Electrons in a Crystal Lattice

Lecture 2. Band structure, metals and isolators, electrons, holes, and excitons. Some funny lessons from very simple and practical cases. Electrons in a Crystal Lattice E Isolated atom, or potential well - quantum dot: bound states , discrete energy levels completely or partiallyfilled by electrons Periodic array of atoms: continuous bands of energy levels, also completely or partially filled V(r) r E V(x) r

Insulators : Wide band gaps between allowed bands: Filled - valent and empty - conduction bands Empty conduction band Forbidden energy gap [Eg] Full valance band Full and empty bands do not take part in electrical conduction.

Metals : • No gap between valance band and conduction band CB CB VB VB Touching VB and CB Overlapping VB and CB

Metals, heavily doped semiconductors, semimetals, zero-point semimetal/semiconductor E E p EF EF -pF pF EF p p -pF pF Semimetal, Ne=Nh(Bi, Sb, graphite, TTF-TCNQ) E Zero point metal (graphene, some nanotubes) Hole metal Electron metal Overall shape for small number of particles: m* - effective mass

Energy Conduction band Impuritylevels: donors-acceptors Valence band g E Metal Semiconductor E F -kF kF -kF kF E EF Creation of charge carriers in the conduction band of semiconductor via doping by impurities – donors or acceptors.

E E Semiconductors: gap between HOMO and LUMO continuum p p Indirect gap Direct gap

Indirect-band gap material; the minimum of the CB and maximum of the VB lie at different k-values. • When an electronand a hole recombine in an indirect-band gap s/c, phonons must be involved to conserve momentum. CB e- E Eg k + VB

Band electron and hole Eg=2Δ E band curvature p E m1* , light mobile m* m2* >>m1* heavy 1/m3* =0, immobile p p

Calculated band structure of GaAs

Band structure of Si

HOMO LUMO Conduction band Eg Valence band Long range Coulomb interactions and excitons Long chains: e and h can go far away from each other Exciton - Coulomb bound e-h pair Free electron and hole Ee+Eh=Eg Eex = Ee+Eh -e2/Rex<Eg Optical transitions: band - to band : E=Eg Free e, h – photoconductivity exciton : E=Eex <Eg photoluminescence

Bound states – semiconducting regime. Attractive point impurity, arbitrary dimension d=1, 2, 3. V(r)=-V0adδ(r) Assumption: the bound state localization length l>>a l -V0 a Wave function: Minimization over l: Energy: optimal y , i.e. optimal l d=1 E d=1 either collapse l→0 or unbinding l→∞ d=2 1/l d=3 E=Ay2-By3stable unbound state y=0, l=∞ d=3

electron hole Long range Coulomb attraction ε – dielectric constant • Particle and the dopant • Electron and hole Minimum at effective Bohr radius aB Hidden mistake: even state, d=1: B→∞ Ln– factor : “capacitance” of elongated cigar of charges logarithmically divergent at both small and large x Cutoffs: fixed a – lattice constant, self-consistent l Result: 4- fold increase of binding for parameters of polymers

Fundamentaledge structure. WKB approximation Exciton sequenceat E<Eg Interaction of unboundparticlesat E<Eg absorption E Eex Eg

Quantization: Normalization: Light absorption to the given level: Averaging over the condensed sequence of levels -1/n2

Band particles – unbound e-h pair : E>Eg With the cutoff at smallest x~a Optical absorption to E>Eg : The Coulomb attraction suppresses the meeting probability, Cancelling the DOS singularity ! The remnant constant value meets the probability accumulated from condensed sequence of excited states at E<Eg

e h e h Suppression of the light emission by the long range Coulomb attraction for free e, h Wrongintuition: attraction keeps e and h together, thus enhancing their recombination (emission probability) But : attraction accelerates the velocities and particles pass by without hand shaking

Optical absorption, long rang Coulomb effects absorption E Eg Eex

Electrons in a Crystal Lattice