Solid State Physics PHONON HEAT CAPACITY

Solid State Physics

A.H. Harker Classical equipartition of energy gives speciﬁc heat of 3pR per mole, Physics and Astronomy where p is the number of atoms in the chemical formula unit. For UCL elements, 3R = 24.94 J K−1 mol−1. Experiments by James Dewar showed that speciﬁc heat tended to decrease with temperature.

4.5 Experimental Speciﬁc Heats

Element Z A Cp Element Z A Cp JK−1mol−1 JK−1mol−1 Lithium 3 6.94 24.77 Rhenium 75 186.2 25.48 Beryllium 4 9.01 16.44 Osmium 76 190.2 24.70 Boron 5 10.81 11.06 Iridium 77 192.2 25.10 Carbon 6 12.01 8.53 Platinum 78 195.1 25.86 Sodium 11 22.99 28.24 Gold 79 197.0 25.42 Magnesium 12 24.31 24.89 Mercury 80 200.6 27.98 Aluminium 13 26.98 24.35 Thallium 81 204.4 26.32 Einstein (1907): “If Planck’s theory of radiation has hit upon the Silicon 14 28.09 20.00 Lead 82 207.2 26.44 heart of the matter, then we must also expect to ﬁnd contradictions Phosphorus 15 30.97 23.84 Bismuth 83 209.0 25.52 between the present kinetic molecular theory and practical experi- Sulphur 16 32.06 22.64 Polonium 84 209.0 25.75 ence in other areas of heat theory, contradictions which can be re- moved in the same way.”

2 4 Einstein’s model If there are N atoms in the solid, assume that each Limits vibrates with frequency ω in a potential well. Then ~ω 2 k T N~ω ~ω e B E = Nhni~ω = , C = Nk . ~ω V B ω 2 k T kBT ~ e B − 1 ekBT − 1 and ∂E ~ω CV = T → 0. Then ekBT >> 1 and ∂T V ~ω ~ω Now 2 kBT − ~ω e −2 kBT ∂ ω ω CV → NkB 2→ T e . ~ ~ kBT ~ω ! = − 2, k T ∂T kBT kBT e B so ∂ ~ω ω ~ω kBT ~ kBT Convenient to deﬁne Einstein temperature, ΘE = ~ω/kB. e = − 2e , ∂T kBT and ~ω 2 k T ~ω e B CV = NkB 2. kBT ~ω ekBT − 1

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Limits

~ω 2 k T ~ω e B CV = NkB 2. kBT ~ω ekBT − 1

~ω ~ω T → ∞. Then ~ω → 0, so ekBT → 1 and ekBT − 1 → ~ω , and kBT kBT 2 ~ω 1 CV → NkB k T 2= NkB. B ~ω kBT This is the expected classical limit.

6 8 • Einstein theory shows correct trends with temperature. • For simple harmonic oscillator, spring constant α, mass m, ω = pα/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.

Systematic deviations from Einstein model at low T. Nernst and Lin- demann 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.

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4.6 Debye Theory

Walther Nernst, working towards the Third Law of Thermodynam- Based on classical elasticity theory (pre-dated the detailed theory of ics (As we approach absolute zero the entropy change in any process lattice dynamics). tends to zero), measured speciﬁc heats at very low temperature.

10 12 The assumptions of Debye theory are Taking, as in Lecture 10, an average sound speed v we have for each • the crystal is harmonic mode V ω2 g(ω) = , • elastic waves in the crystal are non-dispersive 2π2 v3 so • the crystal is isotropic (no directional dependence) Z ωD V ω2 • there is a high-frequency cut-off ω determined by the number of N = dω D 2π2 v3 degrees of freedom 0 V ω 3 = D 6π2 v3 6Nπ2 ω 3 = v3 D V

Equivalent to Debye frequency ωD is ΘD = ~ωD/kB, the Debye temperature.

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4.6.1 The Debye Frequency 4.6.2 Debye speciﬁc heat

The cut-off ωD is, frankly, a fudge factor. Combine the Debye density of states with the Bose-Einstein distri- If we use the correct dispersion relation, we get g(ω) by integrating bution, and account for the number of branches S of the phonon over the Brillouin zone, and we know the number of allowed values spectrum, to obtain of k in the Brillouin zone is the number of unit cells in the crystal, so ~ω 2 2 k T R ωD V ω ~ω e B we automatically have the right number of degrees of freedom. CV = S 2 3 kB 2dω. 0 2π v kBT ~ω ! In the Debye model, deﬁne a cutoff ωD by ekBT −1 Z ωD N = g(ω)dω, 0 Simplify this by writing where N is the number of unit cells in the crystal, and g(ω) is the ~ω kBT x ~ωD x = , so ω = , xD = , density of states in one phonon branch. kBT ~ kBT and Z xD 2 2 2 x V kBT x 2 e kBT CV = S 2 2 3 kBx 2 dx 0 2π ~ v (ex − 1) ~ 3 3 Z xD 4 x V kBT x e = SkB 2 3 3 2dx 2π ~ v 0 (ex − 1)

14 16 4.6.3 Debye model: high T V k3 T 3 Z xD x4ex B 3 x 4 x CV = SkB 2 3 3 2dx. T Z D x e 2π ~ v 0 (ex − 1) C = 3NSk dx. V B 3 x 2 Remember that ΘD 0 (e − 1) 6Nπ2 At high T, x = ω /k T is small. Thus we can expand the inte- ω 3 = v3, D ~ D B D V grand for small x: so x 3 e ≈ 1, V 3N 3N~ = = and 2π2v3 ω 3 k3 Θ 3 D B D (ex − 1) ≈ x and so 3 k3 T 3 4 x C = Sk 3N~ B R xD x e dx, Z xD x4ex Z xD x3 V Bk3 Θ 3 3 0 x 2 2 D B D ~ (e −1) 2dx ≈ x dx = . 3 4 x x 3 T R xD x e 0 (e − 1) 0 = 3NSkB 3 0 x 2dx. ΘD (e −1) The speciﬁc heat, then, is As with the Einstein model, there is only one parameter – in this case T 3 x3 C ≈ 3NSk D, ΘD. V B 3 ΘD 3

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Improvement over Einstein model. 3 3 T xD CV ≈ 3NSkB 3 , ΘD 3 but ~ωD ΘD xD = = kBT T so CV ≈ NSkB. This is just the classical limit, 3R = 3NAkB per mole. We should have expected this: as T → ∞, CV → kB for each mode, and the Debye frequency was chosen to give the right total number of oscillators.

Debye and Einstein models compared with experimental data for Sil- ver. Inset shows details of behaviour at low temperature.

18 20 4.6.4 Debye model: low T 4.6.5 Successes and shortcomings T 3 Z xD x4ex Debye theory works well for a wide range of materials. C = 3NSk dx. V B 3 x 2 ΘD 0 (e − 1) At low, xD = ~ωD/kBT is large. Thus we may let the upper limit of the integral tend to inﬁnity. Z ∞ x4ex 4π4 2dx = 0 (ex − 1) 15 so T 3 4π4 CV ≈ 3NSkB 3 ΘD 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 CV ≈ 1944 ΘD But we know it can’t be perfect. which is accurate for T < ΘD/10.

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Roughly: only excite oscillators at T for which ω ≤ kBT/~. So we expect: • Very low T: OK • Low T: real DOS has more low-frequency oscillators than Debye, so CV higher than Debye approximation. • High T: real DOS extends to higher ω than Debye, so reaches classical limit more slowly. 22 24 Use Debye temperature ΘD as a ﬁtting parameter: expect: • Very low T: good result with ΘD from classical sound speed; • Low T: rather lower ΘD; • High T: need higher ΘD.