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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�
1098-0121/2008/77�4�/045325�8� ©2008 The American Physical
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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
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80
120
160
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100
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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
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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
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045325-3
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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-
PEDERSEN et al. PHYSICAL REVIEW B 77, 045325 �2008�
045325-4
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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.
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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.
PEDERSEN et al. PHYSICAL REVIEW B 77, 045325 �2008�
045325-6
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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.
SPIN QUBITS IN ANTIDOT LATTICES PHYSICAL REVIEW B 77, 045325
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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.
1 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 �1998�.2 G.
Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59,
2070 �1999�.3 F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I.
T. Vink, K. C.
Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K.
Vander-sypen, Nature �London� 442, 766 �2006�.
4 J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A.
Yacoby,M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.
Gossard,Science 309, 2180 �2005�.
5 J. M. Elzerman, R. Hanson, L. H. W. van Beveren, B. Witkamp,L.
M. K. Vandersypen, and L. P. Kouwenhoven, Nature �Lon-don� 430, 431
�2004�.
6 D. P. DiVincenzo, Fortschr. Phys. 48, 771 �2000�.7 J. M.
Taylor, H. A. Engel, W. Dur, A. Yacoby, C. M. Marcus, P.
Zoller, and M. D. Lukin, Nat. Phys. 1, 177 �2005�.8 C. Flindt,
N. A. Mortensen, and A. P. Jauho, Nano Lett. 5, 2515
�2005�.9 K. Ensslin and P. M. Petroff, Phys. Rev. B 41, 12307
�1990�.
10 D. Weiss, K. Richter, A. Menschig, R. Bergmann, H.
Schweizer,K. von Klitzing, and G. Weimann, Phys. Rev. Lett. 70,
4118�1993�.
11 A. Dorn, E. Bieri, T. Ihn, K. Ensslin, D. D. Driscoll, and A.
C.Gossard, Phys. Rev. B 71, 035343 �2005�.
12 Y. Luo and V. Misra, Nanotechnology 17, 4909 �2006�.13 J. I.
Martín, J. Nogués, K. Liu, J. L. Vicent, and I. K. Schuller, J.
Magn. Magn. Mater. 256, 449 �2003�.14 We have used the COMSOL
Multiphysics 3.2 package for all finite-
element calculations. See www.comsol.com. Convergence
withrespect to the mesh size has been ensured.
15 G. Lehmann and M. Taut, Phys. Status Solidi B 54, 469
�1972�.16 J. Hama, M. Watanabe, and T. Kato, J. Phys.: Condens.
Matter 2,
7445 �1990�.17 J. Pedersen, C. Flindt, N. A. Mortensen, and
A.-P. Jauho, 28th
International Conference on the Physics of Semiconductors —
ICPS 2006, �AIP, Melville, NY, 2007�, p. 821.18 J. Pedersen,
Master’s thesis, Technical University of Denmark,
Kongens Lyngby, Denmark, 2007.19 N. A. Mortensen, Opt. Lett. 30,
1455 �2005�.20 N. A. Mortensen, J. Eur. Opt. Soc., Rapid Publ. 1,
06009 �2006�.21 L. I. Glazman, G. K. Lesovik, D. E. Khmelnitskii,
and R. I. Shek-
ter, JETP Lett. 48, 238 �1988�.22 In practice, reliable
single-electron filling may pose a serious ex-
perimental challenge which could require further optimization
ofthe architecture presented here.
23 G. M. Nikolopoulos, D. Petrosyan, and P. Lambropoulos,
J.Phys.: Condens. Matter 16, 4991 �2004�.
24 G. M. Nikolopoulos, D. Petrosyan, and P. Lambropoulos,
Euro-phys. Lett. 65, 297 �2004�.
25 M. Skorobogatiy, K. Saitoh, and M. Koshiba, Opt. Lett. 31,
314�2006�.
26 M. Skorobogatiy, K. Saitoh, and M. Koshiba, Opt. Express
14,1439 �2006�.
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.
28 D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K.
B.Whaley, Nature �London� 408, 339 �2000�.
29 J. Pedersen, C. Flindt, N. A. Mortensen, and A.-P. Jauho,
Phys.Rev. B 76, 125323 �2007�.
30 M. Helle, A. Harju, and R. M. Nieminen, Phys. Rev. B
72,205329 �2005�.
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.
PEDERSEN et al. PHYSICAL REVIEW B 77, 045325 �2008�
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