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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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Presentation Transcript
slide1

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

slide2

Drift: Driven by Electric Field

vd = mE

Electric field

(V/cm)

Velocity

(cm/s)

Mobility

(cm2/Vs)

E

Which has higher

drift?

x

slide3

DRIFT

ECE 663

slide4

Why does a field create a velocity

rather than an acceleration?

Drag

Terminal

velocity

Gravity

slide5

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)

slide6

Why does a field create a velocity

rather than an acceleration?

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

mn= qtn/mn*

mp= qtp/mp*

slide8

From mobility to drift current

drift

drift

Jp = qpv = qpmpE

Jn = qnv = qnmnE

(A/cm2)

mn= qtn/mn*

mp= qtp/mp*

slide9

Resistivity, Conductivity

drift

drift

Jp = spE

Jn = snE

sn= nqmn = nq2tn/mn*

sp= pqmp = pq2tp/mp*

r= 1/s

s = sn + sp

slide10

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?

slide11

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

which determines t)

Recall 1/t = svthNT

slide12

Can we engineer these properties?

  • What changes at the nanoscale?
what causes scattering
What causes scattering?
  • Phonon Scattering
  • Ionized Impurity Scattering
  • Neutral Atom/Defect Scattering
  • Carrier-Carrier Scattering
  • Piezoelectric Scattering

ECE 663

some typical expressions
Some typical expressions
  • Phonon Scattering
  • Ionized Impurity Scattering

ECE 663

combining the mobilities
Combining the mobilities

Matthiessen’s Rule

Caughey-Thomas Model

ECE 663

slide17

Doping dependence of resistivity

rN = 1/qNDmn

rP = 1/qNAmp

mdepends on N too, but weaker..

ECE 663

slide18

Temperature Dependence

Piezo scattering

Phonon Scattering

~T-3/2

Ionized Imp

~T3/2

ECE 663

slide19

Reduce Ionized Imp scattering (Modulation Doping)

Tsui-Stormer-Gossard

Pfeiffer-Dingle-West..

Bailon et al

ECE 663

slide20

Field Dependence of velocity

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

Velocity reduction in GaAs

ECE 663

slide21

Gunn Diode

Can operate around NDR point to get an oscillator

ECE 663

transferred electron devices gunn diode
Transferred Electron Devices (Gunn Diode)

E(GaAs)=0.31 eV

Increases mass

upon transfer under

bias

ECE 663

slide25

DIFFUSION

ECE 663

slide26

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
Drift vs Diffusion

x

x

E2 > E1

E1

t

t

<x2> ~ Dt

<x> ~ mEt

ECE 663

slide28

SIGNS

EC

drift

drift

Jp = qpmpE

Jn = qnmnE

E

Opposite velocities

Parallel currents

vp = mpE

vn = mnE

slide29

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
In Equilibrium, Fermi Level is Invariant

e.g. non-uniform doping

ECE 663

einstein relationship
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

ECE 663

einstein relationship1
Einstein Relationship

Dn= kTtn/mn*

mn= qtn/mn*

½ m*v2= ½ kT

Dn= v2tn = l2/tn

ECE 663

slide33
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

slide36

The equations

At steady state with no RG

.J = q.(nv) = 0

ECE 663

slide38

All the equations at one place

(n, p)

J

E

ECE 663

slide39

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
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
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
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

ECE 663

example 3 1 sided diffusion with traps
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

slide48

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