. C.5 More General Integrals

Подпись: lj Ol,e) Подпись: І (+e)~kl±k (n-e)/(j - к)! k=0 Подпись: (C.21)

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 ‘


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)


/-(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).

Подпись: I ±(п, є1, є2) Подпись: 1 є2 EjdE Г( j +1) Єі exp(E-п)± 1 Подпись: (C.28)

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

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

Подпись: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 )


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.


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

T lifetime

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

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