Solid state Phys.
Chapter 2
Thermal and
electrical
properties
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1
Chapter Four
Thermal and electrical properties
Aim: To provide an overwiev of the description of thermal properties of
solids, particularly insulators, using the dynamics of lattice vibrations.
In this lecture we discuss the specific heat and thermal conductivity of
crystalline solids. We make use of the density of vibrational states and
the concept of phonons that were covered in the previous lecture.
 Classical Model Of The Specific Heat.
 Einstein Model & Debye Model
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Phonon
Vibration of atoms in a crystal
Phonons play a major role in many of the physical properties of solids, including a material's
thermal and electrical conductivities. Hence the study of phonons is an important part of solid
state physics.
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Classical model of the specific heat
Consider a monatomic ideal gas of N atoms. The inner energy is given by the
kinetic energy of the atoms. Each atom has three degrees of freedom, so the
energy is given by,
U=3NkBT/2,
where kB is Boltzmann’s constant and T is temperature. The heat capacity is
given by C = (∂U/∂T). In the case of a crystalline solid, we must add the
potential energy, resulting from three additional degrees of freedom. Hence
we have per mole
U= 3 RT, where R=NAkB is the gas constant.
Taking the derivative gives directly the following result.
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EINSTEIN MODEL
The Einstein solid is a model of a solid based on two assumptions:
 Each atom in the lattice is an independent 3D quantum harmonic
 oscillatorrAll atoms oscillate with the same frequency (contrast with the
Debye model)
N
U  N  n    /
e 1
N oscillators in one dimension
is the thermal avarage of the number of phonons in
an elastic wave of given frequency.
Then the heat capacity of oscillator is
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DEBYE MODEL
Debye model uses wide spectrum of frequencies to describe the complicated pattern
of lattice vibrations.
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Comparision between
Debye and Einstein model
Features of the graph:
(i) The dimensionless heat capacity is zero at absolute zero, and rises to a value of three as the
temperature becomes much larger than the Debye temperature.
Thus, one of the strengths of the Debye model is that it predicts an approach of heat capacity
toward zero as zero temperature is approached, and also predicts the proper mathematical form of
this approach.
(ii) For Einstein solid the value of 3Nk is recovered at high temperatures.
(iii) The horizontal line corresponds to the classical limit of the Dulong-Petit law
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Electrical conductivity
Electrical conductivity is a measure of a material’s ability to conduct an electric
current. The conductivity  is defined as the ratio of the current density (current per
area), J, to the electric field strength, E, i.e.
E
J  E 

Conductivity is the reciprocal of
electrical resistivity, ρ
Thermal conductivity
Q
L
 
t A  T
Units : watt per kelvin-meters (W/m.K)
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Thermal conductivity 
quantity of heat Q
transmitted in time t
thickness L
surface of area A
temperature difference T
E
J  E 

Electrical resistivity is the ohm⋅metre (Ω⋅m).
the electric current
density is measured in
amperes per square
metre
siemens per metre (S⋅m−1) or (Ω −1 ⋅m−1)
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.
Common units of thermal conductivity are W/mK and Btu/hr-ft- F
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o
The free electron theory
In solid-state physics, the free electron model is a simple model for the behaviour
of valence electrons in a crystal structure of a metallic solid. It was developed
principally by Arnold Sommerfeld who combined the classical Drude model with
quantum mechanical Fermi-Dirac statistics and hence it is also known as the
Drude–Sommerfeld model
 The temperature dependence of the heat capacity
 Electrical Conductivities
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Free Electron Model
The allowed values of k therefore form a cubic point lattice in k-space, with
spacing π/L and volume per point (π/L)3. Finding the number of normal
modes of the standing wave wavefunctions with k between k and k+dk is
equivalent to finding the number of lattice points between two spherical
shells of radii k and k+dk in the positive octant of k-space.
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QUESTIONS
1) Why thermal conductivity of solid is greater than liquid?
2) What is the thermal conductivity of aluminum?
3) What is electrical conductivity?
.
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Thermal conductivity is driven by different mechanisms in different types of
materials.
• Metals exhibit high electrical and thermal conductivity as a consequence of
easy transport of electrons.
• FCC metals exhibit the highest electrical and thermal conductivities, e.g.
k(Ag)=430 W.m-1.K-1 at room temperature. Alloying tends to decrease
conductivity.
BCC metals typically exhibit an order of magnitude lower electrical and thermal
conductivity
.
• Electrical conductivity is a very useful probe of solute levels and can be used
to measure the progress of precipitation, especially in Al alloys.
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The heat capacity at constant volume is defined as
Cv= T
(S / T )
V
J·kg-1·K-1
Where S is the entropy, U is the energy, and T is
temperature.
The experimental facts about the heat capacity of solids are these:
1. In room temperature range the value of the heat capacity of nearly all monoatomic
solids is close to 3Nk, or 25 J mol-1 deg -1.
2. At lower temperatures the heat capacity drops rapidly and approaches zero as T3 in
insulators and as T in metals.If metal becomes semiconductor, the drop is faster than T.
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