Summary Solid State Physics
by Ruben Tol
This summary covers all useful theory for the course ”Solid State Physics” given at
the TU Delft. Sections are marked with their associated lecture L#. References
to material (also) found in the book ”The Oxford Solid State Basics” by Steven
H. Simon are marked with [ and the chapter C# in which its found. The first
mention of a theory or quantity is italicised. The most crucial theory is highlighted
in light-gray . Some quantities have their units given between [square brackets].
Phonons and Specific Heat
Einstein Model ( L1)
The heat capacity C of an atom has been observed empirically to be
C = 3kB per atom, ([ C2.0)
where kB is the Boltzmann constant. This law is known as the law of Dulong-Petit.
Boltzmann backed this up using statistical physics by modelling each atom to be in
a harmonic well formed by the interaction which its neighbouring atoms, with the
equipartition theorem telling us that each degree of freedom in a system contributes
kB /2 to the heat capacity C. With three-dimensional (3D) space having three spatial
and three momentum degrees of freedom, this indeed yields the law of Dulong-Petit.
([ C2.0)
However, this law does not always hold. The heat capacity C seems to be dependent
on the temperature T of a solid, with C decreasing as T decreases, and with C = 0
at T = 0.
Einstein then devised a similar model to Boltzmann’s, where he further assumed
every atom to be in an identical harmonic well with an oscillation frequency ω0 ,
known as the Einstein frequency. ([ C2.0)
In one dimension, the energy of the eigenstates of a single harmonic oscillator are
1
En = ℏω0 n + , ([ C2.1)
2
where ℏ is the reduced Planck constant.
When a harmonic oscillator (a bosonic mode) has its wavefunction is in its n-th
excited state, it is said that it is occupied by n bosonic excitations. At a temperature
T , the average occupation number ⟨n⟩ is given by the Bose-Einstein distribution:
1
⟨n⟩ = nB (ℏω0 /kB T ) = .
eℏω0 /kB T −1
1
,This then yields for the expectation of energy ⟨E⟩
1 1 1
⟨E⟩ = ℏω0 ⟨n⟩ + = ℏω0 ℏω0 /k T + . ([ C2.1)
2 e B −1 2
The heat capacity C(T ) is then given by
2
eℏω0 /kB T
∂⟨E⟩ ℏω0
C(T ) ≡ = kB 2 ., ([ C2.1)
∂T kB T (eℏω0 /kB T − 1)
which can be rewritten as
2
eℏω0 /kB T
TE
C(T ) = kB 2,
T (eℏω0 /kB T − 1)
where TE ≡ ℏω0 /kB is the Einstein temperature.
The Einstein model indeed yields a temperature dependent heat capacity C(T )!1
The Einstein temperature TE is the characteristic temperature below which there is
not enough thermal energy to excite a harmonic oscillator above the ground state.
As a consequence, a solid below the Einstein temperature TE has its heat capacity
C start to decrease rapidly. The heat capacity C is then also dependent on the
Einstein frequency ω0 , which itself is a material-dependent parameter.
In the low temperature limit, kB T ≪ ℏω0 , yielding eℏω0 /kB T − 1 ≈ eℏω0 /kB T . This
then gives for the expectation of energy ⟨E⟩ in one dimension
−ℏω0 /kB T 1 1
⟨E⟩ ≈ ℏω0 e + ≈ ℏω0 ,
2 2
which correctly corresponds to the ground state energy of the harmonic oscillator.2
In the high temperature limit, kB T ≫ ℏω0 , yielding eℏω0 /kB T ≈ 1 + ℏω0 /kB T . This
then gives for the expectation of energy ⟨E⟩ in one dimension
kB T 1
⟨E⟩ ≈ ℏω0 + ≈ kB T,
ℏω0 2
where the constant factor is neglected, which then as expected gives back the law
of Dulong-Petit.3
1
Similarly, in two dimensions, we have En = ℏω0 (nx + ny + 1) and find ⟨E2D ⟩ = 2⟨E1D ⟩ and
C2D (T ) = 2C1D (T ). Similarly again, in three dimensions, we have En = ℏω0 nx + ny + nz + 23
and find ⟨E3D ⟩ = 3⟨E1D ⟩ and C3D (T ) = 3C1D (T ). ([ C2.1)
2
Again, in the high temperature limit, ⟨E2D ⟩ ≈ ℏω0 , and ⟨E3D ⟩ ≈ 23 ℏω0 .
3
It again follows that in the high temperature limit, ⟨E2D ⟩ ≈ 2kB T , and ⟨E3D ⟩ ≈ 3kB T .
2
,Debye Model ( L2)
The Einstein model, although extremely successful, still deviates from experimental
results; at low temperatures T , the model underestimates the heat capacity C.
Debye then improved on this model, by realising that the collective motion of atoms
(like sound waves) should be modelled as harmonic oscillators, rather than Einstein’s
assumption of modelling single atoms as harmonic oscillators. ([ C2.2)
These sound waves are characterised by their frequency ω, their wave vector k and
their polarisation. The frequency ω is related to the wave vector k through the
dispersion relation:
ω(k) = vs |k|,
where vs is the sound velocity of a material.
The periodic boundary conditions for three dimensional waves in a cube of volume
L3 restrict the wave vector k to values of4
2π
k= (nx , ny , nz ), ([ C2.2)
L
where nx , ny , nz ∈ Z.
Each volume of size (2π/L)3 is therefore allowed to be occupied by one value of
the wave vector k. When L → ∞, the volume per allowed mode becomes smaller
and smaller. When summing over all possible values of the wave vector k, the
sum over k can be approximated by an integral over three dimensions. Due to
spherical symmetry, this integral can also be converted into spherical coordinates
yielding an integral over only one dimension. Then substituting the dispersion
relation k = ω/vs → dk = dω/vs yields the following cascade:
∞ ∞
L3 4πL3 4πL3
X Z Z Z
2
→ dk → k dk → ω 2 dω. ([ C2.2)
k
(2π)3 (2π)3 0 (2π)3 vs3 0
For each of the three directions of motion, there should be a possible oscillation
mode. Analogous to the Einstein model, the expectation of energy ⟨E⟩ then becomes
X 1 1
⟨E⟩ = 3 ℏω(k) ℏω(k)/k T + . ([ C2.2)
k
e B −1 2
Using the above approximation for the sum over k, we find
Z ∞
4πL3
1 1
⟨E⟩ = 3 ℏω ℏω/k T + ω 2 dω. ([ C2.2)
(2π)3 vs3 0 e B −1 2
4
I’m skipping over the derivation here.
3
,This can be rewritten as
Z ∞
1 1
⟨E⟩ = g(ω)ℏω + dω, ([ C2.2)
0 eℏω/kB T −1 2
where g(ω) is the density of states:5
3
4πω 2
L
g(ω) = 3 . ([ C2.2)
2π vs3
The density of states g(ω) gives us the total number of oscillation modes with
frequencies between ω and ω + dω as g(ω) dω. ([ C2.2)
We will now split up this integral into temperature dependent and temperature
independent parts, as the temperature independent part does not contribute to the
heat capacity C:
Z ∞
3L3 ω3
⟨E⟩ = 2 3 ℏ dω + EZ ,
2π vs 0 eℏω/kB T − 1
where EZ is the zero-point energy.
By substituting x = ℏω/kB T , we can solve the integral:6
3L3 (kB T )4 ∞ x3 3L3 (kB T )4 π 4
Z
⟨E⟩ = 2 3 dx + EZ = + EZ .
2π vs ℏ3 0 ex − 1 2π 2 vs3 ℏ3 15
With C = ∂⟨E⟩/∂T , we find C ∝ T 3 , which is expected for low temperatures T . For
high temperatures T , this should not be the case, as it should level off to C = 3kB
per atom. This is fixed by not considering frequencies ω above a certain threshold
given by the Debye frequency ωD , so
( 3 2
3L ω
2 3 ω ≤ ωD ,
g(ω) = 2π vs
0 ω > ωD .
A three-dimensional system with N atoms has 3N oscillation modes7 , so
Z ωD Z ωD
3L3 L3 ω 3
3N = g(ω) dω = 2 3 ω 2 dω = 2 D3 ,
0 2π vs 0 2π vs
giving us
√
3
vs 6π 2 N
ωD = .
L
5
The 3 in the equation comes from the number of polarisations possible in three dimensions: two
transversal, and one longitudinal polarisation.
6
By using the Riemann-Zeta
R ω function. Rω
7 L
In one dimension, 1N = 0 D g(ω) dω with g(ω) = πv s
; in two dimensions, 2N = 0 D g(ω) dω with
L2 ω
g(ω) = πvs2 . The density of states g(ω) are obtained in a similar matter as in three dimensions.
4
,The corrected expression for the expectation of energy ⟨E⟩ without the contribution
of the zero-point energy EZ then becomes
Z ωD
ℏω
⟨E⟩ − EZ = g(ω) ℏω/k T dω. ([ C2.2)
0 e B −1
√
3
By substituting ωD = vs 6π 2 N /L, x = ℏω/kB T , and with C = ∂⟨E⟩/∂T , we find
3 Z TD /T
T x4 ex
C = 9N kB dx,
TD 0 (ex − 1)2
where the Debye temperature TD is defined as TD ≡ ℏωD /kB .
In the low temperature limit, the expectation of energy ⟨E⟩ becomes ⟨E⟩ ≈ EZ .
In the high temperature limit, we again have nB (ℏω/kB T ) ≈ kB T /ℏω and find
Z ωD
⟨E⟩ = kB T g(ω) dω = 3kB T N, ([ C2.2)
0
which again gives the law of Dulong-Petit: C = ∂⟨E⟩/∂T = 3kB N = 3kB per atom.
Free Electron Model
Drude Model ( L3)
The conduction of electricity in a metal boils down to the fact that electrons are
mobile in metals. Drude then applied Boltzmann’s kinetic theory of gases to create
a model of the motion of electrons within metals. This model made the following
assumptions: ([ C3.0)
• Electrons have a scattering time τ , giving electrons a probability of scattering
withing a time interval dt of dt/τ ;
• Once a scattering event has occurred, the electrons returns to an average
momentum ⟨p⟩ = 0;
• Electrons, with charge −e, respond to an externally applied electric field E
and magnetic field B via the Lorentz force FL = −e(E + v × B).
These scatterings of electrons inside a metal have two causes adding to the scattering
time τ :
• Scattering due to phonons, contributing τph (T ) (with τph → ∞ as T → ∞);
• Impurities/Crystal defects, contributing τ0 .
The reciprocals 1/τ are then added to obtain the total scattering time τt :8
1 1 1
= + .
τt τph τ0
8
The reciprocals can also be denoted with the scattering rate Γ ≡ 1/τ , giving us Γt = Γph + Γ0 .
5
,The average momentum ⟨p⟩ of an electron at at time t+dt then consists of two terms:
a scattering event has occurred with a probability of dt/τ , or the electron undergoes
its original trajectory following the applied Lorentz force FL with a probability of
1 = dt/τ . This yields
dt
⟨p(t + dt)⟩ = 1 − (p(t) + FL dt) + 0 dt/τ. ([ C3.0)
τ
Rearranging the equation then gives us the Drude equation of motion:
dp p dv v
= FL − =⇒ m = −e(E + v × B) − m . ([ C3.0)
dt τ dt τ
Note how the last term on the right-hand side of the equation has the same form as
a drag force, slowing the electron down.
When no magnetic field B is applied, the Drude equation of motion becomes
dv v
m = −eE − m . ([ C3.1)
dt τ
In steady state, dp/dt = 0, so
−eτ
v= E.
m
With j ≡ I/A = −env = σE (definition of current density, with n the electron
density σ the conductivity of a metal), we find for the current density j and the
conductivity σ
e2 τ n e2 τ n
j = −env = E =⇒ σ = . ([ C3.1)
m m
By measuring the conductivity σ of a metal, we can therefore determine the product
of the electron density n and the scattering time τ of the electron. ([ C3.1)
Now, when a magnetic field B is applied, the Drude equation of motion in steady
state can be rewritten as (using p = mv and j = −env)
1 m
E= j×B+ 2 j . ([ C3.1)
ne ne τ
Now, a 3 by 3 resistivity matrix ρ can be defined, relating the current vector j to
the electric field vector E (using ρe = 1/σ):
ρxx ρxy ρxz
E = ρj = ρyx ρyy ρyz j, ([ C3.1)
e ρzx ρzy ρzz
where it follows that
m
ρxx = ρyy = ρzz = ([ C3.1)
ne2 τ
and, assuming B oriented in the ẑ direction,
Bz
ρxy = −ρyx = , ([ C3.1)
ne
with all other components of ρ = 0.
e
6
,The off-diagonal terms ρxy and ρyx are known as the Hall resistivity. These terms
generate a measurable voltage V perpendicular to both the current I and a magnetic
field B applied to a metal.
The Hall coefficient RH is defined as
ρyx −1
RH = = . ([ C3.1)
|Bz | ne
This then allows us to measure the density of electrons n in a metal.
The results of the Drude equation of motion with and without the magnetic field B
then allows us to measure both the electron density n and the scattering time τ of
the electron, by first measuring the Hall resistivity ρyx when a magnetic field B is
applied to obtain n, and then measuring the conductivity σ of the metal when no
magnetic field B is applied to obtain the product τ n. ([ C3.1)
Sommerfeld Model ( L4)
Sommerfield expanded upon Drude’s theories by incorporating Fermi-Dirac statis-
tics. Since electrons are fermions, the occupation of electron states is given by the
Fermi-Dirac distribution (which looks like a ”smeared out” step function):
1
nF (β(ϵ − µ)) = , ([ C4.1)
eβ(ϵ−µ) +1
where β = 1/kB T , ϵ is the eigenenergy of the state, and µ is the chemical potential.
These electrons (always) follow the Schrödinger equation, given by
ℏ2 ∂ 2 ψ
− = ϵψ.
2m ∂x2
Similarly as before, we consider a cube of volume L3 and apply periodic boundary
conditions. This again yields for the wave vector k = (2π)/L(nx , ny , nz ), giving us
an eigenenergy of
ℏ2 |k|2
ϵ(k) = . ([ C4.1)
2m
The total number of electrons in a system N is therefore given by
Z ∞
X 4πV
N = 2s nF (β(ϵ(k) − µ)) = 2s 3
k 2 nF (β(ϵ(k) − µ)) dk, ([ C4.1)
k
(2π) 0
where the subscript s indicates that we’re accounting for spin degeneracy.
The chemical potential µ can be obtained from this equation. A useful concept to
define is the Fermi energy EF , the chemical potential at T = 0. This relation gives
the following equations:
ℏ2 |kF |2 ℏkF
EF = µ(T = 0) =⇒ ϵF = =⇒ vF = .
2m m
7
, The expression for the total energy in a system E is then given by
Z ∞
4πV
E = 2s 3
k 2 ϵ(k) nF (β(ϵ(k) − µ)) dk. ([ C4.2)
(2π) 0
The equations for the total number of electrons N and the total energy
p E in a
system canpbe rewritten for simplicity’s sake. By substituting k = 2ϵm/ℏ2 so
that dk = m/(2ϵℏ2 ) dϵ, and observing that the Fermi-Dirac distribution acts like
a step function, we find that
Z Z r Z
V 2 V 2mϵ m
N = 2 k dk = 2 dϵ = g(ϵ) dϵ,
π π ℏ3 2ϵ
thus we find for the density of states g(ϵ)9
√
dN V m3/2 2ϵ √
g(ϵ) = = ∝ ϵ.
dϵ π 2 ℏ3
The Fermi energy EF sets the amount of electrons N in a system. We find using
the above two equations:
Z EF 2/3 r
ℏ2
2N 3 N
N= g(ϵ) dϵ =⇒ EF = 3π =⇒ kF = 3π 2 ,
0 2m V V
where kF = |kF | is the Fermi wave number.
The Fermi wave number kF allows us to obtain the Fermi wavelength λF ≡ 2π/kF ,
which is telling of the atomic spacing for typical free electron densities in metals.
Now, for T > 0, we find that, again, due to the Fermi-Dirac distribution acting
like a step function, that for any T , EF ≈ µ. This is due to the Fermi temperature
TF ≡ EF /kB always being much larger than conventional temperatures T (like room
temperature) ([ C4.2).
At room temperature, the amount of excited electrons Nexc is approximately given
by Nexc ≈ g(EF )kB T . The total energy of a system E can then be approximated as
E = EF + Nexc kB T ≈ EF + g(EF )kB2 T 2 .
The electron heat capacity Ce (in three dimensions) is then given as
dE T
C= ≈ 2g(EF )kB2 T = 3N kB ∝ T,
dT TF
which is way smaller than the phonon heat capacity Cp ≈ 3N kB T because T ≪ TF .
Cp ∝ kB (T /TD )3 , which becomes smaller than
For T → 0, the phonon heat capacity p
the electron heat capacity Ce at T ≲ TD3 /TF .
9
√
Similar deviation for two dimensions give g(ϵ) = constant , and for one dimension g(ϵ) ∝ 1/ ϵ .
8
