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Semiconductor Physics: Energy Bands The e in a solid exist in certain bands of allowed energy:– valence e exist in the valence bandS. SahaKinetic Energy of electronsEcEgKinetic Energy of holesEv{Similarly, when a hole inthe valence band gainsenergy, it moves down toan E Ev.Conduction BandHole Energy When an e in theconduction band gainsenergy, it moves up to anE Ec.z Electron Energy– conduction e or free electrons exist in the conduction band7.02 x10 4 T 2– energy gap, Eg Ec Ev @ T : E g (T ) 1.160 (1)T 1108Valence BandDistanceHO #2: ELEN 251 - Semiconductor PhysicsPage 1

Intrinsic Semiconductors In a pure single crystalsemiconductor, all thee in the conductionbandare thermallyexcitedfromthevalence band. Then atany temperature, Tn p niwhere n free e- (cm-3)p free holes (cm-3)ni intrinsic carrier (cm-3). Note:– T ni – Eg ni S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 2

Intrinsic Carrier ConcentrationThe product of the number of e and holes is given by:np ni2 (T ) CT 3e E g (T )kTE g (T )3 2 kT2(2)or, ni (T ) 3.9 x1016 T e ni 1.45x1010 cm-3 for silicon @ T 300oK.If T0 known reference temperature (generally, 300oK), thenat any temperature T, we can write:32 T ni (T ) ni (T0 ) e T0 E g (T ) E g (T0 ) 2 kT2 kT0(3)where, Eg(T) is given by (1). Eq. (3) is used in SPICE tocalculate ni at any temperature.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 3

Extrinsic Semiconductors To increase and control the conductivity of semiconductorswe add controlled amounts of “dopants” such as:– N type: P, As, and Sb each has five valence electrons– P type: B, Al, and Ga each has three valence electrons. These atoms occupy substitutional lattice sites and the extrae- or hole is very loosely bound, i.e. can easily move to theconduction or valence bands, respectively. In terms of banddiagrams this can be represented as:EcED1.1eV 0.05eVEvS. SahaDonorEc 0.05eV1.1eVEvEAAcceptorHO #2: ELEN 251 - Semiconductor PhysicsPage 4

Donor and Acceptor LevelsMeasured donor and acceptor levels for various impurities BAlGaIn1.12eVSiEv Acceptor levels are below the gap center and measured fromthe top of the valance band. Donor levels are above the gap center and measured from thebottom of the conduction band band.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 5

Density of StatesFermi LevelZero in forbidden bandEv EcThe probability that an allowed state is occupied is described bythe Fermi-Dirac distribution function:f(E) 1/[1 e(E EF)/kT](4)EF Fermi Level energy at which the probability of occupancy 1/2k Boltzmann constant; T temperature. Again, the probability of a state not being occupied is:(5)1 f(E) 1/[1 e(EF E)/kT] Probability a hole exists.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 6

Fermi Level: Temperature EffectAt T 0oK0.3T 0KT 100 KT 200 KT 300 KT 400 KT 500 KT 600 K0.2E - EF (eV)0.10.0all states EFare occupied. all states EFare empty.At T 0oK-0.1-0.2-0.30.0S. Saha 0.5f(E) some e haveE EF. someholesexist E EF.1.0HO #2: ELEN 251 - Semiconductor PhysicsPage 7

The Boltzmann ApproximationFor energies 3kT ( 75 meV @ room temp) above EF, f(E)can be approximated as:f(E) e (E EF)/kT]1.0f(E)F-D functionM-B function(6) Equation (5) representsMaxwell Boltzmannfunction. This approximation isquite good for manydevice applications, aswe will see later. However, for heavilydoped materials Fermifunction (4) must beused.0.5Approximation0.0-0.30.0E - EFS. Saha0.3HO #2: ELEN 251 - Semiconductor PhysicsPage 8

Summary of Equilibrium Carrier Distribution In an intrinsic or undoped semiconductor:n p ni 3.9x1016T3/2e Eg/2kT(2')EF (1/2)(Ec Ev) Ei(7)Fermi level EF in doped as well as undoped materials is given by:n nie(EF Ei)/kT nieq(φ φF)/kT(8)p nie(Ei EF)/kT nieq(φF φ)/kT(9)where φ Ei/q and φF EF/q. If both n- and p-type dopants are added in intrinsic semiconductor:n NA p ND(10)Then using np ni2, we can show for an n-type semiconductor:[ (1 / 2)[ N] N ] n / Nnn (1 / 2) N D2 4ni2 N D N DpnS. Saha2D 4ni2D2i(11)DHO #2: ELEN 251 - Semiconductor Physics(12)Page 9

Carrier Transport in Semiconductors In thermal equilibrium, mobile (conduction band) e- will be inrandom thermal motion.– avg. velocity of thermal motion, vth 107 cm/sec @ 300oK. In an applied ε field:– motion of e is in the directionopposite to ε.– this process is called drift andcauses a net current flow. Si CrystalεThe average e velocity called the drift velocity is given by:vd µε(13)where µ mobility qτ / m *τ mean free time between collision; m* effective mass of carrierS. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 10

Drift - ε Field Driven Motion vd µε linearrelationshipholds only forsmall fields. Mobility dependson parameterslike:– material– doping– temperature– scatteringmechanisms.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 11

Scattering Mechanisms Phonon scattering. Lattice atoms vibrate, and from QuantumMechanics we know that discrete allowed vibrational statesexist. This can be modeled as discrete particles calledphonons. Transfer of energy from electrons to the lattice iscalled phonon scattering. Ionized impurity atom scattering - important at high dopantconcentrations. Neutral impurity atom scattering - usually negligible. Carrier-carrier (e - e & e - hole) scattering - important athigh carrier concentrations. Surface scattering effects - important in MOSFET devices. Crystal defects - can be minimized by good crystals.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 12

µn Vs. TemperatureIonizedimpurityscattering is moreimportant at low T.Assuming that thedifferentscatteringmechanisms operateindependently:µ 1µLS. SahaHO #2: ELEN 251 - Semiconductor Physics 11µI .Page 13

Drift - ε Field Driven Motion At high fields, vd becomes comparable to vth which is anapproximate upper bound of vd. Thus, vd saturates and iscalled:vsat scattering limited velocity 1 x 107 cm/sec for e in Si 8 x 106 cm/sec for holes in Si.µoε The general expression for vd is: vd The critical field at which vsat occurs is: 2x104 V/cm 2 V/µm. µoε β 1 vsat 1β(14)This is an important physical limit in very small semiconductordevices like MOSFETs.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 14

Drift - CurrentThe drift of carriers under an applied ε results in a current flow:I qAnvd(15)where A area; n number of e per unit volume.Substituting for vd µε and considering both e and holes:I/A (nqµn pqµp)ε(16)Usually, the majority carrier-term in (16) is important. Since fora semiconductor of length L under applied bias V, ε V/L, then:1/(qµnn qµpp) (V/L)/(I/A) (V/I)(A/L) R(A/L) ρwhere R ρ(L/A) resistance of the semiconductor:11 ρ (17)qµ n n qµ p p σwhere ρ resistivity in ohm.cm; σ conductivity in (ohm.cm)-1S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 15

Resistivity Vs. Dopant Density @ T 300 KS. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 16

Diffusion - Gradient Driven MotionDiffusion occurs from high concentration regions towardslow concentration regions.Carrierconcentrationt1Flux ofdiffusioncurrentt2Flux ofdiffusioncurrentt1 t2 t3t30 xThe diffusion flux obeys Fick’s first law:F D(dn/dx)x(18)where F flux of carriersD diffusion constantn carrier density.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 17

Transport EquationsThe diffusion current through a cross-sectional area is:I qADn(dn/dx) qADp(dp/dx)(19)where Dn e diffusion constant 38 cm2/sec in SiDp hole diffusion constant 13 cm2/sec in SiDn and Dp are related to their respective carrier mobilities bythe Einstein relationship:Dp/µp Dn/µn kT/q(20)If both drift and diffusion are important,In qA[µnnε Dn(dn/dx)](21)Ip qA[µppε Dp(dp/dx)](22)We will use these basic equations in deriving devicecharacteristics.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 18

Generation-Recombination Thermal equilibrium is defined by the condition np ni2. Inthermal equilibrium, the equations discussed earlier may beused to calculate EF, n, p etc. However, the equilibrium condition may be disturbed by theintroduction or removal of free carriers.φ Injection: Extra carriers aregenerated by optical orelectrical means, and np ni2.NP S. Saha- Extraction: Here, carriersare removed electrically.And, as a result np ni2.HO #2: ELEN 251 - Semiconductor PhysicsPage 19

Injection of CarriersIf we shine light on the semiconductor with E hν Eg sothat e can be excited into the conduction band.n ni nLp ni pLINJECTIONn & p nizEchν EgEgEv{ν frequencyh Planck’s constantS. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 20

Low Level and High Level InjectionNow, if we shine light on an extrinsic semiconductor with:ND 1015 n this case:nnopnn ND nLp ni2/ND pLND 1015 cm-3Low level: nL NDHigh level: nL NDni for Si@ 300 oKpnoEquilibriumS. SahannnnpnLowlevelHighlevelIn Fig., a low level injection ofexcess carrier 1013 is shown.Since the mathematics for highlevel injection are complex, wewill consider only low levelinjection.HO #2: ELEN 251 - Semiconductor PhysicsPage 21

Direct (Band to Band) RecombinationIf we generate excess carriers ( n, p) at a rate GL due toincident light. Then for low level injection we can show that:z n electrons p n Uτ GLτEcWhere p p poEv{ p holes n n nop and n are total concentrations due to generationno and po are equilibrium concentrationsU net recombination rate GLτ excess carrier lifetime avg. time excess carriers remain free.Thus, τ p/GL p/U n/U(23)For band to band recombination, τn τp as the single phenomenaannihilates e and hole simultaneously.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 22

Direct Recombination Example:Net recombination rate U 1018 carriers cm-3 sec-1Minority-carrier lifetime τ 100 µsec,Then from (23), excess-carrier density: p n (1018) (100x10-6) 1014 cm-3. Summary:For a semiconductor with equilibrium carrier concentrations no andpo the steady state carrier concentrations under light are:S. Sahan no τGLnopo ni2p po τGLnp ni2HO #2: ELEN 251 - Semiconductor Physics(24)Page 23

Indirect Recombination (Trap-assisted)In Si and Ge this is often the dominant mechanism since theyare “indirect band gap semiconductors.” Physically this meansthat the minimum energy gap between Ec and Ev does notoccur at the same point in the momentum space.EEgEEcEvk“Direct band gapSemiconductor” GaAsEgEvEck“Indirect band gapSemiconductor” Si, GeSince momentum (k) must be conserved in any energy leveltransition, GaAs can easily make direct transitions which makesit useful for LEDs. Ep hν Eg. While Si and Ge tend torecombine through intermediate trapping levels.S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 24

Indirect RecombinationEcEtEvLet us consider the following examplewhere an impurity like Au is introducedthat provides a “trapping level” or a setof allowed states at energy Et.The level Et is assumed to act like an acceptor, that is, it canbe neutral or negatively charged. Recombination isaccomplished by trapping an e- and a hole. (The analysiscan be easily extended to the case where the trap acts like adonor, i.e. and neutral charge states).Ec“1” e- capture12“2” e- emissionzzoEt“3” hole captureEiEF“4” hole emission431 3 or 3 1 recombinationEv{S. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 25

Indirect Recombination - SRH Shockley, Read, and Hall showed that for low level injection,the net recombination rate is given by:2vth σ N t ( pn ni )U (25) Et Ei n p 2 ni cosh kT wherevth carrier thermal velocity ( 107 cm/sec)σ carrier capture cross-section ( 10-15 cm2) From Eq (25) we observe:– the “driving force” or the rate of recombination is np ni2,(that is, the deviation from the equilibrium condition).– U 0 when np ni2.– U is maximum when Et Ei. That is, trapping levels near themid-band are the most efficient recombination centersS. SahaHO #2: ELEN 251 - Semiconductor PhysicsPage 26