Solid State Theory Physics 545 The lattice specific heat Statistical thermodynamics of Solids: Kinetic energy Introduction of structured solids Lawwo of Du uolong and Pe tit ( He at c ap ac ity) 1819 Einstein Model of Crystals 1907 Born and von Karman approach 1912 Debye Model of Crystals 1912 Electronic energy Fermi level 1926 Fermi-Dirac distribution Law of Dulong and Petit The crystal stores energy as: - Kinetic energy of the atoms under the form of vibrations. According to the equipartition of energy, the kinetic internal energ y is f½kTf . ½ . k. T where f is the degree of freedom. Each atom or ion has 3 degrees of freedom EK = 3/2 N k T - Elastic potential energy. Since the kinetic energy convert to potential energy and vice versa , the average values are 2 equal Epot = 3 N (½ K x ) = 3 N x (½ k T) Thdlhe stored molar energy i ihs then: E=EE = EK +E+ Epot =3= 3 NA kT=k T = 3 3RT R T Î C=dE/dT=3RC = dE/dT = 3 R Law of Dulong and Petit Within this law, the specific heat is independent of: - temperature - chemical element - crystal structure At low temperatures, all materials exhibit a decrease of their specific heat Classical harmonic oscillator Î Quantum + Statistical mechanics Normal modes and phonons • Description of lattice vibrations has so far been purely classical because we solved classical equations of motion to find the vibrational modes and dispersion relation of the lattice. • In the case of a harmonic potential, the classical approach gives the same modes and dispersion relation as the quantum approach. • Each mode is the mode of vibration of a quantum harmonic oscillator with wave vector k and polarisation s and quantised energy: ⎛ 1 ⎞ 1 E k , s = ⎜ n k , s + ⎟ = ω s ()k , n k , s = = ω ()k / k T ⎝ 2 ⎠ e s B − 1 where n is the number of phonons in the mode k,s. A phonon is a bosonic particle with wave vector k and polarization s • The more phonons in the mode, the greater the amplitude of vibration. Phonon Energy •The linear atom chain can only have N discrete K Æ ω is also discrete Distance • The energy of a lattice vibration mode at frequency ω was found to be ⎛ 1 ⎞ u = ⎜n + ⎟=ω hω ⎝ 2 ⎠ • where ħω can be thought as the energy of a particle called phonon, as an analogue to photon • n can be thought as the total number of phonons with a frequency ω, and follows the Bose-Einstein statistics: 1 n = ⎛ =ω ⎞ Equilibrium distribution exp⎜ ⎟ −1 ⎝ kBT ⎠ Total Energy of Lattice Vibration ⎡ 1⎤ El = ∑ ∑ ⎢ n(ωK, p ) + ⎥=ωK, p p K ⎣ 2⎦ p: polarization(LA,TA, LO, TO) K: wave vector Experimental observations of lattice specific heat preceded inelastic neutron scattering. Model crystal: p atoms per unit cell ) N unit cells ) 3pN harmonic oscillators Thermal energy (quanta) excites cr ystal and an y number of excitations into quantized energy levels for oscillators. BoseBose--EinsteinEinstein Statistics ⎛⎞ 1 Number of excitations n ⎜⎟k = s ⎝⎠ ⎛⎞ =ω ⎜⎟k exps ⎝⎠− 1 iildin particular mode, k, s k T B In harmonic approximation, total energy density ⎛⎞ =ω ⎜⎟k 11⎛⎞ s ⎝⎠ U = 1=ω ⎜⎟k + ∑∑s ⎝⎠ ⎛⎞ VV2 =ω ⎜⎟k kk,s ,sexps ⎝⎠− 1 kT B s: polarization(LA,TA, LO, TO) k: wave vector Specific Heat (at Constant Volume) ⎛ ∂U ⎞ 1 ∂ =ω (k) C = ⎜ ⎟ = s V ⎜ ∂T ⎟ V ∑ ∂T =ω (k) ⎝ ⎠V k,s exp s −1 k BT s: polarization(LA,TA, LO, TO) k: wave vector 8π3 Volume of k-space per allowed k value is Δk = V V F()k = F(k)⋅ Δk ∑ 3 ∑ k 8π k For Δk → 0 (i.e., V → ∞) 1 dk F()k = F()k Lim V ∑ ∫ 3 V→∞ k 8π ∂ dk =ω (k) C = s V ∑∫ 3 s ∂T 8π =ω ()k exps − 1 kBT Density of Phonon States in 1D AliA linear c ha in of N=10 a toms with two ends jointed a Only N wavevectors (K)areallowed() are allowed (one per mobile atom): K= -8π/L -6π/L -4π/L -2π/L 0 2π/L 4π/L 6π/L 8π/L π/a=Nπ/L Only 1 K state lies within a ΔK interval of 2π/L # of states per unit range of K is: L/2π DOS ≡ # of K-vibrational modes between ω and ω+dω : L 1 D(ω) = 2π dω / dK Density of States in 3D 2π 4π Nπ K , K , K = 0;± ;± ;...;± x y z L L L N3: # of atoms Kz Ky K x 2π/L VK 2 1 D(ω) = ; V = L3 2π 2 dω / dK Density of States Define D(ω)suchthatD(ω)dω is total no. of modes with frequencies in range ω to ω +dω per unit volume of crystal. 111ddSω ⋅ Dd()ωω⋅= dk( δωω - ) = dkdS = 333∫∫∫∫∫s n 888πππ∇kω (k) 1 dS D ω = () 3 ∫∫ 8π ∇kω (k) and for any function Q(ωs(k)) dk QdDQω k =⋅ωω ⋅ ω ∑ 3 ( s ()) ∫ () () s 8π ∂ =ω (k) Hence CDdV = ⋅⋅()ω ω ∂T ∫ =ω ()k exp− 1 kBT Lattice Specific Heat 1 ⎡ ⎤ p: polarization(LA,TA, LO, TO) El = ∑ ∑ ⎢ n()ωK, p + ⎥=ωK, p p K ⎣ 2⎦ K: wave vector ⎡ 1⎤ dK ⎡ 1⎤ 4πK 2dK E = n ω + =ω = n ω + =ω l ∑ ∫ ⎢ ()K, p ⎥ K, p 3 ∑ ∫ ⎢ ()K, p ⎥ K, p 3 p ⎣ 2⎦ (2π L) p ⎣ 2⎦ (2π L) Dispersion Relation: K = g(ω ) ⎡ 1⎤ Energy Density: ∈l = ∑ ∫ ⎢ n(ω ) + ⎥=ωD(ω )dω p ⎣ 2⎦ VK 2 1 D(ω) = 2π 2 dω / dK Density of States (Number of K-vibrational modes between ω and ω+dω) d ∈l d n Lattice Specific Heat: Cl = = ∑ ∫ =ωD(ω)dω dT p dT High-Temperature “Classical” Limit: =ω = x « 1 k BT ∂ CkTDdkDdpNk= ⋅()ωω ⋅= () ωω ⋅=3 VB∂T ∫∫ B B which i s th e same as th e cl assi cal resul t ( Dul ong and Peti t l aw: 3 R J/mole/K for a monatomic solid). The reason for this is because at this level of approximation the energy associated with a quantum of lattice vibration, =ω, exactly cancels out and therefore it doesn't matter how biggq that quantum is (includin g)g zero). Low Temperature Limit: Only low-frequency acoustic modes excited. ω ω =ckfhbhhfor each branch, where c = siiiilis initial ss s k slope of the particular phonon dispersion curve. (Note that cs is related to the elastic constant for the mode, e.g., for [100]L elastic waves c c = v = 11 s [100]L ρ where c11 iliis an elastic constant and ρ the density Low Temperature Limit: π 2222 1sin1kkkθ π 1 1ω Ddωθθ= =−==cos ( ) 22∑∑∑∑∫ [ ]0 223 ()22ππssss0 ccs () sss22ππ cc ⎡ ⎤ ω2 1 1 1 3ω2 = ⎢ + + ⎥ = 2π2 ⎢c3 c3 c3 ⎥ 2π2c3 ⎣⎢ L T1 T2 ⎦⎥ ωmax 2 =ω = ∂ =ω 3ω x = dx = ⋅ dω CV = ⋅ ⋅ dω ∂T ∫ =ω 2 3 k BT k BT 0 exp −1 2π c k BT =ωmax 4 kBT 3 2 3 ∂ ()k BT 3 x 2π ⎛ k BT ⎞ CV = ⋅ dx = k B ⎜ ⎟ ∂T 3 2 ∫ x 5 ⎝ =c ⎠ ()=c 2π 0 e −1 at low temperatures the specific heat is proportional to T3. Einstein Model Each molecule in the crystal lattice is supposed to vibrate isotropically about the equilibrium point in a cell delimited by the first neigg,hbors, which are considered frozen. System of N molecules the system can be treated as 3N independent one- dimensional harmonic oscillator Motions in the x, y and z axis are Independent and equivalent Einstein Model System of 1-Dim Harmonic Oscillator Quantized expression of the energy: εv = hυ (v+1/2) v = 0, ,,, 1, 2, ... Partition function (without attributing 0 to the ground state) q = Σ e-(hυ(v+1/2)/kT) = e-(hυ/2kT) Σ e-(hυ/kT) v Considering the vibrational temperature θ = θΕ = hυ/k -θ/2T q = e 1- e-θ/T The molecular internal energy 2 Umolllecular =- d[Ln(q)] / dβ ]NVN, V = kTk T d[Ln(q)] / dT ]NVN, V 1 U = k θ (1/2 + θ ) molecular e /T - 1 Einstein Model energy of the sy stem System of N 3-Dim Harmonic Oscillators 3N Q = q Î U = 3N Umolecular 20000 1 U = 3 N k θ (1/2 +θ ) 18000 e /T - 1 16000gyg 14000 3/2 Nhυ + 3 NkT 12000 al ener nn 10000 3/2 Nhυ 8000 Inter 6000 4000 2000 000.0 050.5 101.0 151.5 202.0 252.5 kT/hυ Einstein Model the heat capacity of the system System of N 3-Dim Harmonic Oscillators The heat capacity of the crystal is then C = dU / dT 3Nk1.0 0.8 pacity 0.6 1 U = 3 N k θ (1/2 + ) θ/T 0.4 e - 1 Heat ca 0.2 θ/T 2 e 0.0 C = 3Nk(θ/T) ––––––θ/T 000.0 050.5 101.0 151.5 202.0 252.5 (e – 1)2 T/θ Einstein Model Assumed model for crystal to be 3n harmonic oscillators, each of ω frequency, E (k BθE = =ωE ) Dnω =−3 δω ω ( ) ( E ) ∂ =ω (k) CDdV = ⋅⋅(ωω) ∂T ∫ =ω (k) Substituting this into Equation exp− 1 kBT 2 ⎛ =ωE ⎞ ⎛ =ωE ⎞ ⎛ θE ⎞ ⎜ ⎟ exp⎜ ⎟ 2 exp⎜ ⎟ ⎝ k BT ⎠ ⎝ k BT ⎠ ⎛ θ ⎞ ⎝ T ⎠ ⎛ θ ⎞ C = 3nk = 3R⎜ E ⎟ = 3R ⋅ F ⎜ E ⎟ V B 2 T 2 E ⎛ ⎛ =ω ⎞ ⎞ ⎝ ⎠ ⎛ ⎛ θ ⎞ ⎞ ⎝ T ⎠ ⎜ ⎜ E ⎟ ⎟ ⎜exp⎜ E ⎟ −1⎟ ⎜exp⎜ ⎟ −1⎟ ⎜ ⎟ ⎝ ⎝ k BT ⎠ ⎠ ⎝ ⎝ T ⎠ ⎠ ⎛ θ ⎞ T » θ , E E FE ⎜ ⎟ → 1, so CV → 3R (the classical high-temperature limit). ⎝ T ⎠ 2 ⎛ θE ⎞ ⎛ θE ⎞ ⎛ θE ⎞ CV dominated by the exponential term, which is not found T « θ , FE ⎜ ⎟ ≈ ⎜ ⎟ exp⎜− ⎟ E ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ experimentally at low temperatures.