Download Fermi-Dirac Statistics

Fermi-Dirac Statistics
• At finite temperature the electrons are no longer entirely in
their ground state configuration
– Fermi-Dirac distribution
describes the probability
that an energy level E
will be occupied in a
system of fermions at
temperature T.
f (E) =
1
exp[(E ! EF ) / k BT ] + 1
States unoccupied at T=0
have electrons in them at
finite temperature
Definitions
• E is the energy we are interested in.
• T is the temperature at equilibrium.
Fermi-Dirac function at Zero Temperature
• Plot the function on the board in the limit of T→0. Show the step-wise nature
of the transition when E=ΕF.
• Note how the step-wise transition at T=0 occurs because E-EF is divided by a
number approaching zero, which is equivalent to multiplying by infinity. Thus if
E-EF is less than zero, the exponential is zero, and f(E)=1, while if E-EF is greater
than zero, then then exponential is infinity, and f(E)=0.
At the transition temperature the Fermi-Dirac function has a width of order kT.
Specifically, the 10%-90% width is 4.4 kT, which for T=300K corresponds to
0.11 eV.
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Electronic Heat Capacity
• Heat Capacity: Coefficient relating the energy added to a
system and the resultant increase in temperature
"U
C!
"T
– The product D(E)f(E)dE gives
the number of occupied states
at a given temperature T.
Only
electrons
near EF
can gain
energy
– Energy at a given temperature
found by evaluating
"
U(T ) = # E ! D(E)! f (E) dE
0
Classical Statistics:
U = 32 NkT
C=
!U 3
= Nk
!T 2
Quantum Statistics: The product D(E)×f(E) gives the number of occupied
states at temperature T. The observed value of the heat capacity is reduced from
the classical value by order T/TF (factor of 0.01).
!2
T2
U=
Nk B
4
TF
"U ! 2
T
C=
=
NkB
"T
2
TF
A two-dimensional model provides a useful means to compute the integral for
U(T) graphically, since D(E)=constant in 2D. We consider that all electrons
within kBT of EF have their energy increased by exactly kBT . All other electrons
do not change their energy, as their occupancy remains unity. Thus, for electrons
which can increase their energy, we have the following
• Energy increase per electron is kT
• Number of electrons increasing their energy is N × kBT/EF=NT/TF
• Total energy change is: energy per electron × number of electrons
= kT × NT/TF=NkT × T/TF (which is the same as the analytic
result bar a factor of π 2/4)
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Comparison with Experiment
• Phonon contribution ∝ T3
• Electron contribution ∝ T
γexpt:γfree
C = !T + AT 3
"
C
= ! + AT 2
T
Li
Na
K
Rb
Cs
Be
2.18
1.26
1.25
1.26
1.43
0.34
For small temperatures the heat capacity is a sum of linear and cubic terms in the
temperature - the linear behaviour here is due to the Fermionic nature of the heat
conduction electrons, while the cubic behaviour arises from the phonons (which
are Bosons). The linear term is expressed in the y-intercept, and the cubic term
in the slope. See that the free electron gas model provides an excellent starting
point for understanding metals.
The coefficient of the linearity is denoted γ
and is known as the Sommerfeld parameter.
For Potassium:
γexpt=2.08 mJ/mol/K2
! free
" 2 Nk B
! =
2 TF
23
$2 3
" 2 (6.02 # 10 )(1.38 # 10 )
=
= 1.67 mJ/mol/K 2
2
24,600
Departures from unity due to the interaction of the conduction electrons with
1) the periodic potential the rigid crystal lattice.
2) the quanta of lattice vibrations (phonons)
3) other conduction electrons.
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