Semiconductor Device Theory[半導(dǎo)體設(shè)備理論](-65)

1、EEE 531: Semiconductor Device Theory IInstructor: Dragica VasileskaDepartment of Electrical EngineeringArizona State UniversityTopics covered:Energy bandsEffective massesEnergy bandsBasic convention:ECEvErefK.E.P.E.+E+VKinetic energy:Potential Energy:Electric field:Energy-wavevector relation for fre

2、e electrons: Definition: de Broglie hypothesis:Energy-wavevector relation for electrons in a crystal:The dispersion relation in a crystal (E-k relation) is obtained by solving the Schrdinger wave equation:Bloch Theorem:If the potential energy V(r) is periodic, then the solutions of the SWE are of th

3、e form:where un(k,r) is periodic in r with the periodicity of the direct lattice and n is the band index.Methods used to calculate the energy band structure: Tight-binding method Orthogonal plane-wave method Pseudopotential method kp method Density functional technique (DFT)Periodic potentialBloch f

4、unctionCell periodic PartPlane wave componentReciprocal Space:A 1D periodic function: can be expanded in a Fourier series:The Fourier components are defined on a discrete set of periodically arranged points (analogy: frequencies) in a reciprocal space to coordinate space.3D Generalization:Where hkl

5、are integers. G=Reciprocal lattice vectorFirst Brillouin Zone (in reciprocal space):First Brillouin Zone for Zinc-Blende and Diamond real spaceFCC latticesThe periodic set of allowed points corresponding to the Fourier (reciprocal) space associated with the real (space) lattice form a periodic latti

6、ce The Wigner-Seitz unit cell corresponding to the reciprocal lattice is the First Brillouin Zone is zone center, L is on zone face in (111) direction, X is on face in (100) directionExamples of energy band structures:SiGaAsBased on the energy band structure, semiconductors can be classified into: I

7、ndirect band-gap semiconductors (Si, Ge) Direct band gap semiconductors (GaAs)Model Energy Bands in III-V and IV Semiconductors:Conduction Band - 3 Valley Model (, L, X minima). Lowest minima: X (Si), L (Ge), (GaAs, most III-Vs)Valence Band - Light hole, heavy hole, spin-split off bandThe energy ban

8、d-gaps decrease with increasing temperature. The variation of the energy band-gaps with temperature can be expressed with a universal function:Eg (eV) 1.12 0.66 1.42 SiGeGaAsEffective MassesCurvature of the band determines the effective mass of the carriers in a crystal, which is different from the

9、free electron mass.Smaller curvature heavier massLarger curvature lighter massFor parabolic bands, the components of the effective mass tensor are calculated according to:SiFrom the knowledge of the energy band structure, one can construct the plot for the allowed k-values associated with a given en

10、ergy = constant energy surfacesSiGeNote: The electron effective mass in GaAs is isotropic, which leads to spherically symmetric constant energy surfaces.Due to the p-like symmetry and mixing of the V.B. states, the constant energy surfaces are warped spheres: The hh-band is most warped The lh- and s

11、o-band are more sphericalValencebandsConstant energysurfacesEEE 531: Semiconductor Device Theory IInstructor: Dragica VasileskaDepartment of Electrical EngineeringArizona State UniversityTopics covered:Counting statesDensity of states functionDensity of states effective massConductivity effective ma

12、ssLet us consider a one-dimensional chain of atoms:According to Bloch theorem, the solutions of the 1D SWE for periodic potential are of the form:The application of periodic boundary conditions, leads to: the allowed k-values are:L =Na length of the chainaaaIntroductory comments -Counting states:Not

13、e on the boundary conditions:If one employs vanishing boundary conditions, it would give as solutions standing waves (sinx or cosx functions), which do not describe current carrying states.Periodic boundary conditions lead to traveling-wave (eikx) solutions, which represent current carrying states.C

14、ounting of the states:Each atom in the 1D chain contributes one state (two if we account for the spin: spin-up and spin-down states).The difference between two adjacent allowed k values is:Length in the reciprocal space associated with one state (2 if we account for the spin)In 3D, the unit volume i

15、n the reciprocal space associated with one state is (not accounting for spin).Calculation of the DOS function:Consider a sphere in k-space with volume:The total number of states we can accommodate in this volume is:The # of states in a shell of radius k and thickness dk is, by similar arguments, equ

16、al to:Use the fact that the number of states is conserved, i.e.whereFor parabolic energy bands, for which E=2k2/2m* # of states per unit length dk # of states per unit volume per unit energy interval dE around E Spin degeneracyDOS effective masses:For single valley and parabolic bands, the DOS funct

17、ion in 3D equals to:for electrons in theconduction bandfor holes in thevalence bandECEVgC(E)gV(E)EFor many-valley semiconductors with anisotropic effective mass, using Herring-Vogt transformation: the expression for the density of states function reduces to the one for the single valley case, except

18、 for the fact that one has to use the density of states effective mass:Si (electrons):Z(# of equivalent valleys)=6, mlm0, mtm0 GaAs (electrons): non-stationary transport velocity overshoot= faster devices (smaller transit time)SiliconVelocity overshoot effectCarrier Mobility:IonizedimpuritiesSi, GaA

19、sneutralimpurities(low T)Si, GaAsAcousticphononsSi, GaAsNon-polaroptical phononsSipolaroptical phononsGaAsPiezoelectric(low-T)GaAsMathiessens rule:Carrier Mobility (Contd):Electron mobilityDrift velocity in Si:Saturation velocity:(A) Electrons:(B) Holes:Hall measurements:Resistivity measurementscarr

20、ier concentration characterization low-field mobility (Hall mobility)The second law of motion for an electron moving in a electric and magnetic field, at low frequencies is of the form:One also has:Hall coefficient:where rn is the Hall scattering factor:Determine nSign=carrier typeThe effective carr

21、ier mobility is obtained in the following manner:1.Calculate the conductivity of the sample:2.Evaluate the Hall mobility:3.Based on the knowledge of the Hall scattering factor,determine the effective mobility using:Diffusion process:+p(x)-n(x)Dn, Dp Diffusion constants for electrons and holesTotal c

22、urrent equals the sum of the drift and diffusion components:Einstein relations (derivation):Assumptions:equilibrium conditionsnon-degenerate semiconductorGeneration-Recombination Mechanisms:Photons and phonons (review):Photons quantum of energy in an electromagnetic wavePhonons quantum of energy in

23、an elastic waveGeneration-Recombination mechanisms:Notation:g generation rater recombination rateR=r-g net recombination rateImportance:BJTs R plays a crucial role in the operation of the deviceUnipolar devices (MOSFETs, MESFETs, Schottky diodes No influence except when investigating high-field and

24、breakdown phenomenaClassification:TwoparticleOne step(Direct)Two-step(indirect)Energy-levelconsiderationPhotogenerationRadiative recombinationDirect thermal generationDirect thermal recombinationShockley-Read-Hall (SRH) generation-recombinationSurface generation-recombinationThreeparticleImpactioniz

25、ationAugerElectron emissionHole emissionElectron captureHole capturePure generation process(1) Direct processesDiagramatic description:EcEvLightE=hfEcEvLightEcEvheatEcEvheatx Photo-generationRadiativerecombinationDirect thermal generationDirect thermal recombinationNot the usual means by whichthe ca

26、rriers are generated orrecombineImportant for:narrow-gap semiconductorsdirect band-gap SCs used for fabricating LEDs for optical communicationsPhotogeneration band-diagramatic description:Momentum and energy conservation:EkEgPhonon emissionPhonon absorptionIndirect band-gap SCsVirtualstatesfinalinit

27、ialphotonfinalinitialphotonEcEVEkDirect band-gap SCsEgphononNear the absorption edge, the absorption coefficient can be expressed as:hf = photon energyEg = bandgapg = constant g=1/2 and 1/3 for allowed directtransitions and forbidden direct transitions g=2 for indirect transitions where phononsare i

28、nvolvedLightintensityDistance1/alight-penetration depthPhotogeneration-radiative recombination mathematical description- Both types of carriers are involved in the process- Limiting cases: (a) Low-level injection: (b) High-level injection:(2) Auger processes:Diagramatic description:EcEvEcEvEcEvEcEvE

29、lectroncaptureHolecaptureElectronemissionHoleemissionRecombination process(carriers near the band edges involved)Generation process(energetic carriers involved)Auger generation takes place in regions with high concent-ration of mobile carriers with negligible current flowImpact ionization requires n

30、on-negligible current flowAuger process mathematical description- Three carriers are involved in the process- Limiting cases (p-type sample): (a) Low-level injection: (b) High-level injection:- Auger Coefficients: (Silvaco)(3) Impact ionization:Diagramatic description identical to Auger generationIo

31、nization rates = generated electron hole-pairs per unit length of travel per carrier- Ionization rates dependence upon the electric field component parallel to the current flow:00.20.40.60.810.30.40.50.60.7Average energy eVDistance along the channel mm0.150.180.250.35(b)Impact ionizationVG=3.3 V, VD

32、 =3.3, 2.5, 1.8 and 1.5 V (4) Shockley-Read-Hall Mechanism:Diagramatic description:Mathematical model:EcEvElectronemissionHoleemissionEcEvElectroncaptureHolecaptureETETRecombinationGenerationEcEvETpTcnennTcpepTwo types of carriersinvolved in the processnTpT- Thermal equilibrium conditions:- Steady-s

33、tate conditions:n1 and p1 are the electron and hole densities when EF=ET- Define carrier lifetimes:- Empirical expressions for electron and hole lifetimes:- Limiting cases: (a) Low level injection (p-type sample): (b) High-level injection:- Generation process (pn 0):tg=generation rate =EEE 531: Semi

34、conductor Device Theory IInstructor: Dragica VasileskaDepartment of Electrical EngineeringArizona State UniversityTopics covered:Description of basic equations for semiconductor device operationConcept of quasi-Fermi levelsSample solution problemsDielectric relaxation time and Debye lengthBasic equa

35、tions for SC device operation:Maxwells equationsCurrent density equationsContinuity equationsPoissons equation(1) Maxwells equations:Any carrier transport model must satisfy the Maxwells equa-tions:(2) Current-Density Equations:For non-degenerate SCs, the carrier diffusion constants and the mobiliti

36、es are related through the Einstains relations:The above equations are valid for low fields. Under high field conditions, the terms mnE and mpE must be replaced with the saturation velocity.Additional terms appear in the presence of a magnetic field.(3) Continuity equations:Derived from Maxwells equ

37、ations:Low-level injection (SRH lifetime dominated by the minori-ty carrier lifetime):(4) Poissons equation:Derived from the Maxwells equations (electrostatics case):Quasi-Fermi levels:In non-equilibrium conditions, one needs to define separate Fermi levels for n and p:Sample problems:Decay of the photo-excited carriersSteady-state injection from one sideSurface-recombination(1) Decay of photo-excited carriers:Consider a sample illuminated with light source until t0. The generation rate equals to G. At t=0 the light source is turned off. Calculate pn(t) for t0 .n-type samplexx=0hfSolut