P honons : The Quantum Mechanics of Lattice Vibrations

1. Phonons: The Quantum Mechanics of Lattice Vibrations

2. The Following Material is Partially Borrowed from the coursePhysics 4309/5304 “Solid State Physics”Taught in the Fall of everyodd numbered year(including 2017!)

3. In any Solid State Physics course, it is shown that the (classical) physicsof lattice vibrationsin a crystalline solid • Reduces to the CLASSICAL • coupled harmonic oscillator problem. • The solution to this problem is to rewrite the total energy in terms of\ • Classical Normal Modes • (uncoupled oscillators). .

4. Thegoalofmuch of the discussion in the vibrational properties chapter in solid state physics is to solve the Classical Problem of • finding the normal mode vibrational frequencies of the crystalline solid.

5. Note: What follows is a discussion of the Vibrational Heat Capacityof crystalline solids using 2 models: • The Einstein Model: • Chapter 7, Section 7 • The Debye Model: • Chapter 10, Sections 1 & 2 • in the book by Reif

6. The CLASSICAL Normal • Mode Problem. • 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. • Next, 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.

7. These quantized normal modes of vibration are called • PHONONS • PHONONSare massless quantum mechanical “particles” which have no classical analogue. • They behave like particles in momentum (k) space. • Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called • “Quasiparticles”

8. “Quasiparticles” Some Examples:

9. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves.

10. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves. • Photons: Quantized Normal Modes of • Electromagnetic Waves.

11. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves. • Photons: Quantized Normal Modes of • Electromagnetic Waves. • Rotons: Quantized Normal Modes of • Molecular Rotational Excitations.

12. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves. • Photons: Quantized Normal Modes of • Electromagnetic Waves. • Rotons: Quantized Normal Modes of • Molecular Rotational Excitations. • Magnons: Quantized Normal Modes of • Magnetic Excitations in Solids.

13. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves. • Photons: Quantized Normal Modes of • Electromagnetic Waves. • Rotons: Quantized Normal Modes of • Molecular Rotational Excitations. • Magnons: Quantized Normal Modes of • Magnetic Excitations in Solids. • Excitons: Quantized Normal Modes of • Electron-Hole Pairs.

14. “Quasiparticles” Some Examples: • Phonons: Quantized Normal Modes of • Lattice Vibrational Waves. • Photons: Quantized Normal Modes of • Electromagnetic Waves. • Rotons: Quantized Normal Modes of • Molecular Rotational Excitations. • Magnons: Quantized Normal Modes of • Magnetic Excitations in Solids. • Excitons: Quantized Normal Modes of • Electron-Hole Pairs. • Polaritons: Quantized Normal Modes of • Electric Polarization Excitations in Solids. • + Many Others!!!

15. Comparison of Phonons & Photons • PHOTONS • Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized Photon Wavelength: λphoton≈ 10-6 m (visible)

16. Comparison of Phonons & Photons • PHONONS • Quantized normal modes of lattice vibrations. The energies & momentaof phonons are quantized: • PHOTONS • Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized Photon Wavelength: λphoton≈ 10-6 m (visible) Phonon Wavelength: λphonon ≈ a ≈ 10-10 m

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

18. Often, we consider Enas being constructed by adding n • excitation quanta of energyħ to the ground state. Ground State (or “zero point”) Energy of the Oscillator. E0 = • 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 ħ. ΔE = (n – n΄) n & n ΄ = integers Phonon Absorption or Emission • In complicated processes, such as phonons interacting • with electrons or photons, it is known that • The number of phonons is NOT conserved. • Phonons can be created & destroyed during such interactions.

19. Thermal Energy &Lattice Vibrations • As is discussed in detail in any solid state • physics course, the atoms in a crystalline solid • vibrate with small amplitude abouttheir • equilibriumpositions. • 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

20. Knowledge of the thermal energyis fundamental to obtaining anunderstanding many properties of solids. • Examples: Heat Capacity, Entropy, Helmholtz Free Energy, • Equation of State, etc. • Question: How is the thermal energy calculated? • For example, we might 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.

21. Most importantly, the thermal energy plays a fundamental role in determining the • Thermal(Thermodynamic) • Properties of a Solid • Knowledge of how the thermal energy changes with temperature gives an understanding of heat energy necessary to raise the temperature of the material. • An important, measureable property of a solid is it’s • Specific Heat or Heat Capacity

22. Lattice Vibrational Contribution to the Heat Capacity • The Thermal Energyis the dominant • contribution to theheat capacity in most solids. • In non-magneticinsulators,it is • the onlycontribution. • Some othercontributions: • Conduction Electronsin metals & semiconductors. • Magnetic ordering in magnetic materials.

23. Calculation of the vibrational • contribution to the thermal energy & • heatcapacity of a solid has 2 parts: • 1. Calculation of thecontribution • of a single vibrational mode. • 2. Summation over thefrequency • distribution ofthe modes.

24. Vibrational Specific Heat of Solids cp Data at T = 298 K

25. Classical Theory of Heat Capacity of Solids We briefly discussed this model in the last class! Summary: Each atom is bound to its site by a harmonic force. When heated, atoms vibrate around their equilibrium sites like a coupled set of harmonic oscillators. By the Equipartition Theorem, the thermal average energy for a 1D oscillator is kT. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3kT. So, the energy per mole is E = 3RT R is the gas constant. The heat capacity per mole is thuss given by Cv (dE/dT)V . This clearly gives:

26. Thermal Energy & Heat Capacity: Einstein Model • We already briefly discussed the Einstein Model! • The following makes use of the • Canonical Ensemble • We’ve already seen that the Quantized Energy • solution to the Schrodinger Equation for a single • 1 D oscillator is: n = 0,1,2,3,.. • If the oscillator interacts with a heat reservoir at • absolute temperature T, the probability Pnof it being • in level n is proportional to:

27. Quantized Energy of a Single Oscillator: n = 0,1,2,3,.. • The probability of the oscillator beingin level n has • the form: Pn  • In the Canonical Ensemble, the average energy of • the harmonic oscillator &therefore of a lattice normal • modeof angular frequencyωattemperature Tis:

28. Straightforward but tedious math manipulation! Thermal Average Energy: Putting in the explicit form gives: The denominator is the Partition Function Z.

29. The denominator is the Partition Function Z. Evaluate it using the Binomial expansion for x << 1:

30. The equation for εcan be rewritten: The Final Result is:

31. (1) • This is the Thermal Average • Phonon Energy.(One oscillator!) • The first term in the aboveequation is called • “The Zero-Point Energy”. • It’s physical interpretation is that, even at • T = 0Kthe atoms vibrate in the crystal & • have a zero-pointenergy. • The Zero Point Energyis the minimumenergy of the system.

32. Thermal Average Phonon Energy: (1) • The first term in (1) is the Zero Point Energy. • Thedenominator of second termin (1) is often written: (2) • (2) is interpreted as the thermal average number of • phonons n(ω) at temperature T & frequency ω. • In modern terminology, (2) is called • The Bose-Einstein Distribution: • or The Planck Distribution.

33. High Temperature Limit: ħω << kBT Temperature dependence of mean energy of a quantum harmonic oscillator. Taylor’s series expansion of ex (x << 1) At high T, <>is independent of ω.Thehigh T limit is equivalent to the classical limit,(energy steps are small compared tototal energy). So, in this case,<>is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem).

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

35. Heat Capacity C(at constant volume) The heat capacity C(for one oscillator) is found by differentiating the thermal average vibrational energy Let

36. where The specific heat in this form Vanishes exponentially at lowT&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. The Einstein Approximation Starts with this form. Area=

37. The specific heatat constant volume Cvdepends • qualitatively ontemperature Tas shown in the figure • below. For hightemperatures,Cv(per mole) is close to 3R • (R= universal gasconstant. R 2 cal/K-mole). • So, at high temperaturesCv6 cal/K-mole The figure shows that Cv= 3R At high temperatures for all substances.This is called the“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!

38. Einstein’s Model of Heat Capacity of Solids The Einstein Model was the first application of quantum theory to solids. He made the(absurd & unphysical)assumption the each of 3N vibrational modes of a solid of N atoms has the same frequency, so that the whole solid has a heat capacity 3N times the heat capacity of one mode.

39. The whole solid has a vibrational heat capacity equal to3N times the heat capacity of one mode. Einstein’s Model • In this model, the atoms are treated as independent oscillators, but their energiesare quantum mechanical. • Itassumes that the atoms are each isolated oscillators, which is not at all realistic. In reality, they are a huge number of coupled oscillators. • But, even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

40. At high temperatures,all crystalline solids have a vibrational specific heatof3R = 6 cal/K per mole; they require 6 calories per mole to raise their temperature 1 K.This agreement 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.

41. Einstein Model for Lattice Vibrations in a SolidCvvs T for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) Points: Experiment Curve: Einstein Model Prediction

42. The Einstein model correctly gives a specific heat tending to zero at absolute zero, but the temperature dependence near T=0 doesnot agree with experiment. However, a model which takes into account the actual distribution of vibration frequencies in a solid is needed in order to understand the observed temperature dependence of Cv at low temperatures: CV = AT3 A= constant

43. Debye Model Vibrational Heat Capacity: Brief Discussion • In general, the thermal average vibrational energy of a solid has the form: • The term in parenthesis is the mean thermal energy for one mode as before: • The function g() is called the density of modes. • Formally, it is the number of modes between  & d • Its form depends on the details of the (k).

44. The Debye Model, assumes that every (k) is an acoustic mode (like ordinary sound waves) with • It can be shown that this results in a density of modes g() which has the form:

45. Use this form & do math manipulation: Longitudinal (L ) & Transverse (T) Sound Velocities

46. More math manipulation & assume low temperatures • kBT << ħ Finally,

47. The Debye model for the heat capacity low temperature, kBT << ħ for one mole

48. Lattice heat capacity in the Debye Model Figure shows the heatcapacity between the limits ofhigh low T predicted by the Debye model. Because it is exact in both high & 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. 1 Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1 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. Solid Ar Na Cs Fe Cu Pb C KCl 93 158 38 457 343 105 2230 235

49. The Debye Model Vibrating Solids

50. Limits of the Debye model