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3
FREE ELECTRON THEORY OF METALS
3.1 Drude Model
The most important characteristic of a metal is its high electrical conductivity.
Around 1900, shortly after J. J. Thompson’s discovery of the electron, people
became interested in understanding more about the mechanism of metallic
conduction. The first work by E. Riecke in 1898 was quickly superseded by
that of Drude in 1900. Drude1 proposed an exceedingly simple model that explained a well-known empirical law, the Wiedermann–Franz law (1853). This
law stated that at a given temperature the ratio of the thermal conductivity
to the electrical conductivity was the same for all metals. The assumptions of
the Drude model are:
(i) a metal contains free electrons which form an electron
­
®gas.
2
(ii) the electrons have some average thermal energy 12 mvT
, but they ­pursue
®
2
random motions through the metal so that hvT i = 0 even though vT
6=
0. The random motions result from collisions with the ions.
(iii) because the ions have a very large mass, they are essentially immovable.
3.2 Electrical Conductivity
In the presence of an electric field E the electrons acquire a drift velocity
vD which is superimposed on the thermal motion. Drude assumed that the
probability that an electron collides with an ion during a time interval dt is
simply proportional to dt
τ , where τ is called the collision time or relaxation
time. Then Newton’s law gives
¶
µ
vD
dvD
+
= −eE,
(3.1)
m
dt
τ
1
P. Drude, Annalen der Physik 1, 566 (1900); ibid., 3, 369 (1900); ibid., 7, 687
(1902).
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3 FREE ELECTRON THEORY OF METALS
where −e is the charge on an electron. Some appreciation of the term of
relaxation time dt
τ can be obtained by assuming that the system acquires a
drift velocity vD in the presence of an electric field E, and then, at time t = 0,
the electric field is turned off. The behavior of vD (t) as a function of time is
given by
vD (t) = vD (0)e−t/τ ,
(3.2)
showing that vD relaxes from vD (0) toward zero with a relaxation time τ .
In the steady state (where v̇D = 0), vD is given by
vD = −
eEτ
.
m
(3.3)
The quantity eτ
m , the drift velocity per unit electric field, is called µ, the
drift mobility. The velocity of an electron including both thermal and drift
components is
eτ E
v = vT −
.
(3.4)
m
The current density caused by the electric field E is simply
X
j = V −1
(−e)v.
(3.5)
all
electrons
But
P
all
electrons
vT = 0, so that
µ
¶
eτ E
j = V −1 N (−e) −
= σE.
m
Here σ, the electrical conductivity, is equal to
electron concentration.
n0 e2 τ
m
(3.6)
where n0 =
N
V
is the
3.3 Thermal Conductivity
The thermal conductivity is the ratio of the thermal current (i.e., the energy
current) to the magnitude of the temperature gradient.­In the ®presence of a
1
2
temperature gradient ∂T
∂x , the average thermal energy 2 mvT will depend
on the local temperature T (x). The electrons sense the local temperature
through collisions with the lattice. Thus, the thermal energy of a given electron
will depend on where it made its last collision. If we choose an electron at
random, the mean time back to its last collision is τ . Therefore, an electron
crossing the plane x = x0 at an angle θ to the x-axis had its last collision
at x = x0 − vT τ cos θ. (See Figure 3.1.) The energy of such an electron is
E(x) = E (x0 − vT τ cos θ). The number of electrons per unit volume whose
direction of motion is in the solid angle dΩ is simply n0 dΩ
4π . (See Figure 3.2.)
3.3 Thermal Conductivity
79
Š–š
GG
Fig. 3.1. An electron crossing the plane x = x0 at an angle θ to the x-axis
The number of such electrons crossing a unit area at x0 is n0 dΩ
4π vT cos θ, giving
for the energy flux through a unit area at x0
Z
dΩ
.
(3.7)
w(x0 ) = E (x0 − vT τ cos θ) n0 vT cos θ
4π
Just as we did for the thermal conductivity due to phonons we expand
E (x0 − vT τ cos θ) and perform the integral over θ from 0 to π. This gives
µ
¶
1
∂E
2
w(x) = − n0 vT
τ
.
(3.8)
3
∂x
But
∂E
∂x
=
∂E ∂T
∂T ∂x ,
so the thermal conductivity κ is given by
κ=
w
1
1 2
2 dE
τ
τ Cv ,
= n0 vT
= vT
−∂T /∂x
3
dT
3
(3.9)
where Cv = n0 dE
dT is the heat capacity per unit volume (or the specific heat).
d Ω = 2π sin θd θ
θ
dθ
Fig. 3.2. Solid angle dΩ in which electrons moving to cross the plane x = x0 at an
angle θ to the x-axis
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3 FREE ELECTRON THEORY OF METALS
3.4 Wiedemann–Franz Law
The ratio of κ to σ is given by
κ
=
σ
1 2
3 v T τ Cv
.
n0 e2 τ
m
(3.10)
2
Now
­ 1 Drude
® 3 applied the classical
¡ 3 ¢gas laws to evaluation of vT and Cv , viz.,
2
2 mvT = 2 kB T and Cv = n0 2 kB . This gave
3
κ
=
σ
2
µ
kB
e
¶2
T.
(3.11)
κ
In addition to agreeing with the Wiedemann–Franz law, the ratio L = σT
had
¡
¢
3 kB 2
−13
the value 2 e
which was equal to 1.24 × 10
esu. The observed values
for L, called the Lorenz number2 , averaged to roughly 2.72 × 10−13 esu.
Drude made an error of a factor of 2 in his original paper and found that
L ∼ 2.48 × 10−13 esu, remarkably close to the experimental value.
3.5 Criticisms of Drude Model
­
®
2
1. If 12 mvT
= 32 kB T , then the electronic contribution to Cv had to be
3
Cv = 2 N kB = 23 R. This is half as big as the lattice contribution and was
simply not observed.
2. Experimentally σ varies as T −1 . This implies that n0 τ ∝ T −1 since e2
and m are constants. In Drude’s picture, the mean free path l ' vT τ was
thought to be of the order of the atomic spacing and therefore independent
of T . Since vT ∝ T 1/2 this would imply that τ ∝ T −1/2 and, to satisfy
n0 τ ∝ T −1 , that n0 ∝ T −1/2 . This did not make any sense.
3.6 Lorentz Theory
Since Drude’s simple model gave some results that agree fairly well with experiment, Lorentz3 decided to use the full apparatus of kinetic theory to investigate the model more carefully. He did not succeed in improving on Drude’s
model, but he did make use of the Boltzmann distribution function and Boltzmann equation which we would like to describe.
2
3
Ludvig Valentin Lorenz (1829-1891).
Hendrik Antoon Lorentz (1853-1928).
3.6 Lorentz Theory
81
3.6.1 Boltzmann Distribution Function
The Boltzmann distribution function f (v, r, t) is defined by
f (v, r, t)d3 rd3 v = the number of electrons in the volume element d3 r centered
at r whose velocity is between v and v + dv at time t.
Boltzmann’s equation says that the total time rate of change of f (v, r, t) must
be balanced by its time rate of change due to collisions, i.e.,
µ ¶
df (v, r, t)
∂f
=
.
(3.12)
dt
∂t c
³ ´
Here ∂f
d3 r d3 v dt is the net number of electrons forced into the volume
∂t
c
element d3 rd3 v (in phase space) by collisions in the time interval dt.
3.6.2 Relaxation Time Approximation
The simplest form of the collision term is
µ ¶
∂f
f − f0
=−
,
∂t c
τ
(3.13)
where f0 is the thermal equilibrium distribution function, f the actual
nonequilibrium distribution function (which differs from f0 due to some external disturbance), and τ is a relaxation time. Once again if f − f0 is nonzero
due to some external disturbance, and if at time t = 0 the disturbanceis turned
off, one can simply write
(f − f0 )t = (f − f0 )t=0 e−t/τ .
(3.14)
3.6.3 Solution of Boltzmann Equation
We are frequently interested in small perturbations away from equilibrium
and can linearize the Boltzmann equation. For example, suppose the external
perturbation is a small electric field E in the x-direction, and a temperature
∂f
gradient ∂T
∂x . The steady state Boltzmann equation ( ∂t = 0) is
µ
¶
∂f
eE
∂f
f − f0
−
+
vx = −
.
(3.15)
∂vx
m
∂x
τ
If f − f0 is small we can approximate f on the left hand side by f0 and obtain
·
¸
eE ∂f0
∂f0
f ' f0 + τ
− vx
.
(3.16)
m ∂vx
∂x
∂T
0
This is linear response since E and ∂f
∂x are already linear in E or ∂x . The
electrical current density and thermal current density are given, respectively,
by
82
3 FREE ELECTRON THEORY OF METALS
Z
(−e)v f (r, v, t) d3 v,
j(r, t) =
and
(3.17)
Z
w(r, t) =
εv f (r, v, t) d3 v.
(3.18)
In Eq.(3.18) ε = 21 mv 2 is the kinetic energy of the electron of velocity v. We
substitute the solution for f given by Eq.(3.16) into Eqs.(3.17) and (3.18) to
calculate j and w.
3.6.4 Maxwell–Boltzmann Distribution
To evaluate j and w it is necessary to know f0 (v). Lorentz used the following
expression
³ m ´3/2
e−ε/Θ .
(3.19)
f0 (v) = n0
2πΘ
Here n0 = N/V , Θ = RkB T , and ε = 21 mv 2 . The normalization constant
has been chosen so that √f0 (v)d3 v = n0 . The reader should check this. (Use
R ∞ 1/2 −x
x e dx = Γ ( 23 ) = 2π .)
0
The use of classical statistical mechanics and the Maxwell–Boltzmann distribution function is the source of the difficulty with the Lorentz theory. In
1925 Pauli4 proposed the exclusion principle; in 1926 Fermi and Dirac5 proposed the Fermi–Dirac statistics, and in 1928 Sommerfeld published the Sommerfeld Theory of Metals. The Sommerfeld theory was simply the Lorentz
theory with the Fermi–Dirac distribution function replacing the Boltzmann–
Maxwell distribution function.
3.7 Sommerfeld Theory of Metals
Sommerfeld6 treated the Drude electron gas quantum mechanically. We can
assume that the electron gas is contained in a cubic box of edge L, and that
the potential inside the box is constant. The Schrödinger equation is
−
h̄2 2
∇ Ψ (r) = EΨ (r),
2m
(3.20)
Ψk (r) = V −1/2 eik·r
(3.21)
and its solution is
Ek =
4
5
6
h̄2 k2
2m .
W. Pauli, Z. Physik 31, 765 (1925).
E. Fermi, Z. Physik 36, 902 (1926); P. A. M. Dirac, Proc. Roy. Soc. London, A
112, 661 (1926).
A. Sommerfeld, Zeits. fur Physik 47, 1 (1928).
3.8 Review of Elementary Statistical Mechanics
83
To avoid difficulties with boundaries, we can assume periodic boundary conditions so that x = 0 and x = L are the same point. Then the allowed values
of kx (and ky and kz ) satisfy kx = 2π
L nx , where nx = 0, ±1, ±2, · · · , and
µ ¶2
¡ 2
¢
h̄2 2π
Ek =
nx + n2y + n2z .
(3.22)
2m L
The functions |ki form a complete orthonormal set with
Z
0
hk|k i = d3 rΨk∗ (r)Ψk0 (r) = δkk0 .
X
|ki hk| = 1 or
k
X
Ψk∗ (r0 )Ψk (r) = δ(r0 − r).
(3.23)
(3.24)
k
Fermi Energy
The Pauli principle states that only one electron can occupy a given quantum
state. In the Sommerfeld model, states are labelled by {k, σ} = (kx , ky , kz ) and σ,
where σ is a spin index which takes on the two values ↑ or ↓. At T = 0, only
the lowest N energy states will be occupied by the N electrons in the system.
Define kF as the value of k for the highest energy occupied state. Then the
number of particles is given by
Z
X
V
N=
1=
2
d3 k.
(3.25)
(2π)3 k<kF
k < kF
σ
The factor of 2 comes from summing over spin. The integration simply gives
4
3
3 πkF , resulting in the relation
kF3 = 3π 2 n0 .
(3.26)
The Fermi
εF (≡ ΘF ), Fermi velocity vF , and Fermi tempera³ energy
´
ΘF
ture TF = kB are defined, respectively, by
εF =
h̄2 kF2
1
= mvF2 = ΘF .
2m
2
(3.27)
For a typical metal, we have n0 = 1023 cm−3 giving εF ' 5 eV, vF '
108 cm/sec, and TF ' 105 K.
3.8 Review of Elementary Statistical Mechanics
Suppose that the states of an ideal Fermi gas are labelled φ1 , φ2 , · · · , φi , · · ·
and that they have energies ε1 , ε2 , · · · , εi , · · · . Then if N is the total number
of Fermions
84
3 FREE ELECTRON THEORY OF METALS
X
ni = N,
(3.28)
i
where ni = 1 if the state φi is occupied and ni = 0 if it is not. The partition
function ZN for this N particle system is defined by
ZN =
0
X
e−β
P
i
ni εi
.
(3.29)
{ni }
P0
−1
In Eq.(3.29) β = (kB T ) and the symbol {ni } means a summation over all
P
sets of values {ni } = {n1 , n2 , · · · , ni , ·} which satisfy the condition i ni = N .
This restriction makes performing the sum to obtain ZN difficult. One can
avoid this difficulty by using the grand canonical ensemble instead of the
canonical ensemble. The grand partition function Q is defined by
Q=
∞
X
eβζN ZN .
(3.30)
N =0
The symbol ζ is is called the chemical potential. When we substitute
Eq.(3.29) into Eq.(3.30), the summation over N removes the restriction on
the sets of values {ni } included in the sum appearing in Eq.(3.29). We can
rewrite the grand partition function as follows:
Q=
∞ X
0
X
e−β
P
i
ni (εi −ζ)
(3.31)
N =0 {ni }
=
1
1
X
X
···
n1 =0 n2 =0
1
X
· · · e−β(ε1 −ζ)n1 e−β(ε2 −ζ)n2 · · · e−β(εi −ζ)ni · · · .
ni =0
It is easy to see that
1
X
e−β(εi −ζ)ni = 1 + e−β(εi −ζ)
ni =0
so that
Q=
i
Yh
1 + e−β(εi −ζ) .
(3.32)
i
The average occupancy of some quantum state l is given by
X
P
n̄l = Q−1
nl e−β i ni (εi −ζ) .
(3.33)
{ni }
For all i 6= l, the factor involving i in the numerator is exactly cancelled by
the same factor in Q−1 leaving us
3.8 Review of Elementary Statistical Mechanics
P
nl e−β(εl −ζ)nl
e−β(εl −ζ)
.
=
−β(εl −ζ)nl
1 + e−β(εl −ζ)
nl e
n̄l = Pnl
85
(3.34)
Thus we find the Fermi–Dirac distribution function of
n̄l =
1
e(εl −ζ)/Θ
+1
.
(3.35)
At Θ = 0 all states whose energy is smaller than εF are occupied; all states
of higher energy empty. Notice that Eq.(3.33) can be written
n̄l = −kB T
∂
ln Q,
∂εl
(3.36)
a form that is sometimes useful.
3.8.1 Fermi–Dirac Distribution Function
At zero temperature the Fermi–Dirac distribution function can be written, as
a function of energy ε, as
½
1 if ε < εF ,
f0 (ε) =
(3.37)
0 if ε > εF .
At a finite temperature
f0 (ε) =
1
.
e(ε−ζ)/Θ + 1
(3.38)
Clearly at ε = ζ, f0 (ε = ζ) is equal to 12 . (See Figure 3.3.) The value of ζ is
determined (as a function of T ) by the condition
X
f0 (εkσ ) = N.
(3.39)
kσ
T
T
Fig. 3.3. Fermi–Dirac distribution function f0 (ε) for two different temperatures
86
3 FREE ELECTRON THEORY OF METALS
3.8.2 Density of States
It is easiest to determine G(ε), the total number of states per unit volume
whose energy is less than ε, and then obtain g(ε) from it.
G(ε + dε) − G(ε) =
dG
dε = g(ε)dε.
dε
(3.40)
2 4π 3
k ,
(2π)3 3
(3.41)
For free electrons we have
X
G(ε) = V −1
1=
kσ
εkσ ≤ ε
where
h̄2 k2
2m
= ε. It is easy to see that
µ
G(ε) = n0
k
kF
¶3
µ
= n0
ε
εF
¶3/2
.
(3.42)
Thus, from Eq.(3.40) we find
g(ε) =
3 n0
2 εF
µ
ε
εF
¶1/2
=
1
2π 2
µ
2m
h̄2
¶3/2
ε1/2 .
(3.43)
For electrons moving in a periodic potential, g(ε) does not have such a simple
form.
At a finite temperature Θ, the number of electrons per unit volume having
energies between ε and ε + dε is simply the product of g(ε)dε and f0 (ε):
n(ε)dε = g(ε)f0 (ε)dε. (See Figure 3.4.) The chemical potential ζ is determined
from
Z ∞
N =V
g(ε)f0 (ε)dε.
(3.44)
0
Fig. 3.4. Particle density n(ε) and the density of states g(ε)
3.8 Review of Elementary Statistical Mechanics
87
3.8.3 Thermodynamic Potential
The thermodynamic potential Ω is defined by
´
X ³
Ω = −Θ ln Q = −Θ
ln 1 + e(ζ−εi )/Θ .
(3.45)
i
Functions that are commonly used in statistical mechanics are:
internal energy U,
Helmholtz free energy F = U − T S,
Thermodynamic potential Ω = U − T S − ζN = −P V,
enthalpy H = U + P V = T S + ζN,
Gibbs free energy G = U − T S + P V = ζN.
(3.46)
These definitions together with Euler’s relation
U = T S − P V + ζN,
(3.47)
and the second law of thermodynamics
dU = T dS − P dV + ζdN
(3.48)
are very useful to remember. By using Eqs.(3.47) and (3.48) and Ω = −P V ,
one can obtain
dΩ = −SdT − P dV − N dζ.
(3.49)
From Eq.(3.49) one can see that the entropy S, pressure P and particle number
N can be obtained from the thermodynamic potential Ω
µ
¶
∂Ω
S=−
,
∂T V,ζ
¶
µ
∂Ω
,
(3.50)
P =−
∂V T,ζ
µ
¶
∂Ω
N =−
.
∂ζ V,T
3.8.4 Entropy
We know that
Ω = −Θ
X
³
´
ln 1 + e(ζ−εi )/Θ .
(3.51)
i
But we can write
1 − n̄i = 1 −
1
1
= (ζ−ε )/Θ
,
i
e(εi −ζ)/Θ + 1
e
+1
(3.52)
88
3 FREE ELECTRON THEORY OF METALS
so that
h
i
ln (1 − n̄i ) = − ln 1 + e(ζ−εi )/Θ .
(3.53)
We can express Eq.(3.51) as
Ω=Θ
X
ln (1 − n̄i ) .
(3.54)
i
Since the entropy is given by S = − ∂Ω
∂Θ , we can obtain
S=−
X
ln (1 − n̄i ) + Θ
i
Evaluating
∂ n̄i
∂Θ
X
(1 − n̄i )
−1
i
∂ n̄i
.
∂Θ
(3.55)
−1
and multiplying by Θ (1 − n̄i ) gives
µ
¶
Θ ∂ n̄i
1 − n̄i
= n̄i ln
.
1 − n̄i ∂Θ
n̄i
(3.56)
Sustituting this result into Eq.(3.55) gives
X
S = −kB
[(1 − n̄i ) ln (1 − n̄i ) + n̄i ln n̄i ] .
(3.57)
i
We have inserted the factor kB into Eq.(3.57); in the derivation we had essentially set it equal to unity. Notice that the expression for S goes to zero as T
goes to zero because n̄i takes on the values 0 or 1 in this limit. In addition we
can write that
µ
¶¸
X·
n̄i
ΘS = −Θ
ln (1 − n̄i ) + n̄i ln
,
1 − n̄i
i
X
X µζ − ε¶
= −Θ
ln (1 − n̄i ) − Θ
n̄i
,
Θ
i
i
X
X
X
= −Θ
ln (1 − n̄i ) − ζ
n̄i +
n̄i εi ,
(3.58)
i
= −Θ
X
i
i
ln (1 − n̄i ) − ζN + U.
i
If we write F = U − T S we have
F = Nζ + Θ
X
i
= Nζ − Θ
X
ln (1 − n̄i ) ,
´
³
ζ−εi
ln 1 + e Θ .
i
If we hold V and T constant, the energy levels εi are unchanged and
(3.59)