SECTION A (Answer ALL questions)

A1.

A one-dimensional solid of lattice spacing a has the dispersion relationship

E(k) = E0 sin2 (ka) where − ≤ k ≤  .

Derive an expression for the effective mass, m∗ (k) for a charge carrier in the band.

[5 marks]

A2.

Figure  1  shows  measurements  of  the  electron  mobility  in  silicon  as  a  function  of  doping concentration. For silicon doped n-type to 1 x 1018 cm-3 operating in the extrinsic region, estimate the resistivity of the material (in units of Ωm).

Figure 1

[5 marks]

A3.

The Gax In1-xAs alloy maintains a direct band gap over its entire composition range. The band gap varies with composition, x, as Eg(x) = 1.42x + 0.36(1 − x) − 0.43x(1 − x) where the energy is in eV.

a.    What are the maximum and minimum band gaps obtainable from this alloy?

b.    Calculate the composition required to achieve a band gap of 1eV.

[5 marks]

A4.

A semiconductor sample has a hole scattering time of 0.5 ps and an effective mass tensor

0.75    0       0

0        0    0.50

where m0 is the free electron (rest) mass.

If an electric field, E = (5, 10, 5) kVcm-1  is applied to the semiconductor, determine the resulting drift velocity vector, vd  of the holes. Hence determine the magnitude of the drift velocity.

[5 marks]

SECTION B (Answer TWO questions)

B1.

Table  1  provides  key  information  for  the  design  of a  germanium  p-n  junction  for  use  in  a photodetector. The parameters are all for a temperature, T=300 K.

Table 1

a.    Using the data provided in the Table (clearly showing your working) :

i.       Calculate the position of the Fermi energy in the p -doped region relative to the valence band edge.

ii.       Calculate the position of the Fermi energy in the n-doped region relative to the conduction band edge.

iii.       Calculate the magnitude of the built-in potential,  bi .

iv.       Calculate the depletion widths on the n- and p-doped sides of the junction and the total depletion width

[10 marks]

b.    Based  on the  information  derived in part  (a),  sketch and label  a  diagram  of the p-n junction showing the key parameters.

[6 marks]

c.    The temperature dependence of the band gap of germanium is described by the Varshni parameters,     Eg(0) = 0.742 eV, a = 5.82x10−4  eVK−1  and F = 235 K.     Calculate     the longest wavelength of light that a germanium photodetector would be able to detect at:

i.       T=300 K

ii.       T=100 K

[4 marks]

B2.

a.    Figure 2 shows the measured optical gain spectrum for a GaAsBi-based semiconductor laser measured at an injection current of 10 mA.

Figure 2

i.       What is the maximum optical gain (in units of cm-1) and at what wavelength does this occur?

ii.       Estimate the optical gain (in units of cm-1) at wavelengths of 850 nm and 865 nm.

[3 marks]

b.    Assuming   no   other   losses,   consider   light   propagating   through   this   material   at wavelengths of 850 nm, 865 nm and at the peak gain wavelength. Assume that the initial light intensity is I0.

i.       Calculate the fractional change in light intensity I(x)/I0 at each of the wavelengths after propagating a distance of 1 mm.

ii.       On the same plot, sketch a graph of the intensity of light versus distance I(x) at each wavelength.

iii.       What is special about light propagation at 865 nm? Hence determine the quasi- Fermi-energy splitting (in units of eV) at this injection current.

[8 marks]

c.    A laser cavity is constructed from this semiconductor with cavity length, L = 0.5 mm, facet reflectivities R1  = R2  = 30 %, an internal loss, ai  = 2 Cm −1 and refractive index, u = 3.5.

i.       Calculate  threshold  gain  for  the  laser  cavity  (in  units  of  cm-1).  What  is  the corresponding threshold current?

ii.       Calculate the mode spacing in the cavity near to the gain peak. Hence estimate how many modes would be supported by the optical gain at an injection current of 10 mA.

iii.       Describe how the laser design could be changed to reduce the number of laser modes to one, i.e. single mode operation.

[9 marks]