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Two-electron quantum dot molecule: Composite particles and the spin phase diagram A. Harju, S. Siljam¨aki, and R.M. Nieminen Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, 02015 HUT, Finland (Dated: February 1, 2008) 2 We study a two-electron quantum dot molecule in a magnetic ﬁeld by the direct diagonalization 0 0 of theHamiltonian matrix. Theground states of themolecule with thetotal spin S =0 and S =1 2 provide a possible realization for a qubit of a quantum computer. Switching between the states is best achieved by changing the magnetic ﬁeld. Based on an analysis of the wave function, we show n that the system consists of composite particles formed by an electron and ﬂux quanta attached to a it. This picturecan also beused to explain the spin phase diagram. J 7 PACSnumbers: 73.21.La,71.10.-w ] l l The progress in semiconductor technology has opened use the one of Ref. [4], namely 1m∗ω2min{(x−d/2)2+ a 2 0 h a rich ﬁeld of studies focused on the fundamental y2,(x + d/2)2 + y2}. This potential separates to two - electron-electroninteractionsandquantumeﬀectsinarti- QDs at large inter-dot distances d, and with d = 0 it s e ﬁcial atoms and molecules [1]. The most striking feature simpliﬁes to one parabolic QD. We use GaAs material m of two-dimensional semiconductor quantum dots (QD) parameters m∗/m = 0.067 and ǫ = 12.4, and the con- e . and quantum molecules (QDM) is that the correlation ﬁnementstrength~ω0 =3.0meV.Aisthevectorpoten- t a andmagneticﬁeldeﬀectsaregreatlyenhancedcompared tial of the magnetic ﬁeld (along the z axis) taken in the m withtheirnormalcounterparts. Thisresultsinarichva- symmetricgauge. OneshouldnotethattheHamiltonian - riety of phenomena in lateral QDMs that have recently is spin-free, and the Zeeman-coupling EZ = g∗µBBSz d been investigated experimentally and theoretically, see (with g∗ = −0.44 for GaAs) of the magnetic ﬁeld to S z n for example Refs. [2, 3, 4, 5]. Also the system param- canbe takeninto accountafterwards. Theeigenstatesof o c eters can easily be changed, unlike in real atoms and the single-particle part of Eq. (1) are easily obtained as [ molecules where the parameters are natural constants. expansions The controllable parameters make it possible to tailor v1 the semiconductor structures and for example to switch ψ(r)= αiφi(r)= αixnx,iyny,ie−21r2 , (2) 6 between diﬀerent ground states. Xi Xi 7 In this Letter we concentrate on a two-electron QDM 0 consisting of two laterally coupled QDs. In addition to where nx and ny are integers, and αi is a complex coef- 1 ﬁcient. We have used the unit of length as in Ref. [8]. theinterestingandfundamentalcorrelationandquantum 0 Fig.1showsexamplesofthenon-interactingchargeden- 2 eﬀects, this system is very important as a candidate for sities, and Fig. 2 displays single-particle energies. It 0 the gate of a quantum computer [7]. A central idea is / to use the total spin (S) of the two-electron QDM as a t a qubit. The ground state spin of the QDM can be either B=4T m B=8T S =0orS =1. OneoftheaimsofthisLetteristostudy 1 - the regions of S = 0 and S = 1 states as a function of d n theimportantsystemparametersintheirrealisticrange, 0.8 o beyond the approximations of Ref. 7 where a change in 0.6 c S was found when the magnetic ﬁeld B was varied. We : v ﬁndasimilarcrossingatweakB,butalsoareappearance 0.4 i of the S = 0 ground state at larger B. This is caused X 0.2 by the correlation eﬀects that are treated accurately in r a our many-body approach. An interesting ﬁnding is that 0 −40 −20 0 20 40 the system naturally consists of composite particles of x [nm] electrons and ﬂux quanta as in the composite-fermion theory [6]. FIG. 1: Conﬁnement potential Vc (dotted line) and non- interacting single-particle density |ψ|2 along y = 0 for d = Wemodelthetwo-electronQDMbya2DHamiltonian 40 nm. The potential is in units of ~ω0 = 3 meV, and the maximum value of each |ψ|2 is scaled to unity. The conﬁne- H = 2 (−i~∇i− ecA)2 +V (r ) + e2 , (1) ment potential is parabolic in the y-direction. Onecan see a (cid:18) 2m∗ c i (cid:19) ǫr localizing eﬀect of large d and strong B. Xi=1 12 whereV istheexternalconﬁningpotential,forwhichwe is interesting to compare the localizing eﬀect of B with c 2 12 10 1.5 V] 8 e E [m 6 1 S=0 V] 0.5 e m 4 E [ 0 S=1 S=0 ∆ 20 2 4 6 8 −0.5 40 B [T] 30 −1 0 20 FIG. 2: Six lowest non-interacting single-particle energies as 2 4 10 afunctionofB. ThedottedlinesarethenormalFock-Darwin 6 states (d = 0), and the solid ones are for d = 20 nm. One B [T] 8 0 d [nm] canseethatford>0,theenergiesareshifteddownandthat the level crossings occur at weaker B. At some of the level- crossings a gap opens, suchs as the one for B ≈1.75 T, E ≈ FIG. 3: Energy diﬀerence between singlet and triplet states 7.4 meV. This is due to the lower symmetry of the problem ∆E as afunction of themagnetic ﬁeldB and inter-dotsepa- ford>0. Ingeneral,thereisareasonablesimilaritybetween ration d. The white lines separate the S = 0 (∆E > 0) and the two sets of energy levels. For suﬃciently large d, the S =1 ground states. energieswouldconvergetodegeneratestatesofisolated dots. All states are on thelowest Landau level after B≈3 T. 0.2 the experimentalﬁndings of Brodskyet al. [3]. They see 0.15 a clear splitting of the QDM electron droplet to smaller 0.1 fragments by a strong B. As they work in a low-density V] e 0.05 lriomleita(swtehaekpcootneﬁntnieaml menitn)i,mimapinurFitiige.s1c.anThheavloecaalsizimatiiloanr ∆ E [m 0 EZ is also related to the formation of Wigner molecules in −0.05 QDs [8], which happens in the low-density limit. 4 Similarly to the single-particle states, the full many- −0.1 5 body wave function with total spin S can be expanded −0.15 6 as 0 1 2 3 4 5 6 7 B [T] Ψ (r ,r ) = α {φ (r )φ (r ) S 1 2 i,j i 1 j 2 FIG. 4: ∆E for a ﬁxed d = 26.736 nm [10]. The eﬀect of Xi≤j theZeemanenergyEZ isalsoshown. Thediﬀerencebetween + (−1)Sφi(r2)φj(r1) (3) expansions with n=5 and 6 is only around 1 µeV. One can (cid:9) see that the second S = 0 state disappears for even a weak which is symmetric for S = 0 and anti-symmetric for EZ. S = 1. Notice that the spin part of the wave func- tion is not explicitly written, and we work with spin- independent wave functions also below. The coeﬃcient see e.g. Ref. [4] for a discussion of this exact property. vector α and the corresponding energy E for the lth However, there is a strong decrease in ∆E as a function l l eigenstate are found from a generalized eigenvalue prob- ofd. Thiscanbeunderstoodfromthefactthatverydis- lem where the Hamiltonian and overlapmatrix elements tant QDs interact only weakly and the energy diﬀerence can be calculated analytically. Details of the computa- of the two spin states is smaller. There is also a strong tional procedure will be published elsewhere [9]. As the decreasein∆E asafunctionofB intheS =0state. For basis functions we have used all the states with both n d=0,∆E isratherlinearuntilthecrossingtotheS =1 x andn ≤n with n=6, andwehavecheckedthe conver- state, but for d > 0 the curves are rounded. The cross- y gence by varying n for all values of d. ing point of the diﬀerent ground states does not depend The energy diﬀerence ∆E between the lowest S = 0 stronglyond; it changesonly from1.6 T to 1.2T as one and S = 1 states is plotted in Figs. 3 and 4. The con- moves from d = 0 to 40 nm. Due to this, changing the vergence of ∆E can be seen in Fig. 4 for large d. One total spin in an experimental setup is not easy by just can see that for weak magnetic ﬁeld values, the ground changing d. On the other hand, around ∆E = 0, the statehasS =0. Weﬁrstconcentrateonthisregime. For slope of ∆E is rather large and changing S by B is the B = 0 the S = 0 state remains lower with arbitrary d, most natural choice. One can achieve a change of S also 3 by changing the strength of V . This changes the ratio the vortices of Ψ. This can be done by ﬁnding the zeros c between the energiesresulting fromthe conﬁnement and of Ψ and studying the change in the phase of Ψ in going electron-electroninteraction. For a weakV , the interac- aroundeachof the zeros. A surprisingﬁnding is that for c tions are stronger and the transition occurs at weaker B bothlargedandd=0therearetwovorticesatbothelec- value. ThusthechangeoftheV canbeseenasachange tron locations, see Fig. 6 for an example. The particles c of the eﬀective value of B. are thus composite particles of an electron and two ﬂux The transition from the weak-B S = 0 state to the quanta, in similar fashion as in the composite fermion S =1stateismostsimplyexplainedbythefactthatthe (CF) theory [12]. The most remarkable feature of the energiesof the two lowestsingle-particlestates approach ﬁnding is the stability of composite particles against the each other as B is made stronger (see Fig. 2). At some change of d. pointthis diﬀerenceissmallerthantheexchange-energy, and the system spin-polarizes. One can see that also for this state, ∆E decreases strongly as a function of d. There exists a second region of S = 0 ground state around B ≈ 6 T at d = 0. The question whether this state terminates at large d remains open, as the energy diﬀerences at d > 40 nm are smaller than the error made in the expansion. The existence of this S = 0 state for small value of d can be understood on the basis of a parabolic two-electron QD, which can be shown to have the exact wave function of the form Ψ=(x12+iy12)mf(r12)e−21(r12+r22), where m is the rela- 50 tiveangularmomentumandf isacorrelationfactor[11]. −50 The simple form is due to the separation of the center- 0 0 of-mass and relative motion of the electrons. For B = 0 the ground state has m=0 (and S =0), and when B is x [nm] 50 −50 y [nm] made stronger, the ground state m has increasing posi- tiveintegervalues. Thesetransitionshappenbecausethe largerm states have smaller Coulombrepulsionbetween FIG. 6: |Ψ[(x,y),(d/2,0)]|2 for the state S =0 at B =6 T and d = 10 nm. The arrows (rotated for clarity) depict the the electrons, and as the single-particle energies group twoﬂuxquantaattheelectronﬁxedatx>0minimumofVc. together to form the lowest Landau level, it is favorable to move to larger m. The second S = 0 region in Fig. 3 We have done a similar analysis for the S = 1 state corresponds to m = 2. It is surprising that the second at B = 3 T and various values of d. We found that the S = 0 region extends to such a large d. For example, at many-body state again consists of composite particles, B = 4.5 T and d = 35 nm the two QDs of a QDM are butthistimethereisonlyoneﬂuxquantumperelectron. rather decoupled (see Fig. 5), but still the S = 0 state One should note that for an odd number of ﬂux quanta remains the ground state. To analyze the structure of per electron, the wave function changes sign when the particles are exchanged, corresponding to S = 1. Simi- larly, for even ﬂux numbers the state has S =0. 25 If one expands |Ψ|2 for small r , one obtains 12 m] C y [n 0 |Ψ|2 ∝r122m+ m+ 1r122m+1+O(r122m+2) , (4) 2 −25 where C is the scaled strength of the Coulomb interac- tion[8],andm≥0isthenumberofﬂuxquantaperelec- −50 −25 0 25 50 x [nm] tron. One can see that for larger m values, the density grows more slowly as a function of r (see Fig. 6). One FIG.5: |Ψ[(x,y),(d/2,0)]|2 forthestateS =0atB=4.5T should note that the same expansion1i2s valid also for the and d = 35 nm. The contour spacing is uniform. We mark larger electron numbers, resulting from the cusp condi- withaplustheﬁxedelectronontheright-handQD.Onecan tion[8]. Thus the electron-electroninteractionis smaller seethattheelectrondensityislocalizedtotheleftQD.There when the number of ﬂux quanta per electron grows. is only a small deformation from circular symmetry. Notice thattheeﬀectiveinter-dotdistanceismuchlargerthanddue OnecanusetheCF-typeapproachtoexplainthephase to theCoulomb repulsion of theelectrons. diagram of Fig. 3. When one moves from B = 0 to stronger values of B, the number of ﬂux quanta in the thisstateforsmallandlargevaluesofd,wehavelocated system increases. At the ﬁrst S = 0 state there is no 4 ﬂuxes in the system, and at the transition point the ﬂux found that the system consists of composite particles of number changesto one per electron. For weakly coupled electrons and the attached magnetic ﬁeld ﬂux quanta. QDs this transition happens at smaller B than in the stronglycoupledones,becauseadistantzeroofthewave This research has been supported by the Academy functionincreasesthekineticenergylessthanacloseone of Finland through its Centers of Excellence Program does. In the following transition points the number of (2000-2005). ﬂux quanta changes again by one per electron. The rea- soningforthed-dependenceoftheﬁrsttransitionapplies to other ones also, and this can be seen in Fig. 3. An interesting prospect resulting from the discussion above is to use the CF-type approach to describe the [1] R. C. Ashoori, Nature(London) 379, 413 (1996). [2] T. H. Oosterkamp, S. F. Godijn, M. J. Uilenreef, Y. V. many-bodystatesofelectronsatstrongB invariouscon- Nazarov, N. C. van der Vaart, and L. P. Kouwenhoven, ﬁning potentials. One should note that after the phase Phys. Rev.Lett. 80, 4951 (1998) structureofthewavefunctionisﬁxed,oneisleftwiththe [3] M. Brodsky, N.B. Zhitenev, R.C. Ashoori, L.N. Pfeiﬀer, bosonic part of the wave function. The quantum Monte and K.W. West,Phys. Rev.Lett. 85, 2356 (2000). Carlo techniques are especially useful for obtaining this [4] A. Wensauer, O. Steﬀens, M. Suhrke, and U. R¨ossler, part [13]. Phys. Rev.B. 62, 2605 (2000). [5] C. Yannouleas and Uzi Landman, Phys. Rev. Lett. 82, If one adds the Zeeman term to the Hamiltonian, the 5325(1999);S.Nagaraja,J.-P.Leburton,andR.M.Mar- energy of the S =1 state is loweredby ∼25 µeV/T and tin, Phys. Rev.B 60, 8759 (1999). the S = 0 energy is unaltered, see Fig. 4. This makes [6] J. K.Jain, Physics Today 53(4), 39 (2000) the high-B S = 0 state to terminate at d value around [7] G. Burkard, D. Loss, and D.P. DiVincenzo, Phys. Rev. 5 nm. One should note that it is possible to lower the B 59, 2070 (1999) Zeemanterm alsoin the experimentalsetup by applying [8] A.Harju,S.Siljam¨aki,R.M.Nieminen,acceptedtoPhys. a tilted magnetic ﬁeld. Most probably the singlet-triplet Rev. B, cond-mat/0105452. separationinenergyforB ≈6Tistoosmallforaqubit, [9] A. Harju, S. Siljam¨aki, R.M. Nieminen, unpublished. but in principle it is possible to changeS by varying the [10] Density-functionaltheoryﬁndstheﬁrsttransitiontoS = 1 at B≈1.25 T, see H. Saarikoski, et al.,unpublished. Zeeman term in this parameter range. [11] A.Harju,B.Barbiellini,R.M.Nieminen,andV.A.Sverd- Inconclusion,wehavedeterminedthetotal-spinphase lov, Physica B 255, 145 (1998). diagram of the two-electron quantum dot molecule as a [12] J. K. Jain and T. Kawamura, Europhys. Lett. 29, 321 functionofthe magnetic ﬁeldandthe inter-dotdistance. (1995). Our results support the possibility to use the system for [13] G. Ortiz,D.M.Ceperley, andR.M.Martin, Phys.Rev. a gate of a quantum computer [7]. In addition, we have Lett. 71, 2777 (1993).

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