Electron-electron interaction mediated indirect coupling of electron and magnetic ion or nuclear spins in self-assembled quantum dots
EElectron-electron interaction mediated indirect coupling of electron and magnetic ionor nuclear spins in self-assembled quantum dots
Udson C. Mendes,
1, 2
Marek Korkusinski, and Pawel Hawrylak
1, 3, 4 Quantum Theory Group, Security and Disruptive Technologies,National Research Council, Ottawa, Canada K1A0R6 Institute of Physics “Gleb Wataghin”, State University of Campinas, Campinas, S˜ao Paulo, Brazil Department of Physics, University of Ottawa, Ottawa, Canada WPI-AIMR, Tohoku University, Sendai, Japan
We show here the existence of the indirect coupling of electron and magnetic or nuclear ion spinsin self-assembled quantum dots mediated by electron-electron interactions. With a single localizedspin placed in the center of the dot, only the spins of electrons occupying the zero angular momentumstates couple directly to the localized spin. We show that when the electron-electron interactions areincluded, the electrons occupying finite angular momentum orbitals interact with the localized spin.This effective interaction is obtained using exact diagonalization of the microscopic Hamiltonianas a function of the number of electronic shells, shell spacing, and anisotropy of the electron-Mnexchange interaction. The effective interaction can be engineered to be either ferromagnetic orantiferromagnetic by tuning the parameters of the quantum dot.
I. INTRODUCTION
There is currently interest in understanding the cou-pling of a localized spin, either magnetic impurity or nu-clear spin, with spins of interacting electrons.
Thisincludes the Kondo effect in metals and quantumdots, the impurity spin in diamond, chargedquantum dots with magnetic ions, and nuclear spinscoupled to fractional quantum Hall states.
Here wefocus on a highly tunable system of quantum dots with asingle magnetic ion and a controlled number of electrons.Such a system is realized in CdTe quantum dots with amagnetic impurity in the center of the dot loaded with acontrolled, small at present, number of electrons. Theinterplay between electron-electron Coulomb interactionsand the electron-Mn exchange interaction has been stud-ied using exact diagonalization techniques and us-ing the mean-field approach.
Other studies focusedon electron-electron interactions in excitonic complexescoupled with localized spins.
Here we focus on the indirect coupling of electron andmagnetic or nuclear ion spins in self-assembled quantumdots (QDs) mediated by the electron-electron interac-tion. With a localized spin placed in the center of thedot, only the spins of electrons occupying the zero an-gular momentum states of the s , d , . . . shells couple di-rectly to the localized spin via a contact exchange in-teraction. The situation is identical to the Kondo prob-lem in metals where only zero angular momentum statesof the Fermi sea are considered as interacting with thelocalized spin. The question arises as to the role ofelectron-electron interactions. Here we show that, inquantum dots, when electron-electron interactions areincluded, the electrons occupying finite angular momen-tum orbitals (e.g., p shell) do interact with the local-ized spin. The effective interaction for p -shell electrons isobtained using exact diagonalization of the microscopicHamiltonian as a function of the number of electronic shells, shell spacing and anisotropy of the exchange in-teraction. The anisotropy of exchange interpolates be-tween the interaction types characteristic for conductionband electrons (Heisenberg-like) and valence band holes(Ising-like). We show that the effective electron-electronmediated exchange interaction can be engineered to be ei-ther ferro- or antiferromagnetic by varying quantum dotparameters.The paper is organized as follows: In Sec. II we de-scribe the model of a self-assembled quantum dot with asingle Mn impurity in its center and a controlled numberof electrons. Section III presents results of exact diag-onalization of the model Hamiltonian for quantum dotsconfining from two to six electrons and the emergence ofthe indirect electron-Mn coupling for QDs with a par-tially filled p shell. Section IV summarizes our results. II. MODEL
We consider a model system of N electrons ( N =2 , . . . ,
6) confined in a two-dimensional (2D) parabolicquantum dot with a single magnetic impurity in the cen-ter. Figure 1(a) illustrates a schematic representation ofthe investigated QD. For definiteness we consider an iso-electronic impurity, a manganese ion with a total spin M = 5 / In the effective massand envelope function approximations, the single-particlestates | i, σ (cid:105) are those of a 2D harmonic oscillator (HO)with the characteristic frequency ω . They are labeledby two orbital quantum numbers, i = { n, m } , and theelectron spin σ = ± /
2. The single-particle states arecharacterized by energy E n,m = ω ( n + m + 1) and an-gular momentum L e = n − m . Figure 1(b) shows thesingle-particle states as a function of angular momentum.We express all energies in units of the effective Rydberg,Ry ∗ = m ∗ e / (cid:15) (cid:126) , and all distances in units of the ef-fective Bohr radius, a ∗ B = (cid:15) (cid:126) /m ∗ e , where m ∗ , e , (cid:15) , and a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y (b) L e = n - ms p d (a) FIG. 1: (Color online) (a) Schematic representation of a CdTequantum dot containing electrons and one Mn spin at its cen-ter. (b) Single-particle states as a function of angular mo-mentum. (cid:126) are respectively the electron effective mass and charge,the dielectric constant, and the reduced Planck constant.For CdTe we take m ∗ = 0 . m and (cid:15) = 10 .
6, where m is the free-electron mass, and Ry ∗ = 12 .
11 meV and a ∗ B = 5 .
61 nm. Unless otherwise stated, we take the HOfrequency ω = 1 .
98 Ry ∗ , consistent with our previouswork. The Hamiltonian of N electrons confined in our QDand interacting with a single Mn spin is written as H = (cid:88) i,σ E i,σ c † i,σ c i,σ + γ (cid:88) i,j,k,lσ,σ (cid:48) (cid:104) i, j | V ee | k, l (cid:105) c † i,σ c † j,σ (cid:48) c k,σ (cid:48) c l,σ − (cid:88) i,j J i,j ( R )2 (cid:104)(cid:16) c † i, ↑ c j, ↑ − c † i, ↓ c j, ↓ (cid:17) M z + ε (cid:16) c † i, ↓ c j, ↑ M + + c † i, ↑ c j, ↓ M − (cid:17)(cid:105) , (1)where c † i,σ ( c i,σ ) creates (annihilates) an electron on theorbital i = { m, n } with spin σ .In the above Hamiltonian, the first term is the single-particle energy and the second term is the electron-electron ( e - e ) Coulomb interaction. The e - e term isscaled by a dimensionless parameter γ : γ = 0 describesthe noninteracting electronic system and γ = 1 describesthe interacting system. The matrix elements (cid:104) i, j | V ee | k, l (cid:105) of the Coulomb interaction are evaluated in the basis of2D HO orbitals in the closed form. The last term of the Hamiltonian describes theelectron-Mn interaction ( e -Mn). It is scaled bythe exchange coupling matrix elements J i,j ( R ) = J C φ ∗ i ( R ) φ j ( R ), where J C = 2 J bulk /d , J bulk = 15 meVnm is the s-d exchange constant for the CdTe bulk ma-terial, d = 2 nm is the QD height, and φ i ( R ) is the am-plitude of the HO wave function at the Mn position R .In particular, we define J ss ( R ) = J C φ ∗ s ( R ) φ s ( R ), whichis the matrix element of an electron on the s shell inter-acting with a magnetic ion. For Mn at the QD center itsvalue is J ss ≈ .
15 meV.The e -Mn interaction consists of two terms. The firstone is the Ising interaction between the electron and Mnspin.The second term accounts for the e -Mn spin-flip in- teractions. The anisotropy of the exchange interaction istuned by the factor ε . By setting ε = 0 we obtain theanisotropic Ising e -Mn exchange Hamiltonian and set-ting ε = 1 we obtain the isotropic, Heisenberg exchangeHamiltonian. In the former case, the spin projections s z and M z are separately good quantum numbers. The totalspin projection of the electrons depends on the numberand polarization of the particles. For the manganese spinwe have M = 5 / M z = − / , . . . , /
2. The isotropic Heisenberg Hamilto-nian, in contrast, conserves the total angular momentum J = M + S and its projection J z = s z + M z . Hence, forthe case ε = 1, one can establish the total spin quantumnumber J of the given manifold of states by consideringits degeneracy g ( J ) = 2 J + 1.Since the elements J i,j depend on the position R ofthe Mn spin, the e -Mn coupling can be engineered bychoosing a specific R . In this work we place the Mnspin in the center of the QD and the only nonzero matrixelements J i,j appear if both orbitals i and j are zeroangular momentum states. The spin of an electron placedon any other HO orbital is not coupled directly to the Mnspin.The eigenenergies and eigenstates of the Hamil-tonian (1) are obtained in the configuration-interaction approach. In this approach, we con-struct the Hamiltonian matrix in the basis ofconfigurations of N electrons and one Mn spin: | ν i (cid:105) = | i ↑ , i ↑ , . . . , i N ↑ (cid:105)| j ↓ , j ↓ , . . . , j N ↓ (cid:105)| M z (cid:105) , where | i σ , i σ , . . . , i Nσ (cid:105) = c † i σ c † i σ , . . . , c † i Nσ | (cid:105) , | (cid:105) is the vac-uum state, and N = N ↑ + N ↓ is the number of electrons,in which N ↑ and N ↓ are the number of electrons withspin up and spin down, respectively. The total numberof configurations depends on the number of electronsand on the number of the HO shells available in the QD.With Mn impurity in the center, the total orbital angularmomentum of electrons L = (cid:80) Ni =1 L ie is conserved by theHamiltonian (1). Moreover, depending on the anisotropyof e -Mn interactions, the Hamiltonian also conservesthe total projections S z and M z of the electron andMn spin separately (the Ising model) or the projection J z = s z + M z of the total spin (the Heisenberg model).Based on these conservation rules, we divide the basis ofconfigurations into subspaces labeled by the numbers L , S z , and M z (for the Ising model) or L and J z (for theHeisenberg model), and diagonalize the Hamiltonian ineach subspace separately.Our model is also suitable for electrons interacting witha single nuclear spin. In the Fermi-contact hyperfineinteraction, the Hamiltonian of electrons interactingwith nuclear spins has the same form as the Hamilto-nian of electrons interacting with Mn spins. Even thoughthe interaction between electrons and single nuclear spinshas not been achieved in self-assembled quantum dots,today it is possible to manipulate a few nuclear spins indiamond, silicon, and carbon nanotubes. The computational procedure adopted in this work isas follows. For a chosen number of electrons N = 2 , . . . , e - e interactions in the Ising and isotropic Heisenbergmodels. By analyzing the degeneracies of the states wefind the total spin of the system. Further, from the or-dering of different states with respect to their total spinwe draw conclusions as to the ferromagnetic or antifer-romagnetic character of the effective e -Mn interactions.By comparing the results for the system with and with-out the e - e interactions ( γ = 1 or γ = 0, respectively)we establish the e - e interaction mediated effective e -MnHamiltonian for electrons not directly coupled to the cen-tral spin. III. SPIN SINGLET CLOSED SHELLSCOUPLED WITH THE MAGNETIC ION
We start with a discussion of a filled s shell with N = 2electrons in the zero angular momentum channel. Eachelectron is directly coupled to the Mn impurity, but thesinglet state couples only via e - e interactions. Here wediscuss the role of anisotropy of the exchange interactionon this indirect coupling. A similar discussion applies toother closed shells, e.g., N = 6.The lowest-energy s -shell spin singlet configurationwith S = 0 and orbital angular momentum L = 0, | s GS z = 0 , M z (cid:105) = c † s, ↑ c † s, ↓ | , M z (cid:105) , is shown schemati-cally in the top left panel of Fig. 2(a). The expectationvalue of the e -Mn Hamiltonian against the configuration | s GS z = 0 , M z (cid:105) is zero.Increasing the number of confined shells to three addsone additional orbital (1 ,
1) with zero angular momentumin the d -shell directly coupled to the Mn spin. Now thetwo-electron triplet states with total angular momentum L = 0 couple to the Mn spin. The triplet with S z = 0, | s Ez = 0 , M z (cid:105) = (1 / √ (cid:16) c † d, ↑ c † s, ↓ − c † s, ↑ c † d, ↓ (cid:17) | (cid:105)| M z (cid:105) . Oneof its components is shown schematically in the top rightpanel of Fig. 2(a), while the bottom left panel of thatfigure shows the spin-polarized triplet | s z = 1 , M z − (cid:105) = c † s, ↑ c † d, ↑ | , M z − (cid:105) , and the bottom right panel shows thetriplet | s z = − , M z + (cid:105) = c † s, ↓ c † d, ↓ | , M z + 1 (cid:105) . Applyingthe e -Mn Hamiltonian to the | s GS z = 0 , M z (cid:105) state, weobtain H e - Mn | s GSz = 0 , M z (cid:105) = − J sd √ M z | s Ez , M z (cid:105)− J sd ε ( β − | s z = 1 , M z − (cid:105) − β + | s z = − , M z + 1 (cid:105) ) , (2)where J sd is the exchange matrix element in which oneelectron is scattered from the s orbital to the d orbital and β ± = (cid:112) ( M ∓ M z )( M ± M z + 1). We find that upon theinclusion of the d shell, the low-energy s -shell singlet two-electron configuration becomes coupled by e -Mn interac-tions to electron triplet configurations, with and withoutflip of the Mn spin. -0.015-0.010-0.0050.000 ( E M n - E e ) / J ss =1 -0.017-0.016-0.015-0.014 N = 2, 3 Shells , = 16 x ( E M n - E e ) / J ss (b)(a)(c) (d) ( E M n - E e ) / Jss
6x 6x 6x6x
Number of ShellsN=2, =1, = 1 zGSz Ms ,0 zEz Ms ,0 zz Ms zz Ms FIG. 2: (Color online) (a) Schematic pictures of two-electron-Mn configurations, GS and electronic triplet states, coupledby the e -Mn interactions. (b) Ground-state energy of the two-electron-Mn system as a function of the number of quantum-dot shells. (c) and (d) Ground-state energies of the two-electron-Mn system for the quantum dot confining three shellsplotted as a function of the strength of electron-electron in-teractions in the Heisenberg e -Mn model (c) and as a functionof the isotropy of the e -Mn Hamiltonian for the fully inter-acting electron system (d). Numbers at the energy level barsrepresent the degeneracy of states. We now diagonalize the two-electron-Mn Hamiltonianand compute the ground-state (GS) energy E Mn of theQD with a manganese ion, and the energy E e of the sys-tem without Mn. Figure 2(b) shows the effect of the Mnion on the ground state energy, ∆ = ( E Mn − E e ) /J ss ,measured from the ground-state energy without the Mnion, as a function of the number of shells for the inter-acting system ( γ = 1) and the isotropic exchange inter-action ( ε = 1). We find that, irrespective of the numberof confined shells, the GS is sixfold degenerate, with thetotal spin J GS = 5 /
2. However, the energy of the GSmarkedly depends on the number of shells. For two con-fined shells we have ∆ = 0, because in this case we cangenerate only one configuration, | s GS z = 0 , M z (cid:105) , which isdecoupled from the Mn spin. The inclusion of the d shelladds an additional L e = 0 orbital into the single-particlebasis, resulting in the scattering of electrons by the local-ized spin and lowering of energy. A further lowering ofthe energy occurs when the fifth shell, containing another L e = 0 single-particle state, becomes confined.Now we fix the number of shells to three, set theHeisenberg form of e -Mn interactions and study the effectof e - e interactions. Figure 2(c) shows the energy ∆ with-out ( γ = 0) and with full Coulomb interactions ( γ = 1).We find that the ground state in both cases is sixfold de-generate but the e - e Coulomb interactions enhance theeffects of the e -Mn coupling, lowering ∆. This is due toa larger contribution of triplet configurations to the GS.We now compare the results for the isotropic couplingversus the anisotropic coupling. For the anisotropic cou-pling, ε = 0, we observe that the GS is split into threeenergy levels labeled by | M z | , each of them twice degener-ate, as shown in Fig. 2(d). In Ising-like coupling the totalangular momentum J is not conserved, and the charac-teristic sixfold degeneracy of the ground state is broken.Comparing the isotropic and anisotropic coupling we ob-serve that ∆ is negative for both couplings and also thatthe Heisenberg-like interaction results in a lower energythan the Ising-like interaction. IV. ELECTRONS IN FINITE ANGULARMOMENTUM CHANNELS
In this section we discuss electrons populating finiteangular momentum channels which are not directly cou-pled with the Mn ion. For N = 3 we show the existence ofan effective coupling mediated by e - e interactions. Simi-lar results are obtained for N = 5. A. One electron on the p shell The lowest-energy configuration in the ground state ofthree electrons is formed by two electrons in the s shelland one electron in the p shell. With Mn in the QD centerthe total angular momentum L of the three electrons isconserved and we show the results for L = 1.Figure 3(a) illustrates the degenerate three-electronconfigurations, | s z = 1 / , M z (cid:105) and | s z = − / , M z + 1 (cid:105) ,with an electron with spin up and Mn in state M z andand electron with spin down on the p orbital and Mn instate M z +1. As the electron-Mn exchange interaction inthe p shell vanishes, J pp = 0, these configurations do notinteract with each other. As a consequence, the GS is 12-fold degenerate, two electron spin configurations timessix Mn spin orientations. In order to understand theeffect of interactions we include configurations coupledwith | s z = 1 / , M z (cid:105) and | s z = − / , M z + 1 (cid:105) by both e - e and e -Mn interactions and diagonalize the Hamiltonianin the L = 1 subspace. The number of three-electron-Mn configurations depends on the number of electronicshells, with 24, 228, 852, and 2520 for two, three, four,and five shells, respectively. ( E M n - E e ) / Jss
Number of Shells12xN=3, =1, = 1(a) (b)(d)(c) ( E M n - E e ) / Jss = 1 e-Mn e-e Indirect e-Mn coupling zz Ms ,21 zz Ms FIG. 3: (Color online) (a) Ground-state three-electron config-urations with the p -shell electron spin up (top) and spin down(bottom). (b) Energy difference ∆ between a three-electronGS in the Mn-doped and undoped QD for both noninteract-ing ( γ = 0) and interacting ( γ = 1) electrons. The numbersindicate the degeneracy of each level. (c) Diagram of couplingbetween electrons in the p shell and Mn. The solid arrow rep-resents a direct coupling via e -Mn coupling or e - e Coulombinteraction, and the dashed arrow illustrates the indirect cou-pling. (d) The energy difference ∆ as a function of the numberof shells for a QD containing three shells and γ = 1. Figure 3(b) shows the result of exact diagonalization ofthe e -Mn Hamiltonian for three confined shells in the QDand an isotropic e -Mn interaction ( (cid:15) = 1), for both non-interacting ( γ = 0) and interacting ( γ = 1) electron sys-tems. For the noninteracting case we observe that the GSis 12-fold degenerate, with the energy lowered by the e -Mn interaction (negative ∆). This behavior is identical towhat was shown for the two electrons in the previous sec-tion, i.e., the two electrons in the s shell are coupled withMn, while the electron in the p shell is only a spectator.However, in the strongly interacting regime, γ = 1, weobserve a splitting of the degenerate GS into two degen-erate shells. The splitting and the degeneracy of levels isconsistent with an effective Hamiltonian H eff = − J eff (cid:126)s · (cid:126)M coupling the p -shell electron spin s with Mn spin M . The effective coupling J eff is mediated by Coulomb in-teractions. In Fig. 3(c) we illustrate the processes whichcouple | s z = 1 / , M z (cid:105) and | s z = − / , M z + 1 (cid:105) states.The e -Mn interaction acting on the | s z = 1 / , M z (cid:105) statescatters the spi- up (blue) electron from the s shell to thespin-down (red) electron on the d shell with a simulta-neous transition of the Mn spin from M z to M z + 1. Inthe next step, the e - e interaction scatters the d -shell and p -shell electron pair into the s -shell and p -shell electronpair, with the spin-down electron on the p shell and thespin-up electron on the s shell. The net result is a spinflip of the p -shell electron and of the Mn spin. We seethat the ground state is sevenfold degenerate, implyingthat the electron spin is aligned with the Mn spin and J eff is hence ferromagnetic.Let us now investigate the dependence of the GS en-ergy on the number of confined shells in the QD. Fig-ure 3(d) shows the evolution of the GS energy as a func-tion of the number of shells for γ = 1 and ε = 1. Weobserve that for two shells there is no splitting , i.e., J eff = 0, while for three and four shells the GS is splitinto two shells. For two shells the GS is 12-fold degen-erate, ∆ = 0, and there is no interaction between Mnand electrons. For three shells the GS is split into twoshells, as discussed above. For four shells the GS is alsosplit into two, but there is an inversion of the degeneracyof the energy levels. This is a consequence of an anti-ferromangetic interaction J eff < | s z = 1 / , M z (cid:105) and | s z = − / , M z +1 (cid:105) whichis mediated via e - e Coulomb and e -Mn interactions be-tween the GS and excited configurations. As the num-ber of shells increases, more excited state configurationsinteract with the GS, stabilizing the antiferromagneticindirect coupling between the electrons and Mn.If the indirect magnetic ordering shown above dependson the number of shells, it also should depend on the QDshell spacing ω . Figure 4(a) shows the dependence of GSenergy on ω for three electrons confined in a Mn-dopedQD containing three shells, γ = 1 and ε = 1. We notethat the exchange coupling changes from ferromagneticto antiferromagnetic for ω ≈ . ∗ . We observe thesame behavior for QDs with four shells, but in this casethe crossing occurs at ω ≈ .
45 Ry ∗ .Next we discuss the effect of anisotropy on the e -Mnexchange interaction. Figure 4(b) shows the GS energyfor three electrons in a QD containing three shells in thestrongly interacting regime as a function of the e -Mn cou-pling. For (cid:15) = 0 the electrons and Mn interact via ananisotropic Ising-like Hamiltonian, and for (cid:15) = 1 the e -Mn interaction is isotropic, Heisenberg-like. For (cid:15) = 0, s z is a good quantum number, and therefore the electronspin degeneracy is preserved. In Fig. 4(b) we observe thatfor (cid:15) = 0 the energy spectrum is split into six doubly de- X X X ( E M n - E e ) / J ss (Ry*) N = 3, 3 Shells, = 1, = 1 X ( E M n - E e ) / Jss =1 (a)(b) FIG. 4: (Color online) (a) Evolution of the energies of three-electron levels with J = 3 and J = 2 as a function of the QDshell spacing ω . (b) Energy difference ∆ for both anisotropic( ε = 0) and isotropic ( ε = 1) e -Mn interactions in a three-shellQD with full interactions ( γ = 1). generate levels. This splitting is due to the e - e Coulombinteraction driving the indirect e -Mn interaction betweenthe p -shell electron and Mn, as was observed in the (cid:15) = 1case. The double degeneracy for the anisotropic couplingarises due to the fact that the state | s z = 1 / , M z (cid:105) has thesame energy as the configuration of | s z = − / , − M z (cid:105) .We also investigated the effect of Mn positions onthree-electron GSs. Moving Mn away from the QD cen-ter couples the electron in the p orbital directly withMn. This coupling is ferromagnetic. Considering a QDcontaining three shells and ω = 1 .
98 Ry ∗ , the indirect e -Mn coupling is also ferromagnetic, and therefore bothdirect and indirect e -Mn interactions add up. As Mn ismoved away from the QD center, the direct coupling be-comes the dominant effect for Mn positions larger than R ≈ . l . Even though the direct e -Mn interaction isdominant for Mn far away of the QD center, the indirect e -Mn coupling is always present. B. Two spin-polarized electrons on the p shell Next we describe the electronic properties of a half-filled p shell. The lowest-energy configuration of the four-electron GS state is formed by two electrons in the s shell and two spin triplet electrons in the p shell. Fig-ure 5(a) illustrates the four-electron configurations, thetriplet | S = 1 , s z = 1 , M z (cid:105) and one of the singlet compo-nent | S = 0 , s z = 0 , M z + 1 (cid:105) configurations. These twoconfigurations have the same total spin projection J z . Inthe presence of an e - e Coulomb interaction the S = 1triplet state is the GS and the singlet is an excited state.For Mn in the QD center the p electrons do not couplewith Mn, the electron spin degeneracy is preserved, andthe degeneracy of the triplet state in a Mn-doped QD is18, while the singlet state is sixfold degenerate.We shall now investigate how the GS of four electronsconfined in a Mn-doped QD is affected by the presence ofthe e - e Coulomb interaction, number of shells, shell spac-ing, and e -Mn coupling. We take advantage of the con-servation of the total angular momentum and diagonalizeour microscopic Hamiltonian in the L = 0 subspace. Thenumber of configurations in this subspace is 30, 498, and3498 for two, three, and four shells, respectively.The e - e mediated coupling of the electronic and Mnspin is interpreted in terms of the effective exchangeHamiltonian. Adding the electron and Mn spins resultsin total spin J = 7 / , / , / e - e interactions ( γ = 1) and the isotropic e -Mncoupling ( ε = 1). The energies of these states are shownrelative to the energy of the ground-state triplet of theundoped QD. The triplet and singlet states split for anynumber of shells due to an e - e exchange interaction. Ina QD with only s and p shells, the effective exchangecoupling for p -shell electrons is zero and the triplet andsinglet states are 18 and six times degenerate, respec-tively. Increasing the number of shells leads to a finiteand ferromagnetic exchange interaction with the tripletstates coupled to the Mn spin and the 18-fold degener-ate shell split into eight-, six- and four-fold degeneratelevels. The character of this exchange interaction de-pends on the number of shells. For three shells we havea ferromagnetic coupling, but for four shells the couplingbecomes antiferromagnetic.Figure 5(c) illustrates the configurations involved inthe indirect coupling of the electrons on the p shell andthe Mn spin. Here, the solid arrows represent the directcoupling between configurations, and the dashed arrowrepresents the indirect interaction between two configu-rations. Let us explain how this indirect coupling arises,starting from the configuration with two spin-up elec-trons in the p shell, which is labeled as | S = 1 , s z =1 , M z (cid:105) [see Fig. 5(c), top left]. This configuration is cou-pled with an excited state in which there are two spin- ( E M n - E e ) / Jss
Number of Shells
N=4, =1, =1 (a) zz Mss ,1,1 (b)(c) e-Mn e-e Indirect e-Mn Coupling zz Mss
FIG. 5: (Color online) (a) Low-energy configurations of fourelectrons in a magnetic QD. (b) Low-energy spectrum of thesystem as a function of the number of shells for interacting( γ = 1) electrons, measured from the respective GS energy E e of a nonmagnetic system. Here the QD shell spacing ω =1 .
98 Ry ∗ . (c) Indirect coupling diagram of two four-electronconfigurations. The solid arrows represent direct interactionsbetween configurations and the dashed arrow represents theindirect e -Mn coupling. down electrons in both L e = 0 orbitals, one in the s shelland the other in the d shell. This coupling occurs via an e -Mn interaction, which scatters the spin-up electron inthe s shell of | S = 1 , s z = 1 , M z (cid:105) to the d shell, flippingthe electron spin down, and the Mn spin up, i.e., M z + 1.This excited state with s z = 0 and M z + 1 is coupledwith one of the | S = 0 , s z = 0 , M z + 1 (cid:105) GS configurationsvia the e - e Coulomb interaction, in which the spin-downelectron in the d shell is scattered to the L e = 1 p orbital,and the spin-up electrons in this orbital are scattered tothe s shell.Figure 6(a) shows the GS energy for both noninteract-ing ( γ = 0) and fully interacting ( γ = 1) electrons. Weconsidered a Mn-doped QD with three confined shells andthe isotropic e -Mn interaction ( ε = 1). For the noninter-acting case there is no triplet-singlet splitting, and as and e - e Coulomb interaction mediates the indirect interactionbetween Mn and the p -shell electrons, the triplet is notsplit either. Therefore, the four-electron GS is 24-fold de- ( E M n - E e ) / Jss N = 4 , 3 S h e lls , = 1 ( E M n - E e ) / Jss N = 4 , 3 S h e lls , = 1 (a)(b)(c) ( E M n - E e ) / J ss (R y *) N = 4 , 3 S h e lls , = 1 , = 1 FIG. 6: (Color online) (a) Energy difference ∆ for noninter-acting ( γ = 0) and interacting ( γ = 1) electrons in the four-electron magnetic dot. (b) GS energy difference as a functionof the QD shell spacing ω for three shells confined in theQD. (c) GS energy difference for the anisotropic ( ε = 0) andisotropic ( ε = 1) e -Mn coupling. generate. Even though the four noninteracting electrontriplet states are not split by the indirect coupling, wesee a negative ∆, which means that electrons lower theirenergy by an exchange interaction with Mn. Turning the e - e Coulomb interaction on results in the singlet-tripletsplitting and a further splitting of the triplet energy shell.The triplet splitting is caused by the indirect interactionbetween Mn and electrons in the p shell, which is medi-ated by the e - e Coulomb interaction.In Fig. 6(a) we show the effect of the e - e Coulomb in-teraction on the low-energy spectrum of the four-electronand Mn complex. We note the appearance of triplet andsinglet energy shells, separated by the e - e exchange in-teraction. The splitting of the triplet shell is governed bythe e - e and e -Mn exchange interactions.Figure 6(b) presents the energy difference ∆, i.e., theeffective exchange coupling, as a function of ω for four interacting electrons ( γ = 1) confined in the Mn-dopedQD with three confined shells. Here we also have aferromagnetic to antiferromagnetic crossing as a func-tion of the QD shell spacing. For QDs with four shellsthe ferromagnetic to antiferromagnetic crossing occurs at ω ≈ .
04 Ry ∗ .Now we show the effect of the symmetry of the e -Mncoupling on the four-electron GS. In Fig. 6(c) we comparethe effects of the anisotropic ( ε = 0) and isotropic ( ε = 1)coupling for a Mn-doped QD with three confined shellsand in the presence of a full e - e Coulomb interaction ( γ =1). For the anisotropic coupling the triplet state is splitinto nine doubly degenerate levels. In this case, both s z and M z are good quantum numbers, and therefore, s z =1 and s z = − s z = 1 and M z is equalto the energy of the state s z = − − M z , these sixstates are double degenerate. The s z = 0 configurationssplit into three, where the degeneracy is given by M z ,i.e., the s z = 0 configurations are degenerate and labeledby | M z | , as for the two electrons interacting with the Mnvia an anisotropic e -Mn interaction. The singlet state isalso split into three doubly degenerate levels.One way to probe the indirect e -Mn interaction is byperforming a circularly polarized photoluminescence ex-periment of quantum dots containing a single Mn spinand confining three or more electrons. In this case, theindirect e -Mn coupling gives rise to a fine structure ofboth initial and final states of the emission process. V. CONCLUSION
In conclusion, we presented a microscopic model of in-teracting electrons coupled with a magnetic ion spin lo-calized in the center of a self-assembled quantum dot. Weshowed that the electrons occupying finite angular mo-mentum orbitals interact with the localized spin throughan effective exchange interaction mediated by electron-electron interactions. The effective interaction for p -shellelectrons is obtained using exact diagonalization of themicroscopic Hamiltonian as a function of the number ofelectronic shells, shell spacing, and anisotropy of the ex-change interaction. It is shown that the effective inter-action can be engineered to be either ferro- or antiferro-magnetic, depending on the quantum-dot parameters. ACKNOWLEDGMENT
The authors thank NSERC and the Canadian Institutefor Advanced Research for support. UCM thanks J. A.Brum for fruitful discussions and acknowledges the sup-port from CAPES-Brazil (Project No. 5860/11-3) andFAPESP-Brazil (Project No. 2010/11393-5). PH thanksY. Hirayama, WPI-AIMR, Tohoku University for hospi-tality. P. Hawrylak, in
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