Lecture 16: Free Electrons (Optical Properties)

Electron bands originate from atomic levels:

Three band structure scenario

Dielectrics Semiconductors CB CB Metals CB E > 3 eV E < 3 eV g EF g EF EF

VB VB CB: A lot of available CB: A lot of available CB: A lot of electrons states, no electrons states, no electrons at T = 0 and available states ⇒ VB: A lot of electrons, no VB: A lot of electrons, no free electron model with states available states available strong conductivity Materials Application of the Lorentz oscillator model to free electron systems: • Metals • Doped semiconductors

Plasma reflectivity Combination: • Lorentz model of dipole oscillator (No restoring force in plasma) • Drude model of free electron conductivity (τ - scattering time)

Oscillation of a free electron induced by AC electric field E(t) of EM wave:

2 γ d x dx m + m = −eE(t) = −eE e−i t 0 dt 2 0 dt 0 −iωt x = x0e Damping term ω eE(t) 1/γ = τ - mean scattering time x(t) = ω 2 m0 ( + iγω) Macroscopic polarization: P = N(-e)x(t) ε ε 2 ε Ne E D = r 0 E = 0 E + P = 0 E − 2 ε m0 ( + i ) ε ω Ne2 ( ) =1− ω r ε 2 0m0 ( + i ) γω Formally follows from ε ω ε bound e-resonace: ω 2γω ω p r ( ) =1− 2 ω ( + i ) ε Ne2 ω 1 1 γω =1+ ω ω ⎛ Ne2 ⎞ 2 r 2 2 ⎜ ⎟ 0m0 ( j − − iγω) p = ⎜ ⎟ ⎝ ε 0m0 ⎠

Under ωj = 0 - No restoring force

ωp – plasma frequency ε ω ω ω ε ω εr In the limit γ⇒0 ω 1 ε 2 ω ( ) =1− p r 2 0 p ω 2 p 1 =1ε− 2ε ε = 0 2 n 1 ε 1 2 ε 1 1 ⎡ 2 2 ⎤ 1 n = ε+ ( + ) 2 = []+ 2 ⎢ 1 1ε 2 ⎥ 1 1 2 ⎣ ε ⎦ 2 ω 1 1 2 1 1 ⎡ 2 2 ⎤ 1 ε 2 2 ω k = ⎢− 1 + ( 1 + 2 ) ⎥ = []− 1 + ε1 p 2 ω 2 k ⎣ ω ⎦ ω < ⇒ n = 0,k = ω p ε 1

> p ⇒ n = ε1 ,k = 0 ωp ω

Lecture 17: Free Electrons (Optical Properties) Main Effects in relation to Plasma Reflectivity Model Plasma model in relation to 1. High reflectivity in visible and NI regimes dipole oscillator model combined with high transmission in UV • Plasma model without damping force, High R for ω < ωp,, high T for ω> ωp neglecting εr with returning 2. Skin Effect at low frequencies 1 ω force 0 • Plasma model with damping force, p ω Absorption for ω <1/τ, τ - scattering time

3. Interband transitions in metals • Quantum process in E-k band diagram, superimposed on the plasma model

4. Free carrier absorption in semiconductors

• Shift of ωp to much lower frequencies, Plasma model with damping forω > ωp 5. Plasmons

• Plasma model at ω = ωp where εr = 0, Restoring force due to macroscopic polarization High reflectivity in visible and NI regimes combined with high transmission in UV

Aluminium

hνp = h ω p = 15.8 eV

σ = 3.6 107 Ω-1 m-1

2 τ = 1/γ = m0σ/(Ne ) – determines the absorption at low ω, as we will see below, σ - DC conductivity Skin Effect at low frequencies Model of conduction E E = 0 + + + + + + + + + + + + + + + + Electron + + + + + + + + + + + + + + + + Net displacement superimposed Frequent collisions, but on the random thermal motion no net displacement ∆t vx vix τ = <∆t> vx v

t t v = = + (eE/m)<∆t>= (eτ/m)E

i = eNAv= (Ne2τ/m)AE ⇒ j = i/A = σE

v = = 0 where σ= Ne2τ/m

Interband transitions in metals

Explains color of such metals as gold and copper

•Very high density of states for • Inner d orbitals form narrow bands below EF • Transition to half-full s band gives an parallel bands absorption • Hence reflectivity dip at 1.5 eV Free carrier absorption and reflectivity in semiconductors

Plasma frequency is

reduced due to εr > 1 and * m > m0:

2 * 1/2 ωp = (Ne /ε0εrm )

•Control plasma frequency by varying the doping density • Plasma edge in the infrared Plasmons

Longitudinal oscillations of the electron plasma at ωp where εr = 0

d 2u ⎛ N 2e2 ⎞ Nm = −NeE = −⎜ ⎟u 0 2 ⎜ ⎟ dt ⎝ 0 ⎠ ε d 2u ⎛ Ne2 ⎞ +ε⎜ ⎟u = 0 dt 2 ⎜ m ⎟ ω ⎝ 0 0 ⎠ 1 2 2 0 = ()Ne / ε 0m0

Metals: h ω p ~3-20 eV

Doped demiconductors: h ω p ~ 10 meV

Measure by Raman Scattering: hωout = hωin ± nhω p Lectures 18: Interband Absorption Direct and Indirect Absorption The Transition Rate for Direct Absorption π Transition Rate: The matrix element M 2 2 W i → f =ψ M g (ωh ) Density of states g h M = * (r ) H ' (r ) ψ (r ) d 3 r The first-order energy shift for a ∫ f i given state is the expectation value of the perturbing potential ' H = − p e ⋅ E photon Quantum perturbation ± ikr E photon (r ) = E 0 e ' ± ikr H (r ) = e E 0 re 1 ψ ( r ) = u (r ) e ± ik i r i V i

1 ± ik r ψ ( r ) = u (r ) e f f V f Continuing

e −ik r M = u* (r)e f (E ⋅re±ikr )u (r)e−ikir d 3r V ∫ f 0 i hk f − hki = ±hk Required for the integral to be non zero M ~ u* (r)xu (r)d 3r ∫ i f For x-polarized light unit _ cell M – electric dipole moment of the transition k f = ki 3 1 ⎛ 2m* ⎞ 2 1 g(E) = ⎜ ⎟ E 2 Obtained previously in 3-D case, 2 ⎜ 2 ⎟ 2π ⎝ h ⎠ m* instead of m Atomic Physics of Semiconductors

•∆m = -1, 0, +1 • ∆l = ±1

•∆ms = 0

Ge: 4s24p2 Bonding and Antibonding Modes in Bispherical “Atoms”

Illustrated by coupled Whispering Gallery Modes in Dielectric Bispheres

Antibonding Bonding

• Due to different fractions in high (dielectric spheres) and low (air) index media these modes have different energy ⇒ anticrossing splitting Lectures 19: Interband Absorption (Continue)

GaAs Band Structure Four-Band Model Joint Density of States µ

ω E < 3 eV g EF 1 1 1 = + * * VB mωe mh 2 2 2 2 2 2 h k hω k µh k h = Eg + * + * = Eg + ω 2me 2mh 2 For _ < E , _ g( ) = 0 h g hω µ 3 1π⎛ 2 ⎞ 2 1 2 For _ h > Eg , _ g(h ) = ⎜ 2 ⎟ ()hω − Eg 2 ⎝ h ⎠ InAs Band Edge Absorption The Franz-Keldysh Effect • At the classical turning points A and B, ψ change from oscillatory to decaying: ikx ψ ~ uke , where k is imaginary

• With increase of E the distance AB decreases and the overlap of ψ increases

• In the absence of photons AB = d = Eg/qE

• For hν < Eg AB = d’= (Eg - hν)/qE – αtunneling barrier is reduced ⇒ absorption First effect ω •The frequency dependence of α is given:

⎛ * 3 ⎞ 4 2me ( ) ~ exp⎜− ()E − ω 2 ⎟ h ⎜ 3e E g h ⎟ ⎝ h el ⎠ • Transmission coefficient through a triangular barrier

Second Effect • For hν > Eg α displays Franz-Keldysh oscillations Magneto-Absorption in Germanium Cyclotron Motion

Application in particle physics:

2 F = qvB = mar = mv /r rcyc = mv/(qB) • ωcyc is independent of the particle speed fcyc = v/2πr = qB/(2πm)

ω = qB/m • Each time traversing the gap the cyc particle gains kinetic energy e∆V Magneto-Optics in Semiconductors

In quantum physics the energy is quantized. In can be shown that this problem can be formally reduced to a harmonic oscillator potential: U = kx2/2 ⇒

En = (n+1/2)ħωc - Landau Levels

The ground state E0 = ħωc/2 is explained by the uncertainty principle: ∆p∆x~ħ ψ(x) ψ2(x)

H = p2/2m + kx2/2 Band Edge Absorption in a Magnetic Field

2 2 e ⎛ 1 ⎞ ehB h kz En (kz ) = Eg + ⎜n + ⎟ * + * ⎝ 2 ⎠ me 2me 2 2 h ⎛ 1 ⎞ ehB h kz En (kz ) = −⎜n + ⎟ − ⎝ 2 ⎠ m* 2m* ω h e It can be shown that the Landau level number n does not change in transition: 2 2 e h ⎛ 1 ⎞ ehB h kz hω = En (kz ) − En (kz ) = Eg + ⎜n + ⎟ + ⎝ 2 ⎠ 2 ⎛ 1 ⎞ ehB h = Eg + ⎜n + ⎟ µ ⎝ 2 ⎠ µ µ

In a 1-D system the density of states is peaked at kz=0

Explains a series of equally spaced peaks

Blue shift of the absorption edge by ħeB/2µ Lectures 20: Interband Absorption (Continue)

Direct Versus Indirect Absorption

Indirect transitions:

E = E + ω ± Ω Energy conservation f i h h Initial state: (Ei, ki) Final state: (Ef, kf) hk f = hki ± hq Momentum conservation Germanium Band Structure Germanium Band Edge Absorption Silicon Absorption Band Structure Absorption Spectroscopy Photodetectors p-i-n Diodes Lectures 21: Excitons Theory: Frenkel 1931 (Ioffe Institute) – exciton is small and tightly bound Mott and Wannier – exciton is large and weakly bound

Experimental discovery (1951) of hydrogen-like exciton optical spectrum in semiconductor crystals of cuprous oxide, E.F. Gross (Ioffe Institute)

An electron and a hole may be bounded together by their attractive coulomb interaction, just as an electron is bound to a proton to form a hydrogen atom. Properties

• An exciton can move through the crystal and transport energy

• It does not transport charge because it is electrically neutral

• Excitons can be formed by photon absorption at any critical point, for if ∇kεv = ∇kεc the group velocities for electrons and holes are equal and they can be bounded.

• In direct band gap semiconductors the excitons can be formed at k = 0 corresponding to energy Eg

• When the band gap is indirect, excitons near the direct gap can be unstable with respect to decay into a free electron and free hole.

• The energy of the exciton formation in a direct transition at k = 0 is Rx less the binding energy due to the Coulomb interaction:

2 En = Eg – Rx /n , where Rx – exciton binding energy Exciton Levels Conduction

Band, me

Conduction Band Continuum E Exciton g Exciton Exciton Levels Binding E Levels b Energy

Eg-Eb Energy

Gap, Eg

Valence 0 Band, mh Valence Band Continuum

• An excition can have translational kinetic energy • Energy levels of an exciton created in a • Excitions can recombine direct process rediatively Bohr model of hydrogenic atoms

Bohr Postulates:

1. Angular momentum is quantized (stationary orbits): mvr = nh/2π (1) 2. Transitions at hν = E – E’

Let us consider nuclear charge Ze, electron –e, radius r.

Coulomb Force = Centrifugal Force (CGSπ ) _ units : Ze 2 mv 2 = _ Combined _ πwith _(1) : r 2 r 1 n 2 h 2 2 Ze 2 r = _ and _ v = 4 2 Ze 2 m hn 1 Ze 2 2π 2 e 4 Z 2 m The _ energy : _ E = mv 2 − = − 2 r h 2 n 2 Weakly Bound (Wannier-Mott) Excitons

For Z = 1 E

4 0 e m0 R = 2 =13.6eV − Rydberg _ Const 2h n=2 2 R a = h = 5.29⋅10−9 cm ≈ 0.5Α m e2 0 n=1

For an exciton in a solid the “Rydberg Constant” can be calculated taking

into account polarization of lattice (εr) and effective mass (µ). Such a “Rydberg Constant” is the exciton binding energy Rx.

µ For Ge, Si and AIIIBV (GaAs) and AIIBVI semiconductors: Rx = R 2 εm0 r ε µ = memh/(me + mh) ~ me~0.1m0, εr ~ 10 ⇒ r m0 ax = a µ Rx ~ 10 meV << Eg , ax ~ 10 nm >> Lattice Constant More detailed data on Rx and ax

• As Eg increases Rx increases and ax decreases

• This can be explained by

decreasing εr and increasing µ for larger Eg

• Rx is smaller than kT at Eg room temperature for all semiconductors

• In insulators Eg ≥ 5 eV ax is comparable to the lattice constant ⇒ Wannier-Mott model is no longer valid Frenkel Excitons • A Frenkel exciton is essentially an excited state of a single atom, but the excitation can hop from one atom to another by virtue of the coupling between neighbors. • Can be present in extremely wide band gap materials. • Stable at room temperature

Phenomenological Treatment Based on interaction constant (T), see Kittel ϕ ψ g = u1u2 ⋅⋅⋅uN −1uN Consider a crystal of N noninteracting atoms

Describe a single excited atom and N-1 atoms jϕ= u1u2 ⋅⋅⋅u j−1v ju j+1 ⋅⋅⋅uN ϕ If there is an interaction between excited atom ϕ and its close neighbors when T measures the ϕ rate of transfer of the excitation. Here E is the Hψj = E j +T ( j−1 +ϕ j+1) free atom excitation energy. ϕ If T = 0 when the solution is ϕj. ψ k = ∑exp(ijka) j If T ≠ 0 when use a Bloch form… j ϕ ϕ ijka ijka Let H operate on ψ H k = ∑e H j =∑e []E j +T ( j−1 +ϕ j+1) k j j Rearrange the right-hand side to obtain: ψ ijka ika −ika H k = ∑e [E +T (e + e )] j =(E + 2T cos ka) k j E = E + 2T cos ka So, the energy eigenvalues are found k ϕ 1 1 1 k = 2πs / Na, _ s = − N,− N +1,⋅⋅⋅ N −1 2 2 2 ψ

Large Eg:

Rare Gas Crystals Ne, Ar, Kr, Xe

Alkali halides: KI, KBr, KCl , KF NaI,…

Molecular thin films and organic crystals Lectures 22: Excitons

Free Exciton Absorption Excitons in Bulk GaAs

The dominating mechanisms of exciton broadening are impurity scattering (low T) and scattering on LO phonons (high T). Since ELO = 35 meV>>kT the occupancy of LO phonons is small ⇒ collisional broadening is small even at high temperatures. Field Ionization in GaAs

Excitonic effects do not play a large part in the physics of bulk semiconductor diodes. The dominant effect is Franz-Keldysh effect. This however is not true for QWs! Magnetic Fields ω eB ω – exciton cyclotron frequency = c h c h µ µ - reduced electron-hole effective mass

Weak Field Limit Strong Field Limit

RX << ħωc R >> ħω X c Landay Levels for individual The magnetic field is a electron and holes perturbation on excitons. Coulomb interaction is a small For a ground state (n=1) it perturbation will be a diamagnetic shift: δ In GaAs the transition occurs e2 E = r 2 B2 at T ~ 2T for n=1 12µ n Induced magnetic moment Free Excitons at High Densities

Exciton-Exciton interaction

• Exciton-exciton interactions are controlled by their concentration

• The excitons wavefunction overlap occurs when their

separation is equal to ax. Mott density:

1 N ≈ Mott 4 πr 3 3 n

rn – radius for the excitons in the n-th quantum state.

23 -3 For excitons in GaAs for the n=1 states NMott = 1.1× 10 m Screening by an electron-hole plasma

Electron-hole pairs generated either directly or in the cause of excitons collisions screen the excitons ⇒ broadening and weakening of the excitonic feature Excitons against hydrogen atoms

Exctions Hydrogen atoms

Bound state of two “atoms” Biexcitons Gaseous phase H2 molecule

Electron-hole “Atoms” condense to form a liquid Liquid hydrogen droplets

Bose-Einstein statistics: the occupancy factors are unlimited. At low T system of N “atoms” switches into coherent state.

Criterion: “Atomic” de Broglie wavelength is comparable with the interparticle separation 3 2 ⎛ mkBTc ⎞ N = 2.612⎜ 2 ⎟ ⎝ 2πh ⎠

Bose-Einsten Bose-Einsten condensate in a solid condensate in a trap