Ground state and far-infrared absorption of two-electron rings in a magnetic field

15 Pages·0.27 MB·English

by  Antonio Puente

Ground state and far-infrared absorption of two-electron rings in a magnetic field
Antonio Puente and Llorenc¸ Serra
Departament de F´ısica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain
(May 30, 2000)
1
0
spectraasafunctionofthe magneticfieldhasshownthe
0
Motivated by recent experiments [A. Lorke et al., Phys.
2 appearanceofelectronicexcitationspeculiarofaringge-
Rev.Lett.84,2223(2000)]ananalysisofthegroundstateand
ometry as well as a magnetic-field induced transition in
n far-infraredabsorptionoftwoelectronsconfinedinaquantum
a the ground state angular momentum. The two electron
ringispresented. Theheightoftherepulsivecentralbarrierin
J systemisthe smallestonewithelectron-electroninterac-
theconfining potential is shown to influencein an important
1 tion and its reduced size places it within the reachof es-
way the ring properties. The experiments are best explained
1 assumingthepresenceofbothhigh-andlow-barrierquantum sentially exact methods, such as the Hamiltonian diago-

rings in thesample. The formation of a two-electron Wigner nalizationinabasisoffunctionsmentionedabove. Inthis
]
l molecule in certain cases is discussed. paperwehavefollowedanalternativealthoughalsoexact
l
a method, based on the solution of the Schr¨odinger equa-
h PACS 73.20.Dx, 72.15.Rn
tion by discretizing the coordinate space in a uniform
-
s grid of points. The method is therefore free from basis
e selection and truncation although its limitation lies, ob-
m
I. INTRODUCTION viously,inthenumberofspatialpoints. Thisschemecan
t. also be used to obtain the dynamical properties by solv-
a
ing the time-dependent Schr¨odinger equation and moni-
m The possibility of confining a controlled number of
toring the evolution of relevant physical quantities with
electrons in artificial nanostructures at the interface of
- time. Inthis workwehaveaddressedthe linearresponse
d two semiconductors, achieved in recent years [1], has at-
n tracted much interest. An obvious reason is the poten- regime,forsmallamplitude oscillations. However,oneof
o the potentialities of the real-time approach is the study
tial application of these systems to sub-micron electron-
c of non-linear (large amplitude) dynamical processes. By
ics technology but, from the theoretical point of view,
[
muchimportanceisattributedtotheircharacterofnovel finding the exact solution one can easily quantify the ef-
1 fect of the electron-electron interaction as well as check
quantumsystemsinwhichnew physics,aswellasanalo-
v thevalidityofapproximatedescriptions,suchasdensity-
gieswithatomicclusters,atomsornuclei,maybefound.
3
functional theory. In fact, an analysis of the experiment
6 Themostwidelystudiedsystemshavebeentheso-called
based on the local spin-density approximation (LSDA)
1 ’quantum dots’, in which a certain number of electrons
1 form a two-dimensional compact island, modelled by a inasymmetry-restrictedapproachhasalreadybeenpre-
0 sented in Ref. [10].
confinig potential of parabolic type (for small dots) plus
1 An important input to the theory is the external con-
the mutualelectron-electroninteractions. The electronic
0
fining potential acting on the electrons. Since we aim at
/ islands have been also formed in the shape of rings. In
t adirectcomparisonwithexperiment,thestrategyshould
a fact, mesoscopic rings were produced by Dahl et al. and
m their magnetoplasmon resonances measured for different be to introduce a reasonablemodel potential, depending
only on a few parameters and explore the exact solution
- ring widths [2]. Subsequent theoretical models, using
d as a function ofthis parameters. Our approachhas been
semiclassical methods [3] and density functional theory
n to takethe parabolaparameterω andring radiusR as
[4] provided a good interpretation of the measured ring 0 0
o
guessed in Ref. [9] and explore the effect of the central
c spectra. Therealsoexistapproachesbasedonexactdiag-
repulsive barrier by solving for a low-, an intermediate-
: onalization methods for very small numbers of electrons
v
and a high-barrier ring. With these barriers the system
which have addressed the so called ’persistent current’
i
X changesfromacompactdot-likestructuretoawelldevel-
[5] and the optical excitations for very narrow rings (ap-
r proachingthe1dlimit)aswellasfordotswitharepulsive oped ring one. We show that the barrier height modifies
a theringpropertiesandthattheexperimentsarebestex-
scatterer center [6,7]. Besides, a description of the opti-
plained assuming a mixed sample containing both low-
calpropertiesintwoelectronringsbasedonananalytical
and high-barrier rings.
treatment of the ground state was given by Wendler et
InrecentpapersYannouleasandLandman[11],aswell
al. [8].
as Koskinen et al. [12] have discussed the electronic lo-
Very recently, Lorke and co-workers reported the fab-
calization, in the relative frame, known as the forma-
rication,using self-assemblytechniques, ofelectronrings
tion of Wigner molecules in two electron quantum dots
in the true quantum limit, not affected by random scat-
and in rings with 6 electrons, respectively. Both groups
terers,containigtwoelectrons[9]. Thefar-infrared(FIR)
have addressed the problem by calculating the low-lying
1
statesanddiscussingtheirrotor-likestructure. Thiscrys- is given by the well known singlet and triplet combina-
tallization is in good agreement with the prediction of tion of spinors. Our method to obtain the spatial wave
mean-field (MF) theories like the Hartree-Fock or the function is based on the only assumption that its gra-
Local-Density-Functional approaches. We also comment dient and Laplacian can be obtained by using finite dif-
ontheformationofaWignermoleculeinthehigh-barrier ference formulas in a spatial grid of uniformly spaced
rings by comparing the conditional probability distribu- points. Associating a macroindex I ≡ (x ,y ,x ,y ) to
1 1 2 2
tionwith the LSDA density. Inthis case,the LSDA pre- each point and dicretizing the derivative operators (we
dicts a symmetry-brokengroundstate that nicely agrees use 5 point formulas), the Schr¨odinger eigenvalue equa-
with the electronic distribution built from the condi- tion transforms into the linear problem
tional probability density of the exact solution to the
Schr¨odinger equation. HIJΨJ =EΨI . (3)
The structure of the paper is as follows. Section II is
It is worth to mention that the interaction and poten-
devoted to the exact solution for the ground state and
tial terms are diagonal (local) and only the kinetic con-
Sec. III to the corresponding exact solution for the dy-
tributiongivesnon diagonalelements. Besides, since the
namics. In Sec. IV we discuss the comparison with the
derivativesinvolveonlyafinitenumberofpoints(wehave
mean field approach and the formation of the Wigner
tipically used 5-point formulas) the Hamiltonian matrix
molecule. Finally, the conclusions are presented in Sec.
is very sparse. We have solved Eq. (3) for the ground
V.
stateusingtheiterativeimaginarytime-stepmethod[16].
In principle, one can also obtain excited states by intro-
II. GROUND STATE ducing the constraint of orthogonalization to the lower
eigenvectors,althoughthecomputationaleffortincreases
very rapidly. A very useful optimization is introduced
A. Hamiltonian and resolution method
by considering only symmetric (singlet) or antisymmet-
ric (triplet) wave functions with respect to the exchange
TheHamiltonianofthetwo-electronsystem,inamag-
of the two particle positions, since this allows to reduce
netic field, using effective units [13], is
by a factor two the wave function dimension.
1 The above method has proved to be quite stable and
H = (−i∇ +γA(r ))2+V(r )
i i i the convergence with the number of points very fast.
(cid:20)2 (cid:21)
iX=1,2 Typically, we have used from N ≈20 up to ≈50 points
1
1 γ
+ +g∗m∗ B·S, (1) for the discretization of a single dimension. Notice that
|r1−r2| 2 the wave function dimension (N14) would normally im-
ply that the Hamiltonian matrix (N4 ×N4) exceed the
where A is the vector potential, γ = e/c (we assume 1 1
storagecapability. Thisisavoidedbyfullyexploitingthe
Gaussian magnetic fields) and V(r) is the confining ex-
sparseness of the Hamiltonian and not really dimension-
ternal potential. The electronic positions are restricted
ing such a big matrix. Analogously, the imaginary-time
to the xy plane, i.e., r ≡ (x,y), and we consider a con-
step method allows us to determine the lower eigenstate
stant and perpendicular magnetic field B = Be , de-
z of this matrix which would be unfeasible by direct di-
scribed in the symmetric gauge with a vector potential
agonalization methods. We have also checked the con-
A(r) ≡ B/2(−y,x). The last piece is the Zeeman term,
vergence with the number of points used to discretize
depending on the total spin S = s + s the effective
1 2 the Laplacian operator, e.g., when going from 5 to 9
gyromagnetic factor g∗ and electronic effective mass m∗
point formulas a relative difference less than 1/103 in
[14].
the ground state energy is found.
Theringexternalpotentialismodelledbythefollowing
circularly symmetric piecewise function,
B. Results
1ω2(r−R )2 if r ≥R
2 0 0 0
V(r)= 1ω2(r−R )2[1+ . (2)
 2 c0(r−R 0)+d(r−R )2] if r <R Inthissubsectionwepresenttheresultsfortheground
0 0 0 state of rings having R =1.4 a∗ and ω =1.1 H∗. The
0 0 0
 correspondingphysicalvalues are, approximately,14 nm
In this expression R is the ring radius and ω gives the
0 0
and13 meV and aretakenas reasonablevalues from the
curvature around the minimum at R . Fixing a barrier
0
experiment. Three different barrier heights are consid-
height V at the origin determines the parameters c and
b ered V = 0.75, 3 and 12 H∗. As shown in Fig. 1 these
d, with the additionalrequirementof smootheness at R b
0
three values correspond, going from low to high barrier,
[15].
to decreasingcentraldensities. Inwhat followsthe three
Since Hamiltonian (1) depends on spin only through
rings will be referred to as V1, V2 and V3 for increasing
the Zeeman term, the spin part of the wave function
2
barrierheight,respectively. WhileringV1hasacompact is neglected. Due to the increasing effect of the con-
density because of its rather low barrier, V3 has a real fining potential the difference is higher for V1 than for
ring shape, with a hollow central region. V3. Differences in angular momentum are also higher
Figure 2 displays the expectation value of the angular for the low barrier ring. This is indicating that in well
momentum hL i as well as the energy of the lowest level developedringsthe interactioneffects are less important
z
for both singlet and triplet states as a function of the than in more compact structures, in agreement with the
magnetic field. At B = 0 the ground state is a singlet findings of Ref. [5] and with the obvious expectation of
and with increasing B there is an alternation of triplet a larger V keeping the electrons more apart from each
b
and singlet ground states in the three rings. The first other than a lower barrier.
singlet-triplettransitionoccursapproximatelyat3,2and We also realize that the triplet solution is unfavoured
1 T for V1, V2 and V3, respectively. The triplet-singlet in the non-interacting case. In V1 the gs has always
energydifferencesremainquitesmallafterthefirstcross- S = 0, except for B = 8 T for which singlet and triplet
ing which hints at possible transitions induced by small areessentiallydegenerate. Asimilarthinghappensinthe
non-spin conserving perturbations like spin-orbit inter- otherrings,withthedifferencethatthesinglet-tripletde-
actions or magnetic impurities. We also notice how the generacy points appear at lower B’s. These differences
groundstateincreasesinstepsitsangularmomentum(in withrespecttotheexactresults,inwhichclearersinglet-
absolute value) as a function of B. This reflects the ex- triplet oscillations are seen, indicate the importance of
actpropertythatangularmomentumisagoodquantum the Coulomb interaction for a proper ground state de-
numbersincetheHamiltoniancommuteswiththeL op- scription.
z
erator. The angularmomentumgainwith magneticfield
is faster for the higher barrier rings. In fact, for V3 the
evolution is markedly linear, already indicating a rotor- III. TIME INTEGRATION
like evolution for this ring. This is further discussed in
Sec. IV. The time evolution of the two-electron wave function
is given by the time-dependent Schr¨odinger equation
C. Non-interacting solution i∂ Ψ(r ,r ;t)=HΨ(r ,r ;t), (4)
1 2 1 2
∂t
When neglecting the Coulomb interaction the Hamil-
whereH istheHamiltonianoperator(1). Wehavesolved
tonian (1) separates for each particle and, therefore, a
Eq. (4), in the same spirit as for the ground state, by
solution built from single-particle (sp) orbitals with ra-
discretizing the coordinate space in a uniform grid of
dial n and angular ℓ quantum numbers is adequate [1].
points. Time is also discretized and the evolution from
Figure 3 shows the sp eigenvalues for all, low and high
time(n)to(n+1)isobtainedthroughtheunitaryCrank-
barrier, rings. In fact, from the lowest maximum of
Nicholson algorithm
the capacitance-voltage spectrum Lorke et al. obtained
thesingle-electronground-stateshiftwithmagneticfield. (n+1) (n) ∆t (n) (n+1)
Ψ =Ψ −i H Ψ +Ψ . (5)
This manifests a changein slope atB ∼8T that wasex- I I 2 IJ J J
XJ (cid:16) (cid:17)
plainedasthetransitionfromℓ=0toℓ=−1inamodel
potential. Ofcourse,thisisinagreementwiththesplev- Thisisanimplicitscheme,inwhichthe new Ψ(n+1) may
els forV1, sincethe low-barrierringessentiallycoincides
be obtained by iteration. The above algorithm allows
withthemodelpotentialusedinRef.[9]. Thefirstcross-
to perform the dynamical simulation of Eq. (4) by using
ingforV2is locatedaround4T,while asshowninFig.3
a small time step ∆t. Of course, the feasibility of this
the highbarriercase(V3) has againa change inslope at
approachtothesolutionofEq.(4)reliesontheverysmall
B ∼8T,attributedto theℓ=−1−→ℓ=−2transition,
number of constituents (two electrons) of the system.
which may indicate that the experimental sample could
In this paper we discuss the FIR absorption of two-
contain both low and high barrier rings. This point will
electronringsandthereforeourinterestwillfocussonthe
be further discussed when presenting the FIR spectra.
dipolestatescorrespondingtosmallamplitudechargeos-
Thenon-interactinggroundstatewithtwoelectronsis
cillations. The ground state wave function is perturbed
obtained by filling the lowest orbitals of the sp scheme,
byaninitial(small)rigiddisplacementinacertaindirec-
with the constraint of having a singlet or triplet spin. tion eˆ, representing the electric field direction, and the
The corresponding plots for the evolution of hL i and
z dipole moment evolution is then followed in time. The
the ground state energy as a function of magnetic field
absorption cross section is given by the frequency trans-
are displayed in Fig. 4. By comparing it with Fig. 2 we
form of the dipole signal
noticetheeffectsoftheCoulombinteractionbetweenthe
two electrons. Obviously, the non-interacting case gives ω
σ(ω)≈ dthΨ(t)|D|Ψ(t)i exp(iωt) , (6)
lower ground state energies since the Coulomb repulsion 2π (cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
where the dipole operator is D = eˆ · (r + r ). As a dant discontinuities. We also notice that when the ring
1 2
testofthe method, developedfrompreviouscalculations barrier grows (from V2 to V3) an important amount of
within mean-field approaches, we have checked that the strengthis placed in the interval 15-25meV at low mag-
oscillation frequencies when assuming a pure parabolic netic fields. Actually the V3 peak in this region has a
potential are given, as a consequence of the generalized strength comparable to the one at low energy ≈50%, as
Kohn theorem, by the values [17] foundexperimentally[10],whileourcalculationforV1as
well as the LSDA results of Ref. [10] yield a much lower
ω = ω2+ω2/4±ω /2, (7) ratio.
± 0 c c
q A comparison with experiment of the calculated spec-
where ω =eB/c is the cyclotron frequency. trareproducesineachsinglecasesomeoftheexperimen-
c
tal features but misses others. A plausible explanation
is that the sample contains a mixture of high and low
A. FIR spectra barrier rings. In Fig. 6 we display a superposition of the
V1 and V3 spectra in comparison with the available ex-
Figure 5 shows the absorption spectra for singlet and perimental points, assuming that the system takes the
triplet states corresponding to different magnetic fields. theoreticalgroundstate ateachmagneticfield, although
The intersection of the baseline for each spectrum with as already noticed in many cases the triplet-singlet en-
the horizontal axis indicates the magnetic field in each ergy difference is quite small. With this interpretation,
case. Notice that the horizontal scale for each spectrum the circles are attributed to high barrier rings while the
is arbitrary, and therefore, it is not given by the hori- triangles and rombuses would correspond to low-barrier
zontal axis. The spectra are drawn in linear scale and ones. Although the positions of the experimental peaks
arbitrary units while the vertical scale gives the energies are not perfectly reproduced (this would involve a fur-
inmeV.NotsurprisinglytheV1spectrumisverysimilar ther optimization with respect to ω and R ) the qual-
0 0
tothatofcompactdots,withtwobranchesofoppositeB- itative agreement is good. A richer mixture including
dispersionwhich merge at B =0. This is indeed evident intermediate-barrierringswouldinduce ablurringofthe
bycomparingthespectrawiththeanalyticalexpressions differentbrancheswiththeintroductionofadditional(in-
(7) (dashed lines), where an empirical ω value fixed to termediate)peaks,butwouldalsoyieldtheabovediscus-
0
the lowest B =0 peak energy has been used. Except for sionandcomparisonwithexperimentmoreinvolved. The
somefragmentationdetailsthesetwobranchesarerather crosses in Fig. 6 are not reproduced by any of our ring
similar for both singlet and triplet cases. More impor- calculations and, as hinted by the experimental group,
tantly, we want to stress that essentially no strength is they could be due to pure quantum dots in the sample.
contained in the energy region 15 meV < ω < 25 meV Wealsomentionthatintheexperiment[9]aninsufficient
for B <6 T. This fact does not imply that there are no signal to noise ratio at low energies does not allow the
excitations in this region. In fact they can be seen using detection of the lower branch at ω <10meV.
logarithmicscale,ashasbeendoneintheLSDAapproach
of Ref. [10]. We have sticked however to the linear scale
firstlybecausethereal-timeapproachrequiresextremely B. Non-interacting spectra
long simulation times in order to extract so low intense
frequencies, and secondly we do not think that the ex- Figure7showsthesingletandtripletspectrawhenthe
periments can easily identify such low intensity peaks. Coulomb interaction is neglected as in subsection II.C.
TheV2 spectrumand,moreclearly,theV3 onereflect Therefore, they correspond to pure particle-hole transi-
a behaviour qualitatively different from that of V1. The tions in the single-particle level scheme. On a first look
high energy branch does not merge with the low one for they seem quite similar to the spectra of Fig. 5. How-
vanishing B, deviating from the analytical prediction of ever, after a second inspection important differences ap-
Eq.(7). This behaviouris wellknownfor quantumrings pear. A dramatic one is the softening, i.e., a decrease in
and is a manifestation of the violation of Kohn’s theo- energy, of the lowest branch in all three cases. In fact
reminthese systems[2–8]. ForV2a globalgap≈8−14 for B ≈ 8 T the lowest state has an almost vanishing
meVpersistsatallmagneticfieldsforbothspins,whilein energy. This fact is explained as the formation of quasi
the case of V3 the corresponding gap lies approximately Landau bands, even for such a small system, with the
in the region 7 − 17 meV. Therefore, this gap consti- property that single-particle states of the same band are
tutes a direct manifestation of the barrier height in the quasi-degenerate and therefore, particle-hole transitions
FIR response. The B-evolution of the spectra is char- are nearly gapless.
acterized by small jumps in the peak positions for each There are also sizeable shifts of peak positions and, in
branch which must be attributed to changes in total an- general, a higher fragmentation is present in the single
gular momentum. Since V3 is the one gaining more an- particlepicture. Thisiseasytounderstandsincethecol-
gular momentum its spectrum also displays more abun- lectivity asociated to the Coulomb interaction is absent
4
inthesecondcase,andthereforethedescriptionofcollec- The overall evolution of the total angular momentum is
tive magnetoplasmon states is quite rough. Altogether, also well reproduced in MF, especially for the triplet.
the correct qualitative result must be understood as an Quite strikingly, however,the angular momentum of the
indication that interaction effects are not overwhelming singlet does not show the step-like evolution peculiar of
for the spectrocopic features of a two-electronsystem. theexactsolution. Infactitistakinginmostcasesclear
non-integervalues. This mustbe attributedto a sponta-
neous breaking of the rotational symmetry in the frame
IV. MEAN FIELD THEORY AND of the mean field.
BROKEN-SYMMETRY SOLUTION ThesymmetryoftheMFsolutionismoreeasilyappre-
ciatedinFig.9, thatshowsthe MFgroundstate density
Usually, the exact solution of the many-body ρ = ρ +ρ and magnetization m = ρ −ρ , where ρ ,
↑ ↓ ↑ ↓ ↑
Schr¨odinger equation is not possible and one must con- ρ are the two spin densities, for V1 and V3 at B = 0.
↓
tent withapproximatesolutions. Inthis sense,the mean Both cases correspondto singlet solutions but, while V1
field aproximation allows in many cases the description hasacircularlysymmetricdensityandvanishingmagne-
of relevant physics. Although in principle the mean field tization,V3isshowingaseparationofspinupanddown
provides an accurate picture for a big enough number of densities, leading to an oscillation of the magnetization
particles, it is interesting to compare it with the exact along the ring perimeter (a spin density wave)as well as
solution in the present case for two particles. This will of the total density (a charge density wave). This be-
provide a strong test of the mean field result, that will haviour is in fact common to all singlet states at B > 0
allow us to quantify its validity. We remarkthat for sys- for which hL i takes non-integer values. This peculiar
z
tems confined in an external potential such as atomic or states were predicted by Reimann et al. [26] in narrow
quantum dot electrons the restriction on having a suffi- rings with a bigger number of electrons, and have been
cient number of particles in order to develop the mean also studied by us in Ref. [27].
fieldis not asstrictas inself-boundsystems,suchas liq- The formation of broken-symmetry solutions within
uid drops or atomic nuclei. This is easy to understand MF approaches is a well known phenomenon in nuclear
since in confined systems the fixed external potential is physics[28]. Inthisfield,ithasbeenshownthatthe MF
alreadyanimportantcontributiontothemeanfieldand, is in fact providing the system structure in an intrinsic
therefore it is not solely based on particle-particle inter- reference frame, given by proper intrinsic coordinates,
actions. while there is degeneracy with respect to the remain-
Imposing circular symetry the mean field theory has ing collective coordinates. In a simplifying picture this
indeed been applied to medium-large quantum dots corresponds to assume a certain structure of the system
(N ≈ 20−200), in the Hartree [18], Hartree-Fock [19] relative to its mean field, that in turn may be moving
and LSDA schemes [20,21]. Nevertheless when using thus leading to a restoration of the exact symmetry. In
symmetry-unrestricted approaches, even at the mean fact, asshownin Fig.1, whenperforminganangularav-
field level, it is normally necessary to restrict to smaller erage of the MF density for V3 an excellent agreement
sizes (N ∼ 10) because of the computational require- with the exact result is obtained. As we discuss further
ments [22–25]. in the next subsection, the MF symmetry breaking can
be interpreted as an electronic crystallization, i.e., the
formation of a Wigner molecule.
A. LSDA ground state
WehaveappliedtheLSDAsymmetry-unrestrictedap- B. The Wigner molecule
proach developed by us in Ref. [23] to the present two-
electron rings. Figure 8 summarizes the ground state The symmetry breaking of the MF solution for V3 is
properties, for both singlet and triplet for V1, V2 and pointing an incipient electron localization in the intrin-
V3. The comparison with the energies for the exact so- sic frame of this system. The formation of these local-
lution (Fig. 2) provethat the the mean field is providing ized structures has been recently discussed using exact
a quantitativelygooddescriptionofthe groundstate en- solutions by Yannouleas and Landman for two electron
ergy, with relative energy differences around 2%. We parabolic dots [11] and, for rings with 6 electrons by
notice that the MF description of the triplet is slightly Koskinen et al. [12]. These authors have computed the
better than that of the singlet. In fact, after the first exact excitation spectrum and shown that it adjusts to
singlet-triplet crossing the MF fails to reproduce a sin- rotor bands, thus revealing the existence of the electron
glet state more bound than the triplet one. This can be molecule.
attributed to a more importanteffect of correlationsdue Evenwhentheexactwavefunctionisknownitisnota
to the interactionin the singletthan in the tripletwhich trivial task to notice the existence of a Wigner molecule
areonlyapproximatelytakenintoaccountwithinLSDA.
5
intheintrinsicframe. Itcanbehintedbymeansofintrin- aretheanalogofEq.4withintheLSDA.Wehaveapplied
sicdistributionfunctions,liketheconditionalprobability thistechniquetothepresenttwo-electronringsandshow
density ρ(r ,θ |r = ξ), which gives the probability of the corresponding singlet and triplet spectra for V1, V2
2 21 1
findinganelectronat(r ,θ )whentheotheroneisfixed and V3 in Fig. 11.
2 21
at r = ξ. Therefore, it provides valuable insight on the The TDLSDA provides a rather good description of
1
intrinsic structure of the two-electron system. In Ref. the rings’dipole spectra, as comparedto the exact ones.
[29] Berry reviews the use of the conditional probability We notice that the overall evolution with B and the lo-
density to describe the formation of electron molecules cation of the strength is quite well reproduced for each
inthe doublyexcitedstatesoftheheliumatom. Inwhat branch. However, finer details of the spectra, such as
follows we show that a similar analysis can be used for thefragmentationstructuresarenotreproducedinmany
the two-electron rings. cases. Althoughbasedonasingle-particlepicturetheun-
Figure 10 shows the conditional probability density restrictedTDLSDA correctsthe lowerbranchinstability
when the first electron is placed at its most probable found in Fig. 7. This is a manifestation of the interac-
value for the radial coordinate in both V1 and V3 con- tion effects taken into account by the TDLSDA. From
fining potentials. Since this position is taken as the ori- the present results we conclude that, despite the present
gin for the angle θ this implies that the first electron rings only contain two electrons, the TDLSDA provides
21
is placed in those graphs at (ξ,0), with ξ = 1.5, 1.8 a∗ a rather sensible picture of the dipole oscillations.
0
for V1 and V3, respectively. Notice that the two dis-
tributions yield a peak for the second electron that lies
opposite to the first one, i.e., at (−ξ,0). On a first look V. CONCLUSIONS
both functions look rather similar although a closer in-
spection reveals that the inner hole is more pronounced In this work we have theoretically analyzed two-
in the case of V3. This is in agreement with a stronger electronringsmotivatedbyrecentmeasurementsoftheir
localization of the second electron in this case. In or- FIR absorption. We have found that in order to explain
der to manifest more clearly this result it is helpful to the experimental spectra it is necessary to assume that
construct the density in the intrinsic frame and com- the sample contained a mixture of rings with different
pare it with the MF one. To this end we assume that barrierheigths. As expected,highbarrierringsarechar-
oncethemostprobablerelativepositionsaredetermined, acterizedbyaclearviolationofKohn’stheorem,showing
(x,y)=(ξ,0)and(−ξ,0),theintrinsicframedensitycan asizeableamountofstrengthatenergiesω >15meVfor
approximately be obtained by centering in each position B < 6 T. On the other hand, low barrier rings approxi-
theconditionalprobabilitydensity. Thisprocedureleads mately fulfill Kohn’s theorem, with an effective ω , and
0
to the two lower plots of Fig. 10. Quite interestingly, have lower energy excitation branches more similar to
the V3 result is clearly breaking the circular symmetry themeasuredones. Amixtureofthesetwotypesofrings
and both are in excellent agreement with the mean field has been used to explain the main features of the exper-
densities of Fig. 9. iment. A more detailed analysis would require includ-
Althoughqualitativetheprecedingdiscussionsupports ingintermediatebarrierrings(ideallyknowingtheactual
the interpretation that the MF describes the intrin- barrier-height distribution in the sample), for which the
sic structure of the ring. Consequently, the existence spectra lie in between the preceding two extreme situ-
of a high barrier in V3 produces an incipient Wigner ations. Despite the very small number of constituents,
molecule. However, it remains as an interesting task for these systems possess a rich FIR spectrum with an in-
the future to obtainthe fullsetofexcitedstatesanddis- trincate B-dependence.
cussitsrotor-likecharacter,followingRefs.[11,12]aswell The evolution of angular momentum and total spin
as the analytical expressions of Ref. [8]. withbarrierheightandmagneticfieldhasbeendiscussed
andcontrastedwithindependent particlepredictions. In
agreement with previous studies we find that interac-
C. Unrestricted time-dependent LSDA (TDLSDA) tioneffectsbecomelessimportantforincreasinglynarrow
rings, obviously reflecting a bigger electronic separation
Based on the mean-field picture, the TDLSDA al- thaninacompactstructureandacorrespondingincrease
lows to describe the collective dipole oscillations. These in angular momentum.
scheme has been used to describe both spin and density The comparison with the symmetry-unrestricted
excitations in quantum dots and rings in a symmetry- LSDAhasallowedustodiscusstheformationofanincip-
restrictedapproach[30,10]. InRef. [23]wedescribedthe ientWignermoleculefortheV3caseand,ingeneral,has
application of TDLSDA relaxing the constraint of circu- shown the good quality of this approximation. We have
lar symmetry and using a real-time approach similar to proposed a method to obtain the density in the intrin-
that of Sec. III. Within TDLSDA this implies the solu- sic reference frame, based on the conditional probability
tionofthe time-dependentKohn-Shamequations,which density,thatquantitativelyagreeswiththeLSDAresult.
6
The FIR absorptionin TDLSDA providesa goodoverall [21] M. Koskinen,M. Manninen, S.M. Reimann, Phys. Rev.
strengthdistribution and dispersionwith magnetic field, Lett. 79, 1389 (1997).
althoughitmissessomefinerdetailsofthefragmentation [22] C. Yannouleas and U. Landman, Phys. Rev. Lett. 82,
patterns. 5325 (1999).
[23] A. Puente and Ll. Serra, Phys. Rev. Lett. 83, 3266
(1999).
[24] C. A. Ullrich and G. Vignale, Phys. Rev. B 61, 2729
ACKNOWLEDGMENTS
(2000).
[25] I.Magnu´sd´ottirandV.Gudmundsson,Phys.Rev.B60,
ThisworkhasbeenperformedunderGrantNo.PB98-
16591 (1999).
0124 from DGESeIC, Spain. [26] S. M. Reimann, M. Koskinen, M. Manninen, Phys. Rev.
B 59, 1613 (1999).
[27] M.Val´ın-Rodr´ıguez,A.Puente,Ll.Serra,LANLpreprint
cond-mat/0006194; Eur. Phys. J. D 2000 (in press).
[28] P. Ring and P. Schuck, The nuclear many-body problem
(Springer-Verlag, NY,1990).
[29] R. S.Berry, Contemporary Phys. 30, 1 (1989).
[1] See, for instance, D.Bimberg, M. Grundmannand N.N. [30] Ll. Serra et al., Phys. Rev. B 59, 15290 (1999); Eur.
Ledentsov,Quantumdotheterostructures, (Wiley,Chich- Phys. J. D 9, 643 (1999).
ester, 1999); L. Jacak, P. Hawrylak, A. W´ojs, Quantum
dots (Springer,Berlin, 1998).
[2] C. Dahl, J. P. Kotthaus, H. Nickel, W. Schlapp, Phys.
0.15
Rev.B 48, 15480 (1993). 12)*
[3] E. Zaremba, Phys. Rev.B 53, 10512 (1996). H
[4] A.Emperador et al.,Phys. Rev.B 59, 15301 (1999). 8 al (
[5] T(1.9C94h)a.krabortyandP.Pietil¨ainen,Phys.Rev.B50,8460 -2*) 0.10 04 potenti
0
a
[6] VLe.tHt.a3lo3n,e3n7,7P(.19P9i6e)t.il¨ainen, T. Chakraborty, Europhys. sity ( 0 1 2radiu3s 4 5
n
[7] K.Niemel¨a,P.Pietil¨ainen,P.Hyv¨onen,T.Chakraborty, de 0.05
Europhys.Lett. 36, 533 (1996).
[8] L.Wendler,V.M.Fomin,A.V.Chaplik,A.O.Govorov,
Phys.Rev.B 54, 4794 (1996).
[9] A.Lorke et al.,Phys. Rev.Lett. 84, 2223 (2000) 0.00
[10] A. Emperador, M. Pi, M. Barranco, A. Lorke, LANL 0 1 2 3 4 5
preprint cond-mat/9911127. radius (a *)
0
[11] C. Yannouleas, U. Landman, LANL preprint cond-
mat/0003245.
[12] M. Koskinen, M. Manninen, B. Mottelson, S. M. FIG.1. Electrondensityforthethreerings(symbols)asa
Reimann,LANL preprint cond-mat/0004095. functionofr. Noticethattheexactsolutionalwaysyieldscir-
[13] Theusualeffectiveunitssystemisdefinedby¯h=e2/ǫ= cularlysymmetricdensities. Inorderofdecreasingdensityat
m = 1, where ǫ is the dielectric constant and m is the r =0 the results correspond to V1, V2 and V3, respectively.
effective electron mass (given in terms of the bare mass Also shown in this plot are the LSDA densities (solid lines),
∗
measm=m me).TakingtheGaAsvaluesǫ=12.4and circularly averaged in thecase of brokensymmetry(V3). In-
∗ ∗
m =0.067wegettheeffectivehartreeunitH ≈12meV set: External potential V(r) for rings V1 (solid), V2 (dash)
and effective Bohr radius a∗0 ≈98 ˚A. and V3 (dot-dash).
∗
[14] For bulk GaAs the effective gyromagnetic factor is g =
−0.44.
[15] For a barrier height Vb the condition of continuity up to
thesecondderivativeofthepotentialatR0givesc=(2−
4ǫb)/R0 and d=(1−3ǫb)/R02, where ǫb =2Vb/(ω02R02).
[16] S. E. Koonin, D. C. Meredith, Computational Physics
(Addison-Wesley,1995), Sec. 7.4.
[17] P.A.MaksymandT.Chakraborty,Phys.Rev.Lett.65,
108 (1990).
[18] D. A. Broido, K. Kempa and P. Bakshi, Phys. Rev. B
42, 11400 (1990).
[19] V.GudmundssonandR.R.Gerhardts,Phys.Rev.B43,
12098(1991); V.GudmundssonandJ.J.Palacios, Phys.
Rev.B 52, 11266 (1995).
[20] M. Pi et al., Phys.Rev.B 57, 14783 (1998).
7
FIG.3. Dependenceofthesingleparticleenergylevelswith
2.5 SS == 01 Egs << LLzz>> -10 B for the three potentials. The different branches, taken in
vertical order at B = 1T, correspond for all three cases to
-2
2.0 -3 (nℓ)=(10),(1−1),(11),(1−2),(1−3),wheren(radial)and
-4 ℓ (z-component of angular momentum) are the usual quan-
V1
1.5 -5 tumnumbers. Eachbranchpresentsasmallspinsplittingfor
increasing B, with the spin up states placed below the spin
0
down ones.
-1
2.5
-2
-3
2.0 V2 -4 2.0 S = 0 Egs < Lz> 0
-5 S = 1 -1
-2
0 1.5
3.0 -3
-1
-4
-2 1.0 V1
-5
2.5 -3
2.5
V3 -4 0
V
3 -5 -1
2.0 2.0
-2
0 2 4 6 8 10 12 0 2 4 6 8 10 12
-3
B (T) B (T)
1.5 -4
V2
-5
3.0
0
FIG.2. Evolution with magnetic field ofthelowest singlet
-1
and triplet states for the three rings. Left panels show the 2.5
∗ -2
state energy (in H ) and right ones its corresponding total
-3
aunngituslaorf ¯hm.omentum in the perpendicular direction hLzi, in 2.0 V V3 -4
3 -5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
B (T) B (T)
2.0 V1
FIG.4. Same as Fig. 2 for thenon-interacting case.
1.5
FIG. 5. Evolution of the FIR spectrum for singlet and
1.0
triplet ground states in the three rings. The results for V1,
V2andV3aredisplayedinthethreerowsfromtoptobottom
0.5
whileleft andright columnscorrespond tosinglet andtriplet
states, respectively. For comparison, the dashed lines show
2.0 V2
the analytical predicition from Kohn’s theorem (7), see Sec.
)
*H 1.5 III.A.
(
y
g
r 1.0
e
n FIG.6. Superposition of V1 (solid lines) and V3 (dashed)
E
spectra at different magnetic fields. The experimental data
0.5
are taken from Lorke et al. [9]. See text (Sec. III.A) for
2.5
details.
V3
2.0
1.5 FIG.7. Same as Fig. 5 for thenon-interacting case.
1.0
0.5
0 2 4 6 8 10 12
B (T)
8
2.5 SS == 01 Egs < Lz> -10
-2
2.0 -3
-4
V1
1.5 -5
3.0
0
-1
2.5 -2
-3
2.0 V2 -4
-5
0
3.0 -1
-2
-3
2.5
-4
V3 -5
2.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12
B (T) B (T)
FIG.8. Same as Fig. 2 within theLSDA.
FIG. 9. Results obtained within the LSDA for B = 0 T:
LowerplotsdisplaythedensityρandmagnetizationmforV3
while the upper one shows the density for V1. In the latter
case themagnetization vanishes everywhere.
FIG. 10. Upper plots: conditional probability densities
(CPD) ρ(r2,θ21|r1 = ξ) obtained from the exact wave func-
tion at B = 0 T. The first electron has been fixed at (ξ,0),
∗
indicated by a solid dot, where ξ = 1.5,1.8 a0 is the most
probable radius for a single electron, in V1 and V3 respec-
tively. Lower plots: Intrinsic-frame density built from the
upper conditional probability densities as explained in the
text (Sec. IV.B).
FIG.11. Same as Fig. 5 within theLSDA.
9
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