MTLE-6120: Advanced Electronic Properties of Materials Intrinsic and - - PowerPoint PPT Presentation
1 MTLE-6120: Advanced Electronic Properties of Materials Intrinsic and extrinsic semiconductors Contents: Band-edge density of states Intrinsic Fermi level and carrier concentrations Dopant states and ionization Extrinsic Fermi
◮ Band-edge density of states ◮ Intrinsic Fermi level and carrier concentrations ◮ Dopant states and ionization ◮ Extrinsic Fermi level and carrier concentrations ◮ Mobility: temperature and carrier density dependence ◮ Recombination mechanisms
◮ Kasap: 5.1 - 5.6
◮ Metals: partially filled band(s) i.e. bands cross Fermi level ◮ Semiconductors / insulators: each band either filled or empty (T = 0) ◮ Drude formula applicable, mobility µ = eτ m∗ ◮ Effective mass m∗ = 2[∇ k∇ kEn(
◮ Filled band does not conduct: eτ
◮ Metals conduct due to carriers near Fermi level σ = g(EF )e2v2 F τ/3 ◮ Semiconductors: g(EF ) = 0 (will show shortly) ⇒ no conduction at T = 0
◮ HOMO = Valence Band Maximum (VBM) with energy Ev
◮ HOMO-LUMO gap Eg = Ec − Ev ≈ 1.1 eV ◮ HOMO and LUMO at different
◮ Diamond: similar band structure, much larger gap (≈ 5.5 eV) ⇒ insulator ◮ Valence electrons/cell = 8 (even), configuration: 3s23p2 (two Si/cell)
◮ HOMO-LUMO gap Eg = Ec − Ev ≈ 1.4 eV ◮ HOMO and LUMO at same
◮ Valence electrons/cell = 8 (even), configuration: Ga(4s24p1), As(4s24p3)
◮ Can calculate numerically from band structure ◮ Parabolic band approximation valid for narrow energy range near gap
1 2 3 4
1 2 3 4 5 E [eV] g(E) [eV-1nm-3] Valence band, m* = -0.3 Conduction band, m* = 0.5
◮ Parabolic bands near each band edge, with different effective masses ◮ Overall DOS reduces with reduced effective mass magnitude ◮ Set Ev = 0 conventionally (overall energy not well-defined) ◮ Conduction band edge Ec = Eg ◮ Where is the Fermi level?
◮ At T = 0, valence band fully occupied ⇒ f(Ev = 0) = 1 ⇒ EF > 0 ◮ At T = 0, conduction band fully empty ⇒ f(Ec = Eg) = 0 ⇒ EF < Eg ◮ Therefore, at T = 0, 0 < EF < Eg i.e. Fermi level is in the band gap ◮ Chemical potential µ → EF as T → 0 ◮ In semiconductor physics, typically refer to EF (T) instead of µ(T) ◮ Therefore, Fermi functions will be
kBT
◮ Given Fermi level EF and density of states g(E) ◮ Number of electrons in conduction band is Ne =
Eg dEg(E)f(E) ◮ Number of holes in valence band is Nh = −∞ dEg(E)(1 − f(E)) ◮ Total number of electrons cannot change with T ⇒ Ne = Nh −∞
Eg
−∞
kBT
kBT
Eg
kBT
−∞
Eg
ε kBT
ε kBT
◮ Given density of states as a function of energy away from band edge
ε kBT
◮ In parabolic band approximation g(ε) =
2m∗ 2π
◮ Therefore band-edge effective of density states:
c/v
3
ε kBT
c/v
3
c/vkBT
3 ◮ Nc/v ∝ (m∗ c/v)3/2 (steeper g(ε) parabola) ◮ Nc/v ∝ T 3/2 (climb higher up the g(ε) parabola)
◮ Charge neutrality imposes
◮ Solve for Fermi level position:
◮ At T → 0, EF is exactly at the middle of the band gap ◮ At finite T, EF moves away ∼ kBT ≪ Eg (still close to gap center) ◮ Which way does the Fermi level move with increasing T? ◮ For electrons in metals (and classical gases), µ ↓ with T ↑ ◮ For semiconductors, EF (T) ↓ with T ↑ iff Nc > Nv
◮ Number density of electrons n ≡ Ne = Nce−(Eg−EF )/(kBT ) ◮ Number density of holes p ≡ Nh = Nve−EF /(kBT ) ◮ Which one is larger? So far, they are equal: charge neutrality! ◮ Note product np = NcNve−Eg/(kBT ) ≡ n2 i , independent of EF ◮ Neutral pure semiconductor, n = p = ni, intrinsic carrier density ◮ If EF ↑, then n ↑ and p ↓ (more electrons than holes) ◮ If EF ↓, then n ↓ and p ↑ (more holes than electrons) ◮ But np = n2 i , constant in all these cases ◮ This is an equilibrium constant, eg. [H+][OH−] = 10−14 M2 in water ◮ How do you change EF ? Doping! (Also later, gating)
◮ Fermi level far from band edges ⇒ Boltzmann statistics in both bands ◮ Velocity distribution: Maxwell-Boltzmann distribution (classical gases) ◮ Internal energy of electrons n · (Eg + 3kBT/2) ◮ Internal energy of holes −p · (−3kBT/2) (holes are missing electrons!) ◮ Net internal energy n · (Eg + 3kBT/2) − p · (−3kBT/2) ◮ Drude theory conductivity σ = neµe + peµh
e/me
h/me
◮ Note that meff for Nc/v is an average of longitudinal / transverse values
L m2/3 T ; for values see Table 5.1 in Kasap) ◮ Nc and Nv increase with meff ◮ ni drops exponentially with increasing Eg
◮ Valence s and three p orbitals ⇒ four sp3 hybrid orbitals ◮ Orbitals point towards vertices of regular tetrahedron ◮ Si, C, Ge: 4 valence electrons each ◮ Form covalent bonds with four neighbours (8 shared electrons/atom) ◮ Bonding orbitals → valence band, anti-bonding orbitals → conduction band ◮ Tetrahedral network: FCC lattice with two atoms per cell
◮ Valence s and three p orbitals ⇒ four sp3 hybrid orbitals ◮ Orbitals point towards vertices of regular tetrahedron ◮ Combine Ga,In (3 electrons) with As,Sb (5 electrons) ◮ Form covalent bonds with four neighbours (8 shared electrons/atom) ◮ Bonding orbitals → valence band, anti-bonding orbitals → conduction band ◮ Tetrahedral network: FCC lattice with two atoms per cell ◮ With Al and N, tend to form closely related Wurtzite structure
◮ Extra / impurity Group III atoms: one less electron per atom ◮ Extra / impurity Group V atom: one extra electron per atom ◮ Covalent bonding theory: atoms want 8 (filled-shell) of shared electrons ◮ Group III ‘acceptor’: pick up electron from solid ⇒ hole in valence band ◮ Group V ‘donor’: give electron to solid ⇒ electron in conduction band
◮ Density Na of acceptor atoms: charge −eNa ◮ Density Nd of donor atoms: charge +eNd ◮ Charge neutrality −en + ep − eNa + eNd = 0 ⇒ n − p = Nd − Na ◮ Change in n and p due to shift in EF , but np = n2 i ◮ Solve for n and p, then find EF = Eg 2 + kBT 2
pNc = EF 0 + kBT 2
p ◮ Even simpler picture: usually Nd, Na ≫ ni ⇒ either p ≫ n or n ≫ p
◮ Donor impurities dominate Nd > 0 (Na = 0 or < Nd) ◮ Typically Nd − Na ≫ ni ⇒ n ≫ p (since p = n2 i /n) ◮ Therefore n ≈ Nd − Na, p ≈ n2 i /(Nd − Na) ◮ EF = EF 0 + kBT 2
p = EF 0 + kBT ln Nd−Na ni
◮ Current predominantly carried by electrons
◮ Acceptor impurities dominate Na > 0 (Nd = 0 or < Na) ◮ Typically Na − Nd ≫ ni ⇒ p ≫ n (since n = n2 i /p) ◮ Therefore p ≈ Na − Nd, n ≈ n2 i /(Na − Nd) ◮ EF = EF 0 + kBT 2
p = EF 0 − kBT ln Na−Nd ni
◮ Current predominantly carried by holes
◮ Simple picture: donor atom donates an electron, becomes positively
◮ Positively charged donor ion can bind electrons: like a hydrogen atom ◮ Binding energy of pseudo-hydrogenic atom Eb = ǫ−2 r m∗ me Ryd ∼ 0.05 eV ◮ Donor level: Ed = Ec − Eb (electrons bound relative to CBM) ◮ Exact argument for acceptors and holes, with charges swapped ◮ Acceptor level: Ea = Ev + Eb (holes bound relative to VBM) ◮ Levels in Si: note some impurities introduce multiple levels
◮ For GaAs, Group II or Group VI are shallow dopants ◮ For GaAs, Group IV can be donor and acceptor dopants: how?
◮ For each donor atom, degenerate donor levels typically with gd = 2 (spin) ◮ Electron occupation: zero or one for the whole atom (repulsions) ◮ Probability of occupation zero ∝ 1 ◮ Probability of occupation one ∝ gd exp EF −Ed kBT ◮ Normalized probability of ionized donor (occupation zero):
d =
kBT ◮ Therefore number density of ionized donors:
d =
kBT
◮ For each acceptor atom, degenerate acceptor levels typically with ga = 4
◮ Hole occupation: zero or one for the whole atom (repulsions) ◮ Probability of hole occupation zero ∝ 1 ◮ Probability of hole occupation one ∝ ga exp Ea−EF kBT
◮ Normalized probability of ionized donor (hole occupation zero):
a =
kBT ◮ Therefore number density of ionized acceptors:
a =
kBT
d − eN − a
Eg−EF kBT
kBT
EF −Ed kBT
Ea−EF kBT
◮ If Na − Nd ≫ ni, then p ≫ n (p-type) ◮ If Nd − Na ≫ ni, then n ≫ p (n-type) ◮ Previous simple analysis holds if:
◮ Net doping is stronger than ni (one of the two regimes above), and ◮ Doping is small enough that EF is far above Ea and far below Ed
Nd/a ni )
◮ Cross-over point from + to − is neutral EF
◮ At moderate doping level, far from mid-gap and acceptor levels
◮ At moderate doping level, far from mid-gap and donor levels
◮ Donors may be partially ionized (simple model no longer works)
◮ Acceptors may be partially ionized (simple model no longer works)
◮ High-enough n doping: Fermi level enters conduction band (‘n+’) ◮ High-enough p doping: Fermi level enters valence band (‘p+’) ◮ One of our approximations breaks down for n+ case with EF > Eg,
◮ Similarly, for p+ case with EF < 0:
◮ These are the Fermi theory expressions with kF = √2m∗εF /
◮ Important: partial donor / acceptor ionization in this regime
◮ Consider a p-type material (Nd = 0) with EF = Ea
◮ Around this EF , acceptors are partially ionized ◮ When exactly does this occur? ◮ Since EF far from conduction band, neglect n ≪ p ◮ Charge neutrality yields
kBT =
kBT
kBT =
Na ◮ Therefore, this happens for high Na when Ea ≪ kBT ◮ But also, for kBT lower than Ea: dopant freeze-out ◮ Importance of shallow donor/acceptor levels!
◮ If T much smaller than ionization threshold, neutrality:
kBT =
kBT
kBT ≈ Na
kBT
◮ Very similar to intrinsic case, except Na/ga replaces Nc ◮ Effective gap between valence band and acceptor level! ◮ In this regime, p2 = p · N − a = (NvNa/ga) exp −Ea kBT ◮ Similar behavior for frozen-out donors in ionization regime for n-type
1/50 1/100 1/300 1/1000 1/T [K
1]
0.0 0.2 0.4 0.6 0.8 1.0 EF [eV] Na/d = 1012 Na/d = 1014 Na/d = 1016 Na/d = 1018
◮ Doping dominates at low T, approach intrinsic level at high T ◮ At low T, Fermi level decided by donor/acceptor level ◮ Note using Ea/d = 0.1 eV from band edges to exaggerate effect
1/50 1/100 1/300 1/1000 1/T [K
1]
1010 1012 1014 1016 1018 1020 n [cm
3]
I
i z a t i
Extrinsic Intrinsic Na/d = 1012 Na/d = 1014 Na/d = 1016 Na/d = 1018
◮ Ionization regime at low T upto threhsold which increases with Nd/a ◮ Constant concentration in extrinsic regime; threshold increases with Nd/a ◮ At high T, dopants don’t matter: intrinsic regime
◮ Drude theory: mobility µ = eτ/m∗ ◮ Typically semiconductor m∗ ∼ 0.1 − 1 me, so expect higher µ than metals ◮ At room temperature, µe i-Si ∼ 1400 cm2/(Vs) and µAg ∼ 60 cm2/(Vs) ◮ Effective mass alone does not explain it! ◮ Remember τ −1 e-ph ∝ g(E)T ◮ For metals, g(E) → g(EF ) since most carriers near Fermi level ◮ For semiconductors, carriers within few kBT of band edge
◮ Averaged over carriers, g(E) ∝
◮ Therefore, τ −1 e-ph ∝ T 3/2 and µ ∝ τ ∝ T −3/2
◮ Doped semiconductor contains ionized donors / acceptors ◮ Charged impurities cause electron PE ∝ 1/r near them ◮ Electrons with KE ≫ PE not scattered significantly ◮ Electrons with KE ≪ PE scattered most strongly ◮ Effective cross-section ∝ r2 c, where PE(rc) ∼ KE ∼ kBT ◮ Therefore rc ∝ T −1 and cross-section σcs ∝ T −2 ◮ Scattering time τI = (Na/dσcsv)−1 ◮ Average velocity v ∝
◮ Therefore τI ∝ N −1 a/dT 3/2
◮ Intrinsic scattering time τe-ph ∝ T −3/2 ◮ Dopant / impurity scattering time τI ∝ N −1 a/dT 3/2 ◮ Net scattering time τ ∝
◮ Therefore mobility µ ∝
◮ At high T, e-ph scattering dominates (intrinsic regime) ◮ At high doping concentration (or low T), impurity scattering dominates
◮ Conductivity σ = neµe + peµh: similar dependence as n
◮ µ effect visible mainly in extrinsic regime where n is constant
◮ Excite electrons and holes in semiconductor to higher energy ◮ e-ph scattering brings electrons and holes to band edges ◮ If np = n2 i , equilibrium: nothing further happens ◮ What if you make more electron-hole pairs (eg. using light)
i ? ◮ Electrons and holes will recombine to restore equilibrium
E in eV
6 −12 L Λ Γ Δ Χ Σ Γ Ev Ec −6
◮ Direct gap: electrons and holes at band edges at same k ◮ Indirect gap: band edge carriers at different k ◮ Which will recombine faster? ◮ Direct gap: momentum conservation, recombine and emit light (usually) ◮ Indirect gap: cannot directly recombine: momentum not conserved
◮ Recombination rate proportional to np − n2 i ◮ Radiative / direct recombination (direct gap materials) ◮ Trap-assisted (Shockley-Read-Hall recombination)
◮ Trap level in gap captures electron (hole), becoming − (+) charged ◮ Later captures hole (electron), becoming neutral ◮ Energy from recombination emitted to phonons ◮ Probability of each capture ∝ Boltzmann factor of trap depth from band
◮ Net rate ∼ sech2 Et−Eg/2
2kBT
◮ Strongest for mid-band-gap states!
◮ Auger recombination
◮ Energy and momentum of e-h pair go to excite another e or h ◮ Need e or h to excite, so rate ∝ n, p ◮ Dominates at very high carrier concentrations
◮ Recombination rate = α(np − n2 i ) ◮ Minority carrier lifetime = 1/(max(n, p)α)