Solid state Phys. Chapter 2 Thermal and electrical properties 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

Phonon 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.

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=3Nk B T/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=N A k B is the gas constant. Taking the derivative gives directly the following result. Classical model of the specific heat

EINSTEIN MODEL is the thermal avarage of the number of phonons in an elastic wave of given frequency. Then the heat capacity of oscillator is N oscillators in one dimension 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)

DEBYE MODEL Debye model uses wide spectrum of frequencies to describe the complicated pattern of lattice vibrations.

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 Comparision between Debye and Einstein model

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. Conductivity is the reciprocal of electrical resistivity, ρ Thermal conductivity Thermal conductivity  quantity of heat Q transmitted in time t thickness L surface of area A temperature difference T Units : watt per kelvin-meters (W/m.K)

Electrical resistivity is the ohm ⋅ metre (Ω ⋅ m). siemens per metre (S ⋅ m −1 ) or (Ω −1 ⋅ m −1 ) the electric current density is measured in amperes per square metre

Common units of thermal conductivity are W/mK and Btu/hr-ft- o F.

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

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.

1)Why thermal conductivity of solid is greater than liquid? 2)What is the thermal conductivity of aluminum? 3)What is electrical conductivity?. QUESTIONS

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.

C v = T Where S is the entropy, U is the energy, and T is temperature. The heat capacity at constant volume is defined as J·kg -1 ·K -1 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 At lower temperatures the heat capacity drops rapidly and approaches zero as T 3 in insulators and as T in metals.If metal becomes semiconductor, the drop is faster than T.