Transcript Slide 1

Stanford University
MatSci 152: Principles of Electronic Materials and Devices
Spring Quarter, 2009-2010
Homework #5
Issued: May 5, 2010
Due: May 12, 2010 (at the start of class)
Suggested reading: Kasap, Chapter 5, Sections 5.1-5.7
Problems:
1.
Kasap Problem 5.3 (Fermi level in intrinsic semiconductors)
2.
Kasap Problem 5.10 (Temperature dependence of conductivity)
3.
Kasap Problem 5.17 (Photoconductivity and speed)
4.
Kasap Problem 5.18 (Hall effect in semiconductors)
5.
Kasap Problem 5.22 (Direct recombination and GaAs)
1.
Kasap Problem 5.3 (Fermi level in intrinsic semiconductors)
Using the values of the density of states effective masses me* and mh* in Table 5.1, find the
position of the Fermi energy in intrinsic Si, Ge and GaAs with respect to the middle of the
bandgap (Eg/2).
2.
Kasap Problem 5.10 (Temperature dependence of conductivity)
An n-type Si sample has been doped with 1015 phosphorus atoms cm-3. The donor energy level for P
in Si is 0.045 eV below the conduction band edge energy.
a. Calculate the room temperature conductivity of the sample.
b. Estimate the temperature above which the sample behaves as if intrinsic.
c. Estimate to within 20 percent the lowest temperature above which all the donors are ionized.
d. Sketch schematically the dependence of the electron concentration in the conduction band on the
temperature as log(n) versus 1/T, and mark the various important regions and critical temperatures.
For each region draw an energy band diagram that clearly shows from where the electrons are
excited into the conduction band.
e. Sketch schematically the dependence of the conductivity on the temperature as log() versus 1/T
and mark the various critical temperatures and other relevant information.
3.
Kasap Problem 5.17 (Photoconductivity and speed)
Consider two p-type Si samples both doped with 1015 B atoms cm-3. Both have identical dimensions of
length L (1 mm), width W (1 mm), and depth (thickness) D (0.1 mm). One sample, labeled A, has an
electron lifetime of 1 s whereas the other, labeled B, has an electron lifetime of 5 s.
a. At time t = 0, a laser light of wavelength 750 nm is switched on to illuminate the surface (L  W) of
both the samples. The incident laser light intensity on both samples is 10 mW cm-2. At time t = 50 s,
the laser is switched off. Sketch the time evolution of the minority carrier concentration for both
samples on the same axes.
b. What is the photocurrent (current due to illumination alone) if each sample is connected to a 1 V
battery?
4.
Kasap Problem 5.18 (Hall effect in semiconductors)
The Hall effect in a semiconductor sample involves not only the electron and hole concentrations n
and p, respectively, but also the electron and hole drift mobilities, e and h. The hall coefficient of
a semiconductor is (see Chapter 2),
RH 
where b 
e
p  nb
2
e  p  nb 
2
h
a. Given the mass action law, pn = ni2, find n for maximum (negative and positive RH). Assume
that the drift mobilities remain relatively unaffected as n changes (due to doping). Given the
electron and hole drift mobilities, e = 1350 cm2 V-1 s-1, h = 450 cm2 V-1 s-1 for silicon determine n
for maximum in terms of ni.
b. Taking b = 3, plot RH as a function of electron concentration n/ni from 0.1 to 10.
5.
Kasap Problem 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

where B is a proportionality constant, called the direct recombination capture coefficient. The
recombination lifetime τr is defined by
 n p
t

n p
r
a. Show that for low-level injection, npo  np  ppo, τr is constant and given by
r 
1
Bp
1

BN
po
a
b. Show that under high-level injection, Δnp  ppo,
 n p
t
  B  p p  n p   B  n p 
2
so that the recombination lifetime τr is now given by
r 
1
Bp p

1
Bn p
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
G ph  
 t



 recombinat

 B n p p p  n po p po

ion
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?