State Physics Lecture 5 Last week: Phys 446: (Ch. 3) • Solid State Physics / Optical Properties • Today: Einstein and Debye models for thermal capacity Lattice vibrations: Thermal, acoustic, and optical properties HW2 discussion

Fall 2007

Lecture 5 Andrei Sirenko, NJIT 1 2

Material to be included in the test •Factors affecting the diffraction amplitude: Oct. 12th 2007 Atomic scattering factor (form factor): f = n(r)ei∆k⋅rl d 3r reflects distribution of electronic cloud. a ∫ • Crystalline structures. r0 sin()∆k ⋅r In case of spherical distribution f = 4πr 2n(r) dr 7 crystal systems and 14 Bravais lattices a ∫ n 0 ∆k ⋅r • Crystallographic directions dhkl = 2 2 2 1 2 ⎛ h k l ⎞ 2πi(hu j +kv j +lw j ) and Miller indices ⎜ + + ⎟ •Structure factor F = f e ⎜ a2 b2 c2 ⎟ ∑ aj ⎝ ⎠ j • Definition of reciprocal lattice vectors: •Elastic stiffness and compliance. Strain and stress: definitions and relation betweenσ them in a linear regimeε (Hooke's law):

ij = ∑Cijklε kl ij = ∑ Sijklσ kl • What is kl kl 2 2 C •Elastic equation: ∂ u C ∂ u eff • Bragg formula: 2d·sinθ = mλ ; ∆k = G = eff x sound velocity v = ∂t 2 ρ ∂x2 ρ 3 4 • Lattice vibrations: acoustic and optical branches Summary of the Last Lecture In three-dimensional lattice with s per unit cell there are ™ Elastic properties – crystal is considered as continuous anisotropic 3s branches: 3 acoustic, 3s - 3 optical medium • Phonon - the quantum of lattice vibration. ™ Elastic stiffness and compliance tensors relate the strain and the Energy ħω; momentum ħq stress in a linearσ region (small displacements,ε harmonic potential) • Concept of the phonon of states Hooke's law: ij = ∑Cijklε kl ij = ∑ Sijklσ kl • Einstein and Debye models for lattice . kl kl ™ Elastic ∂ 2u C ∂ 2u sound velocity C Debye temperature = eff x v = eff ∂t 2 ρ ∂x2 ρ Low and high temperatures limits of Debye and Einstein models ™ Model of one-dimensional lattice: linear chain of atoms • Formula for thermal conductivity 1 ™ More than one in a unit cell – acoustic and optical branches K = Cvl 3 ™ All crystal vibrational waves can be described by wave vectors • Be able to obtain scattering or frequency from within the first Brillouin zone in reciprocal space geometry and data for incident beam (x-rays, neutrons or light) What do we need? 3D case consideration 5 6 Phonons.

Three-dimensional lattice In simplest 1D case with only nearest-neighbor interactions we had equation of motion solution Vibrations in three-dimensional lattice. ∂ 2u M = F = C(u − u ) + C(u − u ) u(x,t) = Aei(qxn −ωt) Phonons ∂t 2 n n+1 n n−1 n In general 3D case the equations of motion are: Phonon Density of states α 2 α ∂ un m Specific heat β M 2 = Fn ∑ α ∂t m,β (Ch. 3.3-3.9) N unit cells, s atoms in each → 3N’s equations Fortunately, have 3D periodicity ⇒

Forcesα depend only on difference m-n Write displacements as α α 1 i(q⋅rn −ωt) un i (x,t) = u i (q)e 7 M 8 α

β Phonons substitute into equation of motion,β get α α Quantum mechanics: energy levels of the harmonic oscillator are quantized β • 2 1 m j iq⋅(rm −rn ) −ω u i (q) − ∑∑ Fn i e uβj (q) = 0 • Similarly the energy levels of lattice vibrations are quantized. , j m M M  • The quantum of vibration is called a phonon α βj D αi (q) - dynamical matrix (in analogy with the - the quantum of the electromagnetic wave) β α Allowed energy levels of the harmonic oscillator: −ω 2u (q) + β D j (q)u (q) = 0 i ∑ i βj where n is the quantum number , j phonon dispersion curves in Ge A normal vibration mode of frequency ω is given by u = Aei(q⋅r−ωt) βj 2 ⇒ Det{D αi (q) −ω 1}= 0 mode is occupied by n phonons of energy ħω; momentum p = ħq - Number of phonons is given by : 1 3s solutions – dispersion branches (T – temperature) nT(,)ω = e=ω kT −1 3 acoustic, 3s - 3 optical direction of u determines The total vibrational energy of the crystal is the sum of the energies of the individual phonons: (longitudinal, transverse or mixed) (p denotes particular 9 10 Can be degenerate because of phonon branch)

11 12 Density of states Consider 1D longitudinal waves. Atomic displacements are given by: u = Aeiqx Boundary conditions: external constraints applied to the ends Periodic boundary condition: Then eiqL =1 ⇒ condition on the admissible values of q: 2π q = n where n = 0, ±1, ± 2, ... L ω regularly spaced points, spacing 2π/L dω L Number of modes in the dq interval dq in q-space : 2π Number of modes in theω frequency range (ω, ω + dω): ω L dq q D( )d = dq 2π D(ω) - density of states 13 14 determined by dispersion ω = ω(q)

Density of states in 3D case Few notes: Now have Equation we obtained is valid only for an isotropic solid, iqx L iq y L iqz L • Periodic boundary condition: e = e = e =1 (vibrational frequency does not depend on the direction of q)

⇒ l, m, n - integers • We have associated a single mode with each value of q. This is not quite true for the 3D case: for each q there are 3 different modes, one longitudinal and two transverse. Plot these values in a q-space, obtain a 3D cubic mesh • In the case of lattice with basis the number of modes is 3s, number of modes in the spherical where s is the number of non-equivalent atoms. shell between the radii q and q + dq: They have different dispersion relations. This should be taken into account by index p =1…3s in the density of states.

V = L3 – volume of the sample

⇒ Density of states 15 16 Lattice specific heat (heat capacity) dQ Defined as (per mole) C = If constant volume V • assumes that the acoustic modes give the dominant contribution dT to the heat capacity The total energy of the phonons at temperature T in a crystal: • Within the Debye approximation the velocity of sound is taken a constant independent of polarization (as in a classical elastic (the zero-point energy is chosen as the E = ∑ nqp =ω p (q) = 0 continuum) q, p origin of the energy). The dispersionω relation: ω = vq, v is the velocity of sound. 1 n = - Planck distribution Then π e=ω kT −1 In this approximation the ωdensity of states is given by: Vq2 1 Vq2 1 Vω 2 replace the summation over q by an integral over frequency: D( ) = = π = 2 2 d dq 2 2 v 2π 2v3 Need to know the limits of integration over ω. The lower limit is 0. How about the upper limit ? Assume N unit cells is the crystal, only Then the lattice heat capacity is: ω π one atom in per cell ⇒ the total number of phonon modes is 3N ⇒ 2 3 1 3 Debye ⎛ 6 v N ⎞ 1 3 ⇒ = ⎜ ⎟ = v()6π 2n frequency 17 D ⎜ ⎟ 18 Central problem is to find the density of states ⎝ V ⎠

The cutoff wave vector which corresponds to this frequency is The total phonon energy is then

modes of wave vector larger than qD are not allowed - number of modes with q ≤qD where N is the number of atoms in the crystal and xD ≡θD/T exhausts the number of degrees of freedom To find heat capacity, differentiate Then the thermal energy is

where is "3" from ?

So, →

where x ≡ħω/kBT and xD ≡ħωD/kBT ≡θD/T

In the limit T >>θD, x << 1, ⇒ Cv = 3NkB - Dulong-Petit law Debye temperature:

19 20 Einstein model Opposite limit, T <<θD : let the upper limit in the integral xD →∞ The density of states is approximated by a δ-function at some ω : Get E D(E) = Nδ(ω – ωE) where N is the total number of atoms – simple model for optical phonons

⇒ within the Debye model at low Then the thermal energy is temperatures C ∝ T3 v The heat capacity is then

The Debye temperature is normally determined by The high temperature limit is the same as that for the Debye model: fitting experimental data.

Cv = 3NkB - the Dulong-Petit law Curve Cv(T/θ) is universal – it is the same for -ħω/k T 3 At low temperatures C ~e B - different from Debye T law different substances v Reason: at low T acoustic phonons are much more populated ⇒ the 21 22 Debye model is much better approximation that the Einstein model

Summary Real density of vibrational states is much more complicated than those described by the Debye and Einstein models. ™ In three-dimensional lattice with s atoms per unit cell there are 3s phonon branches: 3 acoustic, 3s - 3 optical This density of states must be taken into account in order to obtain quantitative description of experimental data. ™ Phonon - the quantum of lattice vibration. Energy ħω; momentum ħq ™ Density of states is important characteristic of lattice vibrations; The density of states for Cu. It is related to the dispersion ω = ω(q). ω Vq2 1 Simplest case of isotropic solid, for one branch: D( ) = π The dashed line is the Debye 2 2 dω dq approximation. ™ Heat capacity is related to the density of states. The Einstein approximation ™ Debye model – good when acoustic phonon contribution dominates. would give a delta peak at At low temperatures gives C ∝ T3 some frequency. v ™ Einstein model - simple model for optical phonons (ω(q) is constant)

™ At high T both models to the Dulong-Petit law: Cv = 3NkB

23 ™ Real density of vibrational states is more complicated 24 Thermal Conductivity Elementary kinetic considerations: Temperature gradient in a material → heat flow from the hotter to the cooler if c is the heat capacity of the single particle, then moving from region end. with T+∆T to a T, particle will give up energy c∆T Heat current density j (amount of heat flowing across unit area per unit time) is proportional to the temperature gradient ( ): ∆T between the ends of the free path length l : dT dT dT/dx x ∆T = l = v τ dx x dx x dT where τ is the average time between collisions ju = −K K - thermal conductivity dx The net energy flux (n – concentration) : • In the heat is carried both by and phonons; τ contribution is much larger dT 1 dT j = −n v 2 c = − n v2 cτ • In insulators, there are no mobile electrons ⇒ heat is transmitted entirely u x dx 3 dx by phonons Heat transfer by phonons for phonons, v is constant. nc = C; l=vτ •phonon : in every region of space there are phonons traveling randomly in all directions, much like the molecules in an ordinary gas 1 dT 1 - phonon thermal •phonon concentration is larger at the hotter end →they move to the cooler end ⇒ j = − Cvl ⇒ K = Cvl u 3 dx 3 conductivity •the advantage of using this gas model: can apply familiar concepts of the 25 26 kinetic theory of

Dependence of the thermal conductivity on temperature Suppose that two phonons of vectors q1 and q2 collide, and produce a third phonon of vector q3. – Cv dependence on temperature has already been discussed Momentum conservation: q = q + q – Sound velocity v essentially insensitive to temperature 3 1 2 q3 may lie inside the Brillouin zone, or not. If it's inside → – The mean free path l depends strongly on temperature momentum of the system before and after collision is the same.

Three important mechanisms are to be considered: This is a normal process. It has no effect at all on thermal resistivity, since it has no effect on the flow of the phonon system as a whole. (a) collision of a phonon with other phonons (b) collision of a phonon with imperfections in the crystal If q3 lies outside the BZ, we reduce it to (c) collision of a phonon with the external boundaries of the crystal equivalent q4 inside the first BZ: q3 = q4 + G The phonon-phonon scattering is due to the anharmonic interaction. Momentum conservation: q1 + q2 = q4 + G If interatomic forces are purely harmonic – no phonon-phonon interaction. The difference in momentum is transferred At high temperature atomic displacements are large ⇒ stronger to the center of mass of the lattice. anharmonism ⇒ phonon-phonon collisions become more important This type of process is is known as the umklapp process At high T the mean free path l ∝ 1/T : number of phonons n ∝ T at high T • highly efficient in changing the momentum of the phonon 27 28 collision frequency ∝ n ⇒ l ∝ 1/n • responsible for phonon scattering at high temperatures The second mechanism - phonon scattering results from defects. Anharmonism Impurities and defects scatter phonons because they partially destroy the periodicity of the crystal. At very low T, both phonon-phonon and phonon-defect collisions become So far, lattice vibrations were considered in harmonic ineffective: approximation. Some consequences: – there are only a few phonons present, – Phonons do not interact; no decay low T: – the phonons are long-wavelength ones ⇒ K ~ T3 – No thermal expansion not effectively scattered by defects, due to Cv which are much smaller in size – Elastic constants are independent of and temperature In the low-temperature region, the primary – Heat capacity is constant at high T (T>>θ ). scattering mechanism is the external D boundary of the specimen - so-called size or geometrical effects. high T: Anharmonic terms in potential energy: Becomes effective because the phonon K ~ 1/T 2 3 4 wavelengths are very long - comparable due to l U(x) = cx –gx –fx x - displacement from equilibrium to the size of the sample L. separation at T = 0 The mean free path here is l ~ L Thermal conductivity of NaF ⇒ independent of temperature. (highly purified) 29 30

Thermal expansion Techniques for probing lattice vibrations Calculate average displacement using Boltzmann distribution function: • Inelastic X-ray scattering ∞ xe−U (x) kBT dx • ∫−∞ x = ∞ −U (x) k T Infrared e B dx • ∫−∞ • Brillouin and Raman scattering If anharmonic terms are small, can use Taylor expansion for exponent Inelastic X-ray scattering 4 5 ∞ 2 π −cx k T ⎛ gx fx ⎞ 1 2 k assumed Ω(q)<<ω dxe B ⎜ x + + ⎟ 3 g 3 2 k = k ± q =ω = =ω ± =Ω(q) 0 ∫−∞ ⎜ ⎟ 5 2 ()kBT q 0 0 - true for x-rays: ⎝ kBT kBT ⎠ 3g x = = 4 c = k T θ 4 ∞ 1 2 2 B ω0 ħΩ < 100 meV; ħω0 ~10 eV −cx2 kBT k0 dxe ⎛ π ⎞ 1 2 4c q = 2k0 sinθ = 2n sinθ n – index of refraction ∫−∞ ⎜ ⎟ ()kBT c ⎝ c ⎠ measuring ω - ω0 and θ sin one can determine dispersion Ω(q) thermal expansion main disadvantage – difficult to measure ω - ω0 accurately Origin of thermal expansion – asymmetric potential This difficulty can be overcome by use of neutron scattering 31 Energy of "thermal" neutrons is comparable with ħΩ (80 meV for 32λ≈1Å) Brillouin and Raman spectroscopy Raman scattering in crystalline Inelastic light scattering mediated by the electronic polarizability of the medium Not every crystal lattice vibration can be probed by Raman • a material or a molecule scatters irradiant light from a source scattering. There are certain Selection rules: Most of the scattered light is at the same wavelength as the source • 1. Energy conservation: (elastic, or Raileigh scattering) =ω = =ω ± =Ω; • but a small amount of light is scattered at different wavelengths (inelastic, i s or Raman scattering) 2. Momentum conservation: Elastic 4πn ks k ћΩ I q ≈ 0 s q (Raileigh) ki = k s ± q ⇒ 0 ≤ q ≤ 2k ⇒ 0 ≤ q ≤ ћΩ β λ ki Scattering i ki α α q ≈ 2k β ћωs ks ki λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0 ћω Anti- ћωi s ћω Stokes i Stokes ω Stokes ωi Anti-Stokes 0 0 ⇒ only small wavevector (cloze to BZ center) phonons are seen in Raman Raman st Scattering Scattering the 1 order (single phonon) Raman spectra of bulk crystals Raileigh ωi- Ω(q) ωi+ Ω(q) Analysis of scattered light energy, polarization, relative intensity 3. Selection rules determined by crystal symmetry 33 34 provides information on lattice vibrations or other excitations

Summary 1 ™ Phonon thermal conductivity K = Cvl 3 ™ Mechanisms of phonon scattering affecting thermal resistivity: • umklapp processes of phonon-phonons collision – important at high T • collision of a phonon with defects and impurities in the crystal • collision of a phonon with the external boundaries of the crystal – important at low T SOME USEFUL SLIDES ™ Anharmonism of potential energy is responsible for such effects as: FROM Physics-I • phonon-phonon interaction • thermal expansion ™ Techniques for probing lattice vibrations: Inelastic X-ray scattering, Neutron scattering, Infrared spectroscopy, Brillouin and Raman scattering Electron energy loss spectroscopy

35 36 Avogadro’s number and Ideal Gases The ; Mole

Ideal Gases Avogadro's Number

Ideal Gas at Constant Temperature

R is a gas constant” M is molar mass; m is molecular mass 37 38

Pressure, Temperature, and The Distribution of Molecular Speeds Speed of molecules

mNA is the molar mass M Root-mean-square speed

39 40 View at the molecular theory of an Heat Capacity

Specific Heat

41 42

Heat Transfer Mechanisms

• Convection • Radiation • Conduction

Q=kA(∆T/L)t, Where k is the thermal conductivity

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