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Unconventional mechanisms of resonance tunneling through complex quantum dots
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Physica E 17 (2003) 149 – 151 www.elsevier.com/locate/physe
Unconventional mechanisms of resonance tunneling through complex quantum dots K. Kikoin∗ , T. Kuzmenko, Y. Avishai Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Abstract It is shown that the resonance tunneling through a complex quantum dot with even occupation reveals the hidden dynamical SO(n) symmetry of this dot. This symmetry is manifested in unconventional mechanisms of Kondo resonance formation, where the gate voltage may tune the rank n gradually. In a speci1c case of triple quantum dot, the regimes with n = 3; 4; 5; 7 can be achieved. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Kondo tunneling; Dynamical symmetry
Dynamical symmetry groups and corresponding Lee algebras provide a universal tool for description of energy spectra and transitions between eigenstates of various quantum systems. Their usefulness was demonstrated by Fock who revealed the hidden SO(4) symmetry of hydrogen atom. Further development is connected with the problem of classi1cation of elementary particles in terms of unitary SU(3) group, description of quantum oscillators, rotators, etc. (see Ref. [1] for the history of the problem and general theory). Recently a novel development of this method in a context of the theory of tunneling through double quantum dots (DQD) with even occupation N was o?ered [2]. It was shown that unlike the conventional Kondo tunneling through the dot with odd N , whose spin state is completely described by SU(2) rotation
∗ Corresponding author. Tel.: +972-3-950-0081; fax: +972-8647-2904. E-mail address: [email protected] (K. Kikoin).
group, the hidden SO(4) symmetry is an inherent property of DQD with even N . This symmetry of spin rotator arises because the lowest spin states of DQD are always singlet (S) and triplet (T). The tunneling processes between DQD and metallic leads may change the spin state of DQD. As a result, the rotational symmetry of DQD is described by two vectors. One of these vectors is a spin vector ˜S of a triplet ˜ describes the spin one state. The second vector M S–T transitions within the manifold. Both these vectors are involved in Kondo processes. As a result, the Kondo tunneling is possible even when the ground state of isolated DQD is singlet. The Kondo e?ect can be induced by external magnetic 1eld [3] or arise as a result of dynamical renormalization of the states within spin manifold due to the tunnel contact with metallic leads [2]. In this paper more general con1gurations are considered, when the spin manifold consist of several singlet and triplet states. For example, the low-energy spectrum of a triplet quantum dot (TQD) shown in Fig. 1, consists of two S- and two T-states for N = 4.
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00718-X
150
K. Kikoin et al. / Physica E 17 (2003) 149 – 151
s Vl
l
Vr
Wl
l
Vg
Wr
r
r
Vg
f Vr
Vl d
Fig. 1. Triple quantum dot with N = 4. Strong Coulomb blockade prevents two-electron occupation of a central dot. Dashed lines indicate tunneling channels. Vgl; r are the gate voltages.
Fig. 2. Phase diagram of TQD. Central spot is the state of highest symmetry.
This system is described by the spin Hamiltonian ˜ a ·˜s) H = E a X a a + (JaT˜S a ·˜s + JaST M +Jlr
a
a
(P˜S a + Ba ) ·˜s +
kb
+ kb ckb ckb :
(1)
This Hamiltonian is obtained from initial tunneling Hamiltonian by means of elimination of charge degrees of freedom using the renormalization group (RG) procedure [4]. Here the 1rst and the last term describe the states in the TQD and metallic leads, respectively. = S; T; a enumerates operators and states related to the left (l) and right (r) dots; k; , are momentum, and spin indices for source and drain electrons marked by b = s; d. The e?ective interaction is driven by the Schrie?er–Wol? (SW) coupling constants describing the dot-lead exchange in T-channel (JaT ), the S–T mixing (J ST ) and the exchange between l and r dots (Jlr ). Pis the scalar l–r permutation operator. In general case the operators entering the Hamiltonian (1) obey the on algebra determined by the following commutation relations: [Saj ; Sak ] = iejkm Sam ;
[Maj ; Mak ] = iejkm Sam ;
[Baj ; Sak ] = iejkm Bam ;
[Baj ; Bak ] = iejkm Sam ;
[Maj ; Sak ] = iejkm Mam ;
[Baj ; Mak ] = P jk ;
[P; Baj ] = Maj ; [P; Saj ] = 0:
[Maj ; P] = Baj ; (2)
(j; k; m are Cartesian indices). Relative positions of S and T levels may be tuned by gate voltages applied to l and r dots. As a result the Kow trajectories of Haldane RG procedure [4] may cross in various ways. If the lowest renormalized states in the SW limit are Sa and Ta , one deals with SO(4) symmetry. The multiplet (Sl ; Tl ; Sr ) possesses SO(5) symmetry, the multiplet (Sl ; Tl ; Tr ) corresponds to SO(7) symmetry, etc. The complete phase diagram of TQD in coordinates y = l =r ; x = !l =!r is presented in Fig. 2. Here l; r are positions of the electron levels in the l, r dots relative to the position c of the electron in the central dot, !a ∼ "0 Va2 is the tunneling rate between the dot a and the leads with the electron density of states "0 . In all regions of the phase diagram except two hatched sectors, Kondo tunneling exists. All three vectors forming the group SO(n) are involved in formation of a Kondo resonance, and Kondo temperatures, TK , are di1erent for di1erent symmetries. Besides, they are explicit functions of a . To conclude, the hidden dynamical symmetry of TQD with even N results in essentially more reach picture of Kondo tunneling, than that observed in conventional spin 1=2 dots. The symmetry of TQD can be changed in a given Coulomb window by tuning the gate voltage. This change results in a sharp change of TK , and a jump in the di?erential tunnel conductance arises as a result.
K. Kikoin et al. / Physica E 17 (2003) 149 – 151
This research is supported by ISF, BSF and DIP funds. References [1] A.I. Malkin, V.I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems, Nauka, Moscow, 1979.
[2] K. Kikoin, Y. Avishai, Phys. Rev. Lett. 86 (2001) 2090; K. Kikoin, Y. Avishai, Phys. Rev. B 65 (2002) 115 329. [3] M. Pustilnik, et al., Phys. Rev. Lett. 84 (2000) 1756. [4] F.D.M. Haldane, Phys. Rev. Lett. 40 (1978) 416.
151
Physica E 17 (2003) 149 – 151 www.elsevier.com/locate/physe
Unconventional mechanisms of resonance tunneling through complex quantum dots K. Kikoin∗ , T. Kuzmenko, Y. Avishai Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Abstract It is shown that the resonance tunneling through a complex quantum dot with even occupation reveals the hidden dynamical SO(n) symmetry of this dot. This symmetry is manifested in unconventional mechanisms of Kondo resonance formation, where the gate voltage may tune the rank n gradually. In a speci1c case of triple quantum dot, the regimes with n = 3; 4; 5; 7 can be achieved. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Kondo tunneling; Dynamical symmetry
Dynamical symmetry groups and corresponding Lee algebras provide a universal tool for description of energy spectra and transitions between eigenstates of various quantum systems. Their usefulness was demonstrated by Fock who revealed the hidden SO(4) symmetry of hydrogen atom. Further development is connected with the problem of classi1cation of elementary particles in terms of unitary SU(3) group, description of quantum oscillators, rotators, etc. (see Ref. [1] for the history of the problem and general theory). Recently a novel development of this method in a context of the theory of tunneling through double quantum dots (DQD) with even occupation N was o?ered [2]. It was shown that unlike the conventional Kondo tunneling through the dot with odd N , whose spin state is completely described by SU(2) rotation
∗ Corresponding author. Tel.: +972-3-950-0081; fax: +972-8647-2904. E-mail address: [email protected] (K. Kikoin).
group, the hidden SO(4) symmetry is an inherent property of DQD with even N . This symmetry of spin rotator arises because the lowest spin states of DQD are always singlet (S) and triplet (T). The tunneling processes between DQD and metallic leads may change the spin state of DQD. As a result, the rotational symmetry of DQD is described by two vectors. One of these vectors is a spin vector ˜S of a triplet ˜ describes the spin one state. The second vector M S–T transitions within the manifold. Both these vectors are involved in Kondo processes. As a result, the Kondo tunneling is possible even when the ground state of isolated DQD is singlet. The Kondo e?ect can be induced by external magnetic 1eld [3] or arise as a result of dynamical renormalization of the states within spin manifold due to the tunnel contact with metallic leads [2]. In this paper more general con1gurations are considered, when the spin manifold consist of several singlet and triplet states. For example, the low-energy spectrum of a triplet quantum dot (TQD) shown in Fig. 1, consists of two S- and two T-states for N = 4.
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00718-X
150
K. Kikoin et al. / Physica E 17 (2003) 149 – 151
s Vl
l
Vr
Wl
l
Vg
Wr
r
r
Vg
f Vr
Vl d
Fig. 1. Triple quantum dot with N = 4. Strong Coulomb blockade prevents two-electron occupation of a central dot. Dashed lines indicate tunneling channels. Vgl; r are the gate voltages.
Fig. 2. Phase diagram of TQD. Central spot is the state of highest symmetry.
This system is described by the spin Hamiltonian ˜ a ·˜s) H = E a X a a + (JaT˜S a ·˜s + JaST M +Jlr
a
a
(P˜S a + Ba ) ·˜s +
kb
+ kb ckb ckb :
(1)
This Hamiltonian is obtained from initial tunneling Hamiltonian by means of elimination of charge degrees of freedom using the renormalization group (RG) procedure [4]. Here the 1rst and the last term describe the states in the TQD and metallic leads, respectively. = S; T; a enumerates operators and states related to the left (l) and right (r) dots; k; , are momentum, and spin indices for source and drain electrons marked by b = s; d. The e?ective interaction is driven by the Schrie?er–Wol? (SW) coupling constants describing the dot-lead exchange in T-channel (JaT ), the S–T mixing (J ST ) and the exchange between l and r dots (Jlr ). Pis the scalar l–r permutation operator. In general case the operators entering the Hamiltonian (1) obey the on algebra determined by the following commutation relations: [Saj ; Sak ] = iejkm Sam ;
[Maj ; Mak ] = iejkm Sam ;
[Baj ; Sak ] = iejkm Bam ;
[Baj ; Bak ] = iejkm Sam ;
[Maj ; Sak ] = iejkm Mam ;
[Baj ; Mak ] = P jk ;
[P; Baj ] = Maj ; [P; Saj ] = 0:
[Maj ; P] = Baj ; (2)
(j; k; m are Cartesian indices). Relative positions of S and T levels may be tuned by gate voltages applied to l and r dots. As a result the Kow trajectories of Haldane RG procedure [4] may cross in various ways. If the lowest renormalized states in the SW limit are Sa and Ta , one deals with SO(4) symmetry. The multiplet (Sl ; Tl ; Sr ) possesses SO(5) symmetry, the multiplet (Sl ; Tl ; Tr ) corresponds to SO(7) symmetry, etc. The complete phase diagram of TQD in coordinates y = l =r ; x = !l =!r is presented in Fig. 2. Here l; r are positions of the electron levels in the l, r dots relative to the position c of the electron in the central dot, !a ∼ "0 Va2 is the tunneling rate between the dot a and the leads with the electron density of states "0 . In all regions of the phase diagram except two hatched sectors, Kondo tunneling exists. All three vectors forming the group SO(n) are involved in formation of a Kondo resonance, and Kondo temperatures, TK , are di1erent for di1erent symmetries. Besides, they are explicit functions of a . To conclude, the hidden dynamical symmetry of TQD with even N results in essentially more reach picture of Kondo tunneling, than that observed in conventional spin 1=2 dots. The symmetry of TQD can be changed in a given Coulomb window by tuning the gate voltage. This change results in a sharp change of TK , and a jump in the di?erential tunnel conductance arises as a result.
K. Kikoin et al. / Physica E 17 (2003) 149 – 151
This research is supported by ISF, BSF and DIP funds. References [1] A.I. Malkin, V.I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems, Nauka, Moscow, 1979.
[2] K. Kikoin, Y. Avishai, Phys. Rev. Lett. 86 (2001) 2090; K. Kikoin, Y. Avishai, Phys. Rev. B 65 (2002) 115 329. [3] M. Pustilnik, et al., Phys. Rev. Lett. 84 (2000) 1756. [4] F.D.M. Haldane, Phys. Rev. Lett. 40 (1978) 416.
151