P honons : The Quantum Mechanics of Lattice Vibrations

1. NOTE!!! The Following Material is Similar to Chapter 10, Sections 10.1 & 10.2 in the book by Reif.It logically follows from the discussion of the Einstein Model for the heat capacity of a solid

2. Phonons: The Quantum Mechanics of Lattice Vibrations

3. The Following Material is Partially Borrowed from the coursePhysics 4309/5304 “Solid State Physics”Taught in the Fall of everyodd numbered year

4. 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 vibrational energy in terms of • Classical Normal Modes • (uncoupled oscillators). .

5. 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.

6. 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

7. 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 thermodynamic & some othe properties of the solid, • it is necessary to QUANTIZE • these normal modes.

8. 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”

9. “Quasiparticles” Some Examples:

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

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

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.

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.

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.

15. “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!!!

16. 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)

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

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

19. 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. ħ E0 = ħ • 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 ħ • 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.

20. 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

21. Knowledge of the thermal energyis fundamental to obtaining anunderstanding many properties of solids. • Examples: Thermodynamic Properties: 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.

22. 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

23. 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.

24. 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.

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

26. Classical Theory of Heat Capacity of Solids We briefly discussed this model already! Summary: Each atom is bound to its site by a harmonic force. When heated, atoms vibrate at low amplitude around their equilibrium sites like a coupled set of harmonic oscillators. By the Equipartition Theorem, the classical 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 thus given by Cv (dE/dT)V . This clearly gives:

27. 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,.. ħ • If the oscillator interacts with a heat reservoir at • absolute temperature T, the probability Pnof it being • in level n is proportional to:

28. Quantized Energy of a Single Oscillator: ħ 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:

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

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

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

32. (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.

33. 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.

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

35. LowTemperature Limit: ħω >> kBT Temperature dependence of mean energy of a quantum harmonic oscillator. ħ ħ ħ “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 ħ

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

37. 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=

38. 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!

39. 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:

40. 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.

41. 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.

42. 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 For diamond the Einstein Temperature TE = 1320 K

43. 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

44. 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).

45. 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: Or g(ω)dω Cω2dω C = constant

46. Use this form & do math manipulation: g(ω)dω Cω2dω C = constant Longitudinal (L ) & Transverse (T) Sound Velocities

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

48. The Debye Model for the Heat Capacity ΘD  Debye “Temperature” Low temperature, kBT << ħ for one mole

49. Lattice heat capacity in the Debye Model The 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 ofthe heat capacity of most solids, eventhough the actual phonon-density ofstates curve may differ appreciablyfrom the Debye assumption. 1 Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1 • The Debye frequency ωD &the Debye temperature ΘDscale with the velocity • of soundin the solid. So solids with low densities and large elastic moduli have • high values of ΘD for various solids in the table. The Debye energy ħωDisan • estimate of the maximum phonon energy in a solid. Solid Ar Na Cs Fe Cu Pb C KCl 93 158 38 457 343 105 2230 235

50. Debye Model of the Vibrational Heat Capacity of Solids