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Berkeley ELENG 130 - Lecture Notes

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1EE130 Lecture 3, Slide 1Spring 2003Lecture #3OUTLINE• Thermal equilibrium• Fermi-Dirac distribution– Boltzmann approximation• Relationship between EF and n, p• Temperature dependence of EF, n, pFinish reading Chapter 2EE130 Lecture 3, Slide 2Spring 2003Review: Energy Band Model and Doping2EE130 Lecture 3, Slide 3Spring 2003EE130 Lecture 3, Slide 4Spring 2003Important Constants• Electronic charge, q• Permittivity of free space,εo• Boltzmann constant, k• Planck constant, h• Free electron mass, mo• Thermal voltage kT/q3EE130 Lecture 3, Slide 5Spring 2003Thermal Equilibrium• No external forces applied:– electric field = 0– magnetic field = 0– mechanical stress = 0• Thermal agitation –> electrons and holes exchange energy with the crystal lattice and each other⇒ Every energy state in the conduction and valence bands has a certain probability of being occupied by an electronEE130 Lecture 3, Slide 6Spring 2003Analogy for Thermal Equilibrium• There is a certain probability for the electrons in theconduction band to occupy high-energy states under the agitation of thermal energy (vibrating atoms, etc.)DishVibrating tableSand particles4EE130 Lecture 3, Slide 7Spring 2003Fermi FunctionkTEEFeEf/)(11)(−+=EFis called the Fermi energy or the Fermi levelThere is only one Fermi level in a system at equilibrium.Probability that an available state at energy E is occupied:EE130 Lecture 3, Slide 8Spring 2003Effect of Temperature on f(E)5EE130 Lecture 3, Slide 9Spring 2003Boltzmann Approximation() )( ,3 IfkTEEFFeEfkTEE−−≈>− 1)( ,3 IfkTEEFFeEfkTEE−−≈>−EE130 Lecture 3, Slide 10Spring 2003Density of StatesEEcEvEcEv∆Ε() 2)(32**hEEmmEgcnncπ−=() 2)(32**hEEmmEgvppvπ−=gc(E)gv(E)E ≥ EcE ≤ Evg(E)dE = number of states per cm3in the energy range between E and E+dENear the band edges:6EE130 Lecture 3, Slide 11Spring 2003Equilibrium Distribution of Carriers• Obtain n(E) by multiplying gc(E) and f(E)• Obtain p(E) by multiplying gv(E) and 1-f(E)EE130 Lecture 3, Slide 12Spring 2003• Integrate n(E) over all the energies in the conduction band to obtain n• By using the Boltzmann approximation, and extending the integration limit to ∞, we obtainEquilibrium Carrier Concentrations∫=band conduction of top)()(cEcdEEfEgn2/32*/)(22 where==−−hkTmNeNnnckTEEcFcπ7EE130 Lecture 3, Slide 13Spring 2003• Integrate p(E) over all the energies in the valence band to obtain p• By using the Boltzmann approximation, and extending the integration limit to -∞, we obtain[]∫−=vband valenceof bottom)(1)(EvdEEfEgp2/32*/)(22 where==−−hkTmNeNppvkTEEvvFπEE130 Lecture 3, Slide 14Spring 2003Intrinsic Carrier Concentration nikTEvcigeNNn2/−=2innp =kTEEcFceNn/)( −−=kTEEvvFeNp/)( −−=andMultiplykTEvckTEEvcgvceNNeNNnp//)(−−−==Recall that8EE130 Lecture 3, Slide 15Spring 2003Intrinsic Fermi Level• To find EFfor an intrinsic semiconductor (n = p = ni):peNeNnkTEEvkTEEciVCi===−− /)(/)(++=CVVCiNNkTEEE ln22++=**ln432npVCimmkTEEEEE130 Lecture 3, Slide 16Spring 2003n(ni, EF) and p(ni, EF)9EE130 Lecture 3, Slide 17Spring 2003Shifting the Fermi LevelEE130 Lecture 3, Slide 18Spring 2003Example: Energy-band diagramQuestion: Where is EFfor n = 1x1017 cm-3?10EE130 Lecture 3, Slide 19Spring 2003Carrier Concentration vs. Temperatureintrinsic regimen = ND“freeze-out” regimeln n1/Thigh temp.roomtemperaturecryogenictemperatureEE130 Lecture 3, Slide 20Spring 2003Dependence of EFon Temperature10131014101510161017101810191020EvEcEf,Donor-dopedEf,Acceptor-dopedNAor ND(cm-3)300K400K400K300K11EE130 Lecture 3, Slide 21Spring 2003Dopant IonizationQ: Nd= 1017cm-3. What fraction of the donors are not ionized?Solution: First assume that all the donors are ionized.EcEFEv146 meVEd45meVProbability of non-ionization ≈02.01111meV26/)meV)45146((/)(=+=+−−eekTEEFdTherefore, it is reasonable to assume complete ionization, i.e., n = NDmeV146cm10317−=⇒==−cfDEENnEE130 Lecture 3, Slide 22Spring 2003Thermal equilibrium:Balance between internal processes with no external stimulus (no electric field, no light, etc.)=> Electron-hole pair (EHP) generation rate = EHP recombination rate• Fermi function: probability that a state at energy E is filled with an electron under equilibrium conditions:– Boltzmann approximation:For high E, i.e. E - EF> 3kT:For low E, i.e. EF– E > 3kT:SummarykTEEFeEf/)(11)(−+=kTEEFeEf/)()(−−≅kTEEFeEf/)()(1−−≅−12EE130 Lecture 3, Slide 23Spring 2003kTEEikTEEciFFceneNn/)(/)( −−−==kTEEikTEEvFivFeneNp/)(/)(


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Berkeley ELENG 130 - Lecture Notes

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