Lecture #5Intrinsic Fermi Level, Ein(ni, Ei) and p(ni, Ei)Example: Energy-band diagramDopant IonizationNondegenerately Doped SemiconductorDegenerately Doped SemiconductorBand Gap NarrowingMobile Charge Carriers in SemiconductorsElectrons as Moving ParticlesCarrier Effective MassThermal VelocityCarrier ScatteringCarrier DriftElectron MomentumCarrier MobilityElectron and Hole MobilitiesExample: Drift Velocity CalculationMean Free PathSummaryLecture #5OUTLINE• Intrinsic Fermi level• Determination of EF• Degenerately doped semiconductor • Carrier properties• Carrier driftRead: Sections 2.5, 3.1EE130 Lecture 5, Slide 2Spring 2007Intrinsic Fermi Level, Ei•To find EF for an intrinsic semiconductor, use the fact that n = p:2ln432ln22** /)( /)(vcnpvciicvvcFkTEEvkTEEcEEmmkTEEEENNkTEEEeNeNvFFcEE130 Lecture 5, Slide 3Spring 2007n(ni, Ei) and p(ni, Ei)•In an intrinsic semiconductor, n = p = ni and EF = Ei : /)( /)(kTEEickTEEciicicenNeNnn /)( /)(kTEEivkTEEvivivienNeNnp /)( kTEEiiFenn /)( kTEEiFienpEE130 Lecture 5, Slide 4Spring 2007Example: Energy-band diagramQuestion: Where is EF for n = 1017 cm-3 ?EE130 Lecture 5, Slide 5Spring 2007Dopant IonizationConsider a phosphorus-doped Si sample at 300K with ND = 1017 cm-3. What fraction of the donors are not ionized?Answer: Suppose all of the donor atoms are ionized. ThenProbability of non-ionization meVEnNkTEEcccF150ln 017.01111 26/)45150(/)(meVmeVmeVkTEEeeFDEE130 Lecture 5, Slide 6Spring 2007Nondegenerately Doped Semiconductor•Recall that the expressions for n and p were derived using the Boltzmann approximation, i.e. we assumed The semiconductor is said to be nondegenerately doped in this case.kTEEkTEcFv33 EcEv3kT3kTEF in this rangeEE130 Lecture 5, Slide 7Spring 2007Degenerately Doped Semiconductor•If a semiconductor is very heavily doped, the Boltzmann approximation is not valid.In Si at T=300K: Ec-EF < 3kT if ND > 1.6x1018 cm-3 EF-Ev < 3kT if NA > 9.1x1017 cm-3The semiconductor is said to be degenerately doped in this case.•Terminology:“n+” degenerately n-type doped. EF Ec“p+” degenerately p-type doped. EF EvEE130 Lecture 5, Slide 8Spring 2007Band Gap Narrowing•If the dopant concentration is a significant fraction of the silicon atomic density, the energy-band structure is perturbed the band gap is reduced by EG :N = 1018 cm-3: EG = 35 meVN = 1019 cm-3: EG = 75 meVTNEG300105.33/18EE130 Lecture 5, Slide 9Spring 2007Mobile Charge Carriers in Semiconductors•Three primary types of carrier action occur inside a semiconductor:–Drift: charged particle motion under the influence of an electric field.–Diffusion: particle motion due to concentration gradient or temperature gradient.–Recombination-generation (R-G)EE130 Lecture 5, Slide 10Spring 2007Electrons as Moving ParticlesF = (-q)E = moaF = (-q)E = mn*awhere mn* is the electron effective massIn vacuum In semiconductorEE130 Lecture 5, Slide 11Spring 2007Carrier Effective MassIn an electric field, E, an electron or a hole accelerates:Electron and hole conductivity effective masses:electronsholes*nmqa*pmqa**EE130 Lecture 5, Slide 12Spring 2007Average electron kinetic energy 2*2123thnvmkT cm/s103.2m/s103.2 kg101.926.0J/eV)106.1(eV026.033753119*nthmkTvThermal VelocityEE130 Lecture 5, Slide 13Spring 2007Carrier Scattering•Mobile electrons and atoms in the Si lattice are always in random thermal motion.–Electrons make frequent collisions with the vibrating atoms•“lattice scattering” or “phonon scattering”–increases with increasing temperature–Average velocity of thermal motion for electrons: ~107 cm/s @ 300K•Other scattering mechanisms:–deflection by ionized impurity atoms–deflection due to Coulombic force between carriers•“carrier-carrier scattering”•only significant at high carrier concentrations•The net current in any direction is zero, if no electric field is applied.12345electronEE130 Lecture 5, Slide 14Spring 2007Carrier Drift•When an electric field (e.g. due to an externally applied voltage) is applied to a semiconductor, mobile charge-carriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons:12345electronE•Electrons drift in the direction opposite to the electric field current flowsBecause of scattering, electrons in a semiconductor do not achieve constant acceleration. However, they can be viewed as quasi-classical particles moving at a constant average drift velocity vdEE130 Lecture 5, Slide 15Spring 2007Electron Momentum•With every collision, the electron loses momentum•Between collisions, the electron gains momentum(-q)Emnmn is the average time between electron scattering eventsdnvm*EE130 Lecture 5, Slide 16Spring 2007• p [qmp / mp*] is the hole mobilityCarrier Mobilitymn*vd = (-q)Emn|vd| = qEmn / mn* = n E• n [qmn / mn*] is the electron mobilitySimilarly, for holes:|vd| = qEmp / mp* p EEE130 Lecture 5, Slide 17Spring 2007Electron and hole mobilities of selected intrinsic semiconductors (T=300K)Si Ge GaAs InAsn (cm2/V∙s) 1400 3900 8500 30000p (cm2/V∙s) 470 1900 400 500 sVcmV/cmcm/s2 has the dimensions of v/E :Electron and Hole MobilitiesEE130 Lecture 5, Slide 18Spring 2007a) Find the hole drift velocity in an intrinsic Si sample for E= 103 V/cm.b) What is the average hole scattering time?Solution:a)b)Example: Drift Velocity Calculationvd = n Eqmmqppmppmpp**EE130 Lecture 5, Slide 19Spring 2007Mean Free Path•Average distance traveled between collisionsmpthvlEE130 Lecture 5, Slide 20Spring 2007Summary•The intrinsic Fermi level, Ei, is located near midgap–Carrier concentrations can be expressed as functions of Ei and intrinsic carrier concentration, ni :•In a degenerately doped semiconductor, EF is located very near to the band edge•Electrons and holes can be considered as quasi-classical particles with effective mass m*–In the presence of an electric field , carriers move with average drift velocity , where is the carrier mobility /)( kTEEiiFenn /)(
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