Solid state Phys.

1. Solid state Phys. Chapter 2 Thermal and electrical properties

2. 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

3. PhononVibration of atoms in a crystal Phonons play a major role in many of the physical properties of solids, including a material's thermal and electricalconductivities. Hence the study of phonons is an important part of solid state physics.

5. 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.

6. 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 • oscillatorrAllatoms oscillate with the same frequency (contrast with the Debye model) 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

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

9. Comparisionbetween 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

10. 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)

11. 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)

12. Common units of thermal conductivity are W/mK and Btu/hr-ft-oF.

13. 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 structureof a metallicsolid. It was developed principally by Arnold Sommerfeld who combined the classicalDrudemodel with quantum mechanicalFermi-Dirac statistics and hence it is also known as the Drude–Sommerfeldmodel • The temperature dependence of the heat capacity • Electrical Conductivities

14. 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.

15. QUESTIONS • Why thermal conductivity of solid is greater than liquid? • What is the thermal conductivity of aluminum? • What is electrical conductivity? • .

16. 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-1at 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.

17. The heat capacity at constant volume is defined as Cv= T 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.