PHONON HEAT CAPACITY

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1 Solid State Physics PHONON HEAT CAPACITY Lecture 11 A.H. Harker Physics and Astronomy UCL

2 4.5 Experimental Specific Heats Element Z A C p Element Z A C p J K 1 mol 1 J K 1 mol 1 Lithium Rhenium Beryllium Osmium Boron Iridium Carbon Platinum Sodium Gold Magnesium Mercury Aluminium Thallium Silicon Lead Phosphorus Bismuth Sulphur Polonium

3 Classical equipartition of energy gives specific heat of 3pR per mole, where p is the number of atoms in the chemical formula unit. For elements, 3R = J K 1 mol 1. Experiments by James Dewar showed that specific heat tended to decrease with temperature. 3

4 Einstein (1907): If Planck s theory of radiation has hit upon the heart of the matter, then we must also expect to find contradictions between the present kinetic molecular theory and practical experience in other areas of heat theory, contradictions which can be removed in the same way. 4

5 Einstein s model If there are N atoms in the solid, assume that each vibrates with frequency ω in a potential well. Then E = N n ω = N ω ek ω, B T 1 and ( ) E C V = T Now so and ( T ( ) T C V = Nk B ( ω k B T V ) ω k B T = ω k B T 2, ek ω B T = ω ω k B k B T 2e T, ) 2 ek ω B T ( ) ek ω 2. B T 1 5

6 Limits C V = Nk B ( ω k B T ) 2 ek ω B T ( ) ek ω 2. B T 1 T. Then ω k B T 0, so e ω k B T 1 and e ω k B T 1 ω k B T, and C V Nk B ( ω k B T This is the expected classical limit. ) 2 ( 1 ) 2 = Nk ω B. k B T 6

7 Limits C V = Nk B ( ω k B T T 0. Then e ω k B T >> 1 and C V Nk B ( ω k B T ) 2 ( e ) 2 ek ω B T ( ) ek ω 2. B T 1 e ω k B T ω k B T k B T. ) 2 T 2 e ω Convenient to define Einstein temperature, Θ E = ω/k B. 7

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9 Einstein theory shows correct trends with temperature. For simple harmonic oscillator, spring constant α, mass m, ω = α/m. So light, tightly-bonded materials (e.g. diamond) have high frequencies. But higher ω lower specific heat. Hence Einstein theory explains low specific heats of some elements. 9

10 Walther Nernst, working towards the Third Law of Thermodynamics (As we approach absolute zero the entropy change in any process tends to zero), measured specific heats at very low temperature. 10

11 Systematic deviations from Einstein model at low T. Nernst and Lindemann fitted data with two Einstein-like terms.einstein realised that the oscillations of a solid were complex, far from single-frequency. Key point is that however low the temperature there are always some modes with low enough frequencies to be excited. 11

12 4.6 Debye Theory Based on classical elasticity theory (pre-dated the detailed theory of lattice dynamics). 12

13 The assumptions of Debye theory are the crystal is harmonic elastic waves in the crystal are non-dispersive the crystal is isotropic (no directional dependence) there is a high-frequency cut-off ω D determined by the number of degrees of freedom 13

14 4.6.1 The Debye Frequency The cut-off ω D is, frankly, a fudge factor. If we use the correct dispersion relation, we get g(ω) by integrating over the Brillouin zone, and we know the number of allowed values of k in the Brillouin zone is the number of unit cells in the crystal, so we automatically have the right number of degrees of freedom. In the Debye model, define a cutoff ω D by N = ωd 0 g(ω)dω, where N is the number of unit cells in the crystal, and g(ω) is the density of states in one phonon branch. 14

15 Taking, as in Lecture 10, an average sound speed v we have for each mode g(ω) = V ω 2 2π 2 v 3, so N = ωd 0 = V ω D 6π 2 v 3 V ω 2 2π 2 v 3 dω 3 ω 3 D = 6Nπ2 V v3 Equivalent to Debye frequency ω D is Θ D = ω D /k B, the Debye temperature. 15

16 4.6.2 Debye specific heat Combine the Debye density of states with the Bose-Einstein distribution, and account for the number of branches S of the phonon spectrum, to obtain C V = S ω D 0 Simplify this by writing and x = ω k B T, C V = S xd 0 V 2π 2 ω2 v 3k B V k 3 = Sk B 2π 2 B T 3 3 v 3 ( ω k B T ) 2 ( e e ω k B T ω k B T 1 ) 2 dω. so ω = k BT x, x D = ω D k B T, V k 2 2π 2 B T 2 x 2 2 v 3 k Bx 2 e x k B T (e x 1) 2 xd 0 x 4 e x dx (e x 1) 2dx 16

17 xd V k 3 C V = Sk B 2π 2 B T 3 x 4 e x 3 v 3 0 (e x 1) 2dx. Remember that ω 3 D = 6Nπ2 V v3, so V 2π 2 v 3 = 3N ω 3 = 3N 3 D kb 3 Θ D 3 and C V = Sk 3N 3 k 3 B kb 3 Θ D 3 B T 3 xd x 4 e x 3 0 (e x 1) 2dx, = 3NSk B T 3 Θ D 3 xd 0 x 4 e x (e x 1) 2dx. As with the Einstein model, there is only one parameter in this case Θ D. 17

18 Improvement over Einstein model. Debye and Einstein models compared with experimental data for Silver. Inset shows details of behaviour at low temperature. 18

19 4.6.3 Debye model: high T xd T 3 x 4 e x C V = 3NSk B Θ 3 D 0 (e x 1) 2dx. At high T, x D = ω D /k B T is small. Thus we can expand the integrand for small x: e x 1, and (e x 1) x so xd x 4 e x 0 (e x 1) 2dx The specific heat, then, is xd 0 C V 3NSk B T 3 Θ D 3 x 2 dx = x3 D 3. x 3 D 3, 19

20 but so C V 3NSk B T 3 Θ D 3 x D = ω D k B T = Θ D T C V NSk B. x 3 D 3, This is just the classical limit, 3R = 3N A k B per mole. We should have expected this: as T, C V k B for each mode, and the Debye frequency was chosen to give the right total number of oscillators. 20

21 4.6.4 Debye model: low T xd T 3 x 4 e x C V = 3NSk B Θ 3 D 0 (e x 1) 2dx. At low, x D = ω D /k B T is large. Thus we may let the upper limit of the integral tend to infinity. so 0 x 4 e x 4π4 (e x 1) 2dx = 15 T 3 4π 4 C V 3NSk B Θ 3 D 15 For a monatomic crystal in three dimensions S = 3, and N, the number of unit cells, is equal to the number of atoms. We can rewrite this as ( ) T 3 C V 1944 which is accurate for T < Θ D /10. Θ D 21

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23 4.6.5 Successes and shortcomings Debye theory works well for a wide range of materials. But we know it can t be perfect. 23

24 Roughly: only excite oscillators at T for which ω k B T/. So we expect: Very low T: OK Low T: real DOS has more low-frequency oscillators than Debye, so C V higher than Debye approximation. High T: real DOS extends to higher ω than Debye, so reaches classical limit more slowly. 24

25 Use Debye temperature Θ D as a fitting parameter: expect: Very low T: good result with Θ D from classical sound speed; Low T: rather lower Θ D ; High T: need higher Θ D. 25

Lecture 12: Phonon heat capacity

Lecture 12: Phonon heat capacity Review o Phonon dispersion relations o Quantum nature of waves in solids Phonon heat capacity o Normal mode enumeration o Density of states o Debye model Review By considering