# . C.5 More General Integrals

For integral j, this has a finite (J + 1) number of non-zero terms and the integral simplifies to:

For example,

в (п, є) = £3в) (п-є)/6 + є2Р1(ц-є)/2 + £/?2 (п-є) + в (п-є)

(C.22) where each term on the right can be evaluated using Eq. (C.16).

The inverse of such a function for integral j can be found by expanding к (п )

in a power series in 0 (п):

I±(п) = I± (п) + І а±г[ї0(п)]r (C.23)

r=2 ‘

where

a±:2 = (2k -1)/2k+1 (C.24)

ak 3 = (3k -1)/3k+1 – (2k -1)/2k+1 (C.25)

with terms for larger values of r becoming increasingly complex. This expression shows that, for negative x:

I+ (п)< I+ (п)< I+(п)+[I+ (п)]2(2k -1)/2k+1 (C.26)

and

/-(п) > I-(п) > I-(п)-[I-(п)]2(2k -1)/2k+1 (C.27)

Retaining only the first term in such an expansion, Eq. (C.22) can be written as a quadratic in р0(п-є). solving for p0 and invoking Eq. (C.12) allows п to be evaluated knowing Д – (п, є).

The actual value of п will be bounded by this value and the value found with only the first term p0 retained, as per Eq. (C.27).

An even more general form of the integrals of interest is given by:

The following expression can be derived from Eq. (C.21):

2 j

j (П, еі, Є2) = X X(-1)t+1(+e)i-kl±, (n – Є" )/(j – k)!

"=1 k=0

For example,

в(n>£1 ,£2 ) = e13A)(n ~£1)/6— £2^0(n~£2 )/6 + £12в1ІП~£1)/2

-£22Р1ІП-£2 )/2 + £1^2(П-£1)-£2P2(П — £2 ) +P3(n-£ )-P3(n-£2 )

(C.30)

An approach to finding the inverse in this case may be to note the relationship: во (n-£2) = во (П – £1) – ln [1 – exp (£ – £2) в -1 (П – £1)] (C.31)

or find a similar way of relating the p0 terms with different arguments.

References

Blakemore JS (1982), Approximations for Fermi-Dirac integrals especially the function F1/2(n) used to describe electron density in a semiconductor, Solid – State Electronics 25: 1067.

Blackmore JS (1987), Semiconductor Statistics, Dover, New York.

List Of Symbols

a absorption coefficient

є dielectric constant; normalized energy; energy

I Reiman zeta function

в Bose-Einstein integral; constant

Г Gamma function

X wavelength

H chemical potential; mobility

П efficiency

p density of states

a Stefan-Boltzmann constant; conductivity

Q solid angle; phase volume

A cross-sectional area

c velocity of light in vacuum

D diffusion coefficient; density of states

e base of natural logarithms

E energy

Ec, Ev energies of conduction – and valence-band edges

Ef Fermi level

f frequency; occupancy function; fraction of hemisphere

FF solar cell fill factor

g degeneracy

G generation rate of electron-hole pairs per unit volume

h Planck’s constant

I current; integral

Io diode saturation current

Isc short-circuit current

J current density

Je, Jh electron and hole current densities

k Boltzmann’s constant; momentum

m0 electronic rest mass

n electron concentration; refractive index

Hj intrinsic concentration

NC, NV effective densities of states in conduction and valence bands N photon flux

p hole concentration

P power; probability

q electron charge

Q heat

r position vector

radiation per unit solid angle and per unit area entropy t time

T temperature

u energy per unit volume

U net recombination rate per unit volume

V voltage; volume

Voc open-circuit voltage

W work

x, y,z position co-ordinates