Appl. Phys. A 71, 367–378 (2000) / Digital Object Identifier (DOI) 10.1007/s003390000550 Applied Physics AMaterialsScience & Processing

Effects of Coulomb interactions on spin states in vertical semiconductorquantum dotsS. Tarucha1,2,∗, D.G. Austing2, S. Sasaki2, Y. Tokura2, W. van der Wiel3, L.P. Kouwenhoven3

1 Department of Physics and ERATO Mesoscopic Correlation Project, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan2 NTT Basic Research Laboratories, 3-1, Morinosato Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan3 Department of Applied Physics and ERATO Mesoscopic Correlation Project, Delft University of Technology, P.O. Box 5046, 2600 GA Delft,The Netherlands

Received: 14 April 2000/Accepted: 17 April 2000/Published online: 6 September 2000 – Springer-Verlag 2000

Abstract. The effects of direct Coulomb and exchange inter-actions on spin states are studied for quantum dots containedin circular and rectangular mesas. For a circular mesa a spin-triplet favored by these interactions is observed at zero andnonzero magnetic fields. We tune and measure the relativestrengths of these interactions as a function of the number ofconfined electrons. We find that electrons tend to have paral-lel spins when they occupy nearly degenerate single-particlestates. We use a magnetic field to adjust the single-particlestate degeneracy, and find that the spin-configurations in anarbitrary magnetic field are well explained in terms of two-electron singlet and triplet states. For a rectangular mesa weobserve no signatures of the spin-triplet at zero magneticfield. Due to the anisotropy in the lateral confinement single-particle state degeneracy present in the circular mesa is lifted,and Coulomb interactions become weak. We evaluate thedegree of the anisotropy by measuring the magnetic field de-pendence of the energy spectrum for the ground and excitedstates, and find that at zero magnetic field the spin-singlet ismore significantly favored by the lifting of level degeneracythan by the reduction in the Coulomb interaction. We also findthat the spin-triplet is recovered by adjusting the level degen-eracy with magnetic field.

PACS: 72.20.My; 73.23.Hk; 73.61.Ey

Transport measurements of semiconductor nanostructureshave revealed a variety of phenomena associated with thequantization for both energy level and charge. Resonant tun-neling through a small quantum box, which is weakly coupledto the leads, shows a series of current peaks when the Fermienergy in the emitter lead and discrete zero-dimensional (0D)levels in the box are energetically aligned [1]. This is wellunderstood within the framework of single-particle tunnelingtheory, which does not incorporate the effect of Coulomb in-teractions. On the other hand, when electrons are effectivelytrapped in the box during transport, the box acts as a Coulomb

∗Corresponding author.(Fax: +81-3/5841-4162, E-mail: tarucha@phys.s.u-tokyo.ac.jp)

island, and the effect of Coulomb interactions leads to thequantization of charge [2]. The trapping of an extra electronraises the electrostatic potential of the island by the charg-ing energy. This energy regulates one by one the numberof electrons on the island, and gives rise to oscillations inthe tunneling conductance (Coulomb-blockade oscillations)when the gate voltage is swept. These oscillations are usuallyperiodic and explained using an orthodox Coulomb-blockadetheory when the number of electrons or the island size is“large” [2]. However, in a “small” dot holding just a few elec-trons, both effects of energy and charge quantization signifi-cantly influence the electronic states (“many-particle states”).Such a system can be regarded as an artificial atom [3]. Re-cently we have fabricated vertical quantum dots in circularmesas [4]. Associated with the rotational symmetry, as wellas the parabolicity in the in-plane two-dimensional (2D) har-monic confinement potential, atom-like properties such asshell-filling and obeyance of Hund’s first rule are all ob-served [5]. The high cylindrical symmetry of the circular dotleads to maximal level degeneracy of single-particle statesfor the 2D parabolic confinement. Consecutive filling of eachset of degenerate 0D states is directly responsible for thecharacteristic shell structure. On the other hand, Hund’s firstrule is associated with the interaction effects for the electronfilling amongst nearly degenerate states. Ferromagnetic fill-ing is then favored amongst half-filled degenerate states ineach shell. Breaking the circular symmetry by deforming thelateral confining potential, lifts the degeneracies present ina circular dot, and destroys the distinctive shell structure forthe circle, and modifies other atomic-like properties [6, 7].

In this paper we study the effects of Coulomb interactionson the spin states in the circular and elliptically deformedquantum dots. Adding electrons to small quantum boxes costsa certain energy and simultaneously changes the configura-tion of quantum states. A change in spin is also associatedwith a certain change in energy to minimize the energy ofthe many-particle ground state. For example, exchange en-ergy is gained when electrons are added with parallel spins ascompared to antiparallel spins. Depending on the characteris-tics of the box, a large total spin value (parallel spin filling)or a minimum total spin value (antiparallel spin filling) is

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favored. In semiconductor quantum dots alternating spin fill-ing [5] as well as spin-polarized filling [8] have been reported.However, it is usually difficult to assign the way of spin fillingfrom the electronic configurations of the quantum states be-cause of the uncertainty in the characteristic of the quantum-mechanical confinement. On the other hand, in our circularquantum dots single-particle states are well defined for a 2Dharmonic potential, and using these single-particle states, weare able to assign the many-particle states from measurementsof magnetic field dependence of Coulomb oscillations [5].Furthermore, this allows us to discuss the way of spin fillingfor a given charactersitic of the quantum states. Here, we puta special emphasis on the spin states influenced by interac-tions between two electrons. When the single-particle statesin the dot are separated by a large energy,∆E, an antiparal-lel spin filling is favored. For small∆E, parallel spin fillingis observed, which is in line with Hund’s first rule. We usea magnetic field,B, to tune∆E(B). This allows us to alter theway of spin filling. We start by using a quantum dot in a circu-lar mesa to study the interaction effects on the spin filling. Weexperimentally distinguish the contributions between directCoulomb (DC) and exchange (EX) to Hund’s first rule at zeroand nonzero magnetic fields and also as a function of numberof electrons in the dot. We show that Hund’s first rule for eachshell at zero magnetic field is only associated with EX effects,whereas Hund’s first rule in the presence of magnetic fieldis influenced by both of DC and EX effects. The obtainedenergies of DC and EX compare well to calculation for two-interacting electron singlet and triplet states. We then moveon the rectangular mesa to discuss the effect of anisotropy inthe lateral confinement on the spin states. Anisotropy in thelateral confinement gives rise to lifting of the single-particlelevel degeneracy present in the circular dot, and reduces theinteraction effect. We measure excitation spectra of the rect-angular mesa to study modifications to the spin-singlet andtriplet states due to the anisotropy, and find that antiparal-lel spin filling is readily favored at zero magnetic field evenfor a small anisotropy in the lateral confinement. We calcu-late the energies for the level degeneracy lifting and the DCand EX interactions as a function of anisotropy in the lateral

Fig. 1. a Schematic diagram of a quantum dot in a vertical device. The quantum dot is located inside the 0.5-µm-diameter pillar and is made from a double-barrier structure [2].b Scanning electron micrographs showing submicron-sized circular and rectangular mesas.c Effective characteristic energy of lateralconfinement,hω0, evaluated for various numbers of electrons in the circular dot. The derivation ofhω0 is explained in the text

confinement. This calculation indicates that a spin-singlet issignificantly favored in an elliptic dot at zero magnetic fieldby the lifting of level degeneracy rather than by the reductionin the Coulomb interactions. In the presence of a magneticfield, however, we find that parallel spin filling is recoveredwhen the single-particle level degeneracy is adjusted.

1 Quantum dot in circular mesa

1.1 Devices

We use a double-barrier structure (DBS) to fabricate verticalquantum dots [4]. The DBS consists of an undoped 12-nmIn0.05Ga0.95As well and two undoped Al0.22Ga0.78As barriersof thickness 9.0 and 7.5 nm, and this is processed to forma circular mesa with a nominal top-contact diameter,D, of0.54µm. Figure 1a shows a schematic diagram of the device.Above and below the DBS there is ann-doped GaAs contact(source and drain), and the circular dot is located between thetwo heterostructure barriers. A single Schottky gate is placedon the side of the mesa, so it is wrapped around the dot. Fig-ure 1b shows typical scanning electron micrographs of a cir-cular mesa, and a rectangular mesa (discussed later) takenimmediately after the deposition of the gate metal. The dot isstrongly confined in the vertical direction by the heterostruc-tures, whereas it is softly confined in the lateral direction bythe Schottky-gate-induced depletion region. This depletionregion is well approximated by a harmonic potential, and thecharacteristic energy,hω0, is evaluated from measurementsof the magnetic field dependence of the Coulomb oscilla-tions [5]. The values ofhω0 obtained for various numbers ofelectrons in the dot are shown in Fig. 1c (The derivation ofhω0 is described in Sect. 1.2). This figure clearly shows thatthe effective lateral confinement becomes weak as the numberof electrons in the dot increases, due to the effect of Coulombscreening.

The current,I , flowing vertically through the dot is meas-ured as a function of gate voltage,Vg, in response to a dc volt-age,V , applied between the contacts. For an arbitrarily small

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V , a series of current peaks (i.e. Coulomb oscillations) resultsfrom changing the number of electrons in the dot,N, one byone [5]. The position of a current peak for the transition fromN −1 to N measures the ground state (GS) electrochemicalpotential,µ(N). When no current flows (Coulomb blockade),the number of electrons in the dot,N, is well defined. Tocharacterize the width of the Coulomb blockade or the spac-ing between peaks we define∆(N) = µ(N +1)−µ(N). TheGSs and excited states (ESs) are both measured whenV is setto a value sufficiently greater than the excitation energy [9].For these measurements the sample is cooled down to about100 mK.

1.2 N-electron ground states in the presence of magneticfield

In real atoms, electrons are so strongly trapped that theirquantum mechanical properties cannot be strongly modifiedunder normal experimental conditions, for example, by ap-plying a magnetic field. In contrast, the electrons in our quan-tum dots are bound in a relatively large region of the orderof 100 nm. This allows us to use readily accessible magneticfields, not only to identify the quantum mechanical states, butalso to induce transitions in the GSs whose counterparts inreal atoms can never be tested on earth.

The eigenstates for a 2D harmonic quantum dot in thepresence of magnetic field parallel to the current are theFock–Darwin (FD) states [10], and are expressed by:

En,� = −�

2hωc +

(n + 1

2+ 1

2|�|

)h√

4ω20 +ω2

c , (1)

wheren and � are the radial quantum number and angularmomentum quantum number, respectively, andhωc = eB/m∗is the cyclotron energy. Figure 2a showsEn,� versus Bcalculated forhω0 = 2 meV. Spin splitting is neglected so

Fig. 2. a Fock–Darwin states calculatedfor hω0 = 2 meV. Dashed circles indi-cate the last crossing of the Fock–Darwinstates.b B-field dependencies of currentpeak positions fromN = 0 to 15. Ovalsalong theB = 0 T axis indicate groups ofpeaks associated with the filling of thefirst to the fourth shells, respectively

each state is two-fold degenerate. The orbital degeneracy atB = 0 T is lifted on initially increasingB, reflecting the firstterm of (1). AsB is increased further, new crossings or newdegeneracies can occur. The last crossing occurs along thebold line in Fig. 2a. Beyond this crossing the down-going FD-states merge to form the lowest Landau level.

Figure 2b shows theB-field dependence of the positionof the current oscillations measured for the circular mesa dis-cussed in [5]. We take into account the interaction energy aswell as the FD diagram to explain the experimental data. Firstof all, we see a large gap (i.e. large∆(N)) between peaksfor N = 2, 6, and 12 (also forN = 20 but not shown here)at B = 0 T. These are all signatures of complete filling of thefirst, second, third, and fourth shells, respectively (marked byovals along the gate-voltage axis). The respective shells aremade from degenerateEn,� states: so including spin degener-acy the first, second, third, and fourth shells respectively aretwofold degenerate withE0,0 (1s orbital), fourfold degener-ate with E0,1 = E0,−1 (2p orbital), sixfold degenerate withE0,2 = E0,−2 (3d orbital) = E1,0 (3s orbital), and eightfolddegenerate withE0,3 = E0,−3 (4 f orbital) = E1,1 = E1,−1(4p orbital). We also see a relatively large gap forN = 4and 9 (also forN = 16 but not shown here). These num-bers correspond to the half filling of the second, third, andfourth shells, respectively and are consistent with Hund’sfirst rule.

It is also clear to see in Fig. 2b that the current peaks gen-erally shift in pairs withB. This pairing is due to the liftingof spin degeneracy. So from the shift of the paired peaks onincreasingB, we assign quantum numbers to the respectivepairs. For example, the lowest, second lowest, and third low-est pairs in the vicinity ofB = 0 T correspond to the filling ofelectrons in the FD states(n, �) = (0, 0), (0, 1), and(0,−1)with antiparallel spins, respectively. Furthermore, the wig-gles or anti-crossings between pairs of peaks correspond tothe crossings of the FD states. For example, the anti-crossing

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at the* label corresponds to the crossing of the FD states(0,−1) and (0, 2). However, from close inspection of thepeak pairing, we find that the pairing is modified here tooin line with Hund’s first rule. This is discussed in detail inSect. 2.2. The last wiggle of each pair of peaks appears alongthe dot-dashed line, which corresponds to the bold line inFig. 2a. This line also identifies the filling factorν = 2. Wenote that theν = 2 line is more vertical in the experiment thanin the calculation. This indicates that the lateral confinementbecomes weaker asN increases. We calculate the FD dia-gram withhω0 as a fitting parameter to reproduce theν = 2line in the experimental data. Thus we obtainedhω0 for even-N ≥ 6 and this is shown in Fig. 1c.hω0 for N = 1 is evaluatedfrom the B-field dependence of the lowest peak followingthe (n, �) = (0, 0) FD state, and that forN = 2 is evaluatedfrom a B-field where a singlet–triplet transition (described inthe following paragraph) is observed for the second lowestpeak [9]. Thehω0 value decreases asN increases, due to theeffect of self-screening and screening of the gate metal andleads [5].

Forν < 2, we see various other transitions associated withB-field-enhanced Coulomb interactions such as spin-flip tran-sitions betweenν = 2 and 1 [11], a spin singlet–triplet tran-sition when N = 2 at ν = 1 [9, 12], and the formation ofthe so-called maximum density droplet (MDD) forN ≥ 2 atν = 1 [13]. For 2> ν > 1, a sequence ofN/2 spin-flips isexpected to lead to a sequential increase in the total spin ofthe N-electron GS from 0 toN/2, i.e. resulting in full spinpolarization of theN-electron dot atν = 1. A self-consistentcharge distribution in the dot results in a compressible center(second lowest Landau level partially filled) being separatedfrom a compressible edge (first lowest Landau level partiallyfilled) by an incompressible ‘ring’ in which the lowest Lan-dau level is completely filled. Sequential depopulation occursas electrons transfer across the ring from spin-down sites atthe center to spin-up sites at the edge. Atν = 1 MDD isformed. This is spatially the most compact state of the spin-polarized electron droplet. AllN-electrons are in the low-est Landau level, and occupy sequentially the up-spin states(n, �) = (0, 0), (0, 1), ..., (0, N −1) without a vacancy. Thedetailed experiments on the spin-flip phase, and the MDDphase in our circular dot are described in [11] and [13],respectively.

Fig. 3. a Schematic diagram of twosingle-particle states with energiesEaand Eb crossing each other at a mag-netic field B = B0. b Electrochemicalpotential, µi(2) = Ui(2)−U(1), for twointeracting electrons. Thethick line rep-resents the ground state energy whereasthe thin lines show the excited states

2 Effects of Coulomb interactions on spin states:circular dot

2.1 Model for two interacting electrons

We first discuss a simple model that describes filling of twosingle-particle states with two interacting electrons. Figure 3ashows two, spin-degenerate single-particle states with ener-giesEa andEb crossing each other atB = B0. The GS energy,U(1), for one electron occupying these states, equalsEa forB < B0 and Eb for B > B0 (thick line in Fig. 3a). For twoelectrons we can distinguish four possible configurations witheither total spinS = 0 (spin-singlet) orS = 1 (spin-triplet).(We neglect the Zeeman energy difference betweenSz = −1,0, and 1.) The corresponding energies,Ui(2, S) for i = 1 to 4,are given by:

U1(2, 0) = 2Ea +Caa , (2a)U2(2, 0) = 2Eb +Cbb , (2b)U3(2, 1) = Ea + Eb +Cab −|Kab| , (2c)U4(2, 0) = Ea + Eb +Cab +|Kab| . (2d)

Here, Cij (i, j = a, b) is the DC energy between two elec-trons occupying states with energiesEi and Ej , and Kab isthe EX energy (Kab < 0) between two electrons occupyingEaand Eb. The electrochemical potential is defined for a two-electron GS asµ(2) = U(2, S )−U(1). For eachUi(2, S )we obtain potentials:µi(2) = Ui(2, S )− Ea for B < B0 andUi(2)− Eb for B > B0 (see Fig. 3b). The GS hasS = 0 awayfrom B0. NearB0, the lowest energy isµ3(2), so S = 1. Thedownward cusp in the thick line identifies this spin-triplet re-gion. The transition in the GS fromS = 0 to 1 andS = 1 to0, respectively, occurs whenµ1 = µ3 for B < B0 (labeled “S-T”) and whenµ2 = µ3 for B > B0 (labeled “T-S”). We definetwo energies,∆1 and∆2, to characterize the size of the down-ward cusp in the GS atB = B0:

∆1 = µ1 −µ3 = Caa −Cab +|Kab| , (3a)∆2 = µ2 −µ3 = Cbb −Cab +|Kab| , (3b)∆1 −∆2 = Caa −Cbb . (3c)

If the states withEa and Eb have the same type of orbitals,EX interactions only contribute to the downward cusp since

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Caa = Cbb = Cab. This is typically the case for Hund’s firstrule in the second shell atB = 0 T, which is observed in thecircular dot [5]. However, FD states at the crossings at fi-nite fields usually have different types of orbitals, and thusCaa �= Cbb �= Cab, so both EX and DC interactions can signifi-cantly contribute to the downward cusp.

2.2 Spin singlet–triplet–singlet transition in the presence ofmagnetic field: experimental

In Fig. 2b we see the wiggles of paired peaks correspondingto the crossings of FD states in the lowB-field range. Thismeans that these crossings of two single-particle states, asshown in Fig. 3a, can be obtained by tuning theB-field. Toexamine in detail the experimental data in the vicinity of sucha crossing point we show in Fig. 4a a magnified plot for thecurrent peaks betweenN = 6 and 16 forB = 0 to 3 T. We seeclearly the paired peaks indicating the antiparallel spin fill-ing of a single orbital state by two electrons. Modificationsto the pairing are observed for the peaks labeled by� at 0 T,and in each of the dashed ovals connecting pairs of peaks atnon-zero field. These are all signatures of Hund’s first rule;i.e. spin-polarized filling.

To demonstrate the resemblance between the data and themodel of Fig. 3b, we show expansions of the evolution ofthe N = 8 andN = 24 peaks in Fig. 4b. The downward cuspsare clearly seen. The dashed lines form a parallelogram, fromwhich we obtain parameters∆1 and∆2.

To compare the two-electron model with larger electronnumbers, we assume that other states are far away in en-ergy so that they can be neglected. Then, the downward cuspsshould occur for all higher even-electron numbers, whereas

Fig. 4. a Magnified plot of theB-field dependencies of current peak pos-itions from N = 7 to 16 shown in Fig. 2b. Thebars along the gate voltageaxis show 1-meV energy scales calibrated at−1.26 and −0.85 V. Thelong dashed curve indicates the last crossing between single-particle states.Dashed ovals group pairs of ground states for odd and even electron num-bers. Spin transitions in the ground states are indicated by� at B = 0 Tand occur inside theovals for B �= 0 T. b Magnified plots of theN = 8and 24 current peaks vs. magnetic field. Thedashed lines illustrate how theinteraction-energy parameters,∆1 and∆2 are determined

they should be absent for all higher odd-electron numbers.This is clearly observed from the ovals in Fig. 4a. For in-stance, theB-field dependence of the 9th peak compares wellto the thick line in Fig. 3a, i.e. a transition between two single-particle states occupied by one electron. TheB-field depen-dence of the 10th peak compares well to the thick line inFig. 3b, i.e. an extra downward cusp where the GS has twoelectrons with parallel spins (S = 1), occupying two differ-ent orbital states which are nearly degenerate. Other pairs ofeven and odd numbered peaks show the same behavior. Thisjustifies our assumption that the other electron states can beneglected and that we can simplify the many-electron systemto just one or two electrons. We note that this type of modifi-cation to peak pairing has been observed before without beingidentified [5, 12].

More detailed agreement of our interacting electronmodel with the experiment is obtained for the data of ex-citation spectrum. Figure 5 shows a dI/dVg plot, taken forV = 2 mV. This larger voltage opens a sufficiently wide trans-port window between the Fermi levels of the source and drain,that both the GS and first few ESs can be detected. The GSand ESs forN = 7 to 9 can be assigned from the magneticfield dependence of the red or dark-red lines. Solid blue lineshighlight the GSs whereas the ESs are indicated by dashedblue lines. The set of GS and ES lines forN = 7 shows a sin-gle crossing similar to that in Fig. 3a. The spectrum forN = 8compares well to Fig. 3b and we can clearly distinguish theparallelogram formed by the GS and first ES. The downwardcusp in the GS forN = 8 (labeled�) is at a slightly higherB-field than the upward cusp in the first ES (labeled�). Thisasymmetry implies that∆1 > ∆2, i.e.Caa > Cbb in (3c). Thesame type of asymmetry is always observed along the dashedline in Fig. 4a, implying thatCaa > Cbb for all N. Note thatthe GS forN = 9 appears as an upward cusp (labeled∇) iden-tical in form to the first ES in the spectrum forN = 8. Thisindicates that the electrochemical potential for theN = 9 GSis closely linked to the electronic configuration of the first ESfor N = 8. A similar but more detailed argument is reportedpreviously [8].

As illustrated in Fig. 4b, we can derive the experimentalvalues for∆1 and∆2 for differentN. These values are plottedas solid symbols in Fig. 6. We find that∆1 is larger than∆2for all electron numbers, again implying thatCaa > Cbb. AsN increases from 6 to 12,∆1 first increases and then slowlydecreases, whilst∆2 slightly decreases.

The above discussion has been kept general for crossingsbetween any type of single-particle states. To calculate DCand EX energies we use FD states calculated for the effectivelateral confinement energyhω0, which changes as a func-tion of N (see Fig. 1c)1. As can be seen from Fig. 2a the lastcrossing is always a crossing between just two FD-states. Theup-going state is always(n, �) = (0,−1), whereas the down-

1 Cij andKij are the usual wavefunction overlap integrals for an unscreenedCoulomb potential:

Cij =∫ ∫

|Ψi(r1)|2 e2

4πε0 |r1 − r2|∣∣Ψj(r2)

∣∣2 dr1dr2 ,

Kij = −∫ ∫

Ψ ∗i (r1)Ψj(r1)

e2

4πε0 |r1 − r2|Ψ∗j (r2)Ψi(r2)dr1dr2

Ei and Ej are eigenvalues for the eigenfunctionsΨi and Ψj , respectively,and r1 and r2 represent the positions of the two electrons.

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Fig. 5. dI/dVg in the plane ofVg and B for N = 7 to 9 measured forV = 2 mV. dI/dVg > 0 for red or dark red and dI/dVg ≤ 0 for white.The solid blue lines indicate the evolution of the GSs with magneticfield whereas thedashed blue lines show the ESs. Thetwo arrows in-dicate singlet–triplet (S-T) and triplet–singlet (T-S) transitions in the GSfor N = 8

Fig. 6. Experimental values for the energy parameters∆1(•) and ∆2(�),on a log scale, versus electron number derived from data as shown inFig. 2a. The uncertainty in the determination of the experimental values is±10% or less. Thedashed and solid curves are calculated from the FD-wavefunctions at 2 T, for an unscreened (unscr) and screened (scr) Coulombinteraction. The calculated exchange energy|Kab| between states with ener-gies Ea = E0,−1 and Eb = E0,N/2−1 decreases quickly withN. Above themain figure the absolute squares of the wavefunctions are shown for therelevant quantum numbersn = 0 and� = 0, ±1, ±2, ±3. As the angularmomentum quantum number� increases, the average radius increases

Fig. 7. Exchange energy|Kab|, on a log scale, associated with spin-tripletsformed when each new shell is filled by just two electrons atB = 0 T. Thesolid circles are the experimental values whose uncertainty in their deter-mination is±10% or less. Theinset shows an expansion for the filling ofthe first two electrons into the third shell (i.e.N = 7 and 8). Theverti-cal double-arrow represents|Kab| in units of gate voltage which is thenconverted to energy. The calculated curves in themain figure are for theunscreened (dashed) and screened (solid) cases

Fig. 8. dI/dVg in the plane ofVg and B for N = 7 and 8 in the vicinityof B = 0 T measured forV = 1.2 mV. dI/dVg > 0 for red or dark red anddI/dVg 0 for white. Thesolid blue lines indicate the evolution of the GSswith magnetic field whereas thedashed blue lines show the ESs. Thearrowindicates a triplet–singlet transition in the GS forN = 8

going state,(0, � > 1), has an increasing angular momentumfor states with higher energy. (The relation with Fig. 5a is:Ea = E0,−1 and Eb = E0,�>1.) Note that the last crossingsalso correspond to the thick line in Fig. 2a.

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From the electron distributions of the FD-states we calcu-late the DC and EX energies for two electrons occupying twodegenerate states. We takehω0 = 2 meV to reproduce the datafor N > 8 and obtain∆1 and∆2 with (3a)–(3c). The dashedcurves in Fig. 6 show∆1 and∆2 when we neglect screeningof the interactions within the dot by electrons in the leads andin the gate. In this case the Coulomb potential falls off as 1/r,wherer is the distance between the electrons1. For the solidcurves we have approximated the screening effects by replac-ing the Coulomb potential by exp{−r/d}/r. We have takend = 10 nm which is roughly the thickness of the tunnel barri-ers. Figure 6 shows that screening considerably reduces∆1 tovalues much closer to the experimental values. Screening alsoremoves the minimum in∆2, which is also in better agree-ment with the experiment. For allN, we find that∆1 > ∆2and thusCaa > Cbb. Since the average radius of the wave-functions increases with angular momentum, two electronsare closer together when they both occupy(0,−1) comparedto when they both occupy(0, � = N/2−1) for even-N > 4 (or� > 1), so the DC interaction is stronger in the former thanthe latter. This explains our observation thatCaa > Cbb forall N. The overlap between different wavefunctions,(0,−1)and (0, � = N/2−1), decreases for even-N > 4 (or � > 1).This results in a decrease in bothCab and |Kab| with N. Itthen follows from (3a) that∆1 increases until it saturates ata value equal toCaa. The gradual decrease of experimental∆1 for N > 12 is probably related to the decrease in the lateralconfinement withN [5] and thus the decrease inCaa.

2.3 Spin singlet–triplet transition near B = 0 T

We now discuss the interaction effects for theN = 4, 8, and14 peaks nearB = 0 T: the N = 4 peak labeled� in Fig. 2band theN = 8 and 14 peaks labeled� in Fig. 4a. These peakscorrespond to the GS electro-chemical potentials for addingthe second electron to the second, third, and fourth shells, re-spectively. The inset to Fig. 7 demonstrates the resemblanceto the model of Fig. 3b forN = 8 near B = 0 T. Compar-ing the data to the FD spectrum, we assign the states suchthat: Ea = E0,−2 and Eb = E0,2. Likewise, for N = 4 wehaveEa = E0,−1 andEb = E0,1 [5] and for N = 14 we haveEa = E0,−3 andEb = E0,3

2. These assignments are confirmedfrom measurements of the excitation spectrum. The data, i.e.dI/dV vs. B − Vg, measured forN = 7 and 8 nearB = 0 Tare shown in Fig. 8. (The data forN = 3 and 4 are shownin Fig. 12a.) The color setting for the dI/dV is the same asthat in Fig. 5. The solid blue lines highlight the GSs, and theESs are indicated by the dashed blue lines. The GS and firstES for N = 7 are degenerate atB = 0 T, and move down andup, respectively, asB is initially increased. The GS and firstES lines forN = 8 are exchanged atB = 0.18 T, indicatinga transition in the ground state. Note that the states here as-signed toEa andEb correspond to wavefunctions with a com-plete overlap. Also, forB = B0 = 0 T the two crossing stateshave the same orbital symmetry implying∆1 = ∆2 = |Kab|;i.e. only EX effects contribute to the downward cusp (Notethat the asymmetry (∆1 �= ∆2) can only occur forB0 �= 0.).

2 For N = 14 it is difficult to identify the corresponding state since theB-dependence is not clear enough to distinguish betweenE0,−3 andE1,−1. Weassume it isE0,−3, and continue to evaluate|Kab|

We derive|Kab| as illustrated in the inset to Fig. 7 from theintersection with theB = 0 T axis of the dashed line and themeasured data. Figure 7 shows that the EX energy quickly be-comes smaller for higher lying shells. For comparison we alsoshow the calculated screened and unscreened values for theEX energy. The screened case provides the best quantitativeagreement for our realistic choices of the confining energyand the screening distance.

Our general model used for the calculations in Figs. 6and 7 provides a clear identification of effects due to EX andDC interactions. An important simplification is the reductionof a many-electron system to just two interacting electrons.We note that there are more advanced calculations supportingour analyses [14].

3 Effects of deformation in the lateral confinement

3.1 Deformed dots in rectangular mesa

A rectangular mesas with top contact area (L × S ), 0.55×0.4µm, is prepared from the same DBS wafer.L(S ) is thenominal dimension of the longest (shortest) side of the topcontact [6, 15]. A simple way to classify the rectangularmesa is to define a geometric parameter,β, to be the ratioL/S (= 1.375). Due to a slight undercut, the area of themesa is a little less than that of the top contact. Figure 9ashows schematically the slabs of semiconductor between thetwo tunneling barriers, and the resulting dot bounded by theshaded depletion region.

As illustrated in Fig. 9a, the lateral confining potentialof the dot inside the rectangular mesa is expected to be ap-proximately elliptical due to rounding at the corners pro-vided the number of electrons in the dot is not too large,or too small [6, 15]. Assuming the confining potential isstill perfectly parabolic:(1/2)m∗(ω2

xx2 +ω2y y2), we choose

to characterize the ‘ellipticity’ by a deformation parameter,δ = Ex/Ey. Here, Ex(Ey) is the confinement energy at 0 Talong the major (minor) axis (Ex = hωx > Ey = hωy). The2D states in the elliptical dot are labeled by the quantum num-bers (nx , ny), wherenx (ny) is the principal quantum num-ber (= 0, 1, 2, ...) associated with the energy parabola alongthe major (minor) axis. The energy of single-particle state(nx, ny) is (nx +1/2)Ex + (ny +1/2)Ey.

For a perfectly circular mesa, we trivially generalize ourdefinition of δ so thatδ = β = 1. On the other hand, for therectangular mesa, there is no simple correspondence betweenβ, a ratio of lengths characteristic of the top metal contact,andδ, a ratio of energies characteristic of the dot in the mesa,andβ is dependent onVg (or equivalentlyN). Nevertheless,although we are not saying thatδ = β, we assume thatβ isa “measure” ofδ.

3.2 Magnetic field dependence

Figure 9b–d shows theB-field evolution up to 6 T of the first10 single-particle energy levels for elliptical dots (δ = 1.5, 2,3.2) calculated according to the model described by Madhavand Chakraborty [7] with a fixed confinement energy ofEQ =3 meV [15]. The confinement energies for the elliptical dot aresimply derived from the relationEx Ey = EQ EQ . Quantumnumbers (nx, ny) for some of the states are indicated.

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Fig. 9. a Schematic diagram of the lateral confinement potential assumedfor a dot in the rectangular mesa.b–d CalculatedB-field dependence ofthe first ten single-particle energy levels for elliptical dots withδ = 1.5, 2,and 3.2

As the deformation is gradually increased, Figs. 9b–d,degeneracies of the single-particle states at 0 T are gener-ally lifted, although accidental degeneracies can occur atcertain ‘magic’ deformations, provided the confining po-tential remains perfectly parabolic. A weakB-field par-allel to the current can also induce degeneracies of thesingle-particle levels in a similar manner to that for a cir-cular dot. As discussed before, ‘wiggles’ in the positionof pairs of current peaks are expected at weak field be-cause of these crossings. The first lowest energy ‘wiggle’originates from the crossing marked by� in Figs. 9b–d,and this crossing moves to higher energy (N) and higherB-field with increasingδ. The evolution of each single-particle energy level on initial application ofB-field is weakerfor larger deformation. The spectra also highlight that thesingle-particle energy level spacing generally decreases asδincreases.

Figure 10 shows theB-field dependence of the Coulomboscillation peak positions measured for the rectangularmesa [15]. The data consists of current vs.Vg traces taken ata very small bias at differentB-fields. Peaks are paired, andthere are no obvious deviations close to 0 T forN = 4 due toHund’s rule. Quantum numbers (nx , ny) of the single-particlestates are assigned, and the first up-moving pair of peaks ismarked by the solid arrow. In the single-particle picture, thefirst up-moving state is(nx, ny) = (1, 0).

Fig. 10. B-field dependences of the current peak positions measured forN = 0 to 10 for the rectangular mesa. The bias voltageV is sufficientlysmall that only the ground states are detected

We also prepared rectangular mesas with largerβ, andfrom the same measurements as shown in Fig. 10, we find thatthere is a reasonable correspondence between the measuredposition of the first up-moving pair of peaks and lowest en-ergy wiggle, and trends predicted from the calculation shownin Fig. 9. This leads to an estimate ofδ in the rectangular mesawe discuss here: 1< δ < 2. Detailed discussion on the rela-tion between the values ofβ andδ, in different rectangularmesas are described in [15].

3.3 Influence on spin states

In Fig. 10 we see no clear modification to the peak pairingfor N = 0 to 10 nearB = 0 T. In particular this indicates thatthe spin-state forN = 4 is a singlet. This spin-singlet is alsoconfirmed from measurements of the Zeeman shift [6]. Fora circular dot atN = 4 the(n, �) = (0, 1) and(0,−1) statesare degenerate at 0 T and their wavefunctions have the samesymmetry, but with deformation this degeneracy is lifted andthese states become the(nx, ny) = (1, 0) and(0, 1) states inan elliptical dot (which are split at 0 T). This energy split-ting, γ , increases withδ. At the same time, the effect of DCand EX interactions are both reduced since overlap of thewavefunctions becomes smaller. Note thatγ depends morestrongly on the deformation than the interaction energies (seeFig. 12b). For a small deformation, ifγ < EEX (EEX is the en-ergy gain due to EX in the circular dot) at 0 T, EX can stilloperate to lower the energy, and thus theN = 4 GS remainsa spin-triplet. On the other hand, ifγ > EEX at 0 T, the GS isa spin-singlet, so normal pairing is expected [7, 15]. Thus, wecan expect a triplet–singlet transition at some critical defor-mation. The absence of deviations to the normal peak pairing

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at N = 4 in Fig. 10 apparently confirms thatδ is greater than1.2−1.3 in line with previous calculations [7, 15].

Detailed information on the effect of deformation in therectangular mesa can be obtained from measurements of theexcitation spectra. Figure 11 shows the excitation spectrumfor the circular mesa (a) and for the rectangular mesa (b). GSsand ESs forN = 3 and 4 are indicated in dI/dVg vs. B − Vgplotted in the same way as for Fig. 8. For the circular mesa,the GS and first ES lines forN = 3 follow the B-field depen-dence of FD states(n, �) = (0, 1) and(0,−1), respectively,and the spectrum forN = 4 is consistent with a crossing ofthe FD states(0, 1) and(0,−1) at 0.4 T. This crossing identi-fies the singlet–triplet transition in the GS. In contrast, thereis no signature of the single–triplet transition forN = 4 inthe spectrum of the rectangular mesa. In addition, for both

Fig. 11a,b. dI/dVg in the plane of Vgand B for N = 3 and 4 measured forthe circular mesa (a) and the rectangularmesa (b). V is set at 1.6 mV. dI/dVg >

0 for red or dark red and dI/dVg ≤ 0for white. The solid blue lines indicatethe evolution of the GSs with magneticfield whereas thedashed blue lines showthe ESs

Fig. 12. a Calculation of energies neededfor adding an extra electron into twodifferent configurations from the two-electron ground state of an elliptic quan-tum dot. The lower- and higher-lyinglines are the ground and first excitedstates, respectively, and the correspond-ing configurations are pictorially illus-trated. The bold arrow indicates the“added electron”.b Contributions fromlifting the level degeneracy (dot-dashedline) and changes in interaction energies(dashed line) to the energy difference(solid line) between the two differentconfigurations shown ina

N = 3 and 4, the GS and first ES are widely spaced at 0 T,and follow theB-field dependence of states(nx, ny) = (0, 1)(down-moving state) and(1, 0) (up-moving state), resepec-tively (see Fig. 9b). The splitting between the GS and firstES is 0.8 meV for N = 4 at 0 T. This energy is smaller by0.8 meV than the same splitting forN = 3, mainly due to thecontribution of EX energy.

To investigate the effect of deformation on the GS and ESobserved in Fig. 11 we calculate the contributions of the lift-ing of level degeneracy and change in Coulomb interactionsusing the same assumptions as employed for the calculationsin Figs. 6 and 7. Figure 12a shows the calculated total energyneeded for adding an extra electron to theN = 2 GS as a func-tion of deformationδ. Here we usehω0 of 3 meV to reproducethe confining energy whenN = 3 (see Fig. 1c). This total en-

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ergy incorporates the energy,γ , lifting of level degeneracyand the energy of interaction between the added third elec-tron, and the residing two electrons forN = 2 GS, which ismade from an orbital state(nx, ny) = (0, 0) occupied by twoantiparallel spin electrons. Note that in our previous calcula-tion of the electrochemical potentialµi(2) for two-electronsin the circular dot (see Fig. 3b) we neglected the influenceof the existing electrons on the different two-electron config-urations (i = 1 to 4) since there is no big difference in thespatial form of these two-electron configurations. However,this assumption becomes invalid when the dot contains a largedeformation in the lateral confinement. The GS, and first ESfor N = 3 are formed by adding an extra electron to a state(nx, ny) = (0, 1) and (1, 0), respectively. The wavefunctionfor state(nx, ny) = (1, 0) (or (0, 1)) is extended along thex(or y) direction as compared to that for state(nx, ny) = (0, 0).However, the degree of the extension is significantly smallerfor state(nx, ny) = (1, 0). So the interaction effect betweenthe third electron and the residing two electrons in theN = 2GS is stronger for the state(nx, ny) = (1, 0) than that forthe state(nx, ny) = (0, 1). In Fig. 12a the calculated ener-gies are degenerate for the GS and ES atδ = 0 and becomesmore widely spaced as the deformation increases. This energyspacing is a measure of the energy difference between the GSand first ES, obtained in the experiment of excitation spec-trum, and thus from comparison with the experimental datafor N = 3 at 0 T we can estimate a deformation ofδ = 1.5 forour rectangular mesa. In Fig. 12b contributions to the split-ting of the GS and ES shown in Fig. 12a are distinguished forthe level degeneracy lifting,γ , and the interaction energy. Asthe deformation increases, the interaction energy decreasesfor the GS, while it increases for the ES. The increase inγwith deformation is significantly stronger than that of the in-teraction energy. So we conclude that the GS forN = 3 in

Fig. 14. a B-field dependencies of cur-rent peak positions measured forN = 5to 12 for the rectangular mesa. The biasvoltage V is sufficiently small to de-tect only the ground states. Thedottedline indicates the last crossing betweensingle-particle states.Dotted ovals grouppairs of ground states for odd and evenelectron numbers. Spin singlet–triplet–singlet transitions in the ground statesoccur at theN labels in the ovals forB �= 0 T. b Schematic of two-single par-ticle states(nx , ny) = (1, 0) with (0, 4)

crossing each other, which accounts forthe B-field dependence of the groundstate forN = 9 in c. c Magnified plots ofthe N = 9 to 11 current peaks vs. mag-netic field. A downward cusp, a signatureof spin transition, is marked byN. Thedashed lines illustrate the excited states(spin singlet) in the region of the down-ward cusp

Fig. 13. Calculation of energies needed for adding two extra electrons intothe different configurations from the two-electron ground state of a quantumdot with anisotropy in the lateral confinement. Thesolid line is a spin-triplet, whereas thedashed lines are all spin-singlets. Thelowest-lying lineis always the ground state and thehigher lying lines are all the excited states

our rectangular mesa is mainly influenced by the lifting of thelevel degeneracy between states(nx, ny) = (0, 1), and(1, 0)and only partially by the reduction in the interaction effect.

For the N = 4 GS and ES we also show the calculationof the energy needed for putting two extra electrons into theGS for N = 2 in Fig. 13. Here we consider four possible

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configurations for a small deformation: one configuration forthe spin-triplet and three configurations for the spin-singlet.The triplet state has two parallel spins in two different or-bital states(nx, ny) = (0, 1) and(1, 0). On the other hand, thesinglet state has two antiparallel spins either in two differentorbital states(nx, ny) = (0, 1) and(1, 0) or in the same orbitalstate of(nx, ny) = (0, 1) (or (nx, ny) = (1, 0)). In this figurethe GS is the triplet for a small deformation, but a transitionfrom triplet to singlet occurs atδ = 1.55. From a comparisonto the excitation spectrum forN = 4 we estimateδ = 1.9. Thevalues obtained from the calculation shown in Fig. 12, (a) forN = 3 and (b) forN = 4 are both consistent with 1< δ < 2evaluated from theB-field depenence of the current peak pos-itions in Fig. 10. Thisδ value for N = 4 is somewhat largerthan that evaulated from the excitation spectrum forN = 3.This discrepancy may reflect a functional dependence ofδ onN, but is more likely to be due to an inaccuracy in the calcu-lation since this calculation becomes less valid asδ becomesgreater than 1.5.

Our calculation for the GSs and ESs in the elliptic dotis very simple, however, it is still useful for understandingthe experimental data. There are more refined calculationsreported elsewhere. Numerical diagonalization has been suc-cessfully employed to calculate basic electronic properties ofdots with anisotropic confining potentials [7]. Such ‘exact’calculations, however, are limited toN ≤ 6. For a larger num-ber of electrons, spin-density functional theory is a powerfultechnique, which explicitly incorporates the electron–spin in-teractions [7, 15]. Both approaches predict changes in theaddition energy spectra, and transitions in the spin-states asthe deformation is varied.

In rectangular mesas, even for a small deformation a spin-singlet is favored by the lifting of level degeneracy atB = 0 T.We can still tune the single-particle levels with a magneticfield to highlight interaction effects related to the spin-tripletin the same way as discussed for the circular mesa. Fig-ure 14a shows the evolution of current peaks with magneticfield for N = 5 to 12. Modifications to the usual pairingof neighboring peaks, i.e., anti-ferromagnetic filling, are ob-served inside each of the dashed ovals connecting pairs ofpeaks at non-zero field. These ovals are located along thelast crossing of the single-particle state(nx, ny) = (1, 0) with(0, ny) (ny ≥ 2). The upper peak inside the dashed oval showsa small but definite downward cusp which is a signature ofthe singlet–triplet–singlet transition. For example, a down-ward cusp is clearly seen forN = 10 at 2 T in the magnifiedplot in Fig. 14c. The size of the downward cusp is smaller inthe rectangular mesa than that for the circular mesa, probablydue to the reduction in both of DC and EX interactions in therectangular mesa.

4 Conclusions

We have used vertically gated circular and rectangular mesasto study the effects of Coulomb interactions on the spinstates in quantum dots. The circular mesa contains a circu-lar quantum dot with a high degree of rotational symmetryand parabolicity in the lateral confinement. This gives riseto systematic sets of degenerate single-particle levels at zeromagnetic field. The filling of these degenerate states leads toatom-like properties such as shell filling and the obeyance of

Hund’s first rule. Application of a magnetic field parallel tocurrent leads to systematic modifications in the energy spec-trum, and these modifications are well reproduced in terms ofthe Fock–Darwin spectra. By adjusting the level degeneracyas a function of magnetic field, we observe parallel spin fillingas favored by the direct Coulomb and exchange interactionsat non-zero magnetic field. This means that electrons tend tohave parallel spins in line with Hund’s first rule when theyoccupy nearly degenerate single-particle states. The influenceof the direct Coulomb and exchange interactions on the spin-configurations are well explained in terms of two-electronsinglet and triplet states. On the other hand, for the rectangu-lar mesa, the quantum dot is subject to elliptical anisotropy inthe lateral confinement. This lifts the level degeneracy and re-duces the interaction energy, both of which favor antiparallelspin filling. Consequently, there are no signatures of parallelspin filling observed for the rectangular mesa at zero mag-netic field. This is attributed mainly to the lifting of the leveldegeneracy rather that the weaker reduction in the interactionenergy. On the other hand, parallel spin filling can still be in-duced when the level degeneracy is adjusted as a function ofweak magnetic field.

Acknowledgements. We thank T. Honda, R. van der Hage, J. Janssen,T. Oosterkamp, H. Tamura, and T. Uesugi for their help and useful discus-sions. We would also thank S.M. Reimann, M. Koskinen and M. Manninenfor their discussions on the effect of anisotropy in the lateral confinement ofa quantum dot. S.T. and L.P.K acknowledge financial support from Grant-in-Aid for Specially Promoted Scientific Research and also for ScientificResearch on the Priority Area of “Single Electron Devices and Their HighDensity Integration” from the Ministry of Education, Science, Sports andCulture in Japan, from CREST-FEMD(JST), from the Dutch Organizationfor Fundamental Research on Matter (FOM), and from the NEDO programNTDP-98.

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