Selected Properties of Solids
Specific Heat Capacity ((T)
The amount of energy necessary to raise the temperature of a given amount of a material by one
degree .
Every type of excitation can absorb heat count the total number of excitations at a temperature T
-
we can
-
Insulators
for a
typical insulator the , specific heat capacity at low temperatures has the form ((T) =
aT
The Th dependence arises from the absorption of heat by lattice vibrations -
phonons .
The constant a has units [JK-"m-3]
Metals
for a
typical metal the specific heat
capacity at low temperatures is
given by ((T) UT
=
,
In this case , phonons are present ,
but at low temperatures their T3 term is much smaller than the
observed linear behaviour .
The principle difference between metals and insulators is the presence of
conducting electrons .
The Constant V has units [JKm-3] and is referred to as the Sommerfeld coefficient.
Another definition of the specific heat capacity is a measure at a
given temperature T of the number of degrees
, ,
of freedom capable of being excited by the transfer of energy from a reservoir with a value of 3/2 ,
corresponding to a single degree of freedom .
C == where S(M T) is the entropy and [(M T) the internal energy .
, ,
Since the entropy be written in terms of the
can grand canonical
free energy s=- it follows
that ((T) C
I
can also be written as = -
The specific heat capacity of a monoatomic ideal
gas is C = kn where n is the particle number
density (volume"]
Magnetic Susceptibility X
The magnetic susceptibility indicates how the magnetisation changes when an external field is applied.
Insulators
Magnetic insulators : Have unpaired electrons on some lattice sites
The susceptibility at high temperatures tends to take on the Curie form X(T)
·
=
The spins of the unpaired electrons behave independently of each other and so
·
,
behave as a
single spin coupled to a heat bath
As the temperature decreases the spins start to interact with each other
·
, if
-
the interaction tends to allign the spins the susceptibility will deviate upwards
,
from the curve -
if the interaction tends to anti-allign the spins ,
the
susceptibility will deviate downwards from the curve .
, The susceptibility tends to remain near zero over a wide
Non-magnetic insulators : ·
range of temperatures-
to magnetise a sample would mean breaking some chemical bonds which
typically have high energies .
Metals
The magnetic susceptibility of metals at low temperatures tends to a constant value -
the Pauli
susceptibility X(T) =
Xo
·
This result is non-zero ,
which means that there must be
magnetic moments in the system.
·
This result is temperature independent ,
which means that there must be energy scale more relevant
an
than the thermal energy governing the result. This is a manifestation that the Pauli exclusion principle in
metals prevents magnetic allignement in metals much better than temperature fluctuations.
The Pauli susceptibility is
typically around 2 orders of magnitude smaller than the Curie susceptibility.
for an isotropic system in a
uniform magnetic field B the ,
magnetisation is
given by M =X(T) B
X (T) is unitless and for small fields is generally field independent
-
Magnetisation is consequence of the adjustment of the system to the externally applied magnetic field.
a
Consequently the grand canonical potential acquires a further field dependence - 1 (M T B)
,
=
, ,
.
The magnetisation density is
given by : M =
- Hence X(T)
: =
Mo =_
Mo
Mo(gMi)2
For a spin S = particle X(T) is the Curie susceptibility Xcurie
, 4kBT : =
The Curie susceptibility increases without limit as T decreases as the spins become , easier to allign
with the field.
Electrical Resistivity , P
This is not athermodynamic observable . The resistivity is related to the resistance of a material by
R = where ( is the length of a cuboid and A the cross-sectional area.
The units of resistivity are (2m]
Insulators
An insulator does not conduct electricity so should have infinite resistivity however at any
-
non-zero
temperature there are thermal excitations that can excite some electrons into conducting
. In many
insulators
the low-temperature resistivity takes the form : p(T) =
Aexp(it) where is the excitation
gap energy
Metals
Metals have some resistivity due to defects in the system When .
a metal is heated the resistivity can take
the form p (T) Po + AT2 + BTS
=
where the constant po is the residual resistivity
↓
L ↳
scattering from scatteringfrom phonons
impurities scattering from electron-electron interactions
, . The
2 Sommerfeld Model
Argument : since electrons in a metal can conduct electricity , they must be able to move around relatively
freely they cannot be too strongly influenced by the electric field of the lattice of positive ions
-
.
lect electron-electron interactions
~ neg
We model the conduction electrons free and independent electrons moving around in
as a cubic box of
length ( -
which represents the interior of the crystal with a positive background. -
↳ maintains
charge neutrality
We impose periodic boundary conditions ,
so that we do not restrict the number of possible periodic
functions ,
and we maintain translational invariance e. ↑(x + L y z) ↑ · ,
=
↑ (x y z) etc.
,
,
free electrons plane
X
&
= wave
Solution to the Single Particle Problem
The particle in the Sommerfeld model is subject to no forces at all ,
giving the Hamiltonian # ,
=
-
p
= -
it
Hence the time-independent Schrodinger equation for the single-particle eigenfunctions is :
-N
E Using separation of variables the solution is a product of plane waves
=
.
:
↑, y z) ,
=
AcilaikyyikzAeThis is normalised by dodyd-
1 1
N =
A "so normalisation
gives A
= 1 => A =
3
We can choose A to be real : A =
3
I
1 for n-dimensions A =
ik r In
Hence the normalised single-particle wave functions are : Th(r) = e
The subscript1 denotes that different values of K distinguish one single-particle state
from another.
?
>k
Solving the TISE we find the energy of a single-particle state with wave vectork : E, =
2m
Applying the boundary conditions eg
. ↑(x 1)
+ =
Tr(x) we obtain k =( .
my ,
nz)
K-Space Representation k =
wavenumber but can be referred to as momentum as
p = tak
Each corresponds to a permitted single-particle state. We can put 2 electrons in each such state.
value of k
A visual representation of these allowedStates is obtained by noticing that in K-space they form a regular ,
cubic lattice of lattice spacing 2/L
A particular k-space lattice pointko occupies
As real-space size increases the lattice points
a volumeUpt =
(2)
,
in k-space become more
densely packed ,
aso this becomes a continuum
Z
·
, The Many fermion Case & the fermi Sphere
To determine the body wave-function assign the Nelectrons to these
many ,
we need to
single-particle states .
from the electron's point of view the temperature inside solids should
try to construct the
is
very low ,
so we
lowest-energy state .
We can use E, =
h Since this
only depends on the magnitude of k ,
the
only parameter that matterson
the is how
determining energy of a
single-particle state far its K-space point is from the origin
-
to make the
lowest should the states with two electrons
energy
state we
single-particle starting at the
origin .
This results in a sphere in K-space -
the fermi sphere and the fermi surface.
fermi number of particles
(k) =
radius
N=
Assuming fixeda number of particles ↓
function telling us the number of single-particle states with K-space distance < K
fermicavena
real-space volume the
L
Number of lattice points within
the sphere of radius k
>
Sphere
(2)3 Gi Gi
To obtain the number of states multiply by 2 to account , for spin : N , (k) = V =
N =
ki =
(in)
where v is the real-space density of the sample .
The energy of the highest occupied single-particle eigenstate is the fermi energy , Ef or
EF
EF =
ht- Ef
EEF
acts
effectively as the zero
are occupied at T = 0
of energy for electrons in the metal : states with
whilst states with EC E + are unoccupied .
Chemical Potential , M
When the number of particles N is fixed the chemical potential is obtained from the Helmholtz free energy F as
,
d+
M dN for discrete N the smallest increment is dN 1 so that M(N) F(N)
=
.
,
F(N-1) =
,
=
-
.
At T where the entropy S 0 the free to the energy F Fo and
0 is equal ground state so
=
energy
= =
,
m(N) =
fo(N) -
Eo(N -
1) =
EF -
At TsO there is no
longer "highest occupied
a state" due to thermal excitations -
so
technically Ef no
longer
exists ,
whereas M(T) remains a well-defined quantity.
Practically ,
Ef is still used as a
pseudonym for Mat Tc O
,
particularly as in most situations KTEF and thus
M= EF
Densities of State
Density of states in I
g,
(k) =
(dN(k) = This is the number of states in a thin spherical shell between radius k and kak , e
the number is given by g, (1) dk
specifically ,
~
sum runs over occupied states
k
=Vog(k)12dk
#1
This can be used to find the ground state energy Er of the filled fermisphere Eo =
&zm zm
If p ~visfixed, To is extensive
hVSVT
= .
=
Specific Heat Capacity ((T)
The amount of energy necessary to raise the temperature of a given amount of a material by one
degree .
Every type of excitation can absorb heat count the total number of excitations at a temperature T
-
we can
-
Insulators
for a
typical insulator the , specific heat capacity at low temperatures has the form ((T) =
aT
The Th dependence arises from the absorption of heat by lattice vibrations -
phonons .
The constant a has units [JK-"m-3]
Metals
for a
typical metal the specific heat
capacity at low temperatures is
given by ((T) UT
=
,
In this case , phonons are present ,
but at low temperatures their T3 term is much smaller than the
observed linear behaviour .
The principle difference between metals and insulators is the presence of
conducting electrons .
The Constant V has units [JKm-3] and is referred to as the Sommerfeld coefficient.
Another definition of the specific heat capacity is a measure at a
given temperature T of the number of degrees
, ,
of freedom capable of being excited by the transfer of energy from a reservoir with a value of 3/2 ,
corresponding to a single degree of freedom .
C == where S(M T) is the entropy and [(M T) the internal energy .
, ,
Since the entropy be written in terms of the
can grand canonical
free energy s=- it follows
that ((T) C
I
can also be written as = -
The specific heat capacity of a monoatomic ideal
gas is C = kn where n is the particle number
density (volume"]
Magnetic Susceptibility X
The magnetic susceptibility indicates how the magnetisation changes when an external field is applied.
Insulators
Magnetic insulators : Have unpaired electrons on some lattice sites
The susceptibility at high temperatures tends to take on the Curie form X(T)
·
=
The spins of the unpaired electrons behave independently of each other and so
·
,
behave as a
single spin coupled to a heat bath
As the temperature decreases the spins start to interact with each other
·
, if
-
the interaction tends to allign the spins the susceptibility will deviate upwards
,
from the curve -
if the interaction tends to anti-allign the spins ,
the
susceptibility will deviate downwards from the curve .
, The susceptibility tends to remain near zero over a wide
Non-magnetic insulators : ·
range of temperatures-
to magnetise a sample would mean breaking some chemical bonds which
typically have high energies .
Metals
The magnetic susceptibility of metals at low temperatures tends to a constant value -
the Pauli
susceptibility X(T) =
Xo
·
This result is non-zero ,
which means that there must be
magnetic moments in the system.
·
This result is temperature independent ,
which means that there must be energy scale more relevant
an
than the thermal energy governing the result. This is a manifestation that the Pauli exclusion principle in
metals prevents magnetic allignement in metals much better than temperature fluctuations.
The Pauli susceptibility is
typically around 2 orders of magnitude smaller than the Curie susceptibility.
for an isotropic system in a
uniform magnetic field B the ,
magnetisation is
given by M =X(T) B
X (T) is unitless and for small fields is generally field independent
-
Magnetisation is consequence of the adjustment of the system to the externally applied magnetic field.
a
Consequently the grand canonical potential acquires a further field dependence - 1 (M T B)
,
=
, ,
.
The magnetisation density is
given by : M =
- Hence X(T)
: =
Mo =_
Mo
Mo(gMi)2
For a spin S = particle X(T) is the Curie susceptibility Xcurie
, 4kBT : =
The Curie susceptibility increases without limit as T decreases as the spins become , easier to allign
with the field.
Electrical Resistivity , P
This is not athermodynamic observable . The resistivity is related to the resistance of a material by
R = where ( is the length of a cuboid and A the cross-sectional area.
The units of resistivity are (2m]
Insulators
An insulator does not conduct electricity so should have infinite resistivity however at any
-
non-zero
temperature there are thermal excitations that can excite some electrons into conducting
. In many
insulators
the low-temperature resistivity takes the form : p(T) =
Aexp(it) where is the excitation
gap energy
Metals
Metals have some resistivity due to defects in the system When .
a metal is heated the resistivity can take
the form p (T) Po + AT2 + BTS
=
where the constant po is the residual resistivity
↓
L ↳
scattering from scatteringfrom phonons
impurities scattering from electron-electron interactions
, . The
2 Sommerfeld Model
Argument : since electrons in a metal can conduct electricity , they must be able to move around relatively
freely they cannot be too strongly influenced by the electric field of the lattice of positive ions
-
.
lect electron-electron interactions
~ neg
We model the conduction electrons free and independent electrons moving around in
as a cubic box of
length ( -
which represents the interior of the crystal with a positive background. -
↳ maintains
charge neutrality
We impose periodic boundary conditions ,
so that we do not restrict the number of possible periodic
functions ,
and we maintain translational invariance e. ↑(x + L y z) ↑ · ,
=
↑ (x y z) etc.
,
,
free electrons plane
X
&
= wave
Solution to the Single Particle Problem
The particle in the Sommerfeld model is subject to no forces at all ,
giving the Hamiltonian # ,
=
-
p
= -
it
Hence the time-independent Schrodinger equation for the single-particle eigenfunctions is :
-N
E Using separation of variables the solution is a product of plane waves
=
.
:
↑, y z) ,
=
AcilaikyyikzAeThis is normalised by dodyd-
1 1
N =
A "so normalisation
gives A
= 1 => A =
3
We can choose A to be real : A =
3
I
1 for n-dimensions A =
ik r In
Hence the normalised single-particle wave functions are : Th(r) = e
The subscript1 denotes that different values of K distinguish one single-particle state
from another.
?
>k
Solving the TISE we find the energy of a single-particle state with wave vectork : E, =
2m
Applying the boundary conditions eg
. ↑(x 1)
+ =
Tr(x) we obtain k =( .
my ,
nz)
K-Space Representation k =
wavenumber but can be referred to as momentum as
p = tak
Each corresponds to a permitted single-particle state. We can put 2 electrons in each such state.
value of k
A visual representation of these allowedStates is obtained by noticing that in K-space they form a regular ,
cubic lattice of lattice spacing 2/L
A particular k-space lattice pointko occupies
As real-space size increases the lattice points
a volumeUpt =
(2)
,
in k-space become more
densely packed ,
aso this becomes a continuum
Z
·
, The Many fermion Case & the fermi Sphere
To determine the body wave-function assign the Nelectrons to these
many ,
we need to
single-particle states .
from the electron's point of view the temperature inside solids should
try to construct the
is
very low ,
so we
lowest-energy state .
We can use E, =
h Since this
only depends on the magnitude of k ,
the
only parameter that matterson
the is how
determining energy of a
single-particle state far its K-space point is from the origin
-
to make the
lowest should the states with two electrons
energy
state we
single-particle starting at the
origin .
This results in a sphere in K-space -
the fermi sphere and the fermi surface.
fermi number of particles
(k) =
radius
N=
Assuming fixeda number of particles ↓
function telling us the number of single-particle states with K-space distance < K
fermicavena
real-space volume the
L
Number of lattice points within
the sphere of radius k
>
Sphere
(2)3 Gi Gi
To obtain the number of states multiply by 2 to account , for spin : N , (k) = V =
N =
ki =
(in)
where v is the real-space density of the sample .
The energy of the highest occupied single-particle eigenstate is the fermi energy , Ef or
EF
EF =
ht- Ef
EEF
acts
effectively as the zero
are occupied at T = 0
of energy for electrons in the metal : states with
whilst states with EC E + are unoccupied .
Chemical Potential , M
When the number of particles N is fixed the chemical potential is obtained from the Helmholtz free energy F as
,
d+
M dN for discrete N the smallest increment is dN 1 so that M(N) F(N)
=
.
,
F(N-1) =
,
=
-
.
At T where the entropy S 0 the free to the energy F Fo and
0 is equal ground state so
=
energy
= =
,
m(N) =
fo(N) -
Eo(N -
1) =
EF -
At TsO there is no
longer "highest occupied
a state" due to thermal excitations -
so
technically Ef no
longer
exists ,
whereas M(T) remains a well-defined quantity.
Practically ,
Ef is still used as a
pseudonym for Mat Tc O
,
particularly as in most situations KTEF and thus
M= EF
Densities of State
Density of states in I
g,
(k) =
(dN(k) = This is the number of states in a thin spherical shell between radius k and kak , e
the number is given by g, (1) dk
specifically ,
~
sum runs over occupied states
k
=Vog(k)12dk
#1
This can be used to find the ground state energy Er of the filled fermisphere Eo =
&zm zm
If p ~visfixed, To is extensive
hVSVT
= .
=
