Solid State Physics Lecture 5 Last week: Phys 446: (Ch. 3) • Phonons Solid State Physics / Optical Properties • Today: Einstein and Debye models for thermal capacity Lattice vibrations: Thermal conductivity 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 ∫ r • Crystalline structures. 0 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 between them in a linear regime (Hooke's law): σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • What is Brillouin zone kl kl 2 2 C •Elastic wave 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 atoms per unit cell there are Elastic properties – crystal is considered as continuous anisotropic 3s phonon 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 linear region (small displacements, harmonic potential) • Concept of the phonon density of states Hooke's law: σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • Einstein and Debye models for lattice heat capacity. kl kl Elastic waves∂ 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 atom 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 wave vector 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. Density of states 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 ∂ 2u Specific heat M nα F mβ α 2 = ∑ nα ∂t m,β (Ch. 3.3-3.9) N unit cells, s atoms in each → 3N’s equations Fortunately, have 3D periodicity ⇒ Forces depend only on difference m-n Write displacements as 1 i(q⋅rn −ωt) unαi (x,t) = uαi (q)e 7 Mα 8 substitute into equation of motion, get Phonons • Quantum mechanics: energy levels of the harmonic oscillator are quantized 2 1 mβj iq⋅(rm −rn ) −ω uαi (q) − ∑∑ Fnαi e uβj (q) = 0 • Similarly the energy levels of lattice vibrations are quantized. β , j m Mα M β • The quantum of vibration is called a phonon βj Dαi (q) - dynamical matrix (in analogy with the photon - the quantum of the electromagnetic wave) Allowed energy levels of the harmonic oscillator: −ω 2u (q) + Dβj (q)u (q) = 0 αi ∑ αi βj where n is the quantum number β , j phonon dispersion curves in Ge A normal vibration mode of frequency ω is given by u = Aei(q⋅r−ωt) βj 2 ⇒ Det{Dαi (q) −ω 1}= 0 mode is occupied by n phonons of energy ħω; momentum p = ħq - dispersion relation 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 polarization 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 symmetry 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 the frequency range (ω, ω + dω): L dq q D(ω)dω = 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) Debye model 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 dispersion relation: ω = vq, v is the velocity of sound. 1 n = - Planck distribution Then e=ω kT −1 In this approximation the density of states is given by: Vq2 1 Vq2 1 Vω 2 replace the summation over q by an integral over frequency: D(ω) = = = 2π 2 dω dq 2π 2 v 2π 2v3 Need to know the limits of integration over ω. The lower limit is 0. Then the lattice heat capacity is: How about the upper limit ? Assume N unit cells is the crystal, only one atom in per cell ⇒ the total number of phonon modes is 3N ⇒ 2 3 1 3 Debye ⎛ 6π v N ⎞ 1 3 ⇒ ω = ⎜ ⎟ = 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). Vq2 1 Simplest case of isotropic solid, for one branch: D(ω) = The dashed line is the Debye 2π 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 lead 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.