Phonons II: Thermal Properties • Heat Capacity of a Crystal • Density of State M.C

Dept of Phys Phonons II: Thermal properties • Heat capacity of a crystal • density of state M.C. Chang • Einstein mode • Debye model • anharmonic effect • thermal conduction

A technician holding a silica fiber thermal insulation tile at 1300 Celsius

See www.youtube.com/watch?time_continue=3&v=Pp9Yax8UNoM Heat capacity, DOS Einstein model, Debye model Anharmonic effect Heat capacity: Heat capacity experiment approaches 3R (~ 25 J/K) at high temperature (Dulong-Petit law, 1819 ) Heat capacity drops to zero at low temperature

After rescaling the temperature by θ (Debye temperature ), which differs from material to material, a universal behavior emerges: Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Debye temperature

In general, a harder material has a higher Debye temperature Heat capacity, DOS Einstein model, Debye model Anharmonic effect

• Dulong-Petit’s law is a result of the equipartition of energy . • Early 1900’s, theory fails at low temperature (for diamond at room temperature , only 1/5 of expected value!) • The classical theory of heat capacity is in trouble, just like the classical theory of thermal radiation.

Planck: Quantization of energy is a mathematical trick. Einstein: Quantization of energy is real. It appears not only in thermal radiation, photoelectric effect, but also in crystal vibration. Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Heat capacity: Quantum theory

• Heat capacity is nothing but the change of U( T) w.r.t. to T: C U/ T V =( ∂ ∂ )V • Internal energy U of a crystal is the summation of vibrational energies (consider an insulator so there’s no electronic energies)

ℏ U() T=∑ ( n ks, + 1/2)ω ks , k, s where s sums over different phonon branc hes (L/T, A/O). • For a crystal in thermal equilibrium, the average phonon number is ( see Kittel, p.107 ) 1 nk, s = ℏ , Bose-Einstein distribution e ωk, s /kT −1 • Therefore, we have ℏ ℏ  ωω ks, ks ,  U( T )= ∑ℏ +  ωk, s / kT k, s e −1 2  Heat capacity, DOS Einstein model, Debye model Anharmonic effect important Connection between summation and integral

b f(x) dxfx( )= lim∑ ∆ xfx ⋅ (i ), or ∫ ∆x → 0 a i b dx ∑ fx(i )≅ ∫ fx ( ). i a ∆x Generalization to 3-dim: a b x d3 x fx()≅ fx () ∑ ∫ 3 x ∆ x d3 k orfk()≅ fk () in solid state ∑ ∫ 3 k ∆ k

2π  3 ∆3k =   L  Heat capacity, DOS Einstein model, Debye model Anharmonic effect important Density of states D(ω) (DOS, 態密度 )

• D(ω)dω is the number of states within the surfaces of constant ω and ω+dω

3 ∫ d k 2π  3 D(ω ω ) d = shell , ∆3k =   ∆3k  L 

d3 k • For example, assume N=16, ∑ ωf()ω ≅ f () × k∫ ∆3k k not for f(k) then there are 2 2=4 states k within the interval dω = ∫ dω ω ω D( ) f ( )

• Once we know the DOS, we can reduce the 3-dim k-integral to a 1-dim ω integral.

Alternative definition: d3 k D ()ω =∫ 3 δ (ωk − ω ) ∆ k • Flatter ω(k) curve, higher DOS. Heat capacity, DOS Einstein model, Debye model Anharmonic effect

DOS: 1-dim dk dk/ d ω ωD(ω ω ) d=2 = 2 d ∆k ∆ k  L 1  for ω ω≤ ∴D(ω ) = π ωd/ dk M  0 otherwise ω ω ω Ex: Calculate D( ) for the 1-dim string with (k)= M|sin(ka /2)| Prob.1(a)

DOS: 3-dim (assume ω(k)= ω(k) is isotropic )

3 3 2 dk4π 2 Lk ω D(ω ω ) d=∫ 3 = 3 k dk = 2 d Shell ∆k ∆ k2π ω d / dk for example, if ω = vk , then D (ω )= V 2 / 2 π 2v 3

Non-dispersive It’s not necessary to memorize the result, just remember the way of deriving it. Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Einstein model (1907 ) Assume that 1. each atom vibrates independently of each other, and ω 2. every atom has the same vibration frequency 0

• The DOS can be written as

D( δω ω ω )= 3 N ( − 0 )

3 dim × number of atoms Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Einstein model

1  ωℏω ℏ UNn=+=3ℏω 3 N0 + 3 N 0   0 ℏ 2  exp(ω0 /kT )− 1 2

2 ℏ ℏ ω0 /kT ω0  e CV=∂( U / ∂ T ) V = 3 Nk   ℏω /kT 2 kT  ()e 0 −1 ℏ ≈e− ω0 /kT as T → 0 K

vibration is “frozen” at low T

Data of diamond from Einstein’s 1907 paper. Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Debye model (1912 ) Based on classical elasticity theory (continuous, prior to the classical theory of lattice dynamics). Vibration produces waves

ℏ ℏ Frequency dispersion U( T )= ∑ n ks, ωω ks , ( ks , /2 neglected) k, s ω Debye 3 ℏω =∑∫ dω ω D s ( ) ℏω/kT s=1 e −1

Debye assumed that Einstein ω ω 0 1. the wave is non-dispersive: = vsk (s= L,T1,T2). 2 2 3 Therefore, D s ( ω π ) = V / 2 v s (quadratic) (Iso-frequency surface is a sphere ) k Heat capacity, DOS Einstein model, Debye model Anharmonic effect

2. The 1st BZ is approximated by a sphere with the same total number of states.

High-frequency cut-off ωD: • A simple estimate

4 3 For every π kD branch 3 = N ∆3k 2 1/3 → π ω D =vk D = v()6 n

3 ωD • or dω ω Ds ( )= 3 N ∑∫0 s=1 3 3 VωD →∑ 2 3 = 3N s=1 6π vs Same areas 33 1 3≡ ∑ 3 vs=1 v s 2 1/3 → πω D =v(6 n ), nNV = /

Debye frequency ω D Heat capacity, DOS Einstein model, Debye model Anharmonic effect Internal energy and heat capacity

3 ωD V 2 ℏω U( T )= ∑ d ωω ℏ 2 3 ∫ ω/kB T s=1 2π vs 0 e − 1 4 Debye temperature 3Vk T  xD x 3 ℏω θ =ℏ B dx , x =D = , k θ ≡ ℏω 2 3 ℏ  ∫ x D B D 2π v  0 e− 1 kTTB 3 T  xD x 3 =9 NkT dx π4 → →∞ B   ∫ x = /15 as T 0 ( xD ) θ  0 e −1

4 3 12 π T  3 3 ∴CV = Nk B   ∝ T as T → 0 (Debye T law ) 5 θ 

At low T, Debye’s curve drops Heat capacity of silver ( θD=215K) slowly because long wavelength vibration can still be excited. Heat capacity, DOS Einstein model, Debye model Anharmonic effect

A simple explanation of the T3-dependence (at low T): ℏ First, define vkT= k B T Suppose that

1. All the phonons with wave vector k

2. Each excited mode roughly has thermal energy kBT

kD kT solid Argon ( θ=92 K)

• Then the fraction of excited modes 3 θ 3 # of normal = ( kT/kD) = ( T/ ) . modes ～ θ 3 • Thermal energy U kBT‧3N(T/ )

～ θ 3 •∴ Heat capacity C 12 Nk B(T/ ) Heat capacity, DOS Einstein model, Debye model Anharmonic effect DOS for general dispersion relation important

An energy shell

dS ω dk ⊥

ω+dω =const Surface ω=const   3 L 3 If v = ∇ω = 0, then there is Dd(ω ω ) =   ∫ dk g k 2π  Shell "van Hove singularity " (1953) ∇ω ⊥ Iso-frequency 3 k d k= dS dk surface ω ⊥ ωd ω ω =∇k ⋅ dk =∇ k dk ⊥

3 dω ∴d k = dS ω ∇k ω L  3 dS ⇒ ω D(ω ) =   ∫ Group 2π  ∇ ω k velocity Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Dispersion relation and DOS

Giannozzi et al, PRB 43, 7231 (1991) Heat capacity, DOS Einstein model, Debye model Anharmonic effect

• Heat capacity of a crystal • density of states • Einstein model • Debye model • anharmonic effect • thermal conduction Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Aharmonic effect in crystals If there is no aharmonic effect, then • No thermal expansion • No phonon-phonon interaction • Thermal conductivity would be infinite (for a pure crystal) • Heat capacity becomes constant at high T •…

No thermal expansion With thermal expansion

physics.stackexchange.com/questions/332524/what-is-thermal-expansion Heat capacity, DOS Einstein model, Debye model Anharmonic effect Thermal conductivity • Thermal current density (Fourier’s law, 1807) ∼ JU =−∇ KT J =−∇σ φ • In metals , thermal current is carried by both electrons and phonons. In insulators , only phonons can be carriers.

• The collection of phonons is similar to an ideal gas

Ashcroft and Mermin, Chaps 23, 24 Heat capacity, DOS Einstein model, Debye model Anharmonic effect

• Thermal conductivity

K=1/3 c v ℓ Dimensional analysis: c: specific heat Assume [K] = [c] a [v] b [ℓ]c v: phonon velocity ‰ a=b=c=1 ℓ: mean free path (Kittel p.122)

• The mean free path of a phonon can be affected by defects, boundary, and other phonons . Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Phonon-phonon scattering A result of the anharmonic vibration

Modulation of elastic const. (～ acoustic grating)

F= − kx + k' x 2 = − (k − kxx ' )

∴ keff () x = kkx − '

• Total momentum of the 2 phonons remains the same during the scattering. No resistance to thermal current? Heat capacity, DOS Einstein model, Debye model Anharmonic effect Phonon-phonon scattering • Normal process : 1st BZ

• Umklapp process (轉向過程 , Peierls 1929 ):

Figs from wiki Heat capacity, DOS Einstein model, Debye model Anharmonic effect

• High T: The number of phonons are proportional to T. The mean free path ~ 1/ Tx (x=1 ～2).

3 • Low T: K ~ c ~ T Highly purified • High T: K ~ ℓ ~ 1/ T NaF

Boer and Pohl, Semiconductor Physics 2018 Heat capacity, DOS Einstein model, Debye model Anharmonic effect

Boundary scattering Isotope scattering

LiF LiF with various Li-6 isotopes (0.02% to 50%)

www.ucl.ac.uk/~ucapahh/teaching/3C25/Lecture12s.pdf Berman and Brock PRS London 1965