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Spin qubits in antidot lattices

Pedersen, Jesper Goor; Flindt, Christian; Mortensen, Niels Asger; Jauho, Antti-Pekka

Published in:Physical Review B Condensed Matter

Link to article, DOI:10.1103/PhysRevB.77.045325

Publication date:2008

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Pedersen, J. G., Flindt, C., Mortensen, N. A., & Jauho, A-P. (2008). Spin qubits in antidot lattices. PhysicalReview B Condensed Matter, 77(4), 045325. https://doi.org/10.1103/PhysRevB.77.045325

https://doi.org/10.1103/PhysRevB.77.045325https://orbit.dtu.dk/en/publications/7b7823ab-de51-4e87-b51f-a2d1ce18f1f7https://doi.org/10.1103/PhysRevB.77.045325

Spin qubits in antidot lattices

Jesper Pedersen,1 Christian Flindt,1 Niels Asger Mortensen,1 and Antti-Pekka Jauho1,21MIC – Department of Micro and Nanotechnology, NanoDTU, Technical University of Denmark, Building 345 East,

DK-2800 Kongens Lyngby, Denmark2Laboratory of Physics, Helsinki University of Technology, P. O. Box 1100, FI-02015 HUT, Finland

�Received 10 August 2007; revised manuscript received 28 September 2007; published 22 January 2008�

We suggest and study designed defects in an otherwise periodic potential modulation of a two-dimensionalelectron gas as an alternative approach to electron spin based quantum information processing in the solid-stateusing conventional gate-defined quantum dots. We calculate the band structure and density of states for aperiodic potential modulation, referred to as an antidot lattice, and find that localized states appear, whendesigned defects are introduced in the lattice. Such defect states may form the building blocks for quantumcomputing in a large antidot lattice, allowing for coherent electron transport between distant defect states in thelattice, and for a tunnel coupling of neighboring defect states with corresponding electrostatically controllableexchange coupling between different electron spins.

DOI: 10.1103/PhysRevB.77.045325 PACS number�s�: 73.22.�f, 75.30.Et, 73.21.Cd, 03.67.Lx

I. INTRODUCTION

Localized electrons spins in a solid state structure havebeen suggested as a possible implementation of a future de-vice for large-scale quantum information processing.1 To-gether with single spin rotations, the exchange coupling be-tween spins in tunnel coupled electronic levels wouldprovide a universal set of quantum gate operations.2 Re-cently, both of these operations have been realized in experi-ments on electron spins in double quantum dots, demonstrat-ing electron spin resonance �ESR� driven single spinrotations3 and electrostatic control of the exchange couplingbetween two electron spins.4 Combined with the long coher-ence time of the electron spin due to its weak coupling to theenvironment, and the experimental ability to initialize a spinand reading it out,5 four of DiVincenzo’s five criteria6 forimplementing a quantum computer may essentially be con-sidered fulfilled. This leaves only the question of scalabilityexperimentally unaddressed.

While large-scale quantum information processing withconventional gate-defined quantum dots is a topic of ongoingtheoretical research,7 we here suggest and study an alterna-tive approach based on so-called defect states that form atdesigned defects in a periodic potential modulation of a two-dimensional electron gas �2DEG� residing at the interface ofa semiconductor heterostructure.8 One way of implementingthe potential modulation would be similar to the periodicantidot lattices9,10 that are now routinely fabricated. Suchlattices can be fabricated on top of a semiconductor hetero-structure using local oxidation techniques that allow for aprecise patterning of arrays of insulating islands, with a spac-ing on the order of 100 nm, in the underlying 2DEG.11 Eventhough the origin of these depletion spots is not essential forour proposal, we refer to them as antidots, and a missingantidot in the lattice as a defect. Alternative fabrication meth-ods include electron beam and photolithography.12,13 In Ref.11 a square lattice consisting of 20�20=400 antidots waspatterned on an approximately 2.5 �m�2.5�m area, and theavailable fabrication methods suggest that even larger antidotlattices with more than 1000 antidots and many defect statesmay be within experimental reach.

The idea of using designed defects in antidot lattices as apossible quantum computing architecture was originally pro-posed by some of us in Ref. 8, where we presented simplecalculations of the single-particle level structure of an antidotlattice with one or two designed defects. Here, we take theseideas further and present detailed band structure and densityof states calculations for a periodic lattice, describe a reso-nant tunneling phenomenon allowing for electron transportbetween distant defects in the lattice, and calculate numeri-cally the exchange coupling between spins in two neighbor-ing defects, showing that the suggested architecture could beuseful for spin-based quantum information processing. Theenvisioned structure and the basic building blocks are shownschematically in Fig. 1.

The paper is organized as follows. In Sec. II we introduceour model of the antidot lattice and present numerical resultsfor the band structure and density of states of a periodic

a b

c d

Antidots2DEG

d

Λ

S w

FIG. 1. �Color online� �a� Schematic illustration of a periodicantidot lattice; antidots may, e.g., be fabricated using local oxida-tion of a Ga�Al�As heterostructure. �b� Geometry of the periodicantidot lattice with the Wigner-Seitz cell marked in gray and theantidot diameter d and lattice constant � indicated. �c� A designeddefect leads to the formation of defect states in which an electronwith spin S can reside. �d� Tunnel coupled defects. The couplingcan be controlled using a split-gate with an effective opening de-noted w.

PHYSICAL REVIEW B 77, 045325 �2008�

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antidot lattice. In particular, we show that the periodic poten-tial modulation gives rise to band gaps in the otherwise para-bolic free electron band structure. In Sec. III we introduce asingle missing antidot, a defect, in the lattice and calculatenumerically the eigenvalue spectrum of the localized defectstates that form at the location of the defect. We develop asemianalytic model that explains the level structure of thelowest-lying defect states. In Sec. IV we consider two neigh-boring defect states and calculate numerically the tunnel cou-pling between them. In Sec. V we describe a principle forcoherent electron transport between distant defect states inthe antidot lattice, and illustrate this phenomenon by wavepacket propagations. In Sec. VI we present numerically exactresults for the exchange coupling between electron spins intunnel coupled defect states, before we finally in Sec. VIIpresent our conclusions.

II. PERIODIC ANTIDOT LATTICE

We first consider a triangular lattice of antidots with lat-tice constant � superimposed on a two-dimensional electrongas �2DEG�. The structure is shown schematically togetherwith the Wigner-Seitz cell in Fig. 1�b�. While experiments onantidot lattices are often performed in a semiclassical regime,where the typical feature sizes and distances, e.g., the latticeconstant �, are much larger than the electron wavelength, wehere consider the opposite regime, where these length scalesare comparable, and a full quantum-mechanical treatment isnecessary. In the effective-mass approximation we thusmodel the periodic lattice with a two-dimensional single-electron Hamiltonian reading

H = −�2

2m*�r

2 + �i

V�r − Ri�, r = �x,y� , �1�

where m* is the effective mass of the electron and V�r−Ri�is the potential of the ith antidot positioned at Ri. We modeleach antidot as a circular potential barrier of diameter d sothat V�r−Ri�=V0 for �r−Ri � �d /2 and zero otherwise. Inthe limit V0→� the eigenfunctions do not penetrate into the

antidots, and the Schrödinger equation may be written as

− �2�r2�n�r� = n�n�r� , �2�

with the boundary condition �n=0 in the antidots, and wherewe have introduced the dimensionless eigenvalues

n = En�22m*/�2. �3�

In the following we use parameter values typical of GaAs,for which �2 /2m*�0.6 eV nm2 with m*=0.067me, althoughthe choice of material is not essential. We have checked nu-merically that our results are not critically sensitive to theapproximation V0→�, so long as the height is significantlylarger than any energies under consideration. All results pre-sented in this work have thus been calculated in this limit,for which the simple form of the Schrödinger equation Eq.�2� applies. In this limit, the band structures presented beloware of a purely geometrical origin. The band structure can becalculated by imposing periodic boundary conditions andsolving Eq. �2� on the finite domain of the Wigner-Seitz cell.We solve this problem using a finite-element method.14 Thecorresponding density of states is calculated using the lineartetrahedron method in its symmetry corrected form.15–17

In Fig. 2 we show the band structure and density of statesof the periodic antidot lattice for three different values of therelative antidot diameter d /�. We note that an increasingantidot diameter raises the kinetic energy of the Bloch statesdue to the increased confinement and that several band gapsopen up. We have indicated the gap eff below which nostates exist for the periodic structure. We shall denote asband gaps only those gaps occurring between two bands, andthus we do not refer to the gap below eff as a band gap inthe following. This is motivated by the difference in the un-derlying mechanisms responsible for the gaps: While theband gaps rely on the periodicity of the antidot lattice, simi-lar to Bragg reflection in the solid state, the gap below effrepresents an averaging of the potential landscape generatedby the antidots, and is thus robust against lattice disorder aswe have also checked numerically.18 The lowest band gap isthus present for d /��0.35 while the higher-energy bandgap only develops for d /��0.45. As the antidot diameter is

��

��

40

80

120

160

��

��

��

��

��

��

��

50

100

150

200

��

��

��

��

��

��

��

��

��

��

100

200

300

��

��

��

��

��

d

Λ

Γ KM

d/Λ = 0.25 d/Λ = 0.5 d/Λ = 0.75

� n

Γ MK Γ g(�) Γ MK Γ g(�) Γ MK Γ g(�)ϑeff{

�

�

FIG. 2. Band structures and densities of states g�� of the periodic antidot lattice for three different values of the relative antidot diameterd /�. Notice the different energy scales for the three cases. On each band structure the gap eff is indicated, below which no states exist forthe periodic lattice. The band gaps and the gap below eff are highlighted as hatched regions. Also shown is the periodic lattice structure withthe Wigner–Seitz cell indicated in gray, as well as the first Brillouin zone �FBZ� with the three high-symmetry points and the irreducible FBZindicated.

PEDERSEN et al. PHYSICAL REVIEW B 77, 045325 �2008�

045325-2

increased, several flat bands appear with �kn�k��0, givingrise to van Hove singularities in the corresponding density ofstates.

III. DEFECT STATES

We now introduce a defect in the lattice by leaving out asingle antidot. Topologically, this structure resembles a pla-nar 2D photonic crystal, and relying on this analogy we ex-pect one or more localized defect states to form inside thedefect.19 The gap eff indicated in Fig. 2 may be consideredas the height of an effective two-dimensional potential sur-rounding the defect, and thus gives an upper limit to theexistence of defect states in this gap. Similar states are ex-pected to form in the band gaps of the periodic structure,which are highlighted in Fig. 2. As defect states decay tozero far from the location of the defect, we have a largefreedom in the way we spatially truncate the problem at largedistances. For simplicity we use a super-cell approximation,but with �=0 imposed on the edge, thus leaving Eq. �2� aHermitian eigenvalue problem which we may convenientlysolve with a finite-element method.14 Other choices, such asperiodic boundary conditions, do not influence our numericalresults. The size of the supercell has been chosen sufficientlylarge, such that the results are unaffected by a further in-crease in size.

In the insets of Fig. 3�a� we show the calculated eigen-functions corresponding to the two lowest energy eigenval-ues for a relative antidot diameter d /�=0.5. As expected, wefind that defect states form that to a high degree are localizedwithin the defect. The second-lowest eigenvalue is twofolddegenerate and we only show one of the correspondingeigenstates. The figure shows the energy eigenvalues of thedefect states as a function of the relative antidot diametertogether with the gap eff. As this effective potential is in-creased, additional defect states become available and wemay thus tune the number of levels in the defect by adjustingthe relative antidot diameter. In particular, we note that ford /��0.42 only a single defect state forms. As the sizes ofthe antidots are increased, the confinement of the defectstates becomes stronger, leading to an increase in their en-ergy eigenvalues. For GaAs with d /�=0.5 and �=75 nmthe energy splitting of the two lowest defect states is approxi-mately 1.1 meV, which is much larger than kBT at sub-Kelvin temperatures, and the level structure is thus robustagainst thermal dephasing.

In Fig. 3�b� we show similar results for defect states re-siding in the lowest band gap of the periodic structure. Whilethe states residing below eff resemble those occurring due tothe confining potential in conventional gate-defined quantumdots, these higher-lying states are of a very different nature,being dependent on the periodicity of the surrounding lattice.For the band gaps, the existence of bound states is limited bythe relevant band edges as indicated in the figure. As the sizeof the band gap is increased, additional defect states becomeavailable and we may thus also tune the number of levelsresiding in the band gaps by adjusting the relative antidotdiameter.

Because the formation of localized states residing below

eff depends only on the existence of the effective potential

surrounding the defect, the formation of such states is notcritically dependent on perfect periodicity of the surroundinglattice, which we have checked numerically.18 Also, the life-times of the states due to the finite size of the antidot latticeare of the order of seconds even for a relatively small num-ber of rings of antidots surrounding the defect.8 However, thelocalized states residing in the band gaps are more sensitiveto lattice disorder, since they rely more crucially on the pe-riodicity of the surrounding lattice. Introducing disorder mayinduce a finite density of states in the band gaps of the peri-odic structure and thus significantly decrease the lifetimes ofthe localized states residing in this region.

0.2 0.4 0.6 0.8

20

40

60

80

0.5 0.55 0.6 0.65 0.760

80

100

120

140

� n� n

d/Λ

d/Λ

ϑeff

�1

�2

�3

�(K)3

�(Γ)2

a)

b)

FIG. 3. �Color online� Energy spectrum for a single defect. The�dimensionless� eigenvalues corresponding to localized states areshown as a function of the relative antidot diameter d /�. For agiven choice of �, the eigenvalues can be converted to meV usingEq. �3�. �a� Energy spectrum for defect states residing in the gapbelow eff. The full line indicates the height eff of the effectivepotential in which the localized states reside. The dotted lines arethe approximate expressions given by Eqs. �4�, �6�, and �7�. Theapproximate results for 1 are in almost perfect agreement with thenumerical calculations. �b� Energy spectrum for the defect statesresiding in the lowest band gap region. The full lines indicate theband gap edges of the periodic structure, 3

�K� and 2��, giving upper

and lower limits to the existence of bound states. The inset in bothfigures show the localized states corresponding to the two lowestenergy eigenvalues indicated by the dashed vertical lines. The ab-solute square is shown.

SPIN QUBITS IN ANTIDOT LATTICES PHYSICAL REVIEW B 77, 045325 �2008�

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In order to gain a better understanding of the level-structure of the defect states confined by eff we develop asemianalytic model for eff and the corresponding defectstates. We first note that the effective potential eff is givenby the energy of the lowest Bloch state at the point of theperiodic lattice. At this point k=0 and Bloch’s theorem re-duces to an ordinary Neumann boundary condition on theedge of the Wigner-Seitz cell. This problem may be solvedusing a conformal mapping, and we obtain the expression20

eff � �C1 + C2C3 − d/�2

, �4�

where C1�−0.2326, C2�2.7040, and C3�1.0181 are givenby expressions involving the Bessel functions Y0 and Y1. Wenow consider the limit of d /�→1 and note that in this casethe defect states residing below eff are subject to a potentialwhich we may approximate as an infinite two-dimensionalspherical potential well with radius �−d /2. The lowest ei-genvalue for this problem is 1

��=�2�0,12 / ��−d /2�2, where

�0,1�2.405 is the first zero of the zeroth order Bessel func-tion. This expression yields the correct scaling with d /�, butis only accurate in the limit of d /�→1. We correct for thisby considering the limit of d /�→0, in which we may solvethe problem using ideas developed by Glazman et al. in stud-ies of quantum conductance through narrow constrictions.21

The problem may be approximated as a two-dimensionalspherical potential well of height �2 and radius �. The low-

est eigenvalues 1��2� of this problem is the first root of the

equation

1��2�J1�1��2��

J0�1��2��= �2 − 1��2�K1

��2 − 1��2��K0��2 − 1��2��

, �5�

where Ji�Ki� is the ith order Bessel function of the first �sec-ond� kind. If the height of the potential well �2 is muchlarger than the energy eigenvalues, the first root would sim-ply be �0,1

2 . Lowering the confinement must obviously shiftdown the eigenvalue, and in the present case we find that

1��2��. By expanding the equation to first order in 1��

2�

around � we may solve the equation to obtain 1��2�

�3.221, which is in excellent agreement with a full numeri-cal solution of Eq. �5�. Correcting for the low-d /� behaviorwe thus find the approximate expression for the lowest en-ergy eigenvalue8

1 � 1�� − lim

d/�→01

�� + 1��2� = 1

��2� +�4 − d/��d/�

�2 − d/��2�0,1

2 .

�6�

A similar analysis leads to an approximate expression for thefirst excited state 2. This mode has a finite angular momen-tum of �1 and a radial J1 solution yields

2 � 2��2� +

�4 − d/��d/��2 − d/��2

�1,12 , �7�

where 2��2��7.673 is the second-lowest eigenvalue of the

two-dimensional spherical potential well of height �2 and

radius �, which can be found from an equation very similarto Eq. �5�. The first root of the first-order Bessel function is�1,1�3.832. The scaling of the two lowest eigenvalues withd /� is thus approximately the same. The approximate ex-pressions are indicated by the dotted lines in Fig. 3, and wenote an excellent agreement with the numerical results. Weremark that the filling of the defect states can be controlledusing a metallic back gate that changes the electron densityand thus the occupation of the different defect states.22

IV. TUNNEL COUPLED DEFECT STATES

Two closely situated defect states can have a finite tunnelcoupling, leading to the formation of hybridized defectstates. The coupling between the two defects may be tunedvia a metallic split gate defined on top of the 2DEG in orderto control the opening between the two defects. As the volt-age is increased the opening is squeezed, leading to a re-duced overlap between the defect states. We model such asplit gate as an infinite potential barrier shaped as shown inFig. 1�d�. Changing the applied voltage effectively leads to achange in the relative width w /� of the opening, which wetake as a control parameter in the following. If we considerjust a single level in each defect we can calculate the tunnelmatrix element as �� = �+−−� /2 where � are the eigenen-ergies of the bonding and antibonding states, respectively, ofthe double defect. In the following, we calculate the tunnelcoupling between two defect states lying below eff, but theanalysis applies equally well to defect states lying in theband gaps.

In Fig. 4 we show the tunnel matrix element �� as afunction of the relative gate constriction width w /� for threedifferent values of d /� in the single-level regime of eachdefect, i.e., d /��0.42. As expected, the tunnel couplinggrows with increasing constriction width due to the increasedoverlap between the defect states. A saturation point isreached when the constriction width is on the order of thediameter of the defect states, after which the overlap is nolonger increased significantly. An electron prepared in one ofthe defect states will oscillate coherently between the twodefect states with a period given as T=�� / ��, which forGaAs with �=75 nm, d /�=0.4, and w /�=0.6 implies anoscillation time of T�0.14 ns. A numerical wave packetpropagation of an electron initially prepared in the left defectstate is shown in Fig. 4�b�, confirming the expected oscilla-tory behavior. With a finite tunnel coupling between two de-fect states, two electron spins trapped in the defects willinteract due to the exchange coupling, to which we return inSec. VI.

V. RESONANT COUPLING OF DISTANT DEFECT STATES

With a large antidot lattice and several defect states it maybe convenient with quantum channels along which coherentelectron transport can take place, connecting distant defectstates. In Refs. 23 and 24 it was suggested to use arrays oftunnel coupled quantum dots as a means to obtain high-fidelity electron transfer between two distant quantum dots.We have applied this idea to an array of tunnel coupled de-


045325-4

fect states and confirmed that this mechanism may be usedfor coherent electron transport between distant defects in anantidot lattice.18 This approach, however, relies on precisetunings of the tunnel couplings between each defect in thearray, which may be difficult to implement experimentally.Instead, we suggest an alternative approach based on a reso-nant coupling phenomenon inspired by similar ideas used tocouple light between different fiber cores in a photonic crys-tal fiber.25,26

We consider two defects separated by a central line of Nantidots and a central back gate Vg in the region between thedefects, as shown in Fig. 5. Again, we consider defect statesresiding below eff, but the principle described here mayequally well be applied to defect states in the band gaps.Using the back gate, the potential between the two defectscan be controlled locally. If the potential is lowered below

eff, a discrete spectrum of standing-wave solutions formsbetween the two defects. In the following we denote theenergy of one of these standing-wave solutions by g, whilethe energy of the two defect states is assumed to be identicaland is denoted d. A simple three-level analysis of this sys-tem, as illustrated in Fig. 5, reveals that by tuning the back

gate so that the levels are aligned, g=d, a resonant couplingbetween the two distant defects occurs, characterized by asymmetric splitting of the three lowest eigenvalues into 0=d and �=d�2 ��, where �� is the tunnel coupling be-tween the defects and the standing-wave solution in the cen-tral back gate region. If an electron is prepared in one of thedefects states, it will oscillate coherently between the twodefects with an oscillation period of T=2�� / ��. By turningoff the back gate at time t=T /2 we may thereby trap theelectron in the opposite defect which may by situated a dis-tance an order of magnitude larger than the lattice constantaway from the other defect.

In Fig. 6 we show the numerically calculated eigenvaluesas a function of the depth �Vg� of the central potential squarewell of the structure illustrated in Fig. 5 for d /�=0.5 and acentral line of N=7 antidots separating the two defects. Con-trary to the simple three-level model, several resonances nowoccur as the back gate is lowered, corresponding to coupling

0 0.5 1 1.5 2

0.01

0.1

1

d/Lam = 0.2d/L = 0.3d/L = 0.4

|τ|

w

w/Λ

d/Λ = 0.2d/Λ = 0.3d/Λ = 0.4

a)

b)

FIG. 4. �Color online� �a� The �dimensionless� tunnel coupling�� as a function of the relative split gate constriction width w /� forthree different values of d /� in the single-level regime. For a givenchoice of �, the tunnel couplings can be converted to meV usingEq. �3�. �b� Time propagation of an electron initially prepared in theleft defect state for d /�=0.4 and w /�=0.6. The absolute square ofthe initial wave function is shown in the upper left panel. The fol-lowing panels show the state after a time span of T /8, 2T /8, and3T /8, respectively, where T is the oscillation period.

-4 -2 0 2 4-4

-2

0

2

4

�g

�d�d

ττϑeff

ϑeff–|Vg|0

(�d − �g)/τ

� n/τ

a) b)

FIG. 5. �Color online� �a� The structure considered for resonantcoupling of distant defect states; two defects separated by a centralline of N=3 antidots, with a central back gate Vg controlling thepotential square well in the region marked with dashed lines. Asimple three-level model of the system is illustrated below. �b� Theeigenvalue spectrum of the three-level model. The dashed linemarks the point of resonance.

15.8 16 16.2 16.4 16.6

8.2

8.25

8.3

8.35

� n

|Vg|FIG. 6. �Color online� Energy eigenvalues as a function of the

magnitude �Vg� of the back gate for the structure illustrated in Fig. 5for d /�=0.5 and a central line of N=7 antidots separating the twodefects. The resonances are marked with dotted lines and character-ized by a symmetric splitting of the eigenvalues.


045325-5

to different standing-wave solutions in the multileveled cen-tral region. The energy splitting at resonance is larger whenthe defect states couple to higher-lying central states due to alarge overlap between the defect states and the centralstanding-wave solution. In Fig. 7 we show a numerical timepropagation of an electron initially prepared in the left de-fect, confirming the oscillatory behavior expected from thesimple model. For GaAs and �=75 nm the results indicatean oscillation period of T�0.16 ns for the time propagationillustrated. The resonant phenomenon relies solely on thelevel alignment g=d and on the symmetry condition thatboth defect states have the same energy and magnitude oftunnel coupling to the standing wave solution in the centralregion. It is in principle independent of the number of anti-dots N separating the two defects, but in practice this range islimited by the coherence length of the sample and the factthat the levels of the central region grow too dense if Nbecomes large.27 We have checked numerically that resonantcoupling of defect levels below eff is robust against latticedisorder.18

VI. EXCHANGE COUPLING

So far we have only considered the single-particle elec-tronic level-structure of the antidot lattice. However, as men-tioned in the Introduction, the exchange coupling betweenelectron spins is a crucial building block for a spin basedquantum computing architecture, and in fact suffices toimplement a universal set of quantum gates.28 The exchangecoupling is a result of the Pauli principle for identical fermi-ons, which couples the symmetries of the orbital and spindegrees of freedom. If the orbital wave function of the twoelectrons is symmetric �i.e., preserves sign under particle ex-change�, the spins must be in the antisymmetric singlet state,

while an antisymmetric orbital wave function means that thespins are in a symmetric triplet state. One may thereby mapthe splitting between the ground state energy ES of the sym-metric orbital subspace and the ground state energy EA of theantisymmetric orbital subspace onto an effective Heisenbergspin Hamiltonian H=JS1 ·S2, where J=EA−ES is the ex-change coupling between the two spins S1 and S2. Theimplementation of quantum gates based on the exchangecoupling requires that J can be varied over several orders ofmagnitude in order to effectively turn the coupling on andoff. In this section we present numerically exact results forthe exchange coupling between two electron spins residingin tunnel coupled defects as those illustrated in Fig. 1�d�.

The Hamiltonian of two electrons in two tunnel coupleddefects may be written as

H�r1,r2� = h�r1� + h�r2� + C�r1,r2� , �8�

where

C�r1,r2� =e2

4�r0

1

�r1 − r2��9�

is the Coulomb interaction and the single-electron Hamilto-nians are

h�ri� =�pi + eA�2

2m*+ V�ri� +

1

2g�BBSz,i, i = 1,2, �10�

where V�r� is the potential due to the antidots and thecoupled defects. As previously, we model the antidots andthe split gate as potential barriers of infinite height, and usefinite-element methods to solve the single-electron problemdefined by Eq. �10�. A Zeeman field Bẑ applied perpendicu-larly to the electron gas splits the spin states, and we choosea corresponding vector potential reading A=B�−yx̂+xŷ� /2.

In order to calculate the exchange coupling J we employ arecently developed method for numerically exact finite-element calculations of the exchange coupling:29 The fulltwo-electron problem is solved by expressing the two-electron Hamiltonian in a basis of product states of single-electron solutions obtained using a finite element method.14

The Coulomb matrix elements are evaluated by expandingthe single-electron states in a basis of 2D Gaussians,30 andthe two-particle Hamiltonian matrix resulting from this pro-cedure may then be diagonalized in the subspaces spannedby the symmetric and antisymmetric product states, respec-tively, to yield the exchange coupling. The details of thenumerical method are described elsewhere.18,29 The resultspresented below have all been obtained with a sufficient sizeof the 2D Gaussian basis set as well as the number of single-electron eigenstates, such that a further increase does notchange the results.31

In Fig. 8 we show the calculated exchange coupling for adouble defect structure. The exchange coupling varies byseveral orders of magnitude as the split gate constrictionwidth is increased, showing that electrostatic control of theexchange coupling in an antidot lattice is possible, similarlyto the principles proposed2 and experimentally realized4 fordouble quantum dots. Just as the tunnel coupling, the ex-change coupling reaches a saturation point when the split

t = 0 t = 1T/8

t = 2T/8 t = 3T/8

t = 4T/8 t = 5T/8

FIG. 7. �Color online� Numerical time propagation of an elec-tron initially prepared in the left defect of the structure illustrated inFig. 5�a� and corresponding to the results of Fig. 6 with �Vg ��16.54. The charge densities ��x ,y� are shown in the upper panels,while the lower panels show �dy��x ,y�. The oscillation period isdenoted T.


045325-6

gate constriction width is on the order of the diameter of thedefect states. This is to be expected since the exchange cou-pling in the Hubbard approximation is proportional to thesquare of the tunnel coupling.2 As illustrated in Fig. 8�b�, theexchange coupling is highly dependent on the lattice con-stant, increasing several orders of magnitude as the latticeconstant is decreased from 60 to 20 nm. This is in part due tothe overall increase in the energies of the eigenstastes and thesplitting between them with increased confinement, but alsodue to a decrease in the ratio of the Coulomb interactionstrength to the confinement strength. As the relative strengthof the Coulomb interaction is decreased, the defect states areeffectively moved closer together, resulting in an increase inthe exchange coupling.

The exchange coupling is also highly dependent on mag-netic fields applied perpendicularly to the plane of theelectrons.2 In Fig. 9 we show the exchange coupling as afunction of �c /�0 where �c=eB /m*c and we define �0= �

2m*�2. For GaAs �c /�0�0.00104 T−1 nm−2 �2B. As ex-

pected, the results of Fig. 9 are very similar to those obtainedfor double quantum dots.2,30 In all cases we note an initialtransition from the antiferromagnetic �J�0� to the ferromag-netic �J�0� regime of exchange coupling, followed by a

return to positive values of the exchange coupling at highermagnetic fields. The initial transition to negative exchangecoupling is caused by long-range Coulomb interactions.2 Asthe magnetic field is increased further, magnetic confinementbecomes dominant, compressing the orbits and thus reducingthe overlap between the single-defect wave functions. Thisleads to a strong reduction of the magnitude of the exchangecoupling. Due to the increased confinement strength forsmaller lattice constants �, these transitions occur at largermagnetic fields. The same is the case for the larger relativeantidot diameters, in which the ratio of magnetic confine-ment to confinement due to the antidots is reduced. We haveonly considered the case of a large constriction width w /�=2, since this regime of relatively large exchange coupling isthe most interesting for practical purposes. For small valuesof w /� we expect to find results similar to those obtained inthe limit of large interdot distances for double quantum dotsystems.2

VII. CONCLUSIONS

In conclusion, we have suggested and studied an alterna-tive candidate for spin based quantum information process-

0.5 1 1.5 2

0.01

0.1

d/Lambda=0.5

0.7

20 30 40 50 60

0.01

0.1

1

10 w/Lambda=1.01.5

2.5

J(m

eV)

J(m

eV)

w/Λ

Λ (nm)

d/Λ = 0.5d/Λ = 0.7

w/Λ = 2.5w/Λ = 1.5w/Λ = 1.0

a)

b)

FIG. 8. �Color online� Exchange coupling J for a double defectstructure. �a� Exchange coupling as a function of the relative splitgate constriction width w /� for two different values of the relativeantidot diameter and a lattice constant �=45 nm. �b� Exchangecoupling as a function of the lattice constant � for three differentvalues of the relative split gate constriction width.

0 5 10 15 20

-0.05

0

0.05

0.1

0.15

d/Lambda=0.5

d/Lambda=0.7

0 5 10 15 20-0.2

0

0.2

0.4 d/Lambda=0.5d/Lambda=0.7

J(m

eV)

J(m

eV)

ωc/ω0

ωc/ω0

Λ = 30 nm

Λ = 45 nm

d/Λ = 0.5d/Λ = 0.7

d/Λ = 0.5d/Λ = 0.7

a)

b)

FIG. 9. �Color online� Exchange coupling J for a double defectstructure as a function of �c /�0, where �c=eB /m*c and �0=� / �2m*�2�. Results are shown for a relative split gate constrictionwidth w /�=2, and two different values of the relative antidot di-ameter d /�. The lattice constant is �a� �=30 nm and �b� �=45 nm.


045325-7

ing in the solid-state, namely, defect states forming at thelocation of designed defects in an otherwise periodic poten-tial modulation of a two-dimensional electron gas, here re-ferred to as an antidot lattice. We have performed numericalband structure and density of states calculations of a periodicantidot lattice, and shown how localized defect states form atthe location of designed defects. The antidot lattice allowsfor resonant coupling of distant defect states, enabling coher-ent transport of electrons between distant defects. Finally, wehave shown that electrostatic control of the exchange cou-pling between electron spins in tunnel coupled defect statesis possible, which is an essential ingredient for spin based

quantum computing. Altogether, we believe that designed de-fects in antidot lattices provide several prerequisites for alarge quantum information processing device in the solidstate.

ACKNOWLEDGMENTS

We thank A. Harju for helpful advice during the develop-ment of our numerical routines, and T. G. Pedersen for fruit-ful discussions during the preparation of this manuscript.A.P.J. is grateful to the FiDiPro program of the FinnishAcademy for support during the final stages of this work.

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27 Another experimental challenge relates to the RC switching timeof the back gate which grows with length, making it harder tocontrol on short time scales.

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31 Because we use an expansion in localized single-particle productstates our method is most reliable for systems with several lo-calized single-particle states. We consequently focus on relativeantidot diameters above the single-level regime of the singledefects �d /��0.42�, see Fig. 3.


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Spin qubits in antidot lattices - DTU Research Databaseantidot lattice. In particular, we show that the periodic poten-tial modulation gives rise to band gaps in the otherwise para-bolic