Phys 446 Solid State Physics Lecture 10 Nov 16 (Ch. 6.8-6.14) Last Time

Phys 446 Solid State Physics Cyclotron resonance and Hall effect in semiconductors Lecture 10 Nov 16 with both types of carriers (Ch. 6.8-6.14) Two cyclotron frequencies: eB eB ωce = - for electrons ω = m* ch * - for holes Last time: Statistics of charge carriers in semiconductors. e mh Electrical conductivity. Mobility. Cyclotron resonance is used to obtain information on effective masses/shape of energy surfaces Suppose the constant energy surface is an ellipsoid Today: High electric field and hot electrons in revolution. Optical properties: absorption, photoconductivity, B is applied at some angle luminescence The cyclotron frequency is 1 2 ⎡cos2 θ sin 2 θ ⎤ ω = eB + c ⎢ 2 ⎥ -depends on effective masses and angle θ ⎣⎢ mt mt ml ⎦⎥ measuring ωc at various angles gives the Lecture 10 effective masses High electric field and hot electrons Lorentz field for electrons: e 1 1 EL = J e B for holes: h ne EL = − J h B pe J = nev = neµeE e h In steady state, no Jy : J y = neµe EL + peµh EL + EH (σ e +σ h ) = 0 d E ⎛ d E ⎞ ⎛ d E ⎞ = ⎜ ⎟ + ⎜ ⎟ dt ⎜ dt ⎟ ⎜ dt ⎟ ()µe Je − µh J h B + EH (neµe + peµh ) = 0 ⎝ ⎠E ⎝ ⎠L E(Te ) − E(T ) nµe pµh = −eEv − = 0 J e + J h = J x Je = J x J h = J x τ E nµe + pµh nµe + pµh τE – energy relaxation time 2 2 v – electron drift velocity ()µ J − µ J B pµh − nµe E h h e e E = J B H = H 2 x 3 3 e(nµe + pµh ) e(nµe + pµh ) E(T ) = k T E(T ) = k T e 2 B e 2 B Electron drift velocity in Ge vs. electric 2 2 2 eτ E µe 2 pµh − nµe - used to determine carrier ⇒ Te = T + E field for different crystallographic ⇒ RH = 3 kB orientations at 300 K (from Landolt- e(nµ + pµ )2 concentration and mobility e h Boernstein - A. Neukermans, G. S. Kino, Phys. Rev. B 7 2693 (1973). Negative differential conductance and Gunn effect Optical absorption processes Conduction band in GaAs 1. The fundamental absorption process • In the lower Γ valley, electrons exhibit a small effective mass and very high mobility, µ1. • In the satellite L valley, electrons exhibit a large effective mass and very low mobility, µ2. • The two valleys are separated by a small energy gap, ∆ E, of approximately 0.31 eV. direct process at k = 0 (zone center) powerful method to determine the energy gap Eg Direct and Indirect Gaps 2. Exciton absorption Exciton absorption in Ge (experiment) fundamental absorption (theory) 3. Free carrier absorption 4. Absorption involving impurities intraband transition – like in metals can occur even when the photon energy is below the bandgap depends on free carrier concentration – more significant in doped semiconductors a) neutral donor → conduction band b) valence band → neutral acceptor c) valence band → ionized donor ionized acceptor → conduction band d) ionized acceptor → ionized donor Photoconductivity Evaluate g per unit volume through absorption coefficient α and slab thickness d: Phenomenon in which a material becomes more conductive due to the absorption of electromagnetic radiation "dark" conductivity: σ 0 = e(n0µe + p0µh ) N(ω) – number of photons incident per unit time: Light absorbed: electron-hole pairs created; Then carrier concentrations increased by ∆n, ∆p ∆n = ∆p new conductivity: ∆σ σ −σ 0 e∆n(µe + µh ) ∆σ αI(ω)τ '(µ + µ ) σ = σ 0 + e∆n(µe + µh ) = = Change in conductivity: = e e h σ σ σ 0 0 0 σ 0 =ωσ 0 Two opposite processes affecting ∆n: dn n − n ∆σ ∆σ • generation of free carriers due to absorption, rate g = g − 0 ∝ α and ∝ I(ω) dt τ ' σ 0 σ 0 • recombination; lifetime of carriers τ' dn numerical estimate: if In steady state = 0 ⇒ ∆n = n - n = gτ' dt 0 (for Ge), get ∆n ~ 5×1014 cm-3 Luminescence Summary Conductivity of semiconductors: Radiative recombination of charge carriers mobility: Classification by excitation mechanisms: - photoluminescence Cyclotron resonance is used to obtain information on effective masses. - electroluminescence pµ 2 − nµ 2 R = h e Hall coefficient: H 2 Hall measurements are used - cathodoluminescence e(nµe + pµh ) - thermoluminescence to determine carrier concentration and mobility. In high electric field, the carriers acquire significant energy and become - chemiluminescence "hot". This affects mobility and can cause current instabilities (e.g. Gunn effect caused by negative differential conductivity due to inter-valley transfer) Same physical processes as for absorption, Mechanisms of optical absorption and luminescence. but in opposite direction Fundamental absorption occurs above the bandgap. photoconductivity – increase of conductivity by generation of additional carriers by electromagnetic radiation Carrier Diffusion In general, total current in a semiconductor involves both Electric field (V - potential) electrons and holes (in the presence of both a concentration gradient and an electric field): By applying a field, all energies will be pushed up by the potential V: The second term in the above equations is the diffusion current (Fick’s law). It arises from non-uniform carrier density. In one dimension, for the negative carrier: At equilibrium, the drift and diffusion currents are equal: ⇒ can write Have Diffusion equation for one carrier type ∂p ⇒ J = −eD + peµ E (holes, one dimension) p p ∂x p Variation of p(x) in time is given by continuity equation: ∂p 1 ∂J = G −U − p ∂t p p e ∂x Substitute this into the diffusion equation, generation recombination flow Assume there is no external excitation, i.e., G =0. Einstein p Get ⇒ relation ⎛ ∂p ⎞ p − p Recombination term: U = ⎜ ⎟ = − 0 τ’p - lifetime p of holes ⎝ ∂t ⎠Recomb τ' p Then ∂p ∂ 2 p ∂ p − p Similarly, for holes 0 = Dp 2 − µ p ()pE − - Diffusion equation ∂t ∂x ∂x τ' p ∂p p 1) Stationary solution for E = 0: = 0 1 ∂t hole E ≠ 0 2 injection ∂ p p − p0 Dp 2 − = 0 ∂x τ ' p E = 0 1 2 0 x −x (Dpτ ' p ) L let p - p0 = p1. Then p1 = p − p0 = Ae D The excess concentration decays exponentially with x. 1) Stationary solution for a uniform field E ≠ 0: 1 2 2 The distance L = ()D τ ' is called the diffusion length ∂ p µ p E ∂p p D p p − 1 − 1 = 0 ⇒ p = Ae−γ x LD 1 2 ∂x2 D ∂x L 2 1 L ⎛ D ⎞ p D v = D = ⎜ ⎟ Effective diffusion velocity: D ⎜ ⎟ µ EL τ ' τ ' 2 p D p ⎝ p ⎠ Where γ = 1+ s − s and s = 2D 1 2 p ⎛ ⎞ ⎜ D ⎟ Diffusion current: J D = ep1vD = ep1 ⎜ ⎟ γ < 1 ⇒ effective diffusion length LD/γ is larger ⎝τ ' p ⎠ Summary of the semiconductors section Summary of the semiconductors section Semiconductors are mostly covalent crystals; They are characterized In a doped semiconductor, many impurities form shallow hydrogen- by moderate energy gap (~0.5 – 2.5 eV) between the valence and like levels close to the conductive band (donors) or valence band conduction bands (acceptors), which are completely ionized at room T: When impurities are introduced, additional states are created in the n = Nd or p = Na gap. Often these states are close to the bottom of the conduction band or top of the valence band Conductivity of semiconductors: Intrinsic carrier concentration: mobility: = p = ni Magnetic field effects: strongly depends on temperature. Cyclotron resonance is used to obtain information on effective Fermi level position in intrinsic semiconductor: eB eB masses. ωce = * - for electrons ωch = - for holes me * mh pµ 2 − nµ 2 R = h e Hall coefficient: H 2 Hall measurements are e(nµe + pµh ) used to determine carrier concentration and mobility. In high electric field, the carriers acquire significant energy and become "hot". This affects mobility and can cause current Basics of selected semiconductor devices: instabilities (e.g. Gunn effect caused by negative differential conductivity due to inter-valley transfer) p-n junctions. Mechanisms of optical absorption and luminescence: Bipolar transistors. band-to-band excitonic Tunnel diodes. free carrier Semiconductor lasers impurity-related Fundamental absorption occurs above the bandgap. photoconductivity – increase of conductivity by generation of additional carriers by electromagnetic radiation Diffusion. Basic relations are Fick's law and the Einstein relation p-n junction In equilibrium, at zero bias, the chemical potential has to be the same at both sides: bending of the conduction and valance bands Charge density near the junction is not uniform: electrons (majority carriers) from the n-side and holes (majority carriers) from the p-side will migrate to the other side through the junction. These migrated particles leave the ionized impurities behind: a charged region is formed The band edge shift across the junction is called the built in voltage Vbi. Recall that for carrier concentration we had N N µ = µ ⇒ E − kT ln c = E − E + kT ln v ⇒ 3 2 n p cn cp g 3 2 ⎛ m kT ⎞ N D N A ⎛ m kT ⎞ µ−E kT µ−E kT e e ()c ()c Nc = 2⎜ ⎟ n = 2⎜ ⎟ e = Nce 2 ⎝ 2π=2 ⎠ ⎝ 2π= ⎠ N N N N V = E − E = E − kT ln c − kT ln v = E + kT ln A D 3 2 3 2 bi cp cn g g m kT N D N A Nc Nv ⎛ h ⎞ ()Ev −µ kT ()Ev −µ kT ⎛ mhkT ⎞ p = 2⎜ ⎟ e = Nve Nv = 2⎜ ⎟ ⎝ 2π=2 ⎠ ⎝ 2π=2 ⎠ 3 2 3 2 For n-type semiconductor (n, ND >> p, NA) n = ND ⇒ Note that ⎛ m kT ⎞ ⎛ m kT ⎞ −E kT −E kT n2 = 4 e h e g = N N e g i ⎜ 2 ⎟ ⎜ 2 ⎟ c v Nc ⎝ 2π= ⎠ ⎝ 2π= ⎠ µn = Ec − kT ln N D For p-type semiconductor (p, NA >> n, ND) p = NA ⇒ N N ⇒ V = kT ln A D bi 2 Nv ni µ p = Ev + kT ln N A Use the above to calculate Vbi Reverse bias Forward bias draws electrons and holes away from the n-side and holes from the “pushes” electrons in the n-side and holes in the p- p-side.

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