Download Lecture notes in Solid State 3 Eytan Grosfeld

Lecture notes in Solid State 3
Eytan Grosfeld
Physics Department, Ben-Gurion University of the Negev
Classical free electron model for metals: The Drude model
Recommended reading:
• Chapter 1, Ashcroft & Mermin.
The conductivity of metals is described very well by the classical Drude formula,
σD =
ne2 τ
m
(1.1)
where m is the electronic mass (as decided by the band-structure) and e is the
electronic charge. The conductivity is directly proportional to the electronic
density n; and, to τ , the mean free time between collisions of the conducting
electrons with defects that are generically present in the system:
• Static defects that scatter the electron elastically, including: static impurities and structural defects. One can define the elastic mean free time
τe .
• Dynamical defects which scatter the electron inelastically (as they can
carry off energy) including: phonons, other electrons, other excitations
(such as plasmons). One can accordingly define the inelastic mean free
time τϕ .
The efficiency of the latter processes depends on the temperature T : we expect it
to increase as T is increased. At very low temperatures the dominant scattering
is elastic, and then τ does not depend strongly on temperature but instead it
depends on the amount of disorder (realized as random static impurities). We
expect τ to decrease as the temperature or the amount of disorder in the system
are increased.
The Drude model
1897 - discovery of the electron (J. J. Thomson). 1900 - only three years later,
Drude applied the kinetic theory of gases to a metal - considering it to be a “gas
of electrons”. Assumptions:
1. Electrons are classical objects: solid spheres of identical shape.
2. There is a compensating positive charge attached to immobile particles,
to ensure overall charge neutrality:
1
2
(a) Each isolated atom of the metallic element has a nucleus of charge
eZa (0 < e = 1.60 × 10−19 C, Za is the atomic number).
(b) Surrounding the nucleus are Za electrons of total charge −eZa , composed of Z weakly bound valence electrons, and Za −Z tightly bound
electrons, the core electrons. In a metal, the core electrons remain
bound to the nucleus and form the (immobile, positively charged)
metallic ion, while the (mobile, negatively charged) valence electrons
are allowed to wander far away from the parent atom. In this context
they are called the conduction electrons.
3. Electrons are almost free (no forces act between electrons), except electrons can collide with ions (more generally, and more correctly, we need
only assume that there is some scattering mechanism, as modern theories
show that static ions on a perfectly ordered lattice do not lead to scattering. We will see that later when we discuss Bloch theorem). Between
collisions they move as dictated by Newton’s laws of motion. Electronelectron interactions are neglected between collisions (independent electron
approximation). Electron-ion interactions are neglected as well (free electron approximation).
Many of the properties of metals can be described using the classical Drude
theory, including
• Conductivity,
• Conductivity at finite magnetic field and Hall conductivity,
• Thermal conductivity (and the historic explanation of the WiedemannFranz law which agrees with experimental results: only about a factor of
2 too small compared to them),
• Thermoelectric effects (Seebeck effect),
• AC conductivity,
• Interaction with the electromagnetic field (reflection below the plasma
frequency, transmission above the plasma frequency, plasma oscillations
at the plasma frequency).
While many of the results turn out to be inaccurate and the mechanism for
scattering remains unexplained, the basic assumption of a free electrons gas
that undergoes scattering remains essentially the same in modern theories.
The modified Newton-like equation describing the average momentum for an
electron can be derived in the following way. An electron experiences a collision
with a probability per unit time of 1/τ , where τ is known as the relaxation
time or the mean free time. So the probability for a collision in a time interval
dt is dt/τ , and this is also the fraction of electrons that collided during that
time. For a macroscopic number of electrons that have collided, the average
momentum immediately following the collision is zero. The average momentum
for the electrons at time t + dt is therefore
dt
dt
[0 + F(t)dt]
+
1−
[p(t) + F(t)dt] ,(1.2)
p(t + dt) =
τ
{z
}
|τ
|
{z
}
electrons that collided electrons that did not collide
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where p(t) is the momentum per electron, F(t) is the external force per electron.
Therefore,
p(t)
p(t + dt) − p(t) = F(t)dt −
dt + O(dt)2 ,
(1.3)
τ
and we get the central equation of motion for the Drude model
dp
p(t)
=−
+ F(t).
dt
τ
(1.4)
Hence the effect of individual electron collisions is to introduce a frictional damping term to the equation of motion.
In more technical terms, following a collision, we will use the Boltzmann
distribution to generate a velocity for the electrons
3/2
mv2
m
exp −
f (v) =
(1.5)
2πkB T
2kB T
so the electrons have an averge speed which is controlled by the (local) temperature, but the average velocity is zero due to the angular averaging.
Conductivity
Ohm’s law: the current I flowing in a wire is proportional to the potential drop
V along the wire
V = IR,
(1.6)
where R is the resistance (measured in Ohms) which depends on the shape of
the wire. One can define the resistivity ρ according to
E = ρj,
(1.7)
where E is the electric field and j is the current density. The quantities are
related in the following way. For a current flowing in a wire of length L and
cross-sectional area A the current density along the wire is j = I/A. Since
V = EL, we get V = (ρI/A) L hence
R = ρL/A.
(1.8)
All the dependence of R on the dimensions of the wire is now explicit, hence ρ
is a material property. The inverse resistivity is the conductivity σ = ρ−1 . The
inverse resistance is the conductance G (measured in 1/Ohms).
In the framework of the Drude model, we can solve explicitly the differential
equation with F = −eE, to get
p(t) = −eτ E + (p0 + eτ E)e−t/τ
(1.9)
when t → ∞ we reach steady state, for which we get a drift velocity
v=−
eEτ
,
m
(1.10)
and a corresponding current density is
j = −nev =
justifying Eq. (1.1).
ne2 τ
E,
m
σD =
ne2 τ
,
m
(1.11)
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Thermal conductivity
The Drude model was able to give an explanation to the empirical law of Wiedemann and Franz (1853), that the ratio of the thermal to electrical conductivity is
directly proportional to the temperature, with a proportionality constant which
is more or less the same for all metals. Hence, one defines the ratio κ/σT , known
as the Lorenz number. To estimate the thermal conducitivity of electrons one
considers a metal bar along which the temperature varies slowly. Fourier’s law
states that
jq = −κ∇T
(1.12)
where jq is the thermal current density (its magnitude is the thermal energy per
unit time crossing a unit area perpendicular to the flow). Electrons arriving at
point r with velocity v have on average underwent a collision at r − vτ and will
carry thermal energy E[T (r − vτ )], hence
jq
=
'
1
− nv {E[T (x − vτ )] − E[T (x + vτ )]}
2
∂E
v · (∇T )
−τ vn
∂T
(1.13)
(1.14)
Averaging, we get
1 2
hv iI
3
(1.15)
dE
3
= nkB
dT
2
(1.16)
1
cV τ hv 2 i
3
(1.17)
κ
1 cV hmv 2 i
=
σT
3 ne2 T
(1.18)
hvvi =
and defining
cV = n
we arrive at the result
κ=
The Lorenz number is
L=
with hmv 2 i = 3kB T one gets
L=
3
2
kB
e
2
= 1.11 × 10−8 (J/CK)
2
(1.19)
Quantum free electron model for metals: The Sommerfeld
model
Recommended reading:
• Chapter 2, Aschroft & Mermin.
A quantum model: a free electron gas. At thermal equilibrium, the number of
electrons having energy E is given by the Fermi distribution
f (E) =
1
,
eβ(E−µ) + 1
(1.20)
in which β = 1/(kB T ) is the inverse temperature (kB is the Boltzmann constant)
and µ is the chemical potential. At zero temperature (β → ∞) the chemical
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potential is equal to the Fermi energy, EF , and the Fermi function describes a
step-function, such that all states with energy E ≤ EF are full and all states
with energy about EF are empty. In the grand-canonical ensemble, where the
chemical potential is fixed, the number of electrons is temperature-dependent.
The electronic density (number of electrons per unit volume) is given by
ˆ ∞
dEΩ(E)f (E),
(1.21)
n=
−∞
where ⊗(E) is the density of states (number of states having energy E per unit
volume).
We can straightforwardly adopt the results of the Drude model for this case,
however, we need to correct the following points:
• The typical velocity squared for the electrons, hv 2 i, is set by the Fermi
energy and not by the temperature
mv 2
= EF ,
(1.22)
2
D 2E
= 32 kB T for the Boltzmann distribution).
(compare with mv
2
• The electronic heat capacity is proportional to the temperature and is
not a constant. The reason is that at finite temperature T there are
∼ Ω(EF )kB T electrons which become excited (here Ω(EF ) ∼ n/EF is
the density of states at the Fermi energy), and they each typically carry
additional energy kB T , leading to an increase in energy of about E ∼
Ω(EF )(kB T )2 compared to the ground state; Hence cV ∝ dE
dT ∝ T .
By using equilibrium results to calculate the various physical quantities, one
can adopt the results from the Drude theory after the following corrections:
• cV → cV kEBFT ,
•
D
mv 2
2
E
→
D
mv 2
2
E
EF
kB T
.
Hence, since L is proportional to the product, it remains about the same, while
Q gets reduced by a factor of kB T /EF to about 1% of its classical value. An
exact calculation within the Sommerfeld model (using a Sommerfeld expansion,
see chapter 2 Ashcroft & Mermin) leads to
L=
π2
3
kB
e
2
2
= 2.44 × 10−8 (J/CK) ,
(1.23)
which is an excellent match with the typical experimental value of L.
A semiclassical way to calculate directly transport coefficients in the Sommerfeld model is provided by the Boltzmann equation, that extends Sommerfeld’s equilibrium theory to nonequilibrium cases.
6
Bloch theorem and second quantization
Translation operator
(TR ψ) (r) = ψ(r + R)
(1.24)
For a perfect crystal
∀R ∈ Lattice
[H, TR ] = 0,
(1.25)
Bloch theorem: there exists a basis of extended Hamiltonian eigenstates of the
form
ψn,k (r)
un,k (r + R)
= eik·r un,k (r)
(1.26)
= un,k (r)
(1.27)
Here n is a band index and k is a lattice momentum.
Example: 1d lattice
X
X
i |iihi| −
t|iihj|
i
(1.28)
hiji
ansatz
|ψi =
X
aj |ji
(1.29)
j
we project schrodinger equation
hi|H|ψi = Ehi|ψi
(1.30)
i ai − t(ai+1 + ai−1 ) = Eai
(1.31)
X
(1.32)
from which we get
notice that
Ta |ψi =
aj |j + 1i
j
and on the other hand, up to a phase the two wavefunctions must be equal
X
X
X
aj |ji = |ψi = eik Ta |ψi = eik
aj |j + 1i = eik
aj−1 |ji
(1.33)
j
j
j
and therefore
aj = eik aj−1
so our wavefunction can be written as
X
|ψi =
eikaj |ji
(1.34)
(1.35)
j
substituting in, we get for i = E = − 2t cos(ak)
(1.36)
leading to a metallic state. There are several instabilities of the metallic state
that makes this prediction fragile.
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Example: a dimerized tight-binding chain
X
H[∆] =
[t(1 + ∆)|i, Aihi, B| + t(1 − ∆)|i + 1, Aihi, B| + h.c.] + u∆2
(1.37)
i
=
X
|i, Ai
t(1 + ∆)
|i, Bi
i
0
1
|i + 1, Ai
+t(1 − ∆)
|i + 1, Bi
1
0
0
0
1
0
hi, A|
hi, B|
hi, A|
hi, B|
+ h.c.
and transform to Bloch waves using
1 X 2ika` |k, Ai
|`, Ai
√
e
=
|k, Bi
|`, Bi
N
(1.38)
k
to get
H=
X
|k, Ai
|k, Bi
H(k)
k
hk, A|
hk, B|
(1.39)
where
H(k) =
0
t(1 + ∆) + t(1 − ∆)e2ika
t(1 + ∆) + t(1 − ∆)e−2ika
0
(1.40)
The Hamiltonian H(k) can also be written as
H(k) = d(k) · σ
(1.41)
where
dx (k)
= t(1 + ∆) + t(1 − ∆) cos(2ak)
(1.42)
dy (k)
= t(1 − ∆) sin(2ak)
(1.43)
dz (k)
=
(1.44)
0
where k ∈ [− πa , πa ]. Diagonalizing the Hamiltonian, we get the energy eigenvalues
p
E± (k) = ± 2t2 (1 + ∆2 ) + 2t2 (1 − ∆2 ) cos(2ak),
(1.45)
so a gap of size 4t|∆| develops around the end points of the BZ. At half-filling,
there is an instability of the metallic state towards an insulating state, known
as P
the Peierls transition. One can plot the energy of the ground state, Egs =
− k E+ (k), to get a double minimum potential with minima at ∆ = ±∆0 .
+ u∆2