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

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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 = 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 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

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?

Drag

Terminal

velocity

Gravity

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 mobility to drift current

drift

drift

Jp = qpv = qpmpE

Jn = qnv = qnmnE

(A/cm2)

mn= qtn/mn*

mp= qtp/mp*

Resistivity, Conductivity

drift

drift

Jp = spE

Jn = snE

sn= nqmn = nq2tn/mn*

sp= pqmp = pq2tp/mp*

r= 1/s

s = sn + sp

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?

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

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Some typical expressions
• Phonon Scattering
• Ionized Impurity Scattering

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Combining the mobilities

Matthiessen’s Rule

Caughey-Thomas Model

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Doping dependence of resistivity

rN = 1/qNDmn

rP = 1/qNAmp

mdepends on N too, but weaker..

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Temperature Dependence

Piezo scattering

Phonon Scattering

~T-3/2

Ionized Imp

~T3/2

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Reduce Ionized Imp scattering (Modulation Doping)

Tsui-Stormer-Gossard

Pfeiffer-Dingle-West..

Bailon et al

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Field Dependence of velocity

Velocity saturation ~ 107cm/s for n-Si (hot electrons)

Velocity reduction in GaAs

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Gunn Diode

Can operate around NDR point to get an oscillator

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Transferred Electron Devices (Gunn Diode)

E(GaAs)=0.31 eV

Increases mass

upon transfer under

bias

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DIFFUSION

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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

E2 > E1

E1

t

t

<x2> ~ Dt

<x> ~ mEt

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SIGNS

EC

drift

drift

Jp = qpmpE

Jn = qnmnE

E

Opposite velocities

Parallel currents

vp = mpE

vn = mnE

SIGNS

diff

diff

Jp = -qDpdp/dx

Jn = qDndn/dx

dn/dx > 0

dp/dx > 0

Parallel velocities

Opposite currents

In Equilibrium, Fermi Level is Invariant

e.g. non-uniform doping

ECE 663

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

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Einstein Relationship

Dn= kTtn/mn*

mn= qtn/mn*

½ m*v2= ½ kT

Dn= v2tn = l2/tn

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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

The equations

At steady state with no RG

.J = q.(nv) = 0

ECE 663

All the equations at one place

(n, p)

J

E

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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
Minority Carrier Diffusion Equations

∂Dnp

∂Dpn

Dpn

Dnp

∂2Dnp

∂2Dpn

= DP

= DN

-

-

+ GP

+ GN

tn

tp

∂t

∂t

∂x2

∂x2

ECE 663

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

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

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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

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