Consider an n -type semiconductor and weak injection conditions. Assume that the minority carrier recombination time,
 h
, is constant (independent of injection─ hence the weak injection assumption). The rate of change of the instantaneous hole concentration,

p n
/

t , due to recombination is given by
 p n
 t
 
 p n h
[5.87]
The net rate of increase (change) in p n
is the sum of the total generation rate G and the rate of change due to recombination, that is, dp n dt

G

 p n h
[5.88]
By separating the generation term G into thermal generation G o
and photogeneration G ph
and considering the dark condition as one possible solution, show that d
 p n dt

G ph

 p n
 h
[5.89]
How does your derivation compare with Equation 5.27? What are the assumptions inherent in
Equation 5.89?
*
Consider recombination in a direct bandgap p -type semiconductor, e.g.
, GaAs doped with an acceptor concentration N a
. The recombination involves a direct meeting of an electron–hole pair as depicted in Figure 5.22. Suppose that excess electrons and holes have been injected ( e.g.
, by photoexcitation), and that Δ n p is the excess electron concentration and Δ p p is the excess hole concentration. Assume Δ n p is controlled by recombination and thermal generation only; that is, recombination is the equilibrium storing mechanism. The recombination rate will be proportional to n p
p p
, and the thermal generation rate will be proportional to n po
p po
. In the dark, in equilibrium, thermal generation rate is equal to the recombination rate. The latter is proportional to n no
p po
. The rate of change of Δ n p is
  n p
 t
 
B
 n p p p
 n po p po

[5.90] where B is a proportionality constant, called the direct recombination capture coefficient. The
recombination lifetime τ r is defined by
  n p
 t
 
 n p
 r
[5.91] a.
Show that for low-level injection , n po
  n p
 p po
, τ r is constant and given by
5.1

 r

1
Bp po

1
BN a b.
Show that under high-level injection, Δ n p
 p po
,
  n p
 t
 
B
 p p
 n p
 
B
  p
2 so that the recombination lifetime τ r is now given by
 r

1
B
 p p

1
B
 n p that is, the lifetime τ r is inversely proportional to the injected carrier concentration.
[5.92]
[5.93]
[5.94] c.
Consider what happens in the presence of photogeneration at a rate G ph
(electron–hole pairs per unit volume per unit time). Steady state will be reached when the photogeneration rate and recombination rate become equal. That is,
G ph



  n p
 t

 recombinat ion

B
 n p p p
 n po p po

A photoconductive film of n -type GaAs doped with 10
13
cm
−3
donors is 2 mm long ( L ), 1 mm wide
( W
), and 5 µm thick (
D ). The sample has electrodes attached to its ends (electrode area is therefore 1 mm × 5 µm) which are connected to a 1 V supply through an ammeter. The GaAs photoconductor is uniformly illuminated over the surface area 2 mm × 1 mm with a 1 mW laser radiation of wavelength λ
= 850 nm (infrared). The recombination coefficient B for GaAs is 7 .
21 × 10 −16
m
3
s
−1
. At λ = 850 nm, the absorption coefficient is about 5 × 10 3
cm
−1
. Calculate the photocurrent I photo
and the electrical power dissipated as Joule heating in the sample. What will be the power dissipated as heat in the sample in an open circuit, where I = 0?
Consider an n -type semiconductor that has a temperature gradient across it. The right end is hot and the left end is cold, as depicted in Figure 5.55. There are more energetic electrons in the hot region than in the cold region. Consequently, electron diffusion occurs from hot to cold regions, which immediately exposes negatively charged donors in the hot region and therefore builds up an internal field and a built-in voltage, as shown in the Figure 5.55. Eventually an equilibrium is reached when the diffusion of electrons is balanced by their drift driven by the built-in field. The net current must be zero. The Seebeck coefficient (or thermoelectric power) S measures
5.2

Figure 5.55: In the presence of a temperature gradient, there is an internal field and a voltage difference. The Seebeck coefficient is defined as dV / dT , the potential difference per unit temperature. this effect in terms of the voltage developed as a result of an applied temperature gradient as
S
 dV dT
a . How is the Seebeck effect in a p -type semiconductor different than that for an n -type semiconductor when both are placed in the same temperature gradient in Figure 5.55? Recall that the sign of the Seebeck coefficient is the polarity of the voltage at the cold end with respect to the hot end
(see Section 4.8.2). b . Given that for an n -type semiconductor,
S n
  k e

 2

( E c

E
F kT
)

 [5.103] what are typical magnitudes for S n
in Si doped with 10
14
and 10
16
donors cm
-3
? What is the significance of S n at the semiconductor device level?
c . Consider a pn junction Si device that has the p -side doped with 10
18
acceptors cm
-3
and the n -side doped 10 14 donors cm -3 . Suppose that this pn junction forms the input stage of an op amp with a large gain, say 100. What will be the output signal if a small thermal fluctuation gives rise to a 1

C temperature difference across the pn junction?
5.3