Leo Kouwenhoven and Charles Marcus

PHYSICS WORLD �� JUNE 1998

By confining electrons in three dimensions inside semiconductors, quantum dots canrecreate many of the phenomena observed in atoms and nuclei, making it possible toexplore new physics in regimes that cannot otherwise be accessed in the laboratory

Quantum dots are man-made "droplets"of charge that can contain anything froma single electron to a collection of severalthousand. Their typical dimensionsrange from nanometres to a few microns,and their size, shape and interactions canbe precisely controlled through the use ofadvanced nanofabrication technology. The physics of quantum dots showsmany parallels with the behaviour ofnaturally occurring quantum systems inatomic and nuclear physics. Indeed,quantum dots exemplify an importanttrend in condensed-matter physics inwhich researchers study man-madeobjects rather than real atoms or nuclei.As in an atom, the energy levels in aquantum dot become quantized due tothe confinement of electrons. With quan-tum dots, however, an experimentalistcan scan through the entire periodic tableby simply changing a voltage. Many of these phenomena can bestudied by allowing single electrons totunnel into and out of the dot, since thisreveals the quantized energy levels of thedevice. Experiments at our labs at theDelft University of Technology in the Netherlands andStanford University in Calfornia have used this techniqueto probe various properties of quantum dots. What ismost exciting is that many of the quantum phenomenaobserved in real atoms and nuclei - from shell structure inatoms to quantum chaos in nuclei - can be observed inquantum dots. And rather than having to study differentelements or isotopes, these effects can be investigated ina quantum dot by simply changing its size or shape.

Symmetric quantum dots as artificial atomsMany properties of quantum dots have been studiedthrough electron tunnelling, a quantum effect that allowselectrons to pass through a classically forbidden potentialbarrier. But electron tunnelling to and from a quantum dotis dominated by an essentially classical effect that arisesfrom the discrete unit of charge on an electron. If thetunnelling to the dot is weak - which happens, for example,when relatively high potential barriers separate the dotfrom a source and drain of

electrons - the number of electrons onthe dot, N, is a well defined integer.

Any movement of electrons throughthe dot requires this number to changeby one. But the Coulomb repulsionbetween electrons means that theenergy of a dot containing N+1electrons is greater than one containingN electrons. Extra energy is thereforeneeded to add an electron to the dot,and no current will flow until thisenergy is provided by increasing thevoltage. This is known as the Coulombblockade.

To see how this works in practice,Seigo Tarucha and colleagues at NTTin Japan and one of us (LK) and co-workers at Delft have studied whathappens in a symmetric quantum-dotstructure (figure 1). The structurecontains a quantum dot a few hundrednanometres in diameter that is 10 nmthick and that can hold up to 100electrons. The dot is sandwichedbetween two non-conducting barrierlayers, which separate it fromconducting material above and below.By applying a negative voltage to ametal gate around the dot, its diametercan gradually be

squeezed, reducing the number of electrons on the dot -one by one - until there are none left.

This makes it possible to record the current flow as thenumber of electrons on the dot, and hence its energy, isvaried. The Coulomb blockade leads to a series of sharppeaks in the measured current (figure 2a). At any givenpeak, the number of electrons on the dot alternatesbetween N and N+1. Between the peaks, the current iszero and N remains constant. The distance betweenconsecutive peaks is proportional to the so-calledaddition energy, Eadd, which is the difference in energybetween dots with N+1 and N electrons.

The simplest model to describe a quantum dot, the so-called constant-interaction model, assumes that the Cou-lomb interaction between the electrons is independent ofN and is described by the capacitance, C, of the dot. Inthis model, the addition energy is given by Eadd = e2l C +∆E, where e is the charge on the electron, and ∆E is theenergy difference between one quantum state and thenext. Adding a single electron to the dot thereforerequires a constant

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charging energy, e2/ C, plus the difference in energy betweenthe quantum states.

Despite its simplicity, this model is remarkably accurateand allows us to describe the measurements in more detail.The first peak on the graph marks the energy at which thefirst electron enters the dot, the second records the entry ofthe second electron and so on. But the spacings between thepeaks are not constant, and significantly more energy isneeded to add the second, sixth and twelfth electrons.

We can picture this in terms of two-dimensional electronorbits, since the shape of the quantum dot restricts electronmotion to this plane (figure 2b). The orbit with the smallestradius corresponds to the lowest energy state. This state hasan angular momentum of zero and - as with atoms - can onlycontain two electrons with opposite spin.

This means that the charging energy, e2/C, is enough toincrease the number of electrons on the dot from one to two.But extra energy, ∆E, is needed to add a third electron, sincethe innermost orbit will be full and the electron must go intoa higher energy state. Electrons in this orbit have an angularmomentum of ± 1 and two spin states, which means that thisshell can contain four electrons. This shell will be full oncethe dot contains six electrons, and so extra energy is neededto add the seventh electron.

The third shell presents a special case if the confiningpotential is parabolic in the radial direction, because thisintroduces a radial quantum number. States in this shellcan have an angular momentum of 0 and a radial quantumnumber of 1, or an angular momentum of ±2 and a radialquantum number of 0. Together with the spin states, thismeans that the third shell can contain six electrons andwill be full when N= 12.

This sequence, N= 2, 6, 12, 20 and so on, provides the"magic numbers" of electrons in a circularly symmetricharmonic potential confined to two dimensions. Theenergy states for such a system were calculated in the1920s by Charles G Darwin and independently byVladimir Fock.

The addition energy (inset to figure 2a) also showssmaller peaks at N= 4, 9 and 16. This substructure reflectshow the interactions between electrons can influence thefilling of energy states. In atomic physics these effects areformulated as Hund's rules, which state that electronsenter a shell with parallel spins until the shell is half full,and then enter with opposite spin. The sequence N= 4, 9and 16 corresponds to a half filling of the second, thirdand fourth shells, where the total spin of the electronsreaches a maximum value.

This picture is summarized in a new "periodic table" oftwo-dimensional elements (figure 2c). The rows areshorter than those of the familiar periodic table becausethe dot is defined in two dimensions rather than three.

But these dots are more useful than simply providinganalogues of real atoms, since their larger and morecontrollable size allows experiments to be carried out inregimes that cannot readily be accessed for real atoms.For example, the electron orbits in both atoms andquantum dots are altered by a magnetic field, but theeffect of a I T magnetic field on a quantum dot iscomparable with the effect of a one million tesla field on areal atom. Such high magnetic fields cannot be producedin the lab.

36 PHYSICS WORLD JUNE 1998

for example, experimental and numerical re-sults indicate that interesting transitionsoccur between the energy states on the dot.These are related to changes in the exchangeenergy, which accounts for the interactionsbetween electrons with parallel spins.

Loss of symmetryIt is worth reiterating that the shell structuresobserved for both the symmetric quantumdots described above and real atoms resultfrom symmetry. For the dots, it is the circularsymmetry of the pillar that leads to theperiodic table in figure 2, while for real atomsthe spherical symmetry of the nuclearpotential leads to the familiar periodic tableof elements. While our familiarity with atomsand shell structure make symmetric quantumsystems seem ubiquitous, in fact just theopposite is true. By far the more commonsituation is that systems lack spatialsymmetry. This is particularly the case forman-made quantum systems. An interestingset of physical laws has recently beenuncovered for quantum systems that lacksymmetry. These asymmetric quantumsystems are characterized by universalstatistics, which were first recognized innuclear physics and studied theoretically byEugene Wigner, Freeman Dyson and othersin the 1950s and 60s. Such universalities arenow a familiar topic in mesoscopic physics.

The basic idea is that all disordered orirregularly shaped quantum systems fall intoa few broad classes that are distinguished byany symmetries that remain in the system,such as symmetry under time-reversal. Thequantum systems within each class sharevarious statistical properties that describe, forexample, their energy-level structure orscattering behaviour.

A surprising aspect of the universal statistics relating toquantum systems is that they seem to be connected withchaotic dynamics observed in the equivalent classicalsystem. For example, the famous "stadium billiard" (atwo-dimensional region bounded by hard walls) is asimple shape, consisting of semicircles connected bystraight edges. Nonetheless, the classical trajectory of aball bouncing in a stadium produces random motion foralmost any initial condition, as proved by LeonidBunimovich (figure 4a). The corresponding quantumsystem (figure 4b) has wavefunctions and energy levelsthat share universal features with other classically chaoticsystems. This example has since become a mainstay ofresearch into the quantum aspects of classical chaos, asubject now known as "quantum chaos".

It is presumably the same universal behaviour ofquantum chaos that leads to the random, but reproducible,fluctuations that are observed in the conductance ofmicron-scale metals at low temperatures. Such"mesoscopic" systems are small enough to exhibitquantum interference, but large enough to contain arandom distribution of scatterers such as impurities anddislocations. These scatterers act as a source of chaos,causing electrons to move along random paths thatdepend on the initial conditions, just like a ball bouncingin a pinball machine.

PHYSICS WORLD JUNE 1998 37

Indeed, studies of the electronic orbits of quantum dotsin a magnetic field have provided further support for theDarwin- Fock energy spectrum. And at high magneticfields it is even possible to study the quantum Hall effectin quantum dots. For example, Ray Ashoori of theMassachusetts Institute of Technology in the US hasmeasured how the electron states evolve as the magneticfield is increased from zero into the quantum HO regime.And in 1991 Paul McEuen - then working with MarcKastner at MIT and now at the University of California atBerkeley - showed that the simple model of constantCoulomb interactions no longer applies at high magneticfields. In this regime, the interactions and confinementshould be treated on an equal footing, and a moresophisticated calculation is needed to get the physicsright.

We have so far focused on electron transport in thelinear regime, where the voltage across the dot is smallcompared with the charging energy and the separationbetween quantum states. But this only allows us to studythe ground-state energies of the artificial atoms. Byincreasing the voltage further, a "transport window" canbe opened across the quantum dot, allowing excited statesto contribute to electron transport (figure 3a).

This makes it possible to measure the energy spectrumof excited states (figure 3b). Such excitation spectra,together with the effect of a magnetic field, can becalculated numerically for up to six electrons. Atmagnetic fields of around 1 T,

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altering the gate voltages. Dots of this type can containany- thing from a few to several thousand electrons.

In 1996 the statistical properties of tunnelling transport inthis type of quantum dot were measured by one of us(CNI) and colleagues at Stanford, and independently byAlbert Chang and co-workers at Bell Labs, New Jersey,US. As in the earlier experiments, the movement ofelectrons is dominated by the Coulomb blockade. Theconductance therefore shows a series of peaks as the gatevoltage is increased (figure 6a). Two aspects of themeasured peaks are expected to follow universal statistics:the distribution in peak heights and the distribution ofspacings between the peaks.

Let us first consider the distribution of peak heights. Atlow temperatures (- 100 millikelvin), the heights of theCoulomb peaks are inversely proportional to temperatureand proportional, on average, to the series conductance ofthe two leads. However, changes in the shape of the dot -or in the number of electrons it contains - lead to randomfluctuations in the coupling between the dot and the leads,and these should result in a universal distribution of peakheights. Rodolfo Jalabert, Doug Stone and Yoram Alhassidat Yale University in the US calculated this distribution in1992, using ideas that lead to the well known Porter-Thomas distribution of scattering widths observed incompound nuclear scattering.

Measurements of the distribution in peak heights can bemade by altering the shape of the quantum dot to collectdata on large numbers of different - but similar - quantumsystems. Distributions measured in this way agree well withtheory (figures 6b). As long as the dot generates chaoticdynamics, the distribution is independent of the size, shapeand transmission of the single-channel leads. It onlychanges in the presence of a magnetic field, since thisbreaks the symmetry under time-reversal, and here againthe theory matches the experimental results (figure 6c).

What about the spacing of the Coulomb peaks? A simpleuniversal prediction can be made for any chaotic quantumdot by simply applying the approach taken earlier for sym-metric dots. In this case the charging energy needed to addelectrons to the dot is simply added to the universaldistribution for spacings between quantum levels, which iswell known from other quantum systems. Indeed, these so-called Wigner-Dyson distributions are the most famousexample of the universality of quantum chaos, and seem todescribe everything from the spacing between resonancesin com- pound nuclear scattering to single-particle energylevel spacings of classically chaotic dynamical systems.

According to the charging-energy model, the distributionof peak spacings should follow a Wigner-Dysondistribution with a width that depends on the properties ofa particular dot. This theoretical distribution shows twomain features: a single spike due to even-numberedelectrons that only require a charging energy to enter thedot, and a broader peak due to odd-numbered electrons thatrequire extra energy to enter the next quantum level (figure6d).

However, this simple scheme requires a series ofquestion- able assumptions. It supposes that the chargingeffects can be described by a single, slowly varyingcharging energy. It also assumes that the electron spinsdictate how the energy levels are filled, and thatfluctuations in the capacitance of the dot are small. Thislast assumption has been called into question by Uri Sivanand colleagues at the Technion in Israel, who haveinvestigated the peak spacings in quantum dots and wires.

PHYSICS WORLD JUNE 1998

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In the case of quantum dots, however, chaos is notgenerated by scatterers, since the mean free path forelastic scattering is several times larger than the dot.Although the interactions between electrons - just likethose between nucleons in the nucleus - can lead toquantum chaos and universal statistics, experiments tostudy these effects do not only rely on these interactionsto generate chaos. Instead, the dots are designed so thattheir shape lacks all spatial symmetry.

In practice, these dots can readily be fabricated bylaterally confining a two-dimensional sheet of electronscreated at the interface between layers of gallium arsenideand aluminium gallium arsenide (figure 5). Several metalgates are used to confine the electrons in an irregularshape, and individual electrons can enter or leave the dotthrough two tunnelling barriers or "leads". Both the shapeof the dot and the movement of electrons through theleads can be controlled by

Based on experimental data and numerical work withRichard Berkovits at Bar-Ilan University in Israel, Sivanand colleagues find that changes in capacitance cansignificantly contribute to, and in some cases dominate,the peak-spacing statistics.

Indeed, the measured distribution of the peak spacings -obtained by the Stanford group for about ten thousandpeaks - disagrees with the simple theoretical model(figure 6d). In particular, the distribution shows no hint ofthe even-odd features seen in the predicted distribution,challenging the assumption that each level fills first withone spin then the other before the next level is occupied.Further work is needed to understand how the spin,Coulomb interactions and chaotic wavefunctions combineto produce this measured distribution.

Several theorists, in particular Michael Stopa, while atRiken in Japan, Berkovits, and Ned Wingreen and KenjiHiroshi at the NEC Research Center in Princeton, US,have recently approached the problem numerically,allowing the possibility of more complicatedarrangements of occupations, and that the total spin mayminimize the energy of the dot. But it remains to be seenwhether there are some universal statistics for theseinteracting systems. Recent experiments by Dan Ralphand Michael Tinkham at Harvard University in the USalso emphasize the importance of spin and inter- action inthe properties of quantum dots. In their case, the dots aremade of aluminium, which leads to superconductivityeffects at low temperatures and magnetic fields.

Future horizons

Much of the physics observed in quantum dots has alsobeen revealed in other quantum systems. For example, themove- ment of single electrons has recently beenmeasured in carbon nanotubes and other molecularsystems, and shows similar features to those observed forquantum dots. These similarities suggest that the physicsof quantum dots applies to many other systems containingconfined electrons.

Meanwhile, the continual development ofnanotechnology will allow a greater range of artificialquantum structures to be studied. More refined theories ofquantum transport will be needed to consider not only thequantization of energy and charge, but also theinteractions between confined and non-confinedelectrons.

Some work has already started in this direction. For ex-ample, a collaboration between MIT and the WeizmannInstitute in Israel and also the group in Delft with one ofus (LK) has recently used quantum dots as artificialmagnetic

impurities embedded between metallic leads. If the dotcontains an odd number of electrons, the total spin will benon- zero and electron tunnelling between the dot and theleads changes the spin on the dot. The coupling betweenthe dot and leads therefore has the effect of a spin-exchange coup- ling. Experiments in this regime haveshown new transport features that resemble the Kondoeffect in metals containing magnetic impurities.

While the Kondo effect owes its existence tointeractions between many electrons, interactions canactually destroy other quantum transport phenomena. Forinstance, in the case of more complex circuits, thequestion arises whether electrons can retain their quantummechanical properties, or whether interactions with theoutside environment will lead to phase decoherence. Inthe last few years Moty Heiblum's group at the WeizmannInstitute has carried out an interesting series ofexperiments aimed at measuring these coherenceproperties, and has observed, for instance, how placing amicro-detector near one arm of the interferometer causesdecoherence. More work along these lines needs to bedone before we can know if it is possible to use quantumdots as the building blocks for quantum circuits.

This is a question with practical importance, since it hasbeen suggested that such quantum circuits could form thebasis of a quantum computer. However, it will take manymore fundamentally interesting experiments before weget to such practical applications.

Further readingR CAshoori 1996 Electrons in artificial atoms Nature 379413

CWJ Beenakker 1997 Random-matrix theory of quantum transport Rev. Mod.

Phys. 69 731; cond-mat/9612179

TGuhr, A Mueller-Groeling and HA Weidenmueller 1998 Random matrix

theories in quantum physics: common concepts Phys. Rep.; cond-mat/9707301

at press

L P Kouwenhoven et al. 1997 Electron transport in quantum dots Mesoscopic

Electron Transport led) L L Sohn, L P Kouwenhoven and G Schoen (NATO

Series, Kluwer, Dordrecht) See also: http://vortex.tn.tudelft.nV-Ieok/papers/

and http-.//www-leland.stanford.edu/group/MarcusLab/grouppubs.html

P L McEtien 1997 Artificial atoms: new boxes for electrons science 273 1682

Leo Kouwenhoven is at the Department of Applied Physics and DIMES,

Delft University of Technology, PO Box 5046,2600 GA Delft, The

Netherlands.

Charles Marcus is at the Department of Physics, Stanford University,

Stanford CA 94305-4060, US

PHYSICS WORLD JUNE 1998 39