TOPICAL REVIEW Single-electron phenomena in ...chem.ch.huji.ac.il/~porath/NST2/Lecture 7/Meirav SET Semi...single-electron tunnelling, or Coulomb-blockade theory. As such this is a

Semicond. Sci. Technol. 11 (1996) 255–284. Printed in the UK

TOPICAL REVIEW

Single-electron phenomena insemiconductors

U Meirav and E B Foxman †Braun Centre for Submicron Research, Weizmann Institute of Science, Rehovot76100, Israel

Received 11 August 1995, accepted for publication 22 August 1995

Abstract. The study of single-electron phenomena associated with tunnelling insemiconductor nanostructures has emerged in recent years as a major forefront ofcondensed matter physics, whose implications range from fundamental physics toelectronic device applications. This paper presents a tutorial review of the subject,with emphasis on the role of single-electron charging in such semiconductor‘quantum dots’. The main purpose is to describe the various phenomena observedin these experiments and to present the theoretical understanding of thesephenomena in an introductory fashion. The paper attempts to explain theunderlying physics at the intuitive level and tries to draw, as much as possible, aunifying perspective on a relatively large body of knowledge acquired within a shorttime by the conjunction of many individual contributions.

1. Overview

This decade has witnessed the emergence of a new branchof solid state and semiconductor physics that studiesthe behaviour of electrons confined in precisely tailoredman-made potentials. This field has developed fromthe confluence of several technologies that now allowfor the routine fabrication and study of semiconductorstructures that entrap small numbers of conduction electrons(< 100) in geometries of size comparable to their deBroglie wavelength (λ ≈ 50 nm). Studying these systemshas proven to be a fertile experimental and theoreticalendeavour, in which the discreteness of charge carried by asingle electron and the interplay of quantum effects becomemanifest in striking ways. These phenomena are broadlyreferred to as single-electron effects in semiconductornanostructures. This article aims to introduce the field andreview its current status.

It is worthwhile taking a moment to clarify the intendedscope of this paper. The field of single-electron chargingevolved initially in the realm of granular metals and metallictunnel junctions. Not only were the first single-electroncharging phenomena observed in such systems, but manyof the pertinent theoretical concepts were developed inthat context. Furthermore, single-electron tunnelling inmetals and in superconductors remains a vibrant field ofresearch today. Indeed, the richness and the difference ofthe phenomena in semiconductors justify a separate review.Although the border between these two branches is at timesarbitrary, this review is clearly restricted tosemiconductor

† Permanent address: Lucas Center for Magnetic Resonance Imaging,Department of Radiology, Stanford University, Stanford, CA 94305, USA.

phenomena and does not attempt to describe the workdone on metals. On the other hand, an attempt has beenmade to present the subject matter, including the theoreticalconcepts, in a self-contained fashion.

Furthermore, even within the specified area of single-electron effects in semiconductors, an exhaustive accountof all the work to date would be an elusive goal. The fieldis rapidly evolving and new results appear on a monthlybasis. Thus, we have chosen to outline the field via aprogression from the more straightforward concepts to whatmay be called the issues currently on the frontiers of ourunderstanding. The objective in doing so is twofold. First,we hope that this paper may serve as a comprehensiveintroduction for workers in semiconductors who are newlybecoming acquainted with single-electron effects; secondly,this review might help those already involved in the subjectin sorting the extensive amount of information accumulatedover time, and perhaps re-cast some familiar issues into anew perspective.

Thus, the paper is organized as follows. Section 2is a survey of the primary experimental facts associatedwith the subject matter, namely a description of thedifferent structures of semiconductor devices which exhibitsingle-electron charging effects and the most commonmeasurements performed on such devices, showing theseeffects. Section 3 is a tutorial on the basics of ‘classical’single-electron tunnelling, or Coulomb-blockade theory. Assuch this is a theory largely ignoring the peculiarities ofthe phenomena in semiconductors, but in our judgment itis a good theoretical starting point upon which to elaborate;this section attempts to clarify the concepts frequentlyencountered in the literature, with emphasis on their

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U Meirav and E B Foxman

applicability to typical semiconductor structures. Section 4is devoted to an exposition of a more realistic quantummechanical view of single-electron tunnelling in quantumdots and demonstrates its necessity for understanding someof the experimentally observed phenomena. In particular,this requires the consideration of the entire (as opposed toelectrostatic) energy of electrons in a quantum dot, andto draw the distinction betweenaddition spectrumandexcitation spectrum. Section 5 is built around the generaltopic of magnetic fields and the role of single-electrontunnelling as a spectroscopic tool. Finally, section 6attempts to round off this review by giving a snap-shotof further issues and experiments that have been drawingthe attention of workers in the field.

2. Semiconductor structures with single-electrontunnelling

Single-electron phenomena have been observed in a widevariety of semiconductor structures. The common featureof all these structures is a collection of free electronsconfined to a small volume of semiconductor material. Thisconfined system of electrons, often referred to as a quantumdot (QD), is then coupled via tunnel barriers to macroscopicelectrical leads, across which electrons tunnel into, andout of, the confined volume. Furthermore, the QD can beaffected by capacitive coupling to nearby electrodes. Theseingredients – a QD, tunnelling and capacitance – are thebasis of the phenomena which are reviewed in this paper.

A variety of approaches have been taken towardsfabricating such devices in semiconductors. In mostschemes the starting point is with GaAs heterostructures,providing confinement of electrons to a two-dimensionallayer. Further confinement is achieved by lithographictechniques.

2.1. Planar quantum dots

One commonly studied structure, which will be referredto as aplanar QD, is created by patterning several metalelectrodes, or gates, on the surface of a two-dimensionalelectron gas (2DEG) heterostructure, usually of GaAs [1–52]. Figure 1 depicts the structure of a representative planarQD schematically. A negative voltage applied to a gateraises the electrostatic potential in its neighbourhood and,typically around−0.5 V, depletes the underlying 2DEGin the vicinity of the gate. Consequently, under suitablebiasing conditions, a small region of 2DEG remains at thecentre of the structure, and is isolated from the remainderof the 2DEG. Numerical methods are commonly usedto model the resulting potential and charge distributionin a self-consistent way [53]. There are two narrowconstrictions, one formed between gates G0 and G1 andthe other between gates G0 and G2, which are depleted ofelectrons, but the potential there is just slightly above theFermi level and thus presents a low-energy barrier acrosswhich an electron can tunnel; in fact, the transparency ofthis tunnel barrier can be tuned by the voltage applied tothese gates.

Figure 1. (a) A typical planar quantum dot (QD), consistingof a GaAs heterostructure with a 2DEG near the surface,and a set of metallic gates which determine the area, totalcharge Q and the tunnelling barriers of the QD. The latterare formed by the constrictions created between G0 andG1 or G2. (b) A schematic representation of a planar QD,namely a small puddle of free charge Q confined by anexternal potential, coupled to two leads via tunnelling and aplunger gate, which capacitively influences the total chargein the puddle.

Before proceeding it should be pointed out that theconstriction in a 2DEG defined by two adjacent gates, oftencalled a quantum point contact, has been studied extensivelyas a system in itself [54–57]. One can distinguishbetween two regimes of transport as a function of gatebias. At weaker bias, ballistic transport occurs throughthe constriction, resulting in the celebrated quantization ofconductance [55, 56] in steps of 2e2/h. At stronger bias,when the potential between the gates is raised above theFermi level of the 2DEG, it forms an energy barrier, underwhich electrons can tunnel [58, 59]. Compared to the morefamiliar tunnel barriers formed by thin layers of insulatingmaterials, point contacts give rise to barriers which arerelatively low in energy and very long. It should notbe taken for granted that well-behaved, tunable tunnellingoccurs though such structures; the fact that the constrictioncan be pinched off continuously is not always a sufficientrequirement for tunnelling to be practically observed. Ifthe slot between the gates is much longer than 100 nm,lithographic and material non-uniformities render it difficultto achieve good tunnelling behaviour. It is fair to saythat many of the phenomena discussed in this paper are anindication, and a result, of the fact that these submicrometreSchottky-gate point contacts form well-controlled tunnelbarriers.

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Single-electron phenomena in semiconductors

A third common, though not universal, feature of planarQDs, besides confinement and tunnelling, is an additionalgate which is capacitively coupled to the confined region.This gate, sometimes referred to as a plunger gate, can bea separate surface electrode on the periphery of the QD[7, 8, 12], shown as GP in figure 1, or a back-gate, namelyan underlying conductive layer [5, 25, 32]. In any case, itsfunction is to modify the electrostatic potentialϕ inside theQD and thus change〈N〉, the average number of carriersconfined within it. To a good approximation, the latteris linear in Vp, the voltage applied to the plunger gate.Although any of the gates will affectϕ, the advantage ofusingVp is that it has less influence on the tunnel barriersthemselves.

Having described the structure of a planar QD, themost fundamental experimental measurements can now bedescribed. With ohmic contacts to the two large 2DEGregions outside the QD, the conductance through the dotis measured as a function ofVp. In other words, a smalldrain–source voltage,Vds , typically no more than a fewµV, is applied between the two sides and the currentI

is measured. The current is due to electrons tunnellingfrom one side into the dot, and out of the dot to the otherside. Often this is done at a very low temperature, inside adilution refrigerator (about 0.1 K).

The low-bias conductanceI/Vds is found to oscillatewith Vp. These oscillations have two striking features:they are approximately periodic inVp, and they can takethe form of very sharp and narrow peaks. The peak-to-valley ratio at low temperatures is noise-limited and isoften well over 100. Such results have been seen bymany groups who have measured planar QDs with plungergates [5–45]. Representative data are shown in figure 2[60]. The number of conductance oscillation periods canbe several hundred. In fact, their precursors were first seenin a system of metallic junctions [61] and subsequently innarrow silicon inversion layers [62, 63]. These oscillations,which are referred to in the literature by names such assingle-electron charging oscillations or Coulomb-blockadeoscillations, have generated great interest and will bedetailed in this review.

The conductance oscillations gradually disappear as thetemperature is increased [6, 63]. The typical temperaturescale is several degrees Kelvin in most structures studiedto date; the reason for this will become clear in the nextsections. In principle, the phenomenon can persist tohigher temperatures with appropriate structures [44, 64–66]. The most salient features of the oscillations can beaccounted for with a relatively simple classical picture,but there are several rather interesting aspects of thetemperature-dependence of the oscillations that require amore sophisticated treatment.

Yet another important type of measurement is the(nonlinear)I–V curve, which is a measurement ofI versusVds , at a fixed value ofVp. In these measurements,one finds a rather distinct threshold voltage beyond whichcurrent begins to flow. An example is given in figure 3,showing the absolute value ofI versusVds in a planar QD.This threshold is associated with single-electron charging.In practice one often measures the differential conductance,

Figure 2. Representative data showing the conductance ofa planar QD versus plunger gate voltage, at very lowtemperature (about 0.1 K). The conductance shows sharppeaks which are approximately equally spaced.

Figure 3. The nonlinear conductance of a planar QD. Theabsolute value of the current is plotted against thedrain–source bias, Vds , at fixed gate voltages. A distinctthreshold voltage is associated with the Coulomb blockade.

G ≡ dI/dVds directly, using simultaneous AC and DCbiasing techniques. In section 4 it will be shown thatcomplex fine structure is found inG versusVds , and howthis structure contains important information on the energyspectrum of the QD.

2.2. Vertical quantum dots

A different approach to realizing single-electron tunnellingin semiconductors is to have current flowing vertically withrespect to the heterostructure layers, relying on AlGaAs, orother large-gap materials, to form tunnel barriers. Liketheir planar counterparts, thesevertical QDs are structuresin which the electrons are confined by the combinationof the heterostructure layers providing vertical confinementand lithography to provide in-plane confinement; however,the details of fabrication are quite different. Here noin-plane tunnelling is required, allowing strong lateralconfinement which is achieved by eliminating all but a

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Figure 4. A schematic sketch of a vertical quantum dot. Anarrow pillar is defined out of a heterostructure where twobarriers are formed by layers of AlGaAs. The QD is formedbetween the layers and electrons can tunnel into and out ofthe dot. The barriers need not be of the same thickness.Measurements of conductance or capacitance on suchstructures are described in the text.

narrow pillar in a standard, or modified, double-barriertunnelling heterostructure. A schematic diagram is shownin figure 4. Several variants of this basic design havebeen explored by workers in the field [67–77]. One of thetechnical challenges is making a separate electrical contactto the top of the pillar. The bottom is usually contacted viathe conductive substrate itself, or a conducting underlayer.

The most straightforward type of measurement is asimple I–V curve. Such measurements have shown non-ohmic features with fine structure related to the energyspectrum of the QD [67–69, 71–90]. Thus in general this isa form of conductance spectroscopy, because tunnelling isenhanced whenever an available energy level of the QDis aligned with the Fermi level of one of the contacts.If single-electron charging affects the tunnelling, then itmay be manifested as steps in theI–V curve. However,to observe this sequential charging effect, the tunnellingelectrons must dwell and accumulate in the dot. Thispossibility is discussed in section 3.

Another important type of measurement is capacitancespectroscopy. This is done in structures which are designedso that there is tunnelling only between the QD and thelower contact layer, namely through the lower AlGaAsbarrier. The top electrode serves only for capacitivecoupling, and hence is also referred to as a gate. Suchmeasurements were first performed on large arrays ofvertical QDs [91, 92], but more recently they have beenperformed on single isolated dots, using more advancedtechniques to provide sufficient measurement sensitivity[70, 93–95]. In practice, the capacitance was measuredbetween an electrode on top of the QD – the gate – anda conducting layer under the dot which is separated fromthe dot by a thin tunnel barrier. In other words both aDC bias and a small additional high-frequency signal areapplied to the gate. The DC bias gradually populates the dotwith electrons, by pulling down its potential with respectto the Fermi energy of the underlying conducting layer.Whenever an additional electron state is aligned with theFermi level of the bottom layer, tunnelling is enhanced,

Figure 5. A plot of capacitance versus gate voltage in avertical QD measured by Ashoori et al [94]. Eachcapacitance peak corresponds to a gate voltage at which asingle additional electron is accumulated in the QD.Reproduced with permission of the authors.

giving rise to an AC current to the dot itself. The resultingcharge modulation in the QD induces a capacitance signalon the gate, due to its proximity to the dot. Thus onefinds peaks in the capacitance which are due to the additionof single electrons to the QD. Such results are shown infigure 5. The precise meaning of this spectrum, as opposedto that which is measured viaI–V curves, will be discussedin section 4.

2.3. Comparison between vertical and planar quantumdots

For some perspective, compare a few distinctive character-istics of planar and vertical QDs.

(i) Vertical QDs have essentially fixed tunnel barriers,which are typically high in energy (a few hundred meV),relatively thin (typically of order 10 nm) and sharp, namelywell modelled by a square barrier. Planar QDs havetunabletunnel barriers, which are usually only a few meV highand of order 100 nm long. These barriers typically have asaddle shape which is broad and smooth on the scale of theelectrons’ Fermi wavelength.

(ii) In vertical QDs tunnelling electrons couple moreor less uniformly to the entire area of the QD, whereas inplanar QDs the electrons tunnel into the edges. This cansignificantly affect the tunnelling rates and lead to strongvariations in heights of conductance resonances (shown infigure 2). Another view of this distinction is that verticaltunnelling approximately conserves in-plane momentum (orvertical angular momentum), whereas in-plane tunnellingconserves vertical momentum.

(iii) Vertical QDs can be empty of electrons and stillenable tunnelling. In fact this is a typical experimentalsituation, which is favourable for probing the non-interacting spectrum of the QD, or the spectrum in the

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presence of a very small number of electrons. In planartunnelling the QD is seldom empty of electrons.

(iv) It is technically more difficult to introduce a plungergate in the vertical QD configuration, although such deviceshave been made [71, 72]. Hence in vertical structuresmost measurements are of two-terminalI–V or C–V type.Perhaps due to (iii) and (iv), Coulomb-blockade oscillationshave not yet been reported in vertical structures.

2.4. Other types of semiconductor quantum dots

There seems to be an unlimited variety of methods toproduce semiconductor QDs in general and specificallyones in which single-electron phenomena can potentially beobserved. This section will not attempt to give a detailedpicture of all the approaches that have been carried out.However, it is important to mention some of these works, ifonly to convey a sense of this variety and of the prevalenceof single-electron phenomena in semiconductor devices.

One class would be QDs created partially orentirely due to random potential fluctuations in a2DEG. Historically, the first observation of single-electroncharging oscillations in semiconductors was in gated narrowchannels in a 2DEG. Such periodic oscillations were seenin silicon [62, 63] and GaAs [96, 97] quasi-one-dimensionalwires, near threshold, in which random, impurity-inducedpotential fluctuations led to the formation of a small isolatedsegment of the wire, which is an instance of a planar QD.The appearance of single-electron charging oscillations indisordered wires near threshold is not rare, although anygiven device will have characteristics which vary each timeit is thermally cycled to room temperature and subsequentlycooled again. The observation of periodic conductanceoscillations in these structures initially motivated the studyof artificially induced planar QDs [5] and the similarityof the phenomena confirmed, in retrospect, the source ofperiodic conductance oscillations in disordered quantumwires. Effects attributed to impurity-induced QDs havebeen reported in a number of other semiconductor structures[71, 74, 76, 98–105]. Although the detailed characteristicsof such devices suffer from being sample-specific andirreproducible, they can present extremely small QDs withcorrespondingly enhanced charging energies.

Another group, which are but a small step from thestandard planar QDs, are planar QDs defined in a 2DEGby unusual gate geometries [106–108], or by techniquesother than Schottky gates. In part the motivation foralternative confinement techniques is the desire to reducethe total capacitance in order to increase the energy scaleand temperature range of single-electron charging. Thesetechniques include etching [109–112] and implantation[113], and even the use of remote [25] or non-metallic[27, 38] gates to modulate the QD’s potential. Anothervariant is the use of delta-doped GaAs as the basis offorming a QD [102, 114, 115], namely a homojunction, asopposed to the more common heterojunction 2DEG. Therehave also been quite a few experiments on planar QDswith no plunger gating, in which single-electron chargingis not apparent, and whose focus was on other phenomena.Furthermore, there has been extensive work on infra-red

spectroscopy of quantum dot arrays [116, 117], which willnot be discussed here.

Single-electron phenomena have been seen in a varietyof semiconductor materials. Apart from GaAs heterostruc-tures, there have been experiments in silicon inversion lay-ers [118–121], other silicon and silicon/germanium struc-tures [122–127] and quite remarkable conductance oscilla-tions in indium oxide wires [126, 128]. An entirely differ-ent class of QDs are those which are formed spontaneouslyduring a chemical or physical deposition process. Theseself-organized dots can be extremely small and relativelyuniform in size. Most notably, InAs self-organized dotshave been the subject of much recent interest; similar dotshave been realized in several other compound semiconduc-tors as well [129–131]. Making contacts to these dots re-mains a challenge, but at present this appears to be one ofthe most promising approaches towards the realization ofvery small QDs. At the extreme end of miniaturization,single-electron charging of isolated molecules, and possi-bly of atoms, would present the ultimate form of a QD[103, 132, 133]. In fact it has been suggested that QDs canbe viewed as ‘artificial’ atoms [134]. However, these tan-talizing possibilities will not be discussed in this review.

3. Classical single-electron charging

In the previous section, the conductance through aquantum dot was shown to exhibit periodic peaks asa function of gate voltage. As hinted above, thisbehaviour in semiconductor devices was first seen ingated quasi-one-dimensional channels [62, 63, 96, 97, 119].In these structures it is believed, in retrospect, thattunnelling barriers, and hence isolated segments or dots,surreptitiously formed out of the random disorder potentialthat is known to exist in the channel [135]. Van Houtenand Beenakker [136] first proposed an explanation forthese early results based on a theory of single-electroncharging, referred to as the Coulomb-blockade theory.The foundations of Coulomb-blockade theory were set byKulik and Shekhter [137, 138], who had been motivatedby phenomena reported yet earlier on transport throughsmall metallic grains [139–143]. The prevalence ofsingle-electron charging in semiconductor quantum dotsturned the Coulomb-blockade concept into an importantpart of modern semiconductor physics. In the Coulomb-blockade model, conductance oscillations are viewed asa manifestation of sequential single-electron tunnellingthrough a system of two tunnel junctions in series [61, 144–150]. This section discusses single-electron chargingeffects in QD structures using the language of the Coulomb-blockade model.

3.1. Periodic conductance oscillations in the classicalCoulomb-blockade theory

The following discussion aims to elucidate the fundamen-tals of the Coulomb-blockade theory by considering howa simple case of this model accounts for the periodic con-ductance peaks observed in a planar QD such as that infigure 2. Begin by considering the abstracted view of a

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planar QD shown in figure 1(b). This schematically de-picts an isolated puddle of electrons, of total chargeQ,that is coupled across tunnel barriers to macroscopic leadson both sides. Also shown is a plunger gate which capaci-tively couples to the small electron puddle. Implicit in theconcept of capacitive coupling is the simple relationship,1Q = Cp1Vp, which relates changes inQ to changes inthe plunger voltage,Vp, via the capacitance between thedot and the plunger,Cp. Somewhat more accurately, theminimum electrostatic energystate is maintained by havingQ follow this relation.

In quantum dot structures, in whichQ is notoverwhelmingly large with respect to the electronic chargee, the relationship betweenVp andQ must take into accountthe discreteness of charge carried by electrons. Doing sois at the heart of the Coulomb-blockade model, and isaccomplished by understanding that, althoughVp may bevaried continuously, the charge in an isolated QD must atall times be an integer multiple ofe, namely Q = Ne.Consequently, the actual charge on the dot is the integermultiple of e that comes closest to the quantity of chargethat would otherwise reside in the dot if charge were notquantized. Therefore, ifVp is increased continuously,the charge in the dot increases incrementally bye atperiodically spaced values ofVp in a stepwise mannerillustrated in figure 6(a).

Given that the charge in the QD is an integer multipleof e, consider the electrostatic energy associated withN

electrons in the dot:

E(N) = (Ne)2

2C− ϕNe. (3.1)

The first term on the right-hand side of this expressionrepresents the capacitive charging energy ofN electronson the dot,C being the total capacitance of the dot (thecapacitance between the QD and the ‘rest of the world’).The second term represents the potential energy of theQD, whereϕ is the electrostatic potential. Section 3.3will treat the case of several gates coupled to the dot andcontributing to its total capacitance, but for the time beingthe assumption is that the plunger gate is the only gatewith significant capacitive coupling to the QD. In this case,C = Cp and ϕ = Vp in equation (3.1). Solving thisrelation for the integer value ofN that minimizesE(N)

for a givenVp reproduces the same staircase relationshiparrived at above in figure 6(a). The plunger gate voltagesat which the charge on the dot increments by one electroncorrespond to situations in whichE(N + 1) = E(N). Thiscondition occurs periodically whenVp = e(N + 1/2)/Cp,indicating that, ifVp is adjusted to a value at which theelectrostatically favoured quantity of charge on the dot is ahalf integer multiple ofe, then the actual lowest energy stateof the dot is degenerate, corresponding either toN or toN +1 electrons, and the dot is energetically free to fluctuatebetween these two states. Midway between these chargedegeneracy points, whenVp = eN/Cp, equation (3.1)indicates that adding (or removing) an additional electronmoves the dot away from its lowest energy state by anenergye2/2C. Here charge fluctuations are suppressed bythe Coulomb energy associated with adding (or removing)

Figure 6. Single-electron charging and the origin ofperiodic conductance peaks. The quantity of charge thatrepresents the minimum electrostatic energy is givenclassically by Q = CVp (more generally, in the presence ofseveral other capacitances, Q = Q0 + CVp) plotted as thebroken line in (a). In the small electron system (a quantumdot), where the discreteness of charge must be taken intoaccount, the actual charge in the QD is the integer multipleof e closest to the continuous quantity Q , shown by the fullline in (a). The schematic plot of conductance peaks in (b)illustrates that periodic conductance peaks correspond toplunger-gate voltages, Vp , associated withcharge-degeneracy points, where the charge in a QD isfree to fluctuate by e.

one electron to the dot. AsVp is swept, this charging energyperiodically attains its maximum value ofe2/2C eachtime Vp is half way between the degeneracy points, andgradually vanishes as the QD approaches the degeneracypoints.

Turn next to consider how the periodic occurrence ofcharge degeneracy points, separated by regions in whichcharge fluctuations are suppressed, give rises to periodicconductance peaks as a function ofVp. In order for currentto flow through a QD, an electron must tunnel across oneof the barriers, momentarily reside in the dot and thentunnel out of the dot through the other barrier. Therefore,unless the two tunnelling events occur simultaneously,conductance is inherently associated with the total chargein the dot fluctuating between two (or more) values,Ne

and (N + 1)e. At low temperatures, these fluctuations areenergetically allowed only at the charge degeneracy pointswhich occur periodically inVp with spacing1Vp = e/Cp

as indicated in figure 6(b). Away from charge degeneracypoints, charge fluctuations in the dot are blocked bythe Coulomb energy involved, and hence the tunnellingconductance is suppressed.

In vertical QD structures in which the top electrode

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acts as a gate [70], this same mechanism accounts forthe periodic peaks in capacitance that are seen as thebias voltage is increased, such as in figure 5. At chargedegeneracy points, an electron is free to tunnel between thedot and the substrate in response to the AC biasing signal.The resulting AC current, by charging and discharging theQD, gives rise to a peak in the measured capacitance. Thecase of DC transport in vertical QDs will be discussed insection 3.4.

This simple picture embodies the essentials ofthe Coulomb-blockade theory of periodic conductanceoscillations. Though it evokes tunnelling to account forelectrons entering and exiting the dot, it is otherwisean entirely classical model in which electron–electroninteractions are fully described by simple capacitivecharging which blocks electron transport onto a QD,except at periodically occurring charge degeneracy points.Working within this model, a rigorous formalism hasbeen developed [150, 151] to describe the conductancequantitatively, taking into account the tunnelling rates, biasvoltages and temperature. The rest of this section willproceed to elaborate this model as well as to point outsome of its limitations.

3.2. Coulomb-blockade oscillations at finitetemperature

A more realistic understanding of the Coulomb blockadetakes into account non-zero temperature, which contributesto transport by allowing some electrons to overcome theCoulomb charging energy. This results in a non-vanishingtemperature-dependent tunnelling conductance between thepeaks. At sufficiently elevated temperatures,kT ∼ e2/C,the Coulomb blockade is completely washed out. At lowerT , the off-peak conductance is non-zero, but it is smallestmidway between conductance peaks. At relatively lowtemperatures,kT � e2/C, this implies that a finite widthis acquired by the conductance peaks, although they stillremain well separated. Including temperature in the modeloutlined above is a straightforward calculation [149], whichyields an expression for the line shape of a conductancepeak. The main result of this calculation is that theamplitude of the conductance peak isT -independent, withexponentially falling tails and a width proportional toT .The actual temperature behaviour of measured conductancepeaks is more complex than the behaviour just described;in fact, this complex behaviour was one of the earliest hintsthat the Coulomb-blockade model does not fully describeconductance in quantum dots [5, 6]. This issue will beelaborated in section 4.

3.3. The Coulomb blockade in the presence of multiplecapacitances

Another simplification in the above outline of the Coulombblockade was that only one gate was shown as beingcapacitively coupled to the QD. In reality, several gatessimultaneously couple to the QD. In addition, there issignificant capacitive coupling between the dot and itsleads, and there are possibly other stray capacitances. In

Figure 7. A scanning electron micrograph of a planar QDstructure, for demonstrating the gate and lead capacitancesin a planar QD. The distance between the two constrictionsis 0.6 µm. The capacitance between the QD in this deviceand its leads and gates was measured [23] as explained inthe text. C1 = 69 aF, C2 = 66 aF, C3 = 35 aF, C4 = 12 aF,C5 = 26 aF and C6 = 76 aF. The total capacitance isC ≡ ∑

Ci = 284 aF (1 aF = 10−18 F). The 2DEG layer inthis structure is located about 85 nm below the surface.

a multiple-gate geometry, capacitance is classically definedby the relation

1Q =∑

i

1ViCi

where 1Q is the change in the equilibrium value ofthe charge, Q, which occurs in response to a change involtages, 1Vi , each applied to the respectiveith gateor lead. By definition,Ci is the capacitance betweenthis gate, or lead, and the QD. This relation implies thatchanging theith gate voltage by1Vi changesϕ (thepotential on the dot) by1Vi(Ci/C), where C ≡ ∑

i Ci

is the total capacitance of the dot. This modifies equation(3.1) only to the extent of associatingϕ with αVp, whereα = Cp/C. As a result, sweeping any gate independentlyyields periodic conductance peaks, with spacing1Vi =e/Ci . This allows one to determine the variousgatecapacitances experimentally with high accuracy. Thecapacitance between the dot and the tunnelling leads canalso be determined quite accurately by biasing the leads, aswill be explained in section 3.4.

Figure 7 shows a representative multiple-gate planarQD for which the individual gate and lead capacitanceswere measured [23]. Carrying out this procedure gavea total capacitance to the dot ofC = 2.84 × 10−16 F.The individual gate and lead capacitances are noted inthe caption to figure 7. This value ofC is typical for

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such devices and compares well with values ofC reportedelsewhere for other planar QDs [10, 18, 22].

The total capacitance,C, is an important parameterin QDs because it determines the temperature range overwhich single-electron charging effects are manifested. Inpractice the criterionkT < e2/C, must be met in orderfor single-electron charging effects to be observable. Forthe case above in whichC = 2.84 × 10−16 F, thisrelationship implies temperatures under 4 K. In verticalQDs, C is comparable or larger. Hence the generalnecessity of measuring both planar and vertical QDs at verylow temperatures (T < 1 K) in order to observe single-electron charging effects clearly.

Various schemes have been suggested for decreasingthe size and capacitance of QDs and thus increasing thetemperature at which single-electron charging effects occur[65]. In fact single-electron charging effects in planarQDs have recently been reported at or above liquid heliumtemperatures [44, 108, 124, 152]. No doubt the objective ofincreasing the temperature range of single-electron chargingeffects will remain high on the research agenda in theforeseeable future.

3.4. Large-bias measurements and the Coulomb gap

Until now the discussion of the Coulomb-blockade modelhas focused exclusively on the zero-bias conductance of aplanar QD, namely conductance in the limit of very smallbias between the drain and source leads. The Coulomb-blockade model is equally applicable towards interpretingthe large-biasI–Vds behaviour of a QD. In figure 3a representative trace ofI versus Vds under Coulomb-blockade conditions shows that no current flows in theneighborhood of zero bias,Vds = 0. This suppression ofcurrent has been referred to as the Coulomb-blockade gap.It arises from the fact that, away from charge degeneracypoints, there is a finite charging energy required to place anadditional electron on a QD. At low temperatures, currentbegins to flow only whenVds is sufficiently large to supplythis charging energy. This defines a threshold voltage,Vth,which is usually determined by extrapolating the high-biasslope of theI–Vds curve to zero current.

The above discussion implicitly assumed that the QDpotential was tuned, for example by the plunger voltage,to the fully blockaded state, midway between conductancepeaks. It is an important exercise to describe the changein the I–V curve asVp is changed. This is illustratedschematically in figure 8. Ignoring the capacitance betweenthe dot and the drain lead, theI–V curve simply shiftswith Vp while essentially preserving its shape. This is adirect result of the fact that the threshold energy to add(remove) an electron from the QD is reduced (increased)proportionally to the positive shift inϕ, as is evident fromequation (3.1). Thus, theI–V curve varies periodicallywith Vp.

Among other things, this implies that theI–V curve ishighly asymmetricfor all but two points in any gate voltagecycle. Interestingly, the total gape/C between positivethreshold and negative threshold is preserved. Right atthe conductance peak, theI–V curve nominally fluctuates

Figure 8. The current versus Vds in a planar quantum dotas a function of plunger-gate voltage, Vp . Schematic I –Vdstraces corresponding to different values of Vp show howthe I –Vds curve shifts as Vds is increased, but otherwise islittle changed. As Vp is increased through acharge-degeneracy point (seen as a peak on the zero-biasconductance) the I –Vds curve is abruptly translated by avoltage equal to the width of the Coulomb gap.

between two curves corresponding to the two degeneratecharge states of the QD: one shifted to the left, the other tothe right.

A somewhat subtle point, which has not alwaysbeen correctly dealt with in the literature, is how finitecapacitance between the dot and its leads effects theI–Vds

relation in the Coulomb-blockade model. When dot–leadcapacitances are negligible, the Coulomb gap measured inthe I–Vds trace is simplye/C, as was the case above.When there is a finite capacitance,Cd , between the dotand the drain (the biased lead), the dot potential is shiftedby the drain voltage,1ϕ = (Cd/C)Vds , thus effectivelydecreasing the actual voltage drop between the drain andthe QD. As a result, a larger voltage is required to overcomethe Coulomb blockade, and the measured gap is equal toe/(C−Cd). Measurements of this type allow determinationof the dot–lead capacitances [23]. Strictly speaking, thisconsideration holds only whenVp is biased sufficiently faraway from a conductance peak,1Vp ≥ eCd/C2. Closer toa peak, the lead biasing can pullϕ into a charge degeneracypoint and cause the measured gap inI–Vds to be smaller,its value depending onVp and the various capacitances.

In vertical quantum dots,I–Vds curves are the most

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Figure 9. An experimental demonstration of the Coulombstaircase in a planar QD [60]. The current, I , as a functionof bias, Vds , between the leads shows a staircaserelationship under conditions described in the text.

common type of measurement, as has been pointed outbefore [72, 73, 81–84, 87–89, 99, 153]. Although here toothe single-electron charging gap should be manifested, amajor practical difference results from the fact that, inmost cases, the QD itself is empty of electrons at zerobias, and a large threshold voltage – typically much largerthan the Coulomb gap – is associated with the first allowedstate into which electrons can tunnel. Thus the thresholdvoltage is usually not a direct measure of the Coulombgap. Nevertheless, single-electron charging can be seen invertical structures via the Coulomb staircase.

3.5. The Coulomb staircase

A particularly striking example of how single-electroncharging can affect theI–Vds characteristics of a QD occurswhen one tunnel barrier is significantly more transmittingthan the other tunnel barrier. In this case theI–V

behaviour of the dot can exhibit what has been referred toas the Coulomb staircase [138, 146, 147], namely a stepwisecurve as seen in figure 9. This celebrated signature ofsingle-electron charging is frequently misunderstood, thusjustifying a somewhat detailed discussion.

The most important point to make is that, unlikethe Coulomb suppression of current in the neighbourhoodof Vds = 0, the staircase is not a universal feature ofthe Coulomb blockade. Rather, it is a special resultof having very different tunnelling rates through the twotunnelling barriers. For simplicity, the lead with the more(less) transparent tunnel barrier will be referred to as thedrain (source). The Coulomb-blockade staircase arisesin the following way. As Vds is increased, eventuallyit becomes sufficiently large to overcome the Coulombcharging energy, and an electron rapidly tunnels into theQD through the drain barrier. The electron then dwellsin the QD for a relatively long time, until it tunnels outthrough the more opaque source barrier. Tunnellingoutto the source lead is the rate-limiting step in transportthrough the QD, and the tunnelling rate in this step isaffected only by the potential difference between the QD

and the source. This potential difference is equal to thecharging energy plus the fraction ofVds that falls acrossthe source barrier, which, since the source is kept at aconstant voltage, is just1ϕ = (Cd/C)Vds , whereCd isthe capacitance between the QD and the drain lead. WhenCd/C is sufficiently small, then the potential differencedriving the rate-limiting step is primarily determined bythe charge state of the QD and it is nearlyindependentof Vds . Consequently, at this point the current that flowsthrough the dot is nearly independent ofVds . In a sense theincreasing bias is falling in the ‘wrong’ place, thus failing toincrease the current. Here is the origin of the first plateauin the Coulomb-blockade staircase. Successive plateauxare repeatedly arrived at asVds becomes large enough tocharge the QD with incrementally more electrons, withcommensurate increase in the dot-to-source potential drop.An experimental curve for a planar QD is shown in figure 9.Thus the plateaux are closely linked to the plateaux ofQ

versusV , in that each one corresponds to a different chargestate of the QD.

While the preceding was intended to give a qualitativeunderstanding of the Coulomb-blockade staircase, theactual current predicted by the Coulomb-blockade modelcan be calculated exactly for an arbitrary set of capacitancesand tunnelling rates across barriers by solving for thestationary state of a set of balanced tunnelling rateequations [138, 144, 146–148, 151, 154]. These works haveshown that the Coulomb staircase will only arise forparticular asymmetric constellations of tunnelling barriersand capacitances.

Finally, special consideration is warranted for the caseof vertical QDs. In these structures the QD is initiallyempty. Charging energy is associated with the temporarytransfer of an electron into the QD. More than one excesselectron can reside in the dotonly if (i) the tunnelling ratesin the barriers are substantially different and (ii) the biasdirection is such that electrons are injected through therelatively transparent barrier. For the other bias polarityno charging will occur since an injected electron will leavethe QD through the other side sooner than the next electronenters. Indeed, the Coulomb staircase has been seen inasymmetric vertical QDs and, as expected, only for onepolarity of theI–Vds curve [73, 77, 81, 82, 88, 89].

3.6. Minimum tunnelling resistance for single-electroncharging

Implicit in the formulation of the Coulomb-blockade modelis the condition that the number of electrons localized inthe dot,N , is a well-defined integer. This is to say, welldefined in the classical sense, as opposed to a quantumdefinition which describesN in terms of an average value〈N〉, which is not necessarily an integer, and time-averagedfluctuations〈δN2〉. The Coulomb-blockade model requiresthat 〈δN2〉 � 1. Clearly, if the tunnel barriers are notpresent, or are insufficiently opaque, nothing will constraina quantized electronic charge to be confined within a certainvolume. The question which has intrigued many workers isthat of whether there is a simple criterion which the tunnelbarriers have to obey in order to validate the fundamental

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premise of the Coulomb-blockade picture. The generalview is that there is a minimumresistance[150, 155] whichthe barriers must exceed in order to have〈δN2〉 � 1, andthis resistance is of the order of the quantum resistanceh/e2 = 25 813�. This should be understood as an order-of-magnitude measure, rather than an exact threshold.

One argument to this end is based on the Thoulesscriterion [156], which proceeds as follows. The condition〈δN2〉 � 1 requires that the time that an electron resides onthe dot,τ , be much greater thanδτ , the quantum uncertaintyin this time. The currentI cannot exceede/τ since (formoderate bias) no more than one extra electron resideson the dot at any instant. The energy uncertainty of theelectron,δE, is no larger than the applied voltage, hencethe condition thatδτ � τ translates into macroscopicvariables usingI ≤ e/τ , δτδE ≥ h and δE < eVds .Doing so gives the minimum tunnelling resistance conditionmentioned above,R = Vds/I ≥ h/e2.

Another popular, though admittedly crude, argumentgoes roughly along the following lines. Given a capacitanceC and a resistanceR by which the capacitance of the dotis charged/discharged, the characteristic time for chargefluctuations is simply δτ ' RC, hence the energyuncertainty isδE = h/δτ = h/RC. The energy gapassociated with a single-electron charge ise2/C. For thisgap to be well defined, that is not to be eliminated by theuncertainty principle, the requirement is thatδE < e2/C

which then reduces toR > h/e2.In fact, more rigorous theoretical studies of this issue

have generally supported this conclusion [155, 157–161].Several experimental tests have also shown this to be anecessary condition for observing single-electron chargingeffects [8, 36, 162–164].

For completeness, it is appropriate to mention a relatedissue which has been discussed in the literature. This isknown as the ‘effect of the electromagnetic environment’on the Coulomb blockade. In essence, the objective isto quantify the time scales associated with the chargefluctuations, which are related to the electromagneticexcitations of theleadsthemselves, or the ‘electromagneticenvironment’ of the tunnel junction. This is a particularlyimportant issue in the context of asingletunnel junction (asopposed to a QD between two junctions), which in principlecan also exhibit Coulomb blockade. The essential physicalpoint is that a low-impedance lead will tend to carry awaycharge more quickly, and thus, roughly speaking, reducethe duration of transient charges and therefore suppress theCoulomb blockade. A detailed discussion of this is outsidethe scope of this paper, but can be found in the literature[151, 163, 165–171].

3.7. Co-tunnelling

In a similar vein, even if the minimum resistance criterionis met and single-electron charging effects are manifested,small quantum fluctuations, or uncertainties, inN are notentirely ruled out. Consider the situation in whichVp isbiased so thatG is between conductance peaks. In theclassical Coulomb-blockade model there is then a fixednumber of electronsN on the QD and atT = 0 the charge

on the QD does not fluctuate. However, the fact that verysmall quantum fluctuation inN may be present correspondsto electrons momentarily tunnelling onto the QD, withan energy deficit on the scale of the classical Coulombcharging energy [155, 172–183]. Essentially, the tunnellingelectron resides on the QD in a virtual charge state for asufficiently brief interval such that the energy uncertaintyof this state is larger than its classical energy deficit,subsequently tunnelling out. This process has been referredto as co-tunnelling or macroscopic quantum tunnelling(MQT) of charge. The rationale behind the latter term isthat the total charge of the system (a macroscopic variable)undergoes a transition through a classically forbiddenintermediate state, in apparent violation of the Coulombblockade.

This mechanism is described by second-order pertur-bation terms of the tunnelling Hamiltonian, of which thereare two distinct types. The first type corresponds to thetunnelling of an electron into a certain energy state andthe tunnelling of an electron from thesamestate out ofthe dot. The end result of the two tunnelling events isthat the state of the QD is unchanged, and as such, thisis referred to aselastic co-tunnelling, which contributesa linear term to theI–V relation. In the second kind ofprocesses, somewhat misleadingly referred to asinelasticco-tunnelling, an electron tunnels into a certain state in thedot and a second electron, from a different state, tunnelsout of the dot. The state of the dot is modified, leaving anelectron–hole excitation. The resulting current is nonlinearin Vds and temperature-dependent. Both mechanisms givea conductance proportional toσ1σ2, namely the product ofthe independent conductances of the two barriers; this ischaracteristic of an off-resonance tunnelling process.

The case of elastic co-tunnelling depends, in principle,on the geometry of the QD. This is because the electroninvolved has to couple to both leads; thus in a sense itmust traverse the dot in a virtual state. One limiting formwhich has been obtained in the literature [182] is

I el = hσ1σ21

8π2e2

(1

E1+ 1

E2

)V (3.2)

where 1 is the average energy separation betweeneigenstates in the QD andE1 (E2) is the charging energyassociated with adding (removing) a single electron to(from) the dot. Note, in particular, that the resultingconductance scales roughly as the ratio between the levelspacing1 and the Coulomb gapU ≡ e2/C.

The case of inelastic co-tunnelling gives the followingwell-known form

I in = hσ1σ2

6e2

(1

E1+ 1

E2

)2[(kT )2 +

(eV

2π

)2]

V (3.3)

namely the sum of two terms: a linear term which dependsquadratically on temperature and a temperature-independentnonlinear termI ∝ V 3. Each of these terms applies inthe limit in which kT or eVds > 1, respectively. Bothresult from increasing the number of possible electron–holeexcitations which can be created in the QD in the courseof the co-tunnelling event.

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Figure 10. The oscillating-barrier single-electron turnstile. Cycling the tunnelling barrier heights by the gate voltages in themanner illustrated causes one electron to be passed through a quantum dot per cycle, thus operating as an electron turnstile.With a constant drain–source bias, (a) the left barrier is lowered, allowing tunnelling of one electron into the dot; the Coulombenergy prevents additional electrons from tunnelling in; (b) the barrier is raised and the electron is trapped inside; (c) theright-hand barrier is lowered, the added electron can leave the dot to the other lead; (d) the barrier is raised, completing acycle in which one electron was passed between the leads. This serves as a frequency-tuned fundamental current source,with I = ef . A larger drain–source bias can cause an integer multiple of electrons to be transferred per cycle, yielding I = nef .

The distinction is made between these two processesbecause their relative contributions to the total net co-tunnelling current depend on the density of states in the QD.In metal QDs, in which the density of states is large, theelastic component of co-tunnelling is usually overwhelmedby the inelastic component. However, in semiconductorQDs, in which the density of states is much smaller than inmetals, both elastic and inelastic terms can contribute to theco-tunnelling current. In practice, co-tunnelling is expectedto modify the classical picture of single-electron chargingin the form of excess current in the region of the Coulomb-blockade gap, in the case ofI–Vds measurements, or excesstunnelling current between conductance peaks in low-biasmeasurements. Experimental observation of co-tunnellinghas been reported in semiconductor quantum dots withrelatively low charging energies,U ≡ e2/C ' 0.2 meV,namely ones in which co-tunnelling effects are enhancedby the low value ofU [7, 16, 26].

3.8. Thermopower

A related transport property in quantum dots is thethermoelectric effect. This effect describes the voltagebuild-up1Vds in response to an imposed thermal gradient,1Tds , with the thermopowerS defined by the relation1Vds = S1Tds . It results from the transfer of electronsfrom (to) the warmer side to (from) the colder side, due tothe different thermal occupation of levels. As such it is notsurprising thatS was found to oscillate periodically withVp in planar QDs, similarly to the conductance [21, 29, 40].These oscillations have been predicted in a direct extensionof Coulomb-blockade theory [184]. Perhaps the most

interesting feature to point out is thatS in a QD undergoesperiodic sawtooth shape oscillations, as opposed to thesymmetric conductance oscillations; furthermore,S ismaximum just before the conductance minimum, at whichit abruptly changes sign and begins another sawtooth rise.

Intuitively, this sawtooth shape can be understood tobe related to the asymmetricI–V curves discussed insection 3.5; the thermopower is a direct result of the factthat it is ‘easier’ for electrons to cross the dot in onedirection than in the other. Roughly speaking, warmerelectrons (or holes) will tend to explore the easier transitionsmore often, and create a positive (negative) charge build-upon their side. Indeed the conductance peaks and valley-centres correspond to symmetricI–V and vanishingS.The theory of Beenakker and Staring [184] gave a detailedderivation of this behaviour.

3.9. Applications of single-electron charging insemiconductor quantum dots

Before concluding this discussion of the classical Coulomb-blockade model, several particularly intriguing applicationsof single-electron charging deserve attention [185]. Theideas discussed so far fall under three categories: currentstandards, sensitive analogue transistors (electrometers) anddigital applications.

Soon after Coulomb-blockade conductance peaks wereobserved it was realized that, by properly cycling thebiasing voltages on a planar QD,one electron per cyclecould be caused to pass through a quantum dot, as hadsimilarly been realized in small metal systems [186–191].In this manner a controlled cycle frequency,f , could lead

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Figure 11. The turnstile I –Vds characteristics of a planarQD, measured by Nagamune et al [111]. Gate voltageswere cycled with an AC signal of f = 10 MHz (1.6 pA) atT = 10 mK. The observed current steps correspond toI = nef = n × 1.6 pA (n is an integer), which are indicatedby dotted lines. Data are reproduced with permission of theauthors.

to a current given precisely byI = ef . An experimentalrealization of this was carried out in a semiconductorquantum dot by Kouwenhovenet al in a structure in which,unlike preceding work in metal tunnel junctions,Vds washeld at a fixed bias and the tunnel barrier heights werecycled [9, 111, 192, 193] . The principle of operation isillustrated in figure 10. The current versusVds observed at acycle frequency of 10 MHz (namely 1.6 pA) is reproducedin figure 11. The first plateau in these traces representsbiases at which only one electron passes through the dotper cycle. The second plateau corresponds to a sufficientlylarge Vds , allowing two electrons per cycle to pass. QDstructures operated in several variants of this mode havebeen referred to as electron turnstile devices and electronpumps, and have been proposed for possible metrologicalapplications, namely, as a frequency-controlled standardcurrent source.

To date, semiconductor turnstile devices have operatedwith an error of approximately 0.3% of the idealcurrent ef . Possible sources of error include limitedresolution of current measurements, leakage across tunnelbarriers, photon-assisted tunnelling from backgroundthermal radiation and unwanted co-tunnelling [65, 194–199]. A partial solution to these problems lies in placingseveral QDs in series and operating at larger bias. Workalong these lines is being pursued both in semiconductorand in metal structures with the hope that current sourcesbased on single-electron charging devices may eventuallyachieve metrological accuracy.

Another group of applications looks at gated QDs asa radically new type of field effect transistor – a single-electron transistor [148, 200] – which can function as ananalogue amplifier or a digital device. As an analogueamplifier, the sensitivity of the conductance of a QD tosmall changes in local electric fields may allow these

devices to serve as electrometers with resolution well belowthe single-electron level [201–209]. It has also beensuggested that single-electron charging devices be operatedas an ultra-high-resolution displacement transducer [210].Quantum dots in the single-electron tunnelling regimefunctioning as low-temperature photodetectors have alsobeen considered [211]. Lastly, several suggestions havebeen put forward that envision the implementation ofsingle-electron devices in the context of electronics circuits.Both memory [115, 212] and voltage-gain [213] deviceshave been demonstrated using various single-electroncharging devices, and on a larger scale, suggestionshave been made for the implementation of single-electroncharging in digital electronics [147, 214–217]. However, atthe same time, others have cautioned against speculatingtoo far afield [65, 218].

4. Single-electron charging and the discrete levelspectrum of a quantum dot

The genesis of the Coulomb-blockade model took placeon the backdrop of single-electron charging phenomenaobserved in small metal particles. In these metal granules,the separation between quantum energy levels is, in general,much smaller thane2/C or kT ; hence the classicaltreatment underlying the Coulomb-blockade model is anexcellent approximation [137, 138]. In a semiconductorQD, however, the situation is different. Consider thesingle-particle density of states in a GaAs 2DEG, which is2.8 × 1013 eV−1 cm−2. For a region of 2DEG comparablein size to a usual QD, the energy separation betweeneigenstates,1ε, is approximately 0.05 meV. This valuecan be compared to the Coulomb gap,U ≡ e2/C, in atypical planar QD which is about 0.5–1 meV. These twoenergies are similar, the charging energy being only anorder of magnitude larger, and they can both be significantlylarger thankT (about 0.01 meV). This observation suggeststhat, in addition to the behaviour attributed to classicalsingle-electron charging, there may be manifestations of thediscrete level spectrum of a QD. In fact, both theoreticalconsiderations and experimental observations have pointedto the importance of this discrete level spectrum [200],which will be the subject of the following discussion. Itwill be shown that, although the Coulomb blockade stillserves as a useful model for interpreting some of the mostsalient features of transport in QDs, the interplay of thediscrete level spectrum and single-electron charging mustbe considered in order to understand the behaviour observedin these systems better.

4.1. Addition spectrum versus excitation spectrum

The single most important message in this section is that,in a QD, there are two energy scales associated with twodistinct experimental perturbations of a QD. The first isa perturbation that adds an electron to the dot, therebychangingN , the number of electrons in the system. Theenergy involved is defined as an addition energy, and the setof energies required for consecutive addition of electronsis collectively referred to as theaddition spectrum. The

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second type of perturbation is one in which the numberof electrons in the dot is held fixed, but some electronsare excited to a higher energy state in the QD. Thecorresponding energies are referred to as theexcitationspectrum, and also loosely termed the spectrum of ‘single-particle’ levels.

These spectra can be thought of as analogues to theionization spectra and internal excitation spectra of atoms.In one case an electron is added or removed from thesystem; in the other case an electron is excited withinthe system. Indeed the analogy between atoms and QDscan be taken quite far [134]. As a preview to thefollowing discussion, several comments can be made aboutthe various experiments concerned. In planar QDs, large-bias measurements generally probe the excitation spectrumof the dot, whereas plunger voltage sweeps at smallVds

probe its addition spectrum. Similarly, in vertical QDs,capacitance spectroscopy measures the addition energies,whereas large-bias conductance can measure excitation oraddition energies, as explained in the following section.

4.2. Transport spectroscopy of vertical QDs

Several groups have experimentally studied the conduc-tance spectrum [67, 72, 73, 77, 82, 84, 87, 153] and capaci-tance spectrum [70, 93–95] of single vertical QDs. As in-dicated in section 3, the conductance of vertical QDs is notnecessarily affected by charging; often there is no accumu-lation of charge in the QD during transport. Such a situationis favourable for measuring the excitation spectrum and isperhaps the most straightforward form of conductance spec-troscopy. On the other hand, the conductance threshold islargely determined by the ‘vertical’ confinement energy re-quired to enter the QD layer. Even if there is a steadyelectron population in the QD, a single-particle spectrumcan be extracted [73, 85, 87]; however, in this case the la-belling of these energies should be qualified, due to theimportance of electron–electron interactions [219].

If the barriers are asymmetric and electrons are injectedfrom the transparent side, they will accumulate in theQD [73, 77, 82, 220]. Here one can measure the modifiedCoulomb staircase (shifted in voltage due to the verticalconfinement energy mentioned above), which is in fact amanifestation of the addition spectrum. This is becauseeach step on theI–V curve corresponds to an increase ofone electron in the QD’s average population. Asymmetricvertical QDs epitomize the two types of spectroscopy:when electrons are injected from the transparent side, theaddition spectrum is measured in terms of the steps inthe I–V curve; when they are injected through the less-transparent barrier, charge does not accumulate in the dotand the structure in theI–V curve corresponds to the truesingle-particle spectrum.

Another spectroscopic technique is the capacitancemeasurements described before [70, 94]. Here it isquite clear that it is the addition spectrum which ismeasured, because each capacitance peak occurs at avoltage corresponding to the increase in the QD populationby one electron. The approximate periodicity in the peaksshown in figure 2 results from the fact that the addition

Figure 12. The differential conductance G ≡ dI /dVds as afunction of Vds measured in a planar QD [23]. The peaksare associated with the excited electron states in the QD,appearing whenever such an excitation is aligned with theFermi level of one of the leads.

spectrum is largely determined by a classical electrostaticenergy, which implies that the charge is approximatelylinear in voltage. Nevertheless, measurable deviations fromperiodicity contain important information on the quantummechanical energy state of electrons in the dot.

4.3. Transport spectroscopy of planar QDs

Two very different forms of conductance spectroscopyhave been employed in planar QDs in the single-electroncharging regime. The first is the measurement of theaddition spectrum via conductance (at smallVds) as afunction of Vp [11, 221]; the second is measurementof the excitation spectrum via large-Vds conductancemeasurements at fixedVp [15, 23, 32, 37, 222]. The formeris based on the notion that a conductance peak occurswhenever the dot potentialϕ, as affected byVp, reachesa charge degeneracy point, namely the threshold for addingan electron to the QD’s equilibrium population. This willbe seen via the formalism introduced below. Its applicationin magnetic fields will be discussed in section 5.

Turning to nonlinear spectroscopy, figure 12 showsG ≡ dI/dVds as a function ofVds measured in a planarQD [23]. This measurement is otherwise similar to thatof figure 3 except that here the differential conductance ofthe structure is plotted in order to reveal higher levels ofdetail. The Coulomb-blockade gap is manifested by theflat region of the trace spanningVds = 0. At the edge ofthe gap, the large peak in differential conductance on eitherside marks the threshold above which electrons can tunnelinto the dot. Beyond these initial rises inG(Vds), a seriesof additional peaks are seen. The characteristic spacing ofthese peaks inVds is about 0.1 mV and can be contrastedwith the Coulomb gap, which is about 0.6 meV in this case.

To interpret these data, consider figure 13, which showstwo schematic representations of the various occupied andempty energy levels of the QD with respect to its leads.The two drawings compare the classical Coulomb-blockade

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Figure 13. An energy-level profile of a QD and its leads,showing both occupied and vacant levels with aCoulomb-charging gap in between. (a) The classical casediscussed in section 3, with a continuous electronicspectrum. (b) The picture with a quantized spectrumsuperimposed upon the Coulomb gap. When one of theleads is gradually biased, current flow will begin when itsFermi level is aligned with the lowest level in the QD.Further biasing will show structure in the differentialconductance each time the Fermi level in a lead is alignedwith one of the levels in the QD.

picture, in figure 13(a), and the more relevant quantumcase, figure 13(b), which will be the focus of the followingdiscussion. Under increasingVds bias condition, the quasi-Fermi level on, say, the left-hand lead, is raised by thebias potential; initially no current flows because electronsat the quasi-Fermi level do not yet have enough energy toovercome the charging energy of the QD. Eventually,Vds

reaches the point at which an electron can tunnel from thelead onto the QD. This initiates current flow and a peak inG is observed. As the quasi-Fermi level is further raised,eventually a second available level becomes energeticallyaccessible. At this point, an electron tunnelling intothe dot from the biased lead can tunnel into either oneof two available states. These states can be differentnot only in energy, but also in their tunnelling rates.However, once one electron has tunnelled to either ofthese states, a second electron cannot tunnel into any otherstate until the first electron leaves the QD. This is becausethe Coulomb-charging energy associated withtwo excesselectrons prevents them from both occupying the QD at thesame time.

This point is an essential feature of nonlinear transportin QDs and can lead to complex structure inG. Forexample, if the dwelling time of an electron in the second

Figure 14. The experimental excitation spectrum of a QDextracted directly from the data of figure 12, withappropriate scaling of voltage to energy as discussed in thetext. Note the two characteristic energy scales, theCoulomb gap and the discrete level spacing.

level is longer than that in the first level, then asVds isincreased the net conductance of the dot decreases whenthe second level comes into play. This mechanism resultsin negative differential conductance,G < 0 [23, 38].

With this understanding in hand, it is clear that peaks,and negative valleys, inG correspond to discrete energylevels in the dot and that a measurement ofG can beexploited to map the excitation spectrum, or single-particlestates, in a QD. Carrying out this procedure on the datain figure 12 gives the spectrum shown in figure 14. Aseries of levels spaced apart by about 0.1 meV is seen,with a roughly 0.6 meV gap containing no states in theneighbourhood ofVds = 0. Following the reasoning ofsection 3.5, the position of differential conductance peaksin Vds has been converted to an energy scale by multiplyingVds by a term 1− Cd/C, which accounts for the fact thatthe potential in the QD partially follows the potential inthe biased lead due to the finite capacitanceCd betweenthe dot and the drain-lead. To make a connection with theclassical Coulomb-blockade model, note that, in the limitin which the spectrum of states becomes continuous, theclassical Coulomb-blockade picture is regained.

Finally, figure 15 shows the large-bias differentialconductance,G, measured at a sequence of plunger-gatevoltages [32]. Note that, in addition to the peaks inG,several valleys withG < 0 are observed. This plot presentsa synopsis of a QD spectrum, in which the features formdiagonal lines describing the evolution of energy levelswith the externally imposed potentials. Two commentsare appropriate for quantitative understanding of this plot.One point has been made above, namely that, as a resultof Cd , there is a non-unity conversion factor betweenVds

and energy. Another related issue is the fact that the gap(the horizontal width of the zero-conductance region) isobviously not constant. This too results from the capacitive

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Figure 15. Left-hand side: a two-dimensional grey-scale plot of G versus Vp and Vds in a planar QD by Weis et al [32].Darker regions correspond to higher G , including slightly negative values for the completely white streaks. Right-hand side:guidelines emphasizing the main features shown on the grey scale, mapping out the entire spectrum of a QD. Broken linesindicate negative G , dotted lines show suppressed features. Reproduced with permission of the authors.

effect of Vds on the dot potential,ϕ, in the following way.Depending on the relative position of the QD levels withrespect to the Fermi level, the conductance can be due notonly to the alignment of the drain voltage with the QDlevels (the usual picture, which would give a fixed gap),but also to alignment of theother lead, which is at fixedvoltage, with the (shifting) levels of the QD. This gives riseto conductance before the otherwise-anticipatedVth.

4.4. The theory of single-electron charging in thepresence of a discrete level spectrum

This section will present in more detail the theoreticaldescription of the implications of a discrete level spectrumfor transport through a QD. Begin by recalling figure 14,which shows the discrete spectrum of levels measuredabove and below the Coulomb gap. This picture, whichsimultaneously embodies the effects of single-electroncharging and discrete levels, will serve as the startingpoint in the following presentation of transport in QDs thatincorporates these two phenomena [223–229].

Aside from the Coulomb gap, the spectrum of statesshown in figure 14 is strongly reminiscent of the energylevels seen in the conventional non-interacting resonanttunnelling picture. The reality, however, is more subtle.Electrons in an actual QD comprise a many-body systemof strongly interacting particles in a complex potential.The spectrum of states in figure 14 belies this complexityand suggests a simplified formulation of transport in thesesystems, which has been embraced by several workers.Essentially, this view assumes that the energy of a QD,

E, is fully described by thesum of a Coulomb-chargingterm, as in the Coulomb-blockade model, plus the energiesof an ‘ad hoc’ set of discrete levels:

E(N) = (Ne)2

2C− ϕNe +

N∑εi (4.1)

whereεi represents the energy of theith eigenstate relativeto the Fermi level in the QD and the summation is over theset of occupied states [228, 229]. The actual values ofεi

need to be calculated numerically for a realistic geometry[53, 230].

Furthermore, each level is associated with a tunnellingrate between the left and right leads,0l

i and 0ri ,

respectively, which can vary substantially from one levelto the other. This implies that these levels have a finitewidth h0i , where 0i (the inverse lifetime) is given by0i = 0l

i + 0ri . For now it will be assumed that ¯h0i is

negligibly small with respect to all other energy scales.Working in this framework, transport through a QD has

been calculated both for large-bias [223–226] and for zero-bias regimes [228, 229]. The approach is straightforward,although the mathematical book-keeping can get quitecumbersome. The idea is to sum the contributions toconductance from all tunnelling processes over all thepossible configurations of the QD, in terms of (i) thenumber of electronsN residing in the dot and (ii) whichlevels they occupy. Each configuration must be given itsappropriate statistical weight, which in itself depends on thebias, temperature and tunnelling rates. The results of thisformulation are most clearly demonstrated in the limit ofsmall source–drain bias. Borrowing from Beenakker [228],

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the conductance in the small-bias case is given by

G = e

kT

∞∑i=1

∞∑N=1

0li0

ri

0li + 0r

i

Peq(N)Feq(εi |N)[1 − f (εi)]

(4.2)where Peq(N) is the equilibrium probability that theQD containsN electrons, Feq(εi |N) is the conditionalprobability that leveli is occupied, given that the QDcontains N electrons, andf (εi) is the Fermi–Diracdistribution with Fermi energy defined as zero.

This equation has two particularly simple limitingforms. One is in the limitkT � 1ε � U , and theother limit is when 1ε � kT � U , where 1ε isthe characteristic energy separation between single-particlelevels. In either casekT � U , which describes theexperimental situation when isolated conductance peaks asa function ofVp are observed.

In the first limit, when kT � 1ε, only tunnellingthrough one quantum state, closest to the Fermi level,contributes significantly to the small-bias conductance ofthe dot. Equation (4.2) reduces to

G = e2

4kT

0li0

ri

0li + 0r

i

cosh−2( εj

2kT

)(4.3)

where εj is the energy of the single-particle state closestto the Fermi level. Recalling that changing the bias on theplunger gate by an amount1Vp shifts the level ofεj (withrespect to the Fermi level) by an amount1ϕ = α1Vp,equation (4.3) is seen to describe the line shape of aconductance peak measured as a function ofVp. Theline shape in this case has an amplitude that scales as1/T , a width proportional to temperature with FWHM≈3.5kT /αe, an integrated area independent ofT , and tailsthat fall off exponentially inVp as exp(−αVp/2kT ). Thisline shape was observed experimentally [5] and is identicalto that of a thermally broadened resonance in the absenceof single-electron charging [231], as expected, since onlyone discrete state contributes to the conductance of the QDin the low temperature limit.

When 1ε � kT � U , equation (4.2) has a secondlimiting form. In this case, the summation over the set ofdiscrete statesεi in equation (4.2) reduces to an integral ofthe form∫ ∞

−∞dε f (ε)[1 − f (ε + εj )] = εj (1 − eεj /kT )−1 (4.4)

and the summation overN need only be evaluated at theone value ofN for which eN is closest to the equilibriumcharge on the QD. In this limit,P(N) is given by theclassical Boltzmann distribution. By further assuming auniform tunnelling rate for all levels,0l and 0r , throughthe left and right tunnel barriers respectively, equation (4.2)reduces to

G = e0l0r

0l + 0r

εj/kT

sinh(εj /kT ). (4.5)

In contrast to (4.3), this expression describes a lineshape with a temperature-independent amplitude and aFWHM = 4.35kT /αe, although the tails still fall off asexp(−αVp/2kT ). Its integrated area increases in proportionto T . This is the same line shape as that predicted by theclassical Coulomb-charging model [138, 149] mentioned insection 3.2.

Figure 16. The temperature dependence of G versus Vp ,showing a comparison between (a) experimentallymeasured data at four different temperatures and (b)numerical calculations [229] based on the theory discussedin this section, plotted in normalized units.

4.5. The position of the conductance peaks

From equation (4.1), the interpretation of the gate-voltageposition of the conductance peaks is revised with respect tothe classical picture. Conductance peaks, which correspondto charge-degeneracy points, are determined at very lowtemperatures by the conditionE(N) = E(N + 1), whichleads toeϕN+1 = (N + 1

2)e2/C + εN+1. In other words,the spacing between conductance peaks is, within a factorof αe (that translatesVp to εϕ), given by e2/C + 1ε,as opposed to juste2/C of the classical picture. Thus,the position of the conductance peaks contains informationabout the single-particle energies. In principle, unless thesingle-particle levelsεi are equally spaced, the conductancepeaks are not exactly periodic inVp. This is true as long askT � 1ε, for which peaks are narrow enough to resolvethe contribution of1ε. However, as long as the chargingenergy is larger than the level spacing, as is often the case inplanar QDs that are a few hundred nm in size, the deviationsfrom periodicity are relatively small.

4.6. Temperature behaviour of conductance peaks

The preceding theory invites a second look at theconductance of a QD as a function of plunger voltage andin particular the temperature behaviour of its conductance.

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Figure 16(a) shows the experimental conductance of aplanar QD plotted againstVp for four temperatures inthe range 0.2–1.25 K. At low temperatures, the heightsof successive peaks inVp vary non-monotonically andadjacent peaks are separated by broad minima. As thetemperature is increased, all the peaks broaden; theiramplitudes decrease in some cases and increase in othercases; in certain cases (the peak shown furthest to theright), the amplitude first decreases then increases as thetemperature is raised [6, 232]. Eventually, at the highesttemperature, peaks overlap significantly and the amplitudeof successive peaks inVp increases monotonically.

This behaviour has been interpreted within theframework of the model described above. At lowtemperatures, when1ε > kT , the conductance of anyparticular peak is entirely determined by the tunnellingrates 0l

i and 0ri of a specific single particle statei.

Large-amplitude peaks are associated with states thatare more strongly coupled to the leads; low-amplitudepeaks with weakly coupled states. As temperature isincreased and1ε ∼ kT , each conductance peak isinfluenced by contributions of tunnelling through severaldiscrete energy states, although still within the constraintof one electron at a time. The monotonic increase inpeak amplitudes at high temperatures simply reflects thetrend towards more strongly coupled levels at higherVp.Figure 16 shows the agreement between the temperaturedata and a numerical calculation incorporating discretelevels [229]. The results of this calculation suggest that1ε ≈ 0.05 meV. Similar temperature measurements [36]have explicitly demonstrated the crossover from the low-temperature resonant tunnelling line shape (4.3) to theclassical Coulomb-blockade line shape (4.5), manifested bythe re-scaling of the FWHM of peaks from 3.5kT to 4.35kT

askT approaches1ε.The entire discussion above ignored the level width ¯h0i ,

which is motivated by the assumption that ¯h0i � kT , evenat the lowest experimentally accessible temperatures. Ifthis were not the case, the conductance line shape wouldbe influenced by the level width. WhenkT � h0, oneexpects to observe a Lorentzian line shape in analogy withthe line shape for resonant tunnelling in the non-interactingcase, namely a Breit–Wigner resonance [233–236]. Infact, conductance peaks with line shapes correspondingto thermally broadened Lorentzians have been observedin planar QDs with strongly transmitting barriers [23].Coupled QDs, which will be discussed in section 6, alsoprovide further insight into line shape and level width [237].

4.7. A perspective on the meaning of the variousenergies

The addition spectrum is a clear and rigorously definedquantity. It is the minimum energy required in order toadd an electron to a closed system, in its ground state,and is essentially the total energy difference between therespectiveN -electron and the (N + 1)-electron groundstates. At a comparable level of rigour, the excitationspectrum corresponds to the energy differences between theground state and the higher-energy collective eigenstates, orexcitations, of a closed system withN electrons.

Where things become more tentative is the associationof the addition spectrum with two separate and additivecomponents, a charging energyU and a quantum energyε,the latter of which is also referred to as confinement energy,kinetic energy or single-particle energy. This is clearlya heuristic view, since, in principle, electron–electroninteractions inside the QD cause the Coulomb and kineticenergies to be irreparably intertwined and inseparable. Yetanother leap of faith is the explicit designation of a fixedvalue to the charging energy part of the addition spectrum,U = e2/C, independent ofN , in extension of the classicalnotion of capacitance. This usually goes hand-in-hand withan implicit assumption that the single-particle spectrum is insome sense unaffected by interactions. However, without asimple assignment of a value toU , the distinction betweenthe two energies becomes of little theoretical utility. Theadvantage of this construct is not only its simplicity, butalso that it links the two spectra intimately, by viewingboth as simple sums of independently definableU and εi

terms. The non-interacting single-particle spectrum in a QDis in itself of significant theoretical interest, and is relatedto extensively studied issues of quantum chaos and levelstatistics [47, 48, 52, 238–244].

The above approach can be appreciated in a historicalperspective. The temperature behaviour of conductancepeaks [6] underscored the limitations of the Coulomb-blockade model, and the hybrid model of ‘ad hoc’discrete levels and constantU = e2/C proved useful inunderstanding the experimental observations. Indeed, itwas understood from the outset that encapsulating electron–electron interactions solely in terms of classical chargingenergy, and then superimposing upon this a discreteset of quantum energy levels, was at best a simplifiedapproximation of the real system. The fact remains thattemperature-dependence and large-bias measurements bothindicate the existence of two energy scales, one of the orderof the classical charging energy and one comparable tothe average separation of single-particle states. The nextsection will demonstrate the concrete implementation ofthese concepts.

5. Transport through quantum dots in highmagnetic fields

In high magnetic fields, the general features of single-electron charging are well preserved. Quasi-periodicconductance or capacitance peaks, the Coulomb gap and theother hallmarks of single-electron charging appear in theirfamiliar form even as quantum dots are exposed to highmagnetic fields ofB ' 10 T. However, closer examinationreveals that a variety of effects emerge in magnetic fields.These usually manifest themselves as systematic evolutionin the voltage position and amplitude of the characteristictransport features, whether gate voltage peaks, capacitancepeaks or the structure in theI–V curves. The magnetic-field-induced behaviour of QDs has been exploited to gaina deeper understanding of QDs in general, whether in afinite or in zero magnetic field. The following discussionaims to give the reader a sense of some of the experimentaland theoretical insights acquired along this new and activeavenue of research.

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Figure 17. The addition spectrum of a QD in a magnetic field, measured by Ashoori et al [94] via capacitance spectroscopy.The grey-scale plot shows capacitance peaks corresponding to addition energies as bright areas, whose voltage (namelyenergy) changes with magnetic field. Thus the white lines trace the addition spectrum versus B . Each line corresponds to Nelectrons in the dot, with (a) showing N = 1–10 and (b) showing N = 6–35. Note the different magnetic field scales.Reproduced with permission of the authors.

5.1. The addition spectrum in a magnetic field

At T = 0, if an electron can freely tunnel into a QD, itindicates that there is an available charge state of the QDaligned in energy with electrons in the leads. This situationcan be identified by measuring the tunnelling current,either by conductance or by capacitance measurements,which show a sharp increase when this alignment occurs.In previous sections it was shown that, when single-electron charging predominates, this alignment occursnearly periodically as a function of gate voltage andcorresponds to the successive addition of electrons to theground state of a QD. In other words, the addition spectrumof a QD consists of a series of almost equally spacedlevels. Consider now this spectrum and its dependenceon magnetic field. Experimentally, this entails measuringthe gate voltage at which conductance or capacitance peaksoccur as a function ofB.

Figure 17 displays the evolution of the capacitancepeaks which were shown in figure 5, with increasingB

up to 10 T [94]. This is a two-dimensional grey-scale plot,in which the vertical axis is the voltage, that is the energyscale of electron addition. The capacitance is highest inthe bright areas; thus the bright lines trace the additionenergies as a function ofB. At B = 0, nearly periodic

peaks are observed, in the form of approximately uniformvertical spacing. AsB is increased, the position of thecapacitance peaks changes continuously and with ostensiblysystematic form. Particularly remarkable is the fact thatthis experiment probes the addition spectrum starting withthe very first electron in the dot. This small-N regime isusually not accessible in planar QDs.

Now consider a related experimental observation, thisone from conductance peaks of planar QDs. Figure 18shows the gate voltage associated with successiveconductance peaks for fields in the range 2.5–5.0 T [60]. Inother words, each point denotes a conductance peak, plottedversusVp andB. Once again, the gate voltage associatedwith a peak is related directly to the addition energyspectrum. An arbitrary vertical cross section through thisplot shows that, at any given field strength, peaks areapproximately periodic inVp, and thus the sequential peakstrace roughly parallel horizontal lines. However, smalldeviations from periodicity are present, of order 10%,which evolve withB and give rise to a variety of structurein the plot. Note the oscillatory fluctuations of peak positionin the neighbourhood ofB = 3.0 T and the relative positionof the fluctuations in adjacent peaks. At higherB thecharacter of the fluctuations changes and on occasion newpeaks emerge between existing ones.

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Figure 18. The addition spectrum of a planar QD in amagnetic field, extracted from the gate-voltage positions ofconductance peaks [60]. Each point on the plot is aconductance peak; thus the vertical scale corresponds toaddition energies of N electrons in the dot, for severalvalues of N .

5.2. Interpretations of the addition spectrum in amagnetic field

Forming an understanding of these experimental observa-tions has been the focus of considerable effort. An earlyinterpretation of the addition spectrum data in quantum dotsworked within the framework of the single-particle energiesmodel presented in section 4.4 [11, 228]. In this scheme,the separation between energy levels (peaks) is the sum oftwo components, a constant charging termU = e2/C and anon-interacting single-particle term,εi(B). Hence the nameof this model, the constant interaction (CI) model. This in-terpretation applied to the addition spectrum in magneticfields suggests that the complex evolution in peak positioncan be attributed to the nature of the energy levelsεi(B)

of non-interacting particles confined within the QD. As amatter of fact, the evolution ofεi(B) with magnetic fieldis relatively well understood and theoretically tractable; inparticular, an exact solution for the idealized case of a cir-cular, parabolic confining potential has been known for overhalf a century [245, 246] and has been reviewed in severalrecent works, such as [247].

At first sight, this picture gave surprisingly goodagreement with the observedB-dependence of the additionspectrum [11]. In particular, it explained the oscillatoryfeatures in figure 18, with additional supporting evidencethat will be mentioned later on. However, while appealingin its simplicity, the CI picture was soon understoodto have serious flaws. For example, it suggests thatZeeman splitting (for electrons with the same levelεi

but opposite spins) should be apparent in the additionspectrum. Experimentally it is not seen [11]. This andother quantitative discrepancies were compelling reasonsto go beyond the CI model.

A second interpretation of the addition spectra datamodels the interacting electron gas in a QD in a self-consistent manner [221]. Although in many ways thismodel is also a simplification, it presents a major change inthe entire view of the energy of a QD. The most important

point is that the energy of the QD is determined by the self-consistent internal Coulomb energy, which in turn dependson the detailed internal organization of electrons in the dot.This organization in itself depends onN, B, the confiningpotential and the electronic interactions, as the QD seeksits lowest energy configuration. It turns out that the lattervaries withB in a fashion which accounts for the observedB-dependence of the addition spectrum. In this sense, theB-dependence of the addition spectrum is a manifestationof changes in Coulomb energy. This is in contrast to the‘naive’ CI model, which viewed the Coulomb energy asconstant and the single-particle energy as solely responsiblefor the detailed structure in the addition spectrum.

A brief review of the main features of the self-consistentmodel is now presented. Details can be found in severalrecent works [248–250] and their references. The smallelectron gas in a QD is described as a confined collectionof fully interacting electrons. As an approximationthe in-plane confinement is assumed to be provided bya circular two-dimensional parabolic potential, namelyVext = 1

2m∗ω20r

2. At B = 0 the electronic chargeeNdistributes itself in a dome-like, or hemispheric, densityρ(r), which is shown schematically in figure 19(a).ρ(r) ishighest in the middle and gradually decreases to zero at adistanceR from the centre, whereR increases roughly asN1/3.

However, in a magnetic field, the quantization ofmagnetic energy into Landau levels (LL) can drasticallymodify this charge distribution. Each spin-polarizedLL can accommodate a certain maximum density ofelectrons, which is given byeB/h ≈ 2.4 × 1010 cm−2

at B = 1 T. The filling factor, ν, defined as the ratiobetween the 2DEG density andeB/h, is a measure ofthe number of occupied LLs. In trying to minimizethe energy by preferring occupation of the lowest LLs,ρ(r) re-distributes in concentric circular strips, alternatingbetween ‘incompressible’ strips where the density is flat(and equal toieB/h such thati LLs are exactly full),and ‘compressible’ strips with a radial density gradient.Although the resulting redistribution of charge will havean increased electrostatic energy, this will be sufficientlycompensated for by the lowering of the magnetic energyof the system. The actual redistribution of charge needsto be calculated numerically. Figure 19 schematicallydepicts this situation for the case of 1< ν < 2, namelyone incompressible strip withi = 1. The self-consistentenergy of the system for a givenB also comes out of thiscalculation, and hence can be used to derive the additionspectrum of a QD as a function ofB. This is at timesreferred to as the ‘wedding cake’ model, due to the step-like profile of the charge density seen in figure 19(b).

Perhaps the most important point to understand is thatthe incompressible strips, although occupied with charge,are effectively insulating regions, since no changes in thecharge density can take place in them. Thus the QD iseffectively turned into several concentric dots (or rings),with mutual interactions, and insulating strips betweenthem. One attractive simplification is to model theirinteraction by mutual capacitances [39, 249] as depicted in

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Figure 19. The self-consistent distribution of chargedensity in a circular QD. (a) At B = 0, ρ(r) is a smoothlydecreasing function of r . (b) At sufficiently high fields, the‘wedding cake’ distribution is formed, as explained in thetext. A plateau in ρ(r) occurs around values correspondingto the filling of complete Landau levels, where ρ = ieB/h(only i = 1 is shown here). (c) A top view of this stateshows the formation of compressible (grey) andincompressible (white) strips. The electrostatic interactionbetween the strips can be approximated by an effectivecapacitance C12, as discussed in the text.

figure 19(c). The key is that this capacitanceCij is B-dependent, due to theB-dependence of the width of thevarious strips.

The number and width of the strips changes withB

and in turn modifies the internal Coulomb energy of theelectrons. AsB increases, the lower LLs can accomodatemore and more electrons in the confined area at theirdisposal. Thus there is a gradual depopulation of the higherLLs (the upper layers of the ‘wedding cake’) into the lowerLLs. Generally, each magnetic-depopulation event – thetransfer of an electron from a higher LL (inner compressibleregion) into a lower LL (outer strip) – corresponds to achange in theB-dependence of the total energy, dE/dB,and hence to one of the wiggles seen in figures 17 or18. This is simply because the energies of higher LLshave a steeper dependence onB. The pronounced dropin the addition energy marked by triangles in figure 17(b),is probably associated withν = 2, namely the completedepopulation of the second orbital LL (still leaving twospin levels). The identification of this point at a givenN

can be used to determineR, the size of the QD [94].Despite the success and simplicity of the self-consistent

model, it still does not fully describe the ground state of aQD. Absent in this model are the effects of exchange and

correlations, which are known to play an important role,for example, in describing spin polarization, particularly inhigh magnetic fields [251–253]. The simplest observationwhich already calls for exchange interaction is seen infigure 17(a), in which the circles indicate a cusp in theaddition spectrum. Similar features have recently beenobserved in conductance measurements as well [77, 90].This cusp has been attributed to exchange-induced spinpolarization [77, 90, 94]. Recent detailed measurement andanalysis of the addition spectrum in planar QDs [254]at ν < 2 has indicated the importance of exchangeinteraction in quantitatively accounting for the magnetic-field-induced spin polarization in a QD with a largernumbers of electrons. There are other features at higherfields [94] which are not accounted for within the simpleself-consistent model.

Eventually, at sufficiently highB, all electrons arein the lowest LL, and the features associated withdepopulation of higher LLs (or spin states) are no longerseen. This is theν = 1 filling factor of the quantumHall effect. As B is further increased, the LL densityof stateseB/h increases and the dot can – in principle– shrink further, thus gaining confinement energy, butincreasing internal Coulomb repulsion. This is the compact,or maximum-density-droplet, phase of the QD; it is a‘maximum-density’ phase in the sense that the electrons arein the most compact filling of states allowed by the Pauliexclusion principle, thus minimizing confining-potentialenergy, but at the expense of Coulomb repulsion energy.The compact phase is predicted to be the lowest energy statefor a certain range ofB values [255]. The correspondingdensity of this state increases withB, and therefore theCoulomb repulsion energy increases. Eventually, the QDwill resist further shrinkage and re-distribute the electronsin complex fashions [255, 256]. Interestingly, non-trivialfeatures have been observed in the addition spectrum inthis regime [94, 254]. It has been speculated that they arerelated to internal reconstruction of the density distributionin the QD. These results and ideas are new and tantalizing,but it is still early to assess their validity and implications.

The study of many-body effects in QDs, usingnumerical and analytical tools, remains a major theoreticalfront [257–272]. In the high-magnetic-field limit, therehave been some specific predictions for the regime of thefractional quantum Hall effect [94, 262, 273, 274] and of thepossible formation of a ‘Wigner molecule’ [275], whichwill not be detailed here. It is clear, however, that afull account of the spectra of QDs eventually requires theconsideration of the entire many-body problem.

5.3. Tunnelling rates through a quantum dot in amagnetic field

The previous section focused only on the addition energy ofa QD which is determined from the position of conductanceor capacitance peaks. Now consider theB-dependence ofthe coupling to the leads, or tunnelling rate,0i , associatedwith the various levels. When temperature is sufficientlylow compared to the spacing between discrete states in adot,kT < 1ε, the coupling strength of the highest occupied

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Figure 20. Conductance peaks versus plunger-gatevoltage in a high magnetic field, from van der Vaart et al[30]. The peak heights vary cyclically, with three peaks percycle, corresponding to three occupied Landau levels (LLs),as explained in the text. The highest peaks correspond tothe outermost states, which belong to the lowest LL. Theinset, taken on the same device under slightly differentconditions, demonstrates that the internal order in eachcycle is not universal. Reproduced with permission of theauthors.

level of a planar QD is manifested in the amplitude of itsassociated conductance peak. Figure 20 shows successiveconductance peaks as a function of plunger voltage inhigh magnetic fields. This striking behaviour, marked byperiodic modulation of the peak amplitudes, has been seenin several QD systems [18, 30, 35, 39, 276]. Depending onthe field regime, the period of the modulation has beenobserved to range between two and as many as eightpeaks per cycle. The period of the modulation envelopeappears always to be an integer multiple of peak spacings,essentially corresponding to the number of occupied LLs.

These observations have been interpreted both withinthe CI model [18, 30] and within the wedding-cake model[39, 276]. In both descriptions, the magnetic field confineselectrons to different sets of levels, some more localizedtowards the centre of the dot and others more towards itsperiphery. It should be pointed out that the single-particlestates can also be assigned a LL index corresponding totheir asymptoticB-dependence at highB. These statescirculate along the QD’s boundary, and are therefore calledcirculating edge states. More will be said about this in thenext section. Generally, at any given energy, higher-indexLLs – if occupied – are confined more towards the interior.In this respect the CI and the ‘wedding cake’ models arequite similar.

In general, peak height reflects thespatial distributionof the highest energy occupied state in the QD, namelythe state into and out of which electrons will repeatedlytunnel, in the course of the conductance measurement.Hence, peak height modulation is related to changes inthe spatial distribution of theN th state, whereN isthe number of electrons in the QD. If this uppermostelectron state belongs to a lower-index Landau level –which would be an outer strip in the self-consistent model,or an outer edge state in the CI model – then it is

closer to the tunnel barriers and hence the tunnelling ratewill be relatively high. The electrons confined to theinterior have a lower tunnelling rate. In other words,because the uppermost electron belongs to a more internal,higher index LL, the conductance peak associated withit will be lower. Furthermore, as electrons are addedto the QD, they populate the various LLs, or wedding-cake layers, in an approximately regular cycle. Hence, inthe course of sequential conductance peaks, the latter arecyclically associated with more strongly (lower LL) andmore weakly (higher LL) coupled states. This explainsthe cyclic modulation of the conductance peak height andrelates its period to the number of compressible strips (orLLs). Both models give the same general features, butthe self-consistent model was more successful in terms ofquantitative agreement with experiments [39, 276]. In anycase the detailed behaviour depends upon the particulargeometry and confining potential [277].

Perhaps even more striking is the evolution of theheight of a specific peak asB is changed. A specific peakcorresponds to a fixed number of electronsN . However,as B is varied and the internal structure evolves, theN thelectron – the only one that contributes to transport – canalternately find itself in different parts of the QD, withresulting variation in the height of theN th conductancepeak. Indeed, such were the experimental observations[6, 11], as shown in figure 21. Heights of each peak arefound to vary by more than an order of magnitude withB and in synchronization with the changing slope dE/dB

in the addition spectrum. The correspondence between thechanging addition energy (peak position) and tunnelling rate(peak height) were a compelling factor in the experimentalobservations and their interpretation.

Although observed peak-height variations are quitelarge – more than an order of magnitude – they are stillmoderate with respect to theoretical expectations, becausethe tunnelling probability into the inner LLs should beexponentially small. A recent experiment [43] has showntemporal switching in the peak positions at high magneticfield, which has been attributed to bistability in the internalcharge distribution of the QD. The characteristic time forthese switching events is on the scale of many seconds.This interpretation suggests extremely slow rates associatedwith tunnelling into the internal part of the dot. Thevery weak coupling between edge electrons of differentLL indices, although still not quantitatively understood,is consistent with observations of long-range adiabatictransport in edge states of macroscopic samples [54, 278].

In vertical tunnelling, the strong relation between radialdistribution and tunnelling rate is not present. Furthermore,in capacitance measurements the peak capacitance is nota direct measure of the tunnelling rate, but rather of theelectronic charge transferred in each AC cycle. Still, ifthe tunnelling rate drops below the AC frequency, thecapacitance signal will be suppressed. Indeed, suppressionof tunnelling in high magnetic fields has been observed toappear in conjunction with particular features in the additionspectrum [94, 95]. Some of these features are associatedwith integer filling factors, but the reason for suppressionof tunnelling rate is not yet understood.

275


Figure 21. The dependence of the tunnelling rate on magnetic field, B , with a fixed number of electrons N in the QD. Thegraph [6] shows peak height for a small range of B , plotted together with the peak position. The tunnelling rate is suppressedwhen the N th electron belongs to a higher LL, since the latter are more strongly confined to the centre of the dot, away fromthe leads. Higher LLs are also associated with a faster rising energy as a function of B , hence the drop in peak heightcoincides with an upward slope in the peak position.

5.4. Adiabatically transmitted edge states andCoulomb blockade in quantum dots

The formation of concentric sets of circulating orbitals, eachset associated with a separate magnetic LL, gives rise to avariety of phenomena in QDs [247]. In particular, magneto-conductance oscillations inB, reminiscent of Aharonov–Bohm oscillations but occurring in a simply connected dot(as opposed to a ring), have been predicted [279] andobserved [2]. These occur in a QD when electrons canenter adiabatically, rather than in the tunnelling regimeassociated with single-electron charging. Referring tothese oscillations as the Aharonov–Bohm effect is perhapsmisleading. In fact the oscillations have to do with themagnetic evolution of the energy levels of electrons in theQD and their cyclic alignment with the Fermi energy.

As the point-contact entry ports to the QD are graduallypinched off, the higher index (higher energy) LLs are thefirst to be depleted in the vicinity of the point contact, thusforming closed orbits inside the QD, while lower LLs maystill be classically transmitted. Ultimately, when the point-contact gates are sufficiently biased, all LLs are confined tothe dot, returning to the ordinary single-electron chargingsituation described above. These various possibilities aredepicted in figure 22.

Interestingly, in the intermediate regime in whichsome edge states are transmitted and others are confined,shown in figures 22(b) and (c), single-electron chargingwill still play a role in transport. Several groups havestudied this regime [13, 17, 19, 24, 28, 33, 250, 280–284].The phenomena can most simply be understood usingthe following general line of thought: closed edge statesbehave like isolated QDs exhibiting charging effects and

energy quantization effects, while the transmitted edgestates play a role similar to current leads. It is true thatthe actual measured current does not consist uniquely oftunnelling current – much of it is simply the contributionof the classically transmitted edge states, but changes inthe conductance as a function ofB or Vp can be attributedto the contribution of tunnelling between the transmittededges to the internal isolated LLs, which in itself isgoverned by single-electron charging effects. Once again,the partially transmitting barrier situation can be discussedin the language of the CI model, as above, or in a morecomprehensive self-consistent approach [250].

The observed phenomena can become relativelycomplex, especially if there are more than two LLs. Forexample, consider the effect of a plunger gate. AsVp (andhence the potentialϕ) is lowered and charge is added tothe QD, the added charge will try to distribute itself amongthe various edge states. Enhanced tunnelling into anyparticular isolated LL will occur only when that LL is onthe threshold for gaining an electron (a charge-degeneracypoint), which occurs less frequently inVp than the generalevent of adding an electron to the entire QD. Roughly, ifthere areν populated LLs, all with comparable capacitivecoupling to the plunger gate, then a full charging cycle willaddν electrons to the entire dot, one per LL. Thus single-electron tunnelling resonances will be of several families– one perclosed LL – and in each family the periodicityin Vp will be approximatelyνe/Cp, as opposed toe/Cp

for the fully isolated QD [13, 281]. Alternatively one cansweepB. Here, the most salient effect [33, 247, 280, 283] isthat the so-called Aharonov–Bohm oscillations of the fullytransmitted edge states are suppressed, or ‘delayed’, as aresult of the single-electron charging energy involved in

276


Figure 22. A quantum dot in high magnetic fields with three edge channels, or Landau levels, with varying degrees ofpinching voltage on the entry port gates: (a) fully formed tunnel barriers and closed orbits for all Landau levels, withtunnelling only; (b) one transmitted edge channel and two closed orbits and (c) two transmitted edge channels. As explainedin the text, single-electron charging affects the tunnelling of electrons into the closed orbits.

the magnetic re-alignment of energy levels in the dot.As a final, perhaps obvious remark, the partially

transmitting barriers in the quantum Hall regime, althoughhaving resistances well below the quantum resistanceh/e2,should not be viewed as contradicting the concept ofminimum resistance for the Coulomb blockade. This isbecause the blockade here is only affecting the contributionof the closed LLs to the current, whose coupling to thecurrent leads does not determine the measured resistance.

5.5. Zeeman spin splitting in the single-particlespectrum

At intermediate magnetic fields, electrons with both spinstates occupy the QD and participate in tunnelling. Ifeach single-particle state is doubly degenerate, pairingshould be observable in the addition and excitation spectracorresponding to the two spin states of each specific level.Generally this is not the case, although some suggestivedata have been observed [31]. This should not come as asurprise, in the light of the above discussion; interactionsare most likely to remove any spin degeneracy. Still, themagnetic-field-dependence of the spectrum should have asmall contribution from the Zeeman energy associated withthe electrons’ spin.

In a perpendicular magnetic field the Zeeman energyis very small compared to the orbital magnetic energy,thus making it quite difficult to observe even within thetheoretical perspective of the simple CI model. However,when B is parallel to the plane of the 2DEG, orbitaleffects are very small whereas the Zeeman energy remainsessentially the same. This fact was used by Weiset al [32]to observe the contribution of parallel-B Zeeman energy tothe electronic energy, as shown in figure 23. The spectrum,extracted from differential conductance peaks in a planarQD, is shown to evolve with the intensity ofB in a waywhich classifies the entire spectrum into two categories,presumably associated with the two possible values of spin.

6. The present outlook

At the time of writing, active research continues on variousaspects of single-electron tunnelling. This final section willattempt to describe some of the additional issues whichare on the agenda. Naturally, this should be taken inthe appropriate perspective, with the understanding that,as in any other field, the ‘frontiers’ are neither constant norobjectively definable.

6.1. Coupled quantum dots

If a quantum dot can be viewed as an artificial atom [134],then the next natural logical step might be to create andstudy artificial molecules, in other words, two or moreQDs in close proximity, allowing electrostatic interactionand tunnelling between them. Like many other aspectsof single-electron tunnelling, work on arrays of metallictunnel junctions preceded work on their semiconductorcounterparts. These were viewed primarily in the context ofclassical charging effects. On the other hand, early interestin chains of semiconductor QDs was motivated by theobjective of realizing an artificial one-dimensional lattice[14, 285].

The case of two coupled QDs has generated increasinginterest in recent years, both theoretical [286–293] andexperimental [14, 22, 220, 237, 294–299], and further workis presently being pursued by several groups. Double-QD systems are already quite complex in their behaviour.Electrostatics by itself can give rise to a richnessof charge states and transport states of this system[286, 287, 296, 297], due to inter-dot capacitance. Stronginter-dot tunnelling will further change the single-particlespectrum as well. The modification of energy levels due tointer-dot tunnelling and Coulomb interactions is essentiallythe analogue of chemical bonding between atoms in theformation of molecules. Furthermore, when two differentQDs are placed in series, a smaller dot can be used as aspectrometer for a larger dot [87, 237].

Larger periodic arrays of QDs have also been realized[300]. These can be viewed as artificial crystals [301] andhave even been considered as potential electronic networks

277


Figure 23. The spectroscopy of a QD in a magnetic field parallel to the plane 2DEG, measured by Weis et al [32]. (a)Differential conductance G in grey scale (black corresponding to G ≥ 1 µS), at fixed Vds = 0.7 mV, mapped as a function ofmagnetic field and plunger-gate voltage. (b) Black circles indicating peaks of G from (a). The fitted curves are all parabolasof two types, differing only by a term linear in B . This linear term has been interpreted as a manifestation of the Zeemanenergy. Reproduced with permission of the authors.

[216]. Generally, the field of coupled QDs is one in whichmuch progress is likely to take place in the near future.

6.2. Coherence

One of the persistent mysteries of resonant tunnellingthrough two or more sequential barriers has been thequestion of whether the process is coherent or not. Inother words, is there a deterministic relation between thequantum mechanical phase of the incoming electron andthe transmitted electron emerging on the other side. Thereason the question is difficult to answer is because themeasured quantity – the conductance – depends only onthe squared amplitude of the transmitted wavefunction, andis insensitive to its phase. For this reason, the line shapesof conductance peaks, or the temperature dependence, forexample, could not be used to distinguish between thetwo a priori conceivable possibilities. The ambiguitycould be removed if the transmission amplitudes of thetwo barriers were known separately, in addition to theircombined transmission amplitude. This was the basis foran early attempt to study the coherence of transmissionthrough planar QDs in high magnetic fields [4]. Morerecently, an interference experiment demonstrated phasecoherence of electrons tunnelling through a planar QD[45]. In this experiment a QD was embedded in onearm of an ‘Aharonov–Bohm ring’. The conductanceof the ring showed oscillations periodic in the magneticflux, namely Aharonov–Bohm interference oscillations,indicating that a significant part of the single-electrontunnelling current transmitted through the QD is coherent.

This is not unreasonable given the characteristic rate ofphase breaking events in a 2DEG at low temperatures,although little is known about phase breakingwithin aQD. Although the present understanding of that experimentis that the geometry was not suitable for measuring theactual phase shift experienced by electrons in the QD[302], the coherence of the tunnelling process appears wellestablished. This new result raises interesting questionsregarding the degree of coherence under various conditions.

6.3. The Kondo effect in quantum dots

The Kondo effect has been a long-standing area ofinvestigation in solid state physics [303]. Essentially,this effects describes the crossover from weak to strongcoupling between a solitary spin (magnetic) impurity and asea of conduction electrons. In bulk metals, this effect isassociated with an anomalous increase in resistance uponcooling to very low temperatures for metals with Kondoimpurities. It has been suggested that QDs may be anovel system in which to observe this effect [304, 305].A QD with an odd number of electrons, and thereforea net spin, is envisioned to form a Kondo transmissionresonance with the electrons in the leads of the dot [304–309]. This effect may manifest itself at low temperatures asa dramatically increased conductance between every otherconductance peak, when an odd number of electrons residesin the QD. Theoretical work has suggested that the zero-bias conductance of a QD may show Kondo enhancement attemperatureskT ' 0.1h0, whereh0 is the non-interactinglevel width described in section 4.4. A value of about

278


10 µeV has been reported for ¯h0 in QDs with low tunnelbarriers [23], which corresponds to a temperature of about0.1 K. This temperature is at the border of what canbe reached with conventional techniques, but it has beensuggested that Kondo effects may in fact be observableat much higher temperatures in large-bias measurements[305, 307, 308, 310–312]. This scenario has been referredto as ‘the Kondo problem out-of-equilibrium’, in referenceto the fact that, unlike the conventional Kondo effect, thisproposed resonance occurs between two electron baths atdifferent chemical potentials. As such, it may serve as newgrounds for studying the Kondo effect both experimentallyand theoretically.

6.4. High-frequency phenomena

Most of the results and experiments discussed so farhave been ones that are associated with steady statetransport, namely DC measurements. Even the turnstileeffect described in section 3.9, though being inherentlyan AC measurement, should, in principle, persist downto arbitrarily low frequencies. On the other hand, thesuppression of tunnelling in the capacitance measurementsof Ashoori et al [94, 95] can be referred to as a high-frequency effect, because the tunnelling rate drops belowthe AC frequency of order 200 KHz.

However, more dramatic frequency-dependent phenom-ena are also under study. One of them is photon-assistedtunnelling, namely tunnelling in the presence of an exter-nally applied microwave or far-infrared field. Tunnellingthrough QDs in the presence of AC fields has become asubject of increasing interest [311, 313–317]. Enhance-ment of tunnelling in QDs has been experimentally ob-served [318] when the energy of one (or several) photon(s)corresponds to the energy difference between the incomingelectron and an available state of the QD, into which it tun-nels. Thus the microwave frequency,f , can be used as anadditional spectroscopic tool to study the energy levels ofQDs, and as such may become more common in the nearfuture. In fact, ‘unintentional’ photon-assisted tunnelling,due to background electromagnetic radiation, has been re-ported in metallic double-junction systems [198, 319, 320].Furthermore, not only photons but also phonons [321] caninfluence tunnelling.

As mentioned before, far-infrared absorption spec-troscopy is an important tool in the study of QDs; practi-cally this requires a large array of nominally identical QDs.Although these do not fall within the scope of the presentreview, they are an important tool for studying the excita-tion spectra of QDs [117].

Concerning another aspect of high frequencies,temporal correlations between single-electron tunnellingevents, as a result of the Coulomb repulsion, have beendiscussed in the literature [147, 150, 322–326]. Thesecould manifest themselves, for example, in the amplitudeand spectral distribution of shot noise associated with aDC tunnelling current. Indirect evidence suggestive oftime-correlated tunnelling in arrays of metal junctions hasbeen obtained from microwave-assisted transport [327],but direct observation of spontaneous temporal correlations

is experimentally much more difficult. To date noexperimental manifestation of temporal correlations insemiconductor QDs has been reported.

Acknowledgments

The list of people who have over the years contributedto our own understanding of the subject would include alarge part of the workers in the field – through privatediscussions, in scientific meetings, and by published work.However, we would like to acknowledge the help thatwe received in the task of preparing the manuscript andin providing us with data that were shown here by RayAshoori, Rolf Haug, Moty Heiblum, Marc Kastner, LeoKouwenhoven, Yasushi Nagamune and David Abusch-Magder. We also acknowledge the support of the BasicResearch Foundation administered by the Israeli Academyof Science and Humanities.

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TOPICAL REVIEW Single-electron phenomena in ...chem.ch.huji.ac.il/~porath/NST2/Lecture 7/Meirav SET Semi...single-electron tunnelling, or Coulomb-blockade theory. As such this is a

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