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- 1. Lecture 5 Junctions! R. Treharne Nov 6th 2014
- 2. Lecture Outline L5 ● Drawing Junctions ● Junction Formation – A Physical Picture ● Abrupt Junction ● Deriving Important Junction parameters ● Deriving diode equation ● Ideality Factor - Recombination
- 3. Metal-Semiconductor L5 metal SC (n-type) qφm qφn qχ EC EF EG EV Schottky Junction
- 4. Metal-Semiconductor L5 metal SC (n-type) qφm qφn qχ EC EF EF EG EV One Rule! ● Fermi level must be in equilibrium
- 5. Metal-Semiconductor L5 “Intimate Contact” metal SC (n-type) qφm qφn qχ EC EF EF EV qφm qφn qχ EC EV metal SC (n-type) EG EG
- 6. Metal-Semiconductor L5 BAND BENDING! metal SC (n-type) qφm qφn qχ EC EF EV metal SC (n-type) EG qφn qχ EC EF EG EV qφm qV qφ bi B “Built-in Voltage” Vbi = φm - φs φB = φm - χ Barrier Height
- 7. Metal-Semiconductor L5 metal SC (p-type) qφm qφp qχ EC EG EF EV
- 8. Metal-Semiconductor L5 metal SC (p-type) qφm qφp qχ EC EG EF EV qφp metal SC (p-type) qχ EC EV EF EG qφm qVbi qφB Vbi = φm - φp φB = EG/q - (φm - χ)
- 9. Metal-Semiconductor L5 φB = φm - χ Why doesn't this rule hold? Metal – SC interface effects ● Lattice Mis-match ● High density of defects ● Interface states in gap
- 10. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction EC
- 11. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction EF EF EC
- 12. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction qφp qχp EF qφn qχn EG EF p n EG EV EC EC EC EV
- 13. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction p n qVbi EF EF
- 14. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction p n qVbi EF EF Ei Ei
- 15. Semiconductor-Semiconductor L5 qφp qχp EG EF p n qφn qχn EF EG EV p-n homojunction p n qVbi EF EF Ei Ei Defining Electrostatic potential, Ψ, for a SC ● Difference between Ei and Ef qΨn qΨp Vbi = Ψn - Ψp Electrostatic Potential, Ψ Electron energy, E KNOW HOW TO DRAW THIS! Coming back here shortly to get a more physical picture
- 16. Semiconductor-Semiconductor L5 Everybody's favourite homojunction - Silicon p-type n-type http://electronics.howstuffworks.com/diode1.htm Every bit of electronics you own is jam packed with silicon homojunctions Extrinsically doped iPhone 5: 1 billion transistors! ~ 45 nm wide Mono-C solar cells ~750 μm thick cell size = 1 cm2
- 17. Semiconductor-Semiconductor L5 p-n heterojunction: Different band gaps qφp qχp EFp p n qφn qχn EFn ΔEC ΔEV wide gap narrow gap EF 1. Align the Fermi level. Leave some space for transition region.
- 18. Semiconductor-Semiconductor L5 p-n heterojunction: Different band gaps qφp qχp EFp p n qφn qχn EFn ΔEC ΔEV wide gap narrow gap EF 2. Mark out ΔEC and ΔEC at mid-way points ΔEC ΔEC
- 19. Semiconductor-Semiconductor L5 p-n heterojunction: Different band gaps qφp qχp EFp p n qφn qχn EFn ΔEC ΔEV wide gap narrow gap EF 3. Connect the C.B. and V.B. keeping the band gap constant in each material Difference in band gaps give rise to discontinuities in band diagrams. This limits the carrier transport by introducing potential barriers at the junction Barrier to hole transport
- 20. Semiconductor-Semiconductor L5 Examples of p-n heterojections Most thin-film PV technologies http://pubs.rsc.org/en/content/articlehtml/2013/ee/c3ee41981a#cit111
- 21. Semiconductor-Semiconductor L5 Examples of p-n heterojections III-V multi-junction devices Being able to control the band offsets of a material can help eliminate barriers
- 22. Physical Picture L5 Drawing band schematics for junctions all day is fun, but what is actually going on here? http://www.youtube.com/watch?v=JBtEckh3L9Q Excellent qualitative description 6:27 – Didactic Model of Junction formation
- 23. Physical Picture L5 Equilibrium Fermi Levels – Explained – our one rule for drawing band schematics! In thermal equilibrium, i.e. steady state: not net current flows J=J (drift )+J (diffusion)=0 http://en.wikipedia.org/wiki/File:Pn-junction-equilibrium.png notice position of “metallurgical” junction. Usually call this x=0
- 24. Physical Picture L5 Lets look at hole current first: J p = J p(drift)+J p(diffusion) = qμp p ξ−qDp dp dx
- 25. Physical Picture L5 Lets look at hole current first: J p = J p(drift)+J p(diffusion) = qμp p ξ−qDp dp dx mobility: hole distribution: e-field: diffusion coefficient μp= e τ mh p=ni e(Ei−EF)/ kB T OK to use Boltzmann as approx to Fermi-Dirac dp dx p (dE= i k T dx B− dEF dx ) ξ=1e dEi dx Dp=(k BT /e )μp Einstein relation will use this shortly
- 26. Physical Picture L5 J p = J p(drift)+J p(diffusion) = qμp p ξ−qDp dp dx = μp p dEi dx −μp p(dEi dx − dEF dx ) 0 = μp p dEF dx Condition for steady state
- 27. Physical Picture L5 dEF dx =0 TA DA! The same is true from consideration of net electron current, Jn
- 28. Abrupt Junction L5 charge density Hook & Hall, “Soild State Physics”, 2nd ed. Chap 6, p. 171 outside depletion region: p = NA and n = ND Space-charge condition NAwp = Ndwn i.e. areas of rectangles must be the same electric field electrostatic potential Max field (at x=0): ξm = qNDxn/εs = qNAxp/εs I've already called this Vbi ξm Area = Vbi Vbi=ϕn−ϕp= k BT q ln(NA ND 2 ) ni
- 29. Abrupt Junction L5 charge density 2 ) Lets derive this! Hook & Hall, “Soild State Physics”, 2nd ed. Chap 6, p. 171 outside depletion region: p = NA and n = ND Space-charge condition NAwp = Ndwn i.e. areas of rectangles must be the same electric field electrostatic potential Max field (at x=0): ξm = qNDxn/εs = qNAxp/εs I've already called this Vbi ξm Area = Vbi Vbi=ϕn−ϕp= k T ln(NNBA D q ni (but skip if I'm being slow)
- 30. Abrupt Junction L5 From Earlier: ϕ =−1e (Ei−EF ) What is in ϕ p and n regions away from junction? Here p=ni e(Ei−EF)/ kB T n=ni e(Ei−EF )/ kB T and ϕp≡−1 e (Ei−EF)|x⩽−x p =− k BT e (NA ni ) ϕn≡−1e (Ei−EF)|x⩾−xn =− k BT e (ND ni ) S. M. Sze, “Semiconductor Devices: Physics and Technology”, Chap 4, p.90
- 31. Abrupt Junction L5 Hence: Vbi=ϕn−ϕp= k BT q ln(NA ND 2 ) ni want to know what this is for Si? Go to: http://pveducation.org/pvcdrom/pn-junction/intrinsic-carrier-concentration Next Question: How do I calculate ϕ in the depletion region? (And why would I want to?) Must solve Poisson's Equation ∇2ϕ=−ρε Go read some electrostatics if you're interested
- 32. Abrupt Junction L5 ∇2ϕ=−ρε x)={−Ne A ρ(+Ne D 0 −wp< x<0 0< x<wn elsewhere ξ=− d ϕ dx ={− N A e ε ( x+wp) ND e ε ( x+wn) −wp< x<0 0< x<wn ϕ( x)={− NA e 2ε (x+wp)2 ND e 2ε (x+wn)2 INTEGRATE ONCE INTEGRATE TWICE −wp< x<0 0< x<wn
- 33. Abrupt Junction L5 Vbi=ϕ(x=wn)−ϕ( x=−wp) Vbi= e 2 ε 2+ND wn 2 (NA wp ) Vbi using result: wn=√ 2 ε NAV bi eND (NA+ND) wp=√ 2 εNDV bi eN A(N A+ND) and with some jiggery pokery: W=wp+wn=√2 ε e (N A+ND N A ND ) NA wp=NDwn
- 34. Under Bias L5 What happens to our ideal p-n junction if we put a forward or reverse bias across it?
- 35. Under Bias L5 p n W 0 wn -wp EF ZERO BIAS Vbi
- 36. Under Bias L5 p n W 0 wn -wp EF FORWARD BIAS Vbi - V EF + - ● Electron energy levels in p side lowered relative to those in n side ● Energy barrier qVbi reduced ● Flow of electrons from n to p increases ● Flow of holes from p to n increases ● Depletion width decreases
- 37. Under Bias L5 p n W 0 wn -wp EF REVERSE BIAS Vbi + V EF - + ● Electron energy levels in p side raised relative to those in n side ● Energy barrier qVbi increase by V ● Flow of electrons from n to p reduced ● Junction width increases
- 38. L5 Under Bias Of course, you already know all about this:
- 39. Ideal dark JV curve L2 J0 = 1 x 10-10 A/cm2 J=J 0 [exp( eV k BT )−1 ] http://pveducation.org/pvcdrom/pn-junction/diode-equation
- 40. Ideal dark JV curve L2 J0 = 1 x 10-10 A/cm2 J=J 0 [exp( eV k BT )−1 ] Lets derive this by considering our band schematic. Fun fun fun! http://pveducation.org/pvcdrom/pn-junction/diode-equation
- 41. L5 Deriving Shockley Equation Steady State again (i.e. no bias) Consider electron drift current – Ieo from p to n
- 42. L5 Deriving Shockley Equation Steady State again (i.e. no bias) Consider electron drift current – Ieo from p to n I e0=e×(generationrate /volume) ×(volume within Le of depletion zone) W Le J e 0=e ( np τ p ) Le
- 43. L5 Deriving Shockley Equation Steady State again (i.e. no bias) Consider electron drift current – Ieo from p to n I e0=e×(generationrate /volume) ×(volumewithin Le of depletion zone) J e 0=e ( np τ p ) Le W Le number of electrons on p side electron lifetime Le=√De τ p diffusion length
- 44. L5 Deriving Shockley Equation Steady State again (i.e. no bias) Consider electron drift current – Ieo from p to n W Le By assuming that all acceptors on p side are ionized: np= n2 i p p = n2 i NA J e 0= 2 eDe ni Le NA can re-write Je0 as:
- 45. L5 Deriving Shockley Equation What happens under forward bias? ● Drift (p to n)? - Nothing. No potential barrier in this direction ● Diffusion (n to p)? - Increases by exponential factor: k BT ) If we assume occupation exp ( eV of states in CB is given by Boltzmann distribution
- 46. L5 Deriving Shockley Equation What happens under forward bias? ● Drift (p to n)? - Nothing. No potential barrier in this direction ● Diffusion (n to p)? - Increases by exponential factor: k BT ) If we assume occupation exp ( eV of states in CB is given by Boltzmann distribution Net electron current: J e =−J e 0+J e 0 exp(eV k BT ) = J e 0[exp(eV k BT )−1]
- 47. L5 Deriving Shockley Equation Can do the same for the net hole current: J h=J h 0 [exp( eV k BT )−1] J h 0= 2 eDH ni Lh ND where
- 48. L5 Deriving Shockley Equation Can do the same for the net hole current: J h=J h 0 [exp( eV k BT )−1] J h 0= 2 eDH ni Lh ND where Therefore: Net Current! J=J e+Jh=J0 [exp( eV kBT )−1] J 0=J e 0+J h0=eni 2( De Ln NA + Dh LhN A ) ideal diode equation reverse saturation current, a.k.a “dark current”
- 49. L5 Deriving Shockley Equation Previous derivation ignores the recombination and generation of carriers within the depletion layer itself! J=J 0[exp( eV n k BT )−1]
- 50. L5 Deriving Shockley Equation Previous derivation ignores the recombination and generation of carriers within the depletion layer itself! J=J 0[exp( eV n k BT )−1] ideality factor – a complete FUDGE Worth reading up about – but don't stress out about it. ● S. M. Sze, “Semiconductor Devices”, 2nd ed. Chap 4, pp 109 – 113 ● Hook & Hall “Solid State Physics”, 2nd ed. Chap 6, pp 178 – 179 I have these books if you want them.
- 51. Lecture Summary L5 ● Drawing Junctions ● Junction Formation – A Physical Picture ● Abrupt Junction ● Deriving Important Junction parameters ● Deriving diode equation ● Ideality Factor - Recombination

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