ECE 663

The story so far..

Drift: Driven by Electric Field

vd = E

Velocity(cm/s) Mobility

(cm2/Vs)

Electric field(V/cm)

E

Which has higherdrift?

x

ECE 663

DRIFT

Why does a field create a velocityrather than an acceleration?

Terminalvelocity

Gravity

Drag

Why does a field create a velocityrather than an acceleration?

Random scatteringevents (R-G centers)

)sec(10

)100(10

sec/10

23

21

12

5

7

2

THz

nmcm

cmv

kTmv

c

th

th

The field gives a netdrift superposed on top

Why does a field create a velocityrather than an acceleration?

mn*(dv/dt + v/n) = -qE

n = qn/mn*

p = qp/mp*

ECE 663

From accelerating charges to drift

From mobility to drift current

n = qn/mn*

p = qp/mp*

Jn = qnv = qnnEdrift

Jp = qpv = qppEdrift

(A/cm2)

Resistivity, Conductivity

n = nqn = nq2n/mn*

Jn = nEdrift

Jp = pEdrift

p = pqp = pq2p/mp*

= 1/

= n + p

Ohm’s Law

Jn = E/ndrift

Jp = E/pdrift

L

AE = V/L

I = JA = V/R

R = L/A (Ohms) V

What’s the unit of ?

So mobility and resistivity depend on material properties (e.g. m*) and sample properties (e.g. NT, which determines )

Recall 1/ = vthNT

• Can we engineer these properties?

• What changes at the nanoscale?

ECE 663

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 resistivityN = 1/qNDn P = 1/qNAp

depends on N too, but weaker..

ECE 663

Phonon Scattering~T-3/2

Ionized Imp~T3/2

Piezo scattering

Temperature Dependence

ECE 663Bailon et al Tsui-Stormer-GossardPfeiffer-Dingle-West..

Reduce Ionized Imp scattering (Modulation Doping)

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 massupon transfer underbias

ECE 663

Negative Differential Resistance

ECE 663

DIFFUSION

ECE 663

J1 = qn(x)vJ2 = -qn(x+l)v

l = v

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

DIFFUSION

diff

ECE 663

Drift vs Diffusion

t

x

t

x

<x> ~ Et <x2> ~ Dt

E1

E2 > E1

SIGNS

EC

E

Jn = qnnEdrift

Jp = qppEdrift

vn = nEvp = pE

Opposite velocitiesParallel currents

SIGNS

Jn = qDndn/dxdiff

Jp = -qDpdp/dxdiff

dn/dx > 0 dp/dx > 0

Parallel velocitiesOpposite currents

ECE 663

In Equilibrium, Fermi Level is Invariant

e.g. non-uniform doping

ECE 663

Einstein Relationship

and D are connected !!

Jn + Jn = qnnE + qDndn/dx = 0diff drift

n(x)= Nce-[EC(x) - EF]/kT = Nce-[EC -EF - qV(x)]/kT

dn/dx = -(qE/kT)n

qnnE - qDn(qE/kT)n = 0Dn/n = kT/q

ECE 663

Einstein Relationship

n = qn/mn*

Dn = kTn/mn*

½ m*v2 = ½ kT

Dn = v2n = l2/n

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…

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

Let’s put all the maths together…

Thinkgeek.com

ECE 663

All the equations at one place

(n, p)

E

J

∫

Simplifications

• 1-D, RG with low-level injection

rN = p/p, rP = n/n

• Ignore fields E ≈ 0 in diffusion region

JN = qDNdn/dx, JP = -qDPdp/dx

ECE 663

Minority Carrier Diffusion Equations

∂np ∂2np

∂t ∂x2

np

n= DN - + GN

∂pn ∂2pn

∂t ∂x2

pn

p= DP - + GP

ECE 663

Example 1: Uniform Illumination

∂np ∂2np

∂t ∂x2

np

n= DN - + GN

Why? n(x,0) = 0n(x,∞) = GNn

n(x,t) = GNn(1-e-t/n)

ECE 663

Example 2: 1-sided diffusion, no traps

∂np ∂2np

∂t ∂x2

np

n= DN - + GN

n(x,b) = 0

n(x) =n(0)(b-x)/b

ECE 663

Example 3: 1-sided diffusion with traps

∂np ∂2np

∂t ∂x2

np

n= DN - + GN

n(x,b) = 0

n(x,t) = n(0)sinh[(b-x)/Ln]/sinh(b/Ln)

Ln = Dnn

Numerical techniques

2

Numerical techniques

ECE 663

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