Table Of ContentSpin-orbit effects on two-electron states in nanowhisker double quantum dots
C. L. Romano,1 P. I. Tamborenea,1 and S. E. Ulloa2
1Department of Physics “J. J. Giambiagi”, University of Buenos Aires,
Ciudad Universitaria, Pabell´on I, C1428EHA Ciudad Aut´onoma de Buenos Aires, Argentina
2Department of Physics and Astronomy, and Nanoscale and Quantum
Phenomena Institute, Ohio University, Athens, Ohio 45701-2979
(Dated: February 2, 2008)
8 We investigate theoretically the combined effects of the electron-electron and the Rashba spin-
0 orbitinteractionsontwoelectronsconfinedinquasi-one-dimensionalAlInSb-baseddoublequantum
0 dots. We calculate the two-electron wave functions and explore the interplay between these two
2 interactions on the energy levels and the spin of the states. The energy spectrum as a function
of an applied magnetic field shows crossings and anticrossings between triplet and singlet states,
n
associated with level mixing induced by the spin-orbit coupling. We find that the fields at which
a
these crossings occur can be naturally controlled by the interdot barrier width, which controls the
J
exchangeintegral in the structure.
1
1
PACSnumbers: 73.21.La,73.21.-b,71.70.Ej
Keywords: electroninteractions,spin-orbitcoupling,Rashbaeffect, quantum dots
]
l
l
a
h I. INTRODUCTION II. THEORETICAL DESCRIPTION
-
s
e Substantial efforts have been devoted to understand- Weinvestigatetheproblemoftwointeractingelectrons
m ing and manipulating the electron spin aiming at poten- inaquasi-one-dimensionaldoublequantumdotstructure
. tialapplications in semiconductorspintronics.1,2,3 While inthe presence ofthe structuralorRashbaspin-orbitin-
t
a the spin-orbit interaction in quantum dots has been teraction. We study two identical 30 nm wide dots, sep-
m extensively studied from the single-electron perspec- arated by an interdot barrier, 3 nm or 5 nm wide. In
- tive, its combined effects with the Coulomb interaction our calculations we consider an Al0.1In0.9Sb-InSb struc-
d in few-particle systems has only recently begun to be ture which has a potential energy depth of 100meV. In
n explored.4,5,6,7 Especially in nanowhiskers, Fasth et al.8 Fig. 1 we show the confining potential in the longitudi-
o
c measured the strength of the spin-orbit interaction in naldirection,Vz(z),andthe eigen-functions un(z)ofthe
[ tlewnog-tehlectro1n20InnAms),cyalninddrPicfaulnddoetts a(dl.i9amsteutderied∼s5p0innrme-, svienrgtliec-apllayrtaicccleorHdainmgilttoonthiaenirHco0r=res2ppmo2z∗nd+inVgz(ezn)e,rdgiysplelavceelsd,
1 ∼
laxation in a similar system. Here we examine two- E .
v n
8 electrons in a double quantum dot10,11,12,13 system with Thenanowhiskerwherethedoubledotstructureisde-
0 only one transverse quantum mode active. The one- fined is assumed to be so thin ( 2 nm) that only the
8 electron spectrum of this thin nanowhisker semiconduc- lowest transverse mode is active.≃As such, the effective
1 torstructurehasbeentreatedbyus previously,14 aswell one-dimensionalHamiltonianoftwointeractingelectrons
1. as the phonon-mediated spin-relaxation.15 with Rashba interaction, in the absence of a magnetic
0 In this paper we calculate and analyze in detail the field, can be written as14
8 two-electronstates insucha system,takinginto account
0 the Rashba spin-orbit coupling and the Coulomb inter- H = H0+H0+H +V , (1)
1 2 1dR int
:
v action between the electrons. We pay special attention
i to the degree of admixture of different two-electron spin where H0 = p2z,i + V (z ), m∗ is the conduction-band
X i 2m∗ z i
wave-functions, which will influence the spin-flip transi- effective mass, z and z are the z-coordinates of the
1 2
ar tions in this system. two electrons, and pz,1 and pz,2 are the z-components of
The paper is organized as follows. In Sec. II we intro- their linear momentum. H and V are the Rashba
1dR int
duce the effective quasi-one-dimensional Hamiltonian of spin-orbit coupling and the electron-electron interaction
two interacting electrons in the presence of the Rashba potential, respectively. The Rashba spin-orbit coupling
interactionanddescribethemethodandapproximations inthe quasi-one-dimensionalstructureconsideredhereis
used in the calculations. In Sec. III we investigate the given by14
effects of the electron-electron interaction on the energy
levelswithandwithoutRashbainteractionandtheir de- 2
γ ∂V
R x
pendence on an applied magnetic field. We monitor the H1dR = ~ (cid:28) ∂x (cid:29)pz,i(σxi −σyi), (2)
mean value of the spin projection as a function of the Xi=1
structuralparameterthatdetermines the strengthofthe
Rashba spin-orbit coupling. In Sec. IV we provide some where γR = 500˚A2,16 σxi,yi are Pauli matrices, and
concluding remarks. ∂Vx is the Rashba field parameter, where the mean
∂x
(cid:10) (cid:11)
2
set ϕ ,i = 1,6 takes into account only the two lowest
i
eige{nstates of H}0, u (z) and u (z) (see Fig. 1). This
1 2
)
V 0 truncated basis simplifies the analysis and presentation
e
m without leaving out essential physics. We have checked
(z u4 that the truncation introduces only small quantitative
V differences, of the order of 1 percent, in the results.
In order to solve the eigenvalue problem of the full
u3 two-electron Hamiltonian given in Eq. (1), we expand
-50 the two-electronwave functions
6
u2 ψi = aijϕj, (5)
Xj=1
u1 where i = 1,..,6, and determine the coefficients a by
ij
-100 numerical diagonalization. It is clear that the Rashba
coupling mixes states with different spin wave functions,
0 20 40 60 80 100 although the mixing depends strongly in structure pa-
Z (nm) rametersandappliedmagneticfield,aswewillseeinthe
next section.
FIG. 1: The Al0.1In0.9Sb-InSb double-well confining poten-
tialinthelongitudinaldirectionofthequasi-one-dimensional
nanowhisker quantum dots. The single-particle eigenfunc-
tions and energies are also shown. 036,,35EE--7718000 Z2 (nm) (GS) 18000 Z2 (nm)
9,7E-7 60 60
value is taken over the ground state Φ of the laterally- 1,3E-6
1,6E-6 40 40
confining potential Vx = Vy (Vx is assumed to be the 22,,03EE--66 20 20
same as V for simplicity). The electron-electron inter-
action is gyiven by13 2,6E-6 00 20 40 60 Z18(0nm1)00 00 20 40 60 Z180(nm1)00
100 m) 100 m)
n (triplet) n
V (z z )= dx dx dy dy 0 80 2 ( 80 2 (
int | 2− 1| Z 1 2 1 2 37,,75EE--77 60 Z 60 Z
e2Φ(x1)2Φ(x2)2Φ(y1)2Φ(y2)2 11,,15EE--66 40 40
, (3) 1,9E-6
whererǫp=(x(x1−,yx,2z)2),+i(=y11−,2y,2a)r2e+th(ez1el−ecztr2o)2npositions, 223,,,260EEE---666 2000 20 ,40 60 Z18(0nm1)00 2000 20 40 60 Z18(0nm1)00
i i i i
and ǫ is the dielectric constant of the material (16.8 for
InSb). In Eq.(3) we approximateΦ by the ground state FIG. 2: (color online) Probability density of two electrons in
(spatialextent 2nm)ofaharmonicoscillatorpotential thedoublequantumdot with Coulomb interaction butwith-
∼
defined over the cross section of the whisker. outRashbacoupling. ψ1: groundstate(GS),ψi=2,...,6excited
As a basis set for the two-electron Hilbert space we states. Notice ψ5 and ψ6 describe states with both electrons
take inthesamedot,whiletheotherscan beseen moreashaving
one electron in each dot.
ϕ = u (z )u (z ) 0,0 ,
1 1 1 1 2
| i
1
ϕ = (u (z )u (z )+u (z )u (z )) 0,0 ,
2 1 1 2 2 2 1 1 2
√2 | i
III. RESULTS
ϕ = u (z )u (z )0,0 ,
3 2 1 2 2
| i
1
ϕ = (u (z )u (z ) u (z )u (z )) 1,1 , We solve the eigenvalue problem of the two-electron
4 1 1 2 2 2 1 1 2
√2 − | i Hamiltonian H given in Eq. (1) with the goal of un-
1 derstanding the interplay between the electron-electron
ϕ = (u (z )u (z ) u (z )u (z )) 1, 1 ,
5 1 1 2 2 2 1 1 2
√2 − | − i Coulombinteractionand the single-particleRashba cou-
1 pling. Inordertogainsomeinitialinsightintothenature
ϕ = (u (z )u (z ) u (z )u (z )) 1,0 . (4)
6 1 1 2 2 2 1 1 2 of the two-electron states, in Fig. 2 we show the proba-
√2 − | i
bility density of the six eigenstates for the two electrons
The two-particle spin wave functions are the usual sin- in a double-dot structure with a 3 nm barrier without
glet S = S = 0,m = 0 and triplet states T+ = Rashba coupling. As expected, in the ground state the
S
1,1|, Ti0 =| 1,0 , T− = 1i, 1 . Thistruncat{e|dbiasis two electrons are essentially localized in different dots,
| i | i | i | i | − i}
3
due to their mutual Coulomb repulsion, and on “bond- S = 0), but around B 2.2T there is a level crossing
z
≈
ing” single-particle states with a non-zero amplitude in anditsspinpartbecomes 1,1 (S =1). Thischangeoc-
z
| i
the interdot barrier region In contrast, states ψ to ψ cursdue to the competitionbetweenthe Zeemanenergy,
2 4
have electrons localized in each dot too, but with an on one hand, and the Coulomb interaction and E E
2 1
−
“antibonding”orbitalwith zero amplitude in the central on the other. At the crossing, the first excited state ψ
2
barrier. Noticealsothatthetwohigherenergystates,ψ goes, naturally, from S = 1 to S = 0. ψ , ψ and ψ
5 z z 2 3 4
and ψ , correspond to singlet states with two electrons are degenerate at B = 0, but having S = 1,0, 1, re-
6 z
−
in each dot. Furthermore, in order to fully characterize spectively,theirdegeneracyisbrokenfornon-zeroB. E
4
the two-electronsysteminwhatcouldbe realisticexper- and E have a level crossing at B 6T, and, finally,
5
≈
imental situations we introduce a magnetic field along E and E cross at B > 7T (not shown). There are no
5 6
the z-direction. The field is chosen small enough (and further crossings at higher magnetic fields.
the whisker so thin) that the x-y orbital wave functions
arenotperturbedsignificantlybyit. Thus,thefieldcon-
tributes only a Zeeman term to the Hamiltonian:
S >
-140
g µ B
HZ = 0 ~B Sz, (6) S > T >
V) -150 S >
e
where µ is the Bohr magneton, g is the Land´e fac- m
B 0 (
tor (g0 = −51 for InSb), and Sz = S1,z + S2,z is the E -160 T > T0 >
z-componentofthetotalspinoperator. InFig.3weplot
ethleecternoenrgeyigaennsdtaetxepsevcst.attiohne avpalpuleiedofmSazg,nfoerticalfiletlhdewtwitho-- -170 TS +>> S > 1
out Rashba interaction, in the double dot with a 3 nm -180 T +> 2
barrier. a) 3
1.0
4
S>z 0.5 b) 5
S > < 0.0
V) -140 S > T > -0.5 6
e
m -150 S > -1.0 B(T)
( 0 1 2 3 4 5 6 7
E
-160 T > 0
T >
FIG. 4: (color online) (a) Energy levels and (b) mean value
-170 T +> S > 1 of Sz = S1,z + S2,z versus applied magnetic field for the
S >
two-electron eigenstates including Coulomb and Rashba in-
-180 T +> 2 teractions. The strength of the Rashba coupling is given by
a) 3 ˙∂∂Vxx¸=1meV/˚A. Barrier width: 3 nm. |Si= singlet state,
-119.00 4 |T±,0i= triplet state.
> b)
Sz0.5 5 We next include the Rashba interaction with a cou-
< 0.0 6 pling parameter ∂Vx = 1meV/˚A. The results for the
∂x
-0.5 energy levels and(cid:10)mea(cid:11)n value of Sz are shown in Fig. 4.
-1.0 B(T) The main differences with Fig. 3 are as follows:
0 1 2 3 4 5 6 7 (i)WithRashbainteraction,atB =0,ψ andψ arenot
2 4
spin eigenstates anymore, as can be seen in Fig. 4(b).
However, we still label the states as if they were pure
FIG. 3: (color online) (a) Energy levels and (b) mean value spin states, as one spin state dominates the admixture
of Sz = S1,z +S2,z vs. applied magnetic field for the two- (far from the avoided crossings discussed next).
electron eigenstates with Coulomb interaction and without (ii) Two of the level crossings in Fig. 3 (E with E at
1 2
Rashba coupling. Barrier width: 3 nm. |Si = singlet state,
B 2.2T, and E with E at B >7T) become avoided
|T±,0i= triplet states. ≈ 5 6
crossings here, as the pair of states involved are coupled
by the Rashba interaction. The other crossing, between
Letuspointoutsomebasicfeaturesoftheresultswith- E and E at B 6T, is shifted slightly upward due to
4 5
≈
outRashbacouplingseeninFig.3. Firstnoticethatsince the effect of the Rashba interaction on each individual
thespatialwavefunctionsdonotdependonthemagnetic level, but it does not become avoided because the levels
field and only the Zeeman energy changes with B, this arenotcoupledtoeachotherthroughtheRashbainterac-
explains the linear field dependence of the energies. At tion. Thislackofmixingarisesfromthe differentspatial
zeromagneticfield,thegroundstateψ isasinglet(0,0 , symmetry of the states and the strong Coulomb interac-
1
| i
4
115 57 38 29 23 19
-140 S > S >
1.0 SO
S > T > a) (nm)
-150 >0.5
eV) -160 T > T0 > <Sz0.0
m
( T +>
E -170 S > S > 1 -0.5 1
T +> 2 -1.0 2
-180 1.0
a) 3 3
<S>z001...050 b) 45 <S>z00..05 b) 45
-0.5 6 -0.5 6
-1.0
-1.0 B(T)
0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0
<dV/dx> (meV/¯)
FIG. 5: (color online) (a) Energy levels and (b) mean value
FIG. 6: (color online) Mean value of Sz = S1,z +S2,z as
of Sz = S1,z + S2,z versus applied magnetic field for the a function of the Rashba parameter ˙∂Vx¸ for the six two-
two-electron eigenstates including Coulomb and Rashba in- ∂x
electron eigenstates. B = 0.5T. Barrier width: 3 nm. (a)
teractions. The strength of the Rashba coupling is given by
WithoutCoulombinteraction;(b)WithCoulombinteraction.
|˙T∂∂±Vxx,0¸i==1trmipelVet/s˚At.atBe.ar(rcioelrorwiodntlhin:e5) nm. |Si= singlet state, λSO is thespin-orbit length.
6(a) and (b). An important difference between (a) and
tion. The symmetry under space reversal(odd vs. even)
(b) is that there is a symmetry around S =0 when the
z
preventsthemixingofastatewithdoubledotoccupancy
Coulomb interaction is absent. The ground state in Fig.
(singlet ψ ) and a state where each dot has one electron
5 6(a) (thick cyan solid line) starts with S = 0 and at a
z
(ψ ), where each state has opposite space-reversal sym-
4 certain (typically large) value of the Rashba parameter
metry. Notice that the width of the avoided crossings is
reachesamaximum. Ontheotherhand,inFig.6(b),this
determinedmainly by the strengthofthe spin-orbitcou-
symmetry about the zero value is lost due to the differ-
pling, and therefore it can be adjusted with a transverse
ent mixing of two-particle orbitals ( ϕ ) in higher- and
i
electric field (gate voltage). { }
lower-lying states produced by the Coulomb interaction.
In Fig. 5 we present the energy levels and the mean
Figure 6 includes an axis(top) in terms ofthe spin-orbit
value ofSz as functions of the magnetic field for a struc- length(λ =~2/m∗γ ∂Vx ),17 whichisinverselypro-
turewitha5nminterdotbarrier. Noticethat,compared SO R ∂x
portionaltotheRashbap(cid:10)aram(cid:11)eter. Thislengthparame-
to the case of a 3 nm interdot barrier (Fig. 4), crossings
terhelpsvisualizethestrengthoftheRashbacouplingin
and anti-crossings shift to lower values of the magnetic
comparisonto thecharacteristicdimensionsofthe struc-
field. This happens because the wider barrier decreases
ture. Itisinterestingtopointoutthatspinmixingisfirst
the bonding-antibonding gap, the wave function overlap
noticeable when λ 60 nm, the size of the two-well
SO
andtheassociatedexchangeintegralthatdeterminesthe ≃
system.
singlet-triple separation at zero field. As the S-T gap is
With a 5 nm barrier width, we find that the ground
smaller,itismoreeasilyovercomebytheZeemanenergy.
state has a mean value of S roughly equal to zero up to
z
Foran8nmbarrier,thebonding-antibondinggapisonly a Rashba constant of 2.2 meV/˚A in the presence of the
slightly over half of the value for the 3 nm barrier, and
Coulombinteraction,unlike thecaseinFig.6,wherethe
the singlet-triplet crossing occurs at a field of 1.2 T, for
groundstateisclearlymixedforRashbaconstantbeyond
example. 1.5 meV/˚A.
InFig.6wepresentthemeanvalueofS asafunction
z
of the Rashba parameter ∂Vx for all the states at a
∂x
given magnetic field B =0(cid:10).5T ((cid:11)using a barrier width of
IV. CONCLUSIONS
3 nm). In this figure, we compare two cases, (a) without
and(b)withCoulombinteraction,inordertoexhibitbet-
ter the role of spin-orbit coupling in the spin mixing. As In this paper, we have investigated the effects of the
expected, without Rashba coupling, i.e. with ∂Vx =0, Coulomb electron-electron interaction on the states, en-
∂x
the spin projection of each of these states natur(cid:10)ally(cid:11)Sz ergy levels, and z-projection of the spin of two electrons
h i
takesthe exactvalues 1,0,and 1,asseeninbothFigs. confinedin a quasi-1Ddouble-quantum-dotnanowhisker
−
5
in the presence of Rashba spin-orbit interaction. As a states, and displayed this reduction as a function of the
function of a magnetic field in the longitudinal direc- Rashba spin-orbit length.
tion, some energy-level crossings become avoided cross-
ingswhentheRashbaspin-orbitisturnedon. Thewidth
of the avoided crossings can be controlled with a lat-
Acknowledgments
eral gate voltage via the intensity of the Rashba param-
eter ∂Vx . The positions of these crossings and avoided
∂x
cross(cid:10)ings(cid:11)as functions of the magnetic field can be se- We acknowledge useful discussions with C. Destefani
lected by changing the width of the interdot barrier. Fi- andL.Meza-Montes,aswellassupportfromCONICET
nally,wefoundthattheCoulombinteractionreducesthe (PIP-5851), UBACyT (X179), and NSF (WMN grant
spin mixing inthe groundstate,as wellas in the excited 0710581). PIT is a researcher of CONICET.
1 G. A. Prinz, Science 282, 1660 (1998). Rev.Lett. 99, 036801 (2007).
2 I. Zˇuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 10 W. Dybalski and P. Hawrylak, Phys. Rev. B 72, 205432
76, 323 (2004). (2005).
3 J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. 11 R.M.Abolfath,W.Dybalski,andP.Hawrylak,Phys.Rev.
Zˇuti´c, arXiv:0711.1461v1[cond-mat.mtrl-sci]. B 73, 075314 (2006).
4 R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, 12 Xuedong Hu and S. Das Sarma, Phys. Rev. A 61, 062301
and L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2000).
(2007). 13 P.I.TamboreneaandH.Metiu,Phys.Rev.Lett.83,3912
5 Y. Weiss, M. Goldstein, and R. Berkovits, condmat (1999).
0710.2772.v1 (2007). 14 C. L. Romano, S. E. Ulloa, and P. I. Tamborenea, Phys.
6 C. F. Destefani, S. E. Ulloa, and G. E. Marques, Phys. Rev.B 71, 035336 (2005).
Rev.B 69, 125302 (2004). 15 C. L. Romano, P. I. Tamborenea, and S. E. Ulloa, Phys.
7 T. Chakraborty and P. Pietilainen, Phys. Rev. B 71, Rev.B 74, 155433 (2006).
113305 (2005). 16 O. Voskoboynikov,C. P. Lee, and O. Tretyak, Phys. Rev.
8 C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and B 63, 165306 (2001).
D. Loss, Phys.Rev.Lett. 98, 266801 (2007). 17 D.V.BulaevandD.Loss,Phys.Rev.B71,205324(2005).
9 A.Pfund,I.Shorubalko,K.Ensslin,andR.Leturcq,Phys.
