PHYSICAL REVIEW B 69, 153308 ~2004!

Quantum dots based on spin properties of semiconductor heterostructures

Manuel Valın-Rodrıguez,1 Antonio Puente,1 and Llorenc¸ Serra1,2

1Departament de Fı´sica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain2Institut Mediterrani d’Estudis Avanc¸ats IMEDEA (CSIC-UIB), E-07122 Palma de Mallorca, Spain

~Received 5 November 2003; revised manuscript received 1 December 2003; published 21 April 2004!

The possibility of a type of semiconductor quantum dots obtained by spatially modulating the spin-orbitcoupling intensity in III-V heterostructures is discussed. Using the effective-mass model we predict confinedone-electron states having peculiar spin properties. Furthermore, from mean-field calculations~local-spin-density and Hartree-Fock! we find that even two electrons could form a bound state in these dots.

DOI: 10.1103/PhysRevB.69.153308 PACS number~s!: 73.21.La

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Low-dimensional semiconductor structures constituteof the most suitable scenarios to observe quantum phenena in condensed-matter physics. In this sense, the atolike properties of quantum dots and the conductance quazation in narrow wires are seminal examples widestudied.1,2 At present, the available fabrication techniquessuch semiconductor structures rely on charge-based conment at length scales where quantum effects are unavoidMore precisely, confinement is obtained through the eneband discontinuities of different materials or by directly crating a potential landscape by means of metallic electroon a semiconductor heterostructure.

The aim of this work is to point out, using a theoreticmodel, the feasibility of a class of mesoscopic semicondustructures where the spin plays a relevant role in the menism that determines the confining properties, without bea pure spin-dependent confinement.3 The mechanism alsodiffers from conventional electrostatic confinement in thadoes not rely on the direct coupling of the electron chawith a horizontal electric field.

In III-V nonmagnetic semiconductor heterostructurthere are two relevant interactions concerning the spin:spin-orbit coupling and the hyperfine interaction betwecarriers and polarized nuclei. The latter mechanism hacharacteristic energy scale usually lower than that ofspin-orbit coupling.3

In semiconductors, spin-orbit coupling stems from trelativistic effect caused by the electric field due to the laof inversion symmetry of certain alloys like III-V heterostructures. Depending on the particular origin of the elecfield two contributions are distinguished; the electric fiecreated by the bulk inversion asymmetry of the mategives rise to the Dresselhaus term,4 while the asymmetry inthe profile of the heterostructure, structural asymmetry, gerates the so-called Bychkov-Rashba term.5 This lattermechanism constitutes the basis of the proposed electrnanostructures.

As mentioned above, the intensity of the Bychkov-Rashspin-orbit coupling depends on the effective electric fieperpendicular to the plane of the quantum well. This implthat it can be, in principle, engineered by adjusting the scific fabrication settings of each heterostructure. Anotpossibility to control the Rashba interaction, which has bedemonstrated experimentally,6–8 is by means of an externaelectric field perpendicular to the plane of the quantum w

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Exploiting the tunability of the Bychkov-Rashba mechnism we consider quantum dots whose main ingredientspace-modulated spin-orbit intensity. This spatial modulatcould be obtained with biased small electrodes on top ofsemiconductor heterostructure.9,10However, such an arrangement would not only modify the vertical electric field butwould also generate an in-plane gradient of electric potenleading to hybrid quantum dots where both mechanismspresent: horizontal electric potential gradient and spin-ointensity gradient. A more refined method to obtain spamodulated spin-orbit intensities requires a deeper insighthe origin of the Rashba intensity. In this respect, de Andret al.11 discussed the relevance of the barrier layers indetermination of the spin-orbit intensity. More recentGrundler8 has shown that for an InAs-based quantum wellapplied vertical electric field of the order of 103 V/cm canlead to a noticeable control of the spin-orbit intensiwhereas the built-in electric field of the sample is of torder of 105 V/cm. This high sensitivity to a relatively weaelectric field has been attributed to the induced variationsthe wave-function penetration into the barriers. The strorelevance of the barrier layers in the determination ofRashba intensity supposes a great advantage that couldto create a quantum well having a space-modulated sorbit coupling through space-engineered wave-function brier penetration~using inhomegeneous barrier layer compsition, thickness, etc.!. This would lead to a fixed~nottunable! substantial step in the spin-orbit intensity capableinducing the confinement.

We model structures confining electrons from the condtion band of the semiconductor and we use the effectimass approximation. We also suppose that the motionpendicular to the well’s plane is restricted to the firtransversal subband, so that the electrons effectively mova two-dimensional region. These simplifications mightquite drastic in some cases and, therefore, the scope ofwork is limited to a discussion of the feasibility of the effecrather than to a detailed quantitative description. In tmodel we include the standard Bychkov-Rashba term

HR5lR~r !

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where px and py represent the components of the in-plaelectron’s momentum and thes ’s are the correspondingPauli matrices.

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BRIEF REPORTS PHYSICAL REVIEW B69, 153308 ~2004!

In Eq. ~1! the inhomogeneity of the spin-orbit intensityincluded through the spatial dependence of the coupling cstantlR(r ), which is assumed of radial type (r and u areused to label the standard polar coordinates in the plane!. Tocharacterize the quantum dot we consider the following slike variation of the spin-orbit intensity:

lR~r !5le1~l i2le!1

11e(r 2R0)/s, ~2!

wherele andl i represent the external and internal constvalues of spin-orbit intensity, respectively. The quantumradius is given byR0 ands is a small diffusivity introducedto avoid discontinuities in the numerical solutions.

The Bychkov-Rashba interaction breaks the rotatiosymmetry of the Hamiltonian. Nevertheless, a symmecombining spatial and spin degrees of freedom is maintaiwhen the Rashba intensity is a purely radial functioNamely, the Hamiltonian is invariant under a combinedtation in real and spin spaces by the same angle, i.e., it cmutes with the generalized angular momentumJz5Lz1Sz .Note that at variance with conventional circular quantudots ~based on electric-potential confinement,1! the Hamil-tonian no longer commutes withLz andSz separately. As aconsequence of theJz symmetry the single-particle solutioncan be expressed, in a general form, as two-compospinors of type

Cn j~r ,u!5S fn j↑~r !ei ( j 21/2)u

fn j↓~r !ei ( j 11/2)uD , ~3!

wheren andj are quantum numbers characterizing the stawith valuesn51,2, . . . , j 561/2,63/2, . . . .Since the an-gular parts are analytical, we can reduce the Schro¨dingerequation to one-dimensional radial equations for the tspinor components,

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wheref↑ andf↓ denotefn j↑(r ) andfn j↓(r ), respectively.The above model has been analyzed using two alterna

numerical procedures:~a! solving the coupled onedimensional equations for the radial components ofeigenspinors, Eqs.~4!; and ~b! solving the full two-dimensional problem in Cartesian coordinates, withoutposing any symmetry restriction, in a uniform grid. Befodiscussing the numerical results it has to be noted thattems with the above spin-orbit interaction are still invariaunder time reversal. Consequently, a twofold degener

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known as Kramers degeneracy, must hold. From the raequations it can easily be seen that the substitutions:f↑→f↓ , f↓→2f↑ and j→2 j lead to a new state~Kramersconjugate! with opposite generalized angular momentum athe same energy eigenvalue («n j5«n2 j ). The expectationvalues ^Sz& and ^Lz& also change sign with the Kramertransformation. It is worth stressing that in the usual qutum dots~with an electric potential confinement of circulasymmetry! the degeneracies are 4 and 2; nonzero~zero! an-gular momentum states being fourfold~twofold! degenerate.Therefore, the reduction in degeneracy from 4 to at mosta genuine effect of the spin-orbit interaction.

Since the spin orientation in Eq.~3! depends on the position r , the eigenstates will show characteristic spin texturThe distribution of vertical and parallel spin read

^sz&~r !5ufn j↑~r !u22ufn j↓~r !u2,

^sW i&~r ,u!52fn j↓~r !fn j↑~r !@cos~u!x1sin~u!y#. ~5!

Note that while the vertical component is a pure radial funtion, the in-plane~horizontal! spin points in the radial direction. The fact that the horizontal spin follows the positiovector can be easily explained from the combined rotatioinvariance in spin and position space: consider two pointsthe dot having the same radius and different azimuthal anu; due to the symmetry of the Hamiltonian withJz , the localin-plane spin orientation in the second point must be thathe first point rotated the angle needed to go from the fipoint to the second in the real space. This property ofeigenstates is of particular interest for determining the rolethe geometrical phase~Berry’s phase! in mesoscopicstructures.12,13

From the numerical simulations we observe that a necsary requirement for confined states is that the spin-orbittensity within the quantum dot must be larger than that ofouter region, i.e.,l i.le . The reason for this behavior cabe understood from bulk considerations: the bulk enebands

«k6bulk5

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display a constant offset2m* le2/2\2 that depends on the

spin-orbit intensity~for a schematic representation of thbulk bands, see Fig. 1!. Therefore a bulk uniform samplewith a big value of spin-orbit intensity has an energy origfor the eigenstates lower than that of a sample havinsmaller intensity. In a finite system this energy mismatch clead to bound states in the region of higher intensity. Nevtheless, it has to be noted that for each spin orientationenergy mismatch is only confining for a certain rangemomenta. This means that the confining mechanism is sselective through the motion of the particles. In the casecircular symmetry, it implies that electrons rotating in osense are confined if they have a given spin orientatwhile electrons rotating in the opposite sense should haveopposite spin orientation to be confined.

We stress that the spin-orbital-dependent~SO-dependent!offset in Eq.~6! is the real cause of confinement since th

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BRIEF REPORTS PHYSICAL REVIEW B69, 153308 ~2004!

term acts as an electrostatic potential when the SO intenis varying in space. Actually, a different SO Hamiltonianwhich this offset were not included would not lead to cofined states.

Another question of relevance refers to the magnitudethe step in the spin-orbit intensity needed to obtain confistates. For a typical size of dot radius of the order ofatomic effective units14 simulation shows that an intensity ithe rangel i.(1.5–2.0)31029 eV cm is required to obtain asingle-particle bound state when the external intensity is z(le50). It is also important to specify the external valule , since the minimum step required to achieve confinstates strongly depends on this quantity. For instance, wthe external intensity takes a value of the order ofle.231029 eV cm the step needed is reduced and the requinternal value is in the rangel i2le.(0.4–0.7)31029

eV cm.Quantum dots satisfying the required conditions sta

above show a level structure analogous to that of convtional quantum dots. For instance, a dot characterizedspin-orbit parameters in the range of the biggest Rashbatensities experimentally measured,8 l i.431029 eV cm andle.231029 eV cm, shows a total of 14 single-particbound states distributed in two subbands~see Fig. 1!. Thefirst one containing ten states characterized by the quannumbersn51, j 529/2,27/2, . . .,7/2,9/2 and the seconone having the remaining four statesn52, j 523/2,21/2,1/2,3/2. Figure 2 shows the spatial representationthe lowest energy state of this dot corresponding ton51,j51/2 and its conjugated staten51,j 521/2. The upper pan-els show the radial profile of the spinorial componentsboth states, the only differences between the two cases

FIG. 1. Structure of the bound states as a function ofgeneralized angular momentumj for a quantum dot havingle50.2H* a0* .231029 eV cm, l i50.4H* a0* .431029 eV cmand a nominal dot radiusR0510 effective atomic units~lowerscale!. Solid lines represent the bulk bands corresponding tosurrounding material characterized byle ~upper scale!.

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the exchange of the spin index and the sign change in onthe components, in agreement with symmetry propertiesrived above. Middle panel shows the spatial distributionthe density which has radial symmetry and is commonboth conjugated states. Finally, the lower panel displaysradial distribution of theSz spin density for the two statesshowing that conjugated states possess opposite spin chters.

Up to now, we have only presented results concerningsingle-particle properties of the dots. There is also the pobility of the spin-orbit-based confinement to be robuagainst electron-electron interaction and, thus, able to sport multielectron bound states. To treat the interacting prlem we impose no symmetry restriction in space andsolve numerically the full two-dimensional problem incluing the interaction within density functional theory, usinggeneralization of the local-spin-density approximationnoncollinear spin densities~see Ref. 15 for a previous applcation of this formalism!. Within this scheme, we find thaeven two-electron states can be confined to the dot. Figushows the ground state density for an interacting twelectron dot characterized by a radius of 15 units andsame spin-orbit parameters of Fig. 2. The density displaycircular ringlike shape although the spin-orbit term breathe rotational invariance. As an additional check of thbound two-electron state, we have also solved the HartFock equations, finding a similar total energy and dens

e

e FIG. 2. Upper panels: radial spinorial components correspoing to the lowest energy pair of conjugated states. Middle patwo-dimensional representation of the probability density corsponding to the above Kramers conjugated states. Lower paradial profile of thesz spin-density distribution corresponding tthe same conjugated states. All coordinates are expressed in tof atomic effective units of distance.

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BRIEF REPORTS PHYSICAL REVIEW B69, 153308 ~2004!

distributions also confined to the dot region. Extendingcalculations to treat three or more interacting electrons wthe same spin-orbit term does not lead to confined state

In summary, we have proposed a mechanism to creelectronic spatial confinement based on the spin propertie

FIG. 3. Two-dimensional representation of the LSDA denscorresponding to an interacting two-electron dot characterizedle50.2H* a0* .231029 eV cm, l i50.4H* a0* .431029 eV cmand a dot radius of 15 units. Atomic effective units of distanceused to represent the Cartesian coordinates.

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III-V semiconductor structures and, more specifically, onBychkov-Rashba spin-orbit interaction and its strong depdence on the barrier layer composition. The eigenstatesfined to these dots show common characteristics with cventional semiconductor quantum dots such as a disclevel structure of the single-particle states and the possibof confined two-electron states. The peculiar origin of tconfinement reflects in the spin properties of the singparticle eigenstates. Not any material is suitable to createtype of structures since, as stated above, Rashba spin-intensities of the order 1029 eV cm and a range of tunabilityof the same magnitude are required. In this sense, InGabased wells7 could be enough to produce the confining effeat mesoscopic scale. However, since the bulk-intensity vale has a strong relevance in the determination of the mmum conditions for confinement, InAs-based wells reveathe best candidates to fabricate these dots. This is becautheir large built-in Rashba intensities, as large as31029 eV cm, and their tunability range of the same orde8

This work was supported by Grant No. BFM2002-032from DGI ~Spain!.

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14Atomic effective units take values in the range of 10–50 nmdistance (a0* ) and 5–10 meV for energy (H* ), using typicalsemiconductor parameters, while atomic effective units of sporbit intensity are more homogeneous for the different materhaving a good approximate value of 131028 eV cm.

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