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Module PHY6002 Inorganic Semiconductor Nanostructures

Assessed questions

Module PHY6002 Inorganic Semiconductor Nanostructures
Assessed questions

· Hints: Be careful with the different units. You will generally need to work in m, m3, kg etc. Also energies given in eV will generally have to be converted into Joules (by multiplying by the electronic charge) before they can be used in the various equations.

Electron Mobility (m
2
V
-1
s
-1
)

0.1

1

10 100

Temperature (K)

1. Using data taken from the above figure calculate the change in the electron scattering time for the clean bulk GaAs sample as the temperature is reduced from ~300K to 5K. What is the corresponding change for the best single heterojunction between ~300K and 1K. Using your results calculate the average distance travelled by electrons between collisions in clean bulk GaAs at 5K and the best heterojunction at 1K. The electron effective mass in GaAs is 0.067m0.

2. The semiconductors AlAs, InSb and GaP are to be grown on an InP substrate. Using the following figure calculate the approximate lattice mismatch between these semiconductors and InP, expressing your results as a percentage of the InP lattice constant. For each case state if the epitaxial layer will be subjected to compressive or tensile strain.

3. The exciton binding energy in a semiconductor is given by the equation

e m4 */32π2ħ2 εr2 ε02

where m* is the carrier effective mass and εr is the relative permittivity of the semiconductor. In a quantum well made from this semiconductor the exciton binding energy is enhanced by a factor of 1.9. Calculate a value for the binding energy of an exciton in this quantum well where m*=0.09mo and εr=10. Calculate the temperature corresponding to this energy.

4. The exciton binding energy of a semiconductor is 8meV and when used to form a quantum wire this binding energy is increased by a factor of three. If the semiconductor has a bulk band gap of 1.520eV and the confinement energies for the lowest electron and hole states are 140 and 25meV respectively, calculate the energy of the lowest excitonic transition.

5. A quantum wire has a rectangular cross section with dimensions 4nm and 6nm. If the effective mass of the electrons is 0.08m0 calculate the energies of the first 6 confined electron states, giving the values of the two quantum numbers for each state.

6. An edge emitting semiconductor laser is found to have facets mirrors of reflectivity R=0.35. If the laser is surrounded by air of refractive index 1 what is the refractive index of the semiconductor?

Useful constants

Charge of an electron e = 1.6×10-19C Mass of an electron m = 9.1×10-31Kg

Planck’s constant h = 6.6×10-34Js

Planck’s constant /2πℏ = 1.0×10-34Js

Boltzmann’s constant k = 1.38×10-23JK-1

Permittivity of free space ε0=8.85×10-12Fm-1

Speed of light c=3.0x108ms-1

1
1
1

Module PHY6002 Inorganic Semiconductor Nano

tructures

Lectures 7, 8, 9 and 10

Lecture 7 – The fabrication of semiconductor
nanostructures I

Introduction
In this lecture we will look at the techniques used to fabricate semiconductor
nanostructures. The well-established epitaxial methods used to produce
quantum wells will be described. The main techniques applied to produce
quantum wires and quantum dots will be discussed, with a comparison of their
relative advantages and disadvantages. In the next lecture we will look in
detail at the most successful technique used to produce quantum dots, self-
organisation.

Epitaxial techniques
There are two well established epitaxial growth techniques used to produce
high quality quantum wells: molecular beam epitaxy (MBE) and metal organic
vapour phase epitaxy (MOVPE).
The following figure shows the main components of an MBE reactor.

The reactor consists of an ultra-high vacuum chamber with a number of
effusion cells, each containing a different element. Each cell has a mechanical
shutter placed in front of its opening. In operation the cells are heated to a
temperature where the elements start to evaporate, producing a beam of
atoms which leave the cells. These beams are aimed at a heated substrate
which consists of a thin wafer of a suitable bulk semiconductor. The incident
beams combine at the surface of the substrate and a semiconductor is
deposited atomic-layer by atomic-layer. The substrate is rotated to ensure
even growth over its surface. By opening the mechanical shutters in front of
certain cells it is possible to control which semiconductor is deposited. For
example opening the shutters in front of the Ga and As cells results in the
growth of GaAs. Shutting the Ga cell and opening the Al cell switches to the
growth of AlAs. Because the shutters can be operated very rapidly in
comparison to the rate at which material is deposited, it is possible to grow

An MBE reactor

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

very thin layers with very sharp interfaces between layers. The following figure
shows a transmission electron microscope image of a quantum well sample
containing five wells of different thicknesses. The thinnest well has a
thickness of only 1nm. Other cells in the MBE reactor may contain elements
used to dope the semiconductor and it is possible to monitor the growth as it
proceeds by observing the electron diffraction pattern produced by the
surface.

The second epitaxial growth technique is metal organic vapour phase epitaxy
(MOVPE). In this technique the required elements are carried, as a
component of gaseous compounds, to a suitable chamber where they mix as
the gases flow over the surface of a heated substrate. The compounds
breakdown to deposit the semiconductor on the surface of the substrate with
the remaining waste gases being removed from the chamber. Valves in the
gas lines leading to the chamber allow the gases flowing into the reactor to be
switched on and off. A suitable switching sequence allows layered structures
to be deposited. Because it is difficult to switch a gas flow quickly, and
because the growth rate with MOVPE is faster than for MBE, the latter
technique is generally capable of growing thinner layers with more abrupt
interfaces. However the faster growth rate of MOVPE has advantages in
commercial production where it is necessary to deposit the material as quickly
as possible. MOVPE has a number of safety implications as the gases are
highly toxic. The following figure shows a schematic diagram of the main
components of a MOVPE system.

A cross sectional transmission electron microscopy (TEM) image of an InGaAs-
InP quantum well structure containing five wells of different thicknesses.

Main components of a MOVPE system (From Davies)

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Requirements for semiconductor nanostructures
Before we look at the various techniques that have been used to produce
quantum wires and dots, it is useful to consider what properties ideal
structures should exhibit. This will help in analysing the relative advantages
and disadvantages of each technique.
The main requirements of a semiconductor nanostructure can be summarised
as follows
• Size. For many applications we require all the electrons and holes to be in

their lowest energy state, implying negligible thermal excitation to higher
states. The amount of thermal excitation is controlled by the ratio of the
energy spacing between the confined states and the thermal energy, given
by kT. At room temperature the thermal energy is 25meV and a rule of
thumb is that the level separation should be at least three times this value
(~75meV). As the spacing between the states is controlled by the size of
the structure (see lecture 5 for the case of a quantum well) this places
requirements on the size of the nanostructure.

• Quality. Defects may increase the probability of carriers recombining non-
radiatively. Structures with a large number of defects may be very
inefficient light producers. For optical applications nanostructures with low
defect numbers are required.

• Uniformity. Devices generally contain a large number of nanostructures.
Ideally all the nanostructures should be identical otherwise they will all emit
light at slightly different energies.

• Density. It should be possible to produce dense arrays of nanostructures.
• Growth compatibility. Industry uses MBE and MOVPE extensively.

Nanostructures will find more applications if they can be produced using
either or both of these techniques.

• Confinement potential. The depth of the potential wells which confine the
electrons and holes must be relatively deep. If this is not case then at room
temperature carriers will be thermally excited out of the nanostructure.

• Electron and/or hole confinement. For electrical applications it is
generally only necessary for either electrons or holes to be trapped
(confined) within the nanostructure. For electro-optical applications it is
necessary for both types of carrier to be confined.

• p-i-n structures. Many applications require the electrical injection of
carriers into the nanostructure or the transfer of carriers, initially created in
a nanostructure, to an external electrical circuit. This can be achieved if the
nanostructure can be incorporated within the intrinsic region of a p-i-n
structure.

Fabrication of semiconductor quantum wires and quantum dots

Lithography and etching
This starts with an epitaxially grown two dimensional system to provide
confinement along the growth direction. Lithography (etch resist, optical
lithography with a mask or electron beam lithography) is then used to define a
pattern on the surface consisting of either wires or dots. These are
subsequently etched using a plasma, resulting in free standing dots or wires.
The structure can subsequently be returned to a growth reactor to be

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

overgrown and incorporated in a p-i-n device. The main stages of this
technique are shown in the following figure. The main disadvantage of this
technique is that the surface is damaged during the etching stage. The
resultant defects produce an optically dead layer where non-radiative
recombination is the dominant electron-hole recombination process. This
dead layer has an almost constant width so becomes increasingly important

as the size of the structure decreases. For the small sizes required for
practical nanostructures the dead layer occupies all of the structure which is
consequently optically dead.

Cleaved edge overgrowth
A quantum well is initially grown and then the sample is cleaved in the growth
reactor along a plane parallel to the growth direction. The sample is then
rotated through 90° and a second quantum well followed by a barrier is grown.
The growth sequence is shown in the following figure.

The two quantum wells form a T-shaped structure. At the intersection of the
two wells the effective well width is slightly larger. Because the confined
energy levels depend on the inverse of well width squared (see Lecture 5) the
intersection region has a slightly lower potential and hence electrons and
holes become trapped there – a quantum wire is formed. If during the initial
growth multiple wells are grown then the overgrowth of the final well results in
a linear array of wires. A second cleave followed by a further overgrowth can
be used to produce quantum dots.
The surfaces produced by cleaving are clean, in contrast to the dirty surface
formed by etching. Hence cleaved edge overgrowth dots and wires have a

(a) (b) (c) (d)(a) (b) (c)

)

The main stages in forming lithographically defined dots. (a) growth of a 2D quantum
well. (b) surface coating with etch resist. (c) exposure of resist to form pattern (d)
etching to form dot or wire.

(a) (b) (c) (d)(a) (b) (c)

(d)

The steps involved in the cleaved edge overgrowth of a quantum wire. (a) initial
quantum well growth (b) cleavage to form a perfect surface (c) rotation (d) growth
of the second quantum well.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

high optical quality. Their main disadvantage is that the potential at the
intersection of the wells is not much smaller than in the wells. The carriers are
only weakly confined in the intersection region and at room temperature their
thermal energy is sufficient to allow them to escape. These structures are
therefore generally suitable for studying physics at low temperatures but not
for device applications, which need to work at room temperature. In addition
the cleaving step is a difficult, non-standard process.

Growth on Vicinal Substrates
Semiconductors are crystalline materials with a periodic structure. Only when
a semiconductor crystal is cut in certain directions will it have a flat surface.
For other directions the surface will consists of a series of steps (think about a
brick wall). Epitaxial growth is usually performed on flat surfaces. However the
use of stepped surfaces (so-called vicinal surfaces) can be used to produce
quantum wires. The size of the steps is determined by the direction along
which the surface is formed but are typically ~20nm or less.

The above figure shows the main steps in the growth of vicinal quantum
wires. Starting with the stepped surface (a) the wire semiconductor is initially
deposited epitaxially (b). Growth tends to occur in the corner of the steps as it
here that the highest density of atomic bonds occurs. As the growth proceeds
the semiconductor spreads out from the initial corner. When approximately
half of the step width has been covered growth is switched to the barrier
material (c) which is used to cover the remainder of the step. Growth can then
be switched back to the wire semiconductor to increase the height of the wire
(d). This growth cycle is repeated until the desired vertical height is obtained.
Finally the wire is overgrown with a thick layer of the barrier material (e).
Although very thin wires can be produced using this technique the growth has
to be very well controlled so that exactly the same fraction of the step is
covered during each cycle. In addition the coverage on different steps varies
and it is difficult to ensure that the original steps are uniform. The resultant
wires tend not to exhibit good uniformity.

Growth on patterned substrates
This starts with a flat semiconductor substrate which is coated with an etch
resist and then exposed using either optical or electron beam lithography to
produce an array of parallel stripes. The regions between the stripes are then
etched in a suitable acid. Because the acid etches different crystal directions
at different rates, a v-shaped groove is obtained. The patterned substrate is
then cleaned and transferred to a growth reactor.

(a) (b) (c) (d) (e)(a) (b) (c) (d) (e)
The main steps in the growth of vicinal quantum wires (a) original stepped surface
(b) growth occurs in corners of steps, sufficient material deposited to cover ~1/2
of step (c) remainder of step filled in with first material (d) more wire material
deposited to increase thickness of wire (e) final over growth of wire.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Quantum wires are usually formed from GaAs, with AlGaAs as the barrier
material. Initially the AlGaAs barrier is deposited. This grows uniformly over
the whole structure and may sharpen the bottom of the groove which, after
the etching, has a rounded profile. Next a thin layer of GaAs is deposited.
Although this again grows over the whole surface, the growth rate at the
bottom of the groove is faster than that on the sides of the grooves due to the
different crystal surfaces. A quantum well is formed with a spatial modulation
of its thickness, being thicker at the bottom of the groove. In a similar manner
to cleaved edge overgrowth, this thicker region results in a potential minimum
forming a quantum wire. A second AlGaAs barrier layer can now be grown;

this re-sharpens the groove after the formation of the wire, after which further
wires can be grown. The main steps of this technique, resulting in v-groove
quantum wires, are shown in the above figure.

The following figure shows a cross sectional transmission electron
microscope image of a multiple v-groove quantum wire structure. The wires
have a crescent cross section.

(a) (b) (c) (d)

The main steps in the formation of v-groove quantum wires (a) original patterned
substrate, (b) growth of barrier semiconductor (c) growth of wire semiconductor,
greater growth at bottom of groove (d) growth of second barrier, re-sharpening of
groove.

A cross sectional transmission electron micrograph of three v-groove quantum
wires. The wires have a maximum thickness of approximately 8nm.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Because the quantum wire is not next to the original etched surface, v-groove
quantum wires exhibit good optical efficiencies. However it is difficult to
control the inplane size of the wires as this is mainly determined by the shape
of the groove. The uniformity of the wire along its length is also influenced by
the original groove quality. For achievable wire sizes the energy level
spacings are typically 20~30meV, some what less than required for room
temperature operating devices. However in some cases careful control of the
groove cross-section has lead to slightly larger level spacings. A further
disadvantage of v-groove quantum wires is their complicated structure. In
addition to the wire there are quantum wells formed on the sides of the groove
(side wall wells) and on the region between the grooves (top wells). These
wells may capture carriers, reducing the fraction which recombine in the wire
and also producing additional features in the emission spectra. Although the
top wells and some of the side wells can be removed by etching after growth
this requires a further fabrication step and the structure may need to be
returned to the reactor to complete the growth of a p-i-n structure.
By initially patterning the substrate not with a single array of stripes but with
two perpendicular arrays to give a two dimensional array of squares, the
subsequent etching forms an array of pyramidal shaped pits. Epitaxial growth
now results in the formation of quantum dots at the bottom of each pit.

Strain induced dots and wires
If a semiconductor is subjected to strain its band structure is modified. In
particular by applying the correct sign of strain the band gap may be reduced.
If strain is only applied to a small region of the semiconductor then a local
reduction of the band gap may occur, resulting in the formation of a wire or
dot. In practise a local strain is produced by depositing a thin layer of a
different material (e.g. carbon) on the surface of the semiconductor. This will
have a very different atomic spacing to the semiconductor so to fit together
both the atomic positions in the carbon layer and the surface region of the
semiconductor will alter. This alteration constitutes a strain. If the carbon layer
is patterned by lithography and then etched to leave only stripes or dots, the
local strain field produces a wire or dot in the underlying semiconductor. The
remaining isolated pieces of carbon are known as stressors. It is necessary to
place a quantum well near to the surface of the semiconductor to provide
confinement along the growth direction. The steps in the production of strain
induced dots and wires are shown in the following figure.

(a) (b) (c)(a) (b)

(c)

Steps in the formation of strain induced nanostructures (a) initial quantum well (b)
deposition of carbon layer (c) formation of stressors by lithography and etching.
The resultant, localised strain field (dashed lines) forms a wire or dot in the
quantum well.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Although this technique involves an etching step, only the carbon layer is
etched, the etching is kept away from the optically active quantum well. Hence
defect formation is not a problem as is the case for the etched dots and wires
described above. However the strain fields only produce a weak modulation of
the band gap and so the confinement potential is relatively small. At room
temperature carriers are thermally excited from the dots or wires.

Electrostatically induced dots and wires
If a thin metal layer is deposited on the surface of a semiconductor (a
Schottky contact) then a voltage can be applied between the metal and the
semiconductor. This voltage has the effect of either raising or lowering the
energies of the conduction and valence bands near the surface, with respect
to their energies deeper in the semiconductor. If the bands are raised then a
potential minimum is created for holes near to the surface. Alternatively if the
bands are lowered a potential minimum for electrons is created. This is shown
in the following figure.

If the metal layer used to make the Schottky contact is patterned using
lithography and etching, then the resultant shapes can be used to locally
modulate the conduction and valence bands, forming quantum wires or
quantum dots. An added sophistication is to form two slightly separated metal
strips on the semiconductor surface, a so-called split gate. By applying
appropriate voltages a potential minimum is created in the region between the
gates, the width of which is determined by the size of the applied voltage.
Hence a wire of variable width is created.
Electrostatically induced nanostructures form clean systems as only the metal
needs to be etched, not the semiconductor. However the potential minima are
not very deep and the spacing between the energy levels is small, they are
hence only suitable for low temperature operation. Their main limitation
however is that only electrons or holes are confined in a given structure, they
are hence not suitable for optical applications.

The effect of applying a voltage to a Schottky contacted semiconductor

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Quantum well width fluctuations
The width of a quantum well is not constant but exhibits a spatial fluctuation
(see the following figure). Because the confined energy levels depend upon
the well width, potential minima are formed for electrons and holes at points
where the well width is above its average value. These fluctuations confine
the carriers within the plane of the dot (the well provides confinement along
the growth direction) to give a quantum dot. Although these dots have good
optical properties their confining potential is very small, as are the spacings
between the confined levels. The inplane size of the dots is virtually
impossible to control (the well width fluctuations are essentially random) and
the spread of dot sizes is very large. These dots have no device prospects.

Thermally annealed quantum wells
A GaAs-AlGaAs well is grown using standard epitaxial techniques. A very
finely focussed laser beam is then used to locally heat the surface. This
produces a diffusion of Al from the AlGaAs into the GaAs well, causing an
increase in the band gap. By scanning the beam round the edges of a square
a potential barrier is produced surrounding the unilluminated centre of the
square. Carriers optically excited within this square are confined by the
potential barrier and the quantum well, forming a quantum dot. Quantum wires
can also be formed by scanning the laser beam along the edges of a
rectangle. Because the minimum size of the focussed laser beam is ~1µm the
minimum size of the dots is fairly large (~100nm). This results in very closely
spaced energy levels and, in addition, the annealing processes can affect the
optical quality of the semiconductor. This technique also requires specialised,
non-standard equipment.

Semiconductor nanocrystals
Very small semiconductor particles, which act as quantum dots, can be
formed in a glass matrix by heating the glass with a small percentage of a
suitable semiconductor. Dots with radii between 1~40nm are formed, the
radius being a function of the temperature and heating time. The main
limitation of these dots is that, because they are formed in an insulating glass
matrix, the electrical injection of carriers is not possible.

Quantum well width fluctuations. The electrons and holes are localised in
regions where the well width is above its average value (blue dashed line).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Colloidal quantum dots
These are formed by injecting organometal reagents into a hot solvent.
Nanoscale crystallites grown in the solution with sizes in the range 1~10nm.
Subsequent chemical and physical processing can be used to select a subset
of the crystallites with good size uniformity. The dots are formed from II-IV
semiconductors, including CdS, CdSe and CdTe. The dots exhibit good
optical properties but as they are free standing the electrical injection of
carriers is not possible.

Summary and conclusions
In this lecture we have looked briefly at the two established epitaxial
techniques (MBE and MOVPE) used to grow two dimensional quantum wells.
We then considered the main requirements for the properties of
semiconductor nanostructures, before discussing the various techniques
which have been developed to produce quantum wires and quantum dots. Of
the techniques used to produce wires the most important are the v-groove
and electrostatic induced ones. Only the former technique has been applied to
room temperature device applications (mainly lasers) although it still has a
number of disadvantages. For quantum dots, growth on patterned substrates,
strain induced structures, electrostatic induced structures, quantum well width
fluctuations, quantum well thermal annealing and colloidal dots have all been
used to study physics in zero-dimensional systems (generally at very low
temperatures). However none of these techniques has so far been suitable for
room temperature device applications. We will see in the next lecture that self-
organised techniques come the closest to producing ideal dots.

Further reading
The epitaxial techniques of MBE and MOVPE are discussed in Davies ‘The
Physics of Low-Dimensional semiconductors’. Bimberg, Grundmann and
Ledentsov ‘Quantum Dot Heterostructures’ discuss some of the requirements
for semiconductor nanostructures. Some of the numerous fabrication
techniques developed to produce wires and dots are described in the
previously mention books and in the book by Weisbuch and Vinter ‘Quantum
Semiconductor Structures’

More information can be obtained from a number of research papers.
Suggestions are
• A close look on single quantum dots, A Zrenner, Journal of Chemical

Physics Volume 112 page 7790 (2000). Provides an overview of many of
the techniques used to prepare quantum dots. Many useful references.

• Photoluminescence from a single GaAs/AlGaAs quantum dot, K Brunner
et al Physical Review Letters Volume 69 Page 3216 (1992). Thermally
annealed dots.

• Quantum size effect in semiconductor microcrystals, A Ekimov et al Solid
State Communications Volume 56 Page 921 (1985). Semiconductor
nanocrystals.

• Luminescence from excited states in strain induced InGaAs quantum dots,
H Lipsanen et al, Physical Review B Volume 51 page 13868 (1995). Strain
induced dots.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

• One-dimensional conduction in the two-dimensional electron gas in a
GaAs-AlGaAs heterojunction, T J Thornton et al, Physical Review Letters
Volume 56 Page 1198 (1986). Electrostatically induced wires.

• Synthesis and characterisation of nearly monodispersive CdE (E=S, Se,
Te) semiconductor nanocrystallites, C B Murray et al, Journal of the
Americal Chemical Society Volume 115 Page 8706 (1993). Colloidal
quantum dots.

• Formation of a high quality two-dimensional electron gas on cleaved
GaAs, L N Pfeiffer et al, Applied Physics Letters Volume 56 Page 1697
(1990). Cleaved edge overgrowth of quantum wires.

• Patterned quantum well heterostructures grown by OMCVD on non-planar
substrates – applications to extremely narrow SQW lasers, R Bhat et al
Journal of Crystal Growth Volume 93 Page 850 (1988). V-groove quantum
wires.

• Molecular beam epitaxy growth of tilted GaAs AlAs superlattices by
deposition of fractional monolayers on vicinal (001) substrates, J M Gaines
et al, Journal of Vacuum Science and Technology B Volume 6 Page 1381
(1988). Growth of quantum wires on vicinal surfaces.

• Self-limiting growth of quantum dot heterostructures on nonplanar {111}B
substrates, A Hartmann et al Applied Physics Letters Volume 71 Page
1314 (1997). Growth of quantum dots on patterned substrates.

• Homogeneous linewidths in the optical spectrum of a single gallium
arsenide quantum dot, D Gammon et al, Science Volume 273 Page 87
(1996). Dots formed from quantum well width fluctuations.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Lecture 8 – The fabrication of semiconductor
nanostructures II

Introduction
In this lecture we will look at the most successful technique developed so-far
to fabricate semiconductor quantum dots – self-assembly. The use of this
technique will be described and some of the properties of resultant dots will
be discussed.

The growth of strained semiconductor layers
Generally when growing quantum wells it is arranged that the well, barrier and
substrate semiconductors have the same atomic spacing (lattice constant).
For example GaAs and AlGaAs have almost identical lattice constants. GaAs
quantum wells with AlGaAs barriers can therefore be grown on GaAs
substrates. If we try to grow a semiconductor which has a very different lattice
constant to that of the substrate, then initially it adjusts its lattice constant to fit
that of the substrate and the semiconductor will be strained. However to strain
a material requires energy. Hence as the thickness of the semiconductor
increases energy will build up. Eventually there is sufficient energy to break
the atomic bonds of the semiconductor and dislocations (a discontinuity of the
crystal lattice) form. Beyond this point the semiconductor can grow with its
own lattice constant, strain energy no longer builds up. The thickness of
semiconductor which can be grown before dislocations form is known as the
critical thickness. The critical thickness is a function of the semiconductor
being grown and also the degree of lattice mismatch between this
semiconductor and the underlying semiconductor or substrate.
Dislocations provide a very efficient mechanism for non-radiative carrier
recombination. Hence a structure which contains dislocations will, in general,
have a very poor optical efficiency. When growing strained semiconductor
layers it is therefore important not to exceed the critical thickness.
A good example of a strained semiconductor system is InxGa1-xAs-GaAs.
When growing quantum wells InxGa1-xAs forms the wells, as it has the smaller
band gap, with GaAs forming the barriers. As the In composition of InxGa1-xAs
increases the lattice mismatch between InxGa1-xAs and GaAs also increases.
Because InxGa1-xAs-GaAs quantum wells are generally grown on a GaAs
substrate the InxGa1-xAs wells are strained to fit the GaAs lattice constant.
For low In compositions (x~0.2) it is possible to grow quantum wells with
thicknesses up to a few 10s nm before the critical thickness is reached.
However for higher x the critical thickness decreases rapidly.

Self-assembled growth of quantum dots
The lattice mismatch between InAs and GaAs is very large (7%) and the
critical thickness for the growth of an InAs layer on GaAs is expected to be
very small (of the order of a few atomic layers). When InAs is first deposited
on GaAs it grows as a highly strained, flat layer (two dimensional growth).
However for certain growth conditions before dislocations start to form the
growth changes to three dimensions in the form of small islands. These
islands form the quantum dots and sit on the original two dimensional layer,
which is known as the wetting layer.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

This behaviour in which the growth transforms from two to three dimensional
is known as the Stranski-Krastanow growth mode. It is caused by a trade off
between elastic and surface energy. All surfaces have an associated energy
because of their incomplete atomic bonds. The surface energy is directly
proportional to the area of the surface. Hence the surface after the islands
start to form has a greater energy than the original flat surface. However
within the islands the lattice constant of the semiconductor can start to shift
back to its bulk value, hence reducing the elastic energy (note this shift is
gradually and increases with distance along the growth direction, there are no
dislocations formed – see following figure). Because the reduction in elastic
energy is greater than the increase in surface energy the transformation to
three dimensional growth represents the lowest energy, and hence most
favourable, state. Following the growth of the dots they are generally
overgrown by the barrier semiconductor GaAs. The following figure shows the
main steps in the formation of self-assembled quantum dots.

InAs
GaAs

(a)

(b)

LHS – change in the lattice spacing for atoms in a self-assembled quantum dot.
RHS the main stages in the formation of a self assembled dot: (a) GaAs substrate
(with buffer layer), (b) initial 2D growth of InAs (c) transformation above critical
thickness to 3D island-like growth (d) over growth of dots with GaAs.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

The Physical Properties of Self-Assembled Dots
The physical properties of self assembled dots (e.g. size, shape and density)
depend to some extent on the conditions used to growth them (e.g.
temperature and growth rate). Typically they have a base size between
10~30nm, a height of 5~20nm and a density of 1×1010~1x1012cm-2. However
values outside this range may be possible by carefully controlling the growth
conditions. Because of their small size the energy separation between their
confined levels is relatively large (40~70meV). They contain no dislocations
and so exhibit excellent optical properties. They have a high two dimensional
density and multiple layers can be grown (see below). They are grown entirely
by an epitaxial process and can easily be incorporated within the intrinsic
region of a p-i-n structure. Their confinement potential is relatively deep (100-
300meV) and both electrons and holes are confined. Uniformity is reasonable
but could be better (see below). The following figure shows a cross-sectional
transmission electron microscope (TEM) image of a typical quantum dot. This
is a bare dot which has not been over grown with GaAs (it is difficult to obtain
similar images of over grown dots as there is very little contrast between InAs
and GaAs in the TEM images).

The following figure shows an AFM image of quantum dot sample. Again the
dots have not been overgrown with GaAs.

A cross-sectional TEM image of an InAs quantum dot grown on GaAs. The base of
the dot is approximately 18nm.

An AFM image of a quantum dot sample. Note the
exaggerated vertical scale.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

The shape and composition of self assembled quantum dots
Although extensively studied there is still considerable uncertainty as to the
precise shape of self assembled quantum dots. Various shapes have been
claimed including pyramids, truncated pyramids, cones and lenses (part of a
sphere). One problem in determining the shape is that it is difficult to study
dots which have been overgrown. Although bare dots can be studied using
AFM and related surface techniques, there is some evidence that the dot
shape may change when they are overgrown. It may be that the shape of self
assembled quantum dots depends upon the precise growth conditions.
A further complication is the composition of the dots. The dots can either be
grown using pure InAs or the alloy InGaAs. However even when grown with
InAs there is evidence that the dots consist of InGaAs indicating the diffusion
of Ga into the dots from the surrounding GaAs. The Ga composition in the
dots is unlikely to be uniform leading to a highly complicated system which is
difficult to model theoretically (see below).

Multiple quantum dot layers
Once one layer of dots has been deposited and overgrown with GaAs a flat
surface is formed upon which a second layer can be deposited. It is hence
possible to grow multiple layers of dots. When the first dot layer is deposited
the positions of the dots are reasonably random. As the InAs in the dots
gradually returns to its bulk lattice constant as the dot height increases, the
initial GaAs deposited on top of the dot will be slightly strained. A strain field
will be produced in the GaAs above each dot, although this will gradually
decrease to zero as the thickness of the GaAs is increased. However if, when
the next dot layer is deposited, these strain fields are still present (only a thin
GaAs layer has been grown) they may act as nucleation sites for the next
layer of dots. In this case the dots are vertically aligned and stacks of aligned
dots may be formed with 10 or more dots in a stack. This alignment only
occurs when successive dot layers are separated by very thin GaAs layers
(<10nm). For thicker GaAs layers the strain field is essentially zero when the next layer is deposited and the dots form at random positions. The following figure shows a cross sectional transmission electron microscope image of a sample containing 10 dot layers with each layer separated by 9nm of GaAs. The vertical alignment of the dots into stacks can be clearly seem. This alignment may be important for the electronic and optical properties as it is possible that electrons and holes may be able to move between the dots in a stack.

A cross sectional TEM image of vertically aligned quantum dots.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Dot uniformity
The growth of self assembled dots is a semi-random process. Dots at different
positions on the surface will start to form at slightly different times as the
amount of InAs deposited will not be totally uniform. This results in the final
shape and size (and possibly composition) varying slightly from dot to dot. As
the energies of the confined energy states are a function of the dot size,
shape and composition these will also vary from dot to dot.
The emission from a single dot will consist of a very sharp line (similar to the
emission from an atom). However most experiments on self assembled
quantum dots probe a large number of dots. For example a typical
photoluminescence experiment will use a laser beam focussed to a diameter
of 250µm. If the dot density is 1x1011cm-2 the area of the laser beam will
contain ~50 million dots, each of which will contribute to the measured
spectrum. As each dot will emit light at a slightly different energy the sharp
emission from each dot will merge into a broad, featureless emission. This is
known as inhomogeneous broadening. Only if the number of dots probed can
be reduced significantly (e.g. by reducing the diameter of the laser beam – see
later lectures) will the individual sharp emission lines be observed.
The non-uniformity of self-assembled quantum dots and the resultant
inhomogeneous broadening of the optical spectra is a disadvantage for a
number of potential device applications. For example the absorption is spread
out over a wide energy range instead of being concentrated at a single
energy. The inhomogeneous broadening also complicates fundamental
physics studies; as will be discussed in later lectures. However there are
some applications (e.g. optical memories) which make use of the
inhomogeneous broadening. The following figure shows photoluminescence
spectra of different numbers of quantum dots. This is achieved by evaporating
an opaque metal mask on the sample surface in which holes of different sizes
are formed. By shining the laser beam through these different size holes,
different numbers of dots can be probed.

Photoluminescence spectra of different numbers of quantum dots.
From Gammon MRS Bulletin Feb. 1998 Page 44

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Theoretical modelling of self-assembled quantum dots
Self assembled quantum dots have a high degree of strain and this strain is
non-uniform. In addition they have a complicated shape. This makes the
calculation of the confined energy levels very difficult. The following figures
show the distribution of strain, calculated for pyramidal shaped dots, and the
shapes of the wavefunctions for the lowest energy electron and hole states.

As we will see in later lectures the optical spectra of the quantum dots are
very complicated and difficult to interpret. Hence it is still not possible to test
the predictions of the various available theoretical models. In addition many of
the input parameters required for the models (e.g. the exact dot size, shape
and composition) are still not well known.

The strain distribution in self assembled quantum dots: (a) through the wetting
layer, (b) through the dot. From Stier et al PRB 59, 5688 (1999).

Electron and hole wavefunctions for the lowest energy confined quantum dot
states. From Stier et al ibid.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Different self assembled quantum dot systems
The most commonly studied self assembled system consists of InAs or
InGaAs dots grown within a GaAs matrix. The band gap of bulk InAs is 0.4eV
but quantum confinement and strain increase this to between 0.95 and 1.4eV,
the precise value being dependent on the shape and size of the dots. This
energy range correspond to wavelengths 1300~900nm, which is in the near
infrared region of the electromagnetic spectrum.
The emission energy can be increased if InAs or InGaAs dots are grown in an
AlGaAs matrix. This allows energies up to ~1.8eV (≡690nm) to be obtained. Al
can also be added to the dots to increase their emission energy (AlInAs-
AlGaAs dots).
Self assembled dots have also been fabricated from other semiconductor
combinations where there is sufficient lattice mismatch. Examples include InP
dots in GaInP (emission energy ~1.6-1.9eV [~775-650nm]), Ge dots in Si and
InSb, GaSb or AlSb dots in GaAs (emission energy ~1.0-1.3eV [~1200-
950nm]). More recently there have been attempts to grow dots in the wide
band gap nitride semiconductors GaN, InN and AlN.

Summary and Conclusions
In this lecture we have looked at the most promising method for producing
quantum dots suitable for electro-optical applications. The main properties of
quantum dots prepared using the self-assembly technique are compared with
other types of dots and wires in the following table. Self-assembled dots
satisfy the majority of requirements for device applications, possibly with the
exception of uniformity. As we will see in later lectures, a number of devices
based on self assembled quantum dots have now been demonstrated.

Further reading
‘Quantum Dot Heterostructures’ by Bimberg et al provides a comprehensive
overview of the self-assembly technique including a discussion of optical,
electrical and structural studies and devices based on these quantum dots.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

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Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Lecture 9 – Modulation doping and transport
phenomena in semiconductor nanostructures

Introduction
Using a technique known as modulation doping it is possible to obtain
extremely high carrier mobilities in semiconductor nanostructures. This has a
number of practical applications and also leads to the observation of a
number of highly novel transport related phenomena.

Modulation Doping
We saw in Lecture 2 that in a bulk semiconductor the carrier mobility is limited
by phonon scattering at high temperatures and scattering from charged
impurity atoms at low temperatures. The temperature dependence of the
electrical mobility hence has the following form.

Although the low temperature mobility can be increased by reducing the
impurity density this lowers the electrical conductivity as it is these impurities
which provide the free carriers (doping).
In a semiconductor nanostructure however it is possible to spatially separate
the dopant atoms and the resultant free carriers, significantly reducing this
scattering mechanism. This leads to very high low temperature carrier
mobilities. This arrangement, which is known as remote or modulation doping,
is shown schematically for n-type doping of a quantum well in the following
figure. In this case the donor atoms are placed only in the wider band gap
barrier material, the quantum well is undoped1. However the electrons
released by the donor atoms in the barrier transfer into the lower energy well
states, resulting in a spatial separation of the free electrons and the charged
donor atoms. The confined electrons in the quantum well are said to form a
two-dimensional electron gas (2DEG); a two-dimensional hole gas can
similarly be formed by doping the barriers p-type. The non-zero charge

1 This is simply achieved during MBE growth by only opening the shutter in front of the cell
containing the dopant atoms during growth of the barriers. In MOVPE the gas carrying the dopant
atoms is similarly switched.

M
ob

ili
ty

Temperature

Phonon
scattering

Impurity
scattering

M
ob
ili
ty
Temperature
Phonon
scattering
Impurity
scattering

Temperature dependence of electrical mobility for a semiconductor

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

present in both the barriers and the well2 adds an electrostatic potential
energy which results in a bending of the band edges, as indicated in figure
(b). This band bending allows the formation of a modulation doping induced
2DEG at a single interface (a single heterojunction) between two different
semiconductors, as shown in figure (c). Here the combined effects of the
conduction band offset and the band bending result in the formation of a
triangular shaped potential well which restricts the motion of the electrons to
two dimensions.

In a modulation doped structure the barrier region immediately adjacent to the
well is generally undoped, forming a spacer layer, which further separates the
charged dopant atoms and the free carriers. By optimising both the width of
this spacer layer and the structural uniformity of the interface, and by

2 The total charge of the structure remains zero but there are equal and opposite charges in the well and
barriers.

(a) (b) (c)

Donor atom Free electron

(a) process of n-type modulation doping in a quantum well, (b) as (a) but also showing the
effects on the band edges of the non-zero space charges, (c) modulation doping of a single
heterostructure.

0.1 1 10

100

1000

1980

1982

1989

GaAs-AlGaAs
single heterojunctions

Clean bulk GaAs

Bulk GaAs

El
ec

tro
n

M
ob
ili
ty

(c
m

2 V
-1
s-1

)

Temperature (K)

Temperature dependence of the mobility of bulk GaAs (standard and clean) and three GaAs-
AlGaAs single heterostructures (numbers give the corresponding years). Data taken from
Stanley et al (Appl. Phys. Lett. 58, 478 (1991)) and Pfeiffer et al (ibid 55, 1888 (1989))

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

minimising unintentional background impurities, it is possible to achieve
extremely high low temperature mobilities. The previous figure compares the
temperature variation of the electron mobility of standard bulk GaAs, a very
clean bulk specimen of GaAs and a series of GaAs-AlGaAs single
heterojunctions. At high temperatures, where mobility is limited by phonon
scattering, the mobilities of the different structures are very similar. At low
temperatures the mobility of bulk GaAs is increased in the cleaner material
where a lower impurity density reduces the charged impurity scattering.
However the absence of doping results in a low carrier density and, as a
consequence, a low electrical conductivity. It is therefore not possible to
achieve both a high conductivity and high mobility in a bulk semiconductor.
Modulation doping however results in both high free carrier densities and low
temperature mobilities more than two orders of magnitude larger than those of
clean bulk GaAs and almost four orders of magnitude larger than ‘standard’
bulk GaAs. The data for the different heterojunctions presented in the figure
demonstrates how the low temperature mobility of a single heterojunction has
increased over time, reflecting optimisation of the structure, the use of purer
source materials and cleaner MBE growth reactors. The ability to produce
2DEGs of extremely high mobility has allowed the observation of a range of
interesting physical processes, a number of which will be discussed later in
this lecture and the following lecture.
Modulation doping is now used extensively to provide the channel of field
effect transistors (FETs), particularly for high frequency applications. Such
devices are known as high electron mobility transistors (HEMTs) or
modulation doped field effect transistors (MODFETs). Although the use of
modulation doping provides negligible enhancement of the room temperature
carrier mobility, the free carriers are confined to a two dimensional sheet in
contrast to a layer of non-zero thickness for conventional doping. This precise
positioning of the carriers results in devices exhibiting more linear
characteristics and, for still unclear reasons, these devices also exhibit lower
noise. III-V semiconductor HEMTs or MODFETs operating up to ~300GHz are
achievable with applications including mobile communications and satellite
signal reception.

The Hall effect in bulk semiconductors
The following figure shows the geometry used to study the Hall effect. A
current Ix flows along a semiconductor bar to give a current density Jx (=Ix/wh).
A magnetic field B applied normal to the axis of the bar produces a magnetic
force on each moving charge carrier given by qvB, where q is the charge and
v the carrier drift velocity. This force causes the carrier motion to be deflected
in a direction perpendicular to both the field and the original motion as shown
in the figure. As a consequence of this deflection there is a build up of the
charge carriers, and hence a non-zero space charge, along the side of the
bar, which results in the creation of an electric field along the y-axis, Ey. This
so-called Hall field produces an electrostatic force (qEy) on the charge carriers
which opposes the magnetic force. Equilibrium is quickly reached where the
two forces balance to give a zero net force.

/( ) / 1/( )y y x y x HqE qvB E vB J B nq or E J B nq R= ⇒ = = =

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

where the last step follows from the relationship Jx=nqv (see Lecture 2). The
ratio Ey/(JxB) is known as the Hall coefficient and has a value 1/(nq). As Ey
produces a voltage between the sides of the bar, given by Vy=wEy, all three
quantities Ey, Jx and B are easily determined allowing RH and hence the
product nq to be found. A Hall measurement of a bulk semiconductor hence
allows the carrier density n to be determined as well as the majority carrier
type (electrons or holes) from the sign of RH.

The Quantum Hall Effect
The Hall effect can also be observed in a nanostructure containing a 2DEG.
Experimentally the electric field along the sample, Ex, can also be determined
by measuring Vx as shown in the previous figure. This allows two resistivities
to be determined, defined as:

ρ ρx

y
x

E
J

= =

Because RH=Ey/(BJx), for a bulk semiconductor ρxy=RHB, which increases
linearly with increasing magnetic field, with ρxx remaining constant. However
for a two-dimensional system a very different behaviour is observed, as
shown in the following figure. In this case although ρxy increase with
increasing field, it does so in a step-like manner. In addition ρxx oscillates
between zero and non-zero values, with zeros occurring at fields where ρxy
forms a plateau. This surprising behaviour of a two-dimensional system is
known as the Quantum Hall effect and was discovered in 1980 by Klaus von
Klitzing, for which he was awarded the 1985 Nobel Physics Prize. The
Quantum Hall effect arises as a result of the form of the density of states of a
two-dimensional system in a magnetic field. This corresponds to that of a fully
quantised system, with quantisation in one direction resulting from the
physical structure of the sample and quantisation in the remaining two
directions provided by the magnetic field. Diagram (a) of the following figure
shows the discrete energy levels for a perfect system. However in any real
system the levels are broadened by carrier scattering events and the energy
levels have the form given by the right hand diagrams. These ‘bands’ of states

VXVY

JX
B

h Ex
Ey

The geometry of the Hall effect

0 1 2 3 4 5 6 7 8 9
0

2000

4000

6000

8000

10000

12000

14000

ρ
XY resistance (h/e

1/7
1/6
1/5

1/4

1/3

1/2

(x60)ρxx
ρ

R
es

is
ta

nc
e

(Ω
)

Magnetic Field (T)

An example of the integer quantum Hall
effect. Data taken from Paalanen et al,
Phys. Rev. B. 25, 5566 (1982)

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

have similarities with the energy bands in a solid (see Lecture 1) and as in
that case the electronic properties are a very sensitive function of how the
charge carriers occupy the bands. Each band formed by the magnetic field is
known as a Landau level and it can be shown that the degeneracy of each
Landau level is given by

eB
h

Hence as the field is increased the degeneracy of each level also increases.
Therefore for a given carrier density in the structure the number of occupied
levels decreases with increasing field. In (c) the Landau level degeneracy is
such that only the lowest two levels are occupied. This corresponds to the
case of an insulator with completely filled bands followed by completely empty
bands. In this case the structure has a zero conductivity (σxx=0). In (b) the
field has been increased so that now the second Landau level is only half
filled. Conductivity is possible for the electrons in this level and hence σxx≠0.
Under conditions of high magnetic field the following relationships relate the
conductivity and resistivity components

1xx
xx xy H

xy xy

R B
σ

ρ ρ
σ σ

≈ ≈ =

The first relationship shows that the zero conductivity values obtained when
exactly an integer number of Landau levels are occupied results in a zero
value for ρxx.
The plateau values of ρxy can be found by noting that if exactly j Landau levels
are fully occupied then

S
eB

N j
h

where NS is the two dimensional carrier density. From the above definition of
ρxy

(a) (b) (c)
Quantised energy levels of a two dimensional system placed in a magnetic field (a) case of
zero level broadening (b) and (c) with level broadening and for different occupations of the
levels up to the dashed line.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

1 25812.8
xy H

B h
R B

N e j e j
ρ = = = = Ω

The plateau values of ρxy are sample independent and are related to the
fundamental constants h and e. Values for ρxy can be measured to very high
accuracy and are now used as the basis for the resistance standard and also
to calculate the fine structure constant α=µ0ce2/2h, where the permeability of
free space, µ0, and the speed of light, c, are defined quantities.
The parameter j is known as the filling factor The quantum Hall effect
discussed previously occurs for integer values of j and is therefore known as
the integer quantum Hall effect. However, in samples with very high carrier
mobilities, plateaus in ρxy and minima in ρxx are also observed for fractional
values of j, giving rise to the fractional quantum Hall effect. The discovery and
theoretical interpretation of the fractional quantum Hall effect, which results
from the free carriers behaving collectively rather than as single particles, lead
to the award of the 1998 Nobel Physics prize to Stormer, Tsui and Laughlin.
An example of the fractional quantum Hall effect is given in the above figure
which was recorded at very low temperatures for a very high mobility GaAs-
AlGaAs single heterostructure. In addition to minima in ρxx and plateaus in ρxy
for integer values of the filling factor, similar features are also observed for
non-integer values, for example 3/5, 2/3, 3/7 etc.

Ballistic Carrier Transport
The carrier transport considered so far is controlled by a series of random
scattering events (see Lecture 2). However the high carrier mobilities which
can be obtained by the use of modulation doping correspond to very long path
lengths between successive scattering events, lengths that can significantly

An example of the fractional quantum Hall effect which where the filling factor j has non
integer values. The integer quantum Hall effect is still observed at low fields. Figure from
R Willet et al Phys. Rev. Lett. 59, 1776 (1987).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

exceed the dimensions of a nanostructure. In this case a carrier can pass
through the structure without experiencing a scattering event, a process
known as ballistic transport. Ballistic transport conserves the phase of the
charge carriers and leads to a number of novel phenomena, two of which will
now be discussed.
When carriers travel ballistically along a quantum wire there is no dependence
of the resultant current on the energy of the carriers. This results from a
cancellation between the energy dependence of their velocity (v=(2E/m*)1/2)
and the density of states, which in one dimension varies as E-1/2 (see Lecture
6). For each subband occupied by carriers, a conductance equal to 2e2/h is
obtained, a behaviour known as quantised conductance. If the number of
occupied subbands is varied then the conductance of the wire will exhibit a
step-like behaviour, with each step corresponding to a conductance change of
2e2/h. Quantum conductance is most easily observed in electrostatically
induced quantum wires (see Lecture 7). The gate voltage determines the
width of the wire, which in turn controls the energy spacing between the
subbands. For a given carrier density, reducing the subband spacing results
in the population of a greater number of subbands and hence an increased
conductance. The following figure shows quantum conductance in a 400nm
long electrostatically induced quantum wire. These measurements are
generally performed at very low temperatures to obtain the very high
mobilities required for ballistic transport conditions. In contrast to the plateau
values observed for ρxy in the quantum Hall effect, which are independent of
the structure and quality of the device, the quantised conductance values of a
quantum wire are very sensitive to any potential fluctuations which result in
scattering events. This sensitivity prevents the use of quantum conductance
as a resistance standard.
The inset to the above figure shows a structure in which a quantum wire splits
into two wires which subsequently rejoin after having enclosed an area A.
Under ballistic transport conditions the wavefunction of an electron incident on
the loop will split into two components which, upon recombining at the far side

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
0

2
4
6
8
10

12 Split gate

2D EG

O hmic co ntacts

Split gate

2D EG
O hmic co ntacts

C
on

du
ct

an
ce

(u
ni

ts
2

e2
/h

)

Split Gate

Bias Voltage (V)

Example of quantum conductance in a
quantum wire defined electrostatically
from a 2DEG. The inset shows the
sample geometry. Data from Hamilton et
al, Appl. Phys. Lett. 60, 2782 (1992).

0 10 20 30 40 50 60 70 80

100

150

200

250

300

R
es
is
ta
nc
e
(Ω
)

Magnetic Field (mT)
An example of the Aharonov-Bohm effect in
an electrostatically defined quantum ring.
The inset shows the sample geometry. Data
from Timp et al, Phys. Rev. B. 39, 6227
(1989).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

of the loop, will interfere. If a magnetic field is now applied normal to the plane
of the loop an additional phase difference is acquired or lost by the
wavefunctions, depending upon the sense in which they traverse the loop.
The phase difference increases by 2π when the magnetic flux through the
loop, given by the area multiplied by the field (BA), changes by h/e. Hence as
the magnetic field is increased the system will oscillate between conditions of
constructive interference (corresponding to a high conductance) and
destructive interference (corresponding to low conductance). The change in
field (∆B) between two successive maxima (or minima) is given by the
condition ∆BA=h/e, resulting in the conductance of the system oscillating
periodically with increasing field. An example of this behaviour, known as the
Aharonov-Bohm effect is shown in the previous figure for a loop of diameter
1.8µm formed from the 2DEG of a GaAs-AlGaAs single heterostructure by
patterning the surface with metal gates defined by electron beam lithography.

Summary and Conclusions
In this lecture we have shown how modulation doping allows the attainment of
very high carrier mobilities at low temperatures. This allows the observation of
a number of novel effects including the integer and fractional quantum Hall
effects. The high mobilities correspond to long average distances between
scattering events and carriers may be able to pass through a nanostructure
ballistically without undergoing a single scattering event. In this case
processes which include quantised conductance and the Aharonov-Bohm
effect are observable.

Further reading
The paper by Pfeiffer et al (Appl. Phys. Lett. 55, 1888 (1989)) describes the
optimisation of the MBE technique to give very high electron mobilities.
Carrier scattering processes are discussed in detail in ‘The Physics of Low
Dimensional Semiconductors’ by J H Davies. The discussion of the integer
quantum Hall effect give in this lecture is relatively non-mathematical. A more
detailed treatment which includes the importance of disorder is given in ‘Band
theory and Electronic Properties of Solids’ by J Singleton (OUP).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

Lecture 10 Tunnelling and related processes in
semiconductor nanostructures

Introduction
Quantum mechanical tunnelling, in which a particle passes through a
classically forbidden region, is the mechanism by which α particles escape
from the nucleus during α decay and electrons escape from a solid in
thermionic emission. Tunnelling can also be observed in semiconductor
nanostructures where the ability to deposit very thin layers permits the easy
production of tunnelling barriers. Tunnelling can be observed either through a
single barrier or through two barriers separated by a quantum well or quantum
dot. A range of novel physical processes are observed with a number of
practical applications.

Tunnelling through a single square barrier
Consider the single square barrier of potential height V0 and thickness a as
shown in the following figure. Such a structure can be easily fabricated by
depositing a thin layer of a wide band gap semiconductor between thicker
layers of a narrower band gap semiconductor. Away from the barrier, and on
both sides, would normally be doped regions to provide a reservoir of carriers.
By fabricating a suitable device an applied voltage can be used to vary the
energy of the carriers and their ability to pass through the barrier is indicated
by the magnitude of current flowing through the device.

The following figure shows the calculated transmission probability for an
electron of energy E incident on a barrier of height 0.3eV and thickness 10nm.
The classical result has a value of zero when the electron energy is less than
the barrier height and one otherwise. In contrast the quantum mechanical
result is non-zero for energies below that of the barrier height indicating that
the electron can quantum mechanically tunnel through the barrier, a region
where classically it would have negative kinetic energy. The oscillations of the
probability for energies which exceed the barrier height are a result of the
interference between waves which are reflected from the two sides of the
barrier.
For electron energies less than the barrier height the transmission probability
T can be approximated to

Schematic diagram of a single barrier tunnelling structure.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

*
0
2

2 ( )16
exp( 2 )

m V EE
T a where

V
κ κ

−
≈ − =

Because of the exponential function the transmission probability is very
sensitive to both the energy of the electron and the width and height of the
barrier.

Double barrier resonant tunnelling structures
Of greater practical interest than a single barrier tunnelling structure is the
case of two barriers separated by a thin quantum well, known as a double
barrier resonant tunnelling structure (DBRTS). A schematic diagram of a
DBRTS is shown in the following figure. Quantised energy levels are formed
in the quantum well as described in Lecture 5.

Calculated transmission coefficient as a function of electron energy for a single barrier of height
0.3eV. taken from J H Davies ‘The Physics of Low-dimensional semiconductors’ CUP

V
I
I
V
V
(a)
(b)
(d)
(c)

A double barrier resonant tunnelling structure.

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

The previous figure also shows a DBRTS for various applied voltages. For the
sign of voltage shown electrons travel from left to right. Electrons are first
incident on the left most barrier through which they must tunnel. However at
low applied voltages their energy when they have tunnelled into the well is
below that of the lowest confined state and the two barriers plus the well
therefore behave as one effectively thick barrier; the tunnelling probability and
hence the current is very low. As the voltage is increased the energy of the
electrons tunnelling through the first barrier comes into resonance with the
lowest state in the well. The effective barrier width is now reduced and it
becomes much easier for the electrons to pass through the structure. As a
result the current increases significantly. For further increase in voltage the
resonance condition is lost and the current decreases. However additional
resonances may be observed with higher energy confined states. The figure
also shows the expected current-voltage characteristic of a DBRTS indicating
the relationship between specific points on the characteristic and the different
voltage conditions.

The previous figure shows experimental results obtained for a DBRTS
consisting of a 20nm GaAs quantum well confined between 8.5nm AlGaAs
barriers. Resonances with five confined quantum well states are observed.
Beyond each resonance a DBRTS exhibits a negative differential resistance,
a region where the current decreases as the applied voltage is increased.
Such a characteristic has a number of applications including the generation
and mixing of microwave signals. Very high frequencies are possible because
of the rapid transit time of the electrons through the structure.
DBRTS can also exhibit hysteresis in their current-voltage characteristics,
particularly when the thicknesses of the two barriers are asymmetrical. A
thinner first barrier allows carriers to tunnel easily into the well but a thicker
second barrier impedes escape, resulting in charge build up in the well. This
charge build up modifies the voltage dropped across the initial part of the
structure and maintains the resonance condition to higher voltages than would

0
10
20
30

0 1 2 3
0

10
20
30
40

x35

C
ur

re
nt

(m
A

)
Bias Voltage (V)

x100

C
ur
re
nt
(m
A
)
Bias Voltage (V)

Measured current voltage characteristics of a double barrier resonant tunnelling
structure. Data supplied by P Buckle and W Tagg (University of Sheffield).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

occur in the case of an empty well. This broadened resonance is only
observed as the voltage is increased allowing charge to accumulate in the
well. If the voltage is taken above the resonance condition the well empties
and decreasing the voltage results in a narrower resonance as there is now
no charge accumulation. For such a structure the current follows a different
path depending upon the direction in which the voltage is varied; the current-
voltage characteristics exhibit a hysteresis. The inset to the previous figure
shows the characteristics of an asymmetrical DBRTS with 8.5 and 13nm thick
Al0.33Ga0.67As barriers and a 7.5nm In0.11Ga0.89As quantum well.
Two important figures of merit for a resonant tunnelling structure are the
widths of the resonance and the ratio of the current at the peak of the
resonance to that immediately after the resonance, the peak-to-valley-ratio.
Once resonance has been reached with the lowest energy confined quantum
well state it might be expected that current would continue to flow for higher
voltages because of the continuum of states which exist as a result of inplane
motion (see Lecture 5). However when an electron tunnels through the first
barrier not only must energy be conserved but also the two components of the
inplane momentum or wavevectors kx and ky. Conservation of kx and ky
prevents tunnelling into higher continuum states as these correspond to high
values of kx and ky whereas the tunnelling electrons will generally have
relatively small inplane wavevectors. In fact the electrons to the left of the first
barrier will have a range of initial energies, a result of their density and the
Pauli exclusion principle, and hence a range of kx and ky values. This range of
inplane wavevectors contributes to the width of the resonance.
That the current immediately after a resonance does not fall to zero indicates
that additional non-resonant tunnelling is occurring. The precise nature of
these additional processes is still unclear but may include tunnelling via
impurity states in the barriers or phonon scattering which allows electrons of
an initially incorrect energy to tunnel via the quantum well states. In general
the peak-to-valley-ratio decreases as the device temperature is increased.

Tunnelling via quantum dots – Coulomb blockade
The quantum well of a double barrier resonant tunnelling structure can be
replaced by a quantum dot. In addition to the modification of the energy level
structure the small size of a typical quantum dot results in a new effect. A
small quantum dot will posses a relatively large capacitance. If a quantum dot
already contains one or more electrons then a significant energy is required to
add an additional electron as a result of the work that must be done against
the repulsive electrostatic force between like charges. This charging energy,
given by e2/2C where C is the dot capacitance, modifies the energies of the
confined dot states which would occur for an uncharged system. Charging
effects are most easily understood by referring to a structure of the form
shown in the inset to the following figure, which consists of a quantum dot
placed close to a reservoir of free electrons. Applying a voltage to the metal
gate on the surface of the structure allows the energy of the dot to be varied
with respect to the reservoir. If a given energy level in the dot is below the
energy of the reservoir then electrons will tunnel from the reservoir into the dot
level. Alternatively if the energy level is above the reservoir then the level will
be unoccupied. Hence by varying the gate voltage the dot states can be
sequentially filled with electrons. This filling can be monitored by measuring

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

the capacitance of the
device which will exhibit a
characteristic feature each
time an additional electron
is added to the dot.
The main part of the
previous figure shows the
capacitance trace recorded
for a device containing an
ensemble of self assembled
quantum dots. These dots
have two confined electron
levels; the lowest (ground
state) able to hold two
electrons (degeneracy of
two) with the excited level
able to hold four electrons
(degeneracy of four). In the
absence of charging effects
only two features would be
observed in the capacitance
trace, one at the voltage

corresponding to the filling of the ground state, the other when the voltage
reaches the value required for electrons to tunnel into the excited state.
However once one electron has been loaded into the ground state charging
effects result in an additional energy, and a higher voltage, being required to
add the second electron. This leads to two distinct capacitance features
corresponding to the filling of the ground state. Similarly four distinct features
are expected as electrons are loaded into the excited state although in the
present case inhomogeneous broadening prevents these being individually
resolved. This charging behaviour is known as Coulomb blockade and is
observed experimentally when the charging energy exceeds the thermal
energy, kT.
Coulomb blockade effects can also be observed in transport processes where
carriers tunnel through a quantum dot. Suitable dots may be formed
electrostatically using split gates to define the dot and to provide tunnelling
barriers between the dot and the surrounding 2DEG which forms a reservoir
of carriers. An additional gate electrode allows the energy of the dot to be
varied with respect to the carrier reservoirs. The relatively large dot size
results in Coulomb charging energies that are much larger than the
confinement energies. The former therefore dominate the energetics of the
system. The inset to the following figure shows a schematic diagram of the
structure where a small bias voltage has been applied between the left and
right two-dimensional carrier reservoirs. The dot initially contains N electrons
resulting in an energy indicated by the lower horizontal line. An additional
electron can tunnel into the dot from the left hand reservoir but this increases
the dot energy by the charging energy. Hence this process is only
energetically possible if the energy of the dot with N+1 electrons lies below
the maximum energy of the electrons in the left hand reservoir. Tunnelling of
this additional electron into the right hand reservoir may subsequently occur

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

2.10

2.12

2.14

2.16

2.18

2.20

2.22

2.24

f r e e e l e c t r o n s

q u a n t u m d o t

b l o c k in g b a r r ie r

g a te

Excited state

Ground state

C
ap

ac
ita

nc
e

(n
F)

Voltage (V)
Structure and results from a device in which a
controllable number of electrons can be loaded on to
a quantum dot. Figure redrawn from Fricke et al
Europhysics Lett. 36, 197 (1996).

Module PHY6002 Inorganic Semiconductor Nanostructures
Lectures 7, 8, 9 and 10

but only if the N+1 dot energy
lies above the maximum
energy of this reservoir. If
these two conditions are
satisfied, requiring that the
N+1 dot energy lie between
the energy maxima of the two
reservoirs, a sequential flow
of single electrons through
the structure occurs; the
system exhibits a non-zero
conductance. As the gate
voltage is used to vary the
dot energy, the condition for
sequential tunnelling will be
satisfied for different values
of N and a series of
conductance peaks will be
observed, an example is
shown in the above figure for
a dot of radius 300nm. This
large dot size results in a

large capacitance and a correspondingly small charging energy (0.6meV for
the present example). Hence measurements must be performed at very low
temperatures in order to satisfy the condition e2/2C>>kT. Two practical
applications of Coulomb blockade will be described in a later lecture.

Summary and Conclusions
In this lecture we have seen that it is possible to fabricate tunnelling structures
based on semiconductor nanostructures. Double barrier resonant tunnelling
structures give very non-linear current-voltage characteristics and display
negative differential resistance. Because the transit time of carriers through
such a structure is very short they have a number of applications including
high frequency microwave oscillators and mixers. Tunnelling structures
containing a quantum dot display an added complication due to the charge of
the carriers; the Coulomb blockade effect.

Further reading
For a fuller, mathematical treatment of Coulomb blockade the following
articles may be useful, ‘Artificial Atoms’ by M A Kastner, Physics Today 24
January 1993 and ‘Single electron charging effects in semiconductor quantum
dots’ by L P Kouenhoven et al Zeitschrift für Physik B Condensed Matter 85,
367 (1991).
The generally mathematics of quantum mechanical tunnelling is described in
quantum mechanics text books and also with respect to the present subject in
‘The Physics of Low-Dimensional semiconductors’ by J H Davies CUP. Finally
‘Low-Dimensional Semiconductors materials, physics, technology, devices’ by
M J Kelly OUP discusses applications of resonant tunnelling structures.

-0.60 -0.58 -0.56 -0.54 -0.52 -0.50
0.0

0.5

1.0
N

N+1

C
on
du
ct
an
ce

(e
2 /h

)

Gate Voltage (V)

Coulomb blockade effect observed for tunnelling
through an electrostatically defined quantum dot.
The measurement temperature is 10mK. The inset
shows the carrier tunnelling steps and the energy
levels of the system. Data redrawn from L P
Kouwenhoven, et al Z. Phys. B. 85, 367 (1991).

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