B12: Semiconductor Devices
Example Sheet 2
1. Starting from the general form of the ambipolar transport equation given
by eq. (5.70) on page 61 of the notes, show that, under low-level injection
conditions, the excess minority carrier hole concentration in a heavily doped
n-type substrate is describe by the differential equation
Dp
∂ 2 p0
p0
∂p0
∂p0
+
G
−
=
.
−
µ
E
p
∂x2
∂x
τp
∂t
Consider the case where there is no applied electric field, and excess carriers are generated uniformly across the substrate at a constant rate (i.e.,
∂ 2 p0 /∂x2 = 0). Derive an expression for the excess minority carrier hole concentration as a function of time and describe, in words, what this solution
means. Now suppose a constant electric field is applied in the positive x
direction, and a finite number of carriers is generated at x = 0 and t = 0,
where G = 0 for t > 0. Show that
(x − µp Et)2
e−t/τp
0
exp −
p (x, t) = p
4Dp t
4πDp t
satisfies the ambipolar transport equation and comment on what happens to
the excess minority carrier hole concentration as time increases. (It is easy
enough to sketch the curve for p0 (x, t), but you may wish to use MATLAB
to explore this function for various values of the arguments.)
2. Draw an energy band diagram for a pn junction with zero bias. Describe
the physical process that leads to equilibrium making reference to the energy
band diagram where appropriate. Now consider the case where a forward bias
of VF volts is applied. Draw an energy band diagram for this case and explain
the carrier action. Based on the mathematical structure of the expressions
that describe the carrier concentrations in the different semiconductor regions
and what you know about carrier transport, argue that it should come as no
surprise that the forward bias current admits the relation
IF ∝ eVF /Vt
1
provided VF Vt . There is no need to resort to precise mathematical
arguments for this exercise, but reference to key equation may be useful.
3. Consider a one-sided diode where the p-type region is very heavily doped
in relation to the n-type region. Draw a simple three-region diagram of
the pn junction (comprised of the p, n and depletion regions), and describe
graphically how the carrier hole and carrier electron current densities vary
as a function of position in the device. Derive the ideal diode equation for
this scenario.
4. Describe how pn junction physics can be used to detect light. How can
light be generated, in an LED for example, and why is the choice of intrinsic
semiconductor important for photon generation?
5. Derive the current-voltage characteristics for a pnp junction transistor. Analyse the emitter efficiency and base transport factor for this transistor, and
draw conclusions about the base region width and the doping concentrations
in the three regions of the device.
6. Consider an npn junction transistor with base-collector depletion region saturation velocity vs and base region diffusion coefficient DB . By analysing the
cutoff frequency fT , determine the base-collector depletion region width wBC
and base region width wB that provide the optimal high frequency response.
You may assume the edge of the base-emitter region is fixed at x = 0, the
B-C depletion region is symmetrical about the corresponding metallurgical
junction and the junction capacitance charge times are negligible. Comment
on possible ways to improve the frequency response of the device based on
your analysis.
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