Introduction to Condensed Matter Physics

Jin Guojun

Nanjing University February 24, 2005

Part II. Wave Behavior in Various Structures

Solid State Physics: Revisit and Extension

5. Waves Propagation in Periodic and Quasiperiodic Structures 6. Dynamics of Bloch Electrons 7. Surface and Impurity E ects 8. Transport Properties 9. Wave Localization in Disordered Systems 10. Mesoscopic Quantum Transport

Like as the waves make towards the pebbled shore, So do our minutes hasten to their end, Each changing place with that which goes before, In sequent toil all forward to contend.— William Shakespeare

Waves always behave in a similar way, whether they are longitudinal or transverse, elastic or electric. Scientists of last century always kept this idea in mind··· . This general philosophy of wave propagation, forgotten for a time, has been strongly revived in the last decade···— L. Brillouin (1946)

Contents

Chapter 5.§5.1

Wave Propagation in Periodic and Quasiperiodic S 5 5 9 14

Unity of the Concept for Wave Propagation . . . . . . . . . 5.1.1 Wave Equations and Periodic Potentials . . . . . . . . 5.1.2 Bloch Waves . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Revival of the Study on Classical Waves . . . . . . . .

§5.2

Electrons in Crystals . . . . . . . . . . . . . . . . . . . . . 5.2.1 Free Electron Gas Model . . . . . . . . . . . . . . . . 5.2.2 Nearly-Free Electron Model . . . . . . . . . . . . . . . 5.2.3 Tight-Binding Electron Model . . . . . . . . . . . . . 5.2.4 Kronig-Penney Model for Superlattices . . . . . . . . 5.2.5 Density of States and Dimensionality . . . . . . . . .

19 20 23 30 35 43 46 46 46 46

§5.3

Lattice Waves and Elastic Waves . . . . . . . . . . . . . . 5.3.1 Dispersion Relation of Lattice Waves . . . . . . . . . . 5.3.2 Frequency Spectrum of Lattice Waves . . . . . . . . . 5.3.3 Elastic Waves in Periodic Composites . . . . . . . . .

§5.4

Electromagnetic Waves in Periodic Structures . . . . . . . . 5.4.1 Photonic Bandgaps in Layered Periodic Media . . . . 5.4.2 Dynamical Theory of X-Ray Di raction . . . . . . . . 5.4.3 Bandgaps in Three-Dimensional Photonic Crystals . . 5.4.4 Quasi Phase Matching in Nonlinear Optical Crystals .

47 47 47 47 47 48 48 48 48 49

§5.5

Quasiperiodic Structures . . . . . . . . . . . . . . . . . . . 5.5.1 One-Dimensional Quasiperiodic Structure . . . . . . . 5.5.2 Two-Dimensional Quasiperiodic Structures . . . . . . 5.5.3 Three-Dimensional Quasicrystals . . . . . . . . . . . .

§5.6

Waves in Quasiperiodic Structures . . . . . . . . . . . . . .

5.6.1 Electronic Spectra in a One-DimensionalQuasilattice .

49

5.6.2 Wave Transmission through Arti cialFibonacci Structures 49 5.6.3 Pseudogaps in Real Quasicrystals . . . . . . . . . . . 49

Chapter 5 Wave Propagation in Periodic and Quasiperiodic Structures

Newton, speed of sound, one-dimensional lattice model Beginning of the 20th century, Cambell, periodic LC circuit theory, lter electric

waves, proposed cut-o frequency, pass band and forbidden band Conventional solid state physics: electrons and lattice vibration Recent years, classical waves: electromagnetic wave and elastic wave in periodic composites In 1980s, quasicrystals

§5.1

Unity of the Concept for Wave Propagation

Formal analogy and similarity between three types of waves 5.1.1 Wave Equations and Periodic Potentials

An electron,λ= 2π/k, p= k Schr¨dinger equation o2 iψ= t 2m 2

+ V (r)ψ

(5.1.1)

V (r+ l)= V (r)

(5.1.2)

Thermal uctuation at nite temperatures u—atomic displacement,Φ—potential energy Newton equation 2 Ms 2 ulsα= t

Φlsα,l sβ ul sβlsβ

(5.1.3)

Force constant—Φlsα,l sβ≡ 2Φ/ ulsα ul sβ|0, periodicityΦlsα,l sβ=Φ0sα,¯β ls¯ with l= l l (5.1.4)

Electromagnetic waves, Maxwell equations×E= ×H= 1 B, c t·B=0· D= 4πρ

1 D 4π+ j, c t c

Four coe cients related to materials:,µ,χ,χm

D= E= E+ 4πP= (1+ 4πχ)E B=µH= H+ 4πM= (1+ 4πχm)H In general,,µ,χ andχm, tensors; Strong elds, nonlinear polarization

Electric displacement vector D, wave equation 1 2D+ c2 t2× 1µ(r)× D (r)=0 (5.1.5)

µ(r) and (r), second-order tensors→ scalarsµ(r+ l)=µ(r), (r+ l)= (r) Three other eld quantities, E, B, H, equivalent (5.1.6)

5.1.2

Bloch Waves

Some common traits for wave propagation in periodic structures Tuning condition: Lattice separation a and Characteristic wavelengthsλ Electrons:λ= ( 2/2mE)1/2, ranges from lattice spacing to bulk size

Lattice vibrations: acoustic branches and optical branches Wavelengths ranges from lattice spacing to bulk size Electromagnetic radiation:γ-ray (< 0.4 ), X-ray (0.4～ 50 ), ultraviolet ray (50～ 4000 ), A A A visible light (4000～ 7000 ), infrared ray (0.76～ 600µm), A to microwave and radio wave (> 0.1 mm)

Periodic structures, Bloch waves, an electron stationary equation2

2

2m

+ V (r)ψ(r)= Eψ(r)

(5.1.7)

ψk (r)= uk (r)fk (r)

(5.1.8)

uk (r+ l)= uk (r)

(5.1.9)

To determine fk (r), consider|ψk (r)|2, note|ψk (r)|2=|ψk (r+ l)|2 get|fk (r+ l)|2=|fk (r)|2 Bloch functionψk (r)= uk (r)eik·r Bloch theoremψk (r+ l)=ψk (r)eik·l (5.1.11) (5.1.10)

Discussed above can be generalized to other cases of wave equations with periodic potentials. Periodicity→ Fourier transformation→ Reciprocal lattice Dispersion relation, Bandgaps, Separated bands Brillouin zones (BZ)–Wigner-Seitz cells of the reciprocal lattice Born-von Karman cyclic boundary conditions