EE201/MSE207 Lecture 14 Particle distributions at 푇 ≠ 0 (quantum statistics) Classical 1. 푁 = const ( of particles distributed 퐸 = const among energy/velocity levels) All microstates are equally probable (postulate)

2. 푁 = const 퐸 ≠ const (exchange of heat) Probability of a state 푃 퐸 ∝ 푒−퐸/푇 (푘퐵 = 1, 푘퐵푇 → 푇) big reservoir Follows from the postulate for microcaconical ensemble (this is how is introduced) Classical statistical mechanics (cont.) 3. 푁 ≠ const (particles can penetrate) 퐸 ≠ const Probability of a state 푃 퐸, 푁 ∝ 푒−(퐸−휇푁)/푇 (two parameters: temperature and ) Chemical potential 휇: average energy cost of bringing an extra particle from big reservoir The formula for 푃(퐸, 푁) also follows from equiprobability in microcanonical ens. From 푃(퐸, 푁) we can derive 푛(휀): average number of particles with energy 휀 휀 − 휇 퐸 = 푛 휀 휀 푛 휀 = exp − Maxwell- 푖 푖 푖 푇

Derivation 푃푘: probability to have 푘 particles with quantized (binned) energy 휀 −푘 휀−휇 /푇 푃푘 = 푃0 푒 푘! (푘! comes from number of combinations) ∞ 푘=0 푃푘 = 1  푃0 = 1 exp[exp(−(휀 − 휇)/푇)]  푘 = exp[− (휀 − 휇) 푇] Quantum statistics Main difference: indistinguishable particles (instead of a question “which one” we are only allowed to ask “how many”)

12 1 2 Example: two particles classical 12 2 on two levels 1 equal probabilities, 1/4 each I I I quantum I I I equal probabilities, 1/3 each Either 0 or 1 particle on a level with energy 휀 (spin increases number of levels, still no 2 particles on the same level) 푃 1 = 푒− 휀−휇 /푇 Still use classical relation 푃 퐸, 푁 ∝ 푒− 퐸−휇푁 /푇 푃0 ⇒ 1 푒− 휀−휇 /푇 푃0 = , 푃 = , 푃0 + 푃1 = 1 1 + 푒− 휀−휇 /푇 1 1 + 푒− 휀−휇 /푇 1 Fermi-Dirac distribution ⇒ (Fermi statistics) 푛 = 푃1 = 휀−휇 /푇 (chemical vs. 1 + 푒 휇 is electrochemical) Quantum statistics (cont.)

푃1 푃2 푃 = 푒− 휀−휇 /푇, = 푒−2 휀−휇 /푇, 3 = 푒−3 휀−휇 /푇, . . . 푃0 푃0 푃0 1 푃 = 1 ⇒ 푃 = = 1 − 푒− 휀−휇 /푇 푛 0 1 + 푒− 휀−휇 /푇 + 푒−2 휀−휇 /푇+ . . . 휀−휇 휀−휇 휀−휇 − − −2 푛 = 0 ∙ 푃0 + 1 ∙ 푃1 + 2 ∙ 푃2 + . . . = 1 − 푒 푇 (1 ∙ 푒 푇 + 2 ∙ 푒 푇 + ⋯ )

휀−휇 휀−휇 휀−휇 − −2 − 휀−휇 푇 푇 푇 − 푒 푒 푒 = 1 − 푒 푇 휀−휇 + 휀−휇 + ⋯ = 휀−휇 − − − 1 − 푒 푇 1 − 푒 푇 1 − 푒 푇

1 Bose-Einstein distribution 푛 = 푒 휀−휇 /푇 − 1 (Bose statistics) 휇 ≤ 0 (if energy starts from 0), otherwise infinity at 휀 = 휇 Particle distributions: summary 2 2

 E e 1 1.5 − 휀−휇 /푇 Maxwell-Boltzmann0E1 푒 1 0.8 (Boltzmann) 1.3 0 0.5 1 1.5 휀 2 1 1 0 E 2 E 0.5 1 e 1 푇 = 0 0E 푛 휀 = Fermi-Dirac 0.5 휀−휇 /푇 1 E 푒 + 1 0.003 e 1 (Fermi) 1.3 0

1 0.5  0.3 휇 휀 E 2 1.5 1 0.5 0 0.5 1 1.5 2 e 1 1  2 E 2 0E 0.5 Bose-Einstein 1 휀−휇 /푇 E 0.002 푒 − 1 e 1 (Bose) 0

 0.3 휇 휀 휇 is Fermi level 0.5 0 0.5 1 1.5 2  0.5 E 2

To find 휇: 푁 = 푛 휀 퐷 휀 푑휀 푛(휀) depends on temperature ⇒ 휇 depends on temperature density of states

Remark 1. Often notation 푓 휀 instead of 푛(휀), especially for Fermi distribution Remark 2. Large-휀 tails of Fermi-Dirac and Bose-Einstein distributions coincide with Maxwell-Boltzmann distribution 2D case (not in textbook)

퐷 휀 푚 퐷 휀 is density of states, 퐴 is area = 2푠 + 1 퐴 2휋ℏ2 푠 is spin, in general 2푠 + 1 is degeneracy

Electrons (Fermi, 푠 = 1 2) spin ∞ spin 푁 푚 1 푚 휇/푇 = 2 2 (휀−휇)/푇 푑휀 = 2 2 푇 ln(1 + 푒 ) 퐴 0 2휋ℏ 푒 + 1 2휋ℏ

No spin factor of 2 in high magnetic field Bosons with 푠 = 0 ∞ 푁 푚 1 푚 휇/푇 = 2 (휀−휇)/푇 푑휀 = 2 푇 ln(1 − 푒 ) 퐴 0 2휋ℏ 푒 − 1 2휋ℏ 3D case

퐷 휀 푚3/2휀1/2 = 2푠 + 1 퐷 휀 is density of states, 푉 is volume 푉 2 휋2ℏ3 푠 is spin, in general 2푠 + 1 is degeneracy (including valleys, etc.) 푁 ∞ 푚3/2휀1/2 1 = (2푠 + 1) 푑휀 2 3 (휀−휇)/푇 푉 0 2 휋 ℏ 푒 ± 1 degeneracy 퐸 ∞ 푚3/2휀1/2 1 = 휀 (2푠 + 1) 푑휀 (e.g., for heat capacity) 2 3 (휀−휇)/푇 푉 0 2 휋 ℏ 푒 ± 1 Fermi: “+”, Bose: “−” Unfortunately, these integrals cannot be calculated analytically.

Simplification if −휇 ≫ 푇, then F-D and B-E distributions reduce to M-B. 1 ≈ 푒− 휀−휇 /푇 푒(휀−휇)/푇 ± 1 when 휀 − 휇 ≫ 푇 Nondegenerate Assume n-type (p-type similar), −휇 ≫ 푇, 2푠 + 1 = 2 conduction band > 3푇 휇 (Fermi level) 푁 ∞ 푚3/2휀1/2 ≈ 푒−(휀−휇)/푇2 푑휀 = . . . 2 3 푉 0 2 휋 ℏ valence band (neglect) 푚푇 3/2 = 2 푒휇/푇 Room temperature: 푇 = 26 meV 2휋ℏ2

3/2 푁 1 2휋ℏ2 휇 = 푇 ln 푉 2 푚푇

degeneracy; can be larger, Si: 26 퐸 ∞ 푚3 2휀1 2 3 푁 ≈ 휀 푒− 휀−휇 푇2 푑휀 = . . . = 푇 2 3 푉 0 2 휋 ℏ 2 푉 3 퐸 = 푇푁 2 Bose-Einstein condensation

For Bose-Einstein distribution usually 휇 < 0 (cannot be 휇 > 0). However, at small enough 푇, it becomes 휇 = 0, then

푁 ∞ 푚3/2휀1/2 1 푚푇 3 = 푑휀 = 2.61 (푠 = 0) 2 3 −휇/푇 2 푉 0 2 휋 ℏ 푒 − 1 2휋ℏ 2/3 2휋ℏ2 푁 Therefore critical temperature 푇 = 푐 푚 2.61 푉

Below 푇푐 particles crowd into the ground state (finite fraction of all particles occupy ground state)

Different calculation: 푁 = 푁 0 + 푛 휀 퐷 휀 푑휀

Examples: superconductivity, , B-E condensation of atoms Massless particles (, phonons) 2휋 휔 휀 = ℏ휔 푘 = = 휆 푐 speed of light or sound velocity

Number of particles is not conserved ⇒ 휇 = 0 (creation of extra particle does not cost extra energy) 1 푛(휔) = (bosons) 푒ℏ휔/푇 − 1

푑푥 푑푘푥 푑푦 푑푘푦 푑푧 푑푘푧 푑푁 푑푘푥 푑푘푦 푑푘푧 DOS: 푑푁 = ⇒ = 2휋 3 푉 2휋 3

2 2 푑푁 4휋푘 푑푘 휔 × 2 for photons (two polarizations) = = 푉 푑휔 2휋 3 푑휔 2휋2푐3 2 1 × 3 for phonons, better 3 + 3 푐⊥ 푐∥ Average energy per d휔 (for photons) 푑퐸 2휔2 1 2ℏ휔3 = ℏ휔 = (Planck’s formula) 푉 푑휔 2휋2푐3 푒ℏ휔/푇 − 1 2휋2푐3(푒ℏ휔/푇 − 1)