IV. Vibrational Properties of the Lattice

1. IV. Vibrational Properties of the Lattice • Heat Capacity—Einstein Model • The Debye Model — Introduction • A Continuous Elastic Solid • 1-D Monatomic Lattice • Counting Modes and Finding N() • The Debye Model — Calculation • 1-D Lattice With Diatomic Basis • Phonons and Conservation Laws • Dispersion Relations and Brillouin Zones • Anharmonic Properties of the Lattice

2. kz m ky kx Classical statistical mechanics — equipartition theorem: in thermal equilibrium each quadratic term in the E has an average energy , so: Having studied the structural arrangements of atoms in solids, we now turn to properties of solids that arise from collective vibrations of the atoms about their equilibrium positions. A. Heat Capacity—Einstein Model (1907) For a vibrating atom:

3. Classical Heat Capacity For a solid composed of N such atomic oscillators: Giving a total energy per mole of sample: So the heat capacity at constant volume per mole is: This law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 300 K). But by the middle of the 19th century it was clear that CV 0 as T  0 for solids. So…what was happening?

4. Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies [later QM showed is actually correct] Einstein Uses Planck’s Work Einstein (1907): model a solid as a collection of 3N independent 1-D oscillators, all with constant , and use Planck’s equation for energy levels occupation of energy level n: (probability of oscillator being in level n) classical physics (Boltzmann factor) Average total energy of solid:

5. Now let Now we can use the infinite sum: To give: Some Nifty Summing Using Planck’s equation: Which can be rewritten: So we obtain:

6. At last…the Heat Capacity! Using our previous definition: Differentiating: Now it is traditional to define an “Einstein temperature”: So we obtain the prediction:

7. 3R CV T/E Limiting Behavior of CV(T) High T limit: Low T limit: These predictions are qualitatively correct: CV 3R for large T and CV 0 as T  0:

8. But Let’s Take a Closer Look: High T behavior: Reasonable agreement with experiment Low T behavior: CV 0 too quickly as T  0 !

9. • 3N independent oscillators, all with frequency  • Discrete allowed energies: Despite its success in reproducing the approach of CV 0 as T  0, the Einstein model is clearly deficient at very low T. What might be wrong with the assumptions it makes? B. The Debye Model (1912)

10. # of oscillators per unit  Einstein function for one oscillator Details of the Debye Model Pieter Debye succeeded Einstein as professor of physics in Zürich, and soon developed a more sophisticated (but still approximate) treatment of atomic vibrations in solids. Debye’s model of a solid: • 3N normal modes (patterns) of oscillations • Spectrum of frequencies from  = 0 to max • Treat solid as continuous elastic medium (ignore details of atomic structure) This changes the expression for CV because each mode of oscillation contributes a frequency-dependent heat capacity and we now have to integrate over all :

11. for wave propagation along the x-direction So the wave speed is independent of wavelength for an elastic medium! we find that is called the dispersion relation of the solid, and here it is linear (no dispersion!) group velocity We can describe a propagating vibration of amplitude u along a rod of material with Young’s modulus E and density  with the wave equation: C. The Continuous Elastic Solid By comparison to the general form of the 1-D wave equation:

12. M a In equilibrium: Longitudinal wave: By contrast to a continuous solid, a real solid is not uniform on an atomic scale, and thus it will exhibit dispersion. Consider a 1-D chain of atoms: D. 1-D Monatomic Lattice p = atom label p =  1 nearest neighbors p =  2 next nearest neighbors cp = force constant for atom p For atom s,

13. Now we use Newton’s second law: For the expected harmonic traveling waves, we can write xs = sa = position of atom s Thus: Or: So: 1-D Monatomic Lattice: Equation of Motion Now since c-p = cp by symmetry,

14. The result is: The result is periodic in k and the only unique solutions that are physically meaningful correspond to values in the range: 1-D Monatomic Lattice: Solution! The dispersion relation of the monatomic 1-D lattice! Often it is reasonable to make the nearest-neighbor approximation (p = 1):

15. In a 3-D atomic lattice we expect to observe 3 different branches of the dispersion relation, since there are two mutually perpendicular transverse wave patterns in addition to the longitudinal pattern we have considered. Dispersion Relations: Theory vs. Experiment Along different directions in the reciprocal lattice the shape of the dispersion relation is different. But note the resemblance to the simple 1-D result we found.

16. A vibrational mode is a vibration of a given wave vector (and thus ), frequency , and energy . How many modes are found in the interval between and ? # modes L = Na s+N-1 s s+1 x = sa x = (s+N)a s+2 E. Counting Modes and Finding N() We will first find N(k) by examining allowed values of k. Then we will be able to calculate N() and evaluate CV in the Debye model. First step: simplify problem by using periodic boundary conditions for the linear chain of atoms: We assume atoms s and s+N have the same displacement—the lattice has periodic behavior, where N is very large.

17. Since atoms s and s+N have the same displacement, we can write: First: finding N(k) This sets a condition on allowed k values: independent of k, so the density of modes in k-space is uniform So the separation between allowed solutions (k values) is: Thus, in 1-D:

18. Now for a 3-D lattice we can apply periodic boundary conditions to a sample of N1 x N2 x N3 atoms: N3c N2b N1a Now we know from before that we can write the differential # of modes as: We carry out the integration in k-space by using a “volume” element made up of a constant  surface with thickness dk: Next: finding N()

19. Rewriting the differential number of modes in an interval: We get the result: N() at last! A very similar result holds for N(E) using constant energy surfaces for the density of electron states in a periodic lattice! This equation gives the prescription for calculating the density of modes N() if we know the dispersion relation (k). We can now set up the Debye’s calculation of the heat capacity of a solid.

20. F. The Debye Model Calculation We know that we need to evaluate an upper limit for the heat capacity integral: If the dispersion relation is known, the upper limit will be the maximum  value. But Debye made several simple assumptions, consistent with a uniform, isotropic, elastic solid: • 3 independent polarizations (L, T1, T2) with equal propagation speeds vg • continuous, elastic solid:  = vgk • max given by the value that gives the correct number of modes per polarization (N)

21. Since the solid is isotropic, all directions in k-space are the same, so the constant  surface is a sphere of radius k, and the integral reduces to: Giving: for one polarization N() in the Debye Model First we can evaluate the density of modes: Next we need to find the upper limit for the integral over the allowed range of frequencies.

22. Giving: max in the Debye Model Since there are N atoms in the solid, there are N unique modes of vibration for each polarization. This requires: The Debye cutoff frequency Now the pieces are in place to evaluate the heat capacity using the Debye model! This is the subject of problem 5.2 in Myers’ book. Remember that there are three polarizations, so you should add a factor of 3 in the expression for CV. If you follow the instructions in the problem, you should obtain: And you should evaluate this expression in the limits of low T (T << D) and high T (T >> D).

23. Debye Model: Theory vs. Expt. Better agreement than Einstein model at low T Universal behavior for all solids! Debye temperature is related to “stiffness” of solid, as expected

24. Debye Model at low T: Theory vs. Expt. Quite impressive agreement with predicted CV T3 dependence for Ar! (noble gas solid) (See SSS program debye to make a similar comparison for Al, Cu and Pb)

25. a M1 M2 M1 M2 G. 1-D Lattice with Diatomic Basis Consider a linear diatomic chain of atoms (1-D model for a crystal like NaCl): In equilibrium: Applying Newton’s second law and the nearest-neighbor approximation to this system gives a dispersion relation with two “branches”: -(k)   0 as k  0 acoustic modes (M1 and M2 move in phase) +(k)   max as k  0 optical modes (M1 and M2 move out of phase)

26. optical acoustic 1-D Lattice with Diatomic Basis: Results These two branches may be sketched schematically as follows: gap in allowed frequencies In a real 3-D solid the dispersion relation will differ along different directions in k-space. In general, for a p atom basis, there are 3 acoustic modes and p-1 groups of 3 optical modes, although for many propagation directions the two transverse modes (T) are degenerate.

27. The optical modes generally have frequencies near  = 1013 1/s, which is in the infrared part of the electromagnetic spectrum. Thus, when IR radiation is incident upon such a lattice it should be strongly absorbed in this band of frequencies. Diatomic Basis: Experimental Results At right is a transmission spectrum for IR radiation incident upon a very thin NaCl film. Note the sharp minimum in transmission (maximum in absorption) at a wavelength of about 61 x 10-4 cm, or 61 x 10-6 m. This corresponds to a frequency  = 4.9 x 1012 1/s. If instead we measured this spectrum for LiCl, we would expect the peak to shift to higher frequency (lower wavelength) because MLi < MNa…exactly what happens!

28. Collective motion of atoms = “vibrational mode”: Quantum harmonic oscillator: Energy content of a vibrational mode of frequency is an integral number of energy quanta . We call these quanta “phonons”. While a photon is a quantized unit of electromagnetic energy, a phonon is a quantized unit of vibrational (elastic) energy. Associated with each mode of frequency is a wavevector , which leads to the definition of a “crystal momentum”: H. Phonons and Conservation Laws Crystal momentum is analogous to but not equivalent to linear momentum. No net mass transport occurs in a propagating lattice vibration, so the linear momentum is actually zero. But phonons interacting with each other or with electrons or photons obey a conservation law similar to the conservation of linear momentum for interacting particles.

29. Schematically: Just a special case of the general conservation law! Photon wave vectors Phonons and Conservation Laws Lattice vibrations (phonons) of many different frequencies can interact in a solid. In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector: Compare this to the special case of elastic scattering of x-rays with a crystal lattice:

30. Remember the dispersion relation of the 1-D monatomic lattice, which repeats with period (in k-space) : 1st Brillouin Zone (BZ) 2nd Brillouin Zone 3rd Brillouin Zone I. Brillouin Zones of the Reciprocal Lattice Each BZ contains identical information about the lattice

31. Wigner-Seitz Cell--Construction For any lattice of points, one way to define a unit cell is to connect each lattice point to all its neighboring points with a line segment and then bisect each line segment with a perpendicular plane. The region bounded by all such planes is called the Wigner-Seitz cell and is a primitive unit cell for the lattice. 1-D lattice: Wigner-Seitz cell is the line segment bounded by the two dashed planes 2-D lattice: Wigner-Seitz cell is the shaded rectangle bounded by the dashed planes

32. The Wigner-Seitz cell can be defined for any kind of lattice (direct or reciprocal space), but the WS cell of the reciprocal lattice is also called the 1st Brillouin Zone. The 1st BZ is the region in reciprocal space containing all information about the lattice vibrations of the solid. Only the values in the 1st BZ correspond to unique vibrational modes. Any outside this zone is mathematically equivalent to a value inside the 1st BZ. This is expressed in terms of a general translation vector of the reciprocal lattice: 1st Brillouin Zone--Definition

33. 1st Brillouin Zone for 3-D Lattices For 3-D lattices, the construction of the 1st Brillouin Zone leads to a polyhedron whose planes bisect the lines connecting a reciprocal lattice point to its neighboring points. We will see these again! bcc direct lattice  fcc reciprocal lattice fcc direct lattice  bcc reciprocal lattice

34. IJ. Anharmonic Properties of Solids Two important physical properties that ONLY occur because of anharmonicity in the potential energy function: • Thermal expansion • Thermal resistivity (or finite thermal conductivity) Thermal expansion In a 1-D lattice where each atom experiences the same potential energy function U(x), we can calculate the average displacement of an atom from its T=0 equilibrium position:

35. independent of T ! IThermal Expansion in 1-D Evaluating this for the harmonic potential energy function U(x) = cx2 gives: Now examine the numerator carefully…what can you conclude? Thus any nonzero <x> must come from terms in U(x) that go beyond x2. For HW you will evaluate the approximate value of <x> for the model function Why this form? On the next slide you can see that this function is a reasonable model for the kind of U(r) we have discussed for molecules and solids.

36. Do you know what form to expect for <x> based on experiment?

37. Usually we write:  = thermal expansion coefficient Lattice Constant of Ar Crystal vs. Temperature Above about 40 K, we see:

38. Classical definition of thermal conductivity high T low T heat capacity per unit volume Thermal energy flux (J/m2s) wave velocity mean free path of scattering (would be  if no anharmonicity) Thermal Resistivity When thermal energy propagates through a solid, it is carried by lattice waves or phonons. If the atomic potential energy function is harmonic, lattice waves obey the superposition principle; that is, they can pass through each other without affecting each other. In such a case, propagating lattice waves would never decay, and thermal energy would be carried with no resistance (infinite conductivity!). So…thermal resistance has its origins in an anharmonic potential energy.

39. deviation from perfect crystalline order To understand the experimental dependence , consider limiting values of and (since does not vary much with T). Phonon Scattering There are three basic mechanisms to consider: 1. Impurities or grain boundaries in polycrystalline sample 2. Sample boundaries (surfaces) 3. Other phonons (deviation from harmonic behavior)

40. Temperature-Dependence of  The low and high T limits are summarized in this table: How well does this match experimental results?

41.  T3 • T-1 ? (not quite) Experimental (T)

42. How exactly do phonon collisions limit the flow of heat? Phonon Collisions: (N and U Processes) 2-D lattice  1st BZ in k-space: No resistance to heat flow (N process; phonon momentum conserved)  Predominates at low T << Dsince  and q will be small

43. Umklapp = “flipping over” of wavevector! What if the phonon wavevectors are a bit larger? Phonon Collisions: (N and U Processes) 2-D lattice  1st BZ in k-space: Two phonons combine to give a net phonon with an opposite momentum! This causes resistance to heat flow. (U process; phonon momentum “lost” in units of ħG.) • More likely at high T >> Dsince  and q will be larger