Page 46 - Photonics Essentials an introduction with experiments

Photodiodes

40 Photonic Devices

In this expression, both n p and n n change to accommodate the bias
voltage V A :

n p0 n p = n p0 + n
and
n n0 n n = n n0 + n (3.3)

In the low-injection limit, which is always true for photodiodes, n n0 +
n n n0 , because n n0 is many orders of magnitude larger than n. We
can use this approximation to derive the excess minority carrier densi-
ty that is induced by the bias voltage at the edge of the depletion region:
n = n n {e –[q(V Bi –V A )/kT] } – n p0 = n p0 e –qV Bi /kT {e –[q(V Bi –V A )/kT] } – n p0
n = n p0 (e qV A /kT – 1) (3.4)

In Eq. 3.4, note the appearance of the term –n p0 . This term is re-
quired to make the current 0 when the applied voltage is 0, and it is
also the origin of the dark current of the photodiode. Current is car-
ried in the diode by both drift and diffusion. However, at the edge of
the depletion region, for example at x p = 0, the current is carried only
by diffusion. If we calculate the I–V characteristic at this point, we
can work with only one equation, the diffusion equation:
d 2 n(x)
n p (x) = De n(x) – + G L (3.5)
t dx 2 e
This equation says that the time rate of change of the excess carrier
concentration is given by the generation rate inside the diode, less
any recombination, and plus any additional carriers generated by
light. We need to write a similar equation for the excess hole minority
carrier density on the n-side of the diode. That equation is completely
analogous to Eq. 3.4, so we can solve 3.4 and deduce the answer for
the n-side of the diode. Equation 3.4 is a second-order differential
equation for n p , which is a function of distance in the diode. The gen-
eration rate of minority carriers from photon absorption is given by
G L , and the minority carrier recombination time is given by e . The
minority carrier diffusion coefficient for electrons in p-type material is
D e . We will first look at steady-state conditions, and this means that:

d 2 n(x)
n p (x) = 0 = D e n(x) – + G L
t dx 2 e
d 2 n(x)
D e n(x) = – G L
dx 2 e
d 2 n(x) G L
n(x) = – (3.6)
dx 2 D e e D e

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