Chap. 6 Free Electron

完全不受拘束的電子(滿足Pauli principle)

1. Heat capacity of metal

2. Electrical conductivity

3. Cyclotron

4. Hall effect

5. Thermal conductivity

電子貢獻,假設所有電子都有貢獻

U=3(kᴃT/2)N=3kᴃTN/2, C_V= 3kᴃN/2=0.54 joul/gK

experimental value C_V=1.23 joul/gK

phonon 貢獻 C_V=3Nk_B=1.08 joul/gK

所以free electron貢獻 C_V=1.23 – 1.08 = 0.15 joul/gK, 與0.54相差太多

Energy level of free electron gas:

Consider a free electron in 1-D Hψ=Eψ, H=P²/2m=(-ℏ²/2m)∙d²/dx²

⸪ P=-jℏ=-jℏ∙d/dx, (-ℏ²/2m)∙d²ψ/dx²=Eψ boundary condition ψ(0)=ψ(L)=0

ψ=Acoscx+Bsincx, ψ(0)=0 → A=0, ψ(L)=0 → BsincL=0, cL=nπ ⸫c=nπ/L n=1,2,3,…

(-ℏ²/2m)∙d²ψ/dx²=(-ℏ²/2m)∙d²[Bsin(nπx/L)]/dx²=(-ℏ²/2m)∙-(nπ/L)²Bsin(nπx/L)=Eψ

⸫ Eₙ=ℏ²/2m∙(nπ/L)², ψ=Bsin(nπx/L)

Fermi energy: the energy of the topmost filled levels in the ground state(at 0 K)

Fermi-Dirac distribution

Give the probability that an orbital at energy ξ will be occupied in an ideal gas in thermal equilibrium.

f(ξ)=1/[e^{(ξ-μ)/kᴃT}+1], ξ : energy level, μ: chemical potential

at 0 K, μ(0)=ξ_F, (i)ξ>μ, f(ξ)=0, (ii)ξ<μ, f(ξ)=1

在任何溫度下,分布情況如何? 據有Fermi energy的電子易被激發

Cₑₗ=∂U∕∂T|ᵥ, U=∫ξf(ξ)D(ξ)dξ

Free electron gas in 3-D

(-ℏ²/2m)∙²ψ=Eψ → (-ℏ²/2m)∙(∂²∕∂x²+∂²∕∂y²+∂²∕∂z²)ψ=Eψ

ψ=Asin(nₓπx/L)sin(n_yπy/L)sin(n_zπz/L)

(-ℏ²/2m)∙²ψ=Eψ → ²ψ+(2mE/ℏ²)ψ=0 let k²= 2mE/ℏ², E=ℏ²k²/2m=P²/2m

²ψ+k²ψ=0 , ψ=Aexp(-jk∙r)

Heat capacity of metal

Total internal energy of free electron gas: U=∫₀^∞ξf(ξ)D(ξ)dξ

Cₑₗ=∂U∕∂T|ᵥ=∫₀^∞ξ(∂f(ξ)/∂T)D(ξ)dξ,

total N=∫₀^∞f(ξ)D(ξ)dξ=const. → ∫₀^∞(∂f(ξ)/∂T)D(ξ)dξ=0, 乘上ξ_F

ξ_F∫₀^∞(∂f(ξ)/∂T)D(ξ)dξ=0 → Cₑₗ=∫₀^∞(ξ-ξ_F)(∂f(ξ)/∂T)D(ξ)dξ

at low temperature, f(ξ)=const. → ∂f(ξ)/∂T=0

在靠近ξ_F 處之電子才有貢獻,使∂f(ξ)/∂T≠0,因此我們可以只算ξ_F 附近的D(ξ_F)的能階數

μ=μ(T) μ → ξ_F

Cₑₗ=∫₀^∞(ξ-ξ_F)(∂f(ξ)/∂T)D(ξ_F)dξ=D(ξ_F)∫₀^∞[(ξ-ξ_F)²/kᴃT²][e^{(ξ-μ)/kᴃT}/(e^{(ξ-μ)/kᴃT}+1)²]dξ

let x=(ξ-ξ_F)/kᴃT, Cₑₗ=D(ξ_F)kᴃ²T∫_-ξf/kᴃT ^∞[x²e^x/(e^x+1)²]dx=D(ξ_F)kᴃ²T∫_-∞^∞[x²e^x/(e^x+1)²]dx

i.e. ∫₀^∞[x/(e^ax+1)]dx=π²/12a² → ∂∫₀^∞[x/(e^ax+1)]dx∕∂a=∂(π²/12a²)/∂a

→ ∫₀^∞[x²e^x/(e^x+1)²]dx=π²/6 (if a=1)

Density of state: D(ξ)=3N(ξ)/2ξ → D(ξ_F)=3N(ξ)/2ξ_F

∵Total number of orbitals, N=2[4πk³/3]/(2π/L)³=Vk³/3π² → k_F=(3π²N/V)^1/3 and ξ_F= ℏ²k_F²/2m N=(V/3π²)( 2mξ/ℏ²)^3/2 → D(ξ)= dN(ξ)/dξ=(V/2π²)( 2mξ/ℏ²)^3/2ξ^1/2

or 取對數再微分 → lnN=3lnξ/2+cost. → dN/N=3dξ/2ξ → dN/dξ=3N/2ξ=D(ξ)

Cₑₗ=D(ξ_F)kᴃ²T∫_-∞ ^∞[x²e^x/(e^x+1)²]dx=[3N(ξ)kᴃ²T/2ξ_F]∙(π²/3)=π²Nkᴃ²T/2ξ_F

Fermi temperature: T_F=ξ_F/kᴃ → Cₑₗ=(π²Nkᴃ/2)∙(kᴃ/ξ_F)T=(π²Nkᴃ/2)∙(T/T_F)

定性分析

N=(V/3π²)( 2mE/ℏ²)^3/2, ∆N=N(ξ_F)-N(ξ_F-kᴃT)=(V/3π²)( 2mξ/ℏ²)^3/2[1-(1-kᴃT/ξ_F)^3/2]=(V/3π²)( 2mξ/ℏ²)^3/2∙(3kᴃT/2ξ_F)=3NT/2T_F

U=∆NkᴃT =3NkᴃT²/2T_F → Cₑₗ=∂U∕∂T|ᵥ=3Nkᴃ(T/T_F)  T

金屬的比熱(at low temperature): C=Cₚₕₒₙₒₙ+Cₑₗ=AT³+rT → C/T=AT²+r

for conductor, T<T_F, T_D

Electrical conductivity σ=1/ρ, 電子運動的平均結果

The electrical resistivity is caused by,

1. The collisions of electrons with phonons

2. The collisions of electrons with lattice imperfections, impurity atoms,

經過l(mean free path)能夠通過某一截面的粒子數:

N=N₀e^-x/l= N₀e^-vt/vτ=N₀e^-t/τ

Number of electrons without suffering collisions during the time interval Δt

N₀e^-∆t/τ=N₀(1-∆t/τ)

Number of electrons with collisions during the time interval Δt: N-N₀= N₀(∆t/τ)

P(t): the average momentum of an electron, P(t)─Δt→P(t+Δt)

P (t+Δt)=1/N₀{[N₀(1-∆t/τ)(P(t)+FΔt)]+[N₀(∆t/τ)FΔt]}=(1-∆t/τ)(P(t)+FΔt)]+(∆t/τ)FΔt

i.e. F(t)=const., P(t)=e^-t/τP(0)+Fτ(1-e^-t/τ)

the displacement of the Fermi sphere in the steady state(t>>) is given by:

P=Fτ=(-eE)τ, dP/dt=P/τ+F

課本(C. Kittel): dP/dt=F=-eE, ∫P =-∫₀^τeEdt → P(τ)-P(0)=ℏk(τ)-ℏk(0)=-eEτ

mv=-eEτ, v=-eEτ/m

J=I/A=N(-e)vA/A=-Nev=-Ne(-eEτ/m)=(Ne²τ/m)E=σE Ohm’s law

I=dq/dt=nv∆tA/∆t= nvA, ⸫ σ=Ne²τ/m, ρ=m/Ne²τ

J= σE → ∫Jdl=σ∫Edl → ∫Idl/A=σV → Il/A=σV → l/σA=V/I=R, R=l/σA=ρl/A

Thermal conductivity of metal k=Cᵥvl/3, k=kₚₕₒₙₒₙ+kₑₗ

kₑₗ=(π²Nkᴃ²T/2ξ_F)∙v_Fl/3=(π²Nkᴃ²τ/3m)T i.e. l=v_Fτ, ξ_F=mv_F²/2

at room temperature: l_ph-ph≈l_ph-el , kₑₗ/kₚₕₒₙₒₙ=10 ⸪ kₑₗ值很大, metal導熱快

k=k_el+k_ph

1. T>>(T>T_D)

   	kel	kph
   C	T	const
   v	const	const
   l	1/T	1/T

 

k=k_el+k_ph 1/T

1. T<< , 與T成正比

   	kel	kph
   C	T	T³
   v	const	const
   l	const	const

Wiedermann-Franz law

Ratio of thermal to electrical conductivity

kₑₗ/σ=[(π²Nkᴃ²τ/3m)T ]/(Ne²τ/m)=π²kᴃ²T/3e²

Lorentz number defined as, L=π²kᴃ²T/3e²

at low temperature, L decrease because _thermal_electrical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

註記: □=ℏ Planck constant