# HW#5

*Due Monday morning of finals week (before noon).*

## Diffusion Length Order of Magnitude (5 pts)

Diffusion length for a minority electron is given by

*L*_e = sqrt(*Dτ*)

where *D* is the diffusion coefficient and τ is minority carrier lifetime. Diffusion coefficient in 3d is given by

*D* = λ<*v*_th>/3

where λ is the scattering length for minority carriers (typically a few nm) and <*v*_th> is the average thermal velocity of minority carriers.

Do an order of magnitude calculation to estimate the room temperature diffusion length of minority electrons in Si. Make sure to write down what doping density, lifetime and effective electron mass you have assumed.

## GaAs solar cell (5 pts)

In class we looked at the frequency-dependent absorption coefficient and doping-dependent diffusion length in silicon.

Do the same thing for GaAs and make your best guess for the thickness and doping levels of GaAs in a high efficiency solar cell. Write a short paragraph explaining your reasoning.

## Maximum J_SC and V_OC (15 pts)

**a)** Use numerical integration of the blackbody spectrum to calculate the maximum short-circuit current density (J_SC) of a single junction solar cell as a function of E_gap, as done in Dr. Cohen's notes. I'm looking for a computer plot of max J_SC as a function of E_gap. Don't forget to label axes with units.

I recommend the rectangle method for numerical integration.

Assumptions:

- Incident radiation follows the shape of a blackbody spectrum (
*T*= 5960 K), and integrated power is 1 kW/m^2. See Dr. Cohen's notes. - Every above-bandgap photon is converted into one e-h pair.

**b)** For a silicon pn junction (standard planar geometry) estimate the open circuit voltage by using the ideal diode equation. Assume dopant densities N_a = N_d = 10^15 cm^-3, and maximal J_SC.

**c)** Follow Shockley & Queiser's argument to find the maximum possible open circuit voltage for a Si pn junction device. Compare to part b.

CLARIFICATIONS FOR NEXT YEAR: For parts a and b assume room temperature operating conditions. For part b, do not assume any particular doping level - the SQ limit gives the best possible V_OC.