Two-electron quantum dot molecule: Composite particles and the spin phase diagram

4 Pages·0.25 MB·English

by  A. Harju

Two-electron quantum dot molecule:
Composite particles and the spin phase diagram
A. Harju, S. Siljam¨aki, and R.M. Nieminen
Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, 02015 HUT, Finland
(Dated: February 1, 2008)
2
We study a two-electron quantum dot molecule in a magnetic field by the direct diagonalization
0
0 of theHamiltonian matrix. Theground states of themolecule with thetotal spin S =0 and S =1
2 provide a possible realization for a qubit of a quantum computer. Switching between the states is
best achieved by changing the magnetic field. Based on an analysis of the wave function, we show
n
that the system consists of composite particles formed by an electron and flux quanta attached to
a
it. This picturecan also beused to explain the spin phase diagram.
J

7 PACSnumbers: 73.21.La,71.10.-w

]
l
l The progress in semiconductor technology has opened use the one of Ref. [4], namely 1m∗ω2min{(x−d/2)2+
a 2 0
h a rich field of studies focused on the fundamental y2,(x + d/2)2 + y2}. This potential separates to two
- electron-electroninteractionsandquantumeffectsinarti- QDs at large inter-dot distances d, and with d = 0 it
s
e ficial atoms and molecules [1]. The most striking feature simplifies to one parabolic QD. We use GaAs material
m of two-dimensional semiconductor quantum dots (QD) parameters m∗/m = 0.067 and ǫ = 12.4, and the con-
e
. and quantum molecules (QDM) is that the correlation finementstrength~ω0 =3.0meV.Aisthevectorpoten-
t
a andmagneticfieldeffectsaregreatlyenhancedcompared tial of the magnetic field (along the z axis) taken in the
m withtheirnormalcounterparts. Thisresultsinarichva- symmetricgauge. OneshouldnotethattheHamiltonian
- riety of phenomena in lateral QDMs that have recently is spin-free, and the Zeeman-coupling EZ = g∗µBBSz
d been investigated experimentally and theoretically, see (with g∗ = −0.44 for GaAs) of the magnetic field to S
z
n
for example Refs. [2, 3, 4, 5]. Also the system param- canbe takeninto accountafterwards. Theeigenstatesof
o
c eters can easily be changed, unlike in real atoms and the single-particle part of Eq. (1) are easily obtained as
[ molecules where the parameters are natural constants. expansions

The controllable parameters make it possible to tailor
v1 the semiconductor structures and for example to switch ψ(r)= αiφi(r)= αixnx,iyny,ie−21r2 , (2)
6 between different ground states. Xi Xi
7 In this Letter we concentrate on a two-electron QDM
0 consisting of two laterally coupled QDs. In addition to where nx and ny are integers, and αi is a complex coef-
1 ficient. We have used the unit of length as in Ref. [8].
theinterestingandfundamentalcorrelationandquantum
0 Fig.1showsexamplesofthenon-interactingchargeden-
2 effects, this system is very important as a candidate for
sities, and Fig. 2 displays single-particle energies. It
0 the gate of a quantum computer [7]. A central idea is
/ to use the total spin (S) of the two-electron QDM as a
t
a qubit. The ground state spin of the QDM can be either B=4T
m B=8T
S =0orS =1. OneoftheaimsofthisLetteristostudy 1
- the regions of S = 0 and S = 1 states as a function of
d
n theimportantsystemparametersintheirrealisticrange, 0.8
o beyond the approximations of Ref. 7 where a change in
0.6
c S was found when the magnetic field B was varied. We
:
v findasimilarcrossingatweakB,butalsoareappearance 0.4
i of the S = 0 ground state at larger B. This is caused
X 0.2
by the correlation effects that are treated accurately in
r
a our many-body approach. An interesting finding is that 0
−40 −20 0 20 40
the system naturally consists of composite particles of x [nm]
electrons and flux quanta as in the composite-fermion
theory [6]. FIG. 1: Confinement potential Vc (dotted line) and non-
interacting single-particle density |ψ|2 along y = 0 for d =
Wemodelthetwo-electronQDMbya2DHamiltonian
40 nm. The potential is in units of ~ω0 = 3 meV, and the
maximum value of each |ψ|2 is scaled to unity. The confine-
H = 2 (−i~∇i− ecA)2 +V (r ) + e2 , (1) ment potential is parabolic in the y-direction. Onecan see a
(cid:18) 2m∗ c i (cid:19) ǫr localizing effect of large d and strong B.
Xi=1 12
whereV istheexternalconfiningpotential,forwhichwe is interesting to compare the localizing effect of B with
c
2
12
10
1.5
V] 8
e
E [m 6 1 S=0
V] 0.5
e
m
4 E [ 0 S=1 S=0
∆
20 2 4 6 8 −0.5 40
B [T] 30
−1
0 20
FIG. 2: Six lowest non-interacting single-particle energies as 2
4 10
afunctionofB. ThedottedlinesarethenormalFock-Darwin 6
states (d = 0), and the solid ones are for d = 20 nm. One B [T] 8 0 d [nm]
canseethatford>0,theenergiesareshifteddownandthat
the level crossings occur at weaker B. At some of the level-
crossings a gap opens, suchs as the one for B ≈1.75 T, E ≈ FIG. 3: Energy difference between singlet and triplet states
7.4 meV. This is due to the lower symmetry of the problem ∆E as afunction of themagnetic fieldB and inter-dotsepa-
ford>0. Ingeneral,thereisareasonablesimilaritybetween ration d. The white lines separate the S = 0 (∆E > 0) and
the two sets of energy levels. For sufficiently large d, the S =1 ground states.
energieswouldconvergetodegeneratestatesofisolated dots.
All states are on thelowest Landau level after B≈3 T.
0.2
the experimentalfindings of Brodskyet al. [3]. They see 0.15
a clear splitting of the QDM electron droplet to smaller 0.1
fragments by a strong B. As they work in a low-density V]
e 0.05
lriomleita(swtehaekpcootnefintnieaml menitn)i,mimapinurFitiige.s1c.anThheavloecaalsizimatiiloanr ∆ E [m 0 EZ
is also related to the formation of Wigner molecules in
−0.05
QDs [8], which happens in the low-density limit.
4
Similarly to the single-particle states, the full many- −0.1 5
body wave function with total spin S can be expanded −0.15 6
as 0 1 2 3 4 5 6 7
B [T]
Ψ (r ,r ) = α {φ (r )φ (r )
S 1 2 i,j i 1 j 2 FIG. 4: ∆E for a fixed d = 26.736 nm [10]. The effect of
Xi≤j
theZeemanenergyEZ isalsoshown. Thedifferencebetween
+ (−1)Sφi(r2)φj(r1) (3) expansions with n=5 and 6 is only around 1 µeV. One can
(cid:9) see that the second S = 0 state disappears for even a weak
which is symmetric for S = 0 and anti-symmetric for EZ.
S = 1. Notice that the spin part of the wave func-
tion is not explicitly written, and we work with spin-
independent wave functions also below. The coefficient see e.g. Ref. [4] for a discussion of this exact property.
vector α and the corresponding energy E for the lth However, there is a strong decrease in ∆E as a function
l l
eigenstate are found from a generalized eigenvalue prob- ofd. Thiscanbeunderstoodfromthefactthatverydis-
lem where the Hamiltonian and overlapmatrix elements tant QDs interact only weakly and the energy difference
can be calculated analytically. Details of the computa- of the two spin states is smaller. There is also a strong
tional procedure will be published elsewhere [9]. As the decreasein∆E asafunctionofB intheS =0state. For
basis functions we have used all the states with both n d=0,∆E isratherlinearuntilthecrossingtotheS =1
x
andn ≤n with n=6, andwehavecheckedthe conver- state, but for d > 0 the curves are rounded. The cross-
y
gence by varying n for all values of d. ing point of the different ground states does not depend
The energy difference ∆E between the lowest S = 0 stronglyond; it changesonly from1.6 T to 1.2T as one
and S = 1 states is plotted in Figs. 3 and 4. The con- moves from d = 0 to 40 nm. Due to this, changing the
vergence of ∆E can be seen in Fig. 4 for large d. One total spin in an experimental setup is not easy by just
can see that for weak magnetic field values, the ground changing d. On the other hand, around ∆E = 0, the
statehasS =0. Wefirstconcentrateonthisregime. For slope of ∆E is rather large and changing S by B is the
B = 0 the S = 0 state remains lower with arbitrary d, most natural choice. One can achieve a change of S also
3
by changing the strength of V . This changes the ratio the vortices of Ψ. This can be done by finding the zeros
c
between the energiesresulting fromthe confinement and of Ψ and studying the change in the phase of Ψ in going
electron-electroninteraction. For a weakV , the interac- aroundeachof the zeros. A surprisingfinding is that for
c
tions are stronger and the transition occurs at weaker B bothlargedandd=0therearetwovorticesatbothelec-
value. ThusthechangeoftheV canbeseenasachange tron locations, see Fig. 6 for an example. The particles
c
of the effective value of B. are thus composite particles of an electron and two flux
The transition from the weak-B S = 0 state to the quanta, in similar fashion as in the composite fermion
S =1stateismostsimplyexplainedbythefactthatthe (CF) theory [12]. The most remarkable feature of the
energiesof the two lowestsingle-particlestates approach finding is the stability of composite particles against the
each other as B is made stronger (see Fig. 2). At some change of d.
pointthis differenceissmallerthantheexchange-energy,
and the system spin-polarizes. One can see that also for
this state, ∆E decreases strongly as a function of d.
There exists a second region of S = 0 ground state
around B ≈ 6 T at d = 0. The question whether
this state terminates at large d remains open, as the
energy differences at d > 40 nm are smaller than the
error made in the expansion. The existence of this
S = 0 state for small value of d can be understood
on the basis of a parabolic two-electron QD, which can
be shown to have the exact wave function of the form
Ψ=(x12+iy12)mf(r12)e−21(r12+r22), where m is the rela-
50
tiveangularmomentumandf isacorrelationfactor[11].
−50
The simple form is due to the separation of the center- 0
0
of-mass and relative motion of the electrons. For B = 0
the ground state has m=0 (and S =0), and when B is x [nm] 50 −50 y [nm]
made stronger, the ground state m has increasing posi-
tiveintegervalues. Thesetransitionshappenbecausethe
largerm states have smaller Coulombrepulsionbetween FIG. 6: |Ψ[(x,y),(d/2,0)]|2 for the state S =0 at B =6 T
and d = 10 nm. The arrows (rotated for clarity) depict the
the electrons, and as the single-particle energies group
twofluxquantaattheelectronfixedatx>0minimumofVc.
together to form the lowest Landau level, it is favorable
to move to larger m. The second S = 0 region in Fig. 3
We have done a similar analysis for the S = 1 state
corresponds to m = 2. It is surprising that the second
at B = 3 T and various values of d. We found that the
S = 0 region extends to such a large d. For example, at
many-body state again consists of composite particles,
B = 4.5 T and d = 35 nm the two QDs of a QDM are
butthistimethereisonlyonefluxquantumperelectron.
rather decoupled (see Fig. 5), but still the S = 0 state
One should note that for an odd number of flux quanta
remains the ground state. To analyze the structure of
per electron, the wave function changes sign when the
particles are exchanged, corresponding to S = 1. Simi-
larly, for even flux numbers the state has S =0.
25
If one expands |Ψ|2 for small r , one obtains
12
m] C
y [n 0 |Ψ|2 ∝r122m+ m+ 1r122m+1+O(r122m+2) , (4)
2
−25 where C is the scaled strength of the Coulomb interac-
tion[8],andm≥0isthenumberoffluxquantaperelec-
−50 −25 0 25 50
x [nm] tron. One can see that for larger m values, the density
grows more slowly as a function of r (see Fig. 6). One
FIG.5: |Ψ[(x,y),(d/2,0)]|2 forthestateS =0atB=4.5T should note that the same expansion1i2s valid also for the
and d = 35 nm. The contour spacing is uniform. We mark
larger electron numbers, resulting from the cusp condi-
withaplusthefixedelectronontheright-handQD.Onecan
tion[8]. Thus the electron-electroninteractionis smaller
seethattheelectrondensityislocalizedtotheleftQD.There
when the number of flux quanta per electron grows.
is only a small deformation from circular symmetry. Notice
thattheeffectiveinter-dotdistanceismuchlargerthanddue OnecanusetheCF-typeapproachtoexplainthephase
to theCoulomb repulsion of theelectrons. diagram of Fig. 3. When one moves from B = 0 to
stronger values of B, the number of flux quanta in the
thisstateforsmallandlargevaluesofd,wehavelocated system increases. At the first S = 0 state there is no
4
fluxes in the system, and at the transition point the flux found that the system consists of composite particles of
number changesto one per electron. For weakly coupled electrons and the attached magnetic field flux quanta.
QDs this transition happens at smaller B than in the
stronglycoupledones,becauseadistantzeroofthewave This research has been supported by the Academy
functionincreasesthekineticenergylessthanacloseone of Finland through its Centers of Excellence Program
does. In the following transition points the number of (2000-2005).
flux quanta changes again by one per electron. The rea-
soningforthed-dependenceofthefirsttransitionapplies
to other ones also, and this can be seen in Fig. 3.
An interesting prospect resulting from the discussion
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5325(1999);S.Nagaraja,J.-P.Leburton,andR.M.Mar-
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