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Chapter 5
5.21 Excess minority carrier concentration 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,  pn/ t, due to recombination is given by
pn
p
 n
t
h
[5.87]
The net rate of increase (change) in pn is the sum of the total generation rate G and the rate of
change due to recombination, that is,
dpn
p
G n
dt
h
[5.88]
By separating the generation term G into thermal generation Go and photogeneration Gph and
considering the dark condition as one possible solution, show that
dpn
p
 G ph  n
dt
h
[5.89]
How does your derivation compare with Equation 5.27? What are the assumptions inherent in
Equation 5.89?
*
5.22 Direct recombination and GaAs Consider recombination in a direct bandgap p-type
semiconductor, e.g., GaAs doped with an acceptor concentration Na. 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 Δnp is the excess electron concentration
and Δpp is the excess hole concentration. Assume Δnp is controlled by recombination and thermal
generation only; that is, recombination is the equilibrium storing mechanism. The recombination rate
will be proportional to np pp, and the thermal generation rate will be proportional to npo ppo. In the dark,
in equilibrium, thermal generation rate is equal to the recombination rate. The latter is proportional to
nno ppo. The rate of change of Δnp 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 τr is defined by
n p
t

n p
[5.91]
r
a. Show that for low-level injection, npo  np  ppo, τr is constant and given by
5.1
r 
1
1

Bp po BN a
[5.92]
b. Show that under high-level injection, Δnp  ppo,
n p
t
  Bp p n p   Bn p 
2
[5.93]
so that the recombination lifetime τr is now given by
r 
1
1

Bp p Bn p
[5.94]
that is, the lifetime τr is inversely proportional to the injected carrier concentration.
c. Consider what happens in the presence of photogeneration at a rate Gph (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,
 n p 

Gph  
 B n p p p  n po p po

t

 recombination


A photoconductive film of n-type GaAs doped with 1013 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 m3 s−1. At λ = 850 nm,
the absorption coefficient is about 5 × 103 cm−1. Calculate the photocurrent Iphoto 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?
*5.29 Seebeck coefficient of semiconductors and thermal drift in semiconductor
devices 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
[5.102]
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,
( E  EF ) 
k
S n   2  c

e
kT

[5.103]
what are typical magnitudes for Sn in Si doped with 1014 and 1016 donors cm-3? What is the
significance of Sn at the semiconductor device level?
c. Consider a pn junction Si device that has the p-side doped with 1018 acceptors cm-3 and the n-side
doped 1014 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