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arXiv:cond-mat/0507488v2 [cond-mat.str-el] 10 Feb 2006

Magnetically tunable Kondo – Aharonov-Bohm effect in triangular quantum dot

T. Kuzmenko1, K. Kikoin1and Y. Avishai1,2

1Department of Physics,

Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

2Ilse Katz Center for Nano-Technology,

The role of discrete orbital symmetry in nanoscopic physics is manifested in a system consisting

of three identical quantum dots forming an equilateral triangle. Under a perpendicular magnetic

field, this system demonstrates a unique combination of Kondo and Aharonov-Bohm features due to

an interplay between continuous (spin-rotation SU(2)) and discrete (permutation C3v) symmetries,

as well as U(1) gauge invariance. The conductance as a function of magnetic flux displays sharp

enhancement or complete suppression depending on contact setups.

PACS numbers: 72.10.-d, 72.15.-v, 73.63.-b

Experimental analysis of the Kondo effect in simple

quantum dots (QD) [1] treats the electron as a local spin

1/2 magnetic moment devoid of orbital degrees of free-

dom. These are absent also in theoretical discussions of

the Kondo effect in composite structures consisting of

two or three dots [2, 3, 4, 5]. However, orbital effects,

which play a crucial role in real metals [6, 7], become

relevant also in mesoscopic physics, e.g., when a QD

is fabricated in a ring geometry, having discrete point

symmetries. At low temperature it can serve both as a

Kondo-scatterer and as a peculiar Aharonov-Bohm (AB)

interferometer, since the magnetic flux affects the nature

of the QD ground and excited states. The simplest such

system (three dots forming a triangle) has been realized

experimentally [8, 9]. Triangular trimer of Cr ions on a

gold surface was also studied [10]. The orbital symmetry

of triangle is discrete. It results in additional degenera-

cies of the spectrum of trimer, which may be the source

of non Fermi liquid (NFL) regime [11].

In the present work we analyze the physics of tunnel-

ing through a triangular triple quantum dot (TTQD) in a

magnetic field with one electron shared by its three iden-

tical constituents (see Fig. 1). It exhibits an interplay

between continuous SU(2) electron spin symmetry, dis-

crete point symmetry C3v and U(1) gauge invariance of

electron wave functions in an external magnetic field. Its

conductance is characterized by an unusual dependence

on the magnetic flux Φ through the triangle, displayed

by sharp peaks or narrow dips, depending on contact ge-

ometry. In a 3-terminal geometry (Fig. 1a) the sharp

peaks arise since the magnetic field induces a symme-

try crossover SU(2) → SU(4). In a 2-terminal geometry

(Fig. 1b) the Kondo tunneling is modulated by AB in-

terference, which blocks the source-draincotunneling am-

plitude at certain flux values. This Kondo-AB interplay

should not be confused with that in mesoscopic struc-

tures with QD as an element in the AB loop [12].

A symmetric TTQD in contact with metallic leads is de-

scribed by the Hamiltonian H = Hd+ Hlead+ Ht, ex-

pressed in terms of dot and lead operators djσ,cjσ, with

j = 1,2,3, and σ =↑,↓. Hddescribes an isolated TTQD,

3

?

+Q′?

Here ?jl? = ?12?,?23?,?31?, Q and Q′are intradot and

interdot charging energies (Q ≫ Q′), and W is the in-

terdot tunneling amplitude. Hleaddescribes electrons in

the respective electrodes,

Hd= ǫ

j=1

?

njσnlσ′ + W

σ

d†

jσdjσ+ Q

?

?

j

nj↑nj↓

(1)

?jl?

?

σ

?jl?

?

σ

(d†

jσdlσ+ H.c.).

Hlead =

?

jkσ

ǫjkc†

jkσcjkσ,(2)

and Htis the tunneling Hamiltonian

Ht= V

?

jkσ

(c†

jkσdjσ+ H.c.).(3)

The dot energy ǫ is tuned by gate voltage in such a way

ddd

111222

333

sss

??

?

?

111

222

333

?

(a)(a)(b)(b)

??

?

?

??

?

?

??

FIG. 1: Triangular triple quantum dot (TTQD) in three-

terminal (a) and two-terminal (b) configurations.

that the ground-state occupation of the isolated TTQD

is N = 1. Consider first a TTQD with three leads and

three identical channels (Fig. 1a). Assuming V ≪ W,

the tunnel contact preserves the rotational symmetry of

the TTQD, which is thereby imposed on the itinerant

electrons in the leads. It is useful to treat the Hamil-

tonian in the special basis which respects the C3v sym-

metry, employing an approach widely used in the theory

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2

of Kondo effect in bulk metals [6, 13]. The Hamiltonian

Hd+ Hleadis diagonal in the basis

3σ)/√3 ,

d†

d†

A,σ= (d†

E±,σ= (d†

1σ+ d†

2σ+ d†

(4)

1σ+ e±2iϕd†

2σ+ e±iϕd†

?

2kσ+ e±iϕc†

3σ)/√3 ;

/√3 ,

c†

A,kσ=

?

c†

1kσ+ c†

2kσ+ c†

3kσ

(5)

c†

E(±),kσ=

?

c†

1kσ+ e±2iϕc†

3kσ

?

/√3 .

Here ϕ = 2π/3, while A and E form bases for two irre-

ducible representations of the group C3v. The Hamilto-

nian of the isolated TTQD in this charge sector has six

eigenstates |DA?,|DE?. They correspond to a spin dou-

blet (D) with fully symmetric ”orbital” wave function

(A) and a quartet doubly degenerate both in spin and

orbital quantum numbers (E). The corresponding single

electron energies are,

EDA= ǫ + 2W,EDE= ǫ − W.(6)

To describe the orbital effect of an external magnetic field

B (perpendicular to the TTQD plane and inducing a flux

Φ through the triangle), one rewrites the spectrum as

EDΓ(p) = ǫ − 2W cos

?

p −Φ

3

?

.(7)

such that for negative W and for B

0, 2π/3, 4π/3 correspond respectively to Γ = A,E±.

Fig. 2 illustrates the evolution of EDΓ(Φ) induced by B.

Variation of B between zero and B0(the value of B corre-

sponding to the quantum of magnetic flux Φ0through the

triangle) results in multiple crossing of the levels EDΓ.

The accidental degeneracy of spin states induced by

the magnetic phase Φ introduces new features into the

Kondo effect. In the conventional Kondo problem, the

effective low-energy exchange Hamiltonian has the form

JS·s, where S and s are the spin operators for the dot and

lead electrons respectively [14]. Here, however, the low-

energy states of TTQD form a multiplet characterized by

both spin and orbital quantum numbers. The effective

exchange interaction reflects the dynamical symmetry of

the Hamiltonian Hd[3, 15]. The corresponding dynami-

cal symmetry group is identified not only by the opera-

tors which commute with the Hamiltonian but also by op-

erators inducing transitions between different states of its

multiplets. Hence, it is determined by the set of dot en-

ergy levels which reside within a given energy interval (its

width is related to the Kondo temperature TK). Since

the position of these levels is controlled by the magnetic

field, we arrive at a remarkable scenario: Variation of

a magnetic field determines the dynamical symmetry of

the tunneling device. Generically, the dynamical symme-

try group which describes all possible transitions within

the set {DA,DE±} is SU(6). However, this symmetry

is exposed at too high energy scale ∼ W, while only the

=0, p=

low-energy excitations at energy scale TK ≪ W are in-

volved in Kondo tunneling. It is seen from Fig. 2 that the

orbital degrees of freedom are mostly quenched, but the

ground state becomes doubly degenerate both in spin and

orbital channels around Φ = (2n+1)π,

Next we analyse the field dependent Kondo effect vari-

able degeneracy. It is useful to generalize the notion

of localized spin operator Si= |σ?ˆ τi?σ′| (employing

Pauli matrices ˆ τi (i = x,y,z)) to Si

in terms of the eigenvectors (4).

tion applies for the spin operators of the lead electrons:

si

ΓΓ′ =?

scale. The only vector, which is involved in Kondo co-

tunneling through TTQD is the spin SAA≡ S. Applying

Schrieffer-Wolff (SW) procedure, the effective exchange

Hamiltonian reads,

(n = 0,±1,...).

ΓΓ′ = |σΓ?ˆ τi?σ′Γ′|,

Similar generaliza-

kk′c†

Γ,kσˆ τicΓ′,k′σ′. In zero field, Φ = 0, the ro-

tation degrees of freedom are quenched at the low-energy

HSW = JE

?S · sE+E++ S · sE−E−

?+ JAS · sAA

(8)

The exchange vertices JΓare

JE= −2V2?∆−1

JA= 2V2?3∆−1

Q′ − ∆−1

+ ∆−1

Q

Q+ 2∆−1

?/3,(9)

1

Q′

?/3,

with ∆1= ǫF−ǫ, ∆Q= ǫ + Q− ǫF, ∆Q′ = ǫ + Q′− ǫF.

Note that JA > 0 as in the conventional SW trans-

formation of the Anderson Hamiltonian. On the other

hand, JE< 0 due to the inequality Q ≫ Q′. Thus, two

out of three available exchange channels in the Hamil-

tonian (8) are irrelevant. As a result, the conventional

Kondo regime emerges with the doublet DA channel and

a Kondo temperature,

T(A)

K

= Dexp{−1/jA},(10)

where jA= ρ0JA, ρ0being the density of electron states

in the leads.

At Φ = (2n+1)π, when the ground state of TTQD be-

comes spin and orbital doublet the symmetry of Kondo

center is SU(4). This kind of orbital degeneracy is differ-

ent from that of occupation degeneracy studied in double

quantum dot systems [16]. The 15 generators of SU(4)

include four spin vector operators SEaEbwith a,b = ±

and one pseudospin vector T defined as

T+=

?

1

2

σ

|E+,σ??E−,σ|, T−= [T+]†,(11)

Tz=

?

σ

(|E+,σ??E+,σ| − |E−,σ??E−,σ|).

Its counterpart for the

?

lead electrons is

kσ(c†

τ+

=

kσc†

E+kσcE−kσ,τz=1

2

?

E+kσcE+kσ-c†

E−kσcE−kσ).

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3

The SW Hamiltonian is [17],

HSW= J1(SE+E+· sE+E++ SE−E−· sE−E−)

+J2(SE+E+· sE−E−+ SE−E−· sE+E+)

+J3(SE+E++ SE−E−) · sAA

+J4(SE+E−· sE−E++ SE−E+· sE+E−)

+J5(SE+E−· (sAE−+ sE+A) + H.c.) + J6T · τ,

where the coupling constants are J1 = J4 = JA, J2 =

J3= J5= JE defined in (9) and J6= V2(∆−1

Thus, spin and orbital degrees of freedom of TTQD in-

terlace in the exchange terms. The indirect exchange

coupling constants include both diagonal (jj) and non-

diagonal (jl) terms describing reflection and transmission

co-tunneling amplitudes. The interplay between spin and

pseudospin channels naturally affects the scaling equa-

tions obtained within the framework of Anderson’s ”poor

man scaling” procedure [14]. The system of scaling equa-

tions has the following form:

(12)

1

+ ∆−1

Q′).

dj1/dt = −[j2

dj2/dt = −[j2

dj3/dt = −[j2

dj4/dt = −[j4(j1+ j2+ j6) + j6(j1− j2)],

dj5/dt = −j5[j1+ j2+ j3− j6]/2.

Here ji = ρ0Ji, and t = lnρ0D. Analysis of solutions

of the scaling equations (13) shows that the symmetry-

breaking vertices j3and j5are irrelevant, and the vertex

j2, whose initial value is negative evolves into positive

domain and eventually enters the Kondo temperature,

1+ j2

2+ j2

3+ j2

4/2 + j4j6+ j2

4/2 − j4j6+ j2

5], dj6/dt = −j2

5/2],

5/2],

6,

(13)

T(E)

K

= Dexp?− 2??(jA(1 +

We see from (14) that both spin and pseudospin ex-

change constants contribute on an equal footing. Unlike

the Kondo Hamiltonian for N = 3 with Jabcd= J dis-

cussed in Ref. [11], the NFL regime is not realized for

N = 1 with HSW (12). The reason of this difference

is that starting with the Anderson Hamiltonian with fi-

nite Q,Q′in (1), one inevitably obtains the anisotropic

SW exchange Hamiltonian for any N. As a result, two of

three orbital channels become irrelevant. However TKis

enhanced due to inclusion of orbital degrees of freedom,

and this enhancement is magnetically tunable. It follows

from (7) that the crossover SU(2) → SU(4) → SU(2)

occurs three times within the interval 0 < Φ < 6π and

each level crossing results in enhancement of TK from

(10) to (14) and back.[18] These field induced effects may

be observed by measuring the two-terminal conductance

Gjlthrough TTQD (the third contact is assumed to be

passive). Calculation by means of Keldysh technique (at

T > TK) similar to that of Ref. [19] show sharp max-

ima in G as a function of magnetic field, following the

maxima of TK(lower panel of Fig. 2).

√2) + jE+ 2jτ)??. (14)

2.557.51012.5 1517.5

?

0.005

0.01

0.015

0.02

0.025

0.03

0.035

G

?G0

2.557.51012.515 17.5

?

?2

?1

1

2

E

??

???? ? ????????? ? ?

W

FIG. 2: Upper panel: Evolution of the energy levels EA(solid

line) and E± (dashed and dash-dotted line, resp.)

panel: corresponding evolution of conductance (G0 = πe2/?).

Lower

So far we have studied the influence of the magnetic

field on the ground-state symmetry of the TTQD. In

a two-lead geometry (Fig. 1b) the field B affects the

lead-dot hopping phases thereby inducing an additional

AB effect [20]. The symmetry of the device is thereby

reduced since it looses two out of three mirror reflec-

tion axes. The orbital doublet E splits into two states,

but still, the ground state is |DA?. In a generic situ-

ation, the total magnetic flux is the sum of two com-

ponents Φ = Φ1+ Φ2. In the chosen gauge, the hop-

ping integrals in Eqs. (1), (3) are modified as, W →

W exp(iΦ1/3),V1,2 → Vsexp[±i(Φ1/6 + Φ2/2)], and

the exchange Hamiltonian now reads,

H = JsS · ss+ JdS · sd+ JsdS · (ssd+ sds). (15)

Cumbersome expressions for the exchange constants

Js(Φ1,Φ2), Jd(Φ1,Φ2) and Jsd(Φ1,Φ2) will be presented

elsewhere. They depend on the pertinent domain in pa-

rameter space of phases Φ1,2. Applying poor man scaling

procedure on the Hamiltonian (15) yields TK,

TK= Dexp

?

−

2

js+ jd+?(js− jd)2+ 4j2

sd

?

, (16)

and the conductance at T > TKreads [19],

G

G0

=

3

4

j2

sd

(js+ jd)2

1

ln2(T/TK). (17)

The conductance G(Φ1,Φ2) (17) obeys the Byers-Yang

theorem (periodicity in each phase) and the Onsager con-

dition G(Φ1,Φ2) = G(−Φ1,−Φ2). We choose to display

the conductance along two lines Φ1(Φ),Φ2(Φ) in param-

eter space of phases, namely, G(Φ1 = Φ,Φ2 = 0) and

G(Φ1 = Φ/2,Φ2 = Φ/2) (figure 3 left and right panels

respectively). The Byers-Yang relation implies respective

periods of 2π and 4π in Φ. Experimentally, the magnetic

flux is applied on the whole sample as in figure 1b, and

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4

123456

?

0.005

0.01

0.015

0.02

G

?G0

24681012

?

0.005

0.01

0.015

0.02

G

?G0

FIG. 3: Conductance as a function of magnetic field for Φ2 =

0 (left panel) and Φ1 = Φ2 = Φ/2 (right panel).

the ratio Φ1/Φ2is determined by the specific geometry.

Strictly speaking, the conductance is not periodic in the

magnetic field unless Φ1and Φ2are commensurate.

The shapes of the conductance curves presented here

are distinct from those pertaining to a mesoscopic AB in-

terferometer with a single correlated QD and a conduct-

ing channel[12, 21] (termed as Fano-Kondo effect [21]).

For example, G(Φ) in Fig. 3 of Ref. [21] (calculated in

the strong coupling regime) has a broad peak at Φ = π/2

with G(Φ = π/2) = 1. On the other hand, G(Φ) dis-

played in Fig. 2 (pertinent to Fig. 1a and obtained in

the weak coupling regime), is virtually flux independent

except near the points Φ = (2n+1)π (n integer) at which

the SU(4) symmetry is realized and G is sharply peaked.

The phase dependence is governed here by interference

effects on the level spectrum of the TTQD. The three

dots share an electron in a coherent state strongly cor-

related with the lead electrons, and this coherent TTQD

as a whole is a vital component of the AB interferom-

eter. In the setup of Fig. 1b, the Kondo cotunneling

vanishes identically on the curve Jsd(Φ1,Φ2) = 0. The

AB oscillations arise as a result of interference between

the clockwise and anticlockwise ”effective rotations” of

TTQD in the tunneling through the {13} and {23} arms

of the loop (Fig. 1b), provided the dephasing in the leads

does not destroy the coherence of tunneling through the

two source channels [22]. On the other hand, in the calcu-

lations performed on Fano-Kondo interferometers, G(Φ)

remains finite [21]. Another kind of Fano effect due to the

renormalization of electron spectrum in the leads induced

by the lead-dot tunneling similar to that in chemisorbed

atoms [23] is beyond the scope of this paper.

To conclude, we have shown that spin and orbital

degrees of freedom interlace in ring shaped quantum

dots thereby establishing the analogy with the Coqblin-

Schrieffer model in real metals. The orbital degrees of

freedom are tunable by an external magnetic field, and

this implies a peculiar AB effect, since the magnetic field

affects the spectrum and the tunneling amplitudes. The

conductance is calculated in the weak coupling regime

at T > TK in three- and two-terminal geometries (Figs

1a,b). In the former case it is enhanced due to change

of the dynamical symmetry caused by field-induced level

crossing (Fig. 2). In the latter case the conductance can

be completely suppressed due to destructive AB inter-

ference in source-drain cotunneling amplitude (Fig. 3).

These results promise an interesting physics at the strong

coupling regime as well as in cases of doubly and triply

occupied TTQD [5]. It would also be interesting to gen-

eralize the present theory for quadratic QD [24], which

possesses rich energy spectrum with multiple accidental

degeneracies.

This research is supported by grants from Clore founda-

tion (T. K.), ISF (K. K., Y. A.) and DIP project (Y.

A.). Critical comments by O. Entin-Wohlman are highly

appreciated.

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