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ECE 663 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 R = (np - n i 2 )/[ p (n+n 1 ) + n (p+p 1 )] 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. The story so far..

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

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ECE 663 DRIFT

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

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

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Why does a field create a velocity rather than an acceleration? m n *(dv/dt + v/ n ) = -q E n = q n /m n * p = q p /m p *

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ECE 663 From accelerating charges to drift

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From mobility to drift current n = q n /m n * p = q p /m p * J n = qnv = qn n E drift J p = qpv = qp p E drift (A/cm 2 )

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Resistivity, Conductivity n = nq n = nq 2 n /m n * J n = n E drift J p = p E drift p = pq p = pq 2 p /m p * = 1/ = n + p

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Ohm’s Law J n = E/ n drift J p = E/ p drift L A E = V/L I = JA = V/R R = L/A (Ohms) V What’s the unit of ?

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So mobility and resistivity depend on material properties (e.g. m*) and sample properties (e.g. N T, which determines ) Recall 1/ = v th N T

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Can we engineer these properties? What changes at the nanoscale?

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

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ECE 663 Some typical expressions Phonon Scattering Ionized Impurity Scattering

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

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ECE 663 Doping dependence of mobility

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ECE 663 Doping dependence of resistivity N = 1/qN D n P = 1/qN A p depends on N too, but weaker..

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

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

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ECE 663 Field Dependence of velocity Velocity saturation ~ 10 7 cm/s for n-Si (hot electrons) Velocity reduction in GaAs

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

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ECE 663 GaAs bandstructure

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

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ECE 663 Negative Differential Resistance

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ECE 663 DIFFUSION

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ECE 663 J 1 = qn(x)v J 2 = -qn(x+l)v l = v J n = q(l 2 / )dn/dx = qD N dn/dx DIFFUSION diff

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ECE 663 Drift vs Diffusion t x t x ~ E t ~ Dt E1E1 E 2 > E 1

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SIGNS ECEC E J n = qn n E drift J p = qp p E drift v n = n E v p = p E Opposite velocities Parallel currents

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SIGNS J n = qD n dn/dx diff J p = -qD p dp/dx diff dn/dx > 0dp/dx > 0 Parallel velocities Opposite currents

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

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ECE 663 Einstein Relationship and D are connected !! J n + J n = qn n E + qD n dn/dx = 0 diff drift n(x)= N c e -[E C (x) - E F ]/kT = N c e -[E C -E F - qV(x)]/kT dn/dx = -(q E /kT)n qn n E - qD n (q E /kT)n = 0 D n / n = kT/q

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ECE 663 Einstein Relationship n = q n /m n * D n = kT n /m n * ½ m*v 2 = ½ kT D n = v 2 n = l 2 / n

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ECE 663 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 !!! So…

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ECE 663 Relation between current and charge

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ECE 663 Continuity Equation

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

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

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

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Simplifications 1-D, RG with low-level injection r N = p/ p, r P = n/ n Ignore fields E ≈ 0 in diffusion region J N = qD N dn/dx, J P = -qD P dp/dx

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ECE 663 Minority Carrier Diffusion Equations ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N ∂pn∂pn ∂2pn∂2pn ∂t ∂x 2 pnpn pp = D P -+ G P

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ECE 663 Example 1: Uniform Illumination ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N Why? n(x,0) = 0 n(x,∞) = G N n n(x,t) = G N n (1-e -t/ n )

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ECE 663 Example 2: 1-sided diffusion, no traps ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N n(x,b) = 0 n(x) = n(0)(b-x)/b

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ECE 663 Example 3: 1-sided diffusion with traps ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N n(x,b) = 0 n(x,t) = n(0)sinh[(b-x)/L n ]/sinh(b/L n ) L n = D n n

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Numerical techniques 2

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ECE 663 At the ends…

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ECE 663 Overall Structure

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ECE 663 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

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