carrier transport ndc f'15 (1)
DESCRIPTION
semiconductorTRANSCRIPT
1
Lecture 1: Semiconductor Crystals
ECE5590: Nanoscale Devices and circuitsMostafizur Rahman
ECE 663
• So far, we looked at equilibrium charge distributions. Theend result was np = ni
2
• When the system is perturbed, the system tries to restoreitself towards equilibrium through recombination-generation
R-G processes
3
Outline
• Recombination Generation• Drift• Diffusion• Conclusions
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Real spaceEnergy space
Direct Band-to-band recombination
The direct annihilation of a conduction band electron and a valence band hole, the electron falling from an allowed conduction band state into a vacant valence band state; Radiative. Exampels: Lasers, LEDs
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R-G Center Recombination
• Defects give rise to deep-level states– Introduces new energy level in the midgap region
• Both carriers get attracted to mid-level; electrons are annhilated
ECE5590 Fall 2015 MR 6
Recombination via Shallow Levels
• Like R-G centers, donors and acceptor sites can also function as intermediaries in the recombination process
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Energy space
Direct Excitonic Recombination
Organic Solar cells, CNTs, wires (1-D systems)
• Electron and a hole can bound together into a hydrogen-atom-like arrangement which moves as a unit in response to applied forces. This coupled electron-hole pair is called an exciton.
• Excitons can be trapped in Shallow-level sites.
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Phonon
Energy space
Auger Recombination
Solar Cells, Junction Lasers, LEDs
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• Band-Band recombination or trapping at a band center occurs simultaneously with the collision between two like carriers.
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Generation
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Band to Band Generation
• Opposite process to Recombination
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Recombination
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R-G Center Generation
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Impact Ionization
• Collision results in electron-hole pair generation. • Occurs in the high field regions
• Ex. Avalanche breakdown in pn junctions.
• Equilibrium distribution of charges in a semiconductor
np = ni2, n ~ ND for n-type
• The system tries to restore itself back to equilibrium when perturbed, through RG processes
R = (np - ni2)/[tp(n+n1) + tn(p+p1)]
• Next-> The processes that drive the system away from equilibrium.• Electric forces will cause drift, while thermal forces (collisions)
will cause diffusion.
Recap
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Drift
• Charge carrier motion in response to an applied electric field• When E is applied, +q charges move in the positive
direction, -q in the opposite• Carrier motion is interrupted by scattering, ionized
impurities, thermally agitated lattice, or other scattering centers
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Drift
• Microscopic drifting of single carrier is complex• Macroscopic observable: drift velocity (vd); averages over all
electrons or holes at the same time.
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Drift Current
• Drift Current
What is the equation of Current?
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Drift Current
Where is the contribution from Electric field?
The hole mobility, is the constant of proportionality between Vd and E
ECE5590 Fall 2015 MR 18
Mobility
Central parameter determining performance of many devices
Electron mobility in Si?Hole mobility in Si?
GaAs electron mobility?GaAs hole mobility?
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Mobility is a measure of the ease of carrier motion within a semiconductor crystal. The lower the mobility of carriers within a given semiconductor, the greater the number of motion-impeding collisions
Scattering Events
• Phonon Scattering• Ionized Impurity Scattering• Neutral Atom/Defect Scattering• Carrier-Carrier Scattering• Piezoelectric Scattering
ECE5590 Fall 2015 MR 20
Impact of Scattering• Phonon Scattering- collision between the carriers and thermally
agitated lattice atoms. (good/bad?)• Ionized Impurity Scattering- Coulombic attraction/repulsion
between charged carriers and ionized donors/acceptors (good/bad?)
• Neutral Atom/Defect Scattering (bad/bad)• Carrier-Carrier Scattering-collision between same carrier
(good/bad/doesn’t matter)– Randomizes carrier
• Piezoelectric Scattering- displacement of the component atoms from lattice site gives rise to electric field (Good/Bad)
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Mobility
For µi, Increasing temperature reduces time spent near vicinity of ionized donor; increasing mobility
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Temperature Dependence
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Phonon Scattering~T-3/2
Ionized Imp~T3/2
Piezo scattering
Temperature Dependence
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Revisiting Drift Velocity
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Velocity saturation ~ 107cm/s for n-Si (hot electrons)Velocity reduction in GaAs
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High Field Effects
Velocity Saturation:• Drift velocity of carrier reaches field independent constant
value– Analogous to free falling object
• Intervalley Carrier Transfer
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Ballistic Transport
Velocity Overshoot:• If the total length a carrier travels is shorter than mean
distance between scattering events– No Scattering – Ballistic transport
• Ballistic transport was supposed to be seen at L~0.1um
• Can we engineer these properties?
• What changes at the nanoscale?
Diffusion
Diffusion is a process whereby particles tend to spread out or redistribute as a result of their random thermal motion, migrating on a macroscopic scale from regions of high particle concentration into regions of low particle concentration
SIGNS
EC
E
Jn = qnmnEdrift
Jp = qpmpEdrift
vn = mnEvp = mpE
Opposite velocitiesParallel currents
SIGNS
Jn = qDndn/dxdiff
Jp = -qDpdp/dxdiff
dn/dx > 0 dp/dx > 0
Parallel velocitiesOpposite currents
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In Equilibrium, Fermi Level is Invariant
e.g. non-uniform doping
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Einstein Relationship
m and D are connected !!
Jn + Jn = qnmnE + qDndn/dx = 0diff drift
n(x)= Nce-[EC(x) - EF]/kT = Nce-[EC -EF - qV(x)]/kT
dn/dx = -(qE/kT)n
qnmnE - qDn(qE/kT)n = 0Dn/mn = kT/q
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Einstein Relationship
mn = qtn/mn*
Dn = kTtn/mn*
½ m*v2 = ½ kT
Dn = v2tn = l2/tn
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• 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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Relation between current and charge
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Continuity Equation
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The equations
At steady state with no RG
.J = q.(nv) = 0
Let’s put all the maths together…
Thinkgeek.com
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All the equations at one place
(n, p)
E
J
∫
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
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Minority Carrier Diffusion Equations
∂Dnp ∂2Dnp
∂t ∂x2
Dnp
tn= DN - + GN
∂Dpn ∂2Dpn
∂t ∂x2
Dpn
tp= DP - + GP
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Example 1: Uniform Illumination
∂Dnp ∂2Dnp
∂t ∂x2
Dnp
tn= DN - + GN
Why? Dn(x,0) = 0Dn(x,∞) = GNtn
Dn(x,t) = GNtn(1-e-t/tn)
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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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Example 3: 1-sided diffusion with traps
∂Dnp ∂2Dnp
∂t ∂x2
Dnp
tn= DN - + GN
Dn(x,b) = 0
Dn(x,t) = Dn(0)sinh[(b-x)/Ln]/sinh(b/Ln)
Ln = Dntn
Numerical techniques
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Numerical techniques
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At the ends…
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Overall Structure
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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