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Home , Depletion region, Schottky barrier

1 MTLE-6120: Advanced Electronic Properties of Materials

Metal- junctions: Schottky and Ohmic contacts Contents: I Contact potential I I Debye screening I Schottky characteristics I Ohmic contacts Reading: I Kasap 5.9 - 5.10 2 Metal-semiconductor contact potential

Metal Semiconductor Metal Semiconductor

I Fermi levels relative to vacuum (Φ) not in same position I Distinction for semiconductor: Φ 6= I Consider case when metal level below n-type semiconductor I Fermi levels line up for equilibrium I Electron transfer to metal in current example I height ΦB = ΦM − χ I Contact potential: V0 = ΦM − Φn 3 Depletion region

+ ++ --- Metal 1 Metal 2 Metal Semiconductor

I Region of semiconductor near interface where charge transfer occurs I Number of charge carriers reduce, hence depletion region I In metal-metal case, bands on both sides respond I In metal-semiconductor case, only semiconductor responds: why? 4 Debye screening

I Poisson equation for variation of electrostatic potential: ρ(~r) ∇2φ(~r) = − 0

I Charge density ρ(~r) includes external and bound charge I Previously considered bound charge due to P~ response to E~ = ∇φ I What if material responds directly to potential as ρ = ρ(φ)? I Specifically consider small change from neutral potential φ0,

0 ρ(~r) ≈ ρ (φ0)(φ(~r) − φ0)

I Poisson equation becomes (using  to account for dipolar reponse): −ρ0(φ ) ∇2(φ − φ ) − 0 (φ − φ ) = 0 0  0

±x/λD with 1D solutions of the form φ(x) = φ0 + e p 0 where λD = /(−ρ (φ0)) is the Debye screening length 5 Debye screening length

p 0 I λD = /(−ρ (φ0)): length scale over which potential restores to neutral I Depletion region non-neutral → becomes neutral over λD length scale I Therefore width of depletion region ∼ λD 0 2 I In metals, −ρ (φ0) = e g(EF ) q −12 8.85×10 F/m ˚ I Typical λD ∼ (1.6×10−19 C)2·1047 J-1m-3 ∼ 0.6 A I Metals restore potential within an atomic layer! 0 2 I In , −ρ (φ0) = e nmaj/(kBT ), where nmaj ≈ Na/d = majority carrier density (larger of n, p) 14 3 I For 10 /cm doped Si at room temperature, λD ∼ 400 nm 18 3 I For 10 /cm doped Si at room temperature, λD ∼ 4 nm −1/2 I In all cases, λD ∝ n , where n = free carrier density I λD in metals  in semiconductors ⇒ depletion region entirely in semiconductor! 6 Applied bias

Metal Semiconductor Metal Semiconductor

I Neutral case: band bending in semiconductor = contact potential V0 I Apply potential bias: change band bending in semiconductor I Forward bias ≡ reduce band bending to V0 − V Rate of electrons M → SC ∝ exp −φB I kB T Rate of electrons SC → M ∝ exp −e(V0−V ) I kB T 7 : I − V characteristics

−e(V0−V ) −φB Net current density j = j2 exp − j1 exp I kB T kB T

−eV0 −φB At equilbrium V = 0, j = 0 ⇒ j2 exp = j1 exp = j0 (say) I kB T kB T I Therefore:  eV  j = j0 exp − 1 kBT

I Exponentially increasing current in forward bias (V > 0) I Current saturates to −j0 in reverse bias I j0 is given by Richardson-Dushman equation, just different Be! 8 Schottky photovoltaic devices

Metal Semiconductor

I Contact potential ⇒ built-in field at interface I Photon absorption → e-h pair, separated by field I In n-type junction, e relaxes to band edge and driven into bulk SC I h recombines at interface (met by e current in metal) I Below band-gap, interfacial generation also possible 9

Metal Semiconductor Metal Semiconductor

I Similar geometry to Schottky diode, but different level alignment I Consider metal lines up inside conduction band (or analogously for p-type material, inside valence band) I Bands bend in opposite direction to equalize Fermi levels I No barrier ΦB any more, electrons free to flow I j(V ) dominated by linear resistance of bulk semiconductor I I-V follows Ohm’s law ⇒ Ohmic contact 10 Peltier effect in Ohmic contacts

I Electrons flowing to the left pick up energy Ec − EF I Electrons flowing to the right lose energy Ec − EF I Energy drawn/dissipated from/as thermal energy at junction I Combine p and n-type junctions to get cooling in same metal I Reverse operation: thermoelectric generator (heat to electricity)

Metal Semiconductor