C hapter 6: Free Electron Fermi Gas
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Transcript of C hapter 6: Free Electron Fermi Gas
Chapter 6: Free Electron Fermi Gas
Free Electron Theory• Conductors fall into 2 main classes; metals &
semiconductors. Here, we focus on metals.• A metal is loosely defined as a solid with valence
electrons that are not tightly bound to the atoms but are relatively easily able to move through the whole crystal.
• The room temperature conductivity of metals is in the range
and it increases on the addition of small amounts of impurities. The resistivity normally decreases monotonically with decreasing temperature & can be reduced by the addition of small amounts of impurity.
6 8( ) ;10 10metalsRT m
Question! Why do mobile (conducting) electrons occur in some solids & not in others?•When the interactions between electrons areconsidered,this becomes a
Very Difficult Question to Answer.D
Some Common Physical Properties of Metals:Great Physical Strength, High Density, Good
Electrical and Thermal Conductivity, etc.•Here, we outline a surprisingly simple theorywhich does a very good job of explaining many ofthese common properties of metals.
“The Free Electron Model” •We begin with the assumption that
conductionelectrons exist & that they consist of thevalence electrons from the metal atoms.•Thus, metallic Na, Mg & Al are assumed to
have1, 2 & 3 mobile electrons per atom, respectively.•This model seems at first as if it is
Way Too Simple!•But, this simple model works remarkablywell for many metals & it will be used to helpto explain many properties of metals.
• According to this
Free Electron Model (FEM)the valence electrons are responsible for the conduction of electricity, & for this reason these electrons are called
“Conduction Electrons”.• As an example, consider Sodium (Na). • The electron configuration of the Free Na Atom is:
1s2 2s2 2p6 3s1
• The outer electron in the third atomic shell(n = 3, ℓ = 0)
is the electron which is responsible for the chemical properties of Na.
Valence Electron(loosely bound)Core Electrons
• When Na atoms are put together to form a Na metal,
• Na has a BCC structure & the distance between nearest neighbors is 3.7 A˚• The radius of the third shell in Na iis 1.9 A˚
• In the solid the electron wavefunctions of the Na atoms overlap slightly. From this observation it follows that a valence electron is no longer attached to a particular ion, but belongs to both neighboring ions at the same time.
Na metal
• Therefore, these conduction electrons can be considered as moving independently in a square well of finite depth& the edges of the well correspond to the edges of the sample.
• Consider a metal with a cubic shape with edge length L: Ψ and E can be found by solving the Schrödinger equation:
0 L/2
V
-L/2
22
2E
m
0V
Since
Use periodic boundary conditions& get the Ψ’s as travelling plane waves.
( , , ) ( , , )x L y L z L x y z
• The solutions to the Schrödinger equation are plane waves,
where V is the volume of the cube, V=L3
• So the wave vector must satisfy
where p, q, r taking any integer values; +ve, -ve or zero.
( )1 1( , , ) x y zi k x k y k zik rx y z e e
V V
Normalization constant
Na p 2,where k
2Na p
k
2 2
k p pNa L
2xk p
L
2yk q
L
2zk r
L
• The wave function Ψ(x,y,z) corresponds to an energy
• The corresponding momentum is:
• The energy is completely kinetic:
2 2
2
kE
m
22 2 2( )
2 x y zE k k km
( , , )x y zp k k k
2 221
2 2
kmv
m
2 2 2 2m v k
p k
• We know that the number of allowed k values inside a spherical shell of k-space of radius k is:
2
2( ) ,
2
Vkg k dk dk
• Here g(k) is the density of states per unit magnitude of k.
Number of Allowed States per Unit Energy Range?
• Each k state represents two possible electron states, one for spin up, the other is spin down.
( ) 2 ( )g E dE g k dk ( ) 2 ( )dk
g E g kdE
2 2
2
kE
m 2dE k
dk m
2
2mEk
( )g E 2 ( )g kdk
dE2
22
V
kk
2
2mE
2
m
k3/ 2 1/ 2
2 3(2 )
2( )
Vm Eg E
Ground State of the Free Electron Gas
• Electrons are Fermions (s = ± ½) and obey the Pauli exclusion principle; each state can accommodate only one electron.
• The lowest-energy state of N free electrons is therefore obtained by filling the N states of lowest energy.
• Thus all states are filled up to an energy EF, known as The Fermi energy, obtained by integrating the density of states between 0 and EF, The result should equal N. Remember that
• Solving for EF (Fermi energy);2/32 23
2F
NE
m V
3/ 2 1/ 22 3
(2 )2
( )V
m Eg E
3/ 2 1/ 2 3/ 22 3 2 3
0 0
( ) (2 ) (2 )2 3
F FE E
F
V VN g E dE m E dE mE
• The occupied states are inside the Fermi sphere in k-space as shown below; the radius is Fermi wave number kF. 2 2
2F
Fe
kE
m
kz
ky
kx
Fermi SurfaceE = EF
kF
From these two equations, kF can be found as,
1/323F
Nk
V
The surface of the Fermi sphere represents the boundary between occupied & unoccupied k states at absolute zero for the free electron gas.
2/32 23
2F
NE
m V
• Typical values may be obtained by using monovalent potassium metal (K) as an example; for potassium the atomic density and hence the valence electron density N/V is 1.402x1028 m-3 so that
• The Fermi (degeneracy) Temperature TF is given by
193.40 10 2.12FE J eV 10.746Fk A
F B FE k T
42.46 10FF
B
ET K
k
• It is only at a temperature of this order that the particles in a classical gas can attain (gain) kinetic energies as high as EF .
• Only at temperatures above TF will the free electron gas behave like a classical gas.
• Fermi momentum
• These are the momentum & velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.
• So, the Fermi Sphere plays an important role in the behavior of metals.
F FP k F e FP mV
6 10.86 10FF
e
PV ms
m
2/32 232.12
2F
NE eV
m V
1/3213
0.746F
Nk A
V
6 10.86 10FF
e
PV ms
m
42.46 10FF
B
ET K
k
Typical values for monovalent potassium metal;
Free Electron Gas at Finite Temperature
• At a temperature T the probability of occupation of an electron state of energy E is given by the Fermi distribution function
• The Fermi distribution function determines the probability of finding an electron at energy E.
( ) /
1
1 F BFD E E k Tf
e
20
Fermi-Dirac Distribution & The Fermi-Level
Main Application: Electrons in a Conductor •The Density of States g(E) specifies
how many states exist at a given energy E. •The Fermi Function f(E) specifies how many of theexisting states at energy E will be filled with electrons.
•I
•The Fermi Function f(E) specifies, underequilibrium conditions, the probability that anavailable state at an energy E will be occupied by anelectron. It is a probability distribution function.
EF = Fermi Energy or Fermi Level
k = Boltzmann Constant
T = Absolute Temperature in K
Fermi-Dirac Statistics
EF is called The Fermi Energy.Note the following:
• When E = EF, the exponential term = 1 and FFD = (½).
• In the limit as T → 0:L
L
• At T = 0, Fermions occupy the lowest energy levels.• Near T = 0, there is little chance that thermal agitation
will kick a Fermion to an energy greater than EF.
The Fermi Energy EF is essentially the same as theChemical Potential μ.
β (1/kT)
Fermi-Dirac DistributionConsider T 0 K
For E > EF : 0)(exp1
1)( F
EEf
1)(exp1
1)( F
EEf
E
EF
0 1 f(E)
For E < EF :
A step function!
EFE<EF E>EF
0.5
fFD(E,T)
E
( ) /
1
1 F BFD E E k Tf
e
Fermi Function at T=0 & at a Finite Temperature
fFD=? At 0°K
a. E < EF
b. E > EF
( ) /
11
1 F BFD E E k Tf
e
( ) /
10
1 F BFD E E k Tf
e
Fermi-Dirac DistributionTemperature Dependence
25
If E = EF then f(EF) = ½ . If then
So, the following approximation is valid:That is, most states at energies 3kT above EF are empty.
If then lSo, the following approximation is valid:So, 1 f(E) = probability that a state is empty, decays to zero.
So, most states will be filled.
kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small in comparison to other
energies relelevant to the electrons.
kTEE 3F
1exp F
kT
EE
kT
EEEf
)(exp)( F
kTEE 3F
1exp F
kT
EE
kT
EEEf Fexp1)(
Fermi-Dirac Distribution: Consider T > 0 K
Fermi-Dirac DistributionTemperature Dependence
T = 0
T > 0
As the temperature increases from T = 0,The Fermi-Dirac Distribution “smears out”.
T = 0
The Fermi “Temperature” is defined asTF ≡ (EF)/(kB).
T > 0
T = TF
• As the temperature increases from T = 0,The Fermi-Dirac Distribution “smears out”.
• When T >> TF, FFD approaches a decaying exponential.
T >> TF
At T = 0 the Fermi Energy EF is the energyof the highest occupied energy level.•If there are a total of N electrons, then is easy to show that EF has the form:
3/23
23/2
1
2/3
1
)3
(8
)3
(
)(3
1
L
N
m
hNEE
E
EN
F
F
Fermi-Dirac Distribution FunctionFermi-Dirac Distribution Function
0.00 0.25 0.50 0.75 1.00 1.25 1.500.0
0.2
0.4
0.6
0.8
1.0
1.2n(k)
(k)/F
F/kBT
T = 0 50 20 10
( ) /
1( )
1Bk k Tn k
e
0 1 2 3
-4
-2
0
T lnT
Temperature
TT-Dependence of the Chemical Potential-Dependence of the Chemical Potential
Classical
T>0
T=0
n(E,T)
E
g(E)
EF
• n(E,T) number of free electrons per unit energy range is just the area under n(E,T) graph.
( , ) ( ) ( , )FDn E T g E f E T
• Number of electrons per unit energy range according to the free electron model?
• The shaded area shows the change in distribution between absolute zero and a finite temperature.
• The Fermi-Dirac distribution function is a symmetric function; at finite temperatures, the same number of levels below EF are emptied and same number of levels above EF are filled by electrons.
T>0
T=0
n(E,T)
E
g(E)
EF
kkyy
kkxx
kkzz
Fermi SurfaceFermi Surface
1/ 323Fk n
2 / 32 3
2 8F
h n
m
/F F BT k
3
53
5
F
F
NU
u
Fermi-Dirac Distribution FunctionFermi-Dirac Distribution Function
0.00 0.25 0.50 0.75 1.00 1.25 1.500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4n(k)
(k)/F
F/kBT = 50
( ) /
1( )
1Bk k Tn k
e dn
d
FF
T
T
Examples of Fermi SystemsExamples of Fermi Systems
Electrons in metals (dense system)
Heat Capacity of the Free Electron Gas
• From the diagram of n(E,T) the change in the distribution of electrons can be resembled into triangles of height (½)g(EF) and a base of 2kBT so (½)g(EF)kBT electrons increased their energy by kBT.
T>0
T=0
n(E,T)
E
g(E)
EF
• The difference in thermal energy from the value at T=0°K
21( ) (0) ( )( )
2 F BE T E g E k T
• Differentiating with respect to T gives the heat capacity at constant volume,
2( )v F B
EC g E k T
T
2( )
33 3
( )2 2
F F
FF B F
N E g E
N Ng E
E k T
2 23( )
2v F B BB F
NC g E k T k T
k T
3
2v BF
TC Nk
T
Heat Capacity ofFree Flectron Gas
Low-Temperature Heat CapacityLow-Temperature Heat Capacity
2
2el BF
TC Nk
T
3
el lat
lat
C C C
C T
Copper
• Total metallic heat capacity at low temperatures
where γ & β are constants found plotting Cv/T as a function of T2
3C T T ElectronicHeat capacity
Lattice HeatCapacity
Transport Properties of Conduction Electrons
• Fermi-Dirac distribution function describes the behaviour of electrons only at equilibrium.
• If there is an applied field (E or B) or a temperature gradient the transport coefficient of thermal and electrical conductivities must be considered.
Transport coefficientsTransport coefficients
σ,Electricalconductivityσ,Electricalconductivity
K,ThermalconductivityK,Thermal
conductivity