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Lecture 1: Semiconductor Crystals

ECE5590: Nanoscale Devices and circuitsMostafizur Rahman

rahmanmo@umkc.edu

ECE 663

• So far, we looked at equilibrium charge distributions. Theend result was np = ni

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• When the system is perturbed, the system tries to restoreitself towards equilibrium through recombination-generation

R-G processes

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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

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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

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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

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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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Doping dependence of mobility

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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