Chap. 5 Phonon II Thermal properties

Gas C_p, C_v and C_p-C_v=R

Solid C_p-C_v=9α²βVT, α:thermal expansion coefficient of linear expansion, β:bulk modulus

能量均分原理: 每一個自由度有相等的能量(½kT)

C_v=(∂U/∂T)ᵥ=3Nk (Dulong-Petit’s law), U=3NkT(動能+位能, 6個自由度)

*只在高溫下適用

model 與experiment比較來證明model

experiment facts about C_v of solids:

1. in the room temperature range, C_v3Nk_B=25 J/moleK

2. at low temperature (<10 k="" span="">

insulator: C_vT³ (Debye’s T³ law)

metal: C_vT

superconducting: C_ve^-A/T

magnetic material: C_vT^3/2

Insulator絕緣體, *觀念: 能量不連續 Einstein’s model

Assume that all atoms in a solid oscillators with the same frequency.

U=[∑_jn]ℏω

Debye’s model

1. the frequency of oscillators in a solid are not the same but some kind of distribution D()

2. the sound velocity is assumed to be constant, n=Planck distribution 低溫

n=∑_n=0^∞nPₙ, Pₙ=Nₙ/∑_sNₛ=e^-Eₙ/kᴃT/∑_s e^-Eₛ/kᴃT → n=∑_n=0^∞ne^-Eₙ/kᴃT/∑_s e^-Eₛ/kᴃT

代入n  e^-Eₙ/kᴃT → Eₙ=nℏω 代入→ n=∑ₛ₌₀^∞se^{-sℏω/kᴃT}/∑_s e^{-sℏω/kᴃT} , 設 x=ℏω/kᴃT

→ n=∑ₛ₌₀^∞se^-sx/∑_s e^-sx =-∂[ln∑_s e^-sx]∕∂x=-∂ln[1/(1-e^-x)]∕∂x=∂ln(1-e^-x)∕∂x=e^-x/(1-e^-x)=1/(e^x-1) Planck distribution

→ U=∫nℏωD()d

density of modes in one dimension

standing wave method, D(k)dk, each vibrating mode is a standing wave.

1. 1. u_s=Ae^i(ska-ωt)+Ae^i(-ska-ωt)∙e^iπ=Ae^-iωt∙2isinska=A(cosωt-isinωt)∙2isinska

real wave function → u_s(x,t)=2Asinωtsinska, u_s  sinska and u_s=0=0, u_s=N=0sinNka=0

Nka=π, 2π , …(N-1)π, Nπ

k=π/Na, 2π/Na, …(N-1)π/Na, π/a

if k=π/a+π/Na,

u_s(x,t)=2Asinωtsinska=2Asinωtsin[s(π/a+π/Na)a]=2Asinωtsin[sπ+sπ/N]

=2Asinωtsin(sπ/N)(-1)^s=2Asinωtsin(sπ/N)

只有N-1個atom可動 → N-1個自由度, k有N個獨立值

if k=π/a, u_s(x,t)=2Asinωtsinska=2Asinωtsinsπ=0測不到此波

⸫獨立k值= π/Na, 2π/Na, …(N-1)π/Na 共N-1個k (N-1 modes)

if Na=L→ k=π/L, 2π/L, …(N-1)π/L

D():每單位頻率有多少值, D(k)dk=D(k)(dk/d)d=D()d

One mode, π/L → D(k)=(π/L)⁻¹=L/π, 0<k<π/a

1. 1. method of periodic boundary condition

u_s=u₀e^i(kx-ωt)=u₀e^i(ska-ωt) , u(sa)=u(sa+L) → e^ikL=1, kL=±2nπ, k=±2nπ/L

⸫ k=0, ±2π/L, ±4π/L,..., ±Nπ/L

L=Na: k與atom數目的關係

ex. N=8, L=Na, k=0, 2/8a, 4/8a, 6/8a, 8/8a(偶數只取”+”), 10/8a(/a2/8a重複)

k=/a u_s=u₀e^{i[s(π/a)a-ωt]}=u₀e^isπe^-iωt=u₀(-1)^se^-iωt

k=-/a u_s=u₀e^{i[s(-π/a)a-ωt]}=u₀e^-isπe^-iωt=u₀(-1)^se^-iωt

k=0 u_s=u₀e^-iωt

ex. N=9, L=Na, k=0, 2/9a, 4/9a, 6/9a, 8/9a, 10/9a[(10/9a)= (18/9a-8/9a)= 2/a-+8/9a]

, 與8/9a相同

there is one mode for every 2π/L, density of mode D(k)= (2π/L)⁻¹=L/2π

1. 1. density of mode in 3-D traveling wave, u_s=u₀e^i(k∙r-ωt)

periodic boundary condition: u(r)=u(r+L)=u(r+Lx)在x方向上k=?

u(r)=u(r+L)=u(r+Lx)=u₀e^i(k∙r-ωt)=u₀e^i(k∙r-ωt)∙e^ikₓ∙Lₓ → e^ikₓLₓ=1,

kₓLₓ=±2nπ, kₓ=0, 2/Lₓ, 4/Lₓ, 6/Lₓ, …, Nₓ/Lₓ| Nₓ modes in x方向上有N個原子

同理得 e^ikyLy=e^ikzLz=1,

k_y=0, 2/L_y, 4/L_y, 6/L_y, …, N_y/L_y| N_y modes

k_z=0, 2/L_z, 4/L_z, 6/L_z, …, N_z/L_z| N_z modes 總共有 N_xN_yN_z個mode

N_total=N_xN_yN_z, k=kₓx+k_yy+k_zz

考慮成正方形, D()dV每單位體積有幾個mode

U=∫nℏωD()d

(2π/L)³的體積中只有含一個mode; there is one mode for every (2π/L)³ in k-space.

Density of state

D(k)=(2π/L)⁻³=L³/8π³=V/8π³

D(k)dk= # of modes with wavevector between k and k+dk

n(k)—represent the total number of modes with wave vector less than k

D(k)dk=n(k+dk)-n(k)

n(k)=[4πk³/3]/(2π/L)³=(V/6π²)k³  k³

D(k)dk=n(k+dk)-n(k)=([n(k+dk)-n(k)]/dk)∙ dk=(dn/dk)∙ dk → D(k)=dn/dk

D(k)=dn(k)/dk=Vk²/2π²

變數轉換k→ω, D(k)dk=D(k)(dk/d)d=D()d

Debye model—“sound speed” is assumed to be constant.

相速v=/k, =vk → d/dk=v group velocity

D(k)dk=D(k)(dk/dω)dω=(Vk²/2π²)∙dω/v=Vω²dω/2π²v³

if there are N atoms in a crystal, there should be N phonon modes for each polarization.

U=∫₀^ωᴅnℏωD(ω)dω

n(k_ᴅ)=N=n(_ᴅ), [4πk_ᴅ³/3]/(2π/L)³=Vω_ᴅ³/6π²v³=N ⸫ ω_ᴅ=[6π²v³(N/V)]^1/3  v

U=∫₀^ωᴅ[1/(e^ℏω/kᴃT-1)]ℏω(Vω²/2π²v³)dω for each polarization 1/vₗ³+1/vₜ₁³+ 1/vₜ₂³

for three polarization  , v_t≠v_l

assume that the phonon velocity is independent of polarization,

 , let x=ℏω/kᴃT → dx=(ℏ/kᴃT)dω and x_ᴅ=ℏω_ᴅ/kᴃT=θ_ᴅ/T

U=3Vℏ/2π²v³∫₀^ωᴅ[ω³/(e^ℏω/kᴃT-1)]dω=3Vℏ/2π²v³∫₀^xᴅ(kᴃT/ℏ)⁴[x³/(e^x-1)]dx

=(kᴃ³V/6π²v³ℏ³N)9NkᴃT⁴∫₀^xᴅ[x³/(e^x-1)]dx=9NkᴃT(T/θ)³∫₀^xᴅ[x³/(e^x-1)]dx

i.e. Define ℏω_ᴅ/kᴃ=θ_ᴅ , Debye temperature

⸪ ω_ᴅ=[6π²v³(N/V)]^1/3 → θ=ℏ[6π²v³(N/V)]^1/3/kᴃ, ⸫ θ³=6π²v³ℏ³N/kᴃ³V

Cᵥ=∂U∕∂T|ᵥ=∂{3Vℏ/2π²v³∫₀^ωᴅ[ω³/(e^ℏω/kᴃT-1)]dω}∕∂T

=3Vℏ/2π²v³∫₀^ωᴅ[ω³(-ℏω/kᴃ)(1/T²)e^ℏω/kᴃT/(e^ℏω/kᴃT-1)]dω

=(3Vℏ/2π²v³)(ℏ/kᴃT²)∫₀^ωᴅ[ω⁴e^ℏω/kᴃT /(e^ℏω/kᴃT-1)]dω

=(3Vℏ²/2π²v³kᴃT²)∫₀^xᴅ(kᴃT/ℏ)⁴[x⁴e^x/(e^x-1)²](kᴃT/ℏ)dx

=(3Vkᴃ⁴T³/2π²v³ℏ³)∫₀^xᴅ[x⁴e^x/(e^x-1)²]dx

=(Vkᴃ³/6π²v³ℏ³N)9NkᴃT³∫₀^xᴅ[x⁴e^x/(e^x-1)²]dx=9Nkᴃ(T/θ)³∫₀^xᴅ[x⁴e^x/(e^x-1)²]dx

1. 1. 1. at high temperature, kᴃT>>ℏω ⸫ x<<1 span="">

U=9NkᴃT(T/θ)³∫₀^xᴅ[x³/(e^x-1)]dx, i.e. e^x=1+x+x²/2!+x³/3!+... as |x|<<1 span="">

U=9NkᴃT(T/θ)³∫₀^xᴅ[x³/(1+x+x²/2!+x³/3!+...-1)]dx≈9NkᴃT(T/θ)³∫₀^xᴅx²dx

= 9NkᴃT(T/θ)³(x_ᴅ³/3)=3NkᴃT → Cᵥ=∂U∕∂T|ᵥ=3Nkᴃ

1. 1. 1. at low temperature, kᴃT<<ℏω ⸫ x>>1 → x_ᴅ=ℏω_ᴅ/kᴃT→∞

1/(e^x-1)=e^-x/(1-e^-x)=e^-x(1+e^-x+e^-2x+e^-3x+...)=e^-x+e^-2x+e^-3x+...=∑ₛe^-sx

∫₀^∞[x³/(e^x-1)]dx=∫₀^∞ x³dx∑ₛe^-sx=∑ₛ∫₀^∞ x³e^-sxdx=∑ₛ6/s⁴=π⁴/15, i.e. ζ(4)= π⁴/90

U=9NkᴃT(T/θ)³∫₀^∞[x³/(e^x-1)]dx=(3π⁴/5)NkᴃT(T/θ)³ → Cᵥ=∂U∕∂T|ᵥ  T³

U=(3π⁴/5)NkᴃT(T/θ)³ → Cᵥ=∂U∕∂T|ᵥ=(12π⁴/5)Nkᴃ(T/θ)³=234Nkᴃ(T/θ)³

“定性”physical understanding of T³-law

k_ᴅ wave vector, k_ᴅ=ω_ᴅ/v=ℏω_ᴅ/ℏv=kᴃθ/ℏv

在低溫只有一部分k值供獻 k_T=ℏω_T/ℏv=(kᴃ/ℏv)T

the function of excited modes, 令密度一樣只須體積比即可

[4πkᴃ³/3]/[4πk_ᴅ³/3]=(kᴃ/k_ᴅ)³=(T/θ)³

U=3NkᴃT(T/θ)³, Cᵥ=∂U∕∂T|ᵥ=12Nkᴃ(T/θ)³  T³

二次項對晶格thermal expansion無貢獻,簡化成U(r)=cx²-gx³-fx⁴

在一度空間上算 x=[∫xP(x)]/P(x), 偏移平衡點的距離 i.e. P(x)  exp[-U(r)/kᴃT]

所測出=r₀+x

x=(∫_-∞^∞xexp[-U(r)/kᴃT]dx)/(∫_-∞^∞exp[-U(r)/kᴃT]dx)

upper part:

∫_-∞^∞xexp[-U(r)/kᴃT]dx=∫_-∞^∞xexp[-(cx²-gx³-fx⁴)/kᴃT]dx

=∫_-∞^∞xexp(-cx²/kᴃT)exp[(gx³+fx⁴)/kᴃT]dx=∫_-∞^∞xexp(-cx²/kᴃT)(1+gx³/kᴃT+fx⁴/kᴃT)dx

=∫_-∞^∞xexp(-cx²/kᴃT)∙(gx⁴/kᴃT)dx=(3√π/4)∙(g/c^5/2)∙(kᴃT)^3/2

lower part:

∫_-∞^∞exp[-U(r)/kᴃT]dx=∫_-∞^∞exp[-(cx²-gx³-fx⁴)/kᴃT]dx

=∫_-∞^∞exp(-cx²/kᴃT)exp[(gx³+fx⁴)/kᴃT]dx=∫_-∞^∞exp(-cx²/kᴃT)(1+gx³/kᴃT+fx⁴/kᴃT)dx

≈∫_-∞^∞exp(-cx²/kᴃT)dx=(π/c)^1/2∙(kᴃT)^1/2

→ x=[(3√π/4)∙(g/c^5/2)∙(kᴃT)^3/2]/[(π/c)^1/2∙(kᴃT)^1/2]=(3g/4c²)kᴃT

thermal conductivity

thermal flux, T₂ > T₁, J=-kdT/dx

3-D, J=-kT

Thermal flux: thermal energy transmitted across unit area per unit time, 由phonon攜帶能量:

Energy flux from the high temperature side:

J₁=½nvₓU(T+l∙∂T/∂x)=½nvₓ[U(T)+l∙∂T/∂x(∂U/∂T)ᵥ]=½nvₓ[U(T)+cᵥ∙l∙∂T/∂x ]

Energy flux from the cold temperature side:

J₂=½nvₓ[U(T)-Cᵥ∙l∙∂T/∂x ]

∑J=J₁-J₂=nvₓcᵥ∙l∙∂T/∂x=Cᵥvₓ∙l∙∂T/∂x=Cᵥvₓ²τ∙∂T/∂x , l=v_xτ

in 3-D ∵vₓ=v²/3 → J=Cᵥv²τ∙T/3=CᵥvlT/3, k=-Cᵥvl/3

Two processes of phonon interaction

1. N-process(Normal process)—three phonons process

ℏk₁+ℏk₂=ℏk₃ 能量一直往前傳遞,沒有阻力

no thermal resistivity or infinite thermal conductivity.

1. U-process(U_mklapp process)

k₁+k₂=k₃+G

Finite thermal resistivity and finite thermal conductivity.

註記: □=ℏ Planck constant