Self-assembled semiconductor quantum dots

Contemporary Physics, 2002, volume 43, number 5, pages 351±364

Self-assembled semiconductor quantum dotsRichard J. WarburtonQuantum dots are nanometre-sized clusters of semiconductor material which con?ne electrons in all three directions.  The physics of quantum dots are dominated by quantization : there are discrete energy levels, as in real atoms.  Quantum dots can now be self-assembled directly in the growth of inorgani c semiconductors, and this discovery has fuelled an explosion in the interest in this ? eld.  A review of some of this work is presented, concentrating on the optical properties of quantum dots, and possible application s for photonic devices. 

1. 

Introduction

Quantum dots combine two contemporary themes.  First, they are nanometre-sized in all three directions.  Secondly, they can be produced by self-assembly, meaning that under certain conditions the dots form spontaneously .  The discovery of the self-assembly of quantum dots has opened up a range of new possibilitie s in semiconductor physics and may well lead to a new range of photonics devices.  The important concept is quantization: for su ciently small quantum dots, there is a series of discrete electronic energy levels.  Nonetheless, a quantum dot is typically buried in a semiconductor environment, and this allows the atom-like properties of quantum dots to be combined with semiconductor heterostructures, stacks of diVerent semiconductors designed for a particular application .  On the one hand, it is possible to design sophisticate d experiments on these arti?cial atoms in order to probe the detailed electronic structure; on the other hand, the dots can be introduced into tried-and-tested semiconductor heterostructures for photonic devices.  Semiconductors are very versatile materials for electronic applications .  Until the 1960s, semiconductor devices consisted essentially of bulk materials, functionalize d by introducing a doping pro?le.  However, it was recognized that there would be considerable advantage s in making semiconductor heterostructures consisting of layers of diVerent semiconductors where each layer is defect free and possibly just a few tens of nanometres thick.  This work was pioneered by Zhores I. Alferov and Herbert Kroemer who were awarded the Nobel prize in 2000 `for developingAuthor’s address: Department of Physics, Heriot±Watt University, Edinburgh EH14 4AS, UK; E-mail: R.J.Warburton@hw.ac.uk

semiconductor heterostructures used in high-speed and opto-electronics’ (see for example, the instructive web site [1]).  The fundamental idea is to exploit electron quantization; when an electron is con?ned between two potential barriers, the continuous energy spectrum breaks up into discrete levels.  The crucial concept is the density of states (see, for example, [2]), the number of availabl e energy levels in a particular small energy range, and the density of states is radically changed in semiconductor heterostructures compared with bulk materials.  In a quantum well for instance, consisting of a thin layer of one material of low bandgap sandwiched between a material of higher band gap, the electrons con?ned to the thin layer are free to move in only two dimensions, their motion being quantized in the third direction, and this has a dramatic eVect on the density of states.  The physics of two-dimensional electron gases in semiconductors has been studied extensively ever since the ? rst samples were availabl e [3, 4], and this has led to both fundamental new discoveries, for instance the quantum Hall eVect [5] and the fractional quantum Hall eVect [6], and much improved devices, for instance laser diodes and low-noise high-speed transistors [7].  The workhorse in the research community is the GaAs material system.  AlAs has almost the same lattice constant as GaAs but a larger band gap, allowing heterostructures to be built up.  In fact, the technique of epitaxy, the growth of one atomic layer at a time, has been perfected in the GaAs system, and materials of unparallele d quality can be produced.  The growth technology for semiconductor heterostructures can be extended to include other materials where the constraint of lattice mismatch is lifted.  For instance, thin

InxGa1-xAs layers can be grown on GaAs with excellent crystal quality even though the InxGa1-xAs lattice constant is considerably larger than that of GaAs.  The InxGa1-xAs layer is obviously strained, and the strain energy is su cient to nucleate dislocation s once a certain thickness has been reached, typically 100±200 nm for a 1% lattice mismatch.  The self-assembly of quantum dots was discovered by growing highly strained layers, for instance InAs on GaAs where the lattice mismatch is some 7%.  After the growth of only a few monolayers , the InAs layer becomes highly dislocated with very poor optical properties.  At smaller coverages however, quantum dots were discovered, nanometre-sized islands of indium-ric h material.  Remarkably, the quantum dots are very uniform, with only small ? uctuations in diameter and height from one to the next.  This discovery has ignited an explosion in research into the physics of quantum dots, and there are now a large number of groups working in this ? eld.  The aims are to understand the fundamental physics of these new systems and to exploit their new properties for photonics devices.  This paper makes an attempt to review the fast-moving ? eld of self-assembled semiconductor quantum dots.  The aim has been to present a coherent story rather than to produce a de?nitive review.  The bibliograph y is therefore very limited, and in no way does justice to the large number of groups which have made valuable contributions to this ? eld.  The emphasis is on the interaction of quantum dots with light, as this is where the quantizatio n is most immediately apparent, and where it might be most easily exploited for devices. 

2. 

Quantization in quantum dots

An individual atom has a set of discrete energy levels.  Conversely, a semiconductor has energy bands through the overlappin g and hybridizatio n of the individual atomic levels (see, for example, [2]).  A quantum dot contains typically hundreds of atoms, and therefore the starting point for a discussion of the quantum dot’s electronic properties is the band structure of the host material rather than the discrete levels of individua l atoms.  In the so-called eVective mass approximation , this amounts to giving both electrons and holes a quadratic dispersion on wave vector k, just as for free electrons, but with an eVective mass, typically about 0.1 for electrons and 0.3 for holes [3].  However, an electron in a quantum dot does not behave like an electron in bulk material because the quantum dot represents a con?ning potential, and this gives rise to quantized states with wavefunctions localized at the quantum dot.  Signi?cantly, self-assembled quantum dots are small enough that quantizatio n eVects are important: the energy separation between the electronic levels is larger than the other important energy scales in the system, and in particular the thermal energy. 

A simpli?ed picture therefore is that of a particle with a particular eVective mass moving in a three-dimensional con?ning potential.  At present, the exact form of the con?ning potential is unknown, and in any case it varies from system to system, and even from quantum dot to quantum dot within the same sample, but hints as to its form can be gleaned by interpreting optical spectroscopy.  For lens-shaped InAs dots on GaAs, the con?ning potential is soft and roughly paraboli c in the plane, and relatively hard in the growth direction.  To a ? rst approximation , the quantum dot represents a two-dimensional harmonic oscillator.  This model works well for lensshaped InAs dots, but less well for others, in particular for dots with facets.  In addition to the level separation, an important parameter is the energy barrier separating a con?ned state from the continuum of energy levels at higher energy.  The continuum arises from the semiconductor material surrounding the quantum dots.  In fact, self-assembled quantum dots sit in a so-called wetting layer, essentially a very thin epitaxial-laye r (epilayer) connecting all the dots.  This behaves like a disordered quantum well; therefore it has a broad density of states and represents the continuum for self-assembled systems.  The barrier height in addition to the dot size determines the number of con?ned levels.  Also, the thermal excitation of electrons or holes out of a quantum dot depends on the ratio of the barrier height to the thermal energy.  This barrier height can be up to several hundred millielectronvolt s (meV).  This means that, at low temperatures, an electron has a very small probabilit y of being thermally excited out of one particular quantum dot and, if a certain number of electrons are excited randomly into an ensemble of quantum dots, they will never come into thermal equilibriu m in the lifetime of a typical measurement [8].  On the other hand, thermal excitation is much faster at room temperature, where the barrier heights are typically not high enough to prevent the system from reaching thermal equilibrium . 

3. 

Self-assembly of quantum dots

When growing one material on top of another, three principal growth mechanisms have been identi?ed: Frank± van der Merwe, Volmer±Weber and Stranski±Krastanov modes.  In the Frank±van der Merwe mode, material is deposited layer by layer, as in the growth of the latticematched pair GaAs and AlAs.  In the Volmer±Weber mode, island formation occurs: the material does not wet the surface because it is energetically unfavourable .  Another possibility arises if there is a lattice mismatch.  In this case, the epilayer is strained, and the strain energy obviously increases with increasing epilayer thickness.  In the Stranski±Krastanow mode, the initial growth is layer by layer, as in the Frank±van der Merwe mode, but beyond a certain

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thickness, the so-called critical thickness, islands form, as shown schematically in ? gure 1.  The island formation limits the strain energy, because some strain relaxation is allowed which is not possible in a thin ? lm, but it also results in an increase in the surface energy.  This means that there is an island size at which the total energy is minimized.  It is now well established that many combination s of semiconductors grow in the Stranski±Krastanow mode, and this can be utilized for the self-assembly of quantum dots.  The most studied example is InAs on GaAs where high quality quantum dots can be grown with both molecular-beam epitaxy (MBE) and metal±organic vapour-phas e epitaxy.  Dots form at a critical thickness of 1.5 ?  monolayers, corresponding to just 4 A of material [9].  For thicknesses less than the critical thickness, the covering is not completely uniform; there are island-lik e structures, elongated along the [011] direction.  However, at the critical thickness, dots form, leaving a thin InAs layer, the wetting layer, between the dots.  As more InAs is added, the density of dots increases with only small changes in the dot size.  As an example, ? gure 2 shows the topography of InAs dots on a GaAs substrate measured with an atomic force microscope.  The dots are roughly 6 nm high and 20 nm in diameter with a density of 10 10 cm72.  The dots grown in the Stranski±Krastanow mode are free of dislocations , which is an essential prerequisite in III±V compounds for a high quantum e ciency for optical emission.  Furthermore, the dots are remarkably homogeneous.  Statistical ? uctuations during growth will always give rise to a distribution in dot size, height and composition.  Nevertheless, the ? uctuations in dot size can be as small as about 10% [9].  The dots are randomly arranged in the lateral plane if the growth proceeds on a ? at substrate, and densities lie typically between 109 and 10 11 cm72.  The shape and composition of the dots depend on the growth parameters.  For instance, InAs/GaAs dots can have facets along particular crystal directions, or they can be lens-shaped without facets, depending on the growth technique, and in particular the growth temperature.  Stranski±Krastanow growth has turned out to be a robust phenomenon for semiconductor materials; quantum dots have been produced in this way for a number of material combinations .  The most important condition is that the deposited material has a substantially larger lattice constant than does the substrate.  InAs forms dots not just on GaAs but also on InP. The InAs/InP dots are particularly interesting because they emit close to the technologicall y important 1.5 mm wavelength [10].  InP can also form the dot material by using GaxInl-xP substrates where the bandgap is pushed up into the red region of the spectrum [11].  Dots can also be grown with II±VI materials, the growth of CdSe on ZnSe is analogou s to the growth of InAs on GaAs [12].  These II±VI dots emit

in the green.  Finally, dots can also be formed in the nitrides, for instance by depositing InxGal-xN on an AlyGal-yN surface.  In fact, the high emission e ciency of GaN, despite the large defect density, has been attributed to the formation of quantum dot-like structures in the active layers [13].  It is a challenge to arrange the dots in the plane of the substrate.  An easy route with self-assembly has not been found to accomplish this.  Nevertheless, dots can be encouraged to grow at particular positions by pre-patterning the substrate.  This obviously limits the separation of the dots in any such array to the limits set by lithography .  However, self-assembly does provide a means to order the dots in the growth direction: if the separation between successive layers is small enough (less than 15 nm for InAs/ GaAs), then the dots grow directly above one another [14], as shown in ? gure 3.  In a new layer, the residual strain ? eld from the dots in the layer below is su cient to seed the dots.  In this way, vertical stacks of dots can be built up.  In fact, it has been observed that the dots tend to become more laterally ordered with each layer [15].  The point is that, if two dots are close together in one layer, their strain ? elds overlap and provide only one seeding centre in the subsequent layer.  It is also a challenge to characterize the shape and composition of self-assembled quantum dots.  The shape is accessible with an atomic force microscope, but only when the dots are on the surface.  For all optical experiments, the dots need to be covered with the host semiconductor, otherwise the surface electric ? eld ionizes the electron±hole pairs, rendering the material incapable of emitting light.  Atomic force microscopy is unfortunately unable to provide any information on covered dots.  It is only recently that signi?cant success has been achieved with other techniques.  A notable experiment is cross-sectional scanning tunnelling microscopy where the sample is cleaved in the chamber of a ultrahigh-vacuu m scanning tunnelling microscope.  The tunnelling current is dependent on the local composition.  For InAs dots on GaAs this has led to the observation of a strong composition pro?le: the dots are indium rich at the top, and indium poor at the bottom [16].  While the dot in this particular experiment takes on the shape of a truncated pyramid, the region of high indium concentration is an inverted pyramid.  This is a surprising result, and it is not yet clear whether this is true also for other quantum dot systems.  Grazing incidence X-ray studies support the conclusion of an indium-ric h apex and an indium-poor base although these measurements were performed on uncapped lens-shaped InAs dots [17].  The truncation of the pyramid is thought to take place during capping, and shape changes during capping have been directly observed in the growth of geranium dots on silicon using an in-situ low energy electron microscope [18]. 

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Figure 1.  A schematic diagram of the Stranski±Krastanow mode.  ML stands for monolayer, and the layer thickness in MLs one would have if the growth proceeded layer by layer is used as a measure of the material deposited.  The material deposited has a substantially larger lattice constant than the substrate, for example InAs on GaAs.  Initially, a thin and uniform layer is deposited.  At the critical thickness (1.5 ML for InAs on GaAs), islands spontaneously form.  Further growth results in an increase in the number of islands.  At large coverages, large and dislocated islands appear. 

Figure 2.  An atomic force micrograph of InAs quantum dots on a GaAs substrate.  The InAs dots were grown by MBE at a growth temperature of 5308C. The vertical scale is expanded relative to the horizontal scale.  The image was taken by Axel Lorke. 

It should be noted that the self-assembly of semiconductors is a complex process, being determined largely by kinetic factors rather than by the condition for thermal equilibrium .  It is now clear that the properties depend on the exact growth conditions .  There are very challengin g issues here, and signi?cant progress can be anticipated through the use of growth chambers with in-situ structural analysis.  It will also be possible to exploit the complexity to grow dots with particular properties.  The extent to which this is possible is largely unknown at present. 

4. 

Optical properties of quantum dots

The main interest in quantum dot physics is the quantization in all three spatial directions, giving rise to atomic-like energy levels.  For a system of self-assembled quantum dots, it is possible to measure the fundamental bandgap, the energy separation between the con?ned states both for electrons and for holes, the barriers to the electron continuum and hole continuum, and the oscillator strengths for the various interband transitions with optical spectroscopy. 

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Figure 3.  A cross-sectional transmission electron micrograph of two layers of InAs quantum dots in GaAs showing how the dots in the upper layer lie directly above the dots in the ? rst.  The image was taken by Gio Medeiros-Ribeiro. 

Perhaps the simplest experiment to perform, if not to understand, is photoluminescenc e in which electron±hole pairs are excited with a laser beam, and the light re-emitted by the sample is detected.  The laser energy can be tuned to lie just above the bandgap of the wetting layer in which case the electron±hole pairs are excited in the wetting layer itself.  Alternatively, if the laser energy is larger than the bandgap of the barrier material, electron±hole pairs are excited also in the barrier.  In both cases, the carriers relax into the dots, and then down the ladder of states in the dot, so that the dot electron and hole ground states are occupied.  Light is emitted when an electron and a hole recombine.  As such, the experiment measures only the energy of the ground-state exciton.  However, if the pump intensity is increased such that electron±hole pairs are generated more rapidly than they can recombine, the excited states of the dots are also occupied and recombination can also occur from excited states [19, 20].  An example of such an experiment is shown in ? gure 4.  At low pump powers, only the ground state emission is detected; at higher pump powers, the ? rst excited state emerges and, at higher powers still, the second excited state also emerges, and so on.  Photoluminescenc e spectra are broadened by the inhomogeneou s broadening , that is ? uctuations from dot to dot, but the broadening is small enough that in the experiment of ? gure 4 the excited state emission can be easily distinguishe d from the ground state emission.  For these particular quantum dots, the energy separation between the ground- and excited-state emission is 15 meV, and this corresponds to the sum of the electron and hole con?nement energies.  The separation between the other emissions is also about 15 meV, implying that the states are approximately equally spaced, implying in turn that the con?ning potential is approximatel y parabolic.  Furthermore in this experiment, emission from the wetting layer can also be seen, and this enables the depth of the electron and hole con?ning potentials to be estimated. 

Figure 4.  Photoluminescence (PL) from an ensemble of quantum dots as a function of laser excitation power.  The measurement temperature was 12 K. The quantum dots are induced in an InxGal-xAs/GaAs quantum well by a strain ? eld generated by InP stressors on the sample surface.  The stressors are grown in the Stranski±Krastanow mode and are 1 nm away from the quantum well.  At low excitation power, only the ground state emission is observed.  As the power increases, emission from excited states is also observed through state ? lling.  The spectra at diVerent powers are oVset vertically for clarity, and the vertical scale has been magni?ed by the factors given (62000, 620, 610, etc.) ( Reprinted with permission from Lipsanen et al. [ 19].  Copyright 1995 American Physical Society.) 

Photoluminescenc e invariably yields the sum of electron and hole energies.  In order to measure just the electronic

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properties independent of the holes, it is possible to occupy the dots with, say, two electrons (so that in each dot the twofold degenerate ground state is fully occupied) and to excite electric dipole transitions between the electron levels.  This has been accomplishe d with so-called charge-tuneable heterostructures, with which electrons can be controllabl y loaded into the dots from an electron reservoir [21].  It has also been accomplished with permanently n-doped samples [22].  The interlevel absorption lies in the far infrared region of the spectrum; for InAs/GaAs quantum dots for instance, the transition is at a wavelength of 25 mm (an energy of 50 meV) [22, 23], as shown in ? gure 5.  The results of this experiment also point to an approximatel y parabolic con?ning potential from the fact that as additiona l electrons are added to the dots, the energy of the far infrared absorption does not change much. ( In the limit of a perfect parabolic potential, a famous result known as the

Figure 5.  Transmission in the far infrared of an ensemble of InAs/GaAs quantum dots where each quantum dot is occupied with two electrons.  The data were taken at a temperature of 4.2 K for a range of magnetic ? elds, and the spectra have been shifted vertically for clarity.  The shaded transmission minima correspond to absorption from the quantum dots.  The transitions are between the electron ground state and the ? rst excited state.  Two absorption peaks are observed even at zero magnetic ? eld, suggesting that the dots are slightly oval in shape.  In a magnetic ? eld, the two peaks move apart, which is a consequence of the Zeeman eVect.  The strong feature at 45 meV arises from an electronic interaction with an interface phonon. ( Reprinted with permission from Fricke et al. [ 23].  Copyright 1996 European Physical Society.) 

Normalized transmission

generalized Kohn theorem states that the long-wavelengt h absorption should show no energy dependence on the electron occupation [24].)  In self-assembled quantum dots, the con?ning potential can be thought of as very steep in the growth direction, and relatively shallow in the lateral plane.  In other words, the dots are essentially two-dimensional discs.  The electron ground state is, in analogy with atomic physics, s-like, and the excited state p-like.  However, because of the disc-like nature of the dots, the p-state is fourfold degenerate, and not sixfold degenerate as in a conventional atom.  One way to demonstrate this is to perform interband absorption experiments on charge-tuneabl e structures [25], as shown in ? gure 6.  The ? rst transition corresponds to the transition from the hole s-state to the electron s-state, and disappear s when the dots are occupied with two electrons.  This is simply because the Pauli principle forbids the transition once the ? nal level is fully occupied.  The second transition corresponds to the transition from the hole p-state to the electron p-state and disappears when the dots are occupied with four electrons, con?rming the degeneracies of the con?ned states.  This experiment, unlike a photolumines cence experiment, also yields a value for the oscillator strength [25].  The result can be understood simply in terms of the overlap integral between the hole and electron states.  In other words, these dots are in the so-called strong-

Figure 6.  Transmission in the near infrared of an ensemble of InAs/GaAs quantum dots at 4.2 K. Spectra are shown for diVerent values of N, the electron occupation.  At N=0, there are three transitions corresponding to transitions from a valence state to an electron state.  These are the s±s, p±p and d±d transitions, where the states are labelled in analogy to atomic physics.  At N=2, the electron ground state is fully occupied, and the s±s transition disappears because it is blocked, a consequence of the Pauli principle.  Similarly, the p±p transition is blocked at N=6 when both the s and the p states of the electron are fully occupied. ( Reprinted with permission from Warburton et al. [ 25].  Copyright 1997 American Physical Society.) 

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con?nement regime where the con?nement energies are larger than the exciton binding energy.  In the other limit, the weak-con?nement regime, the exciton binding energy dominates over the con?nement energies, and it makes sense only in this limit to think of an exciton moving as a coherent entity throughout the dot.  In the weak-con?nement regime, the oscillator strength is actually larger than in the strong-con?nement regime [26].  An important issue in interpreting photoluminescenc e experiments, relevant also for devices such as the quantum dot laser, is the rate at which a highly excited electron±hole pair can relax to the ground state of a quantum dot.  Relaxation in the continuum is known to occur on a picosecond time scale: electrons and holes relax by emitting longitudina l optical (LO) phonons if possible and, if not, acoustic phonons, and if there are many carriers, thermal equilibriu m is established in both conduction and valence bands through carrier±carrier scattering.  Capture of carriers into a quantum dot from the continuum also occurs quickly (apart from in low-density samples where capture may take a few hundred picoseconds).  It was initiall y thought that relaxation of carriers once captured by a quantum dot would be hindered by the discrete nature of the density of states [27].  The argument is that the energy separation between the levels is too large for e cient acoustic phonon emission; and there is no reason why the separation between the levels should match the LO phonon energy (which has a very weak dispersion).  Nonetheless, experiments have shown that relaxation from excited dot states to the ground state is rapid, occurring on a picosecond-time scale [28, 29], very nearly as for quantum wells.  There are a number of explanation s for the absence of the `phonon bottleneck’.  First, an electron in an excited state may relax to the ground state by scattering oV a hole in the same dot, with the hole then relaxing by phonon emission [29].  The importance of this mechanism depends on the large hole eVective mass, which gives rise to a large number of closely spaced hole levels, and also a strong coupling to the phonons, allowing e cient hole relaxation.  Secondly, Auger processes are possible when the sample is highly excited; an electron in a dot loses energy by promoting an electron in the continuum to a higher energy.  Thirdly, it has been suggested that the dot itself in?uences the phonon spectrum such that a LO phonon can decay into acoustic phonons quite rapidly, broadening the LO phonon spectrum, and allowing fast relaxation by LO phonon emission over a range of energies tens of millielectronvolt s wide [30].  Despite this latter prediction, there is some evidence for phonon bottleneck for quantum dots occupied with just one electron without a hole, and without a populatio n in the wetting layer [31].  Under less extreme conditions, however, it is now clear that relaxation occurs more rapidly than recombination (the recombination time is typically 1 ns for self-assembled quantum dots), just as for

quantum wells.  This is a useful property for interband lasers, but a di culty for potential interlevel lasers where it would be extremely desirable to decrease the relaxation rate.  5.  Single-dot spectroscopy

There are inevitably ? uctuations in both size and composition from one dot to the next, and this leads to inhomogeneousl y broadened transitions in optical spectroscopy.  For self-assembled InAs dots on GaAs, the lowest linewidth of the photoluminescenc e from a weakly excited ensemble of dots is at present about 20 meV. Obviously, this width can be much larger for highly inhomogeneou s samples.  The inhomogeneou s broadening is large compared with other relevant energies.  For instance, the emission from a biexciton (a complex of two excitons) or charged exciton (a complex of an exciton and one additiona l electron) is typically shifted by a few millielectronvolt s from the emission of a neutral exciton.  It is di cult to resolve these shifts clearly with an experiment on an ensemble.  A further energy scale is the homogeneous width which, although as yet unexplored in any great detail, can be as small as a few microelectronvolts [28].  In order to access these eVects on a milliectronvol t or sub-millielectronvol t energy scale, it is necessary to perform optical experiments on single quantum dots rather than ensembles.  The development of this ? eld has been aided by the techniques developed for single-molecule spectroscopy which was ? rst reported in the late 1980s and has subsequently become a routine, if challenging , experimental technique [32].  Single-molecul e spectroscopy is usually performed by spinning a suspension of molecules on to a substrate such that the molecules are separated by several microns.  A molecule is excited by a focused laser beam, with the same lens collecting also its emission.  With a high numerical aperture lens, only the emission from one molecule is collected.  A confocal arrangement, with a pinhole as a spatial ? lter, can increase the signal-to-nois e ratio by rejecting all light originating from outside the focus.  The same technique can also be used for single quantum dot spectroscopy, provided that the dot density is low enough so that there is only ever one dot in the focus.  While samples with a low density have been produced for single dot experiments, usual samples have a larger dot density, 1010 cm72 being typical.  This density translates into 100 dots mm2, the area of the spot size in a good microscope, and in this case the smooth ensemble emission spectrum breaks up into overlapping sharp lines, as shown in ? gure 7.  This is a strong indication that individua l dots give sharp emission lines, but it is di cult to perform detailed experiments when the lines cannot be spectrally distinguished.  Given that the spot size of a confocal microscope is in the best case diVraction limited, it is clear that conventiona l

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Figure 7.  Emission from quantum dots (a) from a 1 mm2 area of the sample and (b) from a 0.07 mm2 area of the sample, both taken at 4.2 K. The quantum dots are generated with Stranski±Krastanow growth of InAs on GaAs, using an annealing step to increase the photoluminescence (PL) energy from about 1.1 eV to about 1.3 eV. The spectrum in (a) was taken with a low temperature confocal microscope; and the spectrum in (b) with a near-?eld technique.  It can be seen how large statistical ? uctuations arise in the spectrum in (a), because the number of rings probed is not su cient to give a smooth distribution function.  In the spectrum in (b) just a few peaks can be discerned above the noise level, each arising from the emission from an individual quantum dot. 

far-?eld optics will not provide the spatial resolution to probe unambiguousl y just one dot.  The diVraction limit can be bypassed by a near-?eld technique in which a subwavelength-size d aperture is positioned in close proximity to the sample.  In this case, the aperture size determines the spatial resolution, not the wavelength of light, but there is inevitably a large reduction in the signal strength.  One implementation of this idea is to use a tapered metallized optical ? bre as a probe, brought controllabl y to within a few nanometres of the sample surface [33].  Another method is to de?ne lithographicall y nanometre-sized apertures in a metallic layer deposited on the sample [34].  This is simpler than the use of the tapered ? bre but has the disadvantag e that the aperture cannot be moved with respect to the sample, prohibitin g any imaging, but at least many apertures can be de?ned in one sample, allowing a large number of individual dots to be addressed.  A further technique is to etch subwavelength-size d pillars in the sample, such that each pillar contains on average about one

quantum dot [35].  A possible problem here is the eVect of the free surface on the quantum dots’ optical properties.  Many of the experiments on single dots need to be carried out at a low temperature and both miniature optical cryostats [35, 36] and top-loadin g bath cryostats [37] have been used.  The photoluminescenc e signal from a single quantum dot in a near-?eld experiment can be very small, possibly just a few hundred photons per second, and a singlephoton detector with a high quantum e ciency and low dark count is essential.  A cooled silicon charge-coupled device (CCD) camera is extremely convenient , oVering in addition to its high detectivity an invaluabl e multiplexin g advantage .  However, the response of a silicon camera tails oV in the visible, and decreases rapidly for wavelengths longer than 1050 nm; so it is by no means a universal solution.  In fact most of the single dot experiments on InAs-based systems have used quantum dots with some gallium incorporation in order to push

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the bandgap up so that the emission can be detected with a silicon camera.  The most striking property of the emission from a single quantum dot at low temperature is its spectral purity [38, 39].  Linewidths as small as 25 meV, corresponding to 6.0 GHz, have been reported for quantum dots formed in localizing potentials in a quantum well [28].  The exact line widths of self-assembled quantum dots are di cult to measure because they are smaller than the spectral resolution of grating spectrometer±CCD array systems.  The linewidth is a measure of the exciton dephasing time which is at most the radiative lifetime.  This limit of lifetime broadening has been observed at low temperature on natural quantum dots but, as the temperature increases, the linewidth increases because of scattering by acoustic phonons [28].  It remains to be seen what the corresponding behaviour is for self-assembled dots.  The homogeneous width is clearly important for futuristic application s of quantum dots as the elements for quantum information processing, as it is essential that the system maintains its phase coherence during the manipulatio n of the quantum mechanical wavefunctions.  The sharp lines in single dot spectroscopy give access to the rich physics on a millielectronvol t energy scale for quantum dots.  This includes small energy shifts through the Coulomb interaction.  When a quantum dot containing several excitons decays, the photon energy is renormalized by the Coulomb interactions between the electrons and holes such that the train of photons emitted when a highly excited dot decays all have diVerent energies [35].  Alternatively, a quantum dot can be occupied with a known number of electrons by designing a suitable voltagetuneable heterostructure [25].  Every time that an electron is added to the dot, the emission red shifts, and the size of the shift reveals a shell structure [37, 40]: whenever an electron can be added with the bene?t of exchange energy, the shift in the emission is small but, whenever an electron is added to complete a subshell, the shift in the emission is relatively large.  Notably, the emission for a doubly charged exciton splits into two, a triplet and a singlet, as shown in ? gure 8.  This is because the two electrons in the ? nal state, after the emission of a photon, have either parallel or antiparalle l spins, con?gurations with diVerent energies because of the exchange interaction.  This is entirely analogous to the excited state of the helium atom: the splitting of the lines of the doubly charged exciton is simply twice the exchange energy. 

6. 

Colloida l quantum dots

The focus of this article is the properties of quantum dots which are self-assembled in the growth of inorganic semiconductors on planar substrates.  However, rapid progress has been made in the last decade in fabricating

inorganic quantum dots with techniques from organic chemistry [41, 42].  The end result is a suspension of quantum dots in an organic, or even aqueous, solution.  There are a number of ways to prepare semiconductor nanocrystals in a chemical solution, most of which rely on fast nucleation followed by relatively slow growth [42].  If the growth proceeds smoothly, a particular crystal size can be obtaine d simply by stopping the growth after a certain time.  One important method is to mix a methyl metal alkyl (e.g. dimethylcadmium ) and a chalcogen source (e.g. tri-noctylphosphin e selenide) in tri-n-octylphosphin e (TOP).  The mixture is then injected into hot tri-n-octylphosphine oxide (TOPO) [43].  The temperature, up to about 3508C, is high enough for nanocrystals to nucleate (CdSe in this example).  The nanocrystals are crystalline and can be made with diameters of between about 2 and 20 nm with a size inhomogeneit y of about 5%.  This method requires injecting metal alkyls at elevated temperatures and is potentially hazardous .  It has been shown, however, that this problem can be avoided by the use of a molecular precursor in which the metal±chalcogen bond is already in place [42].  The nanocrystals have a large proportion of their atoms close to the surface, meaning that the surface plays a very important role.  A crucial aspect of these techniques is that the surface is passivated by a monolayer of the solvent ligands, as shown schematically in ? gure 9, such that the quantum e ciency for optical emission can be as high as 10%.  By way of comparison, commercial dye molecules have quantum e ciencies between 12% and 70%.  Without this passivation, the nanocrystal quantum e ciency would be extremely low.  The quantum e ciency can be increased further, up to 50%, by designing a so-called core±shell structure; the core is coated with a semiconductor with a larger bandgap, which limits the in?uence of the surface by con?ning the carriers to the core region.  An example of a core±shell structure is the CdSe/CdS system.  It is easiest to grow II±VI semiconductor nanocrystals in this way.  III±V semiconductors are more covalent in character and this means that crystal formation is hampered by large energy barriers.  Nevertheless, III±V nanocrystals have been produced by using appropriat e precursors [42].  Quantum con?nement eVects are particularly striking in these materials.  With just one material, a wide range of emission energies is possible simply by varying the size of the nanocrystals.  By additionall y using diVerent materials, emission over a wide band of wavelengths can be demonstrated, as shown in ? gure 10.  This has been extended into the infrared recently, to the all important 1.5 mm telecommunications wavelength, by developing techniques for selfassembling CdTe/HgTe quantum dots [44].  There are several possible application s of this technology.  One is the use of water-soluble nanocrystals for biological labelling where a number of diVerent labels can

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Figure 8.  Photoluminescence (PL) emission spectra from a single quantum dot as a function of charge.  X is the neutral exciton, X17 , the singly charged exciton, X27 the doubly charged exciton, and X37 the triply charged exciton.  The level diagram shows the hole ground state, the electron ground state and the electron excited state.  The states are labelled s and p, and the z-component of angular momentum m as in atomic physics.  The hole is shown as an open circle, and electrons as : and ; depending on the spin.  Each time that an additional electron is added, the emission shifts to the red.  In addition, for the X27, there are two emission lines because there are two possible ? nal states, as shown in the level diagrams, with diVerent energies because of the exchange interaction. 

be employed, each identi?able by the wavelength of the emission [45].  An advantag e of the nanocrystals over conventiona l dye molecules is that all nanocrystals can be excited more or less equally with a short-wavelength source.  The nanocrystals may also ? nd applicatio n in optoelectronic devices, such as solar cells.  There are, however, serious challenges to be met in this ? eld.  The nanocrystals tend to lose their advantageou s optical properties in the course of time.  For instance, TOPO-capped II±VI materials degrade in the presence of light and oxygen: unstable chalcogen oxides form.  The nanocrystals also photobleach ; the emission from individua l nanocrystals turns on and oV randomly, even with uniform cw excitation.  Even in the on periods, the emission from nanocrystals exhibits temporal wanderings in wavelength [46].  These eVects are related to the localization of charge at defects on the surface.  In other words, the deleterious eVects of the surface have not been eliminated entirely.  Also, there are important issues in how to embed the nanocrystals into devices.  However, progress can be expected in all these areas, and these nanocrystals represent a fascinating approach to building novel photonics devices. 

7. 

Applications of quantum dots

Quantum dots are potentially useful for a number of diVerent technologies.  While there is at present no established applicatio n of quantum dots, there are a number of very promising areas. 

7.1. 

Laser diodes

The development of laser diodes over the past three decades has been a success story.  Through improvements in material quality and processing techniques, the threshold currents have gradually decreased.  Signi?cantly, a reduction in the eVective dimensionalit y of the active layer, from three to two, has made a considerable improvement.  This is related to the density of states.  In a three-dimensional band structure, the density-of-states increases with energy E (as E‰ close to the band edge at E=0), and this limits gain at the lasing wavelength; the availabl e carriers are spread over a narrow band of energy, limiting the population inversion at the lasing wavelength because of the ? nite temperature.  In a quantum well, the density of states for each quantized level (or subband) is constant, so that the same number of

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Figure 9. ( a) A schematic diagram of the growth of semiconductor nanocrystals; (b) the surface passivation with ligands; (c) a core±shell structure. 

Figure 10.  Room-temperature emission spectra of surfactantcoated colloidal quantum dots.  The blue series are CdSe nanocrystals, the green series are InP nanocrystals, and the ? ?  red series InAs nanocrystals.  Selection of the size (60 A, 46 A, ? ?  36 A, 28 A, etc.)  has been used to give a particular wavelength in each band. ( After [44].  Reprinted (abstracted/excerpted) with permission from Bruchez et al. [ 44].  Copyright 1998 American Association for the Advancement of Science.) 

carriers are spread over a much larger energy than in the three-dimensional case, increasing the gain at the lasing wavelength.  An additional improvement can be obtained by con?ning not only the electrons and holes but also the light.  To do this, a waveguide structure is incorporated into the heterostructure by exploiting the small refractive indices of wide gap compounds.  This con?nes the light along the growth direction.  Light con?nement is also possible in the lateral directions by post-growth processing steps, for instance the etching of a ridge mesa. 

The fundamental laser characteristic is the threshold current, the current at which optical transparency is achieved, such that at currents higher than the threshold current there is net gain and lasing occurs.  Typically , the threshold current depends exponentially on the temperature, I=I0 eT/T0, where T is the temperature, T0 is the characteristic temperature and I0 is the low-temperature threshold current.  Ideally, a laser has a small I0 and a high T0.  However, in many contemporary application s of semiconductor laser diodes, the crucial quantity is the maximum modulation frequency as this determines the bit rate for data transmission through optical ? bres.  As can be shown by a rate equation analysis , the bandwidth is high in the case of high gain.  A semiconductor laser diode with a large density of perfect quantum dots is predicted to have optimum properties: a low and temperature-insensitiv e threshold current and a large modulation bandwidt h [47].  `Perfect’ in this context implies that the dots have very deep con?nement potentials (on the scale of the thermal energy, 25 meV at room temperature) and that they are all identical.  Again, the fundamental reason for this is related to the density of states.  For a homogeneous ensemble of quantum dots, the density of states is a sharp peak at the interband transition energy.  For each dot, only two electrons must be excited across the gap to establish populatio n inversion.  This is also true at high temperature for a deep con?nement potential.  All this leads to the attractive properties, and these predictions have motivated much of the recent work on quantum dot lasers.  In practice, self-assembled quantum dots are not perfect for application as the gain medium in a laser diode.  A number of issues arise.  First, the dots are inhomogeneou s so that only a fraction of the dots can contribute to the lasing signal; the majority are inverted uselessly.  This corresponds to a penalty on the threshold current.  Second, the dot densities are typically about 1011 cm72 for these applications , implying that in the layer plane there is more unoccupie d space than dot material.  This limits the material gain.  Third, the dots do not have deep enough con?nement potentials; carriers thermally populate excited dot states, and they can also escape out of the dot before contributing to stimulated emission.  Finally, the carriers are electrically injected and must diVuse laterally into the quantum dots, and this can cause gain saturation at modest currents.  It is therefore not obvious whether the inherent advantages of quantum dots can be exploited with selfassembled techniques.  Quantum dot lasers have now been realized and emit a continuous wave at room temperature, and the properties of the diodes have already improved signi?cantly since the ? rst reports of laser action [48].  Low temperature thresholds have been reported which compare very favourably with the best quantum well laser diodes [48].  At low

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temperatures, T0 can be as high as 385 K, and this is clearly a consequence of the carrier con?nement in the quantum dots [49].  However, at room temperature it is fair to say that the dream of an ultrahigh T0 has not been realized: T0 decreases rapidly above, say, 300 K, causing the threshold current to increase markedly.  Even if the intrinsic advantage s of quantum dots are at present oVset by the inhomogeneou s broadening for laser applications , quantum dot lasers already oVer some other advantage s over conventional laser diodes.  The InAs/GaAs quantum dot emission wavelength is signi?cantly higher than that of InxGal-xAs/GaAs quantum wells, and with subtle changes in the growth can be increased to 1.3 mm, corresponding to one of the low-loss windows in optical ? bre.  Lasers emitting at this wavelength are typically fabricated with inferior InP-based technology.  Also, much higher powers have been realized with quantum dot lasers than with quantum well lasers.  In quantum well lasers, device failure occurs at high currents, often owing to degradatio n of the end mirrors.  It would appear that the carrier localizatio n in quantum dots is su ciently strong to prevent diVusion to the end facets, allowing the devices to withstand higher currents. 

7.2. 

Single-photo n sources

There is a pressing need in quantum cryptography for a single photon source.  Quantum cryptography uses either the phase or the polarization of single photons as the means of communication and can achieve almost totally secure transmission [50].  The basic idea is that the sender rotates his or her basis randomly, making it impossible for an eavesdropper to detect a photon and accurately to recreate it.  For this technology to become viable, a single photon emitter is required.  In prototype systems, a highly attenuated laser pulse is used as the source, but this is both ine cient as the majority of pulses are empty, and insecure, as a sizeable fraction of the pulses contain two photons .  A source consisting of a single quantum dot can in principle overcome these limitations.  The advantage of quantum dots over single molecules for these applications is that quantum dots, unlike single molecules, do not suVer from photobleaching , although progress has recently been made in the preparation of single molecules by embedding terrylene molecules in a p-terphenyl molecular crystal [51].  A powerful idea in quantum dot physics is to excite a quantum dot with a laser pulse intense enough that the probabilit y of the dot capturing at least one exciton is very close to one [52].  A highly excited quantum dot decays by emitting a series of photons, which have diVerent energies because of the renormalizatio n through the Coulomb interaction.  This enables the ? nal photon to be selected by spectral ? ltering, and this photon can be used for communication. 

This idea has now been explored experimentally and, at least at low temperatures, photons on demand have been generated from individua l quantum dots [53, 54].  This is often called non-classical light because the temporal ordering of the photons is highly dissimilar from classical light described by Poisson statistics.  Quantum dots emit photons in all possible directions so that only a fraction of the emitted photons can ever be collected and subsequently detected.  In a single dot experiment, this limits the signal-to-nois e ratio and, in a realization of quantum cryptography , it seriously limits the bit rate.  One solution to this problem is to place a quantum dot in a microcavity tuned to the emission of the quantum dot.  The cavity changes the photon density of states.  At resonance, this increases the radiative decay rate and it also forces the photons to be emitted into the cavity modes.  The cavity modes are controllabl e and therefore allow in principle a much increased detection e ciency.  The increase in radiative decay rate, known as the Purcell eVect, has been experimentally observed by monitoring the temporal decay of excitons in quantum dots placed in a Fabry±Pe?  rot etalon [55].  The two mirrors of the etalon are formed from Bragg re?ectors in the heterostructure itself and consist of a stack of AlAs/GaAs pairs, with each layer of optical thickness l/4.  The two mirrors de?ne a microcavity in the vertical direction; to con?ne the optical mode also in the lateral direction, a pillar is etched through the entire structure with a diameter of about a micrometer.  The decay rate has been increased by a factor of 5 with this technique, with a concomitant change in the spatial distribution of the emission.  So far, this remains a low temperature eVect; at high temperatures the quantum dot emission is signi?cantly broadened such that a matching cavity has a low quality factor, giving a weak Purcell eVect.  7.3.  Other possible application s

An individua l quantum dot has the capability of trapping individua l electrons or holes and it might be possible to exploit this as a memory element with a bit represented by a single electron in a single quantum dot.  For this to succeed, it is necessary that the storage time is long enough to be practical.  One of the most important parameters is the height of the energy barrier DE separating the quantum dot level from the continuum at higher energies.  For DE>kBT, the thermal escape rate is small; in the other limit, DE<kBT, the thermal escape rate is large (kB is the Boltzmann constant and T the temperature).  For the self-assembled quantum dots in III±V semiconductors at present available , DE is at most a few hundred millielectronvolts , and this leads to large thermal escape rates at room temperature.  However, at 77 K, the temperature of liquid nitrogen, the thermal escape rate can be much reduced.  Notable here are

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InAs dots on an InP substrate which have a particularly deep con?ning potential for holes [10], leading to almost unmeasurably small thermal emission rates at 77 K [56].  While a device architecture does not yet exist for such devices, it has been suggested that the inhomogeneit y of the quantum dots could be used for storage applications , with each dot colour representing a storage possibilit y [57].  The basic principle of an optical quantum dot-based storage device is to separate the electron and hole in the optically excited electron±hole pair with a strong electric ? eld.  One elegant way to do this is to use vertically oriented quantum dots in a ? eld-eVect heterostructure.  The electric ? eld can be changed simply through a voltage applied to the gate electrode.  On illuminatio n (the write part of the process), a large negative bias is applied to generate a high electric ? eld in the device.  Quantum dots in one layer trap one or more electrons, and dots in the other layer one or more holes.  After illumination , a positive bias is applie d so that the electrons tunnel into the dots containing the holes.  The electrons and holes recombine, generating light (the read step of the process).  This scheme has been implemented using a layer of self-assembled quantum dots and an adjacent quantum well [58].  In this case, the residual strain ? eld generates quantum dots in the quantum well through the eVects of strain on the electronic band structure.  Existing devices exhibit memory eVects which persist up to many seconds at low temperature (10 K), but the eVect disappear s for temperatures above about 130 K. It is also possible to implement this idea in vertically oriented selfassembled quantum dots [59].  These experiments are interesting in the sense that they represent an approach to engineering the radiative lifetime of excitons in III±V semiconductors.  For instance, a storage time of 1 s corresponds to an increase in the radiative lifetime by a factor of 109 beyond that of an ordinary quantum dot.  The experiments demonstrate that it is possible to have both long carrier storage, for which one usually uses silicon, and e cient emission, for which silicon is unsuitable, in the same device.  An alternative approach is to detect the charge stored in a quantum dot electrically.  A powerful idea is to embed the quantum dots a few tens of nanometres away from a twodimensional electron gas [60].  For a ? xed electric charge density, the resistance of the two-dimensiona l electron gas depends on the way that the charge is distributed.  When the quantum dots are neutral, the resistance is low; when however, the quantum dots are charged, the resistance is high because the charged dots both deplete locally the twodimensional electron gas and also act as e cient scatterers.  This enables the charge state of the quantum dots to be determined through the electrical resistance of the device.  The charge in the quantum dots can be altered by illumination , typically with a bias applied.  A variety of device geometries are possible.  For instance, on illumina-

tion with light at the dots’ resonant energy, the dots can be used to store a metastable population of electrons [61].  Alternatively, the two-dimensiona l channel itself can be used as an absorber, in which case hole tunnelling can deplete already occupied quantum dots [62].  In the latter case, for small devices with a gate electrode 1 mm long, the resistance of the device is sensitive to individua l charging events, and therefore to the detection of single photons [62].  It remains to be seen whether this technology can oVer a competitive alternative to the concept of single-photo n detection with avalanch e photodiodes . 

8. 

Conclusions

The self-assembly of quantum dots has revitalized the ? eld of semiconductor heterostructures as it allows the creation of quantum objects which con?ne both electrons and holes in all three spatial directions.  The physics of quantum dots are dominated by the con?nement; in a quantum dot, the band structure of the host material breaks up into a set of discrete levels.  In some ways, the dots take on the properties of atoms, but with highly unusual characteristics.  However, many of the properties, for instance the relaxation of the electrons from one level to the next, are strongly in?uenced by the presence of the solid matrix between the dots.  It is possible to embed quantum dots into complicated semiconductor heterostructures and this is facilitating detailed physics experiments and the development of novel photonic devices, such as quantum dot lasers. 

Richard J. Warburto n (RJW) studied for a DPhil at the Clarendon Laboratory, Oxford University (1987 ± 1991), and was subsequently (1990 ± 1993) a Junior Research Fellow at Christ Church, Oxford.  At the end of 1993, RJW moved to the Faculty of Physics of the Ludwig-MaximiliansUniversity (LMU), Munich, Germany, initially as a von Humboldt Fellow, and subsequently as a Marie-Curie Research Fellow.  RJW became an Assistant Professor at the LMU in 1996, completing his Habilitation in 2000.  Since 2000, he has been at the Department of Physics, HeriotWatt University, Edinburgh.