chap2 - ' $ ECE432 Excess Carriers in Semiconductors...

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Unformatted text preview: ' $ ECE432 Excess Carriers in Semiconductors Chapter 2 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 0 The Ohio State University % ' $ Lecture # 4 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 1 The Ohio State University % ' $ Carriers Concentrations in Equilibrium o T ( K) In equilibrium the carrier concentrations verify: 9 EF ;Ei (x) > > = n0 (x) = ni exp kT ! n0p0 = n2i (T ) E (x);EF > > p0 (x) = ni exp i kT In equilibrium the Fermi level EF is constant through out the device. & Patrick Roblin 2 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Departure from Equilibrium −+ Equilibrium is perturbed when we inject energy in the system: light (Chapter 4) applied voltage (Chapter 5) mechanical stress (piezzo electric e ect) & Patrick Roblin 3 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Excess Carriers Away from equilibrium the carrier concentrations depart from their equilibrium values: 9 F (x);E (x) > > = n = n0 + n = ni exp n kT i ! np 6= n2i (T ) Ei (x);Fp (x) > > p = n0 + p = ni exp kT The Fermi level EF is not constant but we can introduce a spatially varying quasi-Fermi levels Fn(x) and Fp(x) for electrons and holes. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 4 The Ohio State University % ' $ Excess Carriers = n0 + p = n0 + n 9 > = n> > p> ! np 6= n2i (T ) What happens to excess carriers generated? excess electrons and holes can recombine they can di use if there is a gradient of excess carrier & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 5 The Ohio State University % ' $ 4.1 Optical Absorption collision with lattice Ec absorption h ω > Eg recombination Ev The light intensity is: intensity (Power) = I (W ) = h! nphoton = photon energy # of photons sec The absorption of light is proportional to the intensity: dI (x) = I ; ) I = I e; x & dx Patrick Roblin 0 T.H.E OHIO S ATE T UNIVERSITY 6 The Ohio State University % ' $ Frequency Dependence of Absorption Coe cient α InSb GaAs GaP Si CdSe CdS SiC Ge ZnS Eg (eV) 0 Eg E =h ω 1 2 3 4 λ (μ m) 5 Infrared 1 0.5 Visible 0.35 Ultraviolet Infrared: GaAs, Si, Ge and InSb Visible: GaP, Cds, SiC (transparent) & Ultraviolet: ZnS Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 7 The Ohio State University % ' $ 4.2 Luminescense 9 > > > > > = 8 > Photon: ( uorescense, phosphorescense) Photo> > > < Cathodo- > Luminescence ) > Electron: (CRT: cathode ray tube) > > > > > : LED (light emitting diode) > Electro- & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 8 The Ohio State University % ' $ Fluorescence and Phosphorescence collision with lattice collision with lattice Ec absorption h ω > Eg absorption h ω = Eg Ec trap h ω > Eg Et h ω = E t − Ev Ev & Ev Fluorescence Phosphorescence uorescence: excitation ! light emitted (direct bandgap only) phosphorescence: excitation ! delay ! light emitted Patrick Roblin 9 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' & Patrick Roblin $ Alternating Current Thin Film Electroluminescence T.H.E OHIO S ATE T UNIVERSITY 10 The Ohio State University % ' $ 4.3 Carrier Lifetime & Photoconductivity low high low Ω meter high Ω meter Light on + & Light o + Excess carriers generated Recombination of excess carriers + + T.H.E conductivity (t) increases conductivity (t) decreases gradually OHIO S ATE T Patrick Roblin UNIVERSITY 11 The Ohio State University % ' $ Direct Recombination of Electrons & Holes Rate of change in electron concentration: dn(t) = generation rate ;recombination rate {z } | dt optical+thermal = gopt + r |{z} ; r n(t)p(t) n2 i n0 p0 In equilibrium: gopt = 0 and np = n0p0 = n2. i In equilibrium the total rate is zero: dn(t) = 0 dt & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 12 The Ohio State University % ' $ Direct Recombination of Electrons & Holes Introduction excess carriers n and p: n(t) = n0 + n(t) p(t) = p0 + p(t) n= p (created by optical generation) in the rate equation we have: d n(t) = g + n2 ; dt n(t)p(t) = gopt + r n2 ; r (n0 + n)(p0 + p) i = gopt ; r (n0 + p0) n ; | r{zn2 (for n small) } opt ri r !0 For low injection level ( d n(t) n = gopt ; & dt Patrick Roblin n n small): with 1 n= r (n0 + p0 ) T.H.E OHIO S ATE T UNIVERSITY 13 The Ohio State University % ' $ Steady State and Relaxation Toward Equilibrium For low injection level ( n small): d n(t) = g ; n with dt opt n= n 1 r (n0 + p0 ) n In steady state ( d dt(t) = 0) under continuous illumination (gopt 6= 0): d n(t) n = 0 = gopt ; ) n = ngopt dt n After the illumination is switched o (gopt(t 0) = 0), the semiconductor relaxes toward equilibrium: n d n(t) =; ) n(t) = n(0)e;t= n & dt Patrick Roblin n T.H.E OHIO S ATE T UNIVERSITY 14 The Ohio State University % ' $ Indirect Recombination & Trapping Impureties which introduce deep energy levels (Et ) can act as recombination centers: EC EC EC EF Et EF Et EF Et EV EV EV step 1 step 2 EHP recombined Example: Gold (Au) impureties in silicon & The capture time for electron and holes is di erent Patrick Roblin 6 n= p T.H.E OHIO S ATE T UNIVERSITY 15 The Ohio State University % ' $ Rate Equations for Indirect Recombination EC step 2 τn step 1 τp EF Et EV For low injection level ( n and d n(t) =g ;n & dt d p(t) dt Patrick Roblin opt = gopt ; p small): n p p with n 6= p T.H.E OHIO S ATE T UNIVERSITY 16 The Ohio State University % ' $ Photoconductivity low high low Ω meter high Ω meter Light on Light o + + n = ngopt and p = pgopt & + = q n(n0 + n) + q p (p0 + p) =q n n+q p p Patrick Roblin n(t) = n e;t= n and p(t) = p e;t= p (t) = q + n n0 + n(t)] + q p p0 + p(t)] T.H.E OHIO S ATE T UNIVERSITY 17 The Ohio State University % ' $ Quasi-Fermi Levels Ec Fn Ei Fp Ev Away from equilibrium, the Fermi level EF is not constant but we can introduce a spatially varying quasi-Fermi levels Fn (x) and Fp (x) for electrons and holes. & n = n0 + n = ni exp p = n0 + p = ni exp Patrick Roblin Fn (x);Ei (x) kT Ei (x);Fp (x) kT 9 = ! np 6= n2 (T ) i T.H.E OHIO S ATE T UNIVERSITY 18 The Ohio State University % ' $ Excess Carrier & Di usion δn Diffusion Diffusion gopt τ x gopt τ /2 L= D τ −d/2 d d/2 x d & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 19 The Ohio State University % ' $ Lecture # 5 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 20 The Ohio State University % ' $ Di usion of Carriers Concentration l & 0 t 2t 3t time x T.H.E OHIO Mean free time between collision: t Patrick Roblin 4t Mean free path between collision:S ` T ATE UNIVERSITY 21 The Ohio State University % ' $ Di usion Applet http://www.geocities.com/SiliconValley/Campus/2811/browni/Difus.html & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 22 The Ohio State University % ' $ Di usion Equation Flux for di usion process: n= !; = n!v! ; n v = 1 n(x) ` ; 1 n(x + `) ` 2 t2 t `2 n(x + `) ; n(x) =; 2t ` `2 dn(x) =; 2t dx Electron & Hole di usion uxes: & n p = ;Dn dn(x) dx = ;Dp dp(x) dx Patrick Roblin and and `2 Dn = ; 2tn n 2 ` Dp = ; 2tp p (electron di usion constant) (hole di usion constant) OHIO T.H.E S ATE T UNIVERSITY 23 The Ohio State University % ' $ Di usion Fluxes and Currents Di usion Fluxes: dn(x) n = ;Dn dx = ;D dp(x) p p dx Di usion currents: Jn = ;q n = Dn dn(x) dx J = q = ;D dp(x) p p p dx What are the directions of the electron and hole uxes & currents ? n(x) & Patrick Roblin p(x) φn (diff) φp (diff) Jn (diff) Jp (diff) x 24 T.H.E x OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Di usion Fluxes and Currents Di usion Fluxes: = ;Dn dn(x) n dx = ;D dp(x) p p dx Di usion currents: Jn = ;q n = Dn dn(x) dx J = q = ;D dp(x) p p p dx n(x) & Patrick Roblin p(x) φn (diff) φp (diff) Jn (diff) Jp (diff) x 25 T.H.E x OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Di usion and Drift of Carriers Total electron and hole current: dn(x) Jn = qn n E + Dn dx dp(x) J = qp E ; D p p | {z } drift p dx | {z } diffusion These equations will be our bread and butter. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 26 The Ohio State University % ' $ Drift, Di usion and Ballistic Transport Consider the simple pastoral scene of a lake with a waterfall on one side and a small creek on the other. There is clearly a continuous ow of water from the waterfall, through the lake and into the creek. However if the lake is very wide, the water drift might not even be perceptible to a sherman on its bank shing for trouts. However a y- sherman shing in the creek will see his dry y quickly drift away and will need to recast his line often. If a red dye or some kind of liquid trout food is poured in the lake at one spot we can expect it to slowly di use and spread throughout the lake. A drift di usion model applies therefore well to the lake area. On the .other T H.E OHIO hand, no dye is expected to be able to di use up the waterfall into its upper reservoir. Clearly S ATE T the waterfall is operating under a ballistic mode. UNIVERSITY & Patrick Roblin 27 The Ohio State University % ' $ Quasi-Fermi Level Derivation Non-equilibrium thermodynamic: Prigogine (Nobel prize) demonstrated that the entropy production rate is minimized: J = qn E ! J = n dFn(x) n n n n dx p( Jp = qp pE ! Jp = p p dFdxx) where n(x) and p(x) are given by (assuming local equilibrium): ! Fn(x) ; Ec(x) n(x) = Nc exp kT ! ; p(x) = Nv exp Ev (x)kT Fp(x) expressed in terms of the quasi-Fermi levels Fn(x) and Fp (x) and with Ec(x) = Ec(0) ; qV (x) and Ev (x) = Ev (0) ; qV (x) & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 28 The Ohio State University % ' Flash Back on Potentiel Energy and Electrostatic Potential Ec (x) = Ec(0) ; qV (x) and ENERGY Ec duct Vale n Ba nce Patrick Roblin Vibrations (heat) and idde Vibrations (heat) = Ev (0) ; qV (x) ion B Forb Ev Ev (x) B A Con & $ E − qV(x) nd c Band E − qV(x) v x x A B x position T.H.E OHIO S ATE T UNIVERSITY 29 The Ohio State University % ' $ Drift-Di usion Equation Derived from Quasi-Fermi Level From the carrier statistics we get: ! n(x) = N exp Fn(x) ; Ec(x) ! ( Fn(x) = Ec(x) + kT Ln nNx) kT c Using Ec (x) = Ec (0) ; qV (x) the quasi Fermi level (electro-chemical potential) is ! n(x) + kT Ln F n (x) = Ec(0) ; qV (x) {z } | N | {z c } electrostatic potential energy chemical potential The total electron current is given by n( Jn = n(x) n dFdxx) " !# dV (x) + n(x)kT d Ln n(x) = ;qn(x) n n dx dx Nc = qn(x) nE + kT n dn(x) dx = qn nE + qDn dn using the Einstein relation: Dn = kT OHIO dx q c & Patrick Roblin ) T.H.E S ATE T UNIVERSITY 30 The Ohio State University % ' Electron and Hole Currents & Equilibrium Jn = nn Jp = pp In equilibrium: dFn dx dFp =0 dFn (x) dx dFp (x) dx ) ) dn(x) E + Dn dx = qp pE ; Dp dp(x) dx = qn Jn n =0 =0 Jp = 0 the quasi-Fermi levels are constant. When electrons and holes are also in equilibrium together EF = Fn = Fp . dx & $ Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 31 The Ohio State University % ' $ Di usion Coe cients & Einstein Relation For Instrinsic Semiconductors at room temperature: Dp Dn Dp Dn p n n p (cm2 /sec) (cm2 /V/sec) (cm2/sec) (cm2 /V/sec) Si 35 1350 0.0259 12.5 480 0.026 Ge 100 3900 0.0256 50 1900 0.0263 GaAs 220 8500 0.0259 10 400 0.025 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 32 The Ohio State University % ' $ 4.4.3 The Continuity Equation φ (x) φ (x+ Δx) p(x) x x+ Δx Δx Conservation of particles using p(x) = p0 (x) + p(x t): @p = @ p = ; p + p (x) ; (x + x) = ; p ; @ p(x) @t @t x p p Continuity Equations for electrons and holes: @ p = ; p ; 1 @Jp(x) using J = q p p @t q @x p @ n = ; n + 1 @Jn(x) using J = ;q n @t q @x & Patrick Roblin n @x (drift+di usion) n (drift+di usion) OHIO T.H.E S ATE T UNIVERSITY 33 The Ohio State University % ' $ Di usion Equation when Electric Field is Neglegible in Uniform Material When the built-in electric eld is negligible the current is carried by di usion only: (x (x Jp = ;Dp @p@x t) =;Dp @ p@x t) for uniform material: p(x t) = p0 + p(x t) J = D @n(x) = D @ n(x) for uniform material: n(x t) = n + n(x t) n @x @x Substituting the di usion currents in the continuity equations: @ p = ; p ; 1 @Jp(x) @t q @x p @ n = ; n + 1 @Jn(x) @t q @x n n 0 n we obtain the Di usion Equations: @ p = D @ 2 p(x) ; p & p @t @x2 p @ n = D @ 2 n(x) ; n n @t @x2 n Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 34 The Ohio State University % ' $ 4.4.4 Steady State Carrier Di usion Length Assumptions: current is carried strictly by di usion (assumes built-in electric eld is negligible): doping is uniform steady state @p @t = @n = 0 @t The Di usion Equations reduce to: @ p = 0 = D @ 2 p(x) ; p ) @ 2 p(x) = p = p with L = qD p p pp @t @x2 @x2 Dp p L2 p p @ n = 0 = D @ 2 n(x) ; n ) @ 2 n(x) = n = n with L = qD nn n n @t @x2 @x2 Dn n L2 n n q p where Lp = Dp p and Ln = Dn n are the di usion lengths. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 35 The Ohio State University % ' $ Di usion of Injected Carrier p(x) diffusion Δp Solving the 2nd order equation @ 2 p(x) = p p0 @x2 L2 p we obtain the solution: recombination Qp x 0 Lp p(x) = C1ex=Lp + C2e;x=Lp with 8 < p(0) = p : p(1) = 0 ) p(x) = pe;x=Lp The di usion current at position x is: Jp(x) = qDp d p(x) = qDp p(x) Lp dx Lp The current at the interface is then: Jp (0) = qDp p = Qp = total excess hole charge OHIO Lp recombination time p & Patrick Roblin T.H.E S ATE T UNIVERSITY 36 The Ohio State University % ' $ Example δ n =δ p Diffusion Diffusion gopt τ x gopt τ /2 L= D τ Recombination −d/2 d d/2 x d See derivation in 2001 midterm # 1 posted in class webpage. Assumption: negligible electric eld which is only rigorously valid if Ln = Lp = L ) n(x) = p(x) & T.H.E What is the total current? Patrick Roblin OHIO S ATE T UNIVERSITY 37 The Ohio State University % ' $ Haynes-Shockley Experiment First performed by Haynes-Shockley Experiment in 1951 at Bell Labs Permits to measure the minority carrier mobility: for a n-type material v L p = Ed = td E with L the sample length and td the drift time. Permits to measure the minority carrier di usion constant Dp by measuring the carrier spread δ p(x,t) drift time t d/4 + − Light pulse at t=0 t d/2 3t /4 d td n−type & +− Patrick Roblin x L 0 σ = 2 Dp t L T.H.E OHIO S ATE T UNIVERSITY 38 The Ohio State University x % ' $ Theory for Haynes-Shockley Experiment Starting from the Di usion Equations (no drift) and assuming for simplicity that p is larger than td (not necessarily veri ed since d is typically on the order of s and td ms): @ p = D @ 2 p(x) ; p ) @ p = D @ 2 p(x) p p @t @x2 @t @x2 p admits for solution the following time-dependent Gaussian distribution: p(x t) = q P e;x2 =(4Dpt) 2 Dpt with P the number of holes per area created at t = 0. q The width the pulse 2Dp t increases with time & In the presence of an electric eld the pulse will also drift Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 39 The Ohio State University % ' $ Java Applet for Haynes-Shockley Experiment http://jas.eng.buffalo.edu/education/semicon/diffusion/diffusion.html & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 40 The Ohio State University % ' $ Ballistic Transport Energy Ballistic Electrons Ec a) E xo N P v Position Normalized electron distribution at x o f(v ) x 1 Ballistic peak b) & Patrick Roblin vx -1 0 41 1 Normalized velocity T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode Energy Position OFF EF1 EC1 EF2 E C2 a) ON EF1 EC1 -qV EF2 E b) & Patrick Roblin EF1 EC1 C2 OFF -qV c) EF2 E C2 T.H.E OHIO S ATE T UNIVERSITY 42 The Ohio State University % ' $ Resonant Tunneling Diode: Potential Conduction band edge 1.5 1 Energy (eV) 0.5 0 VD=0 V −0.5 & −1 VD=0.8 V T.H.E −1.5 Patrick Roblin 0 20 40 60 80 100 120 Position (monolayer) 43 140 160 180 200 OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: Transmission Coe cient Transmission coefficient for various biases 0 10 −2 VD=0.8 V 10 VD=0 V Transmission coefficient −4 10 −6 VD=0.8 V 10 −8 10 −10 VD=0 V 10 & Patrick Roblin T.H.E 0 0.1 0.2 0.3 Energy (eV) 44 0.4 0.5 0.6 OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: IV Characteristic 9 3.5 I−V characteristic x 10 300 K Maximum charge in well 77 K 3 Current density (A/m2 2.5 2 1.5 1 0.5 & 0 T.H.E −0.5 Patrick Roblin 0 0.1 0.2 0.3 0.4 0.5 Voltage (V) 45 0.6 0.7 0.8 0.9 OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: Charge Distribution 5 16 Charge distribution x 10 14 3 Charge density (C/m ) 12 VD=0.8 V 10 8 6 VD=0 V 4 & Patrick Roblin 2 T.H.E 0 0 20 40 60 80 100 120 Position (monolayer) 46 140 160 180 200 OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Inte Material 2 r aye el rfac Interface Material 1 Interface Roughness Scattering Z Y NIR NIR X (A) Side view & Patrick Roblin Energy E c (x) VB E c (x) NIR (B) 47 X=na/2 For <100> T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Scattering Assisted Tunneling Energy E0x and E1x Interface roughness E1x(max)=E0 Incident wave Transmitted incident wave E0x Scattering assisted resonant tunneling Continuum of scattered waves Quasi bound-state & Patrick Roblin Position T.H.E OHIO S ATE T UNIVERSITY 48 The Ohio State University % 209 6.10 Results for resonant tunneling structures Fig. 6.4. Equilibrium (no bias) transmission coefficient TF ( E 0x , E 0⊥ = 0) versus E 0x in the ˚ presence of polar scattering for 50/50/50 A diode at the lattice temperature of 4.2 K (dashed line), 100 K (dashed-dotted line), and 300 K (dotted line). Also shown is the transmission coefficient in the absence of polar scattering (full line). (P. Roblin and W. R. Liou, Physics Review B, Vol. 47, No. 4, Pt II, pp. 2146–2161, January 15 1993. Copyright 1993 by the American Physical Society.) shown in Figures 6.5(a) and (b) is seen to induce both a decrease in the peak current and an increase in the valley current. Polar scattering therefore contributes to the reduction of the peak-to-valley current ratio. At 4.2 K the phonon emission peak of the transmission coefficient has introduced a secondary peak in the I –V characteristic at around VD = 0.6 V. At higher temperature (here 100 K) the variation of the Fermi–Dirac occupation is more gradual around the Fermi energy, and a phonon peak is usually not resolved because its small contribution is smoothed out in the current integration. Note that the detection of a phonon peak in the I –V characteristic is facilitated when plotting higher order derivatives of the I –V characteristic. The reader is referred to [7] for results on acoustic phonon scattering which usually induces a weak scattering-assisted tunneling component to the diode current. The next phonon-scattering process considered is – X intervalley scattering induced by LOX phonons. The transmission coefficient shown in Figure 6.6(a) is seen to be subjected to a self-energy shift. 5% of the current remains carried by the valley. Intervalley scattering is seen to leave the valley current unchanged but does effectively increase the classical diode leakage current at high voltages. 210 Scattering-assisted tunneling Fig. 6.5. I –V characteristic in the presence (dashed line) and absence (full line) of polar scattering ˚ ˚ calculated for (a) a 50/50/50 A diode at 4.2 K and (b) a 34/34/34 A diode at 100 K. (P. Roblin and W. R. Liou, Physics Review B, Vol. 47, No. 4, Pt II, pp. 2146–2161, January 15 1993. Copyright 1993 by the American Physical Society.) Next we consider interface roughness scattering. We show in Figure 6.7 the equilibrium (no bias applied) transmission coefficient TF ( E 0x , E 0⊥ ) plotted versus ' $ I-V Characteristics in the Presence of Scattering 8 8 x 10 7 Current density (Amps/m/m) 6 5 4 3 2 1 & Patrick Roblin 0 0 0.1 0.2 0.3 0.4 0.5 Applied voltage (Volts) 0.6 0.7 0.8 0.9 T.H.E OHIO S ATE T UNIVERSITY 50 The Ohio State University % ' $ Inter-Band Resonant Tunneling Diode EC e-2DEG p+ InAlAs EF EF n+ InAlAs InGaAs InGaAs h-2DHG InAlAs EV EF GaSb AlSb n+InAs AlSb EF EC EV a) b) n+InAs EC Sb spike doped EC EV & Patrick Roblin c) B spike doped EV T.H.E Si n+ Si (or Si0.5Ge 0.5 ) Si p+ OHIO S ATE T UNIVERSITY 51 The Ohio State University % ...
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This note was uploaded on 01/11/2012 for the course ECE 432 taught by Professor Lu during the Fall '08 term at Ohio State.

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