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

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

DRIFT ECE 663

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

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)

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

From accelerating charges to drift ECE 663

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

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

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?

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

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

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

Some typical expressions Phonon Scattering Ionized Impurity Scattering ECE 663

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

Doping dependence of mobility ECE 663

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

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

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

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

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

GaAs bandstructure ECE 663

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

Negative Differential Resistance ECE 663

DIFFUSION ECE 663

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

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

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

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

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

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

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

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

Relation between current and charge ECE 663

Continuity Equation ECE 663

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

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

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

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

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

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

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

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

Numerical techniques 2

Numerical techniques

At the ends… ECE 663

Overall Structure 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 ECE 663