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The Kondo effect in Bohm-Aharonov resonant tunneling devices
ELSEVIER
Physica B 230-232 (1997) 478-481
The Kondo effect in Bohm-Aharonov resonant tunneling devices J.R. I g l e s i a s a'*, E . V . A n d a b, M a r i a A. D a v i d o v i c h c, G. C h i a p p e d a lnstituto de Fisica, Universidade Federal do Rio Grande do Sul, C.P. 15051, 91501-970 Porto Alegre, Brazil b Instituto de Fisica, Universidade Federal Fluminense, 24210-340 Niter6i, Brazil e Departamento de Fisica, Pontificia Universidade Cat6liea do Rio de Janeiro, CP. 38071, 22452-970, Rio de Janeiro, Brazil d Departamento de Fisica, Universidad de Buenos Aires, C.P. 1428, Nu~ez, Buenos Aires, Argentina
Abstract
The Bohm-Aharonov effect in a mesoscopic ring containing a double-barrier structure in its upper part is investigated. The interference between electrons propagating along different sides of the ring is modified by the presence of a double barrier. A local Coulomb interaction, U, is included to describe the highly localized electrons at the well and an AbrikosovSuhl (AS) resonance appears at low temperatures. So, in this energy range the current is flux dependent. Above the Kondo temperature TK the AS resonance disappears and the flux-independent behavior is recovered. The current is studied as a function of magnetic flux, gate potential and position of the Fermi level. Keywords." Mesoscopic ring; Bohm-Aharonov effect; Kondo resonance; Kondo effect
The Bohm-Aharonov effect [1] has been the subject of many experimental and theoretical studies in the realm of mesoscopic transport. The wave functions describing the currents propagating along two branches of a mesoscopic ring suffer a phase shift under the effect of an external magnetic field. According to this shift there is a constructive or destructive interference at the right contact of the device, which is reflected in the conductance as cycles of period 49o = he/e, depending on the magnetic flux encircled by the ring. The interference pattern produced by the dephasing of the two waves can be exploited to study the nature of transport in a particular system. Recently, there has been experimental [2] and theoretical [3] studies of the resonant tunneling of an electron going through a dot which belongs to one of the arms of a ring linked to two contacts. In the experiment mentioned [2] the ring is threaded by an external field and the dot is subjected to a gate * Corresponding author.
potential. The system exhibits pronounced oscillations of the conductance when the gate potential on the dot is modified. The Coulomb blockade of an electron when the dot is already occupied by another electron provides the theoretical explanation for the conductance oscillations. However, at low temperatures and with low external applied potentials, another Coulomb repulsion effect, the Kondo [4] effect, shows up [5]. For temperatures below the Kondo temperature, TK, the I - V characteristics should reflect the Kondo regime. This happens when the Fermi energy has an intermediate value between the energy of the localized level, in this case the gate potential: V0, and the energy the extra electron, V0 + U. The aim of this paper is to study the current going along a quantum dot in two different situations: at low temperatures when the system is in the Kondo regime, so the electron-electron interaction gives rise to an Abrikosov-Suhl resonance, and at higher temperatures, above TK, the flowing charges are dominated by the resonant tunneling condition and a Coulomb
0921-4526/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 6 3 4 - 5
479
J.R. 191esias et aL / Physica B 230-232 (1997) 478-481
blockade behavior is expected. As we show below, these two situations are clearly established because for a large region of the parameter space (gate potential, Coulomb repulsion and temperature) the Kondo effect is characterized by a current which depends strongly upon the magnetic flux q~. This is due to the interference of the waves going up and down along the ring. In the Coulomb blockade regime there is no dependence if the dot is out of the resonant tunneling condition. The system is described by an Anderson-impurity first-neighbor tight-binding Hamiltonian defined over a ring and two contacts such that H = Vo Z
no~ + UnoTno+
+t'~-~(c~ci~+c.c.)+~-]tiyc]~cj~, i~
(1)
ijc;
where i,j denote the atomic sites in the ring and contacts, U and V0 correspond to the electronic repulsion and the gate potential on the dot which, for the sake of simplicity, has been restricted to the site 0 only, tij is the first-neighbor hopping matrix element and F represents the connection of the dot to the upper arm of the ring. The local energy in all the sites, with the exception of the dot, is taken to be 0 and we also assume that the Coulomb interaction U is restricted to the dot where, due to quantum confinement, the electrons interact more strongly. The applied electric potential is taken to be negligibly small so that its contribution to the site energies can be neglected. The transport is supposed to be ballistic in nature so that the Hamiltonian includes neither dissipative mechanism nor other degrees of freedom but electrons. The external magnetic field producing the flux is incorporated in the first-neighbor matrix elements, tij, of the ring as t U = te -O~'/N)~¢'/¢°),
(2)
where t is the first-neighbor hopping, q~ is the quantum flux crossing the ring, ~b = f A d/, where A is the vector potential and 2N is the number of atoms of the ring with lattice parameter a. The current circulating along the ring is calculated using the Keldysh formalism [6]. In addition to the usual advanced and retarded Green's functions, two
other correlation functions are required: G~+(e)) = i
/5
{c~G(t) c/~(0))e i~')tdt,
(3)
and its complex conjugate, where the mean values are calculated on a nonequilibrium stationary state of the system under the effect of an applied electric potential. Gi~:+(e)) gives the spectral representation of the electronic occupation at site i, corresponding to this current flowing stationary state. To calculate the transport properties the system is partitioned into two semi-infinite supports so that the two subsystems are in thermodynamic equilibrium, the electric current is zero, and the corresponding Green's functions can be obtained by standard techniques. The dressed Green's functions for the nonequilibrium situation are calculated through a Dyson equation obtained by reestablishing the connection between the two subsystems, following the Keldysh formalism. The average current in the system due to the application of an external bias is given by = T
d o. (2<)
(4)
To obtain the total current circulating along the contacts and the current on the upper arm of the ring we use Eq. (4) with the site i being, respectively, a point in the contacts or in the upper arm of the ring. The Green's functions appearing in Eq. (4) are calculated using a cumulant diagrammatic expansion where the nondiagonal matrix elements linking the dot with the rest of the system is taken to be the perturbation. The expansion is restricted to the "chain approximation" [7] which has shown to be a suitable scheme to describe the Kondo and Coulomb blockade effects [8]. The dot is modeled by a cluster of three sites represented by the the first three terms of the Hamiltonian H (Eq. (1)) and for simplicity the Coulomb repulsion U is taken to be infinite. The eigenvalues and eigenvectors for all possible numbers of particles are then calculated. To obtain the propagators for the total system we calculate first the Green's function for the cluster from its spectral decomposition using the grand canonical ensemble, and within the "chain approximation" the propagators'for the total system are obtained solving a Dyson-type equation [8].
480
J.R. 191esias et al./ Physica B 230-232 (1997) 478~181
Using the above formalism we study the current circulating along a ring of 27 atoms connected to two semi-infinite linear chains as a function of the magnetic flux, for different values of the gate potential V0. At high temperatures, by varying V0 we change the energy of the dot resonant level to make it to coincide with the Fermi level. In this situation, which we call the resonant tunneling condition, the current circulates along the upper arm of the ring, where the dot is located, as well as along the lower arm, producing interference at the right contact. The interference pattern depends on the magnetic flux encircled by the ring since the magnetic field introduces a phase shift between the currents circulating along the two arms of the ring. As a result the total current shows a strong dependence on the magnetic flux. On the other hand, when the upper arm is taken out of resonance by increasing or reducing the gate potential, the electrons circulate only along one arm and the interference does not take place. The current is then flux independent. These features are shown in Fig. 1 where the current is represented as a function of the magnetic flux for high temperature ( k a T = 0.2 eV) and five values of the gate potential. There are oscillations for V0 equal to -0.2t, 0.0 and 0.2t, which are within the region of resonance, while for V0 equal to - 1 . 2 t and -0.6t, represented in the two lower curves, the currents do not present any magnetic flux dependence. By reducing the temperature below TK and choosing the gate potential such that the system is in the Kondo regime, /7o < ef, we obtain the curves shown in Fig. 2, where the current is depicted as a function of the magnetic flux for k a T = 0.015 eV and the same values of the gate potential as in Fig. 1. One can see that even for the values of V0 for which the dot level is out of resonance the currents show oscillations. This is a consequence of the appearence of an Abrikosov-Suhl resonance at the Fermi level providing a channel for the electron to go along the upper arm of the ring through the dot, given rise to a strong interference with the current circulating along the other arm of the ring. We have shown that from a theoretical view point the Bohm-Aharonov effect permits to make a very clear distinction between the low- and hightemperature behavior in the Kondo regime. We hope this could be a motivation for an experimental verification of these results.
HT
t~
~5
-
(c
.
.
.
.
.
!
(a)
G)
~4
0
I
I
0.5
1
1.5
Magnetic Flux (units of he/e) Fig. l. Current density as a function of magnetic flux for high temperature and different values of the gate potential Volt: (a) -1.2; (b) -0.6; (c) -0.2; (d) 0.0; (e) 0.2.
-
",{b)
/
t~
(e) u
0
, 0
,
,
,
I 0.5
,
,
,
1 Magnetic Flux (units of hc/e)
1.5
Fig. 2. Current density as a function of magnetic flux for low temperature and different values of the gate potential Volt: (a) -1.2; (b) -0.6; (c) -0.2; (d) 0.0; (e) 0.2.
J.R. 191esias et al./ Physica B 230-232 (1997) 478~81 We acknowledge Brazilian a g e n c i e s C N P q , C A P E S , and F A P E R G S for financial support.
References [1] Y. Aharonov and D. Bohm, Phys. Rex,. 115 (1959) 485. [2] A. Yacobi, M. Heiblum, D. Mahalu and H. Shtrikman, Phys. Rev. Lett. 74 (1995) 4047.
481
[3] A. Levy Yeyati and M. Buttiker, Phys. Rev. B 52 (1995). [4] N.S. Wingreen and Y. Meir, Phys. Rev. B 49 (1994) 11 040. [5] D.C. Ralph and R.A. Buhrrnan, Phys. Rev. Lett. 72 (1994) 3401. [6] L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515, [Sov. Phys. JETP 20 (1965) 1018]. [7] W. Metzner, Phys. Rev. B 43 (1991) 8549. [8] B.-H. Bernhard and J.R. lglesias, Phys. Rev. B 50 (1994) 9522; J. Arispe, B. Coqblin, A.S.R. Sim6es and J.R. Iglesias, J. Phys.: Condens. Matter 6 (1994) 7773.
Physica B 230-232 (1997) 478-481
The Kondo effect in Bohm-Aharonov resonant tunneling devices J.R. I g l e s i a s a'*, E . V . A n d a b, M a r i a A. D a v i d o v i c h c, G. C h i a p p e d a lnstituto de Fisica, Universidade Federal do Rio Grande do Sul, C.P. 15051, 91501-970 Porto Alegre, Brazil b Instituto de Fisica, Universidade Federal Fluminense, 24210-340 Niter6i, Brazil e Departamento de Fisica, Pontificia Universidade Cat6liea do Rio de Janeiro, CP. 38071, 22452-970, Rio de Janeiro, Brazil d Departamento de Fisica, Universidad de Buenos Aires, C.P. 1428, Nu~ez, Buenos Aires, Argentina
Abstract
The Bohm-Aharonov effect in a mesoscopic ring containing a double-barrier structure in its upper part is investigated. The interference between electrons propagating along different sides of the ring is modified by the presence of a double barrier. A local Coulomb interaction, U, is included to describe the highly localized electrons at the well and an AbrikosovSuhl (AS) resonance appears at low temperatures. So, in this energy range the current is flux dependent. Above the Kondo temperature TK the AS resonance disappears and the flux-independent behavior is recovered. The current is studied as a function of magnetic flux, gate potential and position of the Fermi level. Keywords." Mesoscopic ring; Bohm-Aharonov effect; Kondo resonance; Kondo effect
The Bohm-Aharonov effect [1] has been the subject of many experimental and theoretical studies in the realm of mesoscopic transport. The wave functions describing the currents propagating along two branches of a mesoscopic ring suffer a phase shift under the effect of an external magnetic field. According to this shift there is a constructive or destructive interference at the right contact of the device, which is reflected in the conductance as cycles of period 49o = he/e, depending on the magnetic flux encircled by the ring. The interference pattern produced by the dephasing of the two waves can be exploited to study the nature of transport in a particular system. Recently, there has been experimental [2] and theoretical [3] studies of the resonant tunneling of an electron going through a dot which belongs to one of the arms of a ring linked to two contacts. In the experiment mentioned [2] the ring is threaded by an external field and the dot is subjected to a gate * Corresponding author.
potential. The system exhibits pronounced oscillations of the conductance when the gate potential on the dot is modified. The Coulomb blockade of an electron when the dot is already occupied by another electron provides the theoretical explanation for the conductance oscillations. However, at low temperatures and with low external applied potentials, another Coulomb repulsion effect, the Kondo [4] effect, shows up [5]. For temperatures below the Kondo temperature, TK, the I - V characteristics should reflect the Kondo regime. This happens when the Fermi energy has an intermediate value between the energy of the localized level, in this case the gate potential: V0, and the energy the extra electron, V0 + U. The aim of this paper is to study the current going along a quantum dot in two different situations: at low temperatures when the system is in the Kondo regime, so the electron-electron interaction gives rise to an Abrikosov-Suhl resonance, and at higher temperatures, above TK, the flowing charges are dominated by the resonant tunneling condition and a Coulomb
0921-4526/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 6 3 4 - 5
479
J.R. 191esias et aL / Physica B 230-232 (1997) 478-481
blockade behavior is expected. As we show below, these two situations are clearly established because for a large region of the parameter space (gate potential, Coulomb repulsion and temperature) the Kondo effect is characterized by a current which depends strongly upon the magnetic flux q~. This is due to the interference of the waves going up and down along the ring. In the Coulomb blockade regime there is no dependence if the dot is out of the resonant tunneling condition. The system is described by an Anderson-impurity first-neighbor tight-binding Hamiltonian defined over a ring and two contacts such that H = Vo Z
no~ + UnoTno+
+t'~-~(c~ci~+c.c.)+~-]tiyc]~cj~, i~
(1)
ijc;
where i,j denote the atomic sites in the ring and contacts, U and V0 correspond to the electronic repulsion and the gate potential on the dot which, for the sake of simplicity, has been restricted to the site 0 only, tij is the first-neighbor hopping matrix element and F represents the connection of the dot to the upper arm of the ring. The local energy in all the sites, with the exception of the dot, is taken to be 0 and we also assume that the Coulomb interaction U is restricted to the dot where, due to quantum confinement, the electrons interact more strongly. The applied electric potential is taken to be negligibly small so that its contribution to the site energies can be neglected. The transport is supposed to be ballistic in nature so that the Hamiltonian includes neither dissipative mechanism nor other degrees of freedom but electrons. The external magnetic field producing the flux is incorporated in the first-neighbor matrix elements, tij, of the ring as t U = te -O~'/N)~¢'/¢°),
(2)
where t is the first-neighbor hopping, q~ is the quantum flux crossing the ring, ~b = f A d/, where A is the vector potential and 2N is the number of atoms of the ring with lattice parameter a. The current circulating along the ring is calculated using the Keldysh formalism [6]. In addition to the usual advanced and retarded Green's functions, two
other correlation functions are required: G~+(e)) = i
/5
{c~G(t) c/~(0))e i~')tdt,
(3)
and its complex conjugate, where the mean values are calculated on a nonequilibrium stationary state of the system under the effect of an applied electric potential. Gi~:+(e)) gives the spectral representation of the electronic occupation at site i, corresponding to this current flowing stationary state. To calculate the transport properties the system is partitioned into two semi-infinite supports so that the two subsystems are in thermodynamic equilibrium, the electric current is zero, and the corresponding Green's functions can be obtained by standard techniques. The dressed Green's functions for the nonequilibrium situation are calculated through a Dyson equation obtained by reestablishing the connection between the two subsystems, following the Keldysh formalism. The average current in the system due to the application of an external bias is given by = T
d o. (2<)
(4)
To obtain the total current circulating along the contacts and the current on the upper arm of the ring we use Eq. (4) with the site i being, respectively, a point in the contacts or in the upper arm of the ring. The Green's functions appearing in Eq. (4) are calculated using a cumulant diagrammatic expansion where the nondiagonal matrix elements linking the dot with the rest of the system is taken to be the perturbation. The expansion is restricted to the "chain approximation" [7] which has shown to be a suitable scheme to describe the Kondo and Coulomb blockade effects [8]. The dot is modeled by a cluster of three sites represented by the the first three terms of the Hamiltonian H (Eq. (1)) and for simplicity the Coulomb repulsion U is taken to be infinite. The eigenvalues and eigenvectors for all possible numbers of particles are then calculated. To obtain the propagators for the total system we calculate first the Green's function for the cluster from its spectral decomposition using the grand canonical ensemble, and within the "chain approximation" the propagators'for the total system are obtained solving a Dyson-type equation [8].
480
J.R. 191esias et al./ Physica B 230-232 (1997) 478~181
Using the above formalism we study the current circulating along a ring of 27 atoms connected to two semi-infinite linear chains as a function of the magnetic flux, for different values of the gate potential V0. At high temperatures, by varying V0 we change the energy of the dot resonant level to make it to coincide with the Fermi level. In this situation, which we call the resonant tunneling condition, the current circulates along the upper arm of the ring, where the dot is located, as well as along the lower arm, producing interference at the right contact. The interference pattern depends on the magnetic flux encircled by the ring since the magnetic field introduces a phase shift between the currents circulating along the two arms of the ring. As a result the total current shows a strong dependence on the magnetic flux. On the other hand, when the upper arm is taken out of resonance by increasing or reducing the gate potential, the electrons circulate only along one arm and the interference does not take place. The current is then flux independent. These features are shown in Fig. 1 where the current is represented as a function of the magnetic flux for high temperature ( k a T = 0.2 eV) and five values of the gate potential. There are oscillations for V0 equal to -0.2t, 0.0 and 0.2t, which are within the region of resonance, while for V0 equal to - 1 . 2 t and -0.6t, represented in the two lower curves, the currents do not present any magnetic flux dependence. By reducing the temperature below TK and choosing the gate potential such that the system is in the Kondo regime, /7o < ef, we obtain the curves shown in Fig. 2, where the current is depicted as a function of the magnetic flux for k a T = 0.015 eV and the same values of the gate potential as in Fig. 1. One can see that even for the values of V0 for which the dot level is out of resonance the currents show oscillations. This is a consequence of the appearence of an Abrikosov-Suhl resonance at the Fermi level providing a channel for the electron to go along the upper arm of the ring through the dot, given rise to a strong interference with the current circulating along the other arm of the ring. We have shown that from a theoretical view point the Bohm-Aharonov effect permits to make a very clear distinction between the low- and hightemperature behavior in the Kondo regime. We hope this could be a motivation for an experimental verification of these results.
HT
t~
~5
-
(c
.
.
.
.
.
!
(a)
G)
~4
0
I
I
0.5
1
1.5
Magnetic Flux (units of he/e) Fig. l. Current density as a function of magnetic flux for high temperature and different values of the gate potential Volt: (a) -1.2; (b) -0.6; (c) -0.2; (d) 0.0; (e) 0.2.
-
",{b)
/
t~
(e) u
0
, 0
,
,
,
I 0.5
,
,
,
1 Magnetic Flux (units of hc/e)
1.5
Fig. 2. Current density as a function of magnetic flux for low temperature and different values of the gate potential Volt: (a) -1.2; (b) -0.6; (c) -0.2; (d) 0.0; (e) 0.2.
J.R. 191esias et al./ Physica B 230-232 (1997) 478~81 We acknowledge Brazilian a g e n c i e s C N P q , C A P E S , and F A P E R G S for financial support.
References [1] Y. Aharonov and D. Bohm, Phys. Rex,. 115 (1959) 485. [2] A. Yacobi, M. Heiblum, D. Mahalu and H. Shtrikman, Phys. Rev. Lett. 74 (1995) 4047.
481
[3] A. Levy Yeyati and M. Buttiker, Phys. Rev. B 52 (1995). [4] N.S. Wingreen and Y. Meir, Phys. Rev. B 49 (1994) 11 040. [5] D.C. Ralph and R.A. Buhrrnan, Phys. Rev. Lett. 72 (1994) 3401. [6] L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515, [Sov. Phys. JETP 20 (1965) 1018]. [7] W. Metzner, Phys. Rev. B 43 (1991) 8549. [8] B.-H. Bernhard and J.R. lglesias, Phys. Rev. B 50 (1994) 9522; J. Arispe, B. Coqblin, A.S.R. Sim6es and J.R. Iglesias, J. Phys.: Condens. Matter 6 (1994) 7773.