Chapter 8 - Photovoltaics Flashcards
What is the photovoltaic effect?
Direct transformation of sunlight into electrical energy without moving parts.
Draw the Feynman diagram of photovoltaics.
See notes.
How are photons described?
Energy = hf = hbar * omega Momentum = p = hbar * k, with k = 2π/lambda = omega / c = 2πf / c.
What kind of particle are photons?
Bosons with angular momentum J = 1.
How many polarization states exist for photons? What are they?
Three states exist.
1) Right circular polarization.
2) Left circular polarization.
3) Linear polarization.
Derive the photon density of states.
See notes.
How does the photon density of states depend on the refractive index? What is the consequence of this?
It goes as n^3. The consequence is that going from a volume with one refractive index to another with a different refractive index, some of the photons must be reflected at the interface.
What is the absorber material in a solar cell?
A semiconductor (eg. Si).
How does the interactions between light and electron occur?
Through electrical field. Basic principle is acceleration of an electron by the electrical field of the light.
What determines the interaction strength?
It is given by the dipole moment e*x in the absorbing medium.
How is the interaction described quantum mechanically?
The excitation from initial state |i> to final state |f> is governed by the transition probability, W.
W is proportional to |E<i>|^2 = E^2e^2 |<i>|^2 which is proportional to I (intensity).</i></i>
How is the intensity of light described in the wave and particle models respectively?
Wave: I = 1/µ_0 |ExB| = 1/µ_0 * E_0 * B_0 = 1/(µ_0c)E_0^2 * sqrt(eps_0/µ_0) E^2.
Particle: hbar*omega * PHI, with PHI = photon flux density per area and time.
Absorption of a photon is subject to three conserved quantities. Which?
Energy: E_f - E_i = hbar * omega
Momentum: p_f - p_i = hbar * k ≈ 0 (small momentum for photons)
Angular momentum: L_f - L_i = hbar.
What are the consequences of angular momentum conservation in a) atoms and b) solids?
a) leads to dipole selection rule: ∆n = ±1
b) leads to creation of excitons.
How is the complex dielectric function defned?
eps = eps_1 + i*eps_2
How is the complex refractive index defined, and what are the two parameters?
ñ = n + ik,
n = refractive index k = absorption coefficient
What happens to the speed of light in absorbing matter?
It goes from c_0 to c_0/n, where c_0 is speed of light in vacuum and n is the refractive index.
What happens to the wave vector in absorbing matter?
It goes from k = omega / c_0 to k = (omega / c_0)*ñ,
where ñ is the complex refractive index.
How is the law of Lambert-Beer given?
From the expression of the electric field.
E = E_0 exp[i(kx - wt] = E_0 exp[i(omega/c_0 * (n+iK)x - wt)]
= E_0 exp[-omega/c_0 * kx] exp [i(omega/c_0 * nx-wt].
The law of Lambert-Beer is then given from the fact that I is proportional to E^2:
I(x) = I_0 * exp[-2 omega/c_0 * k x] = I_0 exp[-alpha x]
where alpha = 2 (omega/c_0) * k is the absorption coefficient.
What is the consequence of the Lambert-Beer law, in terms of absorption length? How is this length for some materials, and why?
I(x) = I_0 * e^-(alpha*x)
Which means that 95% is absorbed in a length of x = 3/alpha.
For GaAs, as an direct semiconductor, this is about 1µm.
For c-Si, as an indirect semiconductor (so needs help from phonon to make the transition due to momentum conservation) this is about 10µm.
For a-Si this is about 1.5µm (no momentum conservation required since there is no periodic lattice, so all transitions are allowed).
For normal incidence of light from the air (where n ≈ 1, and k ≈ 0), how is the reflectivity given?
R = [(n-1)^2 + k^2]/[(n+1)^2 + k^2]
For normal incidence of light from the air (where n ≈ 1, and k ≈ 0), how is the absorbance given?
A = (1-R) * (1-exp[alpha*d]),
where d is the absorbance thickness. Valid in the limit of strong absorption (alpha*d»_space; 1). Otherwise multiple reflections at the boundaries.
For normal incidence of light from the air (where n ≈ 1, and k ≈ 0), how is the transmission given?
T = (1-R)^2exp[-alphad])
where d is the absorbance thickness. Valid in the limit of strong absorption (alpha*d»_space; 1). Otherwise multiple reflections at the boundaries.
For normal incidence of light form the air (where n ≈ 1, and k ≈ 0), how is then the average volume density of excited electrons generated by a photon flux per unit time given?
G = 1/d * APHI = (1-R) 1/d (1-exp[-alphad]) * PHI
where d is the absorbance thickness. Valid in the limit of strong absorption (alpha*d»_space; 1). Otherwise multiple reflections at the boundaries.
G = average generation rate, cm^-3 * s^-1
Explain three different ways to reduce reflection (antireflection) from surface.
(i) Macroscopic texture by anisotropic etching (especially for monocrystalline Si). Here the rays are reflected not only back, but into another part of the surface, thus decreasing the total amount of reflected light. Here the trenches are a lot larger than the wavelength of light.
(ii) Nanotexturing with structures that are a lot smaller than the wavelength. This means that the refractive index changes gradually from the surface and down to the bulk. At any point, the refractive index is basically the same as the previous point. This means that there will be no reflection.
(iii) lambda/4 - antireflex coating (e.g. Si3N4 on Si). A material with a refractive index in between the air and the absorber with a thickness lambda/4 is placed on top of the absorber. This means that there will be destructive interference (total path lambda/2) for reflected wave at surface and lower surface. This is only optimal for one incident angle and wavelength.
Derive the expression for the emission.
See notes. (page 7)
What is thermalization, and on which time scale does this happen?
Thermalization is when excited electrons and holes with excess energy (that is the incoming light was more than required to excite the exciton) emit phonons (and thus heat). This happens on a time scale of 10^-13 s. (0.1 ps)
Describe thermalization in molecules using energy-configuration diagram.
The excitation is a vertical transition, because electron configuration can be changed much faster that the atom configuration. It will then lose energy by the emittance of phonons, and the reemitted light will be red-shifted (Stokes-shift).
Describe thermalization in a semiconductor using band diagrams.
See notes (page 9)
Will thermalization primarily happen in the conduction band or the valence band? Why?
In the conduction band. Because of the effective mass is smaller in CB, the curvature is greater, and the energy in the CB will be higher for the same k-vector than in the VB.
What are the best electron-phonon coupling realized by?
Longitudnial optical phonon modes near the Brillouin-zone center.
Describe the generation and recombination of electrons and holes mathematically.
Number of electrons: n_0 + ∆n(t)
Number of holes: p_0 + ∆p(t)
dn/dt = dp/dt = G-R. When G = R, then dn/dt = 0 and we have steady state.
Most solar cells are doped. What kind of doping?
p-doping. This is because electrons have a smaller effective mass than holes.
What is the efficiency of a solar cell determined by, in terms of recombination?
The recombination time Tau_min of minority carriers.
How can recombination occur?
Either through emitting photons (radiative recombination) or through emitting phonons (nonradiative recombination)
What does the probability of radiative recombination depend on?
The dipole matrix element (same as for the excitation).
What is the radiative recombination time?
It is given as:
tau = (hbar omega)/P,
where P is the probability of radiative recombination in classical radiation theory.
tau = 6πeps_0 hbar * (c^3)/(omega^3 * e^2 * x^2)
where x is the distance between electron and hole in the exciton.
What can be said for the recombination time of high bandgap semiconductors?
It will be short, since tau goes as 1/omega^3.