Tunneling and magnetic properties of triple quantum dots
Unconventional features of Kondo effect in quantum tunneling through triple quantum dots are reviewed. Special attention is paid to the interplay between continuous rotation symmetries in the spin space and discrete point symmetries in the real space. Specific properties of Kondo effect in linear...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1275122025-06-03T16:29:06Z Tunneling and magnetic properties of triple quantum dots Kikoin, K. Электронные свойства низкоразмерных систем Unconventional features of Kondo effect in quantum tunneling through triple quantum dots are reviewed. Special attention is paid to the interplay between continuous rotation symmetries in the spin space and discrete point symmetries in the real space. Specific properties of Kondo effect in linear, cross, fork and triangular configurations of triple quantum geometries dot are discussed. 2007 Article Tunneling and magnetic properties of triple quantum dots / K. Kikoin // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 153-164. — Бібліогр.: 65 назв. — англ. 0132-6414 PACS: 72.10.Fk, 73.21.La https://nasplib.isofts.kiev.ua/handle/123456789/127512 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электронные свойства низкоразмерных систем Электронные свойства низкоразмерных систем Kikoin, K. Tunneling and magnetic properties of triple quantum dots Физика низких температур |
| description |
Unconventional features of Kondo effect in quantum tunneling through triple quantum dots
are reviewed. Special attention is paid to the interplay between continuous rotation symmetries in
the spin space and discrete point symmetries in the real space. Specific properties of Kondo effect
in linear, cross, fork and triangular configurations of triple quantum geometries dot are discussed. |
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Article |
| author |
Kikoin, K. |
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Kikoin, K. |
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Kikoin, K. |
| title |
Tunneling and magnetic properties of triple quantum dots |
| title_short |
Tunneling and magnetic properties of triple quantum dots |
| title_full |
Tunneling and magnetic properties of triple quantum dots |
| title_fullStr |
Tunneling and magnetic properties of triple quantum dots |
| title_full_unstemmed |
Tunneling and magnetic properties of triple quantum dots |
| title_sort |
tunneling and magnetic properties of triple quantum dots |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
| topic_facet |
Электронные свойства низкоразмерных систем |
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https://nasplib.isofts.kiev.ua/handle/123456789/127512 |
| citation_txt |
Tunneling and magnetic properties of triple quantum dots / K. Kikoin // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 153-164. — Бібліогр.: 65 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kikoink tunnelingandmagneticpropertiesoftriplequantumdots |
| first_indexed |
2025-11-26T13:15:23Z |
| last_indexed |
2025-11-26T13:15:23Z |
| _version_ |
1849858902288498688 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3, p. 153–164
Tunneling and magnetic properties of triple quantum dots
K. Kikoin
Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy
Tel-Aviv University, Tel-Aviv 69978, Israel
E-mail: kikoin@bgu.ac.il
Received November 7, 2006
Unconventional features of Kondo effect in quantum tunneling through triple quantum dots
are reviewed. Special attention is paid to the interplay between continuous rotation symmetries in
the spin space and discrete point symmetries in the real space. Specific properties of Kondo effect
in linear, cross, fork and triangular configurations of triple quantum geometries dot are discussed.
PACS: 72.10.Fk Scattering by point defects, dislocations, surfaces, and other imperfections (including
Kondo effect);
73.21.La Quantum dots.
Keywords: Kondo effect, quantum tunneling, triple quantum geometries.
Introduction
Soon after Kondo’s breakthrough in explanation for
the puzzling shallow minimum in the temperature de-
pendent resistivity of metals doped by magnetic impu-
rities [1], theoreticians started to extend this promising
conceptual framework for other physical situations and
for more complex objects than simple localized mo-
ments. His original idea was formulated within the
framework of a well established Hamiltonian describ-
ing exchange between the impurity spin Sj located on a
given site j and the spin pertaining to a Fermi sea of
conduction electrons. The latter is represented by itin-
erant spin skk k
†
k� � �� c c
� ��� projected on the impurity
site j, namely s s k - k jj kk
kk
� � ��
�
� exp ( )i . Here �� is
the vector of Pauli matrices for a spin 1/2. The so
called sd-exchange Hamiltonian is,
H c c Jsd k� � ���
�
� �
k,
k k j jS s† , (1)
where �k is the energy dispersion of the itinerant elec-
trons and J is the exchange coupling constant. The in-
ternal degrees of freedom involved in Kondo effect
are classified in terms of SU(2) group of continuous
rotations in the spin sector of Fock space.
It was recognized that the Kondo mechanism
should work also in systems exhibiting electron tun-
neling, where two metallic slabs are separated by thin
dielectric layer, which forms a tunnel barrier for elec-
trons in the slabs. It was shown [2,3] that the mag-
netic impurity located somewhere near the tunneling
layer plays the same role in tunnel conductance as
magnetic impurity immersed in a metal (and has ex-
change interaction with Fermi sea electrons) does for
impurity resistance. It was also shown [4] that the
Friedel–Anderson model [5,6] for resonance scatter-
ing of conduction electrons by the electrons occupying
the 3d-levels of transition metal impurities can be
mapped on the exchange Hamiltonian (1) provided
the strong Coulomb interaction in the 3D shell sup-
presses charge fluctuations on the impurity site. In the
next stage of development, orbital degrees of freedom
were incorporated in the Kondo physics. The basic
idea of this generalization was that the magnetic im-
purity being put in the center of the coordinates im-
poses its point symmetry on otherwise translationally
invariant crystal, and the adequate description of scat-
tered waves should be based on the partial wave ex-
pansion (either in spherical waves [5] or in cubic
harmonics [7]). Based on this idea, the generalized
Schrieffer–Wolff model was proposed [8,9], where
the magnetic impurity is described as an effective
N-component moment, but the exchange scattering is
not restricted by the usual spin selection rule
�m � �0 1, for the projection m of this moment. An-
other version of this model allots the impurity and
band states both by spin and orbital (referred to as
«color» in the general case) index. In case of spin
s /� 1 2 and N colors the symmetry of the impurity is
© K. Kikoin, 2007
SU(2N) [10,11]. Appearance of additional degrees of
freedom in the Kondo Hamiltonian might lead to the
interesting scenario of overscreened Kondo effect,
which arises when the number of conduction electron
«colors» exceed that of impurity moment [12]. This ef-
fect is characterized by the non-Fermi-liquid low-tem-
perature thermodynamics unlike the standard Kondo
effect, which only modifies (although radically) the
Fermi liquid properties of undoped metal [13].
A powerful incentive for further extension of the
realm of Kondo physics has been offered in 1988,
when the idea of underbarrier tunneling in presence of
Kondo center was extended on the tunneling between
metallic electrodes and nanoobjects like quantum dots
or small metallic grains [14,15]. Such nanoobject may
serve as a Kondo center (a localized moment) pro-
vided (i) the electron spectrum is discrete due to spa-
tial quantization, so that the level spacing � exceeds
the tunneling rate, (ii) Coulomb blockade prevents
charge fluctuations and (iii) the electron occupation
number is odd, so that the effective spin of the
nanoobject is 1/2. In this case the tunneling Hamil-
tonian may be mapped on the effective spin Hamil-
tonian (1), and the Kondo-like singularity arises as a
zero-bias anomaly (ZBA) in tunnel transparency. This
theoretical prediction was confirmed ten years later
[16–18] in the experiments on planar quantum dots.
Many experimental and theoretical studies then fol-
lowed this experimental breakthrough ever since.
Another facet of Kondo physics in nanoobjects was
unveiled, when the possibility of Kondo effect in
quantum dots with even electron occupation number
was considered in several theoretical publications
[19–22]. In this case the quantum with dot a singlet
ground state may become magnetically active due to
external forces, and Kondo effect arises either at finite
energy [19] or in external magnetic field [20–22].
Later on it was recognized that in many cases the di-
rect mapping of the original Kondo model into such
system is impossible, because the effective symmetry
of the pertinent nanoobject is neither SU(2) nor
SU(2N). The aim of this short review is to describe
various physical situations where the underlying
nanoobject possesses complex (and in some sense un-
usual) symmetries which are characterized by non-com-
pact Lie groups or combinations of such groups with
discrete groups of finite rotations.
Surplus symmetries
Among the sources of surplus symmetries which en-
rich Kondo physics of nanoobjects one may find both
discrete and continuous rotations stemming from com-
plicated geometrical configurations of complex quan-
tum dots, as well as those induced by external fields
used in practical realizations of nanodevices. In this
short review we will refrain from description of great
variety of these devices, which may be found in cur-
rent literature (see, e.g., [23–25]). Fortunately, most
of the relevant physics may be exposed in a relatively
simple situation of electron tunneling through
multivalley quantum dots in contact with metallic
electrodes (leads).
A multivalley dot is an island with electrons con-
fined by electrostatic potential in such a way that the
spatially quantized electrons are distributed between
several valleys. These valleys are coupled with each
other by capacitive interaction and tunneling chan-
nels. Up to now there are several realizations of quan-
tum dots with two and three valleys (double quantum
dot, DQD, and triple quantum dot, TQD, respec-
tively). Experimentally, the first such realizations of
DQDs go back to mid 90-es [26–28]. It was pointed
out that these objects can be treated as some forms of
coupled artificial molecules with each dot playing the
role of an artificial atom [29–32].
Compared with DQD, fabrication of TQD is a
much more difficult experimental task and the first
experimental realizations of these nanoobjects ap-
peared only recently. One may mention in this connec-
tion the realization of TQD with an «open» central
valley [33]. The term «open» here means that the tun-
neling between the side dots and the adjacent leads is
limited by strong Coulomb blockade (see below),
whereas the central dot freely donates and accepts
electrons to and from its own reservoir, so its role in
the device is only to mediate indirect exchange be-
tween the two side dots. Another successful attempt to
fabricate a TQD was recorded in response to a theoret-
ical proposal [34] to use this geometry for realization
of ratchet effect in tunneling through nanoobjects. In
this realization [35] the charge fluctuations were sup-
pressed by the Coulomb blockade mechanism in all
three valleys. The feasibility of filling the TQD with
1, 2 and 3 electrons by changing the gate voltages was
demonstrated quite recently [36].
Theoretical studies of electronics in TQD geometry
were also inspired by possible applications in the field
of quantum information [37,38]. Experimental obser-
vation of molecular trimers Cr3 on gold sublayers [39]
by means of the tunnel electron spectroscopy, gave an
impact to the theory of Kondo effect in TQD [40].
Later on, other properties of molecular trimers such as
the two-channel Kondo effect [41,42] and the inter-
play between the Kondo effect and Aharonov–Bohm
effect in tunnel spectroscopy [43] were considered the-
oretically.
Our main focus of interest is a theoretical modeling
of a device consisting of a triple quantum dot, metallic
154 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
K. Kikoin
electrodes and corresponding gates. The latter regu-
late the electron occupation of any particular valley
and of the dot as a whole. These and properties other
are predetermined by the geometrical position of the
valleys relative to the metallic leads (source and
drain) in the device. Triple quantum dots (TQD)
provide theoreticians (and experimentalists as well)
with a rich variety of geometrical configurations and
possess more symmetries (Fig. 1). Similarly to DQD
these trimers may be oriented both in sequential (ver-
tical) and parallel (lateral) geometries (Fig. 1,a and
1,b, respectively). The natural generalization of the
T-shape geometry presented in Fig. 1,c is the cross ge-
ometry (Fig. 1,c). Besides, TQD may be organized in
a form of a triangle, which may form a closed or open
element in an electronic circuit (Figs. 1,d,e and 1,f,
respectively). In the two latter cases one deals with a
three-terminal tunnel device. We will call the confor-
mations shown in Figs. 1,e and 1,f «ring» and «fork»
configurations.
Let us discuss the discrete symmetry elements char-
acteristic for the above TQD configurations. If all
three valleys are equivalent, the discrete symmetry of
an isolated linear TQD is that of the permutation
group P3. The contact with the leads adds one more
symmetry element, the (s—d)-reflection, provided all
three dots are coupled with the leads. We however
consider the devices, where the central dot (filled
black) is not coupled directly with the leads. Then the
P3 symmetry is lost. The only discrete symmetry ele-
ment, namely (s—d)-reflection is left in a vertical
geometry (Fig. 1,a), whereas both (s—d)- and
(l—r)-reflections (which is equivalent to P2 permuta-
tion) characterize the symmetry of lateral TQD (Fig.
1,b). The same is valid for the cross geometry of Fig.
1,c. The P3 symmetry is inherent in the three-terminal
configuration of Fig. 1,e. In this case it is better to use
the classification of discrete rotation groupC v3 , which
is isomorphic to the permutation group P3. One may
say [43] that in the perfect triangular configuration the
TQD imposes itsC v3 symmetry on the device as a whole
in close analogy with the Coqblin—Schrieffer—Cornut
version [8,9] of the conventional Kondo problem. In the
geometries of Fig. 1d,f the only element which is re-
mained of the original symmetry of triangle is the
(l—r)-symmetry like in the cross geometry. External
magnetic field applied perpendicularly to the plane of
the triangle lowers the symmetry of a device by adding
one more element, that is, chirality of the electron
tunneling from the source to the drain.
Before considering the consequences of these sym-
metries for the Kondo physics, several introductory re-
marks about the derivation of the Kondo Hamiltonian
are in order. The generally accepted approach is based
on the assumption that electrons injected from the
source do not lose coherence when propagating through
the sequence of quantum dots until they leave the
chain for the drain electrode. This approach is valid
only provided the tunneling W between the adjacent
valleys exceeds the tunneling V between the dot and
the metallic leads. Working in this paradigm, one
should derive the effective exchange Hamiltonian Hex
in accordance with the following procedure. The start-
ing Hamiltonian is chosen in the same form as the
original Anderson impurity Hamiltonian [6]:
H H H Ht� � �band dot (2)
where Hdot describes the properties of a chain de-
tached from the lead in terms of its eigenstates |
�:
H E Q Ndot � �� �
� �
�
| | ( � )N . (3)
Here �N is the operator of total electron number in the
dot. The last term in (3) describes the Coulomb
blockade mechanism: the total occupation of the dot
N in neutral and charged states is fixed by the Cou-
lomb blockade parameter Q entering the capacitive
energy of a complex quantum as dot a whole. The
eigenvalues E� are found at fixed occupation N of
DQD or TQD with all tunneling matrix elements W
and interdot capacitive interactions Q� taken into ac-
count. The tunneling Hamiltonian Ht intermixes the
states from adjacent charge sectors N , N � 1 due to
injection of extraction of an electron from the com-
plex dot:
H V c Xt
kb s d
bk bk
� ��
��
����
� �
�
��
��
��
,
† h. c. (4)
where X ��
� � �� �| | are the universal Hubbard opera-
tors [44]. The tunnel parameters V
bk�
��� are usually
approximated by a single parameter V. In the
Hamiltonian (4) these configuration changing opera-
Tunneling and magnetic properties of triple quantum dots
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 155
a b c
d e f
Fig. 1. Triple quantum dot in sequential (a), parallel (b),
cross-shape (c), two-terminal triangular (d), three-termi-
nal triangular (e) and fork (f) geometries.
tors describe transitions between the states belonging
to different charge sectors (one of this sectors corre-
sponds to the neutral CQD and another belong to
positively or negatively charged CQD. The index b
enumerates the leads (b s d� , in the two-terminal con-
figurations).
The Hamiltonian Hband has the standard form
H c ck
bk
bk bkband � ��
�
� �
† . (5)
Comparing to the corresponding term in (1), this
Hamiltonian contains one more index b. In the geome-
tries with (s—d)-reflection symmetry one may rotate
the frame in such a way that the band operators are
classified as even and odd operators relative to this
reflection
c c cek
/
sk dk� � �� ��2 1 2( ),
c c cok
/
sk dk� � ��
�2 1 2( ). (6)
In case of single and double quantum dots this rota-
tion usually excludes the odd combination from the
tunneling Hamiltonian [14]. However, it is not neces-
sarily the case for TQD.
The cotunneling (exchange) Hamiltonian is usually
obtained from (2) by means of the Schrieffer—Wolff
(SW) canonical transformation [4], which excludes
the states |
� belonging to the charge sectors N � 1
from the effective Fock space. At fixed N we are left
solely with spin degrees of freedom. In the conven-
tional situation, the relevant symmetry is SU(2) and
the SW procedure results in an effective Hamiltonian
(1), where exchange constant J is estimated as
J V /EC� 2 , and EC is expressed via addition and ex-
traction energies, i.e. the energy costs to add or sub-
tract an electron on/from the quantum dot:
E E EC
�
�
�
�
�� �1 1 1
E Q Ed F F d
� �� �
�
� � � �, . (7)
In case of even N and geometries including discrete
rotations, the situation is more complicated, and the
SW procedure intermixes the states |
� belonging to
different irreducible representations of the Hamil-
tonian (3). The corresponding terms in the effective
Hamiltonian may be expressed by means of the corre-
sponding Hubbard operators X ��� . In many cases,
combinations of these operators form closed algebras
which generate non-compact groups SO N( ) or
SU N( ) with N � 2, describing the dynamical symme-
try of complex quantum dots. Involvement of these
dynamical symmetries turns the procedure of mapping
the tunnel problem onto an effective exchange prob-
lem to be more complicated than in the simpler situa-
tions which were briefly described in first Section.
New features of the Kondo effect arising as a result of
this procedure were described for the first time using
the T-shaped DQD as an example [32]. Various mani-
festations of dynamical symmetries in physical prob-
lems are described in the monongraphs [45,46]. Some
mathematical aspects of the dynamical symmetries as
applied to the Kondo problem may be found in the re-
cent reviews [47,48].
Kondo physics for linear TQD
From the point of view of basic symmetries there is
much in common between the Kondo tunneling
through DQD and TQD in linear geometry because
both of them possess the permutation symmetry P2. In
case of linear TQD shown in Fig. 1,a—c this the per-
mutation of two equivalent side dots. At present, the
tunneling through this simplest DQD is well under-
stood (see [49] for a review). It turned out that the
most general description of Kondo tunneling through
vertical DQD may be given in terms of SU( )4 and
SO( )4 symmetries of a low-energy multiplets in cases
of odd and even electron occupation N , respectively.
We discuss below the configurations where the cen-
tral differs dot from two side dots in its size (and
hereby by the magnitude of the Coulomb blockade pa-
rameter Qc, but the latter dots are identical, so that
the TQD retains its mirror symmetries. Both conceiv-
able situations, namely Q Qc s�� and Q Qc s�� will
be considered (the indices c s, are used in this section
to indicate the physical quantities related to the cen-
tral and side dots, respectively). In the first experi-
mental realization of TQD [33] the former option was
chosen, whereas the first theoretical study [50] was
devoted to to the second possibility.
In a TQD with «open» central dot and Q Qc s�� ,
its role in formation of Kondo tunneling regime
reduces to providing the channel for indirect
RKKY-type mechanism of exchange between two lo-
calized spins formed in side dots. Thus, from the theo-
retical point of view this problem is in fact may be
mapped onto that for a DQD with even electron occu-
pation N � 2. The Kondo effect may be described in
terms of two-site Kondo Hamiltonian, where the trend
to interdot spin coupling competes with the trend to
individual Kondo coupling between two side dots and
the electrons in the leads [51,52].
One may try to describe this problem by means of
two spin operators sl and s r using the above men-
tioned analogy with the two-site Kondo problem [53].
However, such approach [54] should be used with
some caution. One should take into account that in a
situation, where both triplet and singlet two-electron
states are involved in Kondo effect, these spins are
156 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
K. Kikoin
non-independent because the kinematical constraint is
imposed on the S/T manifold by the Casimir operator
C � �s sl r
2 2. In accordance with the prescriptions of
the theory of dynamical symmetries [45–47], two op-
erators
S s s R s s� � �
l r l r, (8)
should be constructed and the kinematic constraint
C � � �S R2 2 3 (9)
should be imposed on them. Then the three compo-
nents ( , , )S S Sz � � of the vector S describe the states
within the spin triplet and transitions between them,
whereas the three components ( , , )R R Rz � � of the
vector R describe transitions between the singlet S
and the states with spin projections � �
10 1, , of the
triplet T. Six components of the vectors S,R form a
closed algebra
[ , ] ,S S ie S� � �� �
[ , ] ,R R ie S� � �� �
[ , ]R S ie R� � �� � .
(10)
and form a set of generators of SO( )4 group. Here
� � �, , are Cartesian coordinate indices, and e�� is the
anti-symmetric Levi-Civita tensor. Two vector opera-
tors are orthogonal, S R� � 0. Under these constraints,
two vectors sl r, are rather fictitious than real spin op-
erators. If one tunneling channel couples this «spin ro-
tator» with the reservoir of conduction electrons, then
the dynamical group SO( )4 exhausts the spin degrees
of freedom involved in Kondo tunneling. More de-
tailed discussion of interconnections between two rep-
resentations as well as the derivation of these operators
by means of the Hubbard operators X ��� may be
found in [32], where this scenario was realized for the
first time in DQD with the T-shape geometry.
Further reduction of the problem from DQD with two
sites to a single QD with two orbital is also possible.
Generic features of Kondo effect in TQD are seen
in the limit Q Qc s�� for sequential and parallel con-
figurations of Fig. 1,a,b [50]. To demonstrate these
features let us consider TQD with in lateral geometry
in more details. Genesis of spin multiplets may be un-
derstood from a general set-up illustrated by Fig. 2. In
case of N � 3 three electrons are distributed over three
dots in such a way that the state with doubly occupied
central is dot suppressed by strong Coulomb blockade
Qc. In accordance with the Young tableaux for a sys-
tem with (l—r)-permutation symmetry, the spin
multiplet consists of two doublets with S /� 1 2 hav-
ing even and odd symmetry relative to this permuta-
tion and one quartet S /� 3 2 with full orbital symme-
try. In case of N � 4 the spin manifold consists of two
spin singlets Se o, and two triplets Te o, , both even and
odd relative to (l—r)-permutation. Varying the gate
voltages v vgl gr, and playing with tunnel parameters
Vl r, andWl r, , one may break (l—r)-symmetry (Fig. 2)
and change the singlet-triplet splitting, so that the
spin states are classified as Sl r, and Tl r, . The relative
positions of energy levels in spin multiplets evolve as a
function of model parameters and various types of
level crossings occur (see [50] for detailed calcula-
tions). Similar situation arises for vertical configura-
tion of Fig. 1,a.
In accordance with general theory of dynamical
symmetries [47], quasi degeneracy of low-lying states
in spin multiplets within the Kondo energy scale
� TK generates special symmetries of TQD. The sim-
plest example of dynamical symmetry is a S/T
multiplet, where the two vectors R R1 2, generate the
group SO( )4 with the kinematic constraint (9). If the
multiplet of low-lying states within a scale TK con-
sists of two singlets and one triplet, the relevant dy-
namical symmetry is SO( )5 . If this multiplet is formed
by two triplets and one singlet, the corresponding
symmetry is SO( )7 , etc. 10 generators, which form the
o5 algebra for the group SO( )5 are the spin vector S
with three components describing transitions within
the triplet S � 1, two three-component vectors R R1 2, ,
which describe transitions between two singlets and
triplet, and the scalar A describing transitions be-
tween two singlets. The algebra o7 contains 21 genera-
tor for the group SO( )7 . These are two spin vectors
S S1 2, corresponding to the irreducible representations
for two triplets, plus four vectors R1 4� and three sca-
lars A1 3� corresponding to various transitions between
singlet and two triplets. The methods of constructing
these generators my means of Hubbard operators
and/or fermionization procedure are described in de-
Tunneling and magnetic properties of triple quantum dots
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 157
s
d
vgl
l r
Vl Vr
Vl Vr
c
Wl Wr
vgr
�l l+ Q
�c c+ Q
�l
�c
�r r+ Q
�r
Fig. 2. TQD in parallel geometry and energy levels of
each dot �a gaev� at Wa � 0.
tails in the reviews [47,48] and in the thesis [55].
Casimir constraint for SO n( ) group has the form
S Ri
i
i
i
i
i
A n2 2 2 1� � �� � �
. (11)
Dynamical symmetry is activated by the cotun-
neling processes between the TQD and the electron
continuum in metallic leads. Cotunneling transitions
induce transitions between different levels in the spin
multiplet. These transitions are spin flips in conven-
tional theory of Kondo effect, but in the systems with
complex symmetry non-diagonal transitions between
the states belonging to different irreducible represen-
tations of spin-multiplet described by the operators
R i also arise. As a result the SW procedure [4] gener-
ates the generalized exchange Hamiltonian
H J Jij
ij
i j i
i
i jex � � � �� �2 2S s R s~ . (12)
The spin operators for conduction electrons s j are
determined in a usual way [see Eq. (1)], but the index
j stands for even and odd combinations (6) of lead elec-
tron wave functions. Unlike the case of single quantum
dot [14], the odd mode is not eliminated from the tun-
neling Hamiltonian [56]. The number of scaling equa-
tions for the system (12) depends on the spin symmetry
of TQD, and the latter may be changed by varying the
parameters of the basic Hamiltonian (2).
The complex phase diagram characterizing Kondo ef-
fect in TQD may be constructed by means of the
perturbative (high-temperature) renormalization group
(RG) approach. This approach is based on the study of
flow diagrams describing the evolution of effective cou-
pling parameters Ja ( )� due to reduction of the effective
energy scale D of conduction electron kinetic energy
[57] (label a enumerates the vertices in the exchange
Hamiltonian, � � ln D is the scaling variable). The gen-
eral form of the system of scaling equations is
dj
d
c j ja
ab
b
a b�
�
� . (13)
Here j Ja a� � are dimensionless coupling constants
and cab are numerical coefficients. These equations
should be solved under the boundary conditions
j D Ja( )0 0� � , where D0 and Ja0 are the initial con-
duction bandwidth and the bare exchange integrals
entering the Hamiltonian Hex .
As a result, a unique opportunity arises to change
the value of index n characterizing the symmetry
SO n( ) of TQD by varying the gate voltages and other
experimentally controllable parameters of a device.
The phase diagram of vertical TQD with N � 4 calcu-
lated in [50] is shown in Fig. 3.
This diagram shows great variety of phases with
different symmetries from the most symmetric one
P SO SO2 4 4� �( ) ( ) to conventional SO( )3 phase
where the ground state of TQD is spin triplet, or
non-Kondo singlet ground state (shaded areas). Each
phase is characterized by its own Kondo temperature,
which deviates from the the standard equation for TK
given by equation T D / JK �
exp( )1 2� , where D is
the characteristic energy scale for the band electrons
in the leads and � is the density of states at the Fermi
level �F . ZBA in tunnel transparency at T TK� fol-
lows the scaling law
G T /G T/TKmax( ) ln ( )0
2� � (14)
where G e /h0
22� is the limiting value of tunnel con-
ductance in a Kondo regime (unitarity limit, where
the sum of all phase shifts on the Fermi levels �i
equals �/2). The limiting value quantify the tunnel
conductance at T
0. This means that the ZBA in
conductance should follow the change of the Kondo
temperature, so that each crossover from one symme-
try to another is accompanied by the abrupt change of
conductance at given temperature. The physical na-
ture of variation of TK in the process of crossover
from one dynamical symmetry to another will be ex-
plained in the next section where the TQD in «cross»
and «fork» geometries illustrated by Fig. 1c,f will be
considered in more details.
At N � 3 we meet a somewhat unexpected situation
where Kondo tunneling in a quantum dot with odd oc-
cupation demonstrates the exchange Hamiltonian of a
quantum dot with even occupation. The reason for
this scenario is the specific structure of the wave func-
tion of TQD with N � 3. The corresponding wave
functions are vector sums of states composed of a «pas-
sive» electron sitting in the central and dot sin-
glet/triplet (S/T) two-electron states in the l r, dots.
Then using certain Young tableaux [50], one con-
cludes that the spin dynamics of such TQD is repre-
158 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
K. Kikoin
2.5
2
1.5
1
0.5
0
0.5 1 1.5 2
y
SO(4)
SO(5)
SO(7)
SO(3)SO(4)
SO(5)
SO(3)
S
O
(7
)
P SO(4) SO(4)
P SO(3) SO(3)
P
S
O
(3
)
S
O
(3
)
x
Fig. 3. Dynamic symmetries in TQD: phase diagram in co-
ordinates x V/Vl r� ( )2, y /l c r c� � �( ) ( )� � � � .
sented by the spin 1 operator S corresponding to the
l r
triplet, the corresponding operator R and the spin
1/2 operator s c of a passive electron in the central
well. The latter does not enter the effective Hamiltonian
Hex but influences the kinematic constraint via the
Casimir operator
C � � � �S M s2 2 2 15
4c . (15)
The dynamical symmetry is therefore SO SU( ) ( )4 2� ,
and only the SO( )4 subgroup is involved in Kondo tun-
neling. Similar situation, although for different reasons
is realized in fork and cross geometries (see below).
Remarkable symmetry reduction occurs in external
magnetic field [50]. First example of such reduction
was found in a situation where the exchange splitting
of S/T multiplet (symmetry group SO( )4 ) is compen-
sated by the Zeeman splitting [20], so that the up spin
projection |T1� of triplet forms a pseudospin with sin-
glet | S� and the symmetry reduction SO SU( ) ( )4 2�
takes place. In case of TQD with SO( )5 symmetry, due
to the same compensation the system may be left in
a subspace {T S Sl l r1 , , }. The symmetry reduction in
this case is SO SU( ) ( )5 3� , and the Anderson Ha-
miltonian is mapped on a very specific anisotropic
Kondo Hamiltonian involving only operators R i ,
H J R sij
ij
i jex � ��
��
��
� �, (16)
where � �, are cartesian components of scalar product.
Here the Kondo effect is described exclusively in
terms of dynamical symmetry.
Another non-standard manifestation of Kondo map-
ping for linear TQD is the possibility of two-channel
Kondo effect in vertical geometry of Fig. 1,a with
Q Qc s�� at N � 3 with preserved (s—d)-mirror sym-
metry [56]. The strong Coulomb blockade in central
dot prevents direct (s—d)-tunneling. The cotunneling
is possible only because the wave functions of elec-
trons centered on the side dots have small tails on the
central dot. It is crucially important that the standard
rotation (6) does not eliminate the odd channel from
the tunneling Hamiltonian in TQD. In the situation,
where the ground state of TQD is the spin doublet
with even parity | De �, the SW transformation for the
original Anderson Hamiltonian results in anisotropic
two-channel exchange Hamiltonian,
H J J Js s d d sd sd dsex � � � � � � �S s S s S s s( ).
(17)
In accordance with the theory of two-channel Kon-
do effect [12], the presence of nondiagonal vertex
Jsd makes the 2-channel fixed point unattainable,
but due to strong Coulomb blockade in central dot
J V W /Qsd c d�
2 2 2� is extremely small, so that one
may approach the fixed point close enough, and the
predecessor of 2-channel regime may be observed
experimentally as a dip in conductance G as a func-
tion of the difference of gate voltages v vgs gd
ap-
plied to the side dots (this difference controls the
degree of channel anisotropy).
TQD in cross and fork geometry
We have seen above that one may meet the situa-
tion, where the linear TQD with odd occupation and
half-integer spin demonstrates the Kondo physics
characteristic for even occupation with integer spin
due to the fact that one of the electrons in the does dot
not participate in tunneling. Here we will discuss two
more mechanisms of such «disguise» [58].
One of these mechanisms is realized in cross geome-
try at occupation N � 3 under conditionQ Qs d�� for
Coulomb blockade parameters. In this case two side
electrons are passive: the tunneling between source
and drain occurs through the central dot. However,
these passive electrons influence Kondo mapping be-
cause they are responsible for the parity of the 3-elec-
tron wave function relative to the (l—r)-reflection.
Diagonalization of the low-energy spin states shows
that it consists of three spin doublets and one spin
quartet, and the lowest state in this manifold is the
doublet Du , which is odd relative to the mirror reflec-
tion (see [58] for details). Although the wave func-
tions of two passive electrons do not enter explicitly
in the indirect exchange integral arising due to SW
transformation, this integral changes its sign due to
odd parity of the state Du . Thus, in contrast to the
standard paradigm of Kondo mapping, the effective
exchange Hamiltonian corresponds to ferromagnetic
coupling, which is irrelevant to Kondo effect, and the
Kondo-type ZBA does not arise in this case in spite of
the fact that the net spin of quantum dot is 1/2.
However, this is not the end of the story. The ex-
cited states in the spin multiplet which are Kondo ac-
tive, influence the tunnel transparency and conduc-
tance at finite temperature and finite energies of
incident electron due to non-trivial dynamical symme-
try of TQD described above. The states involved in
the Kondo effect are the even spin doublet Dg and the
quartet Q, so that the overall dynamical symmetry of
TQD in cross geometry is SU SU SU( ) ( ) ( )2 2 2� � .
Similar situation is known as a «two-stage» Kondo ef-
fect in DQD with N � 2, on the singlet side of S/T
crossover (see [49] and references therein). At high
enough energy or temperature exceeding the width of
the spin multiplet all spin states are involved in
Kondo screening. With decreasing energy in the
course of the RG renormalization the high-energy lev-
Tunneling and magnetic properties of triple quantum dots
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 159
els are quenched and only the lowest spin state is re-
sponsible for the tunneling at T � 0.
In our case there are three stages of Kondo screen-
ing, where the statesQ and Dg are quenched one after
another with decreasing energy or temperature and
the Kondo screening eventually stops at zero T.
Besides, the level crossing may occur because of the
many-body logarithmic renormalization [59,60],
which is determined by the renormalization group
invariants E�
* , namely
E E D D/� � � �� �* ( ) ( )�
�� �1 ln . (18)
The level crossing occurs because the hierarchy of
tunneling rates � � �Q Dg Du� � takes place. This
crossover shown in Fig. 4 is controlled by the parame-
ters of TQD. Here the scaling variable is chosen in the
form � �� ln( )D/ Q� . The value of D is determined
from the crossover condition D E D� � �� ( ), where
the renormalization (18) changes for the SW regime
with fixed charge, D0 is the initial value of scaling
variable.
Three points (ln , ln , ln ,d d du Q cr ) on the abscissa
axis correspond to three values of the control parame-
ters where the crossover to the SW regime occurs for
the ground states EDu
, EQ and the completely degen-
erate ground state, respectively. By means of appro-
priate variation of the control parameters, the system
may be transformed from a non-Kondo regime with
the ground state EDu to the underscreened Kondo re-
gime with the ground state EQ and spin S /� 3 2. In
accordance with the general theory of Kondo map-
ping, TK is maximum in the point of maximum degen-
eracy. Evolution of TK is shown in Fig. 5. Unlike the
case of singlet/triplet crossover in DQD with N � 2,
here one deals with the crossover from the non-Kondo
spin doublet Du to the Kondo spin quartet Q via the
highly degenerate region of SU SU SU( ) ( ) ( )2 2 2� �
symmetry.
Let us now turn to the fork geometry shown in
Fig. 1,f. In this geometry non-trivial Kondo physics
arises already in the simplest case of odd occupation
N � 1 in a situation with the (l—r)-mirror symmetry.
The fork may be considered as a «quantum pendulum»
[38,61]. Three one-electron eigenvalues are
ED sb a,
� � � 2 2W / ED cn
�, � � , (19)
(� � max{|�s – �c|, |Qc – Qs|}). The eigen modes are
classified as a non-bonding spin doublet Dn (odd
combination of the wave functions centered in the
sites 1,2) and bonding/antibonding pair Db a, of cor-
responding even conbination with the state centered
in the cite 3 (see [58] for details). The latter pair is
the analog of resonant valence bonds (RVB) known
in «natural» molecules. To describe this pendulum
one should introduce the pseudospin vector T
T T T� � �� �� �� | |, [ ] ,†Db Dn
�
� �
T z Db Db Dn Dn� ��
���
1
2
(| | | | )� � � �
�
. (20)
together with similar vector t for conduction electrons
t c c tek
k
ok
� � �� �� �
�
� �† †, [ ] ,
t c c c cz ek
k
ek ok ok�
�
1
2
( ).† †
�
�
� � � (21)
Here the states labeled as e o, are the even and odd
combinations for the electrons in the leads 1,2 similar
to those in Eq. (6). In this situation the effective ex-
change Hamiltonian has the form
H J JSW p� � � ��
���
���
� ��S s T t (22)
160 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
K. Kikoin
lndQ lndcr lndu lnd0
lnd
TK–3.6
–3.2
–2.8
–2.4
–2
–1.6
–1 –0.5 0 0.5 1 1.5
E
,m
e
V
ED
EQ
EDg
u
Fig. 4. Flow diagram for the levels E� determined by the
scaling invariant (18). � � D D Qu g, , , d D Q� � /� . En-
ergy is measured in meV units (see text for further expla-
nations).
0
0.2
0.4
0.6
0.8
1
T /TK K
cr
DQ Dcr Du
D
Fig. 5. Evolution of TK in cross geometry as a function of
a control parameter D .
with � , ,� b n; � �, ,� e o. The five vector operators
S ��� and T constitute the 15 generators of the SU( )4
group.
Like in the cross geometry, the level crossing effect
as a function of control parameter takes place
(Fig. 6). Now TK is nonzero on both sides of the cross-
over and its evolution is described by the bell-like
curve shown in Fig. 7. Maximum TK
cr on this curve
correspond to the degeneracy point on the flow dia-
gram of Fig. 6. However the tunnel conductance is
drastically influenced by the pendulum structure of
the electron wave function. In the three-terminal fork
geometry, one should consider separately the situa-
tions, where the bias voltage is applied between the
leads 1 and 2 and between the leads 1 and 3. We
define the corresponding components of tunnel con-
ductance as G22 and G33, respectively. The Kondo
anomaly inG22 is predetermined by the RVB pair, and
the ZBA roughly follows the evolution of TK through
the crossover. More peculiar behavior is expected
in 1–3 channel because the non-bonding state
D d dn
/
� � �
�
�2 1 2
1 2( )† † is detached from the lead 3.
As a result the Kondo contribution to G33 manifests
itself as a finite bias anomaly (FBA) in a situation
where the ground state of TQD is EDn
. Tunnel con-
ductance as a function of bias voltage in both channels
is illustrated in Fig. 8. The dip in the curve b on the
right panel reminds similar dip in the tunnel conduc-
tance of DQD with N � 2 on the singlet side of S/T
crossover [62,63].
Thus, we see that some manifestations of the Kondo
effect in TQD with odd occupation may mimic those
for DQD with even occupation due to specific influ-
ence of the mirror reflection on the structure of the
electron wave functions in trimers.
Kondo physics in triangular quantum dots
It was noticed in [40] that the basic symmetry of
equilateral triangular triple quantum dot (TTQD,
Fig. 1,e) with odd occupation is SU( )4 due to the in-
terplay between the spin and orbital degrees of free-
dom, similar to that in two-orbital DQD. Special at-
tention was paid to the case N � 3 which models
Tunneling and magnetic properties of triple quantum dots
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 161
lndcr lnd0
lnd
TK
–1.2
–0.4
0
–1 –0.5 0 0.5 1 1.5
E
,m
e
V
EDa
EDn
EDb
–0.8
Fig. 6. Flow diagram for the levels E� of the TQD in
fork geometry.
0.5 1 1.5 2. .
0
0.2
0.4
0.6
0.8
1
T /TK K
cr
Dcr
D
Fig. 7. Evolution of TK in fork geometry.
0
0.02
0.04
0.06
0.08
0.1
0.12
1
b
0
0.05.
0.1.
0.15.
0.2.
0.25.
.
.
.
.
.
.
a
2
–25 –15 –5 5 15 25
3
eV/TK
cr
G
/G
2
2
0
G
/G
3
3
0
–25 –15 –5 5 15 25
eV/TK
cr
1
2
Fig. 8. Tunnel conductance in the channel 2–1 for
D D� cr (1), D D� cr (2) and D D� cr (3) (a). Tunnel
conductance in the channel 3–1 for D D� cr (1) and
D D� cr (2) (b).
triangular Co trimer [42,64]. In this charge sector the
effective spin Hamiltonian contains not only exchange
interaction between the spins in the dots and adjacent
leads, but also the two-site Heisenberg exchange be-
tween spins in the neighboring dots. Magnetic frustra-
tions in triangular geometry affect the spin state and
therefore influences the Kondo-type ZBA in tunneling
spectra. These spectra were calculated by the numeri-
cal RG and quantum MonteCarlo methods. Besides, it
was found [42] that in case of complete channel iso-
tropy the non-Fermi-liquid regime arises from the in-
terplay of magnetic frustrations and Kondo physics.
Another phenomenon, which interplays with the
Kondo physics is the Aharonov–Bohm oscillation of
tunnel transparency in magnetic field directed perpen-
dicularly to the plane of triangle. This effect may be
seen already for the TTQD with N � 1, where there is
no room for exchange interaction between spins local-
ized in neighboring sites and concomitant magnetic
frustrations. The starting point for solving the prob-
lem of interplay between Kondo and Aharonov–Bohm
phenomena [43,65] is the Anderson Hamiltonian (2)
rather than the effective exchange Hamiltonian.
In accordance with the general scheme discussed
above, one should start with the diagonalization of the
Hamiltonian of 3-site ring. The point symmetry of this
equilateral triangle is C v3 . This group describes the
«orbital» degrees of freedom, whereas the continuous
spin symmetry is usual SU(2) symmetry of spin 1/2.
According to the irreducible representations of C v3
group, the spectrum of TTQD consists of three states
� �DA DE, . Here as usual D stands for spin dou-
blet, A is the fully symmetric orbital singlet and E�
are two components of orbital doublet. The energies of
these states in out-of-plane magnetic field B are
E pD� ( ) � �
!
"
#
$
%
&2
3
W pcos
'
. (23)
such that for negative W and for B � 0,
p / /� 0 2 3 4 3, ,� � correspond respectively to
� �A E,
with the ground state DA, so that the orbital degrees
of freedom are quenched at low temperature and the
SW mapping procedure ends with conventional
Kondo Hamiltonian (1). However, using the mag-
netic field as a control parameter, one may initiate
level crossing by varying the magnetic flux ' through
TTQD. This level crossing is shown on the upper
panel of Fig. 9. In each crossing points and its nearest
vicinity the dynamical symmetry of TTQD is SU(4),
so that the symmetry crossovers SU( )2 occur at
' '� �( )n
1
2 0, where '0 is the quantum of magnetic
flux. Each crossover is accompanied by the change of
TK from exp( )
1 2/ J to exp( )
1 4/ J and back. The
ZBA peak in the two-terminal tunnel conductance
changes accordingly (Fig. 9,b).
Although formally there are three tunneling chan-
nels, the non-Fermi liquid Kondo regime cannot arise,
because the non-diagonal components Jij appear in the
exchange Hamiltonian (12). Further diagonalization
should be done by means of rotating frameworks for
Bloch electrons. This diagonalization introduces irre-
movable channel anisotropy, so that the 2-channel
non-Fermi-liquid fixed point cannot be achieved unlike
the case considered in [42], where the channel isotropy
was postulated from the very beginning.
In order to realize the Aharonov–Bohm interference,
one should use the two-terminal geometry shown in Fig.
1,d. In this case there are two paths (1–3) and (2–3) for
single electron tunnelling between source and drain. In-
terference of two waves in the drain results in famous
Aharonov–Bohm oscillations. The field B affects the
lead-hopping dot phases. In the chosen gauge, the hop-
ping integrals are modified as, W W i /� exp( )'1 3
V V i / /s12 1 26 2, exp[ ( )]� � �' ' where '12, are mag-
netic fluxes through the upper and lower loop of the
device. As a result the exchange Hamiltonian reads
H J J J Ks s d d sd sd ds� � � � � � � � �S s S s S s s T t( )
(24)
(the latter term becomes actual when the magnetic
field induces level crossing in accordance with
Fig. 9). Magnetic flux enters the coupling constants
Js , Jd , Jsd , K via SW transformation. As a result the
162 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
K. Kikoin
–2
–1
1
2
E – �
W
G
/G
0
2.5 7.5 12.5 17.5
'
'
2.5 7.5 12.5 17.5
0.01
0.02
0.03
à
b
Fig. 9. Evolution of the energy levels EA (solid line) and
E� (dashed and dash-dotted line, resp.) (a). Corre-
sponding evolution of conductance (G e /0
2
� � �) (b).
constant Jsd ( , )' '1 2 turns into zero at some values
of magnetic flux, so that the Aharonov–Bohm inter-
ference completely blocks Kondo transparency. Thus
TTQD serves simultaneously as a Kondo «pass valve»
and as an Aharonov–Bohm interferometer. It should
be stressed that both the continuous spin degrees of
freedom and discrete «rotations» of triangle are in-
volved in these two phenomena in TTQD.
Concluding remarks
Among many aspects of Kondo tunneling through
complex quantum dots we have chosen for this review
only the symmetry related properties predetermined
by the structure of the low-lying states in the spin
multiplet characterizing the fixed charge sector of
complex quantum dot. New features, which are intro-
duced by the dynamical symmetries in the Kondo
physics are the multistage process of Kondo screening,
symmetry crossovers driven by experimentally tunable
control parameters, interplay between continuous spin
rotation symmetry and discrete point symmetry of
nanodevices. The main tool of experimental monitor-
ing of variable symmetries is study of temperature and
magnetic filed dependence of zero- and finite-bias
anomalies in tunnel conductance.
Among other novel facets of Kondo effect in nano-
structures one should mention non-equilibrium Kondo
effect at finite bias and under light illumination,
where both quantum dots and leads are far enough
from thermodynamic balance. Under this conditions
such phenomena as spin relaxation, dephasing and
decoherence influence the tunnel transport in Kondo
regime. Real atoms and molecules also may be included
in electric circuit by means of advanced experimental
techniques (scanning tunnel spectroscopy, break-junc-
tion method etc). In this case phonon- and photon as-
sisted processes should be taken into account, which re-
sult in interplay of Kondo resonance tunneling with
various «polaronic» and «excitonic» effects.
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