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The story so far.. The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = ni2, n ~ ND for 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

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**Drift: Driven by Electric Field**

vd = mE Electric field (V/cm) Velocity (cm/s) Mobility (cm2/Vs) E Which has higher drift? x

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DRIFT ECE 663

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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?**

The field gives a net drift superposed on top Random scattering events (R-G centers)

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**Why does a field create a velocity rather than an acceleration?**

mn*(dv/dt + v/tn) = -qE mn = qtn/mn* mp = qtp/mp*

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**From accelerating charges to drift**

ECE 663

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**From mobility to drift current**

Jn = qnv = qnmnE drift (A/cm2) Jp = qpv = qpmpE drift mn = qtn/mn* mp = qtp/mp*

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**Resistivity, Conductivity**

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

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**Ohm’s Law Jn = E/rn Jp = E/rp L E = V/L I = JA = V/R A R = rL/A (Ohms)**

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

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**So mobility and resistivity depend on material properties (e. g. m**

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

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**Can we engineer these properties?**

What changes at the nanoscale?

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**What causes scattering?**

Phonon Scattering Ionized Impurity Scattering Neutral Atom/Defect Scattering Carrier-Carrier Scattering Piezoelectric Scattering ECE 663

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**Some typical expressions**

Phonon Scattering Ionized Impurity Scattering ECE 663

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**Combining the mobilities**

Matthiessen’s Rule Caughey-Thomas Model ECE 663

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**Doping dependence of mobility**

ECE 663

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**Doping dependence of resistivity**

rN = 1/qNDmn rP = 1/qNAmp m depends on N too, but weaker.. ECE 663

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**Temperature Dependence**

Piezo scattering Phonon Scattering ~T-3/2 Ionized Imp ~T3/2 ECE 663

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**Reduce Ionized Imp scattering (Modulation Doping)**

Bailon et al Tsui-Stormer-Gossard Pfeiffer-Dingle-West.. ECE 663

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**Field Dependence of velocity**

Velocity saturation ~ 107cm/s for n-Si (hot electrons) Velocity reduction in GaAs ECE 663

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

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GaAs bandstructure ECE 663

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**Transferred Electron Devices (Gunn Diode)**

E(GaAs)=0.31 eV Increases mass upon transfer under bias ECE 663

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**Negative Differential Resistance**

ECE 663

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DIFFUSION ECE 663

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**Jn = q(l2/t)dn/dx = qDNdn/dx**

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

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**Drift vs Diffusion x x t t E2 > E1 E1 <x2> ~ Dt**

<x> ~ mEt ECE 663

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**SIGNS E vp = mpE vn = mnE Jn = qnmnE Jp = qpmpE EC Opposite velocities**

Parallel currents vp = mpE vn = mnE Jn = qnmnE drift Jp = qpmpE

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**SIGNS dn/dx > 0 dp/dx > 0 Jn = qDndn/dx Jp = -qDpdp/dx**

Parallel velocities Opposite currents Jn = qDndn/dx diff Jp = -qDpdp/dx

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**In Equilibrium, Fermi Level is Invariant**

e.g. non-uniform doping ECE 663

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**Einstein Relationship**

m and D are connected !! Jn Jn = qnmnE + qDndn/dx = 0 diff drift 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

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**Einstein Relationship**

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

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**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

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**Relation between current and charge**

ECE 663

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Continuity Equation ECE 663

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

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**Let’s put all the maths together…**

Thinkgeek.com

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**All the equations at one place**

(n, p) E J ∫ ECE 663

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**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

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**Minority Carrier Diffusion Equations**

∂Dnp ∂2Dnp ∂t ∂x2 Dnp tn = DN - + GN ∂Dpn ∂2Dpn Dpn tp = DP + GP ECE 663

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**Example 1: Uniform Illumination**

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

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**Example 2: 1-sided diffusion, no traps**

∂Dnp ∂2Dnp ∂t ∂x2 Dnp tn = DN - + GN Dn(x,b) = 0 Dn(x) = Dn(0)(b-x)/b ECE 663

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**Example 3: 1-sided diffusion with traps**

∂Dnp ∂2Dnp ∂t ∂x2 Dnp tn = DN - + GN Dn(x,b) = 0 Ln = Dntn Dn(x,t) = Dn(0)sinh[(b-x)/Ln]/sinh(b/Ln) ECE 663

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Numerical techniques 2

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Numerical techniques

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At the ends… ECE 663

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Overall Structure ECE 663

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**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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