF ) / kT ] + 1
WF – is the constant called the Fermi level,
…
At 0 K all available states up to the energy
level WF are filled, whereas all levels above
are empty.
Probability of occupancy of the Fermi level
is always 0.5, independent of temperature.
The Fermi level is the highest energy level that cabn be occupied by an electron in
a solid at absolute zero temperature.
At higher temperatures the Fermi level corresponds to the value of energy at
which the Fermi-Dirac distribution function has value 0.5.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
9
2008
Energy distribution of electrons in a metal
The number of electrons in a given energy range depends on the number of
available states in the range and also on the probability of occupation of the
states. Because two electrons with different spin quantum numbers can occupy
every energy level,
d N = 2uf F = 2 g (W ) f F (W ) d W
d N = N Σ f (W ) d W = N (W ) d W
N (W ) = N Σ f (W ) = 2 g (W ) f F (W )
W − Wc
8 2 πm 3 / 2
f (W ) =
;
3
exp[(W − WF ) / kT ] + 1
nh
WF − Wc = WFc
h 2  3n 
=
2m  8π 
2/3
The Fermi energy in a metal depends upon the density of free electrons. The
difference WF – Wc is positive.
… The Fermi level lies in the partially filled allowed energy band.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
10
2008
Statistics of electrons in metals. Problems
1. Using distribution function, derive expression for the Fermi energy in a
metal (find how the Fermi energy depends on the density of conduction
electrons).
2. Find the mean kinetic energy of an electron in a metal at absolute zero
temperature.
3. Supposing that WFc= 7 eV , find the mean kinetic energy of an electron in a
metal at absolute zero temperature.
4. Calculate the Fermi energy and the mean kinetic energy of an electron at
0 K in copper given that there is one conduction electron per atom, that the
density of copper is 8920 kg/m3 and its atomic mass is 63.54.
WFc
VGTU EF ESK
h 2  3n 
=
2m  8π 
2/3
3
W k = WFc
5
3
W k = WFc = ... ≅ 4.2 eV
5
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ELEKTRONIKOS PAGRINDAI
11
2008
Distributions of particles in non-degenerate systems.
Equilibrium condition
Objectives:
1. Analysis and properties of electrons in the non-degenerate systems
(general properties of electrons in semiconductors).
2. Derivation of equilibrium condition.
Steps of distribution analysis:
1. Derivation and analysis of the energy distribution function for electrons in
the non-degenerate systems (electrons in semiconductors).
2. Analysis of the non-degenaration condition.
3. Distribution functions for particles in the non-degenerate systems.
4. Use of the distribution functions for calculation of averaqge values
(parameters characterizing systems).
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
11
2008
Distributions of particles in non-degenerate systems
According to the Fermi-Dirac statistics:
d N = 2uf F = 2 g (W ) f F (W ) d W
g (W ) =
4 2 πV
h3
m 3 / 2 W − Wc
1
f F (W ) =
exp[(W − WF ) / kT ] + 1
N (W ) = N Σ f (W ) = 2 g (W ) f F (W )
W − Wc
8 2 πm 3 / 2
f (W ) =
nh 3
exp[(W − WF ) / kT ] + 1
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
12
2008
Distribution functions for a non-degenerate system
The density of free electrons in semiconductors is small. So, free electrons in a
semiconductor form a non-degenerate collection.
The collection of fermions is non-degenerate if
exp[(W − WF ) / kT ] >> 1
dN =
NΣ =
4πV (2m)
h3
3/ 2
4πV (2m)
h3
f F (W ) ≅ eWF / kT e −W / kT = A e −W / kT = f B (W )
W − Wc eWF / kT e −W / kT d W
∞
∞
3/ 2
e
WF / kT −Wc / kT
e
∫
Wk e
2N Σ
π
(kT )−3 / 2
VGTU EF ESK
−Wk / kT
∫
d Wk
x e − ax dx =
0
0
N Σ 2(2 πmkT ) 3 / 2 WF / kT −Wc / kT
n=
e
e
=
V
h3
dN =
4πV (2m) 3 / 2 WF / kT
NΣ =
e
h3
WF / kT
e
Wk e −Wk / kT d Wk = N Σ f (Wk ) d Wk
=
nh 3
2(2 πmkT )
f (Wk ) =
[email protected]
3/ 2
Wmax
∫
W − Wc e −W / kT d W
Wc
π
2a 3 / 2
eWc / kT
2
π
(kT ) − 3 / 2 Wk e −Wk / kT
ELEKTRONIKOS PAGRINDAI
13
2008
Distribution functions for a non-degenerate system
The density of free electrons in semiconductors is small. So, free electrons in a
semiconductor form a non-degenerate collection.
The collection of fermions is non-degenerate if
exp[(W − WF ) / kT ] >> 1
f F (W ) ≅ eWF / kT e −W / kT = A e −W / kT = f B (W )
N (W ) = N Σ f (W ) = 2 g (W ) f F (W )
g (W ) =
4 2 πV
4πV (2m) 3 / 2
WF / kT −W / kT
dN =
W
−
W
e
e
dW
c
3
h
NΣ =
VGTU EF ESK
4πV (2m)
h3
3/ 2
Wmax
eWF / kT
∫
W − Wc e −W / kT d W
Wc
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h3
m 3 / 2 W − Wc
ELEKTRONIKOS PAGRINDAI
14
2008
Distribution functions for a non-degenerate system
4πV (2m)
h3
3/ 2
4 πV (2m)
NΣ =
h3
3/ 2
NΣ =
Wmax
∫
eWF / kT
W − Wc e −W / kT d W
Wc
∞
∞
WF / kT −Wc / kT
e
e
∫
Wk e
dN =
π
(kT )−3 / 2
x e − ax dx =
0
WF / kT
e
=
nh 3
2(2 πmkT ) 3 / 2
Wk e −Wk / kT d Wk = N Σ f (Wk ) d Wk
f (Wk ) =
VGTU EF ESK
∫
d Wk
0
N Σ 2(2πmkT ) 3 / 2 WF / kT −Wc / kT
n=
=
e
e
3
V
h
2N Σ
−W k / kT
2
π
(kT ) − 3 / 2 Wk e −Wk / kT
[email protected]
π
2a 3 / 2
eWc / kT
ELEKTRONIKOS PAGRINDAI
15
2008
Distribution functions for a non-degenerate system
The distribution function for nondegenerate collection:
2
f (Wk ) =
π
eWF / kT =
(kT ) − 3 / 2 Wk e −Wk / kT
nh 3
2(2 πmkT ) 3 / 2
eWc / kT
exp[(WF − Wc ) / kT ] << 1
WF<Wc– kT
e −(W −WF ) / kT = e − (W −Wc ) / kT e (WF −Wc ) / kT << 1
e (WF −Wc ) / kT
nh 3
=
<< 1
3/ 2
2(2 πmkT )
... electrons form a non-degenerate collection, if their density is small.
This condition is usually satisfied for semiconductors.
The difference WF-Wc is negative in this case and the Fermi level lies below the
bottom of the conduction band.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
15
2008
Distributions of classical particles and electrons in the non-degenerate
systems
2NΣ
dN =
(kT ) −3 / 2 Wk e −Wk / kT dWk
π
p2
pdp
Wk =
, dWk =
2m
m
NΣ 2
p2
− p 2 / 2 mkT
dN =
e
dp
3/ 2
π (mkT )
N ( p ) = ...,
f ( p ) = ...
p = mv, dp = mdv
N ( p ) = ...,
f ( p ) = ...
N (v) = ...,
f (v) = ...
NΣ 2  m 
dN =


π  kT 
VGTU EF ESK
3/ 2
2 − mv 2 / 2 kT
ve
dv
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ELEKTRONIKOS PAGRINDAI
15
2008
The mean and the most probable values of classical particles and electrons
in the non-degenerate systems
∞
x = ∫ xf ( x)dx
Wk =
x=v
2kT
v=2
πm
v~ = vrms = v 2
3kT
~
v=
m
x = Wk
kT
Wkt =
2
x= p
pt = 2mkT
0
∞
x = ∫ x f ( x)dx
2
2
0
f' (x) = 0
3
kT
2
x = Wk
2 kT
vt =
m
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
16
2008
Equilibrium condition
Besides the statistical methods, thermodynamical methods are used for
investigation of particle systems. The state of a system is characterized
using macroscopic variables such as pressure, temperature, and volume.
According to thermodynamics at constant temperature and constant
preasure, the free Gibbs energy variation is given by
dG = µ dN,
where µ is the chemical potential of the system, dN is variation in the
number of oparticles in the system.
dG = -µ1 dN + µ2 dN = 0;
dN
µ1 = µ2
WF1 = WF2
… The parts of a system have the
same Fermi energy level in
equilibrium.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
17
2008
Distribution in a non-degenerate system. Problems
1. Find the mean energy and the root-mean-square speed of an electron in
silicon at T = 0 K and T = 300 K.
2. How many times does the mean energy of electrons in germanium increase
with temperature increasing from 10 to 100 0C?
3. Find the most probable kinetic energy of a particle obeying MaxwellBoltzmann statistics at T = 0.
4. Find the maximal density of electrons in a non-degenerate system at 300 K.
VGTU EF ESK
[email protected]