1 MTLE-6120: Advanced Electronic Properties of Materials Intrinsic and extrinsic semiconductors Contents: I Band-edge density of states I Intrinsic Fermi level and carrier concentrations I Dopant states and ionization I Extrinsic Fermi level and carrier concentrations I Mobility: temperature and carrier density dependence I Recombination mechanisms Reading: I Kasap: 5.1 - 5.6 2 Band structure and conduction I Metals: partially filled band(s) i.e. bands cross Fermi level I Semiconductors / insulators: each band either filled or empty (T = 0) eτ I Drude formula applicable, mobility µ = m∗ ∗ 2 ~ −1 I Effective mass m = ~ [r~kr~kEn(k)] tensorial in general R ∗ −1 I Filled band does not conduct: eτ dk(m ) = 0 for each band 2 2 I Metals conduct due to carriers near Fermi level σ = g(EF )e vF τ=3 I Semiconductors: g(EF ) = 0 (will show shortly) ) no conduction at T = 0 3 Band structure of silicon (diamond-cubic semiconductor) 6 E c E 0 v E in eV −6 −12 LΛΓΔ ΧΣΓ I HOMO = Valence Band Maximum (VBM) with energy Ev and LUMO = Conduction Band Minimum (CBM) with energy Ec I HOMO-LUMO gap Eg = Ec − Ev ≈ 1:1 eV I HOMO and LUMO at different ~k ) indirect band gap I Diamond: similar band structure, much larger gap (≈ 5:5 eV) ) insulator 2 2 I Valence electrons/cell = 8 (even), configuration: 3s 3p (two Si/cell) 4 Band structure of GaAs (zinc-blende semiconductor) I HOMO-LUMO gap Eg = Ec − Ev ≈ 1:4 eV I HOMO and LUMO at same ~k (Γ) ) direct band gap 2 1 2 3 I Valence electrons/cell = 8 (even), configuration: Ga(4s 4p ), As(4s 4p ) 5 Density of states: silicon I Can calculate numerically from band structure I Parabolic band approximation valid for narrow energy range near gap 6 Density of states: parabolic-band semiconductor g(E) [eV-1nm-3] Valence band, m* = -0.3 Conduction band, m* = 0.5 4 3 2 1 0 -3 -2 -1 0 1 2 3 4 5 E [eV] I Parabolic bands near each band edge, with different effective masses I Overall DOS reduces with reduced effective mass magnitude I Set Ev = 0 conventionally (overall energy not well-defined) I Conduction band edge Ec = Eg I Where is the Fermi level? 7 Where is the Fermi level? I At T = 0, valence band fully occupied ) f(Ev = 0) = 1 ) EF > 0 I At T = 0, conduction band fully empty ) f(Ec = Eg) = 0 ) EF < Eg I Therefore, at T = 0, 0 < EF < Eg i.e. Fermi level is in the band gap I Chemical potential µ ! EF as T ! 0 I In semiconductor physics, typically refer to EF (T ) instead of µ(T ) I Therefore, Fermi functions will be 1 f(E; T ) = exp E−EF (T ) + 1 kB T 8 Where is the Fermi level at T > 0? I Given Fermi level EF and density of states g(E) R 1 I Number of electrons in conduction band is Ne = dEg(E)f(E) Eg R 0 I Number of holes in valence band is Nh = −∞ dEg(E)(1 − f(E)) I Total number of electrons cannot change with T ) Ne = Nh Z 0 Z 1 dEg(E)(1 − f(E)) = dEg(E)f(E) −∞ Eg Z 0 exp E−EF Z 1 1 dEg(E) kB T = dEg(E) E−EF E−EF −∞ exp + 1 E exp + 1 kB T g kB T Z 0 Z 1 dEg(E)e−(EF −E)=(kB T ) ≈ dEg(E)e−(E−EF )(kB T ) −∞ Eg 1 1 Z " Z " −EF =(kB T ) − k T (EF −Eg )=(kB T ) − k T e d"g(−")e B ≈ e d"g(Eg + ")e B 0 0 | {z } | {z } ≡Nv ≡Nc Assuming EF ;Eg − EF kBT , " ≡ energy from band edge 9 Band edge effective density of states I Given density of states as a function of energy away from band edge 1 Z " − k T Nc=v ≡ d"gc=v(")e B 0 p 3 2m∗ p I In parabolic band approximation g(") = 4π " for both bands 2π~ ∗ ∗ (but with different m ; for tensor m , above defines DOS meff) I Therefore band-edge effective of density states: 3 0q ∗ 1 1 2m Z c=v p " − k T Nc=v ≡ d" @ A 4π "e B 0 2π~ 3 3 0q ∗ 1 0q ∗ 1 2mc=v 2πmc=vkBT 3=2 = @ A 4πΓ(3=2)(kBT ) = 2 @ A 2π~ 2π~ ∗ 3=2 I Nc=v / (mc=v) (steeper g(") parabola) 3=2 I Nc=v / T (climb higher up the g(") parabola) 10 Fermi level for T > 0 I Charge neutrality imposes −EF =(kB T ) (EF −Eg )=(kB T ) Nve = Nce I Solve for Fermi level position: Eg kBT Nv EF (T ) = + ln 2 2 Nc I At T ! 0, EF is exactly at the middle of the band gap I At finite T , EF moves away ∼ kBT Eg (still close to gap center) I Which way does the Fermi level move with increasing T ? I For electrons in metals (and classical gases), µ # with T " I For semiconductors, EF (T ) # with T " iff Nc > Nv (more DOS in positive m∗ band; negative m∗ pulls µ other way) 11 Electron and hole concentrations −(Eg −EF )=(kB T ) I Number density of electrons n ≡ Ne = Nce −EF =(kB T ) I Number density of holes p ≡ Nh = Nve I Which one is larger? So far, they are equal: charge neutrality! −Eg =(kB T ) 2 I Note product np = NcNve ≡ ni , independent of EF I Neutral pure semiconductor, n = p = ni, intrinsic carrier density I If EF ", then n " and p # (more electrons than holes) I If EF #, then n # and p " (more holes than electrons) 2 I But np = ni , constant in all these cases + − −14 2 I This is an equilibrium constant, eg. [H ][OH ] = 10 M in water I How do you change EF ? Doping! (Also later, gating) 12 Intrinsic semiconductor thermodynamics and transport I Fermi level far from band edges ) Boltzmann statistics in both bands I Velocity distribution: Maxwell-Boltzmann distribution (classical gases) I Internal energy of electrons n · (Eg + 3kBT=2) I Internal energy of holes −p · (−3kBT=2) (holes are missing electrons!) I Net internal energy n · (Eg + 3kBT=2) − p · (−3kBT=2) I Drude theory conductivity σ = neµe + peµh 13 Intrinsic semiconductors: typical values at T = 300 K ∗ ∗ -3 -3 -3 Eg [eV] me=me mh=me Nc [cm ] Nv [cm ] ni [cm ] Ge 0.66 0.04,0.28 1.64,0.08 1:0 × 1019 6:0 × 1018 2:3 × 1013 Si 1.10 0.16,0.49 0.98,0.19 2:8 × 1019 1:2 × 1019 1:0 × 1010 GaAs 1.42 0.082 0.067 4:7 × 1017 7:0 × 1018 2:1 × 106 I Note that meff for Nc=v is an average of longitudinal / transverse values 1=3 2=3 (meff = mL mT ; for values see Table 5.1 in Kasap) I Nc and Nv increase with meff I ni drops exponentially with increasing Eg 14 Diamond-cubic structure: sp3 bonding 3 I Valence s and three p orbitals ) four sp hybrid orbitals I Orbitals point towards vertices of regular tetrahedron I Si, C, Ge: 4 valence electrons each I Form covalent bonds with four neighbours (8 shared electrons/atom) I Bonding orbitals ! valence band, anti-bonding orbitals ! conduction band I Tetrahedral network: FCC lattice with two atoms per cell 15 Zinc-blende structure: sp3 bonding 3 I Valence s and three p orbitals ) four sp hybrid orbitals I Orbitals point towards vertices of regular tetrahedron I Combine Ga,In (3 electrons) with As,Sb (5 electrons) I Form covalent bonds with four neighbours (8 shared electrons/atom) I Bonding orbitals ! valence band, anti-bonding orbitals ! conduction band I Tetrahedral network: FCC lattice with two atoms per cell I With Al and N, tend to form closely related Wurtzite structure (FCC to HCP cell) 16 Doping: acceptors and donors I Extra / impurity Group III atoms: one less electron per atom I Extra / impurity Group V atom: one extra electron per atom I Covalent bonding theory: atoms want 8 (filled-shell) of shared electrons I Group III `acceptor': pick up electron from solid ) hole in valence band I Group V `donor': give electron to solid ) electron in conduction band Simple picture of doping: I Density Na of acceptor atoms: charge −eNa I Density Nd of donor atoms: charge +eNd I Charge neutrality −en + ep − eNa + eNd = 0 ) n − p = Nd − Na 2 I Change in n and p due to shift in EF , but np = ni Eg kB T nNv kB T n I Solve for n and p, then find EF = + ln = EF 0 + ln 2 2 pNc 2 p I Even simpler picture: usually Nd;Na ni ) either p n or n p 17 Doping: p-type and n-type n-type semiconductor: I Donor impurities dominate Nd > 0 (Na = 0 or < Nd) 2 I Typically Nd − Na ni ) n p (since p = ni =n) 2 I Therefore n ≈ Nd − Na; p ≈ ni =(Nd − Na) kB T n Nd−Na I EF = EF 0 + ln = EF 0 + kBT ln (shifted " towards CBM) 2 p ni I Current predominantly carried by electrons p-type semiconductor: I Acceptor impurities dominate Na > 0 (Nd = 0 or < Na) 2 I Typically Na − Nd ni ) p n (since n = ni =p) 2 I Therefore p ≈ Na − Nd; n ≈ ni =(Na − Nd) kB T n Na−Nd I EF = EF 0 + ln = EF 0 − kBT ln (shifted # towards VBM) 2 p ni I Current predominantly carried by holes 18 Doping: a more complete picture I Simple picture: donor atom donates an electron, becomes positively charged I Positively charged donor ion can bind electrons: like a hydrogen atom −2 m∗ I Binding energy of pseudo-hydrogenic atom Eb = Ryd ∼ 0:05 eV r me I Donor level: Ed = Ec − Eb (electrons bound relative to CBM) I Exact argument for acceptors and holes, with charges swapped I Acceptor level: Ea = Ev + Eb (holes bound relative to VBM) I Levels in Si: note some impurities introduce multiple levels Physics of Semiconductor Devices, S.