Novel Phenomena in Small Individual and Coupled Quantum Dotsyingshe/QDRev.pdf · of the quantum-dot...

Novel Phenomena in Small Individual and Coupled Quantum Dots

A. M. Chang∗, H. Jeong∗+, M.R. Melloch†

∗Department of Physics, Purdue University, West Lafayette, IN 47907-1396

†School of Electrical Engineering, Purdue University, West Lafayette, IN 47907

+Current address: Dept. of Electrical Engineering, Princeton University, Princeton, NJ

08544

Abstract

We discuss several novel phenomena observed in individual or coupled quan-

tum dots fabricated in GaAs/AlxGa1−xAs materials, where the lithographic

dot-size ranges from 350nm down to 120nm. These include distinct signa-

tures of “quantum chaos” as evidenced in the highly non-Gaussian distribution

of Coulomb blockade conductance peak heights in individual quantum dots,

which are related to the well-known Porter-Thomas distribution of resonance

widths in the scattering of neutrons from nuclei, spin physics in individual

dots manifested in the even-odd (pairing) effect and in the Kondo effect, and

spin physics in an artificial, double-quantum-dot molecule as evidenced by the

clear observation of the coherent Kondo effect. These significant observations

may have ramifications for future implementation of the coupled-quantum-dot

system for quantum computation.

1

I. INTRODUCTION

The field of semiconductor quantum dot [1] single electron transistor [2–4] has witnessedtremendous advances in the past 10 years. Whether from the perspective of fundamen-tal physics or potential technological applications, the quantum dot has generated a greatamount of excitement as a result of its tunability and controllability. This tunability hasenabled the realization of a remarkable variety of physical phenomena, associated with thecharge and spin degrees of freedom of the electron(s) occupying the “upper” most quantumlevels, as well as potential device applications due to the extraordinary sensitive to chargeand electric fields [9,10]. To date, an impressive variety of phenomena has been observed.These include: atomic shell filling in regularly shaped QD, signatures of quantum chaoticdynamics in irregularly shaped QD manifest in the wavefunction and energy level statistics,evolution of levels in a perpendicular magnetic field (with or without interaction effects),molecular state formation in double and triple QD’s, coherent molecular states, covalent andionic states, higher-order effects such as cotunneling, under strong coupling to leads spineffects such as Kondo physics or mixed valence physics associated with the Anderson model,and interference effects such as Fano resonance line shape due to the presence of resonantand non-resonant channels. These results represent the contributions of numerous workersin the field and excellent reviews are available in the literature [1,5–8]. On the applica-tion side, several very interesting proposals, such as single-electron-transistor (SET) basedlogic, memory [10], quantum-dot electron-spin based qubits for quantum computation [11],are currently being investigated in laboratories around the world. Furthermore, notable anduseful devices have been invented, such as the scanning SET for the detection of local electricfields [12,13], and the RF-SET (radio-frequency single electron transistor) for the detectionof charge at the 10−5e/

√Hz level at 100MHz bandwidth [14]. Although several of these

applications/devices do not necessarily require semiconductor quantum dots, and are thusfar implemented in metallic SET systems, there is no fundamental reason they could notalso be workable in semiconductor, and in particular, GaAs/AlxGa1−xAs quantum dots.

To illustrate the richness of phenomena and potential impact on future technology, inthis review we discuss three contributions in our own work on transport in small GaAs −AlxGa1−xAs quantum dots. These results pertain to three rather diverse aspects of QDphysics and underscore the richness of the observable phenomena. The first topic dealswith the highly non-Gaussian distribution of Coulomb Blockade conductance peak heightsin individual quantum dots. In this phenomenon the spin degree of freedom does not playa significant role. While our work represented the first systematic study [15], results similarto ours are also obtained by Folk et al. [16]. Subsequent to the discovery of the CoulombBlockade (CB) in semiconductor quantum dots [1,17,18], Jalabert, Stone, and Alhassid [19]recognized that the height fluctuations may be related to the well-known Porter-Thomastype distribution of resonance line widths in the elastic scattering of neutrons from complexnuclides such as U235, U238, and Th238, etc. [20]. Using a random matrix theory (RMT)approach which neglected the influence of electron-electron interaction beyond the Hartreecontribution responsible for the classical electrostatic charging energy, U, Jalabert et al.

2

computed the distribution of peak heights in the single-level tunneling limit, where electronstunnel through an individual, thermally resolved quantum level within the quantum dot.Results were obtained for both B=0 corresponding to the Gaussian Orthogonal Ensemble(GOE), and B 6= 0 corresponding to the Gaussian Unitary Ensemble (GUE). In both cases,the most striking feature is the non-Gaussian nature with a prevalence of small valued peaks.for g → 0 where g is the dimensionless peak conductance, while the B 6= 0 distribution attainsa maximum near zero.

The second topic concerns the even-odd (pairing) behavior associated with the spin degreeof freedom in the electron filling behavior of an irregularly-shaped individual quantum dot.The pairing behavior arises within a simple scenario appropriate for the individual quantumdot under appropriate conditions discussed below, and results from the spin degree of freedomand the Pauli exclusion principle for Fermions. Here we report distinct signatures of theeven-odd effect observable in the pairing behavior of CB peak spacing, peak heights andspin status. Furthermore, we found that the peak spacing pairing depends significantly onthe rs value characterizing the Coulomb interaction strength.

The third topic pertains to the Kondo effect in an artificial quantum dot molecule [21],arising from coupling of the quantum dot excess-spins to the spin of the electrons in theleads and a resultant screening of the dot-spins [22,23]. Since a quantum dot with an excessspin may be modeled as a magnetic impurity under appropriate conditions, coupled double-quantum-dots provide an ideal model system for studying interactions between localizedimpurity spins. We report on the transport properties of a series-coupled double quantum dotas electrons are added one by one onto the dots. When the many body molecular states areformed, we observe a splitting of the Kondo resonance peak in the differential conductance.This splitting reflects the energy difference between the bonding and antibonding statesformed by the coherent superposition of the Kondo states of each dot. The occurrence ofthe Kondo resonance and its magnetic field dependence agree with a simple interpretationof the spin status of a double quantum dot. This direct observation of coherent couplingbetween an excess spin on each dot will have potential ramification for the implementationof the quantum-dot systems for quantum computation [11].

Lastly, we will briefly present suggestive evidence for sudden rearrangements of thedouble-dot energy ground state configuration, leading to sharp features in the CB peaks,as well as a functioning quantum dot with lithographic dimensions as small as 120nm.

II. MODELS OF SINGLE AND DOUBLE QUANTUM DOT SYSTEMS

To provide the basis for discussing the phenomena reviewed in this article, we present asummary of the simplest physical scenario characterizing the quantum dot system, includingthe external leads connected to the dot through which current can be injected or removed.In this simplest scenario, the physical system of an isolated single QD is viewed as a sort ofartificial “atom” [1], characterized by several parameters and by simple even-odd filling ofthe individual quantum levels under appropriate conditions.

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FIG. 1. (a) Model of a quantum dot with left (L) and right (R) tunnel junctions to the leads (source,

S, and drain, D). The charge on the dot can be tuned via the gate, Vg . (b) Capactive model of the single

quantum dot.

For this nearly-isolated individual dot case, in general two familiar ingredients controlits characteristics: Coulomb blockade (CB) and the discretized 0-dimensional quantum levelstructure, which in the general case may include the effects of spin and exchange/correlationeffects from the Coulomb repulsion between electrons. This simplest scenario arises whenthe effect of the Coulomb repulsion can be accounted for by the electrostatic charging energyalone, which is represented by the Hartree term with the neglect of exchange and correlationeffects. Within this so-called “constant interaction” model [1,24] the charging energy, U (orEC), of the small electron puddle on the quantum dot, in the limit of zero source-drain biasacross the dot, is given by:

E =Q2

2C− Cg

CQVg, (1)

4

FIG. 2. (a) Energy parabolas of the single electron transistor quantum dot versus the number of charge

on the dot, in the absence of the contribution from quantum levels, at fixed gate voltages correspond-

ing to CgVg/|e| = n, (n + 1/2), and (n + 1), respectively. A Coulomb blockade conductance peak takes

when the energy for the Q/|e| = N and (N + 1) configurations become degenerate, as is the case for

CgVg/|e| = (n + 1/2). (b) An alternative representation of the energy parabolas plotted versus Vg , at fixed

charge number, Q/|e| = N and (N + 1), respectively. At a gate voltage (arrow) where the two parabolas

cross, conduction takes place giving rise to the Coulomb blockade peak. (c) Same as (b) but with the in-

clusion of quantum level spacing, ∆. Note that the gate voltage position of the energy degeneracy point is

shifted for the (N-1) to N case.

where C = CL +CR +Cg is the total capacitance of the dot to the environment, for a resistor-capacitor model of a single quantum dot depicted in Fig. 1. Note that to ensure charge isquantized on the dot, it is necessary for the tunneling resistances, RL and RR to be large. Inthe limit of very large tunneling resistances, the model reduces to the situation depicted inFig. 1(b). Coulomb blockade arises when the chemical potential of the dot is tuned via Vg

to favor an integral number (n, n+1, etc.) of electrons as depicted in Fig. 2(a) (solid curves).Since charge is quantized in units of e, in this case the energy can be rewritten in the form:

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E =(−N |e| + CgVg)

2

2C− C2

g

2CV 2

g , (2)

where we have Q = −N |e|, i.e. charge is quantized in units of −|e|, yielding a Coulombcharging energy of EC = e2

2Cfor the addition or removal of an electron. Here, the second

term quadratic in Vg and independent of Q is irrelevant and suppressed in the figures, sincewhat is important is a comparison of the system energy at different values of the charge, Q,in integral multiples of −|e|. At temperatures below this charging energy scale, kT < Ec,electron transport through the dot is suppressed. To facilitate charge transport, it is thusnecessary to tune the chemical potential towards a regime where a 1/2 integer number ofelectron is favored:

CgVg = (n +1

2)|e|. (3)

Here the N and (N+1) electron states are degenerate and charge can flow freely throughthrough the dot without the cost of a charging energy [2,3,1] as depicted in Fig. 2(a) by thedashed curve. Alternatively, we may plot the energy as a function of Vg at fixed number ofelectrons as shown in Fig. 2(b). At a Vg value where the parabolas for N electrons and N+1electrons cross, Coulomb blockade is lifted and transport proceeds freely through the dot.

Within this scenario, inclusion of the effect of 0-dimensional quantum confinement intro-duces an additional quantum energy level structure on top of the charging energy, EC , asdepicted in Fig. 2(c) by the presence of the quantum level spacing, ∆. In the absence of spindegree of freedom, or for phenomena in which spin does not play a role aside from doublingthe density of states, and when higher-order virtual tunneling processes are neglected thecurrent (I) through the quantum dot in the limit of fully coherent tunneling is given by aconvolution of the difference of the Fermi function at the chemical potentials in the two leadsand the Breit-Wigner resonance formula: [49]

I(kT, eVbias) =e2

h

∫

dE[f(E + eVbias) − f(E)]ΓLΓR

Γ2 + (E − Eo)2. (4)

where f is the Fermi-Dirac distribution function, ΓL,R denote the level broadening due tocoupling to the left and right leads, respectively, and Γ = 1

2(ΓL + ΓR), and Eo the energy of

the resonant level which may be tuned by the plunger gate, Vg. In this simplest scenario,if the dot shape is irregular and therefore without the special symmetries which often giverise to accidental degeneracies in the energy spectrum and shell filling effects [25], the 0-dimensional energy levels are expected to exhibit level repulsion related to the Wigner-Dysontype while at the same time the wave function fluctuation at given spatial points reflectedby a resonance width, Γ, which follows the Porter-Thomas distribution. As it turns out,the resonance line-width does indeed follow the Porter-Thomas type distribution to a largeextent [15,16], as will be reviewed below. However, the level spacing distribution deviatesfrom the naive expectation of a Wigner-Dyson type due to the effect of interaction beyondthe Hartree contribution including the influence of spin and exchange energies [26,27].

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Even within this scenario which excludes exchange and correlation effects, spin plays arole in the filling behavior of the dot. If we fill the dot starting from empty, the first electronwill enter the lowest quantum level, followed by a second electron into the same level, albeitwith opposite spin. The two electron will form a spin singlet to ensure antisymmetrizationof the total wave function including of the orbital and the spin parts. The addition of thissecond electron will cost a Coulomb charging energy, EC . If we next proceed to add a thirdelectron, it will be forced to occupy the next high quantum level (1st excited level) due toPauli exclusion. Therefore the energy required would not only involve a charging energy,EC , but also in addition, the level spacing energy equal to the excitation energy of the firstexcited state. The fourth electron will again be accommodated within this same excited leveland cost an energy, ∆E = EC . The electron addition energy, ∆E, therefore is expected toexhibit an oscillatory behavior of period 2, oscillating between ∆E = EC and ∆E = EC +∆.This is a key signature of the so-called even-odd, pairing effect associated with the spindegree of freedom.

More complex physical effects can be uncovered when higher-order processes are takeninto account, particularly when the quantum dot is tuned to the Coulomb blockade valleys(integer number of electrons). Among the variety of effects which arise one of the mostinteresting is the Kondo resonance which results when an effective spin-interaction occursbetween the quantum dot excess spin and the spin of the electron sea contained in the leads.Through virtual processes, screening of the residual spin on the quantum dot leads to thewell-known Kondo effect [22,23,28,29] familiar in the case of dilute magnetic impurities inmetals. Specifically, starting from the Hamiltonian for the dot-lead system:

H = HQD + HR + HL + HQD−Leads (5)

=∑

iσ

εid†iσdiσ +

∑

i

Uni↑ni↓ +∑

kσ

εRk c†RkσcRkσ

+∑

kσ

εLk c†LkσcLkσ +

∑

i,k,σ,σ′

[V Rik d†

iσcRkσ + V Likd†

iσcLkσ + h.c.],

it can be shown that a Kondo resonance, characterized by a Kondo energy scale, EK =kTK =

√UΓe−π|εo−µ|(U+εo)/UΓ, results from higher order processes when a single spin resides

in the highest, occupied level within the quantum dot [22,23]. Loosely speaking, an electronfrom the left lead biased at a slightly higher chemical potential can tunneling into thishighest, singly-occupied level, while at the same time, either electron can at once tunnel outinto the right lead. Since double-occupancy occurs for a short time, no penalty in terms ofthe charging energy is incurred. Adding up such higher order contributions, including thosewhich the external electron from the lead can tunnel back and forth any number of timesbetween the left lead and the dot before an electron exits into the right lead gives rise to aKondo resonance at the Fermi level.

7

FIG. 3. (a) Model of a series-coupled double quantum dot system with left (L) and right (R) tunnel

junctions to the leads (source, S, and drain, D), and a dot to dot tunnel junction (t). The charge on each dot

can be separately tuned via the gates Vg1 and Vg2. (b) Capactive model of the fully symmetric, series-coupled

double-quantum-dot.

A. Double Quantum Dots

When two quantum-dots are coupled together, the electrostatics naturally become morecomplicated as a result of the mutual capacitance between each pair of conductor, includingthe quantum dot metallic puddles and the various pincher and plunger gates for the formationand control of the dots. Here we also summarize the simplest scenario based on even-oddfilling of each of the two dots. To clearly illustrate the role of the electrostatics, we initiallyconsider a situation of weak coupling to the leads (Γ << ∆). This means we will be ableto for the time being neglect Kondo spin effects. For now we will also neglect the existenceof quantum level spacing as well as higher-order effects from inter-dot coupling, which leadsto an effective anti-ferromagnetic coupling between the excess spins on the two dots ofJ ∼ 4t2c/U . These additional ingredients and their effects will be discussed subsequently.

This energy diagram based again on classical electrostatics can readily be calculated bygeneralizing the capactive-resistive model of a single dot (Fig. 1) to the double dot situationas is depicted in Fig. 3 [32–34,5]. It is most convenient to characterize the system using thecapacitance matrix. Starting from the basic charge-voltage difference relationship on thei-th conduction, the total charge Qi, is given by:

Qi =∑

j

qij =∑

j

cij(Vi − Vj). (6)

Cast this into vector form with Q = (Q1, ..., Qi, ...QN)T, V = (V1, ..., Vi, ...VN)T, and definingthe capacitance matrix C , where Cij = (

∑

k cjk)δij − cij, yield:

8

Q = CV, (7)

and an electrostatic energy for the system, E, of:

E =1

2VTCV =

1

2QTC−1Q. (8)

Inclusion of the charge, Qv, and voltage, Vv, of voltage sources, which can be modeled asbatteries with large capacitances, the voltage, Vc, on the conductors may conveniently berelated to its charge, Qc, and the voltage settings on the sources, and the capacitance sub-matrices between the conductors, C

cc, and between the conductors and the voltage sources,

Ccv

[32–34,5]:

Vc = Ccc

−1(Qc − Ccv

Vv). (9)

For the series-coupled double quantum dot depicted in Fig. 3(b) such consideration leads toan electrostatics energy at zero bias of:

E(N1, N2) =1

2N2

1 EC1 +1

2N2

2 EC2 + N1N2ECt (10)

− 1

|e| [Cg1Vg1(N1EC1 + N2ECt) + Cg2Vg2(N2EC2 + N1ECt)]

+1

e2[1

2C2

g1V2g1EC1 +

1

2C2

g2V2g2EC2 + Cg1Vg1Cg2Vg2ECt],

where ECi = e2

Ci

(

11−C2

t /C1C2

)

, ECt = e2

Ct

(

1C1C2/C2

t −1

)

, and Ci ≡ CL,R + Cgi + Ct is the total

capacitance of the i-th (L or R) dot to its surroundings. Substantial intuition may begained by examining the experimentally relevant and simple case of fully identical dots, eachwith the same capacitive coupling to its respective plunger gate, i.e. CL = CR = Cl andCg1 = Cg2 = Cg. Eq. 11 then reduces to the form:

E(N1, N2) =1

2N2

1 EC +1

2N2

2 EC + N1N2ECt (11)

− 1

|e| [CgVg1(N1EC + N2ECt) + CgVg2(N2EC + N1ECt)]

+1

e2[1

2C2

g (V2g1 + V 2

g2)EC + CgVg1CgVg2ECt],

with C = C1 = C2 and EC1 = EC2 = EC . In the limit of zero interdot-coupling, for whichCt = 0 and ECt = 0, the system behaves as two isolated but identical dots, with energy:

E(N1, N2) =EC

2(−N1|e| + CgVg1)

2 +EC

2(−N2|e| + CgVg2)

2, (12)

where EC = e2/Co and Co = Cl + Cg. In the opposite limit of large interdot coupling anddominant Ct � (Cl + Cg), so that C ≈ Ct, we have ECt ≈ EC = e2/2Co and:

9

E(N1, N2) =EC

2[−(N1 + N2)|e| + Cg(Vg1 + Vg2)]

2, (13)

and the system behaves as a single large dot, albeit with a charging energy, EC , which is onehalf the isolated case.

FIG. 4. The energy diagram for a fully symmetric series-coupled, double quantum dot, for different

values of the interdot capacitance, Ct. The energy at fixed occupancy, (N1, N2), is plotted versus the gate

voltage, Vg = Vg1 = Vg2. For Ct = 0, ECt = 0, and the two dots are isolated (short dashed curve). For

Ct > 0 but small compared to (Cl +Cg), ECt < EC and the single degeneracy point is now split as indicated

by the two short arrows (medium dashed curve). This gives rise to a splitting of the Coulomb peak when

contrasted with the isolated case of Ct = 0. when Ct dominates, ECt ≈ Ec, and the two dots behave as a

single large dot with a doubling in the frequency of occurrence for the Coulomb peak as a function of Vg .

If we were to tie Vg1 and Vg2 together so that Vg1 = Vg2 = Vg, and plot the electrostaticenergy at fixed N1, N2:

E(N1, N2) =EC

2[(−N1 +

CgVg

|e| )2 + (−N2 +CgVg

|e| )2] + ECt(−N1 +CgVg

|e| )(−N2 +CgVg

|e| ),

(14)

versus Vg, for the three cases of: (a) Ct = 0 with ECt = 0, (b) Ct < (Cl + Cg) withECt > 0, and (c) Ct � (Cl + Cg) with ECt ≈ EC , we see behaviors at the charge degeneracypoint(s) corresponding to two identical isolated dots, the development of a splitting due tothe finite ECt, and one large dot, as shown in Figs. 4(a), (b), and (c), respectively, with a

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concomitant doubling of the period of the Coulomb blockade conductance peaks versus gatevoltage compared to the isolated-dots case (Ct = 0), when Vg1 and Vg2 are tied together.Here we have plotted the energy for fixed electron numbers (N2, N2), which denote the excessoccupancy above up-down spin paired occupied quantum levels, with Ni=0,1 indicating theexcess occupancy on dot i, in Fig. 1(a) the parabolas depict the energy curve for the (0,0)empty state, the (1,1) singly-occupied state on each dot, and the (0,1) and (1,0) stateswhere one electron occupies one of the two dots. When the inter-dot coupling is turned off,Ct → 0, the (0,1) and (1,0) parabolas are degenerate so that the condition for which Coulombblockade is lifted occurs at one point in the diagram. This is the case of two identical butisolated dots for which in each dot the Coulomb blockade is removed when it is tuned to favora half-integer number of electrons. Note that here the parabola maybe be shifted in energyby a single particle quantum level spacing, ∆. However, such a shift does not qualitativelychange the picture aside from shifting the gate voltage position where the blockade is lifted.When coupling is gradually introduced, the now non-zero inter-dot coupling , Ct, a loweringof the electrostatic energy of the (0,1) and (1,0) states. The interception of the lower curvewith the (0,0) and (1,1) parabolas at two distinct points signals a splitting of the quantumdot conductance peak into two.

FIG. 5. Charge stability, honeycomb diagrams in the Vg1 versus Vg2 plane, for the three cases of Fig. 4,

Ct = 0, Ct > 0 but small, and Ct � (Cl + Cg), respectively. The situations depicted in Fig. 4 correspond to

a cut along the diagonal in these three diagrams, respectively.

In the more general situation where Vg1 6= Vg2, it is useful to examine the stability

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diagram in the Vg1 versus Vg2 plane. Such a stability diagram can be obtained by firstdefining a chemical potential for each dot, µi:

µi ≡ E(N1, N2) − E(N1 − δi,1, N2 − δi,2) (15)

= EC [(Ni −CgVgi

|e| ) − 1] + ECt(Nj −CgVgj

|e| ),

and the addition energy for adding an electron on either dot, Eadd:

Eadd ≡ µi(N1 + δi,1, N2 + δi,2) − µi(N1, N2) = EC , (16)

where i={1,2} and i 6= j. With the definition of µ = 0 for each lead at zero bias, stability foroccupation (N1, N2) is given by the requirement of µ1 < 0 and µ2 < 0. Such stability diagramin this ideal situation is shown in Fig. 5 for different values of Ct and hence ECt. The threescenarios correspond to the situations depicted in the previous Fig. 5. The presence of thesix-sided polygon for general values of ECt has given rise to the nomenclature “honeycomb”diagram. Inclusion of the quantum energy level, non-ideality in real devices such as residualgate-voltage dependence of EC and ECt, and residual mutual capacitance between gates leadto distortions of such honeycombs such variations in the area of honeycombs and a changein the slope of the domain boundaries. Again, in the limit of very strong inter-dot coupling,the double-quantum-dot behaves as a single large dot in accordance with expectation.

In an experiment the stable configuration (N1, N2) is controlled by Vgi. A plot in the

(Vg1,Vg2

) plane can be obtained and represents an extremely useful way to characterize theDQD system. The above-mentioned limits of no inter-dot coupling and strong coupling areshown in Fig. 5. When ECt increases from 0, within a quantum-mechanical picture wheresuch an increase corresponds to increasing the inter-dot tunneling matrix, the non-zero inter-dot tunneling splits the energy of the symmetric and anti-symmetric orbital levels associatedwith the (0,1) and (1,0) states by an amount proportional to ECt, whenever the correspondingunperturbed energy levels for these states are tuned to degeneracy. Such coherent couplingis essential in the use of quantum dots as qubits for quantum computation. Experimentally,such an energy splitting due to this coherence have been demonstrated in transport as wellas microwave absorption experiments [36,35]. Again this scenario takes place when couplingto the leads is negligible and the properties are single-particle in nature.

As in the case of the individual dot, inclusion of quantum level spacing simply shifts theparabolas by the respective level spacings, ∆. Going beyond single particle properties toaccess regimes exhibiting properties of many-body spin correlation may be accomplished byintroducing coupling to the conduction electrons in the leads by increasing Γ to ∼ ∆. Thefull Hamiltonian of the coupled double quantum dots system with leads is given by:

HDQD =∑

kα∈{L,R},σ

εkαc†kα,σckα,σ (17)

+∑

α∈{L,R},σ

εασd†ασdασ + Vo

∑

kα∈{L,R},σ

(c†kα,σdασ + h.c.)

+tc∑

σ

(d†LσdRσ + h.c.) + U(nL,↑nL,↓ + nR,↑nR,↓),

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The introduction of the inter-dot tunneling matrix, tc, in addition to the charging energy,U = EC , level spacing, ∆, and level width characterizing coupling to the leads, ΓL = ΓR ≡ Γ,leads to additional and extremely interesting new physics. For weak-coupling where Γ << ∆,the usual single-particle coherence between the two coupled-dot quantum levels is expectedand observed. [35,36] When the coupling is strong when Γ ∼ ∆, dramatic new physicsassociation with the many-body Kondo effect has been predicted by theory.

Compared to the single QD case, this DQD model contains one extra parameter, tc,which parameterizes the coupling (tunneling matrix element) between the two dots. Theinclusion of this energy scale gives rise to a rich variety of correlated physics beyond thesingle dot case. In the limit of large U (EC), it turns out to be convenient to re-parameterizethe system with the following energy scale obtainable from those given in the Hamiltonian:

1) ΓL,R ≡ πV 2o ρL,R = Γ, is the quantum level broadening in the two dots (L or R) due to

their respective coupling (Vo) to the left (L) or right (R) lead. Here ρL,R denotes theelectronic density of states in the L- or R- lead.

2) tc, the bare, interdot tunneling matrix element,

3) J = 4t2cU

, the effective anti-ferromagnetic coupling between a single excess (unpaired) spinon each dot, and

4) TK ≈√

UΓexp[−π|µ − εo|(U + εo)/ΓU ], the individual dot Kondo temperature.

Accordingly, based on Slave-Boson Mean Field (SBMF) theory [37–40], numerical Lanc-zos [41] and renormalization group (NRG) calculations [42], or the non-crossing approxima-tion [43], three distinct regions of correlated spin behavior is present when an excess spinoccupies each dot. When tc/Γ < 1, the system can be mapped onto the two-impurity Kondoproblem initially discussed by Jones et al. [44,45], albeit without even-odd parity symme-try [46]. In this situation, when the antiferromagnetic coupling is weak, J/TK < 2.5, thesystem behaves as two separate screen Kondo spins, with screening of each dot spin by theconduction electrons in the respective leads. When J increases beyond the “critical” valuewhere J/TK ≈ 2.5, a cross-over takes place to a state where the two impurity (dot) spinsform a single due to the strong anti-ferromagnetic coupling. A cross-over rather than a truephase transition arises due to the relevant perturbation introduced by the hopping term, tc,which automatically breaks the even-odd parity symmetry [42,40]. Note that in an exper-imental device, J can be increased by increasing tc while keeping U relatively constant. Athird region of strongly correlated behavior occurs with further increase of tc. Depending onwhether one uses an infinite U model [37–39,43] or finite U model [42,40], when either tc/Γexceeds 1, or when tc exceeds U/4 [42], the system forms a coherent superposition of themany-body Kondo state of each dot, leading to a strongly renormalized splitting 2tc and adouble peak in the differential conductance near zero bias, in contrast to the ordinary singlepeak behavior. In addition, dramatic negative differential conductance is also predicted inthis regime.

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III. NON-GAUSSIAN DISTRIBUTION OF COULOMB BLOCKADE PEAK

HEIGHTS IN INDIVIDUAL QUANTUM DOTS: PORTER-THOMAS

DISTRIBUTION OF RESONANCE WIDTHS

For open systems, in a separate work we had experimentally demonstrated a distinc-tion between chaotic and non-chaotic behavior [47]. In nearly isolated systems, however,non-chaotic behavior is practically unrealizable. Any residual disorder or lithographic im-perfections will render the dynamics chaotic on the long trapping time-scale before eventualescape into external leads/reservoirs takes place. There are non-trivial predictions on statis-tical properties of nearly isolated cavities which can be tested in experiments in the transportthrough quantum dots, pertaining to the distribution of energy level spacings and the sta-tistical properties of wave functions. In the case of the quantum dots, the presence of theCoulomb charging energy indicates that electron-electron interaction is present. Howeveras a first approximation, it may be reasonable to assume that in some appropriate limit,e.g. high electron density where screening is effective, the interaction simply contributes aclassical charging energy given by EC = e2/C where C is the capacitance of the quantumdot to its surroundings (see Background Section), and does not seriously affect the RMT dis-tributions. The assumption proves to be largely correct in the GaAs/AlxGa1−xAs quantumdots studied. However, there is suggestive evidence that deviations from the RMT universaldistributions maybe observable in our small (≤ 0.25µm) quantum dots.

The specific predictions of theory which we tested pertain to the distribution of theseCoulomb blockade peak heights at B=0, corresponding to the RMT Gaussian OrthogonalEnsemble (GOE), and at B 6= 0, corresponding to the Gaussian Unitary Ensemble (GUE).In the latter case, the magnetic field, B, is required to be exceed some correlation field,Bc, which characterizes the transition from GOE to GUE statistics. According to theory[19,48], in the single level tunneling limit of thermally broadened conductance peaks, theB=0 distribution is given by:

P(B=0) =√

2/πα e−2α ; (18)

with a square-root singularity near zero. Here, α is related to the Coulomb blockade peakconductance, Gmax, is given by: [49]

Gmax =e2

h

π

2kT

ΓLΓR

ΓL + ΓR≡ e2

h

πΓ

2kTα (19)

where ΓL ( ΓR) is the partial decay width into the left (right) lead. This expression isobtainable from Eq. 4 by taking the voltage derivative, dI/dV, in the limits of zero Vbias andlow temperature– Γ � kT � ∆, where single level tunneling dominates so that electronstunnel through a single quantum level within the quantum dot, and the conductance peakis thermally broadened beyond the natural width Γ. In a magnetic field greater than thecorrelation field, the breaking of time-reversal symmetry reduces the number of nearly zerovalues of Gmax. Nevertheless, the distribution [19,50] is still non-Gaussian and peaked nearzero:

14

FIG. 6. Top–Electron micrograph of the gates defining the quantum dots. Four dots are available on

each sample. Bottom-left–G−1

maxversus T for a representative peak at B = 0. The roughly linear behavior

below ∼300 mK indicates this is a single level tunneling peak. Bottom-right– A fit of the convolution of

−∂f/∂ε with the Breit-Wigner resonant tunneling formula to the peak in the left panel at T = 108 mK.

P(B6=0) = 4α[K0(2α) + K1(2α)] e−2α (20)

where Kn are the modified Bessel functions. For GaAs/AlxGa1−xAs quantum dots, Γ canreadily be tuned to be below 3µV (corresponding to T ∼ 34mK) while typically the levelspacing ∆ ≥ 200µV (T ∼ 230mK). The required conditions of thermally broadened singlelevel tunneling limit is therefore accessible at dilution refrigerator temperatures ∼ 70mK.In contrast, in typical metallic dot of size 50 nm, ∆ ∼ 0.1µV (∼ 1.1mK) and many levelsare accessed. This leads to a convolution of independent single level distributions resultingin a Gaussian distribution of peak heights in accordance with the central limit theorem.

From an experimental perspective, the challenges are to fully access the single leveltunneling limit, and to identify GaAs/AlxGa1−xAs starting material sufficiently free of back-ground traps to allow temporal stability of fabricated quantum dots. To fully access the single

15

level limit, ∆ ≤ 5kT is invariably needed. The unavoidable decoupling of the electrons fromthe lattice at low temperatures renders it difficult to reduce the electron temperature belowthe 50-100mK range [18,51] even after strongly filtering any noise (thermal or pickup) fromthe measurement system. For GaAs/AlxGa1−xAs to satisfy this condition, small quantumdots of order 0.25µm or smaller in its largest dimension must be fabricated. Superior elec-tron beam lithography is requisite to produce the multiple metal gate electrodes used toform the weak-coupling, tunnel barriers to external leads (otherwise known as pinchers) andto form the central quantum dot. In Fig. 6 top panel we present an electron micrograph ofthe metal gate pattern for four dots in series used in our experiment. Each dot is roughly0.3µm × 0.35µm in lithographic size but is reduced to below 0.25µm × 0.25µm after gating.In our experiment, individual dots were separately measured rather than the whole series offour dots. Regarding temporal stability, by nature the Coulomb blockade is sensitive to asingle electron charge. Any motion of background charges in near by trap sites will stronglyaffect the transport both via a change in electrostatic energy, and via a distortion of the dotshape. It is found that crystals grown under varying conditions in different laboratories yieldquantum dots which range from temporally quite stable to extremely unstable. The fabri-cation and sorting out of sufficiently stable dots to ensure reproducible Coulomb blockadepeak height values has made our work and the work of Folk et al. possible. In our devices,approximately 50% of individual dots meet the stability criterion.

FIG. 7. A typical trace showing successive Coulomb blockade conductance peaks versus the center gate

voltage, Vg . B = 0 and T = 75 mK (lower trace) or T = 660 mK (upper trace, displaced by 2 units). Note

that three peaks are missing out of seven, but they emerge at higher temperature. The slight shifting in

peak positions is discussed in the text.

In Fig. 7 we show a representative trace of Coulomb blockade peaks at B=0 for T=75mK(bottom curve) and 660mK (top curve) temperatures. Note the missing peaks at the gatevoltages -733, -753, and -762mV in the 75mK curve which are observable at the higher tem-

16

perature of 660mK. The large difference in height of adjacent peaks and the many small peaks

are our primary experimental observation, and qualitatively demonstrate the outstanding fea-

ture of a prevalence of small peaks predicted by RMT in Eq. 18. To demonstrate we are fullyin the single level tunneling limit of temperature broadened peaks, in Figs. 6 lower-left andright panels, we plot G−1

max versus T and the line shape of a representative peak fitted to thetheoretical cosh−2[(E0 − γeVg)/2kT], respectively. In Fig. 6 bottom-left the roughly lineardependence of G−1

max on T at low temperatures indicates we are clearly in the single levelregime (see Eq. 19). In fact we find this behavior in all of the 8 peaks we studied in detail.

FIG. 8. Magnetic field sweep of four peaks at T = 100 mK. The field range for ∼100% change in Gmax

is ∼500 Gauss. (c) and (d) show the two types of behavior of peaks which are very small at B = 0.

For all peaks, we are able to follow the evolution with magnetic field. In Fig. 8 we showmagnetic field traces of Gmax for four representative peaks. From the fastest variation ofwe estimate a correlation field, Bc, of the order of 500 Gauss, somewhat larger then thetheoretical value [19,50] of 200 Gauss. Panels (c) and (d) depict two peaks which are nearlyzero at B=0. The behavior in (d) where the peak remains small for large stretches of Band only occasionally increases to a large value is observed in roughly 1/3 of peaks whichstart near zero. The type of behavior is not expected in the RMT picture where fluctuationsshould occur on the scale of Bc. The fact the height remains small for large regions in Bsuggests an enhancement of small peak probability above the RMT prediction (Eq. 20) forB 6= 0.

The traces in Fig. 7 are obtained by keeping the pinchers (tunnel barriers to externalleads) gate voltages constant while sweeping the central quantum dot gate voltage, Vg.Sweeping Vg more negative, however, also affects the pinchers by gradually closing themoff, thereby reducing the conductance value. When too many peaks produced by sweepingVg alone while keeping the pinchers fixed are included, skewing of the distribution can occur.

17

Therefore, we impose a window of acceptance corresponding to roughly 5 peaks per pinchersetting as valid data. The details of how the window is set can be found in Chang et al. [15].The distribution we obtain will subsequently be compared to the theoretical distributionsdeduced by averaging Equations 1 and 3 over the pincher window of approximately a factorof 3.5 in the transmission probability. By collecting data at different pincher settings wherecare is taken to ensure the Coulomb blockade peaks are uncorrelated from previous settings,we are able to collect data from 72 independent peaks. This procedure yielded 72 peak heightvalues for B=0, and 216 values for B 6= 0. In the latter case height data are taken at threedifferent magnetic fields well separated by several Bc’s to triple the data set. The resultingdistributions are presented as histograms in Fig. 9. The distributions are normalized to unitarea as for a probability density. Both the B=0 (a) and B 6= 0 (b) distributions are stronglynon-Gaussian, and clearly peaks toward zero values. In the B=0 case, nearly 1/3 of the peaksfall in the lowest bin: 23 out of 72 peaks are less than 0.005e2/h compared to a mean of∼ 0.024e2/h. In contrast, for B 6= 0 only 43 out of 216 peaks are this small. Fig. 9 indicatesthat there is a difference between the two distributions for low values as is born out by theKolmogorov-Smirnov statistical test.

Pro

babi

lity

Den

sity

=B 0(a)

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0 4 8

G (0.01 e /h)max2

=/B 0(b)

0 2 4 6 80.0

0.2

0.4

0.6

0 4 80.0

0.2

0.4

0.6

FIG. 9. Histograms of conductance peak heights for (a) B = 0 and (b) B 6= 0. Data are scaled to unit

area; there are 72 peaks for B = 0 and 216 peaks for B 6= 0; the statistical error bars are generated by

bootstrap re-sampling. Note the non-Gaussian shape of both distributions and the strong spike near zero in

the B = 0 distribution. Fits to the data using both the fixed pincher theory (solid) and the theory averaged

over pincher variation (dashed) are excellent. The insets show fits to χ2

6(α)— a more Gaussian distribution—

averaged over the pincher variation; the fit is extremely poor.

18

The mean decay width needed for comparing to theory is not measured experimentallyand is therefore a fitting parameter. This width should be nearly independent of B; thus weintroduce a single scale parameter and fit simultaneously to the B=0 and B 6= 0 data. Fig. 9shows a fit to the data using both the theory for constant pincher transmission [Eqs. 1 and3 (solid)] and this theory averaged over a variation of the pincher transmission by a factorof 3.5 (dashed). The similarity of the two curves shows that the variation in our pinchertransmission can be neglected.

FIG. 10. The B 6= 0 distribution of Fig. 9(b) replotted with the lowest bin further split into two. The

resulting histogram shows that the probability for small peaks continues to increase toward zero, in contrast

to RMT predictions. This trend is likely a result of electron-electron interaction

The experimental distribution in Fig. 9(b) for B 6= 0 appears slightly higher for the small-est height data point compared to the theoretical dashed curve. Even though the difference iswithin error bar, it is suggestive. Intrigued by the possibility of deviation from the RMT the-oretical result, we further split this lowest bin into two, producing the histogram in Fig. 10.The probability of small peaks continues to increase for height values approaching zero, instark contrast to the RMT result of a maximum at ∼ 0.025e2/h! This excess of small peaksis related to the discussion of Fig. 8(d) where certain peaks which are small when B=0remain small for large regions of B > 0. In fact, recent theoretical calculations aimed ataccounting for electron-electron interactions appear to show exactly this trend [52]. Theinfluence of interaction certainly deserves further investigation. Improved statistics shouldfurther elucidate the role of electron-electron interaction in chaotic systems.

19

IV. SPIN AND PAIRING EFFECTS IN ULTRA-SMALL DOTS

FIG. 11. Scanning electron micrograph of device 1(Top). Schematic diagram of the peak positions as a

function of gate voltage(Bottom). The narrow period corresponds to the change from odd to even numbers

by adding an electron with the opposite spin into the same spin degenerate state, and the broad period to

the change from even to odd numbers, occupying different dot energy levels.

The difference between an even and odd-numbered finite Fermion system, known asthe even-odd parity effect, is a distinct feature reflecting the unique behavior of fermionicparticles in the presence of both orbital and spin degrees of freedom. [53–56]. This parityeffect is expected to appear in artificially fabricated semiconductor quantum dots. Theunprecedented control over experimental conditions in lateral quantum dots allows one to fillelectrons one by one as manifested in the phenomenon of Coulomb Blockade [3] in transport.The peak spacing fluctuation in CB peaks provides unique information about single particleenergy level, many-body interaction effect and the parity of electron numbers. In irregularlyshaped dots without special symmetries the CB peaks are expected to be paired with smallerspacings in the odd electron valleys and larger spacings in the even valleys, reflecting thedot spin status which is 1/2 h for an odd number of electrons and paired to zero for an evennumber (Fig. 11 and discussion above). The importance of unambiguous and differing effects

20

reflecting the spin status of the dot pertains directly to the desire to take the next logical stepand couple dots together for the sake of both fundamental physics (competition between theKondo effect and indirect exchange interaction in the two impurity model [57,58,44,45]) andand for technological reasons in the implementation of the double dot system as prototypequantum qu-bits in quantum computation [11]. Because these potentially important futuredevelopments depend on the success in achieving well-defined and well-controlled spin statuson a dot, the clear observation of the pairing effect as well as the elucidation of the necessaryconditions are of central importance.

In this Section, we describe clear pairing features attributable to the even-odd effect insmall GaAs/AlxGa1−xAs lateral quantum dots some with the smallest rs value measured todate. These signatures are observed in the CB peak spacing, peak height as well as spin. Wefind a quantum dot with smaller rs shows more pronounced peak pairing than high rs dots.Furthermore, qualitatively different behaviors in peak spacings are observable depending onthe coupling strength between the dots and leads.

Three devices are fabricated in different rs regimes. Device 1 containing the numberof electrons, N ≈ 10 and device 2 containing N ≈ 40 are made from a crystal of density,n = 3.5 × 1011cm−2(rs = 0.93) while device 3 with N ≈ 10 − 20 from a high density crystalof n = 9 × 1011cm−2(rs = 0.58). The geometry and size of the devices 1 and 2 (lithographicdiameters of 160nm and 230nm, respectively) were chosen carefully to maximize functionalityin spite of their small sizes (Fig. 11(a)). The geometry of device 3 with a lithographicdiameter of 170nm enables us to control N between 10 − 20(inset in Fig. 14(a)) by theapplication of 6 independent gate voltage settings. The minimum size of small quantumdot is to a large extent limited by the depth of two dimensional electron gas (2DEG). Indevices 1 and 2 all length scales, pincher gap, dot size, gate widths and their positions wereoptimized for a relatively deep 2DEG, 95 nm below the surface. While the lithographic sizeis 160 nm, device 1 is estimated to be 60 nm in diameter after accounting for depletion.The main advantage of our small quantum dot with multiple independent gates is that bysetting different gate voltages, different realizations of a quantum dot can be implemented.Although it is often difficult to precisely determine how many electrons reside on lateralquantum dots, to check whether the dot region is completely depleted tunneling barriers arereduced to compensate increased depletion from the plunger gate while it is swept [59]. Inthis way, we observed a maximum of 8 CB peaks in device 1, close to the estimated electronnumber based on the 60nm size.

In a standard constant interaction model [1], the peak spacing in plunger gate voltage isexpressed as a combination of single particle energy level splitting and charging energy term:

∆Vg =C

eCg

{

∆ +e2

C

}

. (21)

21

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FIG. 12. Evolution of the peak structure for different dot-lead coupling in device 1 measured at T=300

mK. From (a) to (c), the coupling is decreased gradually. In (b), dotted line is T=900 mK trace, illustrating

temperature dependence of Kondo and non-Kondo valleys. By closing the dot, the peak spacing pairing is

destroyed even with this small size dot. Inset in (c): Comparison of peak spacing in ( b)(filled circle) and

(c)(unfilled square) in each valley(x-axis).

Here, ∆ is zero if N=odd and greater than or equal to zero for N=even, reflecting the spindegeneracy on each energy level(Fig. 11(b)), C is the total capacitance of the dot and Cg

is the capacitance between the dot and plunger gate. A direct consequence is that thespacing is smaller between adjacent CB peaks separated by a valley with an odd numberof electrons than that with an even number. When the dot size is large, the average levelspacing ∆ is much smaller than the charging energy EC=e2/C and the above formula impliesrelatively small fluctuations in peak spacing. The spacing appears nearly uniform in thiscase. On the other hand, a small quantum dot can be expected to give more pronounced peak

22

spacing pairing from the increased ∆. Furthermore, in the small-number electron system,the addition or extraction of one electron can change the entire energy spectrum due to thestrong Coulomb interaction, known as Koopman’s theorem, resulting in discrepancy betweenconstant interaction model and experiment.

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FIG. 13. Fluctuation of peak spacing in device 2 at T=75 mK. This device has about 40 electrons inside

the dot, fabricated on a low density crystal (rs ∼ 1). Insets in (a) show several features of Kondo effect. Left

inset: zero bias maximum (ZBM) of valley 3 in -0.1 mV < VDS < 0.1 mV range in differential conductance.

Kondo valleys in this device show Zeeman splitting of ZBM under magnetic field, about ±21µV/Tesla. Right

inset: expanded view (×6) of temperature dependence of the valley 5. Dotted line is at T=150mK. Inset in

(b): Filled circle symbol represents stronger coupling(a) than unfilled square symbol(b).

In our experiment, clear signature of an even-odd pairing can be observed in the CB peakspacing when the coupling between the quantum dot and leads is strong in all devices. On theother hand, as the dot-lead coupling is reduced a marked difference emerges between the highrs devices 1 and 2 and the low rs device 3. In device 1 when the dot is nearly open, two broadpeaks are observed as shown in Fig. 12(a). A decrease of the dot-lead coupling resolves more

23

peaks and pairing is clearly present in Fig. 12(b). When the coupling is further reduced, thepairing is again no longer visible(Fig. 12(c)). Even with this smallsize dot, there is no clearsignature of even-odd effect in weak tunneling regime. Fig. 13 demonstrates similar resultsin device 2. The spacing between peaks straddling a valley (2, 4, 6) with an even numberof electrons (paired spins) is distinctly larger than that of the odd valleys(1, 3, 5) (insetto Fig. 13(b)), while at the same time the valleys 1, 3, and 5 exhibit the Kondo resonanceas a signature of unpaired single electron spin. The pairing gradually disappears when thedot-lead coupling is reduced for valleys 7-10 in Fig. 13(a) and all valleys in Fig. 13(b). Thedisappearance of pairing in the weak coupling regime for these two devices indicates thatthe constant interaction model is beginning to break down and explains the difficulty inobserving peak pairing over a large number of peaks in previous works [28–31,26]. In starkcontrast, in the small rs high density device 3 peak pairing is preserved in all regimes of the

dot-lead coupling as evidenced by the ubiquitous pairing behavior in Fig. 14. In fact pairingis observable under quite different gating configurations. For example, the dot contains moreelectrons with the configuration in Fig. 14(c) than Fig. 14(a) or (b). Nevertheless, pairing is

observable for at least 10 peaks in succession! The relative ease of observing pairing in thisdevice is likely a direct consequence of the low value of rs = 0.58, lower than all previousdevices reported in the literature.

We point out that the results in devices 1 and 2 indicate that the controlling parameterfor observability of the peak spacing pairing is not simply the ratio ∆/EC . By controllingthe transmission coefficient of tunnel barriers, confinement strength of the dot is adjustable,which at the same time modifies ∆ and EC rendering it possible to set different values forthis ratio. In the strongly coupled regime we estimate this ratio to be ∼ 1 in device 1compared to 0.5 in device 2 which is larger in size. On the other hand in the weak couplingregime, we expect an increase in both EC [28] and ∆, yielding a ratio between 0.5-0.6 fordevice 1. Even though still larger than device 2 in the open regime, the even-odd pairing hasdisappeared. Furthermore, the pairing we observe can be attributed mainly to the large ∆ inthese small dots and not to a spacing shift caused by the Kondo resonance, based both on arecent theoretical estimate indicating that the Kondo shift is at most 20% of ∆ [60] and thefact that our results often show large even-odd spacing differences as large as EC ∼ 1meV(Fig. 12(b)).

We turn our attention to the different behaviors of the low and high rs dots. The observedbehavior when the dots are closed (weak coupling) depends on rs. When open, peak pairingis observable in all three devices as shown in Fig. 12(b), 13(a) and 14(a). However whenclosed, in contrast to the high rs devices 1 and 2 pairing is still preserved in the low rs

device 3. This is particularly striking in view of the fact that device 3 and device 1 containa similar number of electrons. The difference in behavior is believed to be a consequence ofthe modification in the electron-electron interaction due to a reduced rs = 0.58 in device 3compared to an rs = 0.93 in devices 1 and 2, and suggests that strong Coulomb interactionplays an important role in deciding peak spacing fluctuation, washing out the peak pairingin the low density high rs regime while preserving it in a robust way in the high density lowrs regime. We propose that when the Fermi energy EF is high enough, it offers a better

24

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FIG. 14. CB peaks spacing fluctuations at T=300mK from device 3. The lithographic size of the dot

is 170 nm(inset in (a)). This dot shows clear even-odd effect in the peak spacing for both (a)strong, and

(b)weak coupling to the leads, as well as (c) an entirely different gating configuration where up to 7 pairs

(14 peaks) in succession are observable. Left inset in (b): magnified view of the first peak. Right inset in

(b): peak spacing change of (a)(filled circle) and (b)(unfilled square).

chance of observing pairing effects since electrons with higher kinetic energy, h2k2F /2m∗,

experience relatively less Coulombic interaction and the constant interaction model maybe expected to have better validity. In fact, recent theoretical [61–63] and experimentalworks [64] indicate exactly this observed trend resulting from the contribution of an energyfluctuation term proportional to rs [61–63]. On the other hand the relative ease of observingpairing in all devices when the dot is open likely results from the presence of spectral rigidityin this regime analogous to the situation in open billiards or disordered conductors where

25

universal conductance fluctuations are present.Aside from the peak spacing, peak height pairing is also visible over several peaks in all

devices, particularly in the weak tunneling regime. In each pair, one is small and the otheris large and this alternating sequence is present over all the peaks in Figs. 12(c), 13(b),and 14(b). However, devices 1 (2) and 3 show reversed peak height sequence. Currenttheories as yet do not satisfactorily explains this behavior, although promising theories havebeen advanced to address peak height fluctuations based on quantum chaos ideas [19] andselection rules from spin and geometry with electron correlation [65]. At present the relationbetween these mechanisms and the pairing effect in peak height variation is not clear.

An independent determination of the net spin for a given CB valley can be obtainedfrom the Kondo effect, in which the coupling between unpaired localized electron spin andconduction electron spin results in enhanced density of states around Fermi level [22,23].If the coupling between the dot and the leads are strong enough, this Kondo correlationfor an unpaired spin enhances the zero-bias differential conductance as was demonstratedrecently [28,29]. Typical temperature dependence of odd and even numbered electron valleyconductance is illustrated in Figs. 12 and 13. In each device, odd-electron number valleysshowed conductance enhancement at low temperatures, zero-bias maximum in nonlinearconductance, and Zeeman splitting under magnetic field as evidences of Kondo effect. Onthe other hand, for even-number electrons in the dot the ground state is a singlet and thereis no net spin to couple to the conduction electrons and the Kondo effect is suppressed.

As final points, we wish to distinguish our results from anomalous pairing effects ob-served by capacitive spectroscopy in vertical quantum dots [6] and shell effects. [25] In largevertical dots there exist pairing effects in the addition spectrum when electrons are succes-sively added with some pairs occurring dramatically right on top of each other (at the sameenergy). Recent works [66,67] indicate these pairing arises from the spatial separation ofa relatively large, nominal single dot into effectively two dots in when disorder is present.The pairing corresponds to the addition of one electron each to the two effective dots, asituation completely different from our case of a single very small dot. Our results are alsodistinct from InGaAs-based circular vertical dots in which shell effects are observable sinceshell effects are present only in regularly shaped dots (notably circular dots) possessing ex-tra degeneracies in the energy spectrum. In contrast in our GaAs/AlGaAs dots, the gatingconfiguration employed to define the dot is completely non-circular, yielding an grossly elon-gated dot. Our claim that shell effect is not present is borned out by the data of Fig. 14(c)where up to 7 pairs are observable. The shell effect predicts special positions at 2, 6, 12 and20, electrons completely at variance with our data. Our work also differs from a recent workby Ciorga et al., [59] in which complete emptying of a lateral dot was achieved. In this work,they explicitly pointed out the absence of an even-odd effect in their data due to conflictingsignatures.

26

V. COUPLING BETWEEN TWO DOTS AND LEADS–COHERENT

MANY-BODY KONDO STATES

[Excerpted with permission from “The Kondo Effect in an Artificial Quantum-Dot Molecule,”H. Jeong, A.M. Chang, and M.R. Melloch, Science 293, 2221 (2001). Copyright, 2001,American Association for the Advancement of Science.]

The recent discovery of the Kondo effect in artificial quantum dot systems [28,29,68–72]has raised much interest in the area of quantum impurity and the associated physics ofelectron spin and strong correlation. By now, the Kondo problem in a quantum dot [22,23]is well understood through experimental and theoretical studies. Further interesting physicalsituations can be found when we consider the inter-dot interaction in multiple quantum dots.

The tunability of artificial quantum dots has brought unprecedented control to the in-vestigation of the single impurity level Kondo effect. It is a natural next step to inquireabout the physics of interaction between multiple localized magnetic spin moments and con-duction electrons in a double quantum dot system. Here, the impurities interact via aneffective antiferromagnetic coupling, J = 4t2/U , where t is the tunable interdot couplingand U the ntradot charging energy. These two new energy scales, t and J, are expectedto introduce qualitatively new behavior. The rich physics of a double quantum dot Kondosystem as a tunable, two quantum impurity system has been studied extensively by theorists[37–39,42,41,40,43]. More intriguingly, the double dot has recently been proposed as a feasi-ble two-qubit system for quantum computation [11]. Here, we review the observation of thecoherent Kondo effect in a double quantum dot which is a direct experimental realization ofa two impurity Kondo model [21].

Our device is fabricated on a GaAs/AlxGa1−xAs heterostructure (Fig. 15A). The sampleis mounted in the mixing chamber of a dilution refrigerator parallel to the axis of a super-conducting magnet. The lattice base temperature is 15 mK while the electronic temperatureis estimated around 40 mK. It is possible to set each quantum dot to show similar character-istics by tunning gate voltages properly. Each dot exhibited two Kondo valleys representedby the pronounced zero bias maximum (ZBM) (Fig. 15B). We observe the Kondo ZBMonly in the odd-electron Coulomb blockade (CB) valleys between CB peaks with a spacingsmaller than the adjacent even-electron valleys. These Kondo valleys exhibit an increasein the conductance with decreasing temperature. ∆ and EC (U) were determined in theclosed dot regime by the standard method of measuring the lever arm, α, and the differen-tial conductance at finite source-drain bias to observe excited states. For the upper (lowerdot), ∆ = 0.45 meV (0.37 meV) and charging energy, EC = 1.92 meV (1.73 meV). Theseare comparable values reported previously by other groups for dots of similar size [28,68,71].To accomplish the delicate settings for forming the double dot, the gating information ofeach dot is utilized since all the gates are needed. For example, by using V1, V2, and V3,the upper dot is first formed. Next V4 is biased up to the proper value and finally V5 isenergized. Due to mutual capacitive coupling, the center tunneling barrier becomes higherafter energizing V4 and V2 then needs to be reduced to maintain the optimum conductioncharacteristics through the center barrier.

27

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FIG. 15. (A) The dots are defined by ten independently tunable gates on a GaAs/AlGaAs heterostruc-

ture containing a two dimensional electron gas (2DEG) located 80 nm below the surface. The low temper-

ature sheet electron density and mobility are n = 3.8 × 1011cm−2 and µ = 9 × 105cm2/V s, respectively.

Lithographic dot size is 180 nm in diameter and each dot contains about 40 electrons inside. To reduce

unnecessary degree of freedom in controlling the double dot, gates sitting on the opposite side are connected

together, giving total five pairs of controllable gates. Gate pair V1 and V5 are used to set tunneling barriers,

while the V3 sets the inter-dot tunnel coupling between the dots. V2 and V4 control the number of electrons

and energy levels in each dot separately. (B) Typical traces of Kondo resonance peaks when each dot is

working as single dots. Upper dot shows larger Kondo resonance than lower one. (C)(D)(E) Three unique

cases of spin states of a double dot, based on a simple electron filling with spin degeneracy. (C) is the case

which has been considered to show split Kondo resonance [37–39,42,41,40,43]. (D) and (E) contain singlet

electrons in one of the two dots and the overall Kondo resonance is not allowed.

28

A

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9

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9�

�P 9�

FIG. 16. (A) Schematic of a periodic structure of the electron spin configuration as a function of two gate

voltages controlling the electron numbers in each dot in a double dot. Lines represent positions of Coulomb

blockade peaks. This zig-zag pattern changes depending on the inter dot coupling strength controlled by

V3. For simplicity, only the spins of last electronic levels are shown. Circled regions contain possible double

Kondo impurity spin status corresponding to the region 1, 3, 4 and 6 in (B). (B) Grey [color] scale plot

of the measured conductance of a double dot as a function of gate voltages V2 and V4. The center gate

voltage is set V3 = -860mV. Brighter (darker) [red (blue)] color signifies higher (lower) conductance. The

numbered valley regions, 1, 3, 4 and 6 show zero bias maximum (see Fig. 19). Note that (A) is meant as an

illustration for the comparison to (B) only. In a real device, the double dot characteristics gradually change

as the plunger gate voltages are swept, and the honeycomb pattern inevitably appears distorted from the

ideal situation.

29

-600 -620 -640 -660-640

-660

-680

-700

-720

V4

(mV

)

V2 (mV)

FIG. 17. Honeycomb pattern similar to Fig. 16(B), but for weaker inter-dot coupling (smaller inter-dot

tunneling).

Assuming simple, even-odd electron filling in each quantum dot, we can expect threeunique spin configurations in a double dot. It is necessary to consider only the uppermostenergy level in each dot since it is the most relevant for Kondo physics (Figs. 15C-E) [73].When both dots contain a single electron on their uppermost level, a coherent Kondo reso-nance can occur, and is the case which has attracted the most theoretical attention. If one orboth of the dots contain a spin-singlet, pair of electrons (Fig. 15D or E), a coherent Kondoresonance cannot occur since Kondo coupling cannot be achieved throughout the double-dotsystem. In a series coupled double dot, we can readily distinguish cases C and D as even if aKondo resonance forms in D in one dot, transport is greatly impaired by the presence of thepaired electrons in the remaining dot. Within this scenario we expect a periodic occurrenceof Kondo resonance (circled region of Fig. 16A) as electrons are added one by one.

By first carefully characterizing individual dots and subsequently the double dot systems,we were able to obtain the honeycomb charging diagram shown in Fig. 16 and Fig. 17 atdifferent coupling values, Γ, to the leads. In the measured double dot conductance as afunction of plunger gate voltages V2 and V4 (Fig. 16B), the center gate V3 is set such thatit gives barely enough honeycomb structure for easy identification of the valley regions andalso enough cotunneling conductance for the Kondo resonance. From the splitting of theCoulomb blockade peaks, we estimate an inter-dot conductance of ∼ 0.8(2e2/h) [74–77]. Inthis strong tunneling regime, however, it is not straightforward to convert the conductanceinto a reliable estimate of t. To find out which valleys are Kondo valleys containing a singleelectron on each dot, the differential conductance, dI/dV, versus voltage bias across thedouble dot is measured in a total of 32 valley regions four of which show zero bias maximum

30

peaks. For example, in the six numbered regions in Fig. 16B, we see the appearance anddisappearance of zero bias maximum Kondo resonance peaks roughly matched to Fig. 16A(Fig. 18).

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FIG. 18. Differential conductance traces from 1-6 in Fig. 16B. Trace 4 and 6 are magnified by factor

2. Occurrence of Kondo resonance peaks is well contrasted. The periodicity is consistent with the diagram

Fig. 16A. A unique feature of the Kondo resonance peaks is their splitting compared to the single peaks

from single dots (e.g., Fig. 15B).

Compared to the single dot case, the striking feature of the Kondo resonance peaks isits splitting into two peaks in zero magnetic field. Several theoretical papers have predictedthat when the many-body molecular bonding and antibonding states are formed, the Kondoresonance shows a double peak structure in a coupled quantum dot [38,39,42,41,40,43]. Whenthe split peaks are symmetric (Fig. 18, valley 4), they are centered about zero bias and forthe non-symmetric cases (Fig. 18, valleys 1, 3, 6), the larger peak is closer to zero bias. Webelieve the prevalence of the asymmetric situation comes from the difficulty in achievingthe same condition on both dots. This is supported by the fact that for the symmetrictrace 4, when either dot-lead tunnel barrier is changed through V1 or V5, we obtain asimilar asymmetric double peak. Attempts to make asymmetrical peaks symmetric wereless successful due to slight differences in the characteristics of the two dots and the mutual

31

capacitance and interdependence of the gates. In small devices such as ours, the gates arepushed closer together and experience stronger interdependence. The observed splittingδ ≈ 45µeV is comparable to the molecular bonding-antibonding splitting of 10 − 120µeVpreviously reported in double quantum dots [35,36]. It is important to emphasize that ourresult represents a clear observation of the formation of many-body bonding-antibondingKondo states. Previous experiments could not determine the spin status of the double dotsand were often configured to suppress Kondo correlation (small Γ) yielding a splitting singleparticle in nature. The gap of double peaks can be adjusted to a certain extent (Fig. 19A),depending on the gate voltage settings, for example by changing the coupling of two dots orcoupling of the dots to the leads. However, we were not able to observe single peaks in theregime that the dot-dot coupling is smaller than the dot-lead coupling mentioned in theories.In case the coupling of two dots is reduced, the overall conductance is decreased too much inthis series-coupled configuration and a clear signature of zero bias maximum was no longerobservable.

All four split Kondo peaks showed qualitatively similar results in magnetic and tempera-ture dependence. The parallel magnetic field dependence (Fig. 19B) of the symmetric peaktrace 4 in Fig. 18 shows that as the magnetic field increases, the two split peaks approacheach other, merge and then split again. When the magnetic field is applied, the Zeemaneffect splits the two many-body molecular states formed around the Fermi levels, giving atotal of four peaks. Two of the four peaks closest to the mid point of the left and rightFermi levels overlap when the source-drain bias, VSD, is applied, cross and split again. Thecontribution from the other two outside peaks should in principle be present but are not ob-served possibly due to spin decoherence at larger bias. Similar behavior is also present in thesingle particle, two level Kondo system [69,72] where only two out of four peaks are clearlyvisible. We can estimate a rough value for the electron magnetic moment g factor based onthe Zeeman energy. We find a value of between 0.3-0.6 compared to the known magnitudeof 0.44. As the temperature increases, both of the split peaks have a tendency to decreaseand finally disappear (Fig. 19C). In this temperature dependence, the zero bias conductanceof the symmetrically split peaks increases slightly at first and decreases as the temperaturegoes up (Fig. 19D). Even the double peak structure disappears, the overall broad peak as asingle one is maintained in higher temperature range. Based on the saturation temperatureof the zero bias peak height, the Kondo temperature is approximately 500 mK.

32

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FIG. 19. (A) Peak splitting changes depending on the coupling of the two dots. Dotted (solid) line

is for V3 = -860mV (-870mV). Solid line is magnified by factor 2. For each case, the splitting is 42µV

and 26µV . (B) Magnetic field dependence of the symmetric Kondo resonance peak 4. Traces are for B =

0, 0.25, 0.5, 0.75, 1.0, and 1.25 T. The curves are offset by 0.02e2/h for clarity. Other peaks also show

qualitatively similar behavior. The Zeeman splitting from two split peaks enhance the conductance at zero

bias as the field increases because of the overlap of the density of states from two peaks. For higher fields,

they are going apart further like a single resonance peak case. (C) Temperature dependence of differential

conductance. From top to bottom, T = 40, 50, 60, 70, 80, 90, 100, 120, 150, 180, 210, 250, 300, 350, 400,

and 500 mK, offset by 0.01e2/h for each line. Overall conductance structure decreases as temperature goes

up. (D) Conductance at zero bias in log (T) scale. By increasing temperature T, the conductance increases

initially and goes down.

From our study, we find that the spin status of multiple dots is consistent with aninterpretation based on electron spin filling in a double quantum dot. The Kondo resonancepeaks in this system showed clear splitting as an indication of the Kondo effect in a quantumdot molecule. A more quantitative analysis of the competition of Kondo singlet energy versusantiferromagnetic coupling energy in a tunable manner with the advancement of quantumdot device technology will elucidate diverse physical phenomena in multiple quantum dotsystems.

33

VI. OTHER ULTRA-SMALL DEVICES AND PHENOMENA

-0.68 -0.64 -0.60 -0.560.0

0.2

0.4

0.6

-0.65 -0.63 -0.61 -0.590.0

0.2

0.4

0.6

-0.64 -0.62 -0.600.0

0.1

0.2

b 01021105

a

V4 (V)

01021107

V4 (V)

c

G (

e2 /h)

G (

e2 /h)

G (

e2 /h)

01021508

V2 (V)FIG. 20. Some of the Coulomb blockade peak traces from a double quantum dot. (a) Asymmetric Fano

line shape when the two single dots are strongly coupled. (b) Paired peaks when the two songle dots are

weakly coupled. (c) A sharp feature observed in the intermediate coupling regime between (a) and (b).

34

In addition to the coherent Kondo effect in multiple quantum dots, other interestingphenomena are observable, such as Fano resonances as well as sharp conductance peakslikely assciated with the sudden reorganization of the electronic ground state in this system.Fano-resonance type CB peaks have been observed in the small devices that exhibited theKondo effect [78]. The asymmetric resonance peak shape was explained to result from aninterference between a resonant and a nonresonant path in the system. However, this non-resonant path was not clearly identified. In our small double-quantum-dots that exhibitedthe coherent Kondo effect, we have also observed Fano line shapes under appropriate con-ditions. Asymmetric Fano line shapes were observed when the DQD was operated either asseperate single dots or a coupled double-dot. A typical trace from a double dot configurationis presented in Fig. 20(a). The degree of asymmetry in the peak shape varies for differentcool-downs. However, proper gate voltage settings always produced Fano shapes as well asKondo resonances within the same cool-down session. It was also possible to change thedirection of asymmetry of peaks continuously by changing gate voltages. Furthermore, otherinteresting peak shapes shown in Fig. 20(b) are also believed to originate from Fano-typeresonances. The first peak pair exhibits the smooth line shape normally observed in multiplequantum dots. In contrast, the second and third pairs have sharp asymmetric shapes. Thecondition for formation of the double-dots was different in (a) and (b). This type of tunabil-ity may prove helpful to further sort out the origin of the Fano resonance in quantum dots.One unusual feature which was not previously reported, but was theoretically predicted, isthe sudden reconfiguration of electronis ground state due to spin-related effects [79]. The re-sultant unusual peak shape observed in our device and presented in Fig. 20(c) had previouslybeen obtained in the theoretical calculations [79]. These diverse and intriguing behaviorswere clearly observed in our versatile, fully tunable and controllable double-quantum-dotdevice, and serve as testaments to the rich variety of physical phenomena avaible in thissystem.

As a final indication of the possibilities offered by small, lateral quantum dots defined byelectron beam lithograph, in Fig. 21 we present the smallest lithographic double QD writtenin GaAs/Al[x]Ga[1-x]As to date (of which we are aware). It is generally believed very shallowtwo-dimensional electron gas (2DEG) is necessary to make small devices since the depletionlength around the biased gates on the surface of the semiconductor material is comparableto the depth of the 2DEG. For example, with the 2DEG at 100 nm below the surface, thedevice size should be larger than the twice of 2DEG depth, 200 nm. We find, however, thatdepends on the density of the features in the structure.

35

-0.8 -0.6 -0.4 -0.2 0.00.0

0.2

0.4

0.6

0.00

0.03

0.07

0.10

4.2K

77K

∆G (

e2 /h)

G (

e2 /h)

VG (V)

FIG. 21. (a) Scanning electron micrograph of a SET with 120 nm single dot size as measured from the

inside diameter for the dot. (b) CB oscillations at 4.2 K (lower trace) and 77 K (upper trace). Bias gate

voltages are slightly different for optimum CB peaks in the two traces. Offset in y axis is changed and a

portion of linear background conductance is subtracted for 77 K trace.

The quantum dot in the Fig. 21(a) is defined on the surface of a GaAs/AlxGa1−xAsheterostructure which is grown by molecular beam epitaxy. The 2DEG is about 60 nm belowthe surface with a carrier concentration of 3.7 × 1011cm−2 at 4.2 K. While the lithographicdot size is 120nm, the actual dimension estimated is about 40 nm by electrostatic depletionafter accounting for the pinchers as well as the plunger gate. To prevent serious proximityeffect, the thick lines for fan-out are half micron away from the dot area. Fig. 21(b) showsthe CB oscillations of the dot as a function of the center plunger gates. The two plungergates on the left and right side were swept simultaneously. The measurement was performedat 4.2 K and 77 K and shows residual CB oscillations at 77 K.

36

From the oscillation period of CB peaks, the capacitance between the dot and a pairof plunger gates can be found , by the following formula, CG = e

∆VG. This dot to center

(plunger) gate capacitance, CG, is about 620 zF(10−21F) in several devices. The exact numberof electrons in the dot cannot be known from the measurement, but rough estimation showsthe dot contains 3-5 electrons. By the fact that the CB oscillations still exist at 77 K, thecharging energy, Ec, is almost comparable to the thermal energy corresponding to 77 K,which is 6.6 meV. This value is the highest one reported upto now in controllable lateralquantum dots made in GaAs materials.Acknowledgement We wish to acknowledge the indispensable contributions of our colleaguesHarold Baranger and T.Y. Chang. We also acknowledge helpful discussions with L. Glazman,A. Kaminski. This work was supported in part by NSF grant No. DMR-9801760.

37

REFERENCES

[1] M.A. Kastner, Rev. Mod. Phys. 64, 849 (1992).[2] D.V. Averin, and K.K. Likharev, J. Low Tmp. Phys. 62, 345 (1986).[3] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. 59, 109 (1987).[4] H. Grabert, and M.H. Devoret, Eds., Single Charge Tunneling: Coulomb Block-

ade in Nanostructures, (Plenum Press and NATA Scientific Affairs Division, NewYork/London), 1992.

[5] W.G. van der Wiel, S. De Franseschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, and L.P.Kouwenhoven, cond-mat/0205350.

[6] R.C. Ashoori, Nature 379, 413 (1996).[7] See articles in Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron

Transport, edited by L.L. Sohn, L.P. Kouwenhoven, and G. Schon (Kluwer Series E345,1997).

[8] S. Tarucha, see Chapter in this book.[9] D.V. Averin, and K.K. Likharev, in Mesoscopic Phenomena in Solids, edited by B.L.

Altshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam), p. 67 (1991).[10] K.K. Likharev, and T Claeson, Sci. Am. 266, 80 (1992).[11] D. Loss, D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).[12] M.J. Yoo, T.A. Fulton, H.F. Hess, R.L. Willet, L.N. Dunkleberger, R.J. Chichester, L.N.

Pfeiffer, and K.W. West, Science 276, 579 (1997).[13] S.H. Tessmer, P.I. Glicofridis, R.C. Ashoori, L.S. Levitov, and M.R. Melloch, Nature

392, 51 (1998).[14] R.J. Schoelkopf, P. Wahlgren, A.A. Kozhevnikov, P. Delsing, and D.E. Prober, Science

280, 1238 (1998).[15] A.M. Chang, H.U. Baranger, L.N. Pfeiffer, K.W. West, and T.Y. Chang, Phys. Rev.

Lett. 76, 1695 (1996).[16] J.A. Folk, S.R. Patel, S.F. Godijn, A.G. Huibers, S.M. Cronenwett, C.M. Marcus, K.

Campman, and A.C. Gossard, Phys. Rev. Lett. 76, 1699 (1996).[17] J.H.F. Scott-Thomas, S.B. Field, M.A. Kastner, D.A. Antoniadis, and H.I. Smith, Phys.

Rev. Lett. 62, 583 (1989).[18] U. Meirav, M.A. Kastner, and S.J. Wind, Phys. Rev. Lett. 65, 771 (1990).[19] R.A. Jalabert, A.D. Stone, and Y. Alhassid, Phys. Rev. Lett. 68, 3468 (1992).[20] C.E. Porter and R.G. Thomas, Phys. Rev. 104, 483 (1956).[21] H. Jeong, A.M. Chang, and M.R. Melloch, Science 293, 2221 (2001).[22] T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988).[23] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452 (1988).[24] K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B54, 5637 (1996).[25] S. Tarucha, D. G. Austing, T. Honda R. J. van der Hage, and L. P. Kouwenhoven, Phys.

Rev. Lett. 77, 3613 (1996).[26] S. R. Patel, S.M. Cronenwett, D.R. Stewart, A.G. Huibers, C.M. Marcus, C.I. Duruoz,

J.S. Harris, K. Campman, and A.c. Gossard, Phys. Rev. Lett. 80, 4522 (1998).[27] G. Usaj, and H.U. Barange, Phys. Rev. B64, art. no. 201319 (2001).

38

[28] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, andM.A. Kastner, Nature(London) 391, 156 (1998).

[29] Sara M. Cronenwett, Tjerk H. Oosterkamp, and Leo P. Kouwenhoven, Science 281, 540(1998).

[30] U. Sivan, R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, G. BenYoseph, Phys. Rev. Lett.77, 1123 (1996).

[31] F. Simmel, T. Heinzel, and D. A. Wharam, Europhys. Lett. 38, 123 (1997).[32] I. Ruzin, V. Chandrasekhar, E.I. Levn, and L.I. Glazman, Phys. Rev. B 45, 13469

(1992).[33] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M.H. Devoret, Europhys. Lett. 17,

249 (1992).[34] D.c. Dixon, Ph.D. thesis (University of California, Berkeley), 1998.[35] T. H. Oosterkamp, T. Fujisawa, W.G. van der Wiel, K. Ishibashi, R.V. Hijman, S.

Tarucha, and L.P. Kouwenhoven, Nature 395, 873 (1998).[36] R. H. Blick, D. Pfannkuche, R. J. Haug, K. v. Klitzing, K. Eberl, Phys. Rev. Lett. 80,

4032 (1998).[37] A. Georges, Y. Meir, Phys. Rev. Lett. 82, 3508 (1999).[38] R. Aguado, D. C. Langreth, Phys. Rev. Lett. 85, 1946 (2000).[39] T. Aono, M. Eto, Phys. Rev. B 63, 125327 (2001).[40] Bing Dong and X. L. Lei, Phys. Rev. B 65, art. no. 241304 (2002).[41] C. A. Bsser, E. V. Anda, A. L Lima, M. A. Davidovich, G. Chiappe, Phys. Rev. B 62,

9907 (2000).[42] W. Izumida, O. Sakai, Phys. Rev. B 62, 10260 (2000).[43] R. Aguado, D. C. Langreth, cond-mat/0207283.[44] B. A. Jones, C. M. Varma, Phys. Rev. Lett. 58, 843 (1987).[45] B. A. Jones, C. M. Varma, J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988).[46] I. Affleck and A.W.W. Ludwig, Phys. Rev. Lett. 68, 1046 (1992).[47] A.M. Chang, H.U. Baranger, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 73, 2111

(1994).[48] V.N. Prigodin, K.B. Efetov, and S.Iida, Phys. Rev. Lett. 71, 1230 (1993).[49] C.W.J. Beenakker, Phys. Rev. B 44, 646 (1991).[50] H. Bruus and A.D. Stone, Phys. Rev. B50, 18275 (1994).[51] E.B. Foxman, U. Meirav, P.L. McEuen, M.A. Kastner, O. Klein, P.A. Belk, D.M.

Abusch, and S.J. Wind Phys. Rev. B 50, 14193 (1994).[52] R. Berkovits, and B.I. Altshuler, unpublished.[53] W. Heisenberg, Z. Phys. 78, 156 (1932).[54] A. Bohr, B. R. Mottelson, and D. Pines, Phys. Rev. 110, 936 (1958).[55] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and M. Tinkham, Phys. Rev. Lett.

69,1997 (1992).[56] T. M. Eiles, John M. Martinis, and Michel H. Devoret, Phys. Rev. Lett. 70, 1862 (1993).[57] C. Jayaprakash, R. Krishnamurthy, and J. W. Wilkins, Phys. Rev. Lett. 47, 737 (1981).[58] R. M. Fye, J. E. Hirsch, and D. J. Scalapino, Phys. Rev. B 35, 4901 (1987).

39

[59] M. Ciorga, A.S. Sachrajda, P. Hawrylak, C. Gould, P. Zawadzki, S. Jullian, Y. Feng,and Z. Wasilewski, Phys. Rev. B 61, R16315 (2000).

[60] A. Kaminski, and L.I. Glazman, Phys. Rev. B61, 15927 (2000).[61] Ya. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii, Phys. Rev. Lett. 78, 2449(1997).[62] Harold U. Baranger, Denis Ullmo, and Leonid I. Glazman, Phys. Rev. B. 61, R2425

(2000)[63] B. I. Halperin, unpublished.[64] S. Luscher, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Phys. Rev. Lett.

86, 2118 (2001).[65] Dietmar Weinmann, Wolfgang Husler, and Bernhard Kramer, Phys. Rev. Lett. 74, 984-

987 (1995).[66] C.M. Canali, Phys. Rev. Lett. 84, 3934 (2000).[67] M. Brodsky, N.B. Zhitenev, R.C. Ashoori, L.N. Pfeiffer, and K.w. West, Phys. Rev.

Lett. 85, 2356 (2000).[68] J. Schmid, J. Weis, K. Eberl, K. v. Klitzing, Physica B 256-258, 182 (1998).[69] S. Sasaki, S. De Franceschi, J.M. Elzerman, W.G. van der Wiel, M. Eto, S. Tarucha,

and L.P. Kouwenhoven, Nature 405, 764 (2000).[70] W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J.M. Elzerman, S. Tarucha, and

L.P. Kouwenhoven, Science 289, 2105 (2000).[71] Y. Ji, M. Heiblum, D. Sprinzak, D. Mahalu, H. Shtrikman, Science 290, 779 (2000).[72] J. Nygard, D. Henry Cobden, P. E. Lindelof, Nature 408, 342 (2000) and references

therein.[73] Note that in these figures 15 and 16, a single up arrow denotes an unpaired electron. It

is not intended to represent the actual direction of spin alignment. In fact in a complexmany-body state, the dot spin most likely contains projections from both upand downspin states.

[74] F.R. Waugh, M.J. Berry, D.J. Mar, R.M. Westervelt, K.L Campman, and A.C. Gossard,Phys. Rev. Lett. 75, 705 (1995).

[75] C. Livermore, C. H. Crouch, R. M. Westervelt, K. L. Campman, A. C. Gossard, Science274, 1332 (1996).

[76] K. A. Matveev, L. I. Glazman, H. U. Baranger, Phys. Rev. B 53, 1034 (1996).[77] J. M. Golden, B. I. Halperin, Phys. Rev. B 53, 3893 (1996).[78] J. Gres, D. Goldhaber-Gordon, S. Heemeyer, M. A. Kastner, H. Shtrikman, D. Mahalu,

and U.Meirav, Phys. Rev. B 62, 2188-2194 (2000).[79] P. G. Silvestrov and Y. Imry, Phys. Rev. B 65, 035309 (2002).

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