Phonons : 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!

3. In any Solid State Physics course, it is shown that the (classical) physicsof lattice vibrationsin a crystalline solid • Reduces to a CLASSICAL • Normal Mode Problem. • Agoalofmuch of the discussion in the vibrational properties chapter in solid state physics is to • find the normal mode vibrational • frequencies of the crystalline solid.

4. Note! The Debye Modelof the Vibrational Heat Capacityisdiscussed in Chapter 10 Sections 1 & 2of Reif’s Book

5. 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. • 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.

6. 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 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”

7. “Quasiparticles” Some Examples:

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

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

10. “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.

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. • Magnons: Quantized Normal Modes of • Magnetic Excitations in Solids.

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. • Excitons: Quantized Normal Modes of • Electron-Hole Pairs.

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. • Polaritons: Quantized Normal Modes of • Electric Polarization Excitations in Solids. • + Many Others!!!

14. 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)

15. 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

16. 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!

17. Often, we consider Enas being constructed by adding nexcitation 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. That is, phonons can be created & destroyed during such interactions.

18. Thermal Energy &Lattice Vibrations As is discussed in detail in any solid state physics course, 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

19. Knowledge of the thermal energyis fundamental to obtaining anunderstanding many properties of solids. • Examples: Heat Capacity, Entropy, Helmholtz Free Energy, Equation of State, etc. • A relevant question is how is this 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. • Most importantly, the thermal energy plays a fundamental role in determining the • Thermal(Thermodynamic) • Properties of a Solid

20. 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

21. 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.

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

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

24. 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 the solid is 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 given by Cv (dE/dT)V . This clearly gives:

25. Thermal Energy & Heat Capacity: Einstein Model • We briefly discussed the Einstein Model in the last class! • The following makes use of the • Canonical Ensemble & • the Boltzmann Distribution! • We’ve already seen that the Quantized Energy solution to • the Schrodinger Equation for a single 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 the BoltzmannFactor:

26. Quantized Energy of a Single Oscillator: n = 0,1,2,3,.. • On interaction with a heat reservoir at T, the probability • Pnof the oscillator beingin level n is proportional to: • In the Canonical Ensemble, the average energy of • the harmonic oscillator &therefore of a lattice normal • modeof angular frequencyωattemperature Tis:

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

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

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

30. (1) • This is the • Thermal AveragePhonon Energy. • 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.

31. 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.

32. Temperature dependence of mean energy of a quantum harmonic oscillator. Taylor’s series expansion of ex for x << 1 High Temperature Limit: ħω << kBT 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).

33. The temperature dependence of the mean energy of a quantum harmonic oscillator. “Zero Point Energy” LowTemperature Limit: ħω >> kBT 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

34. Heat Capacity C(at constant volume) The heat capacity Cis found by differentiating the thermal average vibrational energy Let

35. where This approximation is known as The Einstein Approximation The specific heat in this approximation 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. Area=

36. 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!

37. 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

38. Einstein’s 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 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.

39. At high temperatures,all crystalline solids have a vibrational specific heatof6 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 thatat room temperatures and belowthe specific heat of crystalline solidsis not a universal constant.

40. 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. Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be accounted using one dimensional model of monoatomic lattice

41. 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

42. 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

43. 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 .

44. There are two sets of waves for solution; Running waves Standing waves Running waves: These allowed k wavenumbers correspondto 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 These allowed wavenumbers are uniformly distibuted in k at a density of between k and k+dk. running waves

45. Standing waves: 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; These are the allowed wavenumbers for standing waves; only positive values are allowed. for running waves for standing waves

46. 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 waveshave 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 .

47. 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). modes with frequency from  to +d corresponds to modes with wavenumber from k to k+dk

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

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

50. True density of states by means of above equation constant density of states 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.