CHAPTER 4 Crystal Dynamics I

1. CHAPTER 4Crystal Dynamics I SOUND WAVES LATTICE VIBRATIONS OF 1D CRYSTALS chain of identical atoms chain of two types of atoms LATTICE VIBRATIONS OF 3D CRYSTALS PHONONS HEAT CAPACITY FROM LATTICE VIBRATIONS ANHARMONIC EFFECTS THERMAL CONDUCTION BY PHONONS

2. Crystal Dynamics • Concern with the spectrum of characteristics vibrations of a crystalline solid. • Leads to; • consideration of the conditions for wave propagation in a periodic lattice, • the energy content, • the specific heat of lattice waves, • the particle aspects of quantized lattice vibrations (phonons) • consequences of an harmonic coupling between atoms.

3. Crystal Dynamics These introduces us to the concepts of • forbidden and permitted frequency ranges, and • electronic spectra of solids

4. Crystal Dynamics • In previous chapters we have assumed that the atoms were at rest at their equilibrium position. This can not be entirely correct (against to the HUP); Atoms vibrate about their equilibrium position at absolute zero. • The energy they possess as a result of zero point motion is known as zero point energy. • The amplitude of the motion increases as the atoms gain more thermal energy at higher temperatures. • In this chapter we discuss the nature of atomic motions, sometimes referred to as lattice vibrations. • In crystal dynamics we will use the harmonic approximation , amplitude of the lattice vibration is small. At higher amplitude some unharmonic effects occur.

5. Crystal Dynamics • Our calculations will be restricted to lattice vibrations of small amplitude. Since the solid is then close to a position of stable equilibrium its motion can be calculated by a generalization of the method used to analyse a simple harmonic oscillator.The small amplitude limit is known as harmonic limit. • In the linear region (the region of elastic deformation), the restoring force on each atom is approximately proportional to its displacement (Hooke’s Law). • There are some effects of nonlinearity or ‘anharmonicity’for larger atomic displacements. • Anharmonic effects are important for interactions between phonons and photons.

6. Crystal Dynamics • Atomic motions are governed by the forces exerted on atoms when they are displaced from their equilibrium positions. • To calculate the forces it is necessary to determine the wavefunctions and energies of the electrons within the crystal. Fortunately many important properties of the atomic motions can be deduced without doing these calculations.

7. Hooke's Law • One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law. It takes twice as much force to stretch a spring twice as far. Spring constant k

8. Hooke’s Law • The point at which the Elastic Region ends is called the inelastic limit, or the proportional limit. In actuality, these two points are not quite the same. • The inelastic Limit is the point at which permanent deformation occurs, that is, after the elastic limit, if the force is taken off the sample, it will not return to its original size and shape, permanent deformation has occurred. • The Proportional Limit is the point at which the deformation is no longer directly proportional to the applied force (Hooke's Law no longer holds). Although these two points are slightly different, we will treat them as the same in this course.

9. SOUND WAVES • It corresponds to the atomic vibrations with a long λ. • Presence of atoms has no significance in this wavelength limit, since λ>>a, so there will no scattering due to the presence of atoms. • Mechanical waves are waves which propagate through a material medium (solid, liquid, or gas) at a wave speed which depends on the elastic and inertial properties of that medium. There are two basic types of wave motion for mechanical waves: longitudinal waves and transverse waves. Longitudinal Waves Transverse Waves

10. SOUND WAVES • Sound waves propagate through solids. This tells us that wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The detailed atomic structure is unimportant for these waves and their propagation is governed by the macroscopic elastic properties of the crystal. • We discuss sound waves since they must correspond to the low frequency, long wavelength limit of the more general lattice vibrations considered later in this chapter. • At a given frequency and in a given direction in a crystal it is possible to transmit three sound waves, differing in their direction of polarization and in general also in their velocity.

11. Elastic Waves • A solid is composed of discrete atoms, however when the wavelength is very long, one may disregard the atomic nature and treat the solid as a continous medium. Such vibrations are referred to as elastic waves. ElasticWavePropagation (longitudinal) in a bar • At the point x elastic displacement is U(x) and strain ‘e’ is defined as the change in length per unit length. A x x+dx

12. Elastic Waves • According to Hooke’s law stress S (force per unit area) is proportional to the strain e. • To examine the dynamics of the bar, we choose an arbitrary segment of length dx as shown above. Using Newton’s second law, we can write for the motion of this segment, A C = Young modulus x x+dx Mass x Acceleration Net Force resulting from stresses

13. Equation of motion Elastic Waves Cancelling common terms of Adx; Which is the wave eqn. with an offered sol’n and velocity of sound waves ; • k = wave number (2π/λ) • ω = frequency of the wave • A = wave amplitude

14. The relation connecting the frequency and wave number is known as the dispersion relation. DispersionRelation Continuum ω 0 Discrete k • At small λ k → ∞ (scattering occurs) • At long λ k → 0 (no scattering) • When k increases velocity decreases.As k increases further, the scattering becomes greater since the strength of scattering increases as the wavelength decreases, and the velocity decreases even further. * Slope of the curve gives the velocity of the wave.

15. Speed of Sound Wave • The speed with which a longitudinal wave moves through a liquid of density ρ is C = Elasticbulk modulus ρ = Mass density • The velocity of sound is in general a function of the direction of propagation in crystalline materials. • Solids will sustain the propagation of transverse waves, which travel more slowly than longitudinal waves. • The larger the elastic modules and smaller the density, the more rapidly can sound waves travel.

16. Sound Wave Speed Speed of sound for some typical solids • VL values are comparable with direct observations of speed of sound. • Sound speeds are of the order of 5000 m/s in typical metallic, covalent • and ionic solids.

17. Sound Wave Speed • A lattice vibrational wave in a crystal is a repetitive and systematic sequence of atomic displacements of • longitudinal, • transverse, or • some combination of the two • An equation of motion for any displacement can be produced by means of considering the restoring forces on displaced atoms. • They can be characterized by • A propagation velocity, v • Wavelength λ or wavevector • A frequency  or angular frequency ω=2π • As a result we can generate a dispersion relationship between frequency and wavelength or between angular frequency and wavevector.

18. Lattice vibrations of 1D crystalChain of identical atoms Atoms interact with a potential V(r) which can be written in Taylor’s series. V(R) Repulsive 0 r0=4 Attractive R r This equation looks like as the potential energy associated of a spring with a spring constant : min We should relate K with elastic modulus C:

19. Monoatomic Chain • The simplest crystal is the one dimensional chain of identical atoms. • Chain consists of a very large number of identical atoms with identical masses. • Atoms are separated by a distance of “a”. • Atoms move only in a direction parallel to the chain. • Only nearest neighbours interact (short-range forces). a a a a a a Un-2 Un-1 Un Un+1 Un+2

20. Monoatomic Chain • Start with the simplest case of monoatomic linear chain with only nearest neighbour interaction a a Un-1 Un Un+1 • If one expands the energy near the equilibrium point for the nth atom and use elastic approximation,Newton’s equation becomes

21. Monoatomic Chain The force on the nth atom; a a • The force to the right; • The force to the left; Un-1 Un Un+1 • The total force = Force to the right – Force to the left Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies

22. All atoms oscillate with a same amplitude A and frequency ω. Then we can offer a solution; MonoatomicChain Displaced position Undisplaced position

23. MonoatomicChain Equation of motion for nth atom Cancel Common terms

24. MonoatomicChain Maximumvalue of it is 1

25. Monoatomic Chain w • ω versus k relation; C B A k k –л / a 0 л / a 2 л / a • Normal mode frequencies of a 1D chain • The points A, B and C correspond to the same frequency, therefore they all have the same instantaneous atomic displacements. • The dispersion relation is periodic with a period of 2π/a.

26. Monoatomic Chain Note that: • In above equation n is cancelled out, this means that the eqn. of motion of all atoms leads to the same algebraic eqn. This shows that our trial function Un is indeed a solution of the eqn. of motion of n-th atom. • We started from the eqn. of motion of N coupled harmonic oscillators. If one atom starts vibrating it does not continue with constant amplitude, but transfer energy to the others in a complicated way; the vibrations of individual atoms are not simple harmonic because of this exchange energy among them. • Our wavelike solutions on the other hand are uncoupled oscillations called normal modes; each k has a definite w given by above eqn. and oscillates independently of the other modes. • So the number of modes is expected to be the same as the number of equations N. Let’s see whether this is the case;

27. Establish which wavenumbers are possible for our one dimensional chain. Not all values are allowed because nth atom is the same as the (N+n)th as the chain is joined on itself. This means that the wave eqn. of must satisfy the periodic boundary condition which requires that there should be an integral number of wavelengths in the length of our ring of atoms Thus, in a range of 2π/a of k, there are N allowed values of k.

28. Monoatomic Chain What is the physical significance of wave numbers outside the range of ? un x a un x

29. MonoatomicChain un x a un This value of k corresponds to the maximum frequency; alternate atoms oscillate in antiphase and the waves at this value of k are standing waves. White line : Green line : ω-k relation

30. Monoatomic Chain un x a un • The points A and C both have same frequency and same atomic displacements • They are waves moving to the left. • The green line corresponds to the point B in dispersion diagram. • The point B has the same frequency and displacement with that of the points A and C with a difference. • The point B represents a wave moving to the right since its group velocity (dω/dk)>0. w C B A k k π/a 2π/a -π/a ω-k relation (dispertiondiagram) • The points A and C are exactly equivalent; adding any multiple of 2π/a to k does not change the frequency and its group velocity, so point A has no physical significance. • k=±π/a has special significance • θ=90o Bragg reflection can be obtained at k= ±nπ/a x Forthewholerange of k (λ)

31. At the beginning of the chapter, in the long wavelength limit, the velocity of sound waves has been derived as Using elastic properties, let’s see whether the dispersion relation leads to the same equation in the long λ limit. If λ is very long; so

32. Monoatomic Chain • Since there is only one possible propagation direction and one polarization direction, the 1D crystal has only one sound velocity. • In this calculation we only take nearest neighbor interaction although this is a good approximation for the inert-gas solids, its not a good assumption for many solids. • If we use a model in which each atom is attached by springs of different spring constant to neighbors at different distances many of the features in above calculation are preserved. • Wave equation solution still satisfies. • The detailed form of the dispersion relation is changed but ω is still periodic function of k with period 2π/a • Group velocity vanishes at k=(±)π/a • There are still N distinct normal modes • Furthermore the motion at long wavelengths corresponds to sound waves with a velocity given by (velocity formulü)

33. Chain of two types of atom • Two different types of atoms of masses M and m are connected by identical springs of spring constant K; (n-2) (n-1) (n) (n+1) (n+2) K K K K M a) M M m m a b) Un-1 Un Un+1 Un+2 Un-2 • This is the simplest possible model of an ionic crystal. • Since a is the repeat distance, the nearest neighbors separations is a/2

34. Chain of twotypes of atom • We will consider only the first neighbour interaction although it is a poor approximation in ionic crystals because there is a long range interaction between the ions. • The model is complicated due to the presence of two different types of atoms which move in opposite directions. Our aim is to obtain ω-k relation for diatomic lattice Two equations of motion must be written; One for mass M, and One for mass m.

35. Chain of two types of atom M M M m m Un-1 Un Un+1 Un+2 Un-2 • Equation of motionfor mass M (nth): • mass x acceleration = restoring force • Equation of motion for mass m (n-1)th: -1 -1

36. Chain of two types of atom M M M m m Un-1 Un Un+1 Un+2 Un-2 Offer a solution for the mass M For the mass m; -1 • α : complex number which determinesthe relative amplitude and phase of the vibrational wave.

37. Chain of two types of atom For nth atom (M): Cancel common terms

38. Chain of twotypes of atom For the (n-1)th atom (m) Cancel common terms

39. Now we have a pair of algebraic equations for α and ω as a function of k. α can be found as A quadratic equation for ω2 can be obtained by cross-multiplication Chain of twotypes of atom for M for m

40. Chain of twotypes of atom The two roots are;

41. Chain of two types of atom • ω versus k relation for diatomic chain; w A B C k –л / a 0 л / a 2 л / a • Normal mode frequencies of a chain of two types of atoms. • At A, the two atoms are oscillating in antiphase with their centre of mass at rest; • at B, the lighter mass m is oscillating and M is at rest; • at C, M is oscillating and m is at rest. • If the crystal contains N unit cells we would expect to find 2N normal modes of vibrations and this is the total number of atoms and hence the total number of equations of motion for mass M and m.

42. Chain of twotypes of atom w A • As there are two values of ω for each value of k, the dispersion relation is said to have two branches; B Optical Branch C Upper branch is due to the +ve sign of the root. Acoustical Branch k –л / a 0 л / a 2 л / a Lower branch is due to the -ve sign of the root. • The dispersion relation is periodic in k with a period 2 π/a = 2 π/(unit cell length). • This result remains valid for a chain of containing an arbitrary number of atoms per unit cell.

43. Chain of two types of atom • Let’s examine the limiting solutions at 0, A, B and C. • In long wavelength region (ka«1); sin(ka/2)≈ ka/2 in ω-k. Use Taylor expansion: for small x

44. Chain of two types of atom Taking +ve root; sinka«1 (max value of optical branch) Taking -ve root; (min value of acoustical brach) By substituting these values of ω in α(relative amplitude) equation and using cos(ka/2) ≈1 for ka«1 we find the corresponding values of α as; OR

45. Chain of twotypes of atom Substitute into relative amplitude α This solution represents long-wavelength sound waves in the neighborhood of point 0 in the graph; the two types of atoms oscillate with same amplitude and phase, and the velocity of sound is w A Optical B C Acoustical k –π / a 0 π / a 2 π / a

46. Chain of two types of atom Substitute into relative amplitude we obtain, w A This solution corresponds to pointAin dispersion graph.This value of α shows that the two atoms oscillate in antiphase with their center of mass at rest. Optical B C Acoustical k –π / a 0 π / a 2 π / a

47. Chain of two types of atom The other limiting solutions of equation ω2 are for ka= π , i.e sin(ka/2)=1. In this case (C) OR (B) • At max.acoustical point C, M oscillates and m is at rest. • At min.optical point B, m oscillates and M is at rest.

48. Acoustic/Optical Branches • The acousticbranch has this name because it gives rise to long wavelength vibrations - speed of sound. • The opticalbranch is a higher energy vibration (the frequency is higher, and you need a certain amount of energy to excite this mode). The term “optical” comes from how these were discovered - notice that if atom 1 is +ve and atom 2 is -ve, that the charges are moving in opposite directions. You can excite these modes with electromagnetic radiation (ie. The oscillating electric fields generated by EM radiation)

49. Transverse optical mode for diatomic chain Amplitude of vibration is strongly exaggerated!

50. Transverse acoustical mode for diatomic chain