PHYSICS QUESTIONS

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PX4221

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CARDIFF UNIVERSITY EXAMINATION PAPER

Academic Year:

2015-2016

Examination Period:

Spring

Examination Paper Number:

PX4221

Examination Paper Title:

Low Dimensional Semiconductor Devices

Duration:

2 Hours

Do not turn this page over until instructed to do so by the Senior Invigilator Structure of Examination Paper: There are 5 pages. There are four questions in total. There are no appendices. The maximum mark for the examination paper is 60 (80% of the module mark) and the mark obtainable for a question or part of a question is shown in brackets alongside the question. Students to be provided with: The following items of stationery are to be provided: 1 answer book. The following books are to be provided: Mathematical Formulae and Physical Constants. Instructions to Students: Answer THREE questions.

Calculators which have been pre-programmed and calculators with an alphabetical keyboard are not permitted in any examinations.

The use of translation dictionaries between English or Welsh and a foreign language bearing an appropriate departmental stamp is permitted in this examination.

PX4221

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1 (a) With the aid of a sketch describe the growth technique known as Molecular Beam Epitaxy

(MBE). [6]

Describe, with the use of diagrams if necessary, the use of Reflection High Energy Electron

Diffraction (RHEED) in an MBE system. In the context of layer growth what does RHEED

normally tell you? [3]

(b) Figure 1 below shows a shutter sequence for an active layer growth run on a III-V MBE

reactor, where 1 denotes an open shutter, and 0 denotes a closed shutter. Sketch the band

structure (conduction band and valence band edges) of the heterostructure material

produced by this shutter sequence, annotating the material layers produced. Assume non-

degenerate doping such that Ef is just at the conduction band edge and valence band edge

for n and p type doping respectively. [5]

Figure 1. Molecular Beam Epitaxy (MBE) shutter sequence for

an active device layer growth

What might this structure be used for when a suitable bias is applied? [1]

Assuming the material quality is good, how might you increase the efficiency of such a

device? [1]

If the barrier material is comprised of ln0.7Ga0.3As, estimate using Vegard’s law what the turn

on voltage would be for such a device.

Note : Bandgap (RT) Eg (InAs) = 0.36eV, Eg (GaAs) = 1.42eV [4]

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2. (a) GaAs has a unit cell which is a zinc-blende structure (8 atoms per cubic unit cell, 4 Ga and 4

As), with a lattice constant of a=0.565nm, and a dielectric constant ε =12.9. The electron

effective mass is me=0.067 m0 and the heavy hole effective mass is mhh=0.45 m0 (where m0 is

the rest mass of the electron). For a spherical GaAs crystalline nanostructure with 5nm

radius, showing any assumptions made, estimate:

(i) the approximate number of atoms in the sphere [3]

(ii) the approximate number of atoms within a unit cell distance of the surface. [4]

(b) The Bohr radius is given by

𝑎0 = 4𝜋𝜀0ℏ

2

𝑚0𝑞 2

where 𝑞 is the charge on the electron. By using the concept of reduced mass (𝜇), show that

the exciton Bohr radius 𝑎𝑏 is given by

𝑎𝑏 = 𝜀𝑚0

𝜇 𝑎0

[3]

For the GaAs nanostructure described in part (a) deduce the exciton Bohr radius [2]

Comment on whether this system can be considered strong or weak confinement for the

exciton. [1]

(c) Give a basic description of electron beam lithography. What is the key advantage over

conventional contact photolithography? What is the key disadvantage over

photolithography? [5]

Why would the resist profile shown in Figure 2 be desirable for a lift off process? [2]

Figure 2. A cross sectional image of a lithography resist profile

after development.

Permitivity of free space 𝜀0 = 8.85x10 -12 m-3 kg-1 s4 A2

Rest mass of the electron 𝑚0 = 9.1x10 -31 kg

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3. (a) Using the time independent Schrödinger equation

− ℏ2

2𝑚

𝑑2

𝑑2𝑥 𝜓(𝑥) + 𝑉(𝑥)𝜓(𝑥) = 𝐸𝜓(𝑥)

derive an expression for the quantisation energy of carriers in a simple GaAs quantum well

(QW) assuming infinite barriers. Explain any simple modifications that you have made to this

basic infinite square well problem to account for the QW material used. [3]

Write an expression for the total energy of the confined state in the GaAs QW considering

the in-plane dispersion. [2]

(b) Figure 3 shows an idealised schematic of a PL and PLE spectrum at 4.2K from a GaAs/AlGaAs

multi QW (MQW) structure. Assuming the excitation power is low, and assuming the infinite

well approximation, estimate;

(i) the width of the well in nm [6]

(ii) the exciton binding energy in the well [4]

With reasoning, state whether this calculated well width would be an upper or lower limit to

the well width in a real GaAs/AlGaAs MQW. [2]

This MQW structure is doped. If the material is high quality what evidence indicates doping

in Figure 3, and what is the approximate Fermi energy in meV from the bottom of the first

confined sub-band of the well? [3]

Figure 3. Idealised schematic PL and PLE spectra from a

GaAs/AlGaAs multi quantum well (MQW) structure

Rest mass of the electron m0 = 9.1x10 -31 kg

Electron effective mass (GaAs) me = 0.067 m0

Heavy hole effective mass (GaAs) mhh = 0.45 m0.

Eg (GaAs) (4.2K) = 1.519eV

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4. (a) Show that the electron density of states in two dimensions (2D), such as that associated with

a 2 dimensional electron gas in a GaAs/AlGaAs heterostructure is given by

𝑔(𝐸)𝑑𝐸 = 𝑚∗

𝜋ℏ2 𝑑𝐸

where 𝑚∗ is the effective mass [5]

State, with reasoning, the implication on the threshold current of a semiconductor diode

laser as a result of this modification of the density of states compared to a bulk system. [2]

(b) The energy states of a Landau level are given by

𝐸 = 𝐸𝑛 + (𝑛 + 1

2 ) ℏ𝜔𝑐

Using this and the 2D density of states from part (a) deduce the degeneracy of a Landau Level

[3]

Sketch these Landau levels on a graph of number n(E) verses energy (E) [3]

Sketch the position of the Fermi Energy for a filling factor of 6 [1]

(c) Given that the density of states in one dimension is

𝑔(𝐸)𝑑𝐸 = 2

ℎ𝑣 =

1

𝜋ℏ √

2𝑚∗

𝐸

where 𝑣 is the carrier velocity, consider the current through a 1D conductor between two

reservoirs of charge carriers. Demonstrate that conductance is quantised when the channel

length (l) is much smaller than the ballistic length (). [6]