MTLE-6120: Advanced Electronic Properties of Materials. Semiconductor p-n junction diodes. Reading: Kasap ,

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1 MTLE-6120: Advanced Electronic Properties of Materials 1 Semiconductor p-n junction diodes Reading: Kasap ,

2 Metal-semiconductor contact potential 2 p-type n-type p-type n-type Same semiconductor on both sides, different doping Bands line up perfectly, but Fermi level does not Bands bend to line up Fermi level Equal doping N a = N d symmetric bending In general, contact potential shared between both sides Extreme limits: one side p+ or n+, Schottky junction

3 Depletion region charge 3 p-type n-type p-type n-type Far from junction, E F = E F 0 ρ = 0 Approaching junction, E F first deviates k B T from E F 0 : Small deviation from neutral Debye screening regime Once E F more than few k B T away from E F 0 (towards center of gap): n N d (or p N a ) depletion; they were equal in neutral case Across junction, E F has to cross almost entire gap k B T Therefore, depleted width width where Debye screening applicable Assume ρ = +en d for width w n on n-side, en a for width w p on p-side, and 0 elsewhere

4 Depletion region field and potential 4 p-type n-type p-type n-type p-type n-type Neutrality of junction N a w p = N d w n Solve for electric filed using E = E (x) = ρ(x)/ɛ Solving from left, E(x) = en a (w p + x)/ɛ for x > w p (0 otherwise) Solving from right, E(x) = en d (w n x)/ɛ for x < w n (0 otherwise) Peak field E(0) = en a w p /ɛ = en d w n /ɛ Solve for potential using φ = φ (x)ˆx = E(x)ˆx to get: φ(x) = { ena(x+w p) 2 2ɛ, w p x 0 en aw 2 p 2ɛ + ena(2xwn x2 ) 2ɛ, 0 x w n

5 Depletion region width Total width w 0 w p + w n ; N a w p = N d w n w p = Total potential across region: w0n d N a+n d, w n = w0na N a+n d 5 V 0 = en aw 2 p 2ɛ + en a(2w 2 n) 2ɛ = en an d w 2 0 2ɛ(N a + N d ) But total potential is the contact potential: ( ev 0 = E F 0 + k B T ln N ) ( d E F 0 k B T ln N ) a = k B T ln N dn a n i n i n 2 i }{{}}{{} E F n E F p Therefore depletion region width: 2ɛ(N a + N d )V 0 2ɛ(N a + N d )k B T w 0 = = en a N d e 2 ln N dn a N a N d n 2 i When will depletion region width be set by λ D?

6 Applied bias 6 p-type n-type Barriers for e from n p and holes from p n: e(v 0 V ) Corresponding diffusion current: j 2 exp e(v0 V ) k B T Junction field drives drift current : j 1 Must be balanced at V = 0 j 1 = j 2 exp ev0 k B T j 0 ) Therefore j = j 0 (exp ev k B T 1

7 Magnitude of current 7 So far IV -characteristics exactly like Schottky diode For Schottky diode, we found j 0 exp Φ B k B T i.e. significant current when ev Φ B For pn-junction diode at zero bias: equal drift and diffusion currents = j 0 Minority carrier diffusion current driven by concentration gradient In uniform n-semiconductor ṗ = (p p 0 )/τ h (minority carrier lifetime τ h ) In non-uniform semiconductor: ṗ = D h 2 p (diffusion) Therefore in steady-state: D h 2 p = (p p 0 )/τ p decays exponentially towards p 0 with length scale L h = D h τ h Diffusion current j h = ed h p (x) ed hp 0 L h Total minority diffusion current j 0 = ed hn 2 i L h N d + ed en 2 i L e N a = en c N v Therefore current significant for V > E g /e = ed hn 2 i L h N d [ Dh L h N d + D e L e N a ] exp E g k B T

8 Minority carrier concentration in depletion region 8 p-type n-type In n-depletion region, potential changes from neutral value by (V 0 V )N a /(N a + N d ) (maximum at junction) Assume symmetric doping, potential changes by (V 0 V )/2 Hole concentration at junction p M N a exp e(v 0 V ) 2k B T n i exp ev 2k B T Hole recombination rate p M τh wn 2 (averaged over region)

9 Recombination current Current due to recombination of both e and h: ) exp j r = en i ( wn τ h + w p τ e ev 2k B T 9 Previous IV accounted for e and h transport separately, but not this recombination (except that it is needed for equilibrium) Net current density therefore: ( ) ( ) ev ev j(v ) = j dd0 k B T 1 + j r0 2k B T 1 Frequently approximated as ( ) ev j(v ) = j 0 ηk B T 1 with ideality factor η expected to be between 1 and 2 Ideal diode: no recombination η = 1

10 Additional effects in reverse bias 10 Space charge layer generation Reverse bias increases junction potential Higher field in space charge layer (depletion region) e and h in equilibrium: thermal generation vs recombination Field sweeps carriers away before they recombine current Linearly increasing reverse current instead of saturated j0 Avalanche breakdown Depletion region: large field, few carriers If eeλ > Eg, carriers can excite additional e-h pairs Cascade process leading to sudden increase in current Zener breakdown Highly doped junctions narrow depletion regions Potential larger than Eg: direct band-to-band tunneling Design sharp breakdown at specific potential (Zener diodes)

11 Depletion layer capacitance 11 p-type n-type p-type n-type p-type n-type Charge stored = en a w p = en d w n = ew 0 N a N d /(Na + N d ) (per unit area) Substituting for width of depletion region: q A = en an d 2ɛ(N a + N d )(V 0 V ) 2ɛeN a N d (V 0 V ) = Na + N d en a N d (N a + N d ) Therefore differential capacitance: C d A q A V = ɛen a N d 2(V 0 V )(N a + N d ) Typical value for Si, N a = N d = cm 3, C 0.1 µf/cm 2

12 Light-emitting diode 12 Basic design: just a p-n junction, but in direct band-gap material Current density recombination near junction Fraction of recombination is radiative light Spontaneous emission: light is incoherent and in random direction Efficiency: η = P light /(IV ) (can be > 10% for direct semiconductors) Typically use asymmetric junctions: pn + or np + : why?

13 LED: light spectrum 13 Minimum photon energy: E g Peak photon energy E g + k B T Spectral width 3k B T Due to distribution of both hole and electron energies For GaAs, E g = 1.42 ev, λ = 870 nm (IR) (hc/λ) 0.08 ev λ 50 nm Light emitted by LED can be absorbed by semiconductor Circumvent in hetero-junction LEDs

14 Semiconductor laser 14 Very similar to LED at the junction level Key difference: optical cavity using reflecting surfaces Start with spontaneous emission, sharpened by cavity resonance Stimulated emission builds up intensity in specific mode Require population inversion: e-h pairs waiting to recombine Population inversion achieved by bias: electrical pumping! Cavity resonance + laser amplification, λ 0.1 nm k B T

15 Photodiodes and solar cells 15 LED in reverse? Not quite, can use indirect band gap materials Absorption still occurs right at band gap (how?) eh-pairs created in depletion region swept across by field Loss: recombination (radiative or non-radiative) Modified device characteristic [ j(v ) = j ph + j 0 exp ev ] ηk B T 1 Why non-zero current at V = 0 if it is in equilibrium? Photodiode: operated in reverse bias: why?

16 Photovoltaic efficiencies 16 S. Kurtz and D. Levi, NREL Single junction efficiency limited by band-gap; circumvent using multi-junction devices