Solid State Theory Physics 545

The lattice specific heat Statistical of :

Kinetic energy Introduction of structured solids Lawwo of D uoulon g an d P etit (H eat 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 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 -

At low temperatures, all materials exhibit a decrease of their specific heat Classical harmonic oscillator Î Quantum + Normal modes and • Description of lattice vibrations has so far been purely classical because we solved classical equations of motion to find the vibrational modes and 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 vector k and polarisation s and quantised energy:

⎛ 1 ⎞ 1 E = n + = ω k , n = k , s ⎜ k , s ⎟ s () k , s = ω ()k / k T ⎝ 2 ⎠ e s B − 1 where n is the number of phonons in the mode k,s. A is a bosonic particle with wave vector k and 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

• 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 . 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

⎛⎞ =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 = π s V ∑∫ ∂T 8 3 =ω ()k s exps − 1 kBT Density of Phonon States in 1D

AliA linear ch ai n 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 : ω L 1 D( ) = π 2 dω / dK in 3D

2π 4π Nπ K , K , K = 0;± ;± ;...;± x y z L L L

N3: # of atoms

Kz

Ky

Kx 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⋅= δωω dk( - ) = dkdS = 333∫∫∫∫∫s n 888∇k (k) 1 dS D ω =πππ () 3 ∫∫ 8π ∇kω (k) and for any function Q(ω (k)) π s dk QdDQω k =⋅ωω ⋅ ω ω ∑ 3 ( s ()) ∫ () () s 8 ω ∂ =ω (k) CDd= ⋅⋅()ω ω Hence V ∂T ∫ = ()k exp− 1 kBT ω Lattice Specific Heat ω ω 1 ⎡ ⎤ p: polarization(LA,TA, LO, TO) El = ∑ ∑ ⎢ n()K, p + ⎥=ωK, p p K ⎣ 2⎦ K: wave vector ω

⎡ 1⎤ dK ⎡ 1⎤ ω 4πK 2dK E = n()+ = = n()+ = 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 ). 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 ( including 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

c c = v = 11 s [100]L ρ

where c11 iliis an elastic constant and ρ the density ππ Low Temperature Limit:

1sin1π kkk2222θ 1 1ω Dd(ωθθ)= =−==[ cos ] 22∑∑∑∑∫ 0 223 ()22ssss0 ccs ()sss22ππ cc

⎡ ⎤ ω2 1 1 1 3ω2 π = ⎢ + + ⎥ = 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 ) 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 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 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 ) Dn(ω)=−3 δω( ω ) E ∂ =ω (k) CDd= ⋅⋅(ωω) V ∂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. Einstein Model comparison with experiment

The val ue of θΕ = 1325 K was given to produce an agreement with the experiment at 331 ,1 K .

θΕ or ωΕ = kθΕ/∋ is the parameter that distinguishes different substances: 1/2 ωΕ ≈Α (a Ε/m)

where EiE is Y oung’smodldulus m is atomic mass Comparison of the observed molar and a is the lattice parameter heat capacity of diamond (+) with Einstein model gives also a Einstein’s model. (After Einstein’s qualitatively quite good agreement original paper-1907) on term of θΕ calculated from the elastic properties Einstein Model results and limitation

The Einstein Model of crystals takes into account the alteration of the heat capacity by: - temperature - chemical element - crystal structure This model explained the decrease of the heat capacity at low temperature.

However: This decrease is too fast! The experimental results evolve as T3

Reason is that the Einstein model does not consider the collective motion and only consider one vibrational frequency. Born and von Kármán approach

System of N atoms possess 3N degrees of freedom, all expressing vibrational motion. Thus, the whole crystal has 3N normal modes of vibration characterized by their frequencies υi =ωi/2π

THE LATTICE VIBRATIONS OF THE CRYSTAL ARE EQUIVELENT TO 3N INDEPENDENT OSCILLATORS

3N (hυ /kt) -1 E = Σ hυi (1/2 + (e i -1) ) Propagation of sound wave in solids notion

This propagation could be solved using the classical concepts since the atomic structure (dimensions) can be ignored in comparison to the wavelength of a sound wave.

The 3-D ∇2 φ(r) + k2 φ(r) = 0 where: k is the magnitude of the wave vector k = 2π/λ

Wave phase velocity v = λ υ =λ ω/2π = ω/k Propagation of sound wave in solids standing waves in a box

The 3-D wave equation of motion solved in a cubic box with the side L

Φn1 n2 n3 (r) = A sin(n1π x / L) sin(n2π y / L) sin(n3π z / L)

The wave vector in the Cartesian coordinates is k(πn1/L, πn2/L, πn3/L)

In the k space, formed by the allowed values of k(ni = 1, 2, ...), is composed of cubic point lattice with the separation of π/L and the 3 volume of Vu= (π/L) . Propagation of sound wave in solids Density of states Defining the density of states come to the determination of the number of normal modes of standing waves with the lying magnitude between k and k+dk. f(k) dk = (1/8) (4 πk2)dk/() dk /(π/L)3 = Vk2 dk/(2π2)

In term of circular frequency: f(k) dk = f(ω) dω =(Vk= (Vk2/2π2) (dk/dω)d) dω 2 2 2 = V ω dω /(2 v vg π )

Where vg= dω/dk ihis the group vel liocity Propagation of sound wave in solids Density of states

In a non dispersing medium v g =v= v

f(k) dk = f(ω)d) dω = V ω2 dω /(2 v 3 π2)

The wave vector has three independent modes : 1 longitudinal and 2 transversal modes

2 2 3 3 f(ω) dω = V ω dω /(2 π ) (1/vL + 2/vT ) In an itisotropi iMdic Medium vL = vT =vm

2 3 2 f(ω)d) dω =3V= 3V ω dω /(2 v m π ) Debye Model

Lattice vibrations are regarded as standing waves of the atomic planes´ displacement

It is assumed that all normal mode frequencies satisfy the equation ofthf the d ensit y of st at es

An upper limit for frequencies is, however, set such as ƒωD f(ω) dω = 3N 2 3 Î f(ω)d) dω = 9Nω dω/ωD

Now the sum can be replaced with an integral

3N Σ ..... = ƒωD .....f(ω)d) dω Debye Model

Debye Approximation: ω = v K ω s ω = vsK

2 2 ency,

g (ω) dg ω uu Debye Density of D(ω)= π = States 2 2 dω 2π 2v3

s Freq 4 πK 3 N = 3 D Number of Atoms: 3 ()2π L 1 2 3 0 Wave vector, K π/a Debye cut-off Wave Vector KD = (6π η) Debye Temperature [K] Deby e Cu t-off Freq . ω = v K D ω s D C(dimnd) 1860 Ga 240 Si 625 NaF 492 1 π 2 3 Ge 360 NaCl 321 = D =vs (6 η) Debye Temperature θ D = = B 1250 NaBr 224 kB kB Al 394 NaI 164 Debye Model The energy of the crystal

3N

U = Σ1 εi 3N ħωi = Σ [(1/2)ħω + ⎯ħω⎯i ⎯ ] 1 i e kT -1 ω D ħω = ƒ [(1/2)ħω + ⎯⎯⎯ħω ] D(ω) dω 0 e kT -1

x 9 9 D x3 = ⎯ NkθD + ⎯ NkT ƒ ⎯⎯x dx 3 0 8 xD e -1

Where θD =ħωD/k xD = ħωD/kT x = ħω/kT Debye Model The heat capacity of the crystal 9 9 xD x3 E = Nkθ + NkT dx D 3 ∫ x 8 xD 0 e −1

TÆ∝ TÆ0 XÆ 0 XÆ∝ 3 xD /3 π4/15 ÎE=9NkE = 9NkωD/8 + 3NkT ÎE = 9NħωD/8 Vibrational zero-ppgyoint energy Î Cv = 3Nk

Î Cv = dE/dT]v = 0 Debye Model The heat capacity at low temperature

Cv =(dU/dT) v T enters this expression only in the exponential term (β) x D 4 x Cv= 3Nk {⎯3 ⎯x⎯e⎯ dx } 3 ƒ x 2 x 0 (e -1) D x D 4 x x e 4 When T<<θD Î ƒ ⎯⎯⎯x 2 dx = 4π /15 0 (e -1) 3 124 T Î CV = ⎯ π NkN k ⎯ 5 θD Debye Model-Experiment

The Debye Model gives good fits to the experiment; however, it is only an interpolation formula between two correct limits (T = 0 and infinite) Latticeω Specific Heat ω θ ω ⎡ D ⎤ =ω D ⎡ 1⎤ =π 3 ω ⎢η9 k T x3dx ⎥ Energy Density ∈ = n()+ d = B T 4 x = l ∑ ∫ ⎢ 2⎥ 2 3 ⎢ 3 ∫ x ⎥ kBT p 0 ⎣ ⎦ 2 vs ⎢ θ D 0 e −1⎥ ⎣ ⎦ 3 θD η ⎡ T x 4 ⎤ Specific Heat ⎛ T ⎞ ⎢ e x dx ⎥ Cl = 9 kB ⎜ ⎟ ⎜θ ⎟ ⎢ ∫ x 2 ⎥ ⎝ D ⎠ 0 (e −1) ⎣ ⎦ 10 7 C = 3ηk = 4.7 ×106 J B m3 −K

10 6

3 5 Diamond When T << θD, 10

4 3 -K) (J/m ∈l ∝ T , Cl ∝ T 10 4

C ∝ T 3 10 3 pecific Heat, C SS Classical 10 2 Regime Quantum θD =1860 K

10 1 Regime 10 1 10 2 10 3 10 4 Temperature, T (K) Einstein-Debye Mode ls Lattice structure of Al Cubic Closest Packing

ΘE / Θelst = 0790,79

ΘD / Θelst = 0,95

The lattice parameter a = 0,25 nm The density ρ=2,7 g/cm3 The wave velocity v =3,4 km/s

ÎΘelst Lattice parameter

Cubic close packed, (a) Hexagonal close Body centered cubic packed (a, c) (a) Cu (3.6147) Be (2.2856, 3.5832) Fe (2.8664) Ag (4.0857) Mg (3.2094, 5.2105) Cr (2.8846) Au (4.0783) Zn (2.6649, 4.9468) Mo (3.1469) Al (4. 0495) Cd (2. 9788, 5. 6167) W (3.1650) Ni (3.5240) Ti (2.506, 4.6788) Ta (3.3026) Pd (3.8907) Zr (3.312, 5.1477) Ba (5.019) Pt (3. 9239) Ru (2.7058 , 4. 2816) Pb (4.9502) Os (2.7353, 4.3191) Re (2.760, 4.458) Debye Temperature

x 9 9 D x3 Ε = ⎯ NkθD + ⎯ NkT ƒ ⎯⎯x dx 3 0 8 xD e -1

Where θD =ħωD/k x D = ħωD/kT x = ħω/kT 3 12 4 T CV = ⎯ π NkN k ⎯ 5 θD The limit of the Debye Model

Î The electronic contribution to the heat cappyacity was not considered Electronic contribution Fermi level

At temperature, electrons pack into the lowest available energy, respecting the Pauli exclusion principle “each quant um st ttate can h ave one b btut onl y one parti til“cle“

Electrons build up a Fermi sea , and the surface of this sea is the Fermi Level. Surface fluctuations (ripples) of this sea are induced by the electric and the thermal effects.

So, the Fermi level, is the highest energetic occupied level at zero abltbsolute

41 Electro nic co ntribu tio n Fermi function

The Fermi function f(E), drown from the Fermi-Dirac statistics, express the probability that a given electronic state will be occupied at a given temperature .

1.0 EEE-E < 0 F ) 0.5 f(E

E-E > 0 F 0.0 0 200 400 600 800 1000 Temperature Electronic contribution to the internal energy

Orbitals are filled starting from the lowest levels, and the last filled or orbital will be characterized by the Fermi wave vector K F The total number of electron in this outer orbital is: L3 K F N = 2 f (k)dk =2 k 2dk T ∫ 2 ∫ Because electrons can 3π 0 Adopt 2 spin orientations

V 3 = 3 N T 3 3π 2 N = k F T 3π 2 k F V Electronic contribution to the internal energy

The wavefunction of free electron is: Ψ(x,t) = Aei(kx+ω.t) Its substitution in the Schrödinger equation:

2 2 2 = ∂ Ψ(x,t) EΨ(x,t) = ∇ Ψ(x,t) = − 2 2 2m ∂x = 2 Î E = k 2m Electronic contribution to the internal energy

Fermi Energy 2 2 2 = 2 = ⎡ N ⎤ 3 2 2 = = 3 T [3π ] E F 2m k F 2m ⎣⎢ V ⎦⎥

Fermi Temperature E = F T F k Tempe ratu re e ffec t o n e lec tro ns

Metal K Na Li Au Ag Cu

NT /V (10^22 cm^3) 1.34 2.5 4.6 5.9 5.8 8.5

KF (1/A°) 0.73 0.9 1.1 1.2 1.2 1.35

EF (eV) 212.1 313.1 474.7 555.5 555.5 7

TF (K) 24400 36400 54500 64000 64000 81600

Only electrons near from FillFermi level are aff ffdbhected by the temperature. Electronic contribution in the heat capacity of a metal ∂ C e = [E e ] v ∂T N ,V ∞ e ∂ ⎡ ⎤ Cv = ⎢2∑ Ei f (Ei )⎥ ∂T ⎣ i=1 ⎦ N ,V

∞ e ∂ ⎡ ⎤ Cv = ⎢2 Ei f (Ei )dN⎥ ∂T ∫ ⎣ 0 ⎦ N ,V 3/ 2 e V ⎛ 2m ⎞ 1/ 2 2 Cv = ⎜ ⎟ EF k T 2π 2 ⎝ =2 ⎠ Summary

The nearest model describing the thermodynamic properties of cryypstals at low temperatures is the one where the ener gy is calculated considering the contribution of the lattice vibrations in the Debye approach and the contribution of the electronic motion (t his i s of i mportance wh en metal s are studi ed) . 3 Cv = α.T +γ .T Summary

Terms that replaced the partition function are: Density of state (collective motion) Fermi function (()electronic contribution)