Lecture 10Phys 446 Solid State Physics Lecture 10 Nov 16(Ch. 6.8-6.14)Last time: Statistics of charge carriers in semiconductors. Electrical conductivity. Mobility. Today: High electric field and hot electronsOptical properties: absorption, photoconductivity, luminescenceCyclotron resonance and Hall effect in semiconductors with both types of carriers*ecemeB=ω- for electrons *hchmeB=ω- for holesTwo cyclotron frequencies: Cyclotron resonance is used to obtain information on effective masses/shape of energy surfacesSuppose the constant energy surface is an ellipsoid in revolution. B is applied at some angleThe cyclotron frequency is 21222sincos⎥⎥⎦⎤⎢⎢⎣⎡+=lttcmmmeBθθω-depends on effective masses and angle θmeasuring ωcat various angles gives the effective massesBJneEeeL1=Lorentz field for electrons:for holes: BJpeEhhL1−=In steady state, no Jy:0)( =+++=heHhLheLeyEEpeEneJσσµµ()0)( =++−heHhheepeneEBJJµµµµxheJJJ =+())(heeehhHpneBJJEµµµµ+−=heexepnnJJµµµ+=BJpnenpExheehH222)(µµµµ+−=hehxhpnpJJµµµ+=⇒222)(heehHpnenpRµµµµ+−=- used to determine carrier concentration and mobilityHigh electric field and hot electronsElectron drift velocity in Ge vs. electric field for different crystallographic orientations at 300 K (from Landolt-Boernstein - A. Neukermans, G. S. Kino, Phys. Rev. B 7 2693 (1973).J = nev = neµeE0)()(=−−−=⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=EeLTETEvedtEddtEddtEdτEEτE– energy relaxation timev – electron drift velocityeBeTkTE23)( =TkTEB23)( =⇒232EBeEekeTTµτ+=Negative differential conductance and Gunn effectConduction band in GaAs• 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 energygap, ∆ E, of approximately 0.31 eV.Optical absorption processes1. The fundamental absorption processdirect process at k = 0 (zone center)powerful method to determine the energy gap EgDirect and Indirect Gaps2. Exciton absorptionfundamental absorption(theory)Exciton absorption in Ge(experiment)3. Free carrier absorptionintraband transition – like in metalscan occur even when the photon energy is below the bandgapdepends on free carrier concentration –more significant in doped semiconductors4. Absorption involving impuritiesa) neutral donor → conduction bandb) valence band → neutral acceptorc) valence band → ionized donor ionized acceptor → conduction bandd) ionized acceptor → ionized donor PhotoconductivityPhenomenon in which a material becomes more conductive due to the absorption of electromagnetic radiation"dark" conductivity: )(000 hepneµµσ+=Light absorbed: electron-hole pairs created; carrier concentrations increased by ∆n, ∆p ∆n = ∆pnew conductivity: )(0 heneµµσσ+∆+=0000)(σµµσσσσσhene +∆=−=∆Two opposite processes affecting ∆n: •generation of free carriers due to absorption, rate g•recombination; lifetime of carriers τ''0τnngdtdn−−=0=dtdnIn steady state⇒∆n = n - n0= gτ'Evaluate g per unit volume through absorption coefficient α and slab thickness d:N(ω) – number of photons incident per unit time:Then00)(')(ωσµµτωασσ=heIe+=∆Change in conductivity: ασσ∝∆0)(0ωσσI∝∆andnumerical estimate: if(for Ge), get ∆n ~ 5×1014cm-3LuminescenceRadiative recombination of charge carriersClassification by excitation mechanisms: - photoluminescence- electroluminescence- cathodoluminescence- thermoluminescence- chemiluminescenceSame physical processes as for absorption, but in opposite directionSummaryConductivity of semiconductors: mobility:Cyclotron resonance is used to obtain information on effective masses.Hall coefficient: Hall measurements are 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 instabilities (e.g. Gunn effect caused by negative differential conductivity due to inter-valley transfer)Mechanisms of optical absorption and luminescence. Fundamental absorption occurs above the bandgap.photoconductivity – increase of conductivity by generation of additional carriers by electromagnetic radiation222)(heehHpnenpRµµµµ+−=Carrier DiffusionIn general, total current in a semiconductor involves both electrons and holes (in the presence of both a concentration gradient and an electric field):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:Electric field (V - potential) By applying a field, all energies will be pushed up by the potential V:⇒ can writeHave⇒Substitute this into the diffusion equation, Get⇒Einstein relationSimilarly, for holesDiffusion equation for one carrier typeEpexpeDJpppµ+∂∂−=(holes, one dimension)Variation of p(x) in time is given by continuity equation:xJeUGtpppp∂∂−−=∂∂ 1generation recombination flowAssume there is no external excitation, i.e., Gp=0.pcombRep'pptpUτ0−−=⎟⎠⎞⎜⎝⎛∂∂=()ppp'pppExxpDtpτµ022−−∂∂−∂∂=∂∂Thenτ’p- lifetime of holes- Diffusion equationRecombination term:1) Stationary solution for E = 0:0=∂∂tp0'022=−−∂∂ppppxpDτlet p - p0= p1. Then ()21'01ppDxAepppτ−=−=The excess concentration decays exponentially with x. The distance is called the diffusion length()21'ppDDLτ=Effective diffusion velocity:21''⎟⎟⎠⎞⎜⎜⎝⎛==ppDDDLvττDiffusion current:2111'⎟⎟⎠⎞⎜⎜⎝⎛==pDDDepvepJτLDxp1holeinjection0E = 0E ≠ 01) Stationary solution for a uniform field E ≠ 0:021122=−∂∂−∂∂DppLpxpDExpµDLxAepγ−=1⇒Where ss −+=21γpDpDELs2µ=and γ< 1 ⇒ effective diffusion length LD/γis largerSummary of the semiconductors sectionSemiconductors are mostly covalent crystals; They are characterized by moderate energy gap (~0.5 – 2.5 eV) between the valence and conduction bandsWhen impurities are introduced, additional states are created in the gap. Often these states are close to the bottom of the conduction band or top of the valence bandIntrinsic carrier concentration: strongly depends on temperature. Fermi level position in intrinsic semiconductor:= p =