ece 663 the first few chapters showed us how to calculate the equilibrium distribution of charges in...
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ECE 663
The story so far..
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Drift: Driven by Electric Field
vd = E
Velocity(cm/s) Mobility
(cm2/Vs)
Electric field(V/cm)
E
Which has higherdrift?
x
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ECE 663
DRIFT
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Why does a field create a velocityrather than an acceleration?
Terminalvelocity
Gravity
Drag
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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
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Why does a field create a velocityrather than an acceleration?
mn*(dv/dt + v/n) = -qE
n = qn/mn*
p = qp/mp*
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ECE 663
From accelerating charges to drift
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From mobility to drift current
n = qn/mn*
p = qp/mp*
Jn = qnv = qnnEdrift
Jp = qpv = qppEdrift
(A/cm2)
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Resistivity, Conductivity
n = nqn = nq2n/mn*
Jn = nEdrift
Jp = pEdrift
p = pqp = pq2p/mp*
= 1/
= n + p
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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 ?
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So mobility and resistivity depend on material properties (e.g. m*) and sample properties (e.g. NT, which determines )
Recall 1/ = vthNT
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• Can we engineer these properties?
• What changes at the nanoscale?
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ECE 663
What causes scattering?
• Phonon Scattering• Ionized Impurity Scattering• Neutral Atom/Defect Scattering• Carrier-Carrier Scattering• Piezoelectric Scattering
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ECE 663
Some typical expressions
• Phonon Scattering
• Ionized Impurity Scattering
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ECE 663
Combining the mobilities
Matthiessen’s Rule
Caughey-Thomas Model
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ECE 663
Doping dependence of resistivityN = 1/qNDn P = 1/qNAp
depends on N too, but weaker..
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ECE 663
Phonon Scattering~T-3/2
Ionized Imp~T3/2
Piezo scattering
Temperature Dependence
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ECE 663Bailon et al Tsui-Stormer-GossardPfeiffer-Dingle-West..
Reduce Ionized Imp scattering (Modulation Doping)
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ECE 663
Field Dependence of velocity
Velocity saturation ~ 107cm/s for n-Si (hot electrons)Velocity reduction in GaAs
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ECE 663
Gunn Diode
Can operate around NDR point to get an oscillator
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ECE 663
GaAs bandstructure
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ECE 663
Transferred Electron Devices (Gunn Diode)
E(GaAs)=0.31 eV
Increases massupon transfer underbias
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ECE 663
Negative Differential Resistance
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ECE 663
DIFFUSION
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ECE 663
J1 = qn(x)vJ2 = -qn(x+l)v
l = v
Jn = q(l2/)dn/dx = qDNdn/dx
DIFFUSION
diff
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ECE 663
Drift vs Diffusion
t
x
t
x
<x> ~ Et <x2> ~ Dt
E1
E2 > E1
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SIGNS
EC
E
Jn = qnnEdrift
Jp = qppEdrift
vn = nEvp = pE
Opposite velocitiesParallel currents
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SIGNS
Jn = qDndn/dxdiff
Jp = -qDpdp/dxdiff
dn/dx > 0 dp/dx > 0
Parallel velocitiesOpposite currents
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ECE 663
In Equilibrium, Fermi Level is Invariant
e.g. non-uniform doping
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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
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ECE 663
Einstein Relationship
n = qn/mn*
Dn = kTn/mn*
½ m*v2 = ½ kT
Dn = v2n = l2/n
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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…
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ECE 663
Relation between current and charge
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ECE 663
Continuity Equation
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ECE 663
The equations
At steady state with no RG
.J = q.(nv) = 0
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Let’s put all the maths together…
Thinkgeek.com
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ECE 663
All the equations at one place
(n, p)
E
J
∫
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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
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ECE 663
Minority Carrier Diffusion Equations
∂np ∂2np
∂t ∂x2
np
n= DN - + GN
∂pn ∂2pn
∂t ∂x2
pn
p= DP - + GP
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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)
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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
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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
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Numerical techniques
2
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Numerical techniques
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ECE 663
At the ends…
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ECE 663
Overall Structure
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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