8. Semiconductor Crystals Band Gap Equations of Motion Intrinsic Carrier Concentration Impurity Conductivity Thermoelectric Effects Semimetals Superlattices

Transistors, switches, diodes, photovoltaic cells, detectors, thermistors, … IV: Si, Ge III-V: InSb, GaAs II-VI: ZnS, CdS IV-IV: SiC Strong T dependence Insulator: ρ > Ω cm IIIIVVVI BCNO AlSiPS GaGeAsSe InSnSbTe TlPbBiPo SemiC: 10 9 > ρ > 10 –2 Ω cm

E g = 0.66 eVE g = 1.11 eV Intrinsic temperature range: σ indep of impurities

Band Gap Excitons not shown For γ & e of same energy,For ph & e of same k, ph emitted, low T.

InSb Another way for determining E g : σ i (T) or n i (T) determined from R H.

IIIIVVVI BCNO AlSiPS GaGeAsSe InSnSbTe TlPbBiPo

Equations of Motion Physical Derivation of  k = F Holes Effective Mass Physical Interpretation of the Effective Mass Effective Masses in Semiconductors Silicon and Germanium

Wave packet:group velocity Particle subjected to force F: → Lorentz force: → Particles in contant B field move on surface of constant energy perpendicular to B.

Physical Derivation of  dk/dt = F Plane wave expansion: Electron momentum:

Holes → ε = 0 at top of valence band: → no spin-orbit interaction: Inversion symmetry →

see next section e moves toward –k x ; so does h C.B. V.B.  E E  E E

Effective Mass = effective mass tensor ( of electrons ) Near zone boundary : CB VB CB VB m* < 0 near top of VB U << λ g/2

Physical Interpretation of the Effective Mass PW k + Bragg reflected k−G ( p transferred to lattice) vice versa C 0 / C −1 =  1 → standing wave m* < 0 m* > 0 m* < 0

Effective Masses in Semiconductors m* determined by cyclotron resonance (rf) at low carrier concentration. Condition for complete orbit without collison: cyclotron frequency Landau levels: For m* = 0.1 and ω c = 24GHz, we have B = 852 gauss.

Prob 9.8 → m*  E g for direct-gap crystals For InSb, InAs, InP E g from Table 1

Silicon and Germanium VB at k=0 : p 3/2 + p 1/2

CB of Ge with B in (110). CB edge at L. 4 mass spheroids along [111]; 2 of which are equivalent in (110) plane. m l = 1.59 m, m t = m. θ = angle with longitudinal axis

Si GaAs Direct-gap Spheroids along. CB edges on Δ line near point X. m l = 0.92 m, m t = 0.19 m. A=−6.89, B=−4.5, C=6.2, Δ =0.341 Isotropic m c = 0.067m.

Intrinsic Carrier Concentration Near CB edge: Isotropic band:

Near VB edge: Isotropic band: → np values at 300K: (independent of doping)

Black body radiation: At equilibrium:= const at given T Intrinsic carrier concentration: Carrier compensation: n+p is reduced by increasing either n or p through doping. Pure sample: →

Intrinsic Mobility Mobility μ of single type of carriers: Electrical conductivity σ of semiconductor: μ h < μ e due to interband scattering Ionic crystals: h moves by hopping. Self-trapped via Jahn-Teller effect E g small → m* small → μ large, esp D-G

Impurity Conductivity Donor States Acceptor States Thermal Ionization of Donors and Acceptors Stoichiometric deficiency → Deficit semiconductors Impurities → Doped semiconductors e.g., 10 –5 B → σ = 10 3 σ i for Si at 300K

Donor States Donor = Impurity atom that tends to give up an electron Bohr model: Bohr radius: Valid when a d >> atomic distance. & E d << E g. Anisotropy need be considered for Si & Ge IIIIVVVI BCNO AlSiPS GaGeAsSe InSnSbTe TlPbBiPo

Bohr model Impurity band formed at low impurity concentrations. Mott (metal-insulator) transition. Conduction in impurity band is by hopping. Occurs at lower concentration in compensated materials.

Acceptor States Acceptor = Impurity atom that tends to capture an electron IIIIVVVI BCNO AlSiPS GaGeAsSe InSnSbTe TlPbBiPo Complication: VB degeneracy.

Ultra pure Ge: imp conc < 10 –11 active impurities ~ 2  cm −3 intrinic region Electrically inactive impurities in Ge: H, O, Si, C. Can’t be reduced below – cm –3. at T = 300K with ρ i  43 Ω cm

Thermal Ionization of Donors and Acceptors No acceptors present: Reminder: Extrinsic region:

Thermoelectric Effects Electrical conductivity: Thermal conductivity: Seebeck effect: Peltier effect: S = Seebeck coeff. (Thermal power) b = carrier mobility Π = Peltier coeff. Heat current density J Q : →Steady state → Kelvin relation (derived from thermodynamics)

Thermoelectric Effects: Boltzmann Eq Ref: Haug, IV.B. Kittel, App F Boltzmann eq.: Relaxation time approximation Linearization: semicond metals A-current density:

For isotropic materials, J A is a linear combination of integrals

→ Electrical conductivity: Thermal conductivity: Seebeck effect: Peltier effect: Seebeck coeff. (Thermal power) Kelvin relation (derived from thermodynamics) For spherical energy surfaces: b = carrier mobility

Peltier coefficent of Si

Semimetals 2 Group V atoms in primitive cell → insulator Band overlap → semimetal

Superlattices Superlattice: lattice with long period created by stacking layers of atoms. Ref: J.Singh,”Physics of Semiconductors & Their Heterostructures” (GaAs) 1 (InAs) 1 (GaAs) 2 (InAs) 2

Bloch Oscillator Bloch Oscillator: For a collisionless electron accelerated across a Brillouin zone, the motion is periodic. A = superlattice constant along E Bloch frequency Simple TBM:

Zener Tunneling (field-induced interband tunneling): Tilting of band by E → different bands at same ε → Zener tunneling (breakdown) Heavily doped p-n junction Strong reverse bias → Zener breakdown I-V curve