Multiband Electronic Structure of Magnetic Quantum Dots: Numerical Studies
MMultiband Electronic Structure of Magnetic Quantum Dots: Numerical Studies
D. Rederth, R. Oszwa(cid:32)ldowski, and A. G. Petukhov
South Dakota School of Mines and Technology, Rapid City, USAEmail: [email protected]
J. M. Pientka
St. Bonaventure University, New York, USA (Dated: November 12, 2018)Semiconductor quantum dots (QDs) doped with magnetic impurities have been a focus of contin-uous research for a couple of decades. A significant effort has been devoted to studies of magneticpolarons (MP) in these nanostructures. These collective states arise through exchange interactionbetween a carrier confined in a QD and localized spins of the magnetic impurities (typically: Mn).We discuss our theoretical description of various MP properties in self-assembled QDs. We presenta self-consistent, temperature-dependent approach to MPs formed by a valence band hole. We usethe Luttinger-Kohn k · p Hamiltonian to account for the important effects of spin-orbit interaction. PACS numbers: 73.21.La,73.22.-f,75.50.Pp
I. INTRODUCTION
With spintronics and quantum computing as the driv-ing forces, one of the primary foci of nanomagnetism andsemiconductor spintronics is design and fabrication ofmagnetic Quantum Dots (QDs) with customized prop-erties. These nanostructures, based on Dilute MagneticSemiconductors (DMS), are also interesting from a fun-damental physics point of view; their description requiresa combination of quantum and statistical approaches tosmall systems.The magnetic properties of DMS are introduced by thetransition-metal ions (such as manganese). In bulk sam-ples of DMS, alignment of Mn spins is typically achievedthrough an external magnetic field. An alternative sce-nario may be realized in magnetic QDs charged with car-riers. Owing to their strong confinement, the exchangeinteraction of these carriers with Mn ions is enhanced.This obviates the need of external magnetic field, re-placed by spin-density of the trapped carriers. This effec-tive internal field may be large enough to strongly alignthe Mn spins, resulting in a magnetized quantum dot,Fig. 1. At the same time, the ground-state energy ofthe carrier is lowered, and spin degeneracy of energy lev-els is lifted. This combination of effects is referred to asformation of a magnetic polaron (MP).Evidence of MP formation is provided by numerousoptical experiments on magnetic II-VI QDs embeddedin bulk semiconductor, e.g. the early reports in Refs. 3and 4. The first time-resolved studies of this effect inself-assembled DMS QDs were presented in Refs. 5 and6. Later, formation of magnetic polarons was revealed incolloidal magnetic QDs. In this work, we present a self-consistent, multiband,temperature dependent description of hole magnetic po-larons. We focus on MPs formed by exchange interac-tion of a single hole with multiple Mn spins embeddedin II-VI QDs, as in the experimental studies Refs. 8 and9. We use the Luttinger-Kohn Hamiltonian to describe
FIG. 1. Dilute magnetic semiconductor quantum dot. (a)Situation without a hole: the Mn ions’ spins point in randomdirections. (b) Situation when a hole is confined in the QD:the Mn ions’ spins align anti-parallel to the hole spin forminga magnetic polaron. quantum states of the hole. For temperature dependence,we introduce a well-controlled mean-field approach. Thiscombination of quantum and statistical descriptions al-lows us to formulate a self-consistency condition, whichreflects the mutual influence of the confined carrier andlocalized, paramagnetic Mn spins.Magnetic polarons are typically formed at cryogenictemperatures. The corresponding energy gain is de-stroyed by thermal fluctuations at higher temperatures.We present results showing that our method correctlydescribes this temperature dependence. Our approach,based on the envelope-function approximation, goes be-yond the often employed “muffin-tin” ansatz for a con-fined carrier wavefunction. Thus, we are able to reveal aninteresting effect: localization (“shrinking”) of the holewavefunction in QDs with the above Mn placement.We note that MP formation is known to occur also indoped bulk DMS systems. In that case, a carrier boundto a donor or acceptor aligns the Mn spins within itseffective Bohr radius.
This scenario, called boundmagnetic polaron, has similarity to MPs formed in QDs.However, the degree of freedom offered by the tunabilityof QD confinement is absent in that scenario. a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l II. NON-MAGNETIC HAMILTONIAN
We employ the Luttinger-Kohn Hamiltonian to de-scribe the hole states:ˆ H LK = ˆ P + ˆ Q − ˆ S ˆ R − ˆ S ∗ ˆ P − ˆ Q R ˆ R ∗ P − ˆ Q ˆ S R ∗ ˆ S ∗ ˆ P + ˆ Q , (1)where all the quantities (containing Luttinger parameters γ , , ) and the phase convention are defined in Ref. 12,Cartesian components of wave vector are replaced by par-tial derivatives. We take advantage of the Envelope Function Approx-imation by adding a confining potential, resulting fromthe band offset at semiconductor heterojunctions. Wemodel the potential by V ( r ) I , where I is the 4 × V ( r ) is the same for heavy- andlight-holes. It consists of the infinite-well potential inthe growth direction, z , and in-plane parabolic poten-tial, m ∗ ω ( x + y ), where m ∗ = m γ + γ is the in-planeheavy-hole effective mass. Altogether, the non-magneticpart of the Hamiltonian is ˆ H = ˆ H LK + V ( r ) I . III. SELF-CONSISTENT EXCHANGEHAMILTONIAN
The key element of this work is the non-linear, temper-ature dependent Schr¨odinger equation for the effectivecarrier wavefunction corresponding to the most probablespin fluctuation . This equation can be justified as fol-lows. First, we will consider the Hamiltonian describingthe contact exchange interaction between the fermionsand the magnetic ions. It is convenient to express thisHamiltonian in the second-quantized form:ˆ H = ˆ H + ( β/ (cid:88) j ˆ S jz (cid:16) ψ † j ˆ J z ψ j (cid:17) (2)where ψ † j = (cid:16) ψ † j, / , ψ † j, / , ψ † j, − / , ψ † j, − / (cid:17) is the four-component spinor field of a hole with spin J = 3 /
2, thespin indexes σ = ± / σ = ± / ψ † jσ and ψ jσ are creation and annihilation fermion field operators suchthat ψ jσ ≡ ψ σ ( R j ) where R j is the location of a magneticion, β is the exchange coupling constant, and ˆ S z and ˆ J z are the Mn ( S = 5 /
2) and the heavy-hole ( J = 3 / g -factor anisotropy .We take advantage of the fact that the Ising Hamil-tonian does not contain the double spin-flip processes inwhich a hole and a Mn impurity exchange a unit spin. It means that the system’s wavefunction can be repre-sented as a product of the hole and Mn-spin parts. Theresulting hole Hamiltonian will depend on the set of c -numbers S jz (spin projections) rather than on the set ofthe non-commuting spin operators. Second, we assumethat the time evolution of the Mn spin subsystem is slowenough to treat it as a static non-uniform exchange fieldacting upon the hole spins, and consider only the station-ary states of the thermalized holes for each configurationof the Mn spin projections. Therefore the partition func-tion of the system can be calculated using Gibbs canon-ical distribution: Z = Tr S jz (cid:88) n exp (cid:20) − E n ( { S jz } ) k B T (cid:21) , (3)where k B is the Boltzmann constant, T is temperature,and n is a quantum number labeling the hole eigenvaluesthat, in turn, depend on 6 N c -numbers S jz . Thus, inorder to calculate the partition function in Eq. (3) onewould need to solve 6 N replicas of the hole Schr¨odingerequation, which makes the problem intractable.To overcome this obstacle we will partition the areaof the quantum dot with N Mn spins into a set of N c square blocks (cells), containing few ( N k ) Mn spins (i.e. N = N c N k ), and neglect spatial variation of the holewavefunction (and the spin density) within each cell. Fora particular cell k with N k spins a distribution functionof the average dimensionless magnetization, µ k ≡ ¯ S ( k ) z ,can be expressed as: Y ( µ k ) = Tr S ( k ) jz δ µ k − N − k (cid:88) j S ( k ) jz ∝ exp (cid:20) − G S ( µ k /S ) k B T (cid:21) , (4)where the Gibbs free energy of N k non-interacting spins, G S ( µ k /S ), that can be obtained using Legendre’s trans-formation, reads: G S ( x ) = k B T N k (cid:34) xB − S ( x ) − ln sinh (cid:2) (1+1 / S ) B − S ( x ) (cid:3) sinh (cid:2) (1 / J ) B − S ( x ) (cid:3) (cid:35) (5)Here B − S ( x ) is the inverse of the Brillouin function B S ( x ). We note that the distribution function Y ( µ k )is temperature independent, i.e. purely entropic.Using the distribution functions Y ( µ k ) we can trans-form the partition function of Eq. (3) into a multipleintegral over continuous variables µ k : Z ∝ (cid:88) n (cid:90) N c (cid:89) k =1 dµ k exp (cid:20) − (cid:80) k G S ( µ k /S ) + E n ( { µ k } ) k B T (cid:21) (6)The expression in parentheses of Eq. (6) contains twoterms: the fermion energy E n responsible for the forceexerted by the fermions on the magnetic ions and themagnetic term (cid:80) k G S ( µ k /S ) responsible for the restor-ing force exerted by the ions on the fermions. The lat-ter, so-called entropic (or emerging) force has a ratherpeculiar origin because it is not derived from any physi-cal interaction between the particles, i.e. magnetic ions.Rather it is caused by the tendency of the system toassume the state with the maximum entropy. This isprecisely why the magnetic polarons are fundamentallydifferent from the lattice polarons. The entropic forcescan be efficiently controlled by the temperature, mag-netic or even electric field. On the contrary, the physicalcontrol of the lattice polarons is severely limited.For any particular n the integral in Eq. (6) can beevaluated using the steepest descent method. The saddlepoint equation must be combined with the Hellmann-Feynman theorem − β (cid:104) ˆ J z ( k ) (cid:105) ∂E n ( µ , ..., µ k , ..., µ N c ) ∂µ k , where (cid:104) ˆ J z ( k ) (cid:105) = N − k (cid:80) N k j =1 (cid:104) Ψ | ψ † j ˆ J z ψ j | Ψ (cid:105) is the averagehole spin density of the cell.This leads to the non-linear Schr¨odinger equation : δ (cid:104) Ψ | ˆ H | Ψ (cid:105) δ Ψ + βS (cid:88) k N k B S (cid:34) Sβ (cid:104) ˆ J z ( k ) (cid:105) k B T (cid:35) δ (cid:104) ˆ J z ( k ) (cid:105) δ Ψ = 0 , (7)Here | Ψ (cid:105) is the state vector corresponding to the mostprobable spin fluctuation. The first term in Eq. (7)is a variational form of the standard non-magneticSchr¨odinger equation while the second term describesa non-linear and temperature-dependent contribution ofthe spin fluctuations induced by the paramagnetic ions.Analysis of the single magnetic polaronHamiltonian shows that the ordered solution ofEq. (7) exists at any temperature. It means that theexponent in the integrand of Eq. (6) may be expandedaround the saddle point and the multiple Gaussianintegration with respect to all µ k can be carried over.Remarkably, the result of this integration coincides with the exact integral calculated by Wolff . Moreover,one can generalize this procedure to the case of theHeisenberg exchange. The Gaussian integration in thiscase is more intricate because the Hamiltonian possessescontinuous rotational symmetry and the expansion ofthe exponent in Eq. (6) in the vicinity of the saddlepoint contains two zero-frequency transverse modein accordance with the Goldstone theorem . Theproper elimination of the Goldstone modes allows tocomplete the Gaussian integration, and the result againagrees with those of Wolff and Dietl and Spa(cid:32)lek .Thus the free energy of a magnetic polaron calculatedby means of the steepest descent integration replicatesthe exact results with no signature of spurious phasetransition. This is because the non-linear Schr¨odingerequation (7) contains quantum mechanical rather thanthermal average of the fermion spin density, contraryto the conventional method previously used in manyworks on magnetic quantum dots. The latter approachimposes artificial thermodynamic limit on a nanoscalesystem leading to spurious results. In this case, anattempt to calculate the partition function using thesteepest descent method would lead to a divergence(1 − T /T c ) − / in the vicinity of the critical temperature.The coarse-grained variables of Eq. (7) can be re-placed with continuous variables in a standard way.Thus, the self-consistent approach replaces the Mn spins,ˆ S z , with their thermal average, i.e. with magnetization, m ( r ) = SB S [ Sβρ s ( r ) / k B T ], where the spin density ρ s ( r ) = (cid:104) J z ( r ) (cid:105) and we replaced coarse-grained cell index k with a continuous variable r . The exchange Hamilto-nian reads : ˆ H ex = 13 x Mn | N β | m ( r ) ˆ J z , (8)where x Mn is the Mn (molar) fraction, and N is thecation (number) density. (The x Mn values in this paperare effective, i.e., assume no antiferromagnetic couplingbetween Mn ions.) The quantities m ( r ) and ρ s ( r ) mustbe found self-consistently using a continuous version ofthe nonlinear Schr¨odinger equation (7):ˆ H F ( r ) + 13 x Mn | N β | ˆ J z SB S (cid:20) Sβ k B T F † ( r ) ˆ J z F ( r ) (cid:21) F ( r ) = E F ( r ) , (9)where F ( r ) = (cid:104) r | Ψ (cid:105) is a 4-component spinor: F ( r ) = F / ( r ) F / ( r ) F − / ( r ) F − / ( r ) (10) IV. NUMERICAL APPROACH
We have solved the nonlinear Schr¨odinger Eq. (9) withthe Finite Difference method. The method discretizes theHamiltonian and envelopes F by dividing the QD into acubic mesh. Since the mesh lengths are on order ofthe crystal lattice spacing, the derivatives of the envelopefunctions are well approximated by finite differences. This numerical method can be used for any shape ofquantum dot, double QDs, and QDs on a wetting layer. The self-consistent procedure to model the magneticpolaron follows. We start with the initial magneti-zation m ( r ) = 0, this models mutual cancellation ofrandom Mn spins. This magnetization enters the ex-change Hamiltonian, Eq. 8. We use the envelopes result-ing from Eq. 9 to calculate the local hole spin density, ρ ( i ) s ( r ) = F ( i ) † ( r ) ˆ J z F ( i ) ( r ), which gives a new magneti-zation [cf Eq.(6) of Ref. 14]: m ( i +1) ( r ) = SB S (cid:34) Sβρ ( i ) s ( r )3 k B T (cid:35) (11)This form of magnetization is similar to the customaryone, except that ρ s ( r ) replaces the external magneticfield.Eq. 11 is the self-consistency condition. Our goal isto solve Eq. 9 using a recursive procedure, which loopsbetween the magnetization and spin density. When themagnetization is within a tolerance of the previous iter-ation, then the self-consistency loop ends. Our assump-tion is that after a few iterations, this procedure findsthe actual magnetization and a consistent spin density.In brief, the self-consistent algorithm is:1. Start with zero position-dependent Mn magnetiza-tion, i.e. m i = 0, where i = 12. Employing the finite-difference method, solve( ˆ H + ˆ H ex ( m i )) F ( i ) = EF ( i ) with ˆ H ex from Eq. 8to calculate the next iteration of hole eigenstates(envelope wavefunctions, F σ in Eq. 10)3. Calculate the spin density, ρ ( i ) s , from F ( i )
4. Calculate m ( i +1) in Eq. 11 using ρ ( i ) s
5. Find the maximum, with respect to position, of theabsolute value of differences (cid:12)(cid:12) m ( i ) − m ( i +1) (cid:12)(cid:12) . If it ismore than a specified tolerance (cid:15) , replace m ( i ) with m ( i +1) in the subsequent iteration, ( i → i + 1), goto point 26. Otherwise, the iteration loop ends, effectively solv-ing the nonlinear Schr¨odinger equation Eq. 9We used (cid:15) = 0 . × − . As a result, we obtain: theenergy of the ground state, its envelopes, and the magne-tization profile. These quantities indicate MP formationfor suitable system parameter regimes. V. SELF-CONSISTENT RESULTS
In this section, we present our numerical results ob-tained using the above approach. Our standard systemis a Cd Mn Te QD. The parabolic potential is givenby ¯ hω = 30 meV, corresponding to the in-plane charac-teristic length, ξ = (cid:113) ¯ hm ∗ ω = 4.19 nm, where ¯ h is the Dirac constant. The distance between the infinite barri-ers, i.e., the QD height, is h QD = 3 nm. The Luttingerparameters are γ =5.3, γ =1.62, γ =2.1. The exchangecoupling constant, β , is given by | N β | = 0 .
88 eV. The energy gain due to the magnetic polaron, E MP , isdefined as the difference between the ground-state energyof a non-magnetic and magnetic QD: E MP = E GS ( x Mn =0) − E GS ( x Mn > x M(cid:0) = % x M(cid:0) = % x M(cid:0) = % (cid:1) (cid:2) (cid:3)(cid:1) (cid:3)(cid:2) (cid:4)(cid:1) (cid:4)(cid:2) (cid:5)(cid:1) (cid:5)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:5)(cid:1)(cid:6)(cid:1)(cid:2)(cid:1) (cid:1)(cid:2) (cid:1) [ (cid:7) ] (cid:1) (cid:1) (cid:2) [ (cid:1) (cid:2) (cid:3) ] FIG. 2. The energy gain from the magnetic polaron, E MP ,for different Mn contents. The MP energy is at maximum at T = 0 K. excitation overcomes the exchange energy gain and theQD relaxes into a non-magnetic state. This temperaturedependence, as well as the dependence on x Mn is shownin Fig. 2.Fig. 3 shows our results for the position-dependentmagnetization m ( r ). This quantity has a strong tem- - - -
15 0 5 10 1500.511.522.5 Position in QD [ nm ] M agne t i z a t i on FIG. 3. Mn magnetization profile, m( r ), vs. temperature. At T = 0 K (dotted line), the Mn spins are aligned and parallel,causing the magnetization to be saturated. At T = 0 . T = 8 K (dot-dashed line), the magnetizationis still evident; at T = 40 K (solid line), the magnetization isalmost lost to temperature. perature dependence. It saturates at m = 5 / m is centeredon the QD center, its volume decreases with increasingtemperature.Fig. 4 demonstrates an interesting effect found in oursimulations: “shrinking” of the envelopes in a tempera-ture range. Because the Mn spins coupled to the tail ofthe wavefunction (i.e., on the QD periphery) are moreprone to thermal excitation with rising temperature, thewavefunction localizes to the center, thus maintainingsome of the exchange energy gain through a strongeralignment of Mn spins in the central region. (cid:1) = (cid:2) (cid:3)(cid:1) = (cid:4) (cid:3)(cid:1) = (cid:4)(cid:2) (cid:3) (cid:5) (cid:1)(cid:2) = (cid:6)(cid:7)(cid:8)(cid:6) (cid:9)(cid:10)(cid:5) (cid:1)(cid:2) = (cid:6)(cid:7)(cid:6)(cid:11) (cid:9)(cid:10)(cid:5) (cid:1)(cid:2) = (cid:6)(cid:7)(cid:12)(cid:11) (cid:9)(cid:10) - - - (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:4)(cid:2)(cid:6) (cid:4)(cid:6) (cid:7)(cid:8) [ (cid:6)(cid:9) ] (cid:1) (cid:2) (cid:3)(cid:4) (cid:5) (cid:3) (cid:6) (cid:7)(cid:8)(cid:3) [ (cid:4) (cid:9) - (cid:1) / (cid:2) ] FIG. 4. At T = 0 (dotted line), the wavefunction envelopedoes not localize and its width is very close to that of the non-magnetized case. The localization effect is strong at T = 2 K(solid line). At T = 20 K (dashed line), the envelope startsto relax into a non-magnetic state due to thermal excitationof the system. We quantify the localization effect through the in-plane“envelope width”, L mp , defined as L mp = (cid:115)(cid:90) (cid:88) σ | F σ ( x, y, z ) | ( x + y ) dxdydz (12)For our standard QD, the in-plane envelope localizationis 14 times stronger than out-of-plane (defined analo-gously to Eq.12). The out-of-plane envelope localizationis about 1%, thus negligible to our study.Fig. 5 shows the numerical temperature dependence of L mp . For clarity, we normalize L mp to the non-magneticwidth, L ∞ , which is realized at T → ∞ or, equivalently,for x Mn = 0. At T = 0, the mixing of the light- andheavy-holes leads to a small difference of the envelopewith respect to the high T limit: L mp ( T = 0) /L ∞ =1 . T and high– T envelopes are exactlyequal in the single-band approximation, where the light-and heavy-holes do not mix, so that L ∞ = L mp ( T = 0) = ξ .)At high temperatures, the magnetization is thermallydestroyed. Since there is no more energy gain from theenvelope localization, the system relaxes to a nonmag-netic state, see Fig. 5 inset. (cid:1) (cid:2) (cid:3)(cid:1) (cid:3)(cid:2) (cid:4)(cid:1) (cid:4)(cid:2) (cid:5)(cid:1) (cid:5)(cid:2) (cid:6)(cid:1)(cid:1)(cid:7)(cid:8)(cid:2)(cid:1)(cid:7)(cid:9)(cid:1)(cid:7)(cid:9)(cid:2)(cid:3) (cid:1) [ (cid:10) ] (cid:2) (cid:1)(cid:2) (cid:2) (cid:3) (cid:6)(cid:1) (cid:4)(cid:1)(cid:1) (cid:6)(cid:1)(cid:1) (cid:1)(cid:7)(cid:9)(cid:8)(cid:1)(cid:7)(cid:9)(cid:9)(cid:1)(cid:7)(cid:9)(cid:9)(cid:2)(cid:3) FIG. 5. Localization due to magnetic polaron vs. temperaturefor a Cd − x Mn x Te QD with x = 3% Mn. The envelopesexhibit strong localization for temperatures between 1 and3 K. Inset: localization decreases at high temperatures, sothe ratio becomes 1. VI. DELOCALIZATION: Mn OUTSIDE QD
We have shown above that Mn placed in QDs producesa localization effect on the carrier wave function. In theseminal experiment reported by Seufert et al., as well asothers, the Mn ions were placed (nominally) outside ofQDs i.e., in the barrier. What effect will this Mn positionhave on the carrier wave function confined inside the QD?To study this problem, we consider the saturatedregime ( T = 0 K), where the Mn spins are fully spinpolarized. We use a use a simple single-band model, inwhich the heavy-hole envelope is the ground-state solu-tion of a 2D harmonic oscillator in the x − y plane, timesa normalized sine along the z − axis (consistent with thepotential assumed in Sec. V) F HH ( r, z ) = 1 √ πL mp e − r / L (cid:115) h QD sin (cid:18) πzh QD (cid:19) , (13)where r = x + y . Here, L mp is the variational pa-rameter, unlike in Sec.V. The classical turning radius ofthis oscillator is referred to as R cls = √ ξ . We take thecylindrical surface of radius R cls and height h QD to bethe boundary between the QD and the surrounding layercontaining Mn.The ground state variational energy of the MP is ob-tained by calculating the expectation value of the totalHamiltonian, Eq. 13. For a fixed carrier and Mn spinconfiguration, we obtain E ( L mp ) = ¯ h m ∗ L + ¯ h π m ∗ z h + 12 m ∗ ω L − | N β | x Mn SJ e − R /L , (14)where the first term is the kinetic energy in the x − y plane, the second term is the kinetic energy along the z − axis with effective mass m ∗ z = m / ( γ − γ ), thethird term is the confinement energy and the last termis the exchange energy. For details on the derivationof Eq. 14 see Refs. 26 and 27. To obtain the optimalwidth, we numerically minimize Eq. 14 with respect to L mp and obtain an approximate MP wave function andenergy. In the limit of no Mn ( x Mn → L mp → ξ ).Using this model, we find L MP /ξ = 1 . , .
21 and 1 . x Mn = 1% ,
3% and 5%, respectively. Thus, the mag-netic width increases with increasing x Mn . The heavyhole wave function expands to increase the overlap be-tween the carrier spin and the Mn spins to lower thetotal energy. The exchange interaction gives rise to a delocalization effect.The delocalization effect can also be studied in QDswith multiple occupancies. We model this by consideringa 2D QD with harmonic confinement, and occupied bytwo heavy holes. As before, the Mn ions are distributedcontinuously in the space surrounding the confined re-gion ( R ≥ R cls ). Using the linear variational method, wenumerically diagonalize the QD Hamiltonian (containingCoulomb interaction) and obtain the heavy hole’s ener-gies and wave function. We found that with increasing x Mn , the electronic density of the ground state delocal-izes from the QD center to the QD boundary. Using theabove parameters with a relative permittivity ε = 9 . we find that the weight of the singlet electronic densityoutside R cls to be ≈
23% for x Mn = 0% and ≈
40% for x Mn = 3%, where half of the Mn spins outside of the QDpoint up on one side of the line that divides the QD inhalf, and down on the other side of this line. This indi-cates that it is energetically favorable for the heavy holesto increase their overlap with the magnetic ions outsidethe QD. A comprehensive discussion of our descriptionof multiply occupied QD is in preparation for a separatepublication. VII. COMPARISON TO EXPERIMENT
Detailed comparison of results of the self-consistentmethod to experimental findings for CdMnTe QDs em-bedded in a bulk semiconductor is not straightforward.Magnetic polaron signatures are typically detected inphotoluminescence (PL), where the optical transitionsare between levels derived from the conduction and va-lence bands. The transition energy, h ν , depends on QDgeometry, which typically has significant uncertainties.One should also take into account the Varshni shift whenanalyzing the temperature variation of h ν . Moreover,our calculated E MP values cannot be directly comparedto MP signatures seen in those QDs, where recombi-nation time is smaller or comparable to MP formationtime. Hence, we can only compare general trends and or-ders of magnitude of E MP . For example, Maksimov et al. , obtain an estimate of 10.5 meV at T (cid:39) . Mn . Te QD formed by fluctuation of aquantum-well width. This value corresponds well to therange presented in Fig. 2, taking into account the dif-ferent confinements, and the fact that at the nominal x Mn = 7%, some Mn ions are not magnetically active. K(cid:32)lopotowski et al. obtained E MP from PL of individ-ual, self-assembled Cd − x Mn Mn x Mn Te QDs at T = 8 K. Their values, E MP = 9 . x Mn = 3 .
5% and20%, respectively, are in the range of our results (thelatter high nominal x Mn corresponds to a much smallereffective x Mn ).Finally, we note that the wavefunction localization hasbeen seen experimentally for DMS quantum wells. VIII. DISCUSSION AND CONCLUSION
We have presented preliminary results from a robustnumerical method to calculate electronic and magneticproperties of self-assembled DMS quantum dots, in whichequilibrium magnetic polarons are formed. The methodis based on a well-controlled mean-field approach. Theobtained values of magnetic polaron binding energy infunction of temperature are in the correct range. Ourmethod allows to calculate spatial profiles of Mn-ion mag-netization, as well as localization of wave-function en-velopes.To our knowledge, little effort has been devoted so farto this level of theoretical description of MP formationin quantum dots, accounting for self-consistency. A sim-ilar, but not identical, approach has been presented inRef. 32. Results from the two methods seem to differ atlow Mn concentrations, where the approach of Ref. 32does not predict lifting of ground state degeneracy. Adetailed comparison requires further calculations. A self-consistent mean-field model adapted to spherical Quan-tum Dots has been presented in Ref.33.As a next step of development of the numericalmethod, we will consider “delocalization” of envelopes.Here, we have presented a simulation of this effect bya simple model for singly-occupied QDs at zero temper-ature. We have also reported briefly on the predictionof a related effect in double-occupied QD. This resulthas been obtained with a variational method taking intoaccount Coulomb interaction between confined carriers.The variational method will be discussed in a separatepublication.
ACKNOWLEDGMENTS
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