Magnetization of Quantum Dots: A Measure of Anisotropy and the Rashba Interaction

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by  Siranush Avetisyan

Magnetization of Quantum Dots: A Measure of Anisotropy and the Rashba
Interaction
Siranush Avetisyan,1 Tapash Chakraborty‡,1 and Pekka Pietil¨ainen2
1Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2
5 2Department of Physics/Theoretical Physics, University of Oulu, Oulu FIN-90014, Finland
1
0 The magnetization of anisotropic quantum dots in the presence of the Rashba spin-orbit inter-
2
action has been studied for three interacting electrons in the dot. We observe unique behaviors of

n magnetizationthataredirectreflectionsoftheanisotropyandthespin-orbitinteractionparameters
a independently or concurrently. In particular, there are saw-tooth structures in the magnetic field
J dependence of the magnetization, as caused by the electron-electron interaction, that are strongly

5 modified in the presence of large anisotropy and high strength of the spin-orbit interactions. We
report thetemperature dependenceof magnetization that indicates the temperature beyondwhich

] thesestructuresduetotheinteractions disappear. Additionally,wefound theemergenceof aweak
ll sawtooth structure in magnetization in the high anisotropy and large spin-orbit interaction limit
a that was explained as a result of merging of two low-energy curves when the level spacings evolve
h
with increasing valuesof theanisotropy and the spin-orbit interaction strength.
-
s
e PACSnumbers: 73.21.La,78.67.Hc
m
.
t The magnetization of quantum confined planar elec- anisotropy and large spin-orbit interaction limit that we
a
m tron systems, e.g. quantum dots (QDs), or artificial explain as a result of merging of two low-energy curves
atoms2–4 is an important probe that reflects entirely on when the level spacings evolve with increasing parame-
-
d the properties of the energy spectra. This is a thermo- ters. With the help of the theoretical insights presented
n dynamical quantity that for the QDs has received some here, experimental studies of magnetization will there-
o experimentalattention5–7,particularlyafter the theoret- fore provide valuable information on the inter-electron
c ical prediction that the electron-electron interaction is effects, the Rashba spin-orbit coupling and the degree of
[
directly manifested in this quantity8. In addition to the anisotropy of the quantum dots.

1 large number of theoretical studies reported in the lit- At zero temperature the magnetization M of the QD
v erature on the electronic properties of isotropic quan- isdefinedasM=−∂Eg whereE isthegroundstateen-
5 tum dots, there has been lately some studies on the ergyofthesystem8,9.∂BWehavestugdiedthemagneticfield
2 anisotropic quantum dots, both theoretically10,11 and
0 dependence of M by evaluating the expectation value of
1 experimentally12. Theoretical studies of the magnetiza- the magnetization operator mˆ = −∂H, where H is the
tion of elliptical QDs have also been reported13. Effects ∂B
0 systemHamiltonian. Since the Columb interactionis in-
1. of the Rashba spin-orbitinteraction (SOI)14 on the elec- dependent of B, mˆ would be just a one-body operator,
tronic properties of isotropic15 and anisotropic quantum
0 i.e.,wecanignoretheinteractionpartfromtheHamilto-
dots16havebeeninvestigatedearlier. Anexternalelectric
5 nian. The Hamiltonian of a single-electron system sub-
1 field can induce the Rashba spin-orbit interaction which jected to an external magnetic field with the vector po-
v: cpouulspiolensstihnethdeiffeenreerngtyspspinecsttrautmes15a–n17d.iTnthriosdcuocuepslilnegveilsraen- tential A = 12B(−y,x), the confinement potential, and
i the Rashba SOI is
X importantingradientfortheburgeoningfieldofsemicon-
1 e 2
r ductor spintronics, in particular, for quantum comput- H = p− A + 1m ω2x2+ω2y2
a ers with spin degrees of freedom as quantum bits18,19. 2me (cid:16) c (cid:17) 2 e(cid:0) x y (cid:1)
Three-electronquantumdots areparticularlyrelevantin + α σ×(p− eA) + 1gµ Bσ .
this context20,21. Here we report on the magnetic field ~ h c iz 2 B z
dependence of the magnetization of an anisotropic QD
containing three interacting electrons in the presence of
The first term of the Hamiltonian is the kinetic energy,
the Rashba SOI. Our present work clearly indicates how
which can be written as
the magnetization of the QDs uniquely reflects the in-
fluence of anisotropy and the Rashba SOI, both con- 1 eB e2B2
K = p2 +p2+ (yp −xp )+ (y2+x2) .
currently as well as individually as the strengths of the 2m (cid:18) x y c x y 4c2 (cid:19)
e
SOI and the anisotropy are varied independently. The
temperature dependence of magnetization is also stud- The SOI part (third term) is
ied here, where we noticed the gradual disappearence
α eB eB
of the interaction induced structures in magnetization H = σ (p − x)−σ (p + y) ,
SO ~ (cid:20) x y 2c y x 2c (cid:21)
with increasingtemperature. Another importantfeature
that we found in our present study is the emergence of while the second and the last term correspond to the
a weak sawtooth structure in magnetization in the high confinementpotentialandtheZeemanterm,respectively.
2
wherethepartialderivativeswereevaluated,asexplained
above,astheexpectationvaluesofthemagnetizationop-
eratorintheinteractingstateslabelledbym. Inelliptical
confinements,themutualCoulombinteractionishandled
bythe numericalschemeelucidatedpreviously22,i.e., we
diagonalizethemany-bodyHamiltonianinthebasiscon-
sisting of non-interacting many-body states, which are
constructed by the SO coupled single-particle spinors.
Thesespinorsareinturn,astheresultofthediagonaliza-
tion of the SO Hamiltonian, expressed as superpositions
of the fundamental 2D oscillator spinors
|n ,n ;s i=|n ,n i|s i.
x y z x y z
Here n and n are the oscillator quantum numbers and
x y
|s i stands for the spinors
z
1
|+1i=|↑i=
(cid:18)0 (cid:19)
0
|−1i=|↓i= .
FIG.1: Temperaturedependenceofmagnetizationofathree- (cid:18)1(cid:19)
electron anisotropic dot without the Rashba SOI (α = 0).
The results are for ωx = 4 meV and (a) ωy = 4.1 meV, (b) We see that in the end the magnetization evaluation re-
ωy = 6 meV, (c) ωy = 8 meV, and (d) ωy = 10 meV. The duces to a many-fold summation of the matrix elements
zero-temperaturemagnetizationcurveforthenon-interacting
system is also shown in red. hn′,n′;s′|mˆ|n ,n ;s i
x y z x y z
e ′ ′
= − hn ,n ;s |yp −xp |n ,n ;s i
2m c x y z x y x y z
We then need to evaluate the expectation value of the e
magnetization operator − e2B <n′,n′;s |y2+x2|n ,n ;s i
4m c2 x y z x y z
∂H e
mˆ = − αe ′ ′ ′
∂B + 2c~ <nx,ny;sz|σxx+σyy|nx,ny;szi
= − 1 e (yp −xp )+ eB(y2+x2) − 1gµ hn′,n′;s′|σ |n ,n ;s i,
2m c (cid:18) x y 2c (cid:19) 2 B x y z z x y z
e
eα
+ (σ x+σ y)− 1gµ σ , which is now susceptible to direct numerical evaluation.
2c~ x y 2 B z
with respect to the interacting electron states. We
In our numerical investigations, we chose the InAs
should, however, point out that the energy spectra in
quantumdotthatshowsstrongRashbaeffects15,16,22. In
the present studies were evaluated for the Hamiltonian
with the Coulomb interaction V =e2/ǫr included, as in this case, the relevant parameters are: ǫ = 15.15,g =
ourearlierwork22butnowforthcreeinteractingelectrons. −14,me = 0.042. The energy spectra are shown in Fig.
2,Fig. 4,andFig. 6forvariousvaluesoftheSOcoupling
Here ǫ is the background dielectric constant.
strengthα andfordifferent valuesofthe anisotropy. For
B =0thegroundstatesaretwo-folddegeneratenomat-
ter howstrongthe SO couplingis orhowanisotropicthe
We have also studied the finite-temperature behav- QDbecomes. Interestingly,contrarytoourexpectations,
ior of the magnetization, following the thermodynami- at non-zero magnetic fields most of the level crossingsof
cal model discussed earlier23. Since we are investigating the energy spectra do not turn into anticrossings when
the system with a fixed number of electrons, we use the the SO coupling is turned on. Only for a strong value
canonicalensemble. The temperature dependence of the of the Rashba parameter α (α = 40) [Fig. 6 (a), (b)]
magnetization was therefore evaluated from the thermo- the level crossings transform to level repulsions. How-
dynamic expression ever,when the QD becomes more anisotropicthose level
repulsions reappear as level crossings [Fig. 6 (c),(d)].
∂E
M= m e−Em/kT/ e−Em/kT, (1)
∂B
Xm Xm
3
FIG. 4: Same as in Fig. 2, but for α=20 meV.nm.
FIG.2: Energylevelsofathree-electronanisotropicdotwith-
outtheRashbaSOI(α=0). Theresultsareforω =4meV
x
and(a)ω =4.1meV,(b)ω =6meV,(c)ω =8meV,and
(d) ω =y10 meV. y y retical works8 (and confirmed in our present work), the
y
electron-electron interaction causes saw-tooth structure
in the magnetic field dependence of the magnetization,
whichis aconsequenceofthe changeinthe groundstate
energy from one magic angular momentum to another
(in the case of isotropic QDs)3. An interesting behav-
ior of magnetization that should be pointed out here is
that with increasing strength of the Rashba SO parame-
ter α the jump in magnetization at the level crossingsin
the energyspectra,movestolowermagneticfields,while
increasinganisotropyoftheQDpushesthejumpinmag-
netization to higher magnetic fields. For InAs elliptical
QDs this shift is at most ∼1 Tesla, when α is increased
from zero to 40 meV.nm and ω is varied from 4.1 - 10
y
meV. Therefore, low-field magnetization measurements
of the QDs could be a direct probe of the SO coupling
strength.
The main feature of the temperature dependence of
magnetizationisthat,asthetemperatureisincreasedthe
saw-toothstructureofthemagnetizationcurvegradually
disappears (Fig. 1 and Fig. 3). An important point
to notice here is that the jump in magnetization slowly
FIG. 3: Sameas in Fig. 1, but for α=20 meV.nm. vanishes even in the absence of increasing temperature
for an anisotropic QD [Fig. 5 (a), (b)] which is a result
of large SO coupling. However, an emergent small jump
in magnetization is again visible (at T =0 K) for strong
Our results for the magnetic field dependence of the
anisotropic QDs and large SOI [Fig. 5 (c), (d)] which is
magnetization for anisotropic QDs are shown in Fig. 1,
clearly the result of two low-lying energy levels crossing
Fig. 3 and Fig. 5, calculated both with and without
near 1.2 Tesla [Fig. 6 (c), (d)] (marked by circles).
(red curves) the Coulomb interaction between the elec-
trons for various values of the SO coupling strength and
for various values of anisotropy. A major difference be-
tweenthethenon-interactingsystemandtheinteracting IntheabsenceoftheSOcouplingthemagnetizationis
system can be found in the magnetization results: while invariantunder the time reversal,whichimplies that the
there is no structure present in the non-interacting cases derivative of energy with respect to the magnetic field
(red curves), there are prominent structures for the in- must vanish at B = 0. This means a vanishing mag-
teracting systems. As it was predicted in earlier theo- netization at B = 0. However, non-zero SO coupling
4
breaks the time-reversal symmetry and in that case the
derivative of energy with respect to the magnetic field
can be discontinuous at B = 0, which would imply non-
vanishing magnetization at B =0 [Fig. 4 and Fig. 6].
Tosummarize: wehavereportedhere detailedandac-
curate studies of the magnetization of anisotropic quan-
tum dots with interacting electrons in the presence of
the Rashba SOI. The Coulomb interaction in the pres-
ence of the spin-orbit coupling exhibits a very strong
FIG. 5: Sameas in Fig. 1, but for α=40 meV.nm. effect on magnetization, particularly in the presence of
strong anisotropy by introducing large saw-tooth struc-
tures inthe magnetic fielddependence ofthe magnetiza-
tion, which is weakened by increasing temperature. In-
terestingly,thereisalsotheemergenceofthisstructurein
thehighanisotropyandlargeSOIlimitthatis explained
asduetomergingoftwolow-energycurveswhenthelevel
spacings evolve with increasing parameters. Any further
extension of the present work for a larger system would
be computationally verychallenging,but we expect that
the novel features observed in the present work will be
displayedin that case. That would be the subject of our
future publications. Armed with the theoretical insights
presented here, an experimental probe of magnetization
in anisotropic quantum dots will undoubtedly provide
valuable information about the inter-electron strength,
the strength of the QD anisotropy, as well as the SOI
strength in the quantum dot.
FIG. 6: Same as in Fig. 2, but for α = 40 meV.nm. The
circles in (c) and (d) indicate the level crossings that lead to The work was supported by the Canada Research
new structuresin Fig. 5 (c) and (d) at T =0 K.
Chairs Programof the Government of Canada.
‡ Electronic address: [email protected] Nazarov, N.C. van der Vaart, and L.P. Kouwenhoven,
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