- Email: [email protected]
Coherent transport through T-shaped electrostatically coupled quantum dots
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 2423–2424 www.elsevier.com/locate/jmmm
Coherent transport through T-shaped electrostatically coupled quantum dots S. Lipinski, D. Krychowski Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, Poznan´ 60-179, Poland Available online 16 November 2006
Abstract The Fano–Kondo resonance is studied in a system consisting of a pair of side-coupled double quantum dots attached to quantum wires. The double dots are capacitively coupled. The mean field slave boson approach and the equation of motion method are used. For symmetric dots Kondo effect has two possible sources: spin and charge degeneracies. The conductance as a function of gate voltage is characterized by a Fano asymmetric parameter q. For q ¼ 0 a half-reflection is observed. The same value of transmission is expected in the limit q-N. For intermediate cases transmission line has a general asymmetric Fano-type shape. We show that by tuning the gate voltage or changing magnetic field, the single spin-orbital Kondo channel opens and the system can operate as a spin filter. r 2006 Elsevier B.V. All rights reserved. PACS: 72.15.Qm; 72.25.Dc; 73.63.Kv; 75.20.Hr Keywords: Spin polarized transport; Quantum dot; Kondo effect
Recently there has been substantial interest in the interplay of Kondo effect and interference [1–3]. Kondo effect may also occur in the absence of spin if another quantum number, e.g., charge or orbital degree of freedom, gives rise to degeneracy. Such phenomena were observed in semiconducting double dots [4,5] and in carbon nanotubes [6]. In the present paper, we discuss Fano-orbital Kondo effect. We consider a pair of double quantum dots (DQD) capacitively coupled and side attached to quantum wires. Each of the DQDs consists of the open dot (OQD) contacted by the leads and the interacting dot (IQD). The system is modeled by an extended two-site Anderson Hamiltonian: X X X þ H¼ E kais cþ c þ c c þ E is d þ 0 0is kais is d is 0is kais is
kais
is
X X þ þ t cþ t1 c0is d is þ h:c: kais c0is þ h:c: þ kais
þ
X i
Uniþ ni þ
X
is
U 1 n1s n2s0 ,
ð1Þ
ss0
Corresponding author. Tel.: +48 061 869 127; fax: +48 061 868 4524.
E-mail address: [email protected] (S. Lipinski). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.811
Eis ¼ Ei+gish (we set |e| ¼ mB ¼ 1). The first term describes electrons in the electrodes, the second and third represent the OQDs and IQDs, the next two terms describe tunneling to the leads and interdot IQD–OQD tunneling and the last two terms account for intra (U) and intercoulomb (U1) interactions. Linear response conductance is given by the Landauertype formula: sis ¼
e2 G h 2
qf ðoÞ do Im½G 0is ðoÞ , qo 1
Z
1
(2)
where f is the Fermi distribution function, G the coupling strength to the electrodes (for the rectangular density of states of electrodes 1/2D for |o|o D, G ¼ pt2/D). G0is denotes the Green’s function of OQD. In the following, we restrict to the limit of infinite interactions. The Green’s function is found by the slave boson approach (SBMFA) [7] and complementary, by the equation of motion method (EOM) [8]. The calculations were performed for D ¼ 50 G (G—energy unit), t1 ¼ 0.5. For vanishing magnetic field and symmetric dots (E1 ¼ E2), Kondo effect appears simultaneously in spin and orbital sectors (SU(4)). Interference between ballistic channels through OQDs
ARTICLE IN PRESS
1.0
0.5
0.8
0.8 0.6
q=0
0.4
Polarization of conductance
Transmission
1.0
0.2
Transmission
0.0
-0.02
0.6
0.00
0.02
Energy
q = -1 0.4
q= 1
Polarization of conductance
S. Lipinski, D. Krychowski / Journal of Magnetism and Magnetic Materials 310 (2007) 2423–2424
2424
0.2
∆E = 0.005 OQD2
0.0
-0.2 -0.02 -0.01 0.00 0.01 0.02 h
0.0
∆E = 0 ∆E = 0.005
0.2
-0.5 0.0 -2.0x10-6
2.0x10-6
0.0
4.0x10-6
-0.01
Energy Fig. 1. Transmission of T-shaped DQD calculated by SBMFA for E1 ¼ E2 ¼ 3, q ¼ 71. Inset presents the case of q ¼ 0.
1.0
Conductance [2e2/h]
0.8
0.6
0.4 E = -1/2 (EOM) E = -3 (SBMFA) E = -1/2 (SBMFA)
0.2
0.0 -1
0
OQD1
0.00 h
0.01
Fig. 3. EOM results for the polarization of conductance of the OQD1 as a function of magnetic field for E1 ¼ 3 and two values of DE ¼ E2E!, g1 ¼ g2 ¼ 1. Inset shows PC of OQD2.
q ¼ 1 and to the full transmission for q ¼ 1 (constructive interference). For higher values of Ei the conductance is driven not only by the interference with Kondo resonance channel, but is also influenced by atomic level peak. Fig. 3 illustrates a possibility of control of the spin polarization of conductance (PC(i) (si+si)/(si++si)). For E1 ¼ E2 magnetic field causes crossover from SU(4) state to purely orbital Kondo states for each spin channel (SU(2)). The dip in transmission line splits and conductance becomes strongly spin dependent. Spin filtering for the nondegenerate case E16¼E2 results from magnetic field tuning into orbital degeneracy. For the considered case (g1 ¼ g2), the signs of conductance polarizations at dots are opposite, for g1 ¼ g2 they are the same. The work was supported by the EU grant CARDEQ under contract IST-021285-2 and by MAG-EL-MAT network.
1
q Fig. 2. Conductance vs. asymmetric Fano parameter for different values of IQDs energies.
and the Kondo resonant channels from IQDs leads to a dip structure in the transmission line symmetric for q ¼ e0/G ¼ 0 and asymmetric Fano type for qa0 (Fig. 1). The observed half-reflection for q ¼ 0 is a consequence of p/4 phase shift characteristic for SU(4) symmetry. For the deep Kondo range (Figs. 1 and 2) destructive interference leads to a complete reflection for
References [1] K. Kang, S. Cho, J. Kim, S. Shin, Phys. Rev. B 32 (2001) 113304. [2] I. Maruyama, N. Shibata, K. Ueda, J. Phys. Soc. Japan 73 (2004) 434. [3] M. Sato, H. Aikawa, K. Kobayashi, S. Katsimoto, Y. Iye, Phys. Rev. Lett. 95 (2005) 066801. [4] U. Wilhelm, J. Schmidt, J. Weis, K.V. Klitzing, Physica E 14 (2002) 385. [5] A.W. Holleitner, A. Chudnovsky, D. Pfannkuche, K. Eberl, R.H. Blick, Phys. Rev. B 70 (2004) 07525204. [6] P. Jarillo-Herrero, J. Kim, H.S.J. Van Der Zant, C. Dekker, L.P. Kouvenhoven, S. De Fransechi, Nature 434 (2005) 484. [7] P. Colemann, Phys. Rev. B 29 (1984) 3035. [8] C. Lacroix, J. Phys. F 11 (1981) 2389.
Journal of Magnetism and Magnetic Materials 310 (2007) 2423–2424 www.elsevier.com/locate/jmmm
Coherent transport through T-shaped electrostatically coupled quantum dots S. Lipinski, D. Krychowski Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, Poznan´ 60-179, Poland Available online 16 November 2006
Abstract The Fano–Kondo resonance is studied in a system consisting of a pair of side-coupled double quantum dots attached to quantum wires. The double dots are capacitively coupled. The mean field slave boson approach and the equation of motion method are used. For symmetric dots Kondo effect has two possible sources: spin and charge degeneracies. The conductance as a function of gate voltage is characterized by a Fano asymmetric parameter q. For q ¼ 0 a half-reflection is observed. The same value of transmission is expected in the limit q-N. For intermediate cases transmission line has a general asymmetric Fano-type shape. We show that by tuning the gate voltage or changing magnetic field, the single spin-orbital Kondo channel opens and the system can operate as a spin filter. r 2006 Elsevier B.V. All rights reserved. PACS: 72.15.Qm; 72.25.Dc; 73.63.Kv; 75.20.Hr Keywords: Spin polarized transport; Quantum dot; Kondo effect
Recently there has been substantial interest in the interplay of Kondo effect and interference [1–3]. Kondo effect may also occur in the absence of spin if another quantum number, e.g., charge or orbital degree of freedom, gives rise to degeneracy. Such phenomena were observed in semiconducting double dots [4,5] and in carbon nanotubes [6]. In the present paper, we discuss Fano-orbital Kondo effect. We consider a pair of double quantum dots (DQD) capacitively coupled and side attached to quantum wires. Each of the DQDs consists of the open dot (OQD) contacted by the leads and the interacting dot (IQD). The system is modeled by an extended two-site Anderson Hamiltonian: X X X þ H¼ E kais cþ c þ c c þ E is d þ 0 0is kais is d is 0is kais is
kais
is
X X þ þ t cþ t1 c0is d is þ h:c: kais c0is þ h:c: þ kais
þ
X i
Uniþ ni þ
X
is
U 1 n1s n2s0 ,
ð1Þ
ss0
Corresponding author. Tel.: +48 061 869 127; fax: +48 061 868 4524.
E-mail address: [email protected] (S. Lipinski). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.811
Eis ¼ Ei+gish (we set |e| ¼ mB ¼ 1). The first term describes electrons in the electrodes, the second and third represent the OQDs and IQDs, the next two terms describe tunneling to the leads and interdot IQD–OQD tunneling and the last two terms account for intra (U) and intercoulomb (U1) interactions. Linear response conductance is given by the Landauertype formula: sis ¼
e2 G h 2
qf ðoÞ do Im½G 0is ðoÞ , qo 1
Z
1
(2)
where f is the Fermi distribution function, G the coupling strength to the electrodes (for the rectangular density of states of electrodes 1/2D for |o|o D, G ¼ pt2/D). G0is denotes the Green’s function of OQD. In the following, we restrict to the limit of infinite interactions. The Green’s function is found by the slave boson approach (SBMFA) [7] and complementary, by the equation of motion method (EOM) [8]. The calculations were performed for D ¼ 50 G (G—energy unit), t1 ¼ 0.5. For vanishing magnetic field and symmetric dots (E1 ¼ E2), Kondo effect appears simultaneously in spin and orbital sectors (SU(4)). Interference between ballistic channels through OQDs
ARTICLE IN PRESS
1.0
0.5
0.8
0.8 0.6
q=0
0.4
Polarization of conductance
Transmission
1.0
0.2
Transmission
0.0
-0.02
0.6
0.00
0.02
Energy
q = -1 0.4
q= 1
Polarization of conductance
S. Lipinski, D. Krychowski / Journal of Magnetism and Magnetic Materials 310 (2007) 2423–2424
2424
0.2
∆E = 0.005 OQD2
0.0
-0.2 -0.02 -0.01 0.00 0.01 0.02 h
0.0
∆E = 0 ∆E = 0.005
0.2
-0.5 0.0 -2.0x10-6
2.0x10-6
0.0
4.0x10-6
-0.01
Energy Fig. 1. Transmission of T-shaped DQD calculated by SBMFA for E1 ¼ E2 ¼ 3, q ¼ 71. Inset presents the case of q ¼ 0.
1.0
Conductance [2e2/h]
0.8
0.6
0.4 E = -1/2 (EOM) E = -3 (SBMFA) E = -1/2 (SBMFA)
0.2
0.0 -1
0
OQD1
0.00 h
0.01
Fig. 3. EOM results for the polarization of conductance of the OQD1 as a function of magnetic field for E1 ¼ 3 and two values of DE ¼ E2E!, g1 ¼ g2 ¼ 1. Inset shows PC of OQD2.
q ¼ 1 and to the full transmission for q ¼ 1 (constructive interference). For higher values of Ei the conductance is driven not only by the interference with Kondo resonance channel, but is also influenced by atomic level peak. Fig. 3 illustrates a possibility of control of the spin polarization of conductance (PC(i) (si+si)/(si++si)). For E1 ¼ E2 magnetic field causes crossover from SU(4) state to purely orbital Kondo states for each spin channel (SU(2)). The dip in transmission line splits and conductance becomes strongly spin dependent. Spin filtering for the nondegenerate case E16¼E2 results from magnetic field tuning into orbital degeneracy. For the considered case (g1 ¼ g2), the signs of conductance polarizations at dots are opposite, for g1 ¼ g2 they are the same. The work was supported by the EU grant CARDEQ under contract IST-021285-2 and by MAG-EL-MAT network.
1
q Fig. 2. Conductance vs. asymmetric Fano parameter for different values of IQDs energies.
and the Kondo resonant channels from IQDs leads to a dip structure in the transmission line symmetric for q ¼ e0/G ¼ 0 and asymmetric Fano type for qa0 (Fig. 1). The observed half-reflection for q ¼ 0 is a consequence of p/4 phase shift characteristic for SU(4) symmetry. For the deep Kondo range (Figs. 1 and 2) destructive interference leads to a complete reflection for
References [1] K. Kang, S. Cho, J. Kim, S. Shin, Phys. Rev. B 32 (2001) 113304. [2] I. Maruyama, N. Shibata, K. Ueda, J. Phys. Soc. Japan 73 (2004) 434. [3] M. Sato, H. Aikawa, K. Kobayashi, S. Katsimoto, Y. Iye, Phys. Rev. Lett. 95 (2005) 066801. [4] U. Wilhelm, J. Schmidt, J. Weis, K.V. Klitzing, Physica E 14 (2002) 385. [5] A.W. Holleitner, A. Chudnovsky, D. Pfannkuche, K. Eberl, R.H. Blick, Phys. Rev. B 70 (2004) 07525204. [6] P. Jarillo-Herrero, J. Kim, H.S.J. Van Der Zant, C. Dekker, L.P. Kouvenhoven, S. De Fransechi, Nature 434 (2005) 484. [7] P. Colemann, Phys. Rev. B 29 (1984) 3035. [8] C. Lacroix, J. Phys. F 11 (1981) 2389.