A close look on single quantum dots

A close look on single quantum dotsA. Zrenner Citation: J. Chem. Phys. 112, 7790 (2000); doi: 10.1063/1.481384 View online: http://dx.doi.org/10.1063/1.481384 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i18 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 18 8 MAY 2000

A close look on single quantum dotsA. Zrennera)

Walter Schottky Institut, Technische Universita¨t Munchen, D-85748 Garching, Germany

~Received 10 November 1999; accepted 1 February 2000!

Quantum dots, often referred to as artificial atoms, open the field of quantum resolved spectroscopyto semiconductor physics. The current article is designed to review the field of interband opticalspectroscopy on single semiconductor quantum dots. ©2000 American Institute of Physics.@S0021-9606~00!70116-4#

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I. INTRODUCTION

Within the last 20 years the field of optical spectroscoon semiconductors moved more and more towards nastructures. One of the major driving forces in this evolutiwas certainly the general trend to go towards smallersmaller devices.1 Another even more fundamental aspect wconnected with the new ability to now do quantum resolvspectroscopy on isolated quantum mechanical objectssolid.2–5 This in turn can lead in the future to new classesdevices which offer quantum controlled functions.6,7 Today,at the beginning of the new millennium, high quality quatum dots~QDs! are available to the community, which aregarded to be key elements in future solid-state optoetronics.

Currently optical properties of QDs are intensively ivestigated and a very broad range of different aspectthereby covered. From an experimental point of view sinQD spectroscopy is here the technique of choice. Forfundamental understanding of the optical properties of aone has to get rid of all spurious signals which might coplicate the optical spectra and hence comparisons to theAs we will see in the following sections, few bodinteractions,8–10spin related phenomena,11,12and interactionswith phonons10,13,14 lead to rich and complicated specteven from a single QD. Because of the typically high surfadensity of QDs, the limit of single dot spectroscopy is hoever hard to achieve and remains therefore a challenge.perimental methods with extremely high spatial resolutare required to isolate single constituents for advanced stroscopic analysis. The subject of single QD spectroscoptherefore in many ways related to single molecuspectroscopy.15 Basically the selection of a single constituecan be realized in real space or in the frequency domMicroscopy in general is one way to get access to a sinQD in real space. Far-field methods like confocal microcopy allow us to obtain spatial resolutions in the range omm,16 special versions like the solid immersion lens tecnique helps us to extend this range down to 0.25mm.17,18

Near field methods, either performed within situnanoapertures5 or scanning probes,3 extend this range downto 50 nm in special cases, however at the cost of optthroughput. Also cathodoluminescence19 and STM-inducedluminescence~STML!20–22 have been shown to be powerf

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tools, suitable to reach the limit of single dot spectroscopAnother way to achieve access to a single QD in r

space is connected to sample preparation. Low densitysembles of QDs in the range of 1mm22 can be obtained byspecial growth conditions for the case of self-organizdots11 or, in the case of colloidal species, by dilution. For tstill not widely spread methods of growth on patternsubstrates23 or cleaved edge overgrowth,24 QDs can even beintroduced on predefined positions.

It is the purpose of this contribution to review the fieof optical single QD spectroscopy at the current point. Vaous issues will be addressed in the following sections: FiQDs will be discussed in terms of artificial atoms, followeby topics concerning technical aspects of single QD spectcopy, the status of experiment and theory, and finally a boutlook.

II. QUANTUM DOTS: GENERAL REMARKS

A. Optical linewidth

Semiconductor quantum dots~QDs! are often describedas artificial atoms due to theird-function-like density ofstates. This analogy is nicely reflected in the fact that bfree atoms and QDs exhibit optical line spectra with narrlinewidth. Although optical spectroscopy on atoms is dealwith linewidths which are orders-of-magnitudes smalcompared to those of QDs,25 in semiconductor interbandspectroscopy linewidth below 100meV are in fact quite un-usual. They are readily obtained only from donor or accepbound excitons in extremely pure bulk crystals26 and sinceabout 1994 also from QDs.2–4 In semiconductor QDs, loweband gap material with typical lateral dimensions in the 130 nm regime, corresponding to about 103 to 106 lattice at-oms, is embedded into a matrix of higher band gap mateDue to complete size quantization of the electronic levelsthe QD the associated density of states is discrete. Typintraband level spacings~ground to first excited state! are inthe range between 10 to 100 meV. From this point of viQDs look like promising objects for both applied and fundmental research on zero-dimensional systems.

In structures with higher dimensionality such as quatum wells and wires the density of states is still continuoustranslationary invariant directions. The optical absorptiontherefore continuous as well, with a cutoff at the effectiband gap of the material. The optical emission appears attemperatures at or slightly below the energy of the effect

0 © 2000 American Institute of Physics

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7791J. Chem. Phys., Vol. 112, No. 18, 8 May 2000 A close look on single quantum dots

bad gap, with typical optical linewidth in the range betwe1 and 30 meV. Different sources of disorder like well widand alloy fluctuations lead here to inhomogeneous broading effects and the appearance of band tails. Lateral diffusand relaxation of excitons results on a mesoscopic scaltransient occupancies of sites with different energies wittheir lifetime and hence in enhanced, inhomogeneous lwidth.

The situation in semiconductor QDs is basically notdifferent, except that lateral diffusion in the vicinity of thQD is suppressed and relaxation into the QD is rapid enouso that almost all excitons end up in the ground state ofQD prior to radiative recombination. Even nonresonant ocal excitation of a single QD does therefore result in a nrow emission line at the position of the ground state enerThe excitonic ground state energy is a well-defined quanin a QD, it is a stationary eigenvalue in a confined enviroment. Within this environment mesoscopic disorder is mor less irrelevant as long as the characteristic length scalethe resulting energy fluctuations are below the exciton diaeter. Under such conditions we can expect to find extremnarrow absorption and emission lines from single quantdots. In an ideal low temperature case we can hope tolifetime limited homogeneous linewidths of the order ofm eV for direct gap QDs.27

B. Quantum dots versus atoms

What makes semiconductor QDs so attractive compato real atoms? An answer to this question can be found eafrom the technological point of view. The evolution of semconductor technology took its way from bulk devices twards low-dimensional structures. Driving forces in this ogoing process have been various aspects, such as the geneed for downscaling, the need for the improvement ofvice performance, the search for new phenomena in mecopic systems, and the long term goal to achieve quandevices and functions which operate in a controlled andherent way on the basis of single electrons, excitons, or ptons.

From this point of view we can see a semiconductor Qas a manmade, artificial environment, in which we can ksingle charges or elementary excitations. Once preparedhave this artificial atom available for experiments, in a dfined and stable way, safely contained in a host crystalthis respect the host crystal plays here the role of a long tatom or ion trap. In special configurations it allows us alsodeliberately charge or electrically excite an embedded QDexpose it to high electric fields. Control over the spontaneemission rate can be obtained by embedding QDsmicroresonators,28 which can be also part of the host crystaAll those arrangements become possible through the tremdous efforts which have been put into material science othe last decades. QDs as artificial atoms look therefore, fthe technological point of view, very promising.

There are however also important differences betwfree atoms and QDs. Those originate mainly from the fthat semiconductor QDs typically consist of thousandsatoms arranged in a finite size lattice, which again is ctained in a host crystal with different composition. In co


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trast to free atoms, QDs are therefore subjected to thecific elementary excitations of such composite solidInteractions with phonons in particular have been shownleave their fingerprints in the optical spectra of QDs. In tfield of self-assembled QDs, broad~multi! LO phonon reso-nances have been observed in inhomogeneously broadensembles by near resonance excited photoluminesc~PL! as well as PL excitation~PLE! experiments.13,14 Thoseresonances may be caused partly by phonon-assisted abtion and absorption via excited states followed by resoninter level relaxation. In single self-assembled QDs absotion by excited states and sharp phonon resonances caobserved independently.10,29 Also differences between QDand atoms concerning the accuracies of the eigenenerlevel degeneracies, and selection rules arise finally fromfact that QDs are tiny crystals with partly specific shapes.atom is a Coulomb bound object consisting of an exact nuber of elementary particles. The resulting energy levelstherefore precisely defined. If we start building clusters wincreasing numbers of atoms, for some cases stable obwith specific structure may be formed, like the famous C60

molecule. For larger objects however self-assembly usufails to be accurate. With increasing numbers of atomsenergy differences between various configurations becomore and more insignificant. Nucleation and crystal growhappen not strictly in thermodynamic equilibrium but ivolve also kinetic processes. During the formation of Qsize fluctuations in the ensemble will therefore occur. Sinthe transition energies of QDs are determined by size qutization, those fluctuations lead to inhomogeneous broading effects in the optical spectra of ensembles, which seembe unavoidable so far. Furthermore the transition energieQDs are subjected to thermal drift, which is caused in fiorder by the temperature dependence of the band gaps orespective semiconductors. The shell structure of electrolevels and their degeneracies depend on the shape of theand the band structure of the semiconductor~i.e., valley de-generacy!. For the case of quantum discs with parabolic cofinement potential the degeneracies of the shells are give2n, wheren is the principal quantum number in the syste~n51, 2, 3,...5s, p, d,...!. Optical interband transitions arallowed between states with equal n in the valance and cduction band. One has to add here, that the above assigninto shells accounts for the effective mass picture. For msemiconductors the underlying Bloch functions at thelence~conduction! band edge arep ~s!-like.

C. How to make quantum dots

From the technological point of view the most direct aflexible way to produce QDs is to laterally pattern existitwo-dimensional~2D! quantum well ~QW! structures. Bye-beam lithography and reactive ion or wet chemical etchinanostructures can be produced deliberately down to latdimensions of about 30 nm.30 Alternatively differentschemes employing intermixing by focused laser beams31 orfocused ion beams32 have been demonstrated. So far, hoever, the incorporation of impurities and structural imperfetions during the process of lateral patterning cannotcompletely avoided. As a consequence, nonradia


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7792 J. Chem. Phys., Vol. 112, No. 18, 8 May 2000 A. Zrenner

recombination will be enhanced, and, within an arrayquantum dots, fluctuations in composition and size will apear. In optical experiments, zero-dimensional confinemin patterned structures was demonstrated in single quandots as well as in arrays of dots. The advantage of thoseto bottom approaches is clearly the ability to produce Qwith predefined dimensions, which allow systematic studof quantum confinement effects as a function of structsize.31,33 Both the magnitude of the obtainable interlevel eergy separation and the photoluminescence yield remhowever to be improved to match the performance of alnative schemes described below.

Isolated and well-developed local potential minimaQWs are referred to as so-called natural QDs.2 Disorderedsystems with high amplitude potential fluctuations leadlocalization and center-of-mass quantization of excitons. Tdiscrete energy states of such systems can be well-sepafrom continuum states. Natural quantum dots can beserved, for example, in narrow QWs, in which case thederlying disorder originates from well-width fluctuations. Iterface roughness with an amplitude of at least omonolayer~ML ! in growth direction is known to appear isemiconductor heterostructures.34 The lateral length scale othose fluctuations covers a broad range. Interface fluctuatin the range of one or two monolayers result in huge latevariations of the exciton energy if narrow QWs are consered. Excitons can be localized in regions with locally ehanced well-width. On this basis it is justified to describenarrow QW as a disordered array of quantum dots withbitrary dimensions. The amplitude of those potential vartions are in the range of 50 meV for a 10 ML QW.2 The mainadvantage of natural QDs as compared to lithographicdefined systems arises from the fact that no additionalfects are introduced during their formation. Up until now itthe cleanest zero-dimensional model system availablethe most narrow emission and absorption linewidth obserso far for semiconductor QDs.27

In many ways more advantageous and hence moreevant for applied physics are self-assembled QD formedStranski–Krastanow growth.35 Those are created by a themodynamic instability during the two-dimensional growthstrained layer systems. The basic phenomenon is explahere briefly by the example of InAs growth on a GaAs sustrate. For thin InAs layers below the critical thickness, fia pseudomorphic layer with the lateral lattice constantGaAs is formed. With increasing InAs layer thickness taccumulated compressive strain can no longer be accomdated in a two-dimensional arrangement and the systemunstable against the formation of three-dimensional, cohently strained islands with reduced strain energy buthanced surface energy. This process leads—within cerlimits and under the condition of sufficiently high In surfamobilities—to the formation of approximately equally-sizedislocation-free InAs islands, so-called self-assembled InQDs, which are arranged on top of the remaining part oftwo-dimensional InAs layer, the so-called wetting layer.dividual QDs nucleate typically in disordered arrays waverage nearest-neighbor separations down to 30 nm, csponding to QD surface densities up to about 1011 cm22.


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The diameter~height! of the QDs may be in the range of 1nm ~5 nm! and can be adjusted within certain limits bchoosing appropriate growth conditions. The exact shapthe resulting QDs is still under discussion; depending onmaterial system and growth conditions, it can be pyramidmultifacetted, or more lens-shaped. Furthermore the shcan change during overgrowth by diffusion and segregatFor optoelectronic applications, overgrowth with nominaunstrained barrier material is mandatory in order to avsurface recombination effects. The obtainable interlevelergy separations can reach values of the order of 100 mapplications at room temperature such as lasers are therpossible.36–39 Self-assembled QDs can be realized by mlecular beam epitaxy~MBE! or metal organic chemical vapor deposition~MOCVD! on a variety of different semiconductor material systems, such as In~Ga!As/GaAs,40

InP/InGaP,41,42 GaSb/GaAs,43 InSb/GaSb,44 ~Si!Ge/Si,45,46

InAs/Si,47 InAlAs/AlGaAs,48 and also II/VI ~Ref. 49! andIV/VI ~Ref. 50! materials.

Closely related to Stranski–Krastanow growth are strinduced QDs. Self-assembled islands are deposited herstressors on top of a surface near QW structure. Withstressor in close proximity to the QW, three-dimensionstrain-induced confinement of excitons is obtained herethe QW.51 In a more general way this concept has been deonstrated already earlier with electron beam patterstressors.52

Only briefly discussed here are semiconductor nanoctals, which can be realized by precipitation out of organliquids or incorporation and nucleation in oxides and glassCdSe nanocrystals in particular can be also coated inliquid phase with thin layers of ZnS or CdS. For the resulticore–shell nanocrystals~CdSe–ZnS or CdSe–CdS! the lu-minescence yield can be as high as 80%. By control ofnanocrystal size~typically in the range of 1.5 to 12 nm!, theemission wavelength can be shifted over the entire visrange. For further information the interested reader isferred here to existing overview articles.53

For sophisticated technical applications of QDs, growon predefined positions has to be controlled. To a cerdegree this goal can be achieved by growing stacks ofassembled QDs. In such structures strain driven vertalignment of QDs is observed.54 Attempts to control the firstlayer of QDs have been conducted already by growthweakly patterned substrates.23 In pioneering work, the for-mation of three-dimensional QD crystal was reported, whbecomes possible in host crystals with anisotropic elaproperties.50

Highly-defined incorporation of QDs on predefined psitions can be achieved by epitaxial growth on strongly pterned substrates. Using organometallic chemical vadeposition in inverted tetrahedral pyramids on$111%B GaAssubstrates, the local enhancement of the GaAs layer thness at the tip of the pyramid results in the formation olens-shaped QD structure.55

A totally planar approach to QDs is cleaved-edovergrowth.56 The well-controlled growth of a two-dimensional QW is performed here first on a$100% GaAssurface. Afterin situ cleavage, QW growth is resumed on th


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atomically smooth$110% cleavage plane. With this seconstep of growth a T-shaped quantum wire is formed. Afcleaving the sample for a second time, now along the o$110% cleavage plane, QW growth is again resumed ansecond quantum wire is formed. At the intersection of bwires a single QD is formed with the precision inherentMBE.57 By growing two or more QWs in the first step, alswell defined coupled QDs or QD superlattices canobtained.58 So far, cleaved-edge overgrowth has the bento produce QDs with the best definition and control; the mjor drawback is however the poor QD confinement enerwhich is currently only about 10 meV. A very elaborate aproach towards ensembles with reduced size fluctuationsploys atomic hydrogen assisted MBE growth.59 Step bunch-ing is induced here by atomic hydrogen on patterned hiindex substrates. Self-organized growth on such a subswas shown to result in QD arrays with very small size flutuations.

III. SINGLE DOT SPECTROSCOPY FROM ATECHNICAL POINT OF VIEW

The purpose of single QD spectroscopy is to exploreoptical properties of an isolated dot under various conditioThe methods and instrumentations used for optical sinQD analysis are therefore focused more on spectroscopynot so much on microscopy. As pointed out already inintroduction, spatial resolution is required here primarilyproperly isolate QDs for detailed spectroscopy. Imagingindividual dots by scanning near field microscopy or concal microscopy has been performed in the early dayssingle QD spectroscopy;3,4 in the meantime however AFMand TEM techniques turned out to be more attractive. Thmay be destructive or may require special samples, butare able to also supply information about the topographystructure of QDs.

For the community of single QD spectroscopy the revant length scale and hence the required spatial resolutithe interdot separation and not the size of an individual dTechnical considerations have to include, therefore, thetical system as well as the QD system under investigatWhere the optical system is concerned, far-field methodsconfocal microscopy allow us to obtain, even in cryogeenvironments, spatial resolutions in the range of 1mm;16

special versions like the solid immersion lens techniquelow us to extend this range down to 0.25mm.17,18 The mainadvantage of those techniques is the high collection eciency and the ability to get images without scanning. Ltemperature versions of confocal microscopes can betained, for example, by using small flow cryostats with thwindows and~corrected! long working distance objectivelenses held at room temperature.31 Alternatively, the objec-tive lens can be mounted inside the cryogenic environmtogether with anx-y-z positioner. This approach allows fomore flexibility in choosing the working distance and nmerical aperture of the objective lens. Complete unitsalso be built small enough to fit into commercial supercoducting magnet systems.60

For higher spatial resolution the concept of near-fioptical spectroscopy has to be applied. In the area of exp


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mental QD physics near-field spectroscopy through apertis used in different forms. For special cases the obtainaspatial resolution is extended down to 50 nm, however,the cost of optical throughput. The basic concepts for optnear-field microscopy were established about 15 yeago.61–64 Aperture scanning near-field optical microsco~SNOM! is a technique, which is designed to be complemtary to STM or AFM. Different versions of SNOMs are summarized in Ref. 65. Since the ability to create imagesscanning is not often required for optical spectroscopysingle QDs, also the use of fixed nano-apertures has becvery popular. Those are produced by patterning an opametal film on the sample surface by either electron beam5 orAFM lithography.10 Nano-apertures are typically producein larger matrices, which increases the chance to obtainaperture with exactly one QD underneath. Optical spectrcopy with excitation and collection through the nanaperture can be performed with the above described confsetups. The main advantages of near-field spectroscthrough nanoapertures arise from the inherent and absostability concerning the arrangement of the aperture relato the QD. This is primarily important for investigations asfunction of temperature or magnetic field. Even with an aerture size in the region of 100 nm single QD spectroscocannot be achieved with high density QD arrays in thegime of 300mm22. As pointed out in the introduction, lowdensity ensembles of QDs have to be prepared by spegrowth conditions for the case of self-organized dots11 or, inthe case of colloidal species, by dilution. An example foMBE-grown low density array of self-assembled InGaAQDs is shown in Fig. 1.

Also, cathodoluminescence19 and STM-induced lumi-nescence~STML!20–22 have been shown to be powerfutools, suitable to reach the limit of single dot spectroscoSTML in particular has the potential for extremely high sptial resolution approaching the regime below 50 nm, if cafully designed structures are used, which allow for low eergy charge injection from a STM-tip.21 Compared tooptically excited luminescence, STML has however tdrawback that possible unipolar charging effects during etric excitation cannot be avoided.

IV. CURRENT EXPERIMENTAL RESULTS

This section is intended to provide a brief overvieabout achieved experimental results to date in the areoptical single QD spectroscopy. Within the current pagraph it is of course not possible to reference all existwork in this field. Some important topics are summarizedthe following

The observation of narrow emission lines from isolatQDs was first reported in 1994. The work in this early stawas concerned mostly with the demonstration of quantresolved spectroscopy on single QDs. Spatially resolvedvestigations have been performed on natural QDs using cfocal spectroscopy on coupled GaAs/AlAs QW structure2

and on narrow GaAs/AlGaAs QWs by means of SNOM3 orconfocal spectroscopy.4 At the same time narrow line emission from lithographically defined sub-ı´m mesa structurescontaining self assembled InAs/GaAs QDs w


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demonstrated.66 Soon after, narrow line emission on thsame system was also observed in cathodoluminescexperiments.19 All those experiments revealed optical linwidths in the region of 100meV. At this point it was alreadyclear that quantum resolved spectroscopy had found its pin semiconductor spectroscopy. What followed is no lonreviewed in historical but in topical order.

The properties of QD ground states have been invegated typically in the one-exciton limit. This means, thunder the condition of weak optical excitation, at maximuone exciton per time is contained in the QD. One of the fitopics addressed was the temperature dependence of thtical linewidth. From near-field PL experiments througnano-apertures, the QD lines were shown to be homoneously broadened. In the limit of low temperatures thlinewidth could not be fully resolved experimentally; upplimits could be however set to about 30meV. With risingtemperature the line width was shown to increase andvelop a linear temperature dependence, consistentacoustic phonon scattering.27

FIG. 1. AFM images of self assembled In0.4Ga0.6As QDs grown by MBE ona GaAs substrate at a temperature of 530 °C. Nominally 7.5 monolayeIn0.4Ga0.6As, slightly above the critical thickness for Stranski–Krastangrowth have been deposited under the condition of nonrotated substrateresulting gradient in In0.4Ga0.6As coverage leads to a variation of the Qsurface density across the wafer~upper part:;100mm22, lower part:;20mm22).


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Polarization dependent spectroscopy further revealesome cases a fine-structure splitting of the excitonic grostate, which can be traced back to asymmetries in theshape. At zero magnetic field the ground state emissiofound to be a doublet of different linear polarized lines, serated by the electron hole exchange energy. For highly cfined QD excitons this splitting is enhanced as comparedsystems of higher dimensionality. The splitting was foundbe in the range of 20–50meV for natural QDs5 and 200meVfor self assembled InGaAs/GaAs QDs.11 As a result of an-isotropy mediated mixing of bright (Dm51) and dark exci-ton states (Dm52) the originally dark states gain finite oscillator strength and appear therefore in the spectra. In fimagnetic fields both states undergo a Zeeman splitting wtheir characteristicg-factors,ge1gh for the bright excitonand ge2gh for the dark exciton. Spin-resolved magnetspectroscopy on a QD with reduced symmetry allows thefore the determination of both magnitude and sign ofassociatedg-factors.11

In systematic studies on etched InGaAs/GaAs QDsciton g-factors have been found to depend strongly onQD size. Mixing effects in the valence band have been mresponsible for this phenomenon.67 For the same system alssystematic studies of the diamagnetic shift and the deduexciton binding energy are available.68 Also hyperfine inter-actions of QD excitons with nuclear spins are reported iso far unique study on natural QDs.12 First spin coherenceexperiments exist for CdSe QDs.69

Not many results have been published on electric fiedependent properties of single QDs. First Stark effect msurements are available for InGaAs self-assembled Qwhich suggests the existence of excitons with permanenpole moment in accordance with model calculations.70 Sys-tematic studies of single electron charging effects detecteoptical absorption are available for QD ensembles71

Magneto-PL experiments on a larger number of single Qlines seem to suggest that a substantial fraction of the QDcharged, probably by residual impurities.72 Systematic stud-ies on tuneable single QD structures have been until now,available.

Of particular importance for fundamental research apossible applications of QDs are spectroscopic results infew exciton limit. Clear signatures of the biexciton statesingle QDs have been obtained by power dependent PLresonant two-photon absorption experiments in natuQDs.73 Since then biexciton binding energies of about 3 m~20 meV! have been reported for various types of III/V~II/VI ! based QDs.10,74–77 Higher exciton occupancies leavemore and more complicated fingerprint in the emission sptra of single QDs. In QDs, Pauli blocking allows for ontwo excitons in s-shell; additional excitons have to becommodated in the higher shells. With higher occupancthe associated renormalization effects in the optical transienergies gain more and more importance. Emissions fdifferent shells appear therefore at specific energies, whare different and characteristic for each given occupancythe QD. Multiexciton emission spectra hence consist olarge number of emission lines.9,10 PL data typically displaysalso a time average over many statistically varying confi

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rations. An example for typical power-dependent PL dfrom a single QD~isolated via a nano aperture! is shown inFig. 2.

Excitation spectroscopy allows to determine the bareenergy levels in the one exciton limit. Experimental resuon single QDs10,28,78,79and ensembles reveal, besides thecitonic transitions among the shells, also strong phononlated lines. Until now it is not completely clear in how faphonon assisted absorption or resonant Raman scatterinto be regarded as the dominant interaction process withlattice.

Only recently has the field of coherent nonlinear sptroscopy on single QDs been opened. First results shstrong similarities between natural QDs and atoms to whcoherent optical interactions are concerned.6 Dephasing~30

FIG. 2. Power dependent PL spectra from a single QD isolated by nearspectroscopy through a nano-aperture. At low excitation powerPL only thesingle exciton decay from thes-shell is observed~1X!. At elevatedPL

occupancies with two and more excitons are realized. The sequentialciton decay leads to the appearance of the biexciton line~2→1! in thep-shell ~2X!. The decay of higher occupancies~for example 3→2, 4→3!leads to new emission lines in the spectral region of thep-shell and addi-tional, further renormalized lines in the region of thes-shell.


a

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ps! and energy relaxation times~20 ps! are found to beroughly in the same regime.

V. THE STATUS OF THEORY

The optical properties of semiconductor QDs as quoin the experimental part above have attracted considerinterest in the field of theory. The phenomena to be descriand understood fall into a vast variety of different areas.

Most fundamental in this context is the quantitative dscription of the single particle energy levels in a QD wiarbitrary shape and composition. Solutions for this problcan be obtained from numericalk•p calculations, which areperformed typically in a eight-band version.80 Besides bareband structure parameters and bulk elastic propertiesshape and composition of the QD enters here as main inparameter. Dots shaped like pyramids or truncated pyramaccording to the results of various structural investigatioare typically treated here. Using elastic continuum modand taking into account piezoelectric effects the profilesthe relevant band edges are calculated in a first step in oto get a realistic confinement potential of the QD. The eigstates in the conduction- and valence-band are subsequobtained within the framework of an effective mass eigbandk•p calculation. On the basis of those eigenstates inand intraband transition energies, oscillator strengths asas polarizations for the various transitions, and also excground state binding energies can be obtained. Such olatedk•p approaches have been performed for strainedunstrained QD systems, such as self-assembled In~Ga!As orInP QDs,80–83 nanocrystals,84 and QDs formed by cleavededge overgrowth.85 The general concept of thek•p approachis known to yield accurate results for single particle systemIt is also a well-suited method if the impact of small pertubations such as electric fields or variations in compositionsize has to be explored. As an interesting, but still not widspread method to calculate single particle electron and pton states in a QD in effective mass approximation is alsoboundary element method, which is briefly referenced her86

As method beyond the effective mass approximation,empirical pseudopotential theory is also applied to QDAlso here self-assembled QDs as well as nanocrystals hbeen treated.87–90 One of the most interesting and challening subjects in theoretical QD physics is the descriptionpartially occupied dots. Optical excitation of a QD resuunder the condition of sufficiently high pumping poweroccupancies with several excitons. As compared to the sinparticle transition energies of an empty QD, the optical trasitions in the few particle system appear therefore renormized. Appropriate descriptions of the resulting transition eergies for occupancies with different numbers of excitorequire the diagonalization of an appropriate few partiHamiltonian. For rectangular quantum boxes with infinhigh barriers exact diagonalization techniques have bapplied.91 For quantum discs with cylindrical symmetry boexact numerical diagonalization techniques8,92 and semiana-lytical approaches93 have been performed. As comparedthe above describedk•p approaches, the actual shape of tQD and effects of strain are not considered here. Insteamore simple parabolic confinement potential in radial dire

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tion is assumed and band coupling is neglected. For leshaped QDs, such as self-assembled InGaAs/GaAs dotsresulting accuracy seems to be however sufficient, in partlar if the magnitude of few body corrections to otherwiexperimentally known single particle states has to beplored.

Few-particle effects in QDs have been also studied bdensity-matrix approach that explicitly accounts for excitoexciton, as well as exciton–carrier interactions.94 This ap-proach is in particular well-suited to describe the nonequirium carrier dynamics in semiconductor QDs.

Besides the so far discussed quantum confined stateQDs, also the existence of intrinsic gap states has beenposed theoretically using multibandk•p theory.95 Thesestates are shown to originate from Shockley-type surfstates and they are predicted to appear in some QD systdepending on the band structure of involved semicondumaterials. The detailed analysis of single QD ground starevealed fine-structure in the excitonic emissispectra.5,11,96 For QDs with asymmetric confinement potetial it can be shown that also the radiative decay of dexcitons, which is forbidden in cubic symmetry, contributto the spectra. This additional fine-structure can be descrin terms of a spin Hamiltonian which describes thee–h ex-change energy and the interaction of the exciton spin wthe external magnetic field.96,97

Relaxation processes in QDs have been believed inearly days of QD physics to be slowed down as comparesystems of higher dimensionality. The break down of acotic phonon assisted relaxation in QD structures was shotheoretically to be responsible for the so-called phonon boneck, which would prevent carriers in excited levels frofast relaxation to the QD ground state.98 The physical originfor this phenomenon comes from the reduced possibilifor k-conservation for acoustic phonon scattering in QDslater work however it was shown, that other mechanislike Coulomb scattering, still could guarantee for rapid aefficient relaxation to the ground state.99

In a number of QD systems, like CdSe nanocrystals100

and self-assembled InP dots,101 the PL intensity for cw exci-tation shows random telegraph behavior, which is oftenferred to as blinking. For CdSe nanocrystals this phenoenon was shown theoretically to be consistent withquenching by nonradiative Auger recombination in part-tiionized configurations.102 Recent advances in single Qspectroscopy allow now for direct measurements of themogeneous linewidth of QDs. A description of the tempeture dependence of the obtainable linewidth is an imporissue for both applied and fundamental research. The tperature induced line broadening of the QD emission willlimits to the maximum obtainable gain in QD lasers. Tminimum obtainable line width in QDs is mostly limited bphonon scattering. Primarily LA acoustic but also opticphonons give rise to scattering events among the varlevels of a QD. For large QDs with small interlevel spacinthe LA acoustic phonon scattering rate is dominant and leto a lifetime limited linewidth which is proportional totemperature.27 For small QDs and large interlevel spacinphonon scattering is predicted to be less effective. In


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regime lattice relaxation associated with the optical trantions has been shown to be a dominant broadenmechanism.103

An important issue for near field spectroscopy on sinQDs are tip–QD interactions. Those have been modeledusing Bethe’s theory104 to describe the optical field distribution close to the aperture of the tip and Fermi’s golden rto calculate the transition rates in the QD. As a result lighole vs heavy hole transitions are found to be selectivenhanced, and forbidden transitions are expected to gainsiderable oscillator strength as compared to far-fispectra.105 SNOM is therefore a technique, which is basicacapable to map out exciton wavefunctions in a QD.

VI. FUTURE PROBLEMS AND FINAL REMARKS

Reviews like this can never be complete, and certaione or the other valuable contribution to the field was nincluded. But even to the current extent, the sections abclearly demonstrate that the fast growing field of single Qspectroscopy has made its way to one of the most fascinaareas of semiconductor physics. It is a field which has ovlap with fundamental physics, both experimental and thretical, with material science, and with applied researchwe want to locate future problems, we have to concentratcourse on all of the just mentioned disciplines.

Although material science has provided us with all tQD systems we are currently working on, the need for Qensembles with substantially reduced inhomogeneous brening is quoted here in the first place. Further demands ccern the regular or in any way precise arrangement of Qwhich eventually would allow also for the critical couplinof dots, even in ensembles.

In the field of fundamental physics the many body apects with all the associated nonlinearities are regarded tone of the key issues, as are the sources of decoherewhich are relevant in the area of ultrafast spectroscopy,for all related applications such as quantum computing. Wall those inputs we can expect to get even better QD laand also conceptually new devices such asQ-bits in the formof coupled dots or deterministic photon sources on the bof dots in microresonators. For analysis and for controlphotonic connection to the outside world, spatially resolvspectroscopy will be one of the experimental key techniquwhich allows us to have a really close look at single quantdots.

ACKNOWLEDGMENTS

I would like to thank Evelin Beham, Frank Findeis, anGerhard Abstreiter for their support. Also I would like tacknowledge financial support by the Deutsche Forschungemeinschaft via the SFB 348.

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