1. 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

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

3. DRIFT
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4. Why does a field create a velocity rather than an acceleration?
Terminal
velocity
Gravity
Drag

5. 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)

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

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

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

10. 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?

11. 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

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

13. What causes scattering?
Phonon Scattering
Ionized Impurity Scattering
Neutral Atom/Defect Scattering
Carrier-Carrier Scattering
Piezoelectric Scattering
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14. Some typical expressions
Phonon Scattering
Ionized Impurity Scattering
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15. Combining the mobilities
Matthiessen’s Rule
Caughey-Thomas Model
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16. Doping dependence of mobility
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17. Doping dependence of resistivity
rN = 1/qNDmn
rP = 1/qNAmp
m depends on N too, but weaker..
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18. Temperature Dependence
Piezo scattering
Phonon Scattering
~T-3/2
Ionized Imp
~T3/2
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19. Reduce Ionized Imp scattering (Modulation Doping)
Bailon et al
Tsui-Stormer-Gossard
Pfeiffer-Dingle-West..
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20. Field Dependence of velocity
Velocity saturation ~ 107cm/s for n-Si (hot electrons)
Velocity reduction in GaAs
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21. Gunn Diode
Can operate around NDR point to get an oscillator
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22. GaAs bandstructure
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23. Transferred Electron Devices (Gunn Diode)
E(GaAs)=0.31 eV
Increases mass
upon transfer under
bias
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24. Negative Differential Resistance
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25. DIFFUSION
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26. 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
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27. Drift vs Diffusion x x t t E2 > E1 E1 <x2> ~ Dt
<x> ~ mEt
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28. SIGNS E vp = mpE vn = mnE Jn = qnmnE Jp = qpmpE EC Opposite velocities
Parallel currents
vp = mpE
vn = mnE
Jn = qnmnE
drift
Jp = qpmpE

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

30. In Equilibrium, Fermi Level is Invariant
e.g. non-uniform doping
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31. 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
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32. Einstein Relationship
mn = qtn/mn*
Dn = kTtn/mn*
½ m*v2 = ½ kT
Dn = v2tn = l2/tn
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33. 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 !!!
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34. Relation between current and charge
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35. Continuity Equation
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36. The equations
At steady state with no RG
.J = q.(nv) = 0
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37. Let’s put all the maths together…
Thinkgeek.com

38. All the equations at one place
(n, p) E J ∫ 
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39. 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

40. Minority Carrier Diffusion Equations
∂Dnp
∂2Dnp ∂t
∂x2
Dnp tn
= DN -
+ GN
∂Dpn
∂2Dpn
Dpn tp
= DP
+ GP
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41. 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)
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42. 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
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43. 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)
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44. Numerical techniques 2

45. Numerical techniques

46. At the ends…
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47. Overall Structure
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48. 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