Spin-selective Aharonov-Bohm oscillations in a lateral triple quantum dot.
ABSTRACT We present a theory of spin-selective Aharonov-Bohm oscillations in a lateral triple quantum dot. We show that to understand the Aharonov-Bohm (AB) effect in an interacting electron system within a triple quantum dot molecule (TQD) where the dots lie in a ring configuration requires one to not only consider electron charge but also spin. Using a Hubbard model supported by microscopic calculations we show that, by localizing a single electron spin in one of the dots, the current through the TQD molecule depends not only on the flux but also on the relative orientation of the spin of the incoming and localized electrons. AB oscillations are predicted only for the spin singlet electron complex resulting in a magnetic field tunable "spin valve."
arXiv:0809.1621v1 [cond-mat.str-el] 9 Sep 2008
Spin selective Aharonov-Bohm oscillations in a lateral triple quantum dot
F. Delgado1,2, Y.-P. Shim1, M. Korkusinski1, L. Gaudreau1,3, S. A. Studenikin1, A. S. Sachrajda1, and P. Hawrylak1,2
1Institute for Microstructural Sciences, National Research Council, Ottawa, Ontario, Canada K1A 0R6
2Department of Physics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 and
3R´ egroupement Qu´ eb´ ecois sur les Mat´ eriaux de Pointe,
Universit´ e de Sherbrooke, Qu´ ebec, Canada J1K 2R1
We present a theory for spin selective Aharonov-Bohm oscillations in a lateral triple quantum
dot. We show that to understand the Aharonov-Bohm (AB) effect in an interacting electron system
within a triple quantum dot molecule (TQD) where the dots lie in a ring configuration requires one
to not only consider electron charge but also spin. Using a Hubbard model supported by microscopic
calculations we show that, by localizing a single electron spin in one of the dots, the current through
the TQD molecule depends not only on the flux but also on the relative orientation of the spin of
the incoming and localized electrons. AB oscillations are predicted only for the spin singlet electron
complex resulting in a magnetic field tunable “spin valve”.
PACS numbers: 73.21.La,73.23.Hk
The Aharonov-Bohm (AB) effect results from the
accumulation of phase by a charged particle moving in
a ring threaded by a magnetic flux [2, 3]. AB oscilla-
tions are detected e.g. in the magnetization of a macro-
scopic number of electrons in metallic rings as well
as in the optical emission from a charged exciton in a
nanosize semiconductor quantum ring . The prepa-
ration, manipulation and detection of individual spins
of localized electrons in nanoscale semiconductor sys-
tems are important elements of nano-spintronic appli-
cations [6, 7, 8], with efficient generation and detection
of spin polarized carriers playing a crucial role.
electron spins can be localized in single and coupled
semiconductor quantum dots (QDs) defined and con-
trolled electrostatically[9, 10, 11, 12, 13, 14] with poten-
tial applications as elements of electron-spin based cir-
cuits [15, 16], coded qubits , entanglers , rectifiers
and ratchets [19, 20]. The Spin blockade technique in
a double dot system used for the conversion of spin to
charge information has played an important role in the
development of such applications .
The possibility of the co-existence of spin blockade
with AB oscillations in a lateral TQD in a ring geom-
etry [12, 13, 14] was discussed in Ref. 22. In this paper
we describe a TQD, shown schematically in Fig. 1(a),
where two dots, 1 and 3, are connected to the leads and
in addition to dot number 2. A single electron spin is lo-
calized in dot 2 by lowering the confining potential. The
transport of an additional electron through the TQD will
now depend on the relative orientation of the spin of the
incoming and localized electrons. If the two spins are
anti-parallel, as shown in Fig. 1(b), the additional elec-
tron can tunnel from the left lead to dot 1, and proceed
either directly to dot 3 or through dot 2 to dot 3 and
thus to the right lead. In the presence of the magnetic
field the two paths acquire a different phase and can in-
terfere, resulting in the AB oscillations of the current
amplitude (upper inset). When the spin of the incoming
electron is parallel to the spin of the electron in dot 2, the
Pauli exclusion principle prevents tunneling through dot
2, resulting in a single tunneling path and the absence of
AB oscillations. (lower inset). We present here the the-
ory of these spin selective AB oscillations in transport
through a TQD in a perpendicular magnetic field with a
controlled number of electrons. The electronic properties
of a TQD are treated by a fully microscopic LCHO-CI
approach and by Hubbard and t-J models with exact
many-electron eigenstates obtained using configuration-
interaction (CI) method . The Fermi Golden Rule
and the sequential tunneling approach are used to
calculate the current through the TQD weakly connected
to two non-interacting leads. The current flows when the
chemical potential of the TQD is equal to the chemical
potential of the leads. This can also be understood in
terms of degeneracies of many electron charge configura-
tions (N1,N2,N3), with Nithe number of electrons in dot
i. The degeneracy point described here, referred to as the
quadrupole point (QP), involves the one electron config-
uration (0,1,0) and two-electron configurations (1,1,0),
(0,2,0) and (0,1,1), with one electron always confined
in dot 2, as shown in Fig. 1(b).
For clarity we only present results of the Hubbard
model with one orbital per dot [22, 24]. The Hamiltonian
of the TQD subject to a uniform perpendicular magnetic
field, B = Bˆ z, is given by
where the operators diσ(d†
tron with spin σ = ±1/2 on orbital i (i = 1,2,3).
density on orbital level i. Each dot is represented by a
single orbital with energy Eiσ= Ei+g∗µBBσ+E0, where
iσ) annihilate (create) an elec-
iσdiσand ̺i= ni↓+ ni↑are the spin and charge
FIG. 1: (a) Schematic diagram of the TQD close to the con-
sidered QP. (b) Electrons with antiparallel spins can form
a loop an the corresponding energy levels E experiment AB
like oscillations with the magnetic flux φ while electrons with
parallel spin are spin-blockaded.
g∗is the effective Land´ e g-factor, µB is the Bohr mag-
neton and E0 is the common energy shift of the three
dots measured from the Fermi level of the leads which
is tunable by external gates. The dots are connected
by magnetic field dependent hopping matrix elements
˜tij= tije2πiφij. For the three dots in an equilateral
configuration φ12= φ23= φ31= −φ/3 and φji= −φij,
where φ = BA/φ0is the number of magnetic flux quanta
threading the area A of the triangle, φ0 = hc/e is the
magnetic flux quantum, e is the electron charge, c is the
speed of light and ¯ h is the Planck’s constant. The in-
teracting part of the Hamiltonian is parametrized by the
on-site Coulomb repulsion, Ui, and the interdot direct
repulsion term Vij.
In order to describe transport through the TQD we
first determine the QP of the isolated TQD. We start
by determining the “classical QP” where we neglect the
inter-dot tunneling and require the four configurations
A ≡ (1,1,0), B ≡ (0,2,0), C ≡ (0,1,1) and D ≡ (0,1,0)
to have equal energy. Their energies are ǫA= E1+E2+
V12+2E0, ǫB= 2E2+U2+2E0, ǫC= E2+E3+V23+2E0,
and ǫD= E2+E0. The QP condition without tunneling
requires ǫA = ǫB = ǫC = ǫD+ µL, where µL is the
chemical potential of the leads. This implies that at the
= µL− E0− V12, EQ
3= µL− E0− V23.
Let us consider now the case of finite tunneling matrix
elements. The (0,1,0) charge configuration describes the
= µL− E0− U2 and
FIG. 2: Lowest energy spectrum of the two electron TQD
at the QP (upper panel) and total spin of the ground state
(lower panel) versus the magnetic flux. The QP condition was
found numerically for δ1 = δ3 = 2.44|t| and δ2 = 2.77|t|.
two spin states of an electron localized in dot 2, |2σ? ≡
2σ|0? with energy E2(|0? is the vacuum state). The two
electron classical charge configurations (1,1,0), (0,2,0)
and (0,1,1) correspond to the following quantum spin
singlet configurations: |S1? =
|S2? = d†
Hamiltonian describing the motion of the spin singlet pair
takes the form
2↓|0? and |S3? =
At the classical QP, we have ǫA
into a new basis:
|K2? = 1/√3?|1? + ei2π/3|2? + ei4π/3|3??
1/√3?|1? + e−i2π/3|2? + e−i4π/3|3??
ε1= E − 2|t|cos(2πφ/3), ε2= E − 2|t|cos[2π(φ + 1)/3]
and ε3 = E − 2|t|cos[2π(φ − 1)/3], respectively. Since
one of the electrons is kept in dot 2, the energy spectrum
of a pair of singlet electrons is essentially the same as
that of a single electron added to a resonant TQD,
with the energy levels oscillating with a period of one
flux quantum . Away from the resonance the level
crossing is replaced by anti-crossing.
A pair of spin triplet electrons describes only (1,1,0)
and (0,1,1) charge configurations. The corresponding
two spin triplet configurations for SZ = 1 are |T1? =
√2t13,we can diagonalize the
|K1?1/√3(|1? + |2? + |3?),
and |K3? =
2↑|0? with energy ǫA(B) and |T2? = d†
FIG. 3: Stability diagram of the TQD close to the QP with
charge configurations (1,1,0), (0,2,0), (0,1,1) and (0,1,0)
at (a) φ = 0 and (b), φ = 0.44. The classical QP (t = 0) is
found at Vg1 = Vg2 = 0 while the quantum one at φ = 0.44 is
indicated by the white circle.
energy ǫC(B), with ǫA(B) = ǫA + g∗µBBSz.
eigenenergies of the 2 × 2 triplet Hamiltonian are ε±
By comparing the singlet and triplet eigenvalues we see
that singlet is the ground state at B = 0 and the eigen-
values of the triplet do not oscillate as a function of the
magnetic field. Even at this qualitative level, we obtain a
remarkable result that triplet states do not oscillate with
the magnetic field while singlets do.
In the case of finite tunneling each classical configura-
tion is no longer an eigenstate of the system. Therefore,
we will define QP as the point in the parameter space
where the ground state energies of two and one electrons
differ by µLand the three degenerate two-electron config-
urations are found with the same probabilities. Then, at
the QP Ei= EQ
corrections that are obtained numerically, with δ1= δ3
for the symmetric case described here.
We shall analyze now the magnetic field dependence of
the two electron energy spectrum close to the QP. The
numerical CI calculations include the full Hilbert space
generated from the three orbital levels. Hubbard param-
eters were obtained from the LCHO calculation with an
interdot distance of 61.2 nm: t = −0.23 meV, Ui= 50|t|
and Vij= 10|t|. g∗= −0.44 corresponding to GaAs will
be assumed. The upper panel of Fig. 2 shows the lower
(ǫA(B) − ǫC(B))2+ 4|t13|2?1/2?.
i+δi, where the energies δiare quantum
part of the energy spectrum for E0= −|t|, while the low-
est panel indicates the total spin of the ground state. As
was described above, the singlet (solid line) is the ground
state at B = 0. The lowest energy of a singlet oscillates
with period of one flux quantum while the energy of a
triplet decreases monotonically with increasing magnetic
field due to Zeeman energy. Notice that triplets show
a small oscillation due to a coupling with higher energy
configurations. The oscillating singlet energy and mono-
tonically decreasing triplet energy leads to a number of
transitions between singlet and triplet with increasing
magnetic field. These transitions interrupt the AB os-
cillations of the singlet, and lead to their end at a criti-
cal value of the magnetic field, BC= −∆ST/g∗µB with
φC= ABC/φ0, indicated in Fig. 2. Above φCthe triplet
is the ground state. Hence the presence of a trapped
electron should lead to AB oscillations of the tunneling
electron, interrupted and eventually terminated by the
Figure 3 shows the dominant charge ground state con-
figurations of the TQD at two different values of the mag-
netic flux quantum, φ = 0 (upper panel) and φ = 0.44
(lower panel), versus the voltages Vg1and Vg2for E0=
−|t|. Here it has been assumed that the on-site ener-
gies Ei’s change linearly with the voltages Vg1 and Vg2,
Ei = αiVg1+ βiVg2+ γi, with αi,
experiment in Ref. 13. For the chosen value of E0, the
zero magnetic field stability diagram shows only a triple
point, while at φ the QP is clearly visible.
We now turn to the illustration how these spin selective
AB oscillations can be observed in transport experiment.
Following Ref. 27, the Hamiltonian of the TQD connected
to two leads is given by H = HL+HTQD+HLD, where
HLis the Hamiltonian describing the two non-interacting
leads, HTQDcorresponds to the isolated triple dot where
we assume that the on-site energies change with the ap-
plied bias ∆V as Eiσ → Eiσ− ∆V/2 and HLD is the
tunneling Hamiltonian between the leads and the TQD.
The leads are described with a one-dimensional tight-
binding model with nearest neighbor hopping tL, on-
site energies ǫL and ǫR for the left (right) leads and
coupling strength between dots and leads tLD. The
current through the system is evaluated using a set of
master equations for the occupation probabilities within
the sequential tunneling approximation . In this ap-
proach we neglect higher order processes such as co-
tunneling which is important for high tunnel-coupling
strengths and for temperatures below the Kondo tem-
perature [28, 29]. The occupation probabilities are then
calculated using a detailed balance condition imposed by
the conservation of charge. The spin components of the
current in the linear regime at the lowest order in the
coupling tLD and at zero temperature are then given
by Iσ= eπ/(2¯ h)|tLD|2ρ(εF)∆V Cσδ(εF− (ε2,G− ε1,G))
where ε2,G(ε1,G) is the ground state energy of the two
(one) electrons and ρ(εF) is the density of states in the
βi extracted from
FIG. 4: Spin down (solid line) and spin up (dashed line) com-
ponents of the conductance in units of G′
versus the number of flux quantum φ for the same parame-
ters as in Fig. 2. The inset shows the energy spectrum of the
two electron complex (as in Fig. 2), together with the one
electron lowest levels (thick blue line).
0= e2|tLD|2/¯ h|tL|2
leads at the Fermi level. Here we make the assumption
ρL(εF,L) ≈ ρR(εF,R). Cσ= 1/3 for σ =↓ (singlet ground
state) and Cσ= 1 for σ =↑ (triplet ground state).
Next we present the results for the linear conductance
G = I/∆V . The calculations were done at 50 mK
(kBT = 0.0145|t|), ∆V = 2×10−3|t| and µL= 0. In ad-
dition, |tL| = 23.72 meV ≫ |t|, E0, ∆V . Since transport
through the TQD is allowed whenever the single-particle
ground state and the two-particle ground state are on
resonance, the AB oscillations of the energy spectra lead
to repeated peaks in current. The spin components of the
conductance Gσ= Iσ/∆V are shown in Fig. 4. At low
magnetic fields, the spin down current is dominant and
transport is mainly through the lowest oscillating singlet
state. When the ground state of two particles becomes
triplet, spin up current is dominant until the current is
In summary, the presence of an extra electron localized
in one dot of a ring-like TQD leads to spin selective AB
oscillations as a function of magnetic field. The energy of
the singlet ground state oscillates as a result of the inter-
ference between the two possible paths while the triplet
state does not oscillate since one of the paths is spin
blockaded by the presence of a localized particle. The
magnetic field orients the spin of the localized particle
leading to the transport of electrons with a specific spin
polarization. The AB oscillation of the singlet electron
pair is reflected as peaks in the spin-down polarized cur-
rent. At higher magnetic field, the Zeeman energy causes
a singlet-triplet transition, which results in a change of
the dominant spin component of the current.
The Authors acknowledge support by the Quantum-
Works Network and the Canadian Institute for Advanced
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