Strong Enhancement of Rashba spin-orbit coupling with increasing anisotropy in the Fock-Darwin states of a quantum dot

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by  Siranush Avetisyan

Strong Enhancement of Rashba spin-orbit coupling with increasing anisotropy in the
Fock-Darwin states of a quantum dot
Siranush Avetisyan,1 Pekka Pietil¨ainen,2 and Tapash Chakraborty‡1
1Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2
2
2Department of Physics/Theoretical Physics, University of Oulu, Oulu FIN-90014, Finland
1
0
We have investigated the electronic properties of elliptical quantum dots in a perpendicular ex-
2
ternal magnetic field, and in the presence of the Rashba spin-orbit interaction. Our work indicates

n thattheFock-Darwinspectra displaystrongsignatureof Rashbaspin-orbitcouplingevenfor alow
a magneticfield,astheanisotropyofthequantumdotisincreased. Anexplanationofthispronounced
J effect with respect to the anisotropy is presented. The strong spin-orbit coupling effect manifests

4 itself prominently in the corresponding dipole-allowed optical transitions, and hence is susceptible
2 todirect experimental observation.

]
l
l In recent years our interest in understanding the stronginanellipticalQD[13],whichshouldprovideadi-
a
uniqueeffectsofthespin-orbitinteraction(SOI)insemi- rect route to unambigiously determine (and control) the
h
- conductor nanostructures [2] has peaked, largely due to SOcouplingstrength. Ithasbeenproposedrecentlythat
s
the prospect of the possible realization of coherent spin theanisotropyofaquantumdotcanalsobetunedbyan
e
m manipulation in spintronic devices [3], where the SOI is in-plane magnetic field [14].
destined to play a crucial role [4]. As the SOI couples The Fock-Darwin energy levels in elliptical QDs sub-
.
t the orbital motion of the charge carriers with their spin jected to a magnetic field was first reported almost two
a
m state, an all-electrical control of spin states in nanoscale decadesago[13],whereitwasfoundthatthemajoreffect
semiconductor devices could thus be a reality. In this of anisotropy was to lift the degeneracies of the single-
-
d context the Rashba SOI [5] has received particular at- particle spectrum [15]. The starting point of our present
n tention,largelybecauseinatwo-dimensionalelectrongas study is the stationary Hamiltonian
o
thestrengthoftheRashbaSOIhasalreadybeenshownto
c 1 e 2
[ be tunedbytheapplicationofanelectricfield[6]. While HS = 2m∗ (cid:16)p− cAS(cid:17) +Vconf(x,y)+HSO+Hz
2 tehleecteraornliegrass,tuthdeieastwteenrteiopnrihmaasrniloywinbeaentwfooc-duismedenosniotnhael = H0+HSO+Hz
v
2 role of SOI in a single InAS quantum dot [7]. The quan- where the confinement potential is chosen to be of the
4 tum dot (QD) [8], a system of few electrons confined in form
8 the nanometer region has the main advantage that the V = 1m∗ ω2x2+ω2y2 ,
4 shape andsize of the confinement canbe externally con- conf 2 x y
1. trolled, which provides an unique opportunity to study H = α σ× p− eA (cid:0) is the Rashb(cid:1)a SOI, and H
SO ~ c S z z
0 theatomic-likepropertiesofthesesystems[8,9]. SOcou- istheZeem(cid:2)anco(cid:0)ntributio(cid:1)n(cid:3). Herem∗ istheeffectivemass
2 plinginquantumdotsgeneratesanisotropicspinsplitting of the electron, σ are the Pauli matrices, and we choose
1
[10] whichprovides importantinformationabout the SO the symmetric gauge vector potential A = 1(−y,x,0).
: S 2
v coupling strength. AsinRef.[13],weintroducethe rotatedcoordinatesand
i
X momenta
ExtensivetheoreticalstudiesoftheinfluenceofRashba
ar SOI in circularly symmetric parabolic confinement have x = q1cosχ−χ2p2sinχ,
already been reported earlier [11], where the SO cou- y = q cosχ−χ p sinχ,
2 2 1
plingwasfoundtomanifestitselfmainlyinmultiplelevel
p = p cosχ+χ q sinχ,
x 1 1 2
crossings and level repulsions. They were attributed to
p = p cosχ+χ q sinχ,
an interplay between the Zeeman and the SOI present y 2 1 1
in the system Hamiltonian. Those effects, in particular, where
the level repulsions were however weak and as a result, 1
χ = − 1 Ω2+Ω2 2 , χ =χ−1,
wouldrequireextraordinaryeffortstodetectthestrength 1 2 1 2 2 1
ofSOcoupling[12]inthosesystems. Hereweshowthat, tan2χ = m∗(cid:2)ω(cid:0) 2 Ω2+(cid:1)Ω(cid:3)2 21 / Ω2−Ω2 ,
c 1 2 1 2
by introducing anisotropy in the QD, i.e., by breaking
Ω2 = m∗2 ω(cid:2)2(cid:0) + 1ω2 (cid:1),(cid:3) ω(cid:0)=eB/m(cid:1)∗c.
the circular symmetry of the dot, we can generate a ma- 1,2 x,y 4 c c
(cid:0) (cid:1)
jor enhancement of the Rashba SO coupling effects in a
In terms of the rotated operators introduced above, the
quantum dot. As shown below, this can be observed di-
Hamiltonian H is diagonal [13]
0
rectly in the Fock-Darwin states of a QD, and therefore
1
should be experimentally observable [8, 9]. We show be- H = β2p2 +γ2q2 ,
0 2m∗ ν ν ν ν
low that the Rashba SO coupling effects are manifestly νX=1,2(cid:2) (cid:3)
2

20 20
16 16
eV)12 16 eV)12 16
m 1 2 m 12
E ( 8 E ( 8
4 8 4 8
0 (a) 4 (c) 0 (a) 4 (c)
20 20
16 16
)
V)12 16 eV12 16
E (me 8 1 2 E (m 8 12
4 8 4 8
(b) (d) (b) (d)
0 4 0 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
B (T) B (T) B (T) B (T)
FIG. 1: Magnetic field dependence of the low-lying Fock- FIG. 2: Same as in Fig. 1, but for α=20.
Darwin energy levels of an elliptical dot without the Rashba
SO interaction (α = 0). The results are for (a) ω = 4 meV
x
and ωy = 4.1 meV, (b) ωx = 4 meV and ωy = 6 meV, (c) ωy = 4.1,6,8,10 meV in (a)-(d) respectively. We have
ωx = 4 meV and ωy = 8 meV, and (d) ωx = 4 meV and considered the parameters of an InAs QD [11] through-
ω =10 meV.
y out, because in such a narrow-gap semiconductor sys-
tem, the dominant source of the SO interaction is the
structural inversion asymmetry [16], which leads to the
where
RashbaSOinteraction. Asexpected,breakingofcircular
β2 = Ω21+3Ω22+Ω23, γ2 = 1 3Ω2+Ω2+Ω2 , symmetry in the dot results in lifting of degeneracies at
1 2(Ω2+Ω2) 1 4 1 2 3 B =0,thatisotherwisepresentinacirculardot[13,15].
1 2 (cid:0) (cid:1)
3Ω2+Ω2−Ω2 In Fig. 1 (a), the QD is very close to being circularly
β2 = 1 2 3, γ2 = 1 Ω2+3Ω2−Ω2 ,
2 2(Ω2+Ω2) 2 4 1 2 3 symmetric, and as a consequence, the splittings of the
1 2 (cid:0) (cid:1)
zero-field levels are vanishingly small. As the anisotropy
1
Ω2 = Ω2−Ω2 2+2m∗2ω2 Ω2+Ω2 2 . of the QD is increased [(b) – (d)], splitting of the levels
3 1 2 c 1 2
h(cid:0) (cid:1) (cid:0) (cid:1)i becomes more appreciable.
Since the operator H is obviously equivalent to the As the SO term is linear in the position and momen-
0
Hamiltonianoftwoindependentharmonicoscillators,the tumoperatorsitisalsolinearintheraisingandlowering
states of the electron can be described by the state vec- ladder operators. It is also off-diagonal in the quantum
tors |n ,n ;s i. Here the oscillator quantum numbers number s . As a consequence, the SOI can mix only
1 2 z z
n = 0,1,2,... correspond to the orbital motion and stateswhichdifferinthespinorientation,anddifferby1
i
s =±1 to the spin orientation of the electron. eitherinthequntumnumbern orinn butnotinboth.
z 2 1 2
In the case of rotationally symmetric confinements these
selection rules translate to the conservation of the total
The Rashba Hamiltonian, in terms of the rotated op- angular momentum j = m+sz in the planar motion of
the electron.
erators is now written as,
~
H = σ (sinχχ −cosχω )q
α SO x 1 0 1
− σy(sinχχ1+cosχω0)q2 AtthefieldB =0thegroundstates|0,0;±1iaretwo-
2
− σ (cosχ−sinχω χ )p fold degenerate. Due to the selectionrules,this degener-
y 0 2 1
acy cannot be lifted either by the eccentricity of the dot
+ σ (cosχ+sinχω χ )p ,
x 0 2 2
or by the Rashba coupling. Many of the excited states,
where ω = eB/2c. The effect of the SO coupling is such as |n ,n ;±1i retain their degeneracy no matter
0 1 2 2
readilyhandledbyresortingtothestandardladderoper- how strong the SO coupling is or how eccentric the dot
ator formalism of harmonic oscillators and by diagonal- is,aswecanseeintheFigs.1-3. Atthesametime,many
izing H in the complete basis formed by the vectors otherdegeneraciesareremovedbysqueezingorstreching
SO
|n ,n ;s i. the dot. At non-zero magnetic fields some of the cross-
1 2 z
The Fock-Darwin states in the absence of the Rashba ingsofthe energyspectraareturnedtoanti-crossingsby
SOI (α = 0) are shown in Fig. 1, for ω = 4 meV and the Rashba term in the Hamiltonian. For example, the
x
3

20 9 9
16
V)12 16 meV)6 6
me 12 E (
E ( 8 3 3
4 8 (a) (d)
(a) (c) 0 0
0 4
20 12 12
V)16 eV)9 9
E (me128 1126 E (m36 36
(b) (e)
4 8 0 0
(b) (d) 12 12
0 4
0 1 2 3 4 0 1 2 3 4 eV)9 9
m
B (T) B (T) E (6 6
FIG. 3: Same as in Fig. 1, but for α=40.
3 3
(c) (f)
0 0
0 1 2 0 1 2
second and third excited states in Fig. 2 (a) – Fig. 2 (d) B(T) B(T)
are composed mainly of the states |0,0;1i and |1,0;−1i
2 2 FIG.4: Opticalabsorption(dipoleallowed)spectaofelliptical
which are mixed by the HSO around B = 3T causing a QDs for various choice of parameters: (a) iα = 0, ωx = 4
levelrepulsion. Wecanalsoseethatthesqueezingofthe meV, ω = 6, (b) α = 20, ω = 4 meV, ω = 8 meV, and
y x y
dotenhancestheSOcoupling. Thiscanbethoughtofas (c) α=40, ωx =4, ωy =6. The polarization of the incident
aconsequenceofpushingsomestatesoutoftheway,just radiation is along the x-axis. The parameters for (d)-(f) are
asinourexampleofthestate|1,1;1i. SOImixesitwith thesame,exceptthattheincidentradiationispolarizedalong
2 the y-axis. The areas of the filled circles are proportional to
thestate|1,0;−1icausingthelatterstatetoshiftdown-
2 thecalculated absorption cross-section.
ward in energy thereby reducing the anti-crossing gap.
Squeezingthedot,howevermovesthe stateenergetically
farther away from |1,0;−1i and so weakens this gap re-
2 be neglected as well. So we can simply replace in the
duction effect. It is abundantly clear from the features stationaryHamiltonianH the vectorpotential A with
S S
revealed in the energy spectra that for a combination of the field A = A +A . Discarding terms higher than
S R
strong anisotropyof the dot and higher values of the SO linear order in A leads to the total Hamiltonian
R
coupling strength, large anti-crossinggaps would appear
even for relatively low magnetic fields. H=H +H ,
S R
The effects of anisotropyand spin-orbitinteraction on
the energy spectra above are also reflected in the optical where the radiative part H is given by
R
absorption spectra. Let us turn our attention on the
e e αe
absorption spectra for transitions from the ground state H =− A · p− A − [σ×A ] .
to the excited states. For that purpose we subject the R mec R (cid:16) c S(cid:17) ~c R z
dot to the radiation field
The radiative Hamiltonian, even in the presence of the
A =A ǫˆ ei(ω/c)nˆ·r−iωt+e−i(ω/c)nˆ·r+iωt , Rashba SO coupling can be expressed in the well-known
R 0
(cid:16) (cid:17) form
where ǫˆ, ω and nˆ are the polarization, frequency and e
H =i A ·[x,H ],
the directionofpropagationofthe incidentlight,respec- R c~ R S
tively. Welettheradiationenterthedotalongthedirec-
tion perpendicular to the motion of the electron, that is x being the position operator in the xy-plane.
paralleltothez-axis. Duetothetransversalitycondition
the polarization vector will then lie in the xy-plane.
Our second approximation is the familiar dipole ap-
proximation. Weassumethattheamplitude ofradiation
As usual, we shallmake two approximations. First we canbetakenasconstantwithinthequantumdot,sothat
assumetheintensityofthefieldbesoweakthatonlythe we are allowed to write the field as
terms linear in A has to be taken into account. Then
R
the effect of the radiative magnetic field on the spin can A ≈A ǫˆ e−iωt+eiωt .
R 0
(cid:0) (cid:1)
4
Since the transitionenergiesexpressedin terms ofradia- flip and is therefore strongly forbidden without the SOI.
tion frequences are of the order of THz, the correspond- Because the SOI mixes the state 1,0;1 into the for-
2
ing wavelengths are much larger than the typical size of mer one and the 0,1;−1 into the(cid:12)latter(cid:11)one, the tran-
2 (cid:12)
a dot, thus justifying our approximation. Applying now sition becomes po(cid:12)ssible. (cid:11)The appearance of other new
(cid:12)
theFermiGoldenRuleleadstothedipoleapproximation lines can be explained by analogous arguments. There
form are also additional features involving discontinuities and
anti-crossings in Fig. 4. A comparision with the energy
σ (ω)=4π2α ω |hn|ǫˆ·x|ii|2δ(ω −ω)
abs f ni ni spectra indicates that these are the consequences of the
of the absorption cross section for transitions from the anti-crossings present in the energy spectra.
inital state |ii to the final state |ni. Here α is the fine It is also readily verified that the oscillator strengths
f
structure constant and ω is the frequency correspond- satisfy the Thomas-Reiche-Kuhn sum rule [17]
ni
ing to the transition energy ~ω.
f =1.
The familiar dipole selection rules for oscillator states ni
X
dictatelargelythe featuresseeninFig.4. Intheabsence n
of the SOI, these rules – the spin state is preserved and In terms of the crosssectionthis translatesto the condi-
either n or n is changed by unity – completely deter- tion
1 2
mine the allowed two transitions 0,0;−1 → 1,0;−1
2 2 ∞ 2π2~α
and 0,0;−1 → 0,1;−1 . In c(cid:12)ontrast (cid:11)to th(cid:12)e case o(cid:11)f σ (ω)dω = f.
2 2 (cid:12) (cid:12) Z abs m∗
circu(cid:12)lar dots(cid:11)the ab(cid:12)sorption(cid:11)in the elliptical dot depends −∞
(cid:12) (cid:12)
stronglyonthe polarization. This is explainedby noting The absorptionsvisible in Fig. 4 practically saturate the
that the oscillator strengths sumrule,the saturationbeing,of coursecomplete in the
2m∗ω absence of the SOI in panels (a) and (d). The largest
f = ni |hn|ǫˆ·x|ii|2. fraction (of the order of 1/10) of the cross section either
ni ~
fallingoutsideofthedisplayedenergyscaleorhavingtoo
actuallyprobetheoccupationsofquantumstatesrelated
lowintensity to be discernible in ourpictures is foundat
to oscillations in the direction of the polarizationǫˆ. In a the strongest Rashba coupling in the panels (c) and (f)
circulardotalloscillationdirectionsareequallyprobable
for large magnetic fields, as expected.
at all energies implying that the oscillator strengths are
The results presented here clearly indicate that, the
independent ofthe polarizationanddepend onlyslightly
anisotropyofaQDalonecausesliftingofthedegeneracies
onthetransitionenergyviaω ,andthefinalstatequan-
ni oftheFock-DarwinlevelsatB=0,asreportedearlier[13].
tum numbers n . When the dot is squeezed in the y-
1,2 However, for large SO coupling strengths α, the effects
direction,say,theoscillatorstatesrelatedtothey-motion
of the Rashba SOI, mainly the level repulsions at finite
are pushed up in energy. This means that the polariza-
magneticfields,aremaginifiedrathersignificantlyasone
tion being along x-axis most of the oscillator strength introduces anisotropy in the QD. This is reflected also
comes from transitions to allowed states with lowest en-
in the corresponding dipole-allowed optical transitions
ergies. Similarly,whentheincidentradiationispolarized
wherethedistinctanti-crossingbehaviorisobservedthat
along the y-axis most of the contribution is due to the
is a direct manifestation of the anti-crossings in the en-
transitions to the oscillator states pushed up in the en-
ergy spectra. This prominent effect of the Rashba SOI
ergy. Inellipticaldotstheoscillatorstatesarenotpurex-
predicted here could be confirmed experimentally in op-
and y-oscillators but their superpositions. Therefore in
tical spectroscopy and the Fock-darwin spectra of few-
additiontothemainabsorptionlines,otherallowedfinal
electron QDs [9, 18, 19]. It would also provide a very
states have also non-vanishing oscillator strength. Fur-
usefulstepto controlthe SOcoupling innanostructures,
thermore, as one can see by looking at the phase space
en route to semiconductor spintronics [3].
rotationformulastheexternalmagneticfieldtendstoro-
The work was supported by the Canada Research
tate directions of the oscillator motion causing a shift of
Chairs Programof the Government of Canada.
the oscillator strength from an allowed transition to an-
other. ThisisexactlywhatweseeinFig.4(a)andFig.4
(d).
Even in the presence of the SOI the two allowed final
oscillator states provide major contributions to the cor- [‡] Electronic address: [email protected]
responding correctedstates. Hence we still see two dom- [2] Y.Oreg,P.W.Brouwer,X.Waintal,andB.I.Halperin,in
Nano-Physics&Bio-Electronics: ANewOdyssey,edited
inant absorption lines. However, now many forbidden
by, T. Chakraborty, F. Peeters, and U. Sivan (Elsevier,
transitions have become allowed. The lowest absorption
Amsterdam, 2002).
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2 2
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(cid:12) (cid:12)
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