Unformatted text preview: obability that a photon is absorbed is a constant The number of photons absorbed per increment of distance is propor`onal to the number of incident photons We assume that the propor`onality rela`onship holds for all values of incident photon ﬂux (no satura`on) Differential Equation:
dI ( x )
−
= α I (x)
dx
Solution:
I (x) = ( Io − Ir ) e −α x I 0e −α x Absorption for sample with thickness "l":
I t I o e− α l
Note: We've assumed I R is negligible in this analysis I (x) I0 I ( x ) − α I ( x ) Δx I 0 − α I 0 Δx thickness
Δx ΔI ( x ) = ( I ( x ) − α I ( x )Δx ) − I ( x )
ΔI ( x ) = −α I ( x )Δx
ΔI ( x )
= −α I ( x )
Δx
dI ( x )
⎛ ΔI ⎞
lim ⎜ − ⎟ = −
= α I (x)
Δx→0 ⎝
Δx ⎠
dx 9 Photon Flux Versus Posi`on • The photon ﬂux drops exponen`ally in the sample • The rela`onship for the generated carriers is more complicated due to diﬀusion 0 I
I0 l I ( x ) = I o e− α x x Lower
Absorption Difference due
to IR αê༎ IT αé༎ IR Sample Greater
Absorption I t = I o e− α l 10 Absorp`on Coeﬃcient • The absorp`on coeﬃcient depends upon the incident wavelength λ • Below the bandgap there is minimal absorp`on • The units of absorp`on are typically cm 1 Useful Relationship:
c 1.239
=
λ λ (µm)
A photon with a wavelength of 1 µm has an energy of 1.239 eV
E (eV ) = hv = h 11 Comments on Op`cal Absorp`on • Semiconductors absorb energies greater than their bandgap, which means wavelengths shorter than the wavelength at their bandgap • GaAs, Si, and Ge are in the infrared part of the spectra • GaP is in the visible • GaN pushes into the UV • If I want to absorb light emi5ed by one semiconductor, I generally need to use another with a smaller bandgap 12 Bandgap and Layce Constant of Common Semiconductors 13 Terminology Reminder • Equilibrium: No external perturbing excita`on is present except temperature, and there is no net mo`on of charge. • Steady State: A non equilibrium condi`on in which all perturbing processes are constant. In steady state, opposing processes are present that balance perturbing processes. Equilibrium Genera`on Some electrons have sufficient energy to jump to the conduction band
(phonon interactions) Generation and recombination
are balanced: no(K) ༉ r g (T ) = gi = ri = α r nopo = α n 2
ri An A...
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 Spring '11
 Leburton

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