1. Chapter 5: Phonons II – Thermal Properties

2. What is a Phonon? We’ve seen that the physics of lattice vibrations in a crystalline solid Reduces to a CLASSICAL normal mode problem. The goal of the ENTIRE DISCUSSION so far has been to find the normal mode vibrational frequencies of the crystalline solid. In the harmonic approximation, this is achieved by first writing the solid’s vibrational energy as a system of coupled simple harmonic oscillators & then finding the classical normal mode frequencies & ion displacements for that system. Given the results of the classical normal mode calculation for the lattice vibrations, in order to treat some properties of the solid, It is necessary to QUANTIZE these normal modes.

3. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space.

4. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles”

5. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves.

6. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids

7. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs Polaritons: Quantized Normal Modes of electric polarization excitations in solids + Many Others!!!

8. These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have NO CLASSICAL ANALOGUE. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs Polaritons: Quantized Normal Modes of electric polarization excitations in solids + Many Others!!!

9. Phonon Wavelength: λ phonon ≈ a ≈ 10 -10 m Comparison of Phonons & Photons (visible) Photon Wavelength: λ photon ≈ 10 -6 m >> a PHONONS Quantized normal modes of lattice vibrations. The energies & momenta of phonons are quantized PHOTONS Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized

10. n = 0,1,2,3,.. E Quantum Mechanical Simple Harmonic Oscillator Quantum mechanical results for a simple harmonic oscillator with classical frequency ω: The energy is quantized. EnEn The energy levels are equally spaced!

11. Often, we consider E n as being constructed by adding n excitation quanta of energy to the ground state. If the system makes a transition from a lower energy level to a higher energy level, it is always true that the change in energy is an integer multiple of Phonon Absorption or Emission Oscillator Ground State (or “zero point”) Energy. ΔE = (n – n΄) n & n ΄ = integers In complicated processes, such as phonons interacting with electrons or photons, it is known that The number of phonons is NOT conserved. That is, phonons can be created & destroyed during such interactions. E 0 =

12. Thermal Energy & Lattice Vibrations As was already discussed in detail, the atoms in a crystal vibrate about their equilibrium positions. This motion produces vibrational waves. The amplitude of this vibrational motion increases as the temperature increases. In a solid, the energy associated with these vibrations is called the Thermal Energy

13. A knowledge of the thermal energy is fundamental to obtaining an understanding many of the basic properties (thermodynamic properties & others!) of solids. Examples Heat Capacity, Entropy, Helmholtz Free Energy, Equation of State, etc.... A relevant question is how is this thermal energy calculated? We would like to know how much thermal energy is available to scatter a conduction electron in a metal or a semiconductor. This is important because this scattering contributes to electrical resistance & other transport properties.

14. How is this thermal energy calculated? We would like to know how much thermal energy is available to scatter a conduction electron in a metal or a semiconductor. This is important because this scattering contributes to electrical resistance & other transport properties. Most important, the thermal energy plays a fundamental role in determining the Thermal (Thermodynamic) Properties of a Solid Knowledge of how the thermal energy changes wih temperature gives an understanding of heat energy necessary to raise the temperature of the material. An important, measureable property of a solid is its Specific Heat or Heat Capacity

15. The Thermal Energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Other contributions: Conduction Electrons in metals & semiconductors. Magnetic ordering in magnetic materials. The calculation of the vibrational contribution to the thermal energy & heat capacity of a solid has 2 parts: 1. Evaluation of the contribution of a single vibrational mode. 2. Summation over the frequency distribution of the modes. Lattice Vibrational Contribution to the Heat Capacity

16. Vibrational Specific Heat of Solids c p Data at T = 298 K

17. 17 In 1819, using room temperature data, Dulong and Petit empirically found that the molar heat capacity for solids is approximately Here, R is the gas constant. This relationship is now known as the Dulong-Petit “Law” Historical Background

18. 18 Assume that the heat supplied to a solid is transformed into the vibrational kinetic & potential energies of the lattice. To explain the Dulong-Petit “Law” theoretically, using Classical, Maxwell- Boltzmann Statistical Mechanics, a knowledge of how the heat is divided up among the degrees of freedom of the solid is needed. The Molar Heat Capacity

19. 19 The Dulong-Petit “Law” can be explained using Classical Maxwell-Boltzmann statistical physics. Specifically, The Equipartition Theorem can be used. This theorem states that, for a system in thermal equilibrium with a heat reservoir at temperature T, The thermal average energy per degree of freedom is (½)kT If each atom has 6 degrees of freedom: 3 translational & 3 vibrational, then R = N A k The Molar Heat Capacity The Molar Energy of a Solid

20. 20 By definition, the heat capacity of a substance at constant volume is Classical physics therefore predicts: A value independent of temperature The Molar Heat Capacity Heat Capacity at Constant Volume

21. 21 The Molar Heat Capacity Experimentally, the Dulong-Petit Law, however, is found to be valid only at high temperatures.

22. 22 Einstein Model of a Vibrating Solid In 1907, Einstein extended Planck’s ideas to matter: he proposed that the energy values of atoms are quantized and proposed the following simple model of a vibrating solid: Each atom is independent Each vibrates in 3-dimensions Each vibrational normal mode has energy:

23. 23 In effect, Einstein modeled one mole of a solid as an assembly of 3N A distinguishable oscillators. He used the Canonical Ensemble to calculate the average energy of an oscillator in this model.

24. 24 To compute the average, note that it can be written as Here,   [1/(kT)] Z is called The Partition Function Z has the form:

25. 25 This follows from the geometric series result With   [1/(kT)] and E n = n ,, the partition function Z for the Einstein Model is

26. 26 Differentiating Z with respect to  gives: Multiplying by –1/Z gives: This is the Einstein Model Result for the average thermal energy of an oscillator. The total vibrational energy of the solid is just 3N A times this result.

27. 27 The Heat Capacity in the Einstein Model is given by: Do the derivative & define T E   /k. T E is called The Einstein Temperature Finally, in the Einstein Model, C V has the form:

28. 28 The Einstein Model of a Vibrating Solid Einstein, Annalen der Physik 22 (4), 180 (1907) C V for Diamond

29. Thermal Energy & Heat Capacity: The Einstein Model: Another Derivation The following assumes that you know enough statistical physics to have seen the Cannonical Ensemble & the Boltzmann Distribution! The Quantized Energy of a Single Oscillator has the form: If the oscillator interacts with a heat reservoir at absolute temperature T, the probability P n of it being in quantum level n is proportional to the Boltzmann Factor: P n 

30. In the Cannonical Ensemble, a formal expression for the average energy of the harmonic oscillator & therefore of a lattice normal mode of angular frequency ω at temperature T is given by: The probability P n of the oscillator being in quantum level n has the form: P n  [exp (-β  )/Z] where the partition function Z is given by: Quantized Energy of a Single Oscillator:

31. Putting in the explicit form gives: According to the Binomial expansion, for x << 1 where Now, some straightforward math manipulation! Average Energy:

32. The equation for  ε  can be rewritten: Finally, the result is:

33. This is the Mean Phonon Energy. The first term in (1) is called the Zero-Point Energy. As mentioned before, even at 0ºK the atoms vibrate in the crystal & have a zero-point energy. This is the minimum energy of the system. The thermal average number of phonons n(ω) at Temperature T is given by The Bose-Einstein Distribution, & the denominator of the second term in (1) is often written: (1)

34. By using (2) in (1), (1) can be rewritten: = ћω[n(  ) + ½] In this form, the mean energy looks analogous to a quantum mechanical energy level for a simple harmonic oscillator. That is, it looks similar to: So the second term in the mean energy (1) is interpreted as The number of phonons at temperature T & frequency ω. (1) (2)

35. At high T, is independent of ω. This high T limit is equivalent to the classical limit, (the energy steps are small compared to the total energy). So, in this case, is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem). High Temperature Limit: ħω << k B T Temperature dependence of the mean energy of a quantum harmonic oscillator. Taylor’s series expansion of e x  for x << 1

36. Low Temperature Limit: ħω > > k B T Temperature dependence of the mean energy of a quantum harmonic oscillator. “Zero Point Energy” At low T, the exponential in the denominator of the 2 nd term gets larger as T gets smaller. At small enough T, neglect 1 in the denominator. Then, the 2 nd term is e -x, x = (ħω/(k B T). At very small T, e -x  0. So, in this case, is independent of T:  (½)ħω

37. Einstein Heat Capacity C V The heat capacity C V is found by differentiating the average phonon energy: Let

38. Area = The specific heat C V in this approximation vanishes exponentially at low T & tends to the classical value at high T. These features are common to all quantum systems; the energy tends to the zero- point-energy at low T & to the classical value at high T. where Einstein Heat Capacity C V

39. The figure shows that C v = 3R at high temperatures for all substances. This is the classical Dulong-Petit law. This states that specific heat of a given number of atoms of any solid is independent of temperature & is the same for all materials! The specific heat at constant volume C v depends qualitatively on temperature T as shown in the figure below. For high temperatures, C v (per mole) is close to 3R (R = universal gas constant. R  2 cal/K- mole). So, at high temperatures C v  6 cal/K-mole

40. Einstein Model for Lattice Vibrations in a Solid C v vs T for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) Points: Experiment Curve: Einstein Model Prediction

41. Einstein Model of Heat Capacity of Solids The Einstein Model was the first quantum theory of lattice vibrations in solids. He made the assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency, so that the whole solid had a heat capacity 3N times In this model, the atoms are treated as independent oscillators, but the energies of the oscillators are the quantum mechanical energies. This assumes that the atoms are each isolated oscillators, which is not at all realistic. In reality, they are a huge number of coupled oscillators. Even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

42. At high temperatures, all crystalline solids have a vibrational specific heat of 6 cal/K per mole; they require 6 calories per mole to raise their temperature 1 K. This arrangement between observation and classical theory breaks down if the temperature is not high. Observations show that at room temperatures and below the specific heat of crystalline solids is not a universal constant. In each of these materials (Pb,Al, Si,and Diamond) specific heat approaches a constant value asymptotically at high T. But at low T, the specific heat decreases towards zero which is in a complete contradiction with the above classical result.

43. The Einstein model also gives correctly a specific heat tending to zero at absolute zero, but the temperature dependence near T= 0 does not agree with experiment. Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be accounted using one dimensional model of monoatomic lattice

44. Density of States According to Quantum Mechanics if a particle is constrained; the energy of particle can only have special discrete energy values. it cannot increase infinitely from one value to another. it has to go up in steps. Thermal Energy & Heat Capacity Debye Model

45. These steps can be so small depending on the system that the energy can be considered as continuous. This is the case of classical mechanics. But on atomic scale the energy can only jump by a discrete amount from one value to another. Definite energy levels Steps get small Energy is continuous

46. In some cases, each particular energy level can be associated with more than one different state (or wavefunction ) This energy level is said to be degenerate. The density of states is the number of discrete states per unit energy interval, and so that the number of states between and will be.

47. There are two sets of waves for solution; Running waves Standing waves These allowed k wavenumbers correspond to the running waves; all positive and negative values of k are allowed. By means of periodic boundary condition an integer Length of the 1D chain Running waves: These allowed wavenumbers are uniformly distibuted in k at a density of between k and k+dk. running waves

48. In some cases it is more suitable to use standing waves,i.e. chain with fixed ends. Therefore we will have an integral number of half wavelengths in the chain; Standing waves: These are the allowed wavenumbers for standing waves; only positive values are allowed. for running waves for standing waves

49. These allowed k’s are uniformly distributed between k and k+dk at a density of DOS of standing wave DOS of running wave The density of standing wave states is twice that of the running waves. However in the case of standing waves only positive values are allowed Then the total number of states for both running and standing waves will be the same in a range dk of the magnitude k The standing waves have the same dispersion relation as running waves, and for a chain containing N atoms there are exactly N distinct states with k values in the range 0 to.

50. modes with frequency from  to  +d  corresponds to modes with wavenumber from k to k+dk The density of states per unit frequency range g(  ): The number of modes with frequencies  &  + d  will be g(  )d . g(  ) can be written in terms of  S (k) and  R (k).

51. Choose standing waves to obtain Let’s remember dispertion relation for 1D monoatomic lattice ;

52. Multibly and divide Let’s remember: True density of states

53. constant density of states True density of states by means of above equation True DOS(density of states) tends to infinity at, since the group velocity goes to zero at this value of. Constant density of states can be obtained by ignoring the dispersion of sound at wavelengths comparable to atomic spacing.

54. The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus Mean energy of a harmonic oscillator One can obtain same expression of by means of using running waves. for 1D It should be better to find 3D DOS in order to compare the results with experiment.

55. 3D DOS Let’s do it first for 2D, then for 3D. Consider a crystal in the shape of 2D box with crystal sides L. + + +- - - L 0 L y x Standing wave pattern for a 2D box Configuration in k-space

56. Let’s calculate the number of modes within a range of wavevector k. Standing waves are choosen but running waves will lead same expressions. Standing waves will be of the form Assuming the boundary conditions of Vibration amplitude should vanish at edges of Choosing positive integer

57. + + +- - - L 0 L y x The allowed k values lie on a square lattice of side in the positive quadrant of k-space. These values will so be distributed uniformly with a density of per unit area. This result can be extended to 3D. Standing wave pattern for a 2D box Configuration in k-space

58. L L L Octant of the crystal: k x,k y,k z (all have positive values) The number of standing waves;

59. is a new density of states defined as the number of states per unit magnitude of in 3D.This eqn can be obtained by using running waves as well.  (frequency) space can be related to k-space: Let’s find C at low and high temperature by means of using the expression of.

60. High and Low Temperature Limits This result is true only if at low T’s only lattice modes having low frequencies can be excited from their ground states; Each of the 3N lattice modes of a crystal containing N atoms k  Low frequency long sound waves =

61. depends on the direction and there are two transverse, one longitudinal acoustic branch: and at low T Velocities of sound in longitudinal and transverse direction

62. Zero point energy = at low temperatures

63. Debye Model of Heat Capacity of Solids

64. How good is the Debye approximation at low T? The lattice heat capacity of solids thus varies as T 3 at low temperatures; this is referred to as the Debye T 3 law. The figure illustrates the excellent agreement of this prediction with experiment for a non-magnetic insulator. The heat capacity vanishes more slowly than the exponential behaviour of a single harmonic oscillator because the vibration spectrum extends down to zero frequency.

65. The Debye interpolation scheme The calculation of is a very heavy calculation for 3D, so it must be calculated numerically. Debye obtained a good approximation to the resulting heat capacity by neglecting the dispersion of the acoustic waves, i.e. assuming for arbitrary wavenumber. In a one dimensional crystal this is equivalent to taking as given by the broken line of density of states figure rather than full curve. Debye’s approximation gives the correct answer in either the high and low temperature limits, and the language associated with it is still widely used today.

66. 1. Approximate the dispersion relation of any branch by a linear extrapolation of the small k behaviour: Einstein approximation to the dispersion Debye approximatio n to the dispersion The Debye approximation has two main steps:

67. Debye cut-off frequency 2. E nsure the correct number of modes by imposing a cut- off frequency, above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that

68. The lattice vibration energy of becomes and, First term is the estimate of the zero point energy, and all T dependence is in the second term. The heat capacity is obtained by differentiating above eqn wrt temperature.

69. Let’s convert this complicated integral into an expression for the specific heat changing variables to and define the Debye temperature The heat capacity is

70. The Debye prediction for lattice specific heat where

71. How does limit at high and low temperatures? High temperature x is always small  

72. How does limit at high and low temperatures? Low temperature For low temperature the upper limit of the integral is infinite; the integral is then a known integral of. We obtain the Debye law in the form T < <

73. Lattice heat capacity due to Debye interpolation scheme Figure shows the heat capacity between the two limits of high and low T as predicted by the Debye interpolation formula. Because it is exact in both high and low T limits the Debye formula gives quite a good representation of the heat capacity of most solids, even though the actual phonon-density of states curve may differ appreciably from the Debye assumption. Debye frequency and Debye temperature scale with the velocity of sound in the solid. So solids with low densities and large elastic moduli have high. Values of for various solids is given in table. Debye energy can be used to estimate the maximum phonon energy in a solid. Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1 1 SolidArNaCsFeCuPbCKCl 93158384573431052230235