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The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n

The story so far.. . The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n i 2 , n ~ N D for n-type The last chapter showed how the system tries to restore itself back to equilibrium when perturbed, through RG processes

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The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n

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  1. The story so far.. • The first few chapters showed us how to calculate the • equilibrium distribution of charges in a semiconductor • np = ni2, n ~ NDfor n-type • The last chapter showed how the system tries to restore itself • back to equilibrium when perturbed, through RG processes • R = (np - ni2)/[tp(n+n1) + tn(p+p1)] • In this chapter we will explore the processes that drive the system • away from equilibrium. • Electric forces will cause drift, while thermal forces (collisions) • will cause diffusion. ECE 663

  2. Drift: Driven by Electric Field vd = mE Electric field (V/cm) Velocity (cm/s) Mobility (cm2/Vs) E Which has higher drift? x

  3. DRIFT ECE 663

  4. Why does a field create a velocity rather than an acceleration? Drag Terminal velocity Gravity

  5. Why does a field create a velocity rather than an acceleration? The field gives a net drift superposed on top Random scattering events (R-G centers)

  6. Why does a field create a velocity rather than an acceleration? mn*(dv/dt + v/tn) = -qE mn= qtn/mn* mp= qtp/mp*

  7. From accelerating charges to drift ECE 663

  8. From mobility to drift current drift drift Jp = qpv = qpmpE Jn = qnv = qnmnE (A/cm2) mn= qtn/mn* mp= qtp/mp*

  9. Resistivity, Conductivity drift drift Jp = spE Jn = snE sn= nqmn = nq2tn/mn* sp= pqmp = pq2tp/mp* r= 1/s s = sn + sp

  10. Ohm’s Law drift drift Jp = E/rp Jn = E/rn L E = V/L I = JA = V/R R = rL/A (Ohms) A V What’s the unit of r?

  11. So mobility and resistivity depend on material properties (e.g. m*) and sample properties (e.g. NT, which determines t) Recall 1/t = svthNT

  12. Can we engineer these properties? • What changes at the nanoscale?

  13. What causes scattering? • Phonon Scattering • Ionized Impurity Scattering • Neutral Atom/Defect Scattering • Carrier-Carrier Scattering • Piezoelectric Scattering ECE 663

  14. Some typical expressions • Phonon Scattering • Ionized Impurity Scattering ECE 663

  15. Combining the mobilities Matthiessen’s Rule Caughey-Thomas Model ECE 663

  16. Doping dependence of mobility ECE 663

  17. Doping dependence of resistivity rN = 1/qNDmn rP = 1/qNAmp mdepends on N too, but weaker.. ECE 663

  18. Temperature Dependence Piezo scattering Phonon Scattering ~T-3/2 Ionized Imp ~T3/2 ECE 663

  19. Reduce Ionized Imp scattering (Modulation Doping) Tsui-Stormer-Gossard Pfeiffer-Dingle-West.. Bailon et al ECE 663

  20. Field Dependence of velocity Velocity saturation ~ 107cm/s for n-Si (hot electrons) Velocity reduction in GaAs ECE 663

  21. Gunn Diode Can operate around NDR point to get an oscillator ECE 663

  22. GaAs bandstructure ECE 663

  23. Transferred Electron Devices (Gunn Diode) E(GaAs)=0.31 eV Increases mass upon transfer under bias ECE 663

  24. Negative Differential Resistance ECE 663

  25. DIFFUSION ECE 663

  26. DIFFUSION J2 = -qn(x+l)v J1 = qn(x)v l = vt diff Jn = q(l2/t)dn/dx = qDNdn/dx ECE 663

  27. Drift vs Diffusion x x E2 > E1 E1 t t <x2> ~ Dt <x> ~ mEt ECE 663

  28. SIGNS EC drift drift Jp = qpmpE Jn = qnmnE E Opposite velocities Parallel currents vp = mpE vn = mnE

  29. SIGNS diff diff Jp = -qDpdp/dx Jn = qDndn/dx dn/dx > 0 dp/dx > 0 Parallel velocities Opposite currents

  30. In Equilibrium, Fermi Level is Invariant e.g. non-uniform doping ECE 663

  31. Einstein Relationship m and D are connected !! drift diff Jn + Jn = qnmnE + qDndn/dx = 0 n(x)= Nce-[EC(x) - EF]/kT = Nce-[EC -EF - qV(x)]/kT dn/dx = -(qE/kT)n Dn/mn= kT/q qnmnE - qDn(qE/kT)n = 0 ECE 663

  32. Einstein Relationship Dn= kTtn/mn* mn= qtn/mn* ½ m*v2= ½ kT Dn= v2tn = l2/tn ECE 663

  33. So… • We know how to calculate fields from • charges (Poisson) • We know how to calculate moving charges • (currents) from fields (Drift-Diffusion) • We know how to calculate charge • recombination and generation rates (RG) • Let’s put it all together !!! ECE 663

  34. Relation between current and charge ECE 663

  35. Continuity Equation ECE 663

  36. The equations At steady state with no RG .J = q.(nv) = 0 ECE 663

  37. Let’s put all the maths together… Thinkgeek.com

  38. All the equations at one place (n, p) ∫ J E  ECE 663

  39. Simplifications • 1-D, RG with low-level injection • rN = Dp/tp, rP = Dn/tn • Ignore fields E ≈ 0 in diffusion region • JN = qDNdn/dx, JP = -qDPdp/dx

  40. Minority Carrier Diffusion Equations ∂Dnp ∂Dpn Dpn Dnp ∂2Dnp ∂2Dpn = DP = DN - - + GP + GN tn tp ∂t ∂t ∂x2 ∂x2 ECE 663

  41. Example 1: Uniform Illumination ∂Dnp Dnp ∂2Dnp = DN - + GN tn ∂t ∂x2 Dn(x,0) = 0 Dn(x,∞) = GNtn Why? Dn(x,t) = GNtn(1-e-t/tn) ECE 663

  42. Example 2: 1-sided diffusion, no traps ∂Dnp Dnp ∂2Dnp = DN - + GN tn ∂t ∂x2 Dn(x,b) = 0 Dn(x) = Dn(0)(b-x)/b ECE 663

  43. Example 3: 1-sided diffusion with traps ∂Dnp Dnp ∂2Dnp = DN - + GN tn ∂t ∂x2 Dn(x,b) = 0 Ln = Dntn Dn(x,t) = Dn(0)sinh[(b-x)/Ln]/sinh(b/Ln) ECE 663

  44. Numerical techniques 2

  45. Numerical techniques

  46. At the ends… ECE 663

  47. Overall Structure ECE 663

  48. In summary • While RG gives us the restoring forces in a • semiconductor, DD gives us the perturbing forces. • They constitute the approximate transport eqns • (and will need to be modified in 687) • The charges in turn give us the fields through • Poisson’s equations, which are correct (unless we • include many-body effects) • For most practical devices we will deal with MCDE ECE 663

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