Chap 6 The free and single electron model (Sommerfeld, 1927) 1926: Schrodinger eq., • FD statistics • • specific heat

Note: is most accurate for alkali metals. It gives good result on electron specific heat, electric and thermal conductivities… etc. Dept of Phys

M.C. Chang Electronic degrees of freedom

Free electron Chap 6 Chap 10 Electron in lattice Realistic Chap 7, 8 calculations

Electron-electron interaction Chap 9

Ionic degrees of freedom

Phonon Chap 13

Electrons interact with EM fields, impurities, … etc (next semester) Free electron model z

2 ⎛⎞= 2 G G ⎜⎟−∇+0()ψεψrr = () ⎝⎠2m Lz separable, assume Ly y G Lx ψ ()rxyz= φφφxyz () () () then Advantage: 2 allows travelling d 2 x ()xkφφ+= () x 0, waves dx2 xxx d 2 Periodic Boundary Condition (PBC): ()yk+=2 () y 0, 2 yyyφφ dy φφφφφφxx()LLL=== x (0);() yy y (0);( zz)(0) z 2 2 d 2 ⇒=knnZ, ∈ 2 zzz()zk+= () z 0, xxx dz φ Lx π 2m φ 2 kkk222++= knnZ=∈, x yz =2 xxxL x π G G G 1 ik ⋅r ε 2 ⇒ ()re= knnZ=∈, V xxx φ Lx π Box BC vs Periodic BC

Plane wave solution ψε()xAeBe=+ikx− ikx , () k ==22 k /2 m

• “Box” BC • Periodic BC

k = π/L, 2π/L, 3π/L… k = ±2π/L, ±4π/L, ±6π/L… Box BC vs Periodic BC

box BC periodic BC

• Each point allows 2 electrons (due to spin). After filling in N electrons, the occupied points form a Fermi sphere. Its radius is called the Fermi wave vector, and the energy of the outermost electron is called the Fermi energy.

• Different BCs give the same Fermi wave vector and the same energy

box BC periodic BC Connection between electron density and Fermi energy

4 3 π k 3 F k NV==2 3 F 3 2 ⎛⎞2π 3 ⎜⎟ ⎝⎠L π F 1/3 ⇒=kn()3 2 F π 2 = 2/3 ⇒=επ()3 2n F 2m

For K, the electron density N/V=1.4×1028 m-3, therefore −19 kA= 0.746 −1 ε F = 3.40×10 J = 2.12eV F

• εF is roughly the order of the atomic energy levels -1 • kF is of the order of a . Fermi temperature and Fermi velocity m ε ==kT v2 FBFF2

• The Fermi temperature is of the order of 104 K important

• Some useful formulas:

T=0 T≠0 G 1 Nks==1(,)α Nf= α where f (εα ) = ∑ ∑ ()/α − kT filled α e +1 filled α εμ ε Ef= ε E = ∑ ∑ α α α α filled filled α

• Connection between summation and integral

b

dxh() x=Δ lim x⋅ h ( xi ), or h(x) ∫ Δ→x 0∑ a i b dx hx()≅ h ().x ∑ i ∫ i a Δx Similarly in 2D and 3D, e.g., a b x dk3 fE()GG≅ f ()E ∑G k ∫ 3 k k Δ k important

• Density of states D(ε) (DOS, 態密度)

D(ε)dε is the number of states within the energy surfaces of ε and ε+dε

dk3 3 ∫shell 3 ⎛⎞2π Dd()εε=Δ=3 , k⎜⎟ Δ kL⎝⎠

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

dk3 hdDh(ε G )= εεε ()() ∫∫Δ3k k

• For a 3D Fermi sphere,ππ G dk322G = k εεDd(εε )== 2 ( k ) ∫shell (2 /Lm )3 2 3/2 42kdk2 V⎛⎞ m =⇒= 2π D (ε ) ε 322⎜⎟= (2π /L ) 2 ⎝⎠ • Free electron DOS (per volume) in 1D, 2D, and 3D

• Multiple bound states in 2D

z

⊥ • Thermal distribution of electrons G dk3 Recall 2f (ε GG )→= 2G fdDf (εεεε ) ( ) ( ) ∑G kk∫∫3 k Δ k 1 where ff ( )= ()/εμ− kT , ()= 1/2 εμe +1

μ(0)TK==F

με(0TK>≠) ε F

• Combine DOS D(ε) and thermal dist f(ε,T) D(ε) D(ε) Hotel rooms N =→2 f dDε ( ) f ( ,T) ∑G k ∫ ε k ε tourists ET()=→ 2 fεεεε d D ()(,) f Tε ∑G kk ∫ k money Sommerfeld expansion

Integral of the form (h is arbitrary) εε df − dε ∞ dK K dh()(ε f ), h()ε ≡ ∫0 dε

∞ ∞ df =−Kf d K 0 ∫0 sharp ε ε d ε around μ μ

If K is smooth near εF, then expand K around μ (not εF): ∞ ⎛⎞df dε −=1 2 ∫0 ⎜⎟ dK1 d K 2 ⎝⎠d KK(ε )εε μ= εμ ( εμ ) ε+−+−+ εε() () ... ε 2 ∞ ddμ 2! ⎛⎞df d ()εε−− μ⎜⎟ =0 ∫0 ⎝⎠d εε2 ∞ μ π 2 dh 2 μ ∞ 22df ⇒=++dh()() f dh ()() kTB ... ⎛⎞ ∫∫006 d μ dkT()−−=⎜⎟ε ()B ∫0 εε μ⎝⎠d 3 unknown π ε Chemical potential at finite temperature (n=N/V, g=D/V) Electronic specific heat (u=U/V) Heuristic argument εε ε π

2 D(ε)f(ε) D(ε)f(ε) μ 2 dg(ε ) udg=+( )() kTB + ... ∫0 6 d μ μ F kBT dgε εεεεμεεε dg()+−()F FF g ( ) ～ ∫∫00ε  2 2 − ()kT g' FB6 2 ε 2 ⇒=uu +()() kTg +... 0 6 π BF ε π • Electrons near the Fermi surface are excited by thermal energy kT. Their number ε is of the order of N’= NA（kT/εF）

• The energy absorbed by the electrons is ∝ T 2 U(T)-U(0) ≈ NA (kT) /εF

• specific heat Ce ≈ dU/dT

= 2R kT/εF = 2R T/TF

a factor of T/TF smaller than classical result C γ = e T

z In general CV = Ce + Cp = γT + AT3

Ce is important only at very low T …

L. Hoddeson et al, Out of the crystal maze, p.104