SemiconductorsSemiconductors

• Materials that are neither metallic nor insulator. • Elemental Semiconductors: Si, Ge • Si dominates IC industry: >99% is Si based devices) • Compound Semiconductors – III-V semiconductors : GaAs (most widely used compound semiconductor), InSb, InP – II-VI semiconductors: CdS, ZnSe, ZnO

• Alloy Semiconductors: HgCdTe, AlxGa1-x, As, GaAsxP1-x SemiconductorsSemiconductors SemiconductorSemiconductor ModelsModels –– BondingBonding ModelModel

¾ Lines represent shared ¾ Visualization of a ¾ Breaking of an atom- valence electron vacancy point defect to-atom bond and ¾ Circle represent the freeing of an electron core of Semiconductor (e.g., Si atom)

http://www.play-hookey.com/semiconductors/basicstructure.html Review: Molecular Orbitals by linear combination of atomic orbitals

Quantum mechanics & Schrodinger’s eqn. HΨ = EΨ

www.chembio.uoguelph.ca/educmat/ chm729/band/basic2.jpg SemiconductorSemiconductor ModelsModels –– EnergyEnergy BandBand ModelModel

¾ Energy states bunch up into “bands” and energy states are so close to each other in energy that they are approximately continuous. ¾ Depending on the energy difference between the bonding and antobonding orbitals a gap opens up between the bands corresponding to the bonding and antibonding orbitals where there are no allowed energy levels

www.chembio.uoguelph.ca/educmat/ chm729/band/basic2.jpg EnergyEnergy BandBand ModelModel BandBand gapsgaps ofof commoncommon semiconductorssemiconductors TheThe differencedifference between insulators, semiconductors, andand metalsmetals inin thethe bandband theorytheory frameworkframework

−Eg Few electrons 2 kT Thermal excitation ~ e of electrons is easy

Gap is very wide thermal excitation is No or very narrow easy kT~0.026 eV gap ~ 8 eV for SiO2 ~ 5 eV for diamond ~ 1.12 eV for Si ChargeCharge carrierscarriers andand intrinsicintrinsic semiconductorsemiconductor

¾ Electrons are the charge carriers in he conduction band ¾ Holes are the charge carriers in the valence band Doping and Dopants: Manipulating Charge Carrier Densities in Semiconductors – n-type doping

Semiconductors that are doped such that the majority charge carriers are electrons are called n-type semiconductors

http://www.play-hookey.com/semiconductors/basicstructure.html Doping and Dopants: Manipulating Charge Carrier Densities in Semiconductors – p-type doping

Semiconductors that are doped such that the majority charge carriers are holes are called p-type semiconductors

http://www.play-hookey.com/semiconductors/basicstructure.html VisualizationVisualization ofof DopingDoping usingusing thethe energyenergy bandband modelmodel DensityDensity ofof StatesStates

¾ We talked about formation of bands but did not yet analyze how the allowed states are distributed in energy. ¾ From quantum mechanics, g(E)dE: # of states cm-3 in the range E & E+dE; g(E) has units of cm-3 eV-1

Density of states in the conduction band m* g ( E ) = n 2m* ( E − E ) c π 2 3 n c h

Density of states in the valence band

m* g ( E ) = h 2m* ( E − E ) v π 2 3 h v h FermiFermi Distribution,Distribution, FermiFermi Energy,Energy, andand FermiFermi LevelLevel

¾ What is the probability that an electronic state with energy E is occupied? ¾ Answer: Probability is given by Fermi-Dirac Distribution

= 1 F( E ) − 1+ e( E EF )/ kT

¾ EF is a parameter called the Fermi Energy. It is the chemical potential (µ) of the electrons in the semiconductor.

¾ By definition if E=EF probability of occupation is ½. ¾ Comes from a statistical mechanics calculation. Assuming Pauli exclusion principle, it is the most probable distribution on the condition that the total system energy is given. FermiFermi--DiracDirac DistributionDistribution

1.0 T= 0 K

0.8

Increasing T 0.6 F(E) 0.4

0.2

0.0 1.0 1.2 1.4 1.6 1.8 2.0 E =1.65 eV E (eV) F Fermi level in an intrinsic semiconductor at T= 0 K is in the middle of the band gap

Etop = n ∫ f ( E )gc ( E )dE

Ec

Ev = − p ∫(1 f ( E ))gv ( E )dE

Ebottom CalculationCalculation ofof ChargeCharge CarrierCarrier ConcentrationsConcentrations inin SemiconductorsSemiconductors π η η ()* 3/ 2 ∞ 1/ 2 − = 2 mnkT d η = E Ec n ∫ η−η where 2 3 + F h 0 1 e EF π 3/ 2 η ()* Fermi integral of order 1/2 = 2 mnkT n 2 3 F1/ 2( F ) π h η 2 n = N F ( ) c π 1/ 2 F

 * 3/ 2 =  2 mnkT  “Effective” density of conduction band states Nc 2   h2  η π − = 2 η = Ev EF p Nv F1/ 2( F ' ) where F ' EF ChargeCharge CarrierCarrier ConcentrationsConcentrations inin SemiconductorsSemiconductors –– SimplifiedSimplified ExpressionsExpressions

¾ When EF is away from the band edges (most cases) + < < − Ev 3kT EF Ec 3kT ¾ the expressions for n and p simplify to − = ( EF Ec )/ kT n Nce − = ( Ev EF )/ kT p Nve

¾ For intrinsic semiconductor Ei, the Fermi level for intrinsic semiconductor is close to the midgap and − − = = = ( Ei Ec )/ kT = ( Ev Ei )/ kT n p ni Nce Nve ChargeCharge CarrierCarrier ConcentrationsConcentrations inin SemiconductorsSemiconductors –– SimplifiedSimplified ExpressionsExpressions

¾ Intrinsic carrier concentration, ni − − Ei / kT = Ec / kT nie Nce

Ei / kT = ( Ev )/ kT nie Nve − − −E / 2kT 2 = ( Ec Ev )/ kT ⇒ = g ni Nv Nce ni Nv Nc e

¾ Simplified expressions for n and p in terms of ni and EF − n = n e( EF Ei )/ kT i ⇒ = 2 − np ni = ( Ei EF )/ kT p nie IntrinsicIntrinsic carriercarrier concentrationconcentration vsvs TT inin SiSi,, GeGe,, andand GaAsGaAs

− = Eg / 2kT ni Nv Nc e

Intrinsic carrier concentration is a material property that depends on temperature. CarrierCarrier distributionsdistributions asas aa functionfunction ofof EEF positionposition CarrierCarrier ConcentrationConcentration inin dopeddoped Materials:Materials:

ManipulatingManipulating EEF andand nn andand pp byby dopingdoping

¾ Consider a semiconductor with donor density ND.

¾ Since EC-ED is small almost all donors and acceptors will be ionized at room T. Charge neutrality requires that − + + = qp qn qN D 0 − + = p n qN D 0 = 2 ¾ np ni and elimination of p results in a quadratic equation for n n2 i − n + N = 0 ⇒ n2 − N n − n2 = 0 n D D i 2 2 N  N  ni = D +  D  + 2 ⇒ ≅ p = n ni n N D 2  2  N D ≈ 13 − 21 −3 >> ≈ 10 −3 N D 10 10 cm ni 10 cm ( at room T ) CarrierCarrier ConcentrationConcentration inin dopeddoped Materials:Materials:

ManipulatingManipulating EEF andand nn andand pp byby dopingdoping

¾ Arguments are similar for a p-type semiconductor and

the result is 2 ≅ = ni p N A and n N A ¾ “Power of doping”: We can manipulate carrier concentrations by manipulating dopant concentrations. ¾ Processing implications: Need very pure material (ppb)

≈ × 22 −3 NSi 5 10 cm ≈ 13 − 21 −3 N A or N D 10 10 cm IntrinsicIntrinsic FermiFermi levellevel EEi

n = p − − ( Ei Ec )/ kT = ( Ev Ei )/ kT Nce Nve 3/ 2 +   +  m*  = Ec Ev + kT  Nv  = Ec Ev + kT  p  Ei ln  ln *  2 2  Nc  2 2  mn  +  m*  = Ec Ev + 3  p  Ei kT ln *  2 4  mn  At room T this term is 0.012 eV for Si

This term is 0.56 eV for Si

¾ In an intrinsic semiconductor EF=Ei is ~ @ midgap DependenceDependence ofof EEF onon dopantdopant concentrationconcentration

− = = ( EF Ei )/ kT n N D nie n-type   − =  N D  EF Ei kT ln   ni 

− = = ( Ei EF )/ kT p N A nie p-type   − =  N A  Ei EF kT ln   ni  DependenceDependence ofof EEF onon dopantdopant concentrationconcentration TT dependencedependence ofof nn andand pp