Carrier Concentrations

CarrierConcentrationsPresented by Chanam LeeAugust 22nd

Carrier ConcentrationCarrier PropertiesState and Carrier DistributionsEquilibrium Carrier ConcentrationCarrier Concentration for the Quantum WellDevices

Carrier PropertiesCarrier Movement in Free SpaceCarrier Movement Within the CrystalIntrinsic Carrier ConcentrationExtrinsic n-Type SemiconductorExtrinsic p-Type Semiconductor

Electronic MaterialsTwo-dimensional representation of anIndividual Si atom.Elemental semiconductorsValenceRepresents each valence electronIIIIVVBCAlSiPGaGeAs

SemiconductorsWhen Si (or Ge & GaAs) atoms contact otherSi atoms, they form a tetrahederal2D representation of lattice structure:

Electronic MaterialsTwo dimensionsThree dimensions

Carrier Movement in Free SpaceNewton’s second lawdvF qE m0dt

Carrier Movement Within the CrystalElectrons moving inside a semiconductor crystal will collide withsemiconductor atoms behaves as a “wave” due to the quantummechanical effectsThe electron “wavelength” is perturbed by the crystals periodicpotential

Carrier Movement Within the Crystalm n* (electron effective mass)m *p (hole effective mass)dvF qE mdtdvF qE mdt*p*nMaterialmn* / m0m*p / m0Si1.180.81Ge0.550.36GaAs0.0660.52Density of States Effective Masses at 300 K

Intrinsic Carrier ConcentrationContains an insignificantconcentration of impurity atomsUnder the equilibrium conditions,for every electron is created, ahole is created alson p niAs temperature is increased, thenumber of broken bonds (carriers)increasesAs the temperature is decreased,electrons do not receive enoughenergy to break a bond andremain in the valence band.

Extrinsic n-Type SemiconductorDonors (Group V): The 5th in afive valence electrons is readilyfreed to wander about thelattice at room temperatureThere is no room in the valenceband so the extra electronbecomes a carrier in theconduction bandDoes NOT increase the numberof hole concentration

Extrinsic p-Type SemiconductorAcceptors (Group III) : threevalence electrons readilyaccept an electron from anearby Si-Si bondCompleting its own bondingcreates a hole that can wanderabout the latticeDoes NOT increase the numberof electron concentration

State and Carrier DistributionHow the allowed energy states are distributed in energyHow many allowable states were to be found at any givenenergy in the conduction and valence bands?Essential component in determining carrier distributions andconcentrationDensity of StatesFermi FunctionDopant States

Density of States:Number ofcm 3gc (E) Statesmn* 2mn* ( E Ec )π23,/eVE Ecg c ( E )dE represents the number ofconduction band states/ cm 3 lying inthe energy range between E and E dEgv (E) m *p 2m *p ( Ev E )π23, E Evg v ( E )dE represents the number of3valence band states / cm lying in theenergy range between E and E dEg c ( Ec ) g v ( Ev ) 0

Fermi Function (I)How many of the states at the energy E will be filledwith an electronf ( E) 11 e ( E EF ) / kTf(E), under equilibrium conditions, the probability that anavailable state at an energy E will be occupied by anelectron1- f(E), under equilibrium conditions, the probability thatan available state at an energy E will NOT be occupiedby an electron

Fermi Function (II)If E E F , f ( E F ) 1 / 2If E EF 3k BT ,f ( E ) exp[( EF E ) / k BT ]If E E F 3k BT ,f ( E ) 1 exp[( E EF ) / k BT ]At T 0K (above), No occupation of statesabove EF and complete occupation ofstates below EFAt T 0K (below), occupation probability isreduced with increasing energy f(E EF) 1/2 regardless of temperature.At higher temperatures, higher energy states can be occupied, leavingmore lower energy states unoccupied (1-f(E))

Fermi Function (III)

Dopant Statesn-type: moreelectrons thanHolesIntrinsic:Equal numberof electronsand holesp-type: moreholes thanelectronsDensity ofStatesOccupancyfactorsCarrierDistribution

Equilibrium Carrier ConcentrationFormulas for n and pDegenerate vs. Non-degenerate SemiconductorAlternative Expressions for n and pni and the np ProductCharge Neutrality RelationshipCarrier Concentration CalculationsDetermination of EF

Formulas for n and pn n EtopEcmn* 2mn*π2EcE EckTE Ecwhenn Etop3Letting η Letp g c ( E ) f ( E ) dEETop*nm( E Ec )dE( E E F ) / kT1 eand ηc EbottomNc 2E F EckTgv (E)[1 f (E)]dE2πm kTh*n23/ 2, Nv 22πm*p kTh2, η 0 *n2 32m ( kT )πEvn Nc3/ 2E E0η1/ 21 e(η η c )F1 / 2 (η c )dηp Nv2π2πF1 / 2 (η c )F1 / 2 (η v )3/ 2

Degenerate vs. Non-degenerateSemiconductorE F E c 3k B T , f ( E ) If1 e (η η c )(η η c )1 eF1/ 2 (η c ) π2e ( EF Ec ) / kTn N c e ( E F Ec ) / k B T , p N v e ( E v E F ) / k B TIfE F Ev 3k BT , F1/ 2 (η v ) π2e ( Ev EF ) / kT

Alternative Expressions for n and pn0 N c ep0 N v e ( E v E F ) / k B T( E F Ec ) / k B TWhen n ni, EF Ei, thenni N c e( Ei Ec ) / kBT N v e( Ev Ei ) / k BTN c ni e( E c Ei ) / k B T, N v ni e( Ec Ei EF Ec ) / kBTn0 nie( Ei E v ) / k B T( EF Ei ) / kBT ni ep0 ni e ( Ei Ev Ev EF ) / k BT ni e ( Ei EF ) / k BTn0 p0 ni2

ni and the np Productni N c e( Ei Ec ) / k BTn Nc Nve2i N ve ( Ec Ev ) / k B T( E v Ei ) / k BT Nc Nveni N c N v e E g / 2 k BT E g / k BT

Charge Neutrality RelationshipFor uniformly doped semiconductor:Charge must be balanced under equilibrium conditions otherwisecharge would flow A Dqp qn qN qN 0Thermally generated assume ionization ofdopant additionall dopant sites

Carrier Concentration Calculations( p N A ) ( N D n) 0ni2( N A ) ( N D n) 0nn 2 n( N D N A ) ni2 0n ND N A 2ND N A21/ 22 ni2,p np n i2N A ND 2N A ND21/ 22 ni2

DRelationship for N and NN A NA1 g A( E A E F ) / kT, N D AND1 gD( E F E D ) / kTThe degeneracy factors account for the possibility of electronswith different spin, occupying the same energy level (no electronwith the same quantum numbers can occupy the same state)Most semiconductor gD 2 to account for the spin degeneracy atthe donor sitesgA is 4 due to the above reason combined with the fact thatthere are actually 2 valence bands in most semiconductorsThus, 2 spins x 2 valance bands makes gA 4

Determination of EF (Intrinsic Material)n Nce( Ei Ec ) / kBT Nve( Ev Ei ) / kBT pN c e ( Ei E c ) / k B T N v e ( E v Ei ) / k B TEi Ec Ev k BTN ln v22Nc2πm kTNc 2h*n23/ 2, Nv 22πm kTh*p23/ 2m *pNv Ncmn*1/ 2m*pEc Ev 3k BTEi ln *mn24

Determination of EF (Extrinsic Material)n ni e( E F Ei ) / k B T,p ni e( Ei E F ) / k B TnpE F Ei kT ln kT lnniniorfor ND NA and ND niNE F Ei kT ln Dniorfor NA ND and NA niE F Ei kT lnNAni

Determination of EF (Extrinsic Material)Fermi level positioning in Si at room temperature as a function ofthe doping concentration. Solid EF lines were established using Eq.(previous page)

Carrier Concentration for theQuantum Well DevicesDensity of States 3D vs. 2DCarrier Concentration – 2DCharge Neutrality

Density of States 3D vs. 2D3Dg (E) 2Dg m 2mEπ23mπ23LZEnergy DependentEnergy Independent

Carrier Concentration – 2Dn EtopEcg c ( E ) f ( E ) dEmn* kTn 2 3π Lzp im*p kTπ23Lz[ln 1 e ( EFC Ei ) / kT[ln 1 ei( E FV Ei ) / kT]]

Charge NeutralityPhh Plh N Γ N Χ N L

ReferencesRobert F. Pierret, Semiconductor Fundamentals(VOLUME I), Addison-Wesley Publishing Company,1988, chapter 2Robert F. Pierret, Advanced SemiconductorFundamentals (VOLUME VI), Addison-WesleyPublishing Company, 1987, chapter 4Ben G. Streetman and Sanjay Banerjee, Solid StateElectronic Devices, Prentice Hall, Inc., 2000, chapter 3Peter S. Zory, JR., Quantum Well Lasers, AcademicPress, 1993, chapter 1, 7

Effective Mass of Holes - 3Dgv (E) m*p 2m*p ( Ev E )π23**mhh2mhh( Ev E)π2 3 mlh* 2mlh* ( Ev E)π23* 3/ 2(m *p ) 3 / 2 (mhh) (mlh* ) 3 / 2[m (m )*p* 3/ 2hh]* 3/ 2 2 / 3lh (m )

At T 0K (above), No occupation of states above EF and complete occupation of states below EF . Distribution. Equilibrium Carrier Concentration Formulas for n and p Degenerate vs. Non-degenerate Semiconductor Alternative Expressions for n and p ni and the np Product