- Email: [email protected]
Thermopower and thermal conductance for a Kondo correlated quantum dot
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245 www.elsevier.com/locate/jmmm
Thermopower and thermal conductance for a Kondo correlated quantum dot R. Francoa,, J. Silva-Valenciaa, M.S. Figueirab a
Departamento de Fı´sica, Universidad Nacional de Colombia, A. A. 5997 Bogota´, Colombia Instituto de Fı´sica, Universidade Federal Fluminense (UFF), Avenida litoraˆnea s/n, CEP: 24210-346, Nitero´i, Rio de Janeiro, Brazil
b
Available online 21 February 2008
Abstract We study the thermopower and thermal conductivity of a gate-defined quantum dot, with a very strong Coulomb repulsion inside the dot, employing the X-boson approach for the impurity Anderson model. Our results show a change in the sign of the thermopower as function of the energy level of the quantum dot (gate voltage), which is associated with an oscillatory behavior and a suppression of the thermopower magnitude at low temperatures. We identify two relevant energy scales: a low temperature scale dominated by the Kondo effect and a TD temperature scale characterized by charge fluctuations. We also discuss the Wiedemann–Franz relation and the thermoelectric figure of merit. Our results are in qualitative agreement with recent experimental reports and other theoretical treatments. r 2008 Elsevier B.V. All rights reserved. PACS: 72.15.Jf; 73.21.La; 73.63.Kv; 73.50.Lw; 73.23 Hk Keywords: Quantum dot; Kondo effect; Transport; X-boson; Thermoelectric and thermomagnetic effects
1. Introduction
2. Model and theory
The thermoelectrical effects in nanostructures have gained renewed interest in recent years [1–5]. It happens due to the possibility of using nanoscopic systems to enhance the efficiency of macroscopic devices through the control of the energy transport on a microscopic scale. Recently, it was reported the experimental realization of nanoscopic thermal rectifiers [5], as well as the Coulombian control over a refrigeration process in a mesoscopic system [6]. In this work, we study the thermopower (S), thermal conductance ðkÞ and the thermoelectric figure of merit (Z), in a single quantum dot (QD) as a function of the gate voltage. We employ the X-boson treatment for the impurity Anderson model (AIM), considering very strong Coulombian repulsion inside the QD [7].
We employ the model Hamiltonian H ¼ H L þ H R þ H D þ H T to describe a embedded QD, connected to a paramagnetic leads system. The Hamiltonian for the left (L) and right (R) leads are given by X Ha ¼ E k;s cyk;a;s ck;a;s ða ¼ L; RÞ, (1)
Corresponding author.
E-mail address: [email protected] (R. Franco). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.02.056
k;a;s
where cyk;a;s ðck;a;s Þ is a creation (destruction) operator of an electron with energy E k;s , momentum k and spin s on the lead a. The interacting QD is described by X E d;s X d;ss , (2) HD ¼ s
where we employ the AIM characterized by a localized d level E d;s , associated with the QD, in the representation of the Hubbard operators, that are convenient for working with local state linked with the QD and are defined in general by X d;ab ¼ jd; aihd; bj [7,8]. The tunneling
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
Hamiltonian H T is X X ðV a X yd;0s ck;a;s þ H:c:Þ. HT ¼
4.0
(3)
L22 ¼
pT 2 X D h s
Z qnF orQD;s ðoÞ do, qT
(5)
G sQD ðoÞ ¼
Ds , o E~d þ V 2 Ds Gsc ðoÞ
(7)
with the quantity Ds ¼ hX d;00 i þ nQD;s being responsible for the correlation in the cumulant X-boson approach and E~d ¼ E d þ L, where the parameter L renormalizes the localized level in such a way, that the completeness in the QD occupation numbers is fulfilled. The solution is obtained by the minimization of the Helmholtz free energy. 3. Results and discussion In our calculations we choose the following parameters D ¼ pV 2 =2D ¼ 1, D ¼ 100D, with all the energies expressed in D units. In Fig. 1(a) we represent the linear thermopower (S) vs. the gate voltage of the QD, here represented by the QD energy ðE QD Þ, for different temperatures. The linear thermopower result shows the
S (1/e)
0.0
-2.0 -3.0 -10
-8
-6
-2
-4
0 EQD/Δ
2
4
8
6
10
3 EQD=2.0Δ EQD=Δ
2
EQD=0.0Δ EQD=-Δ EQD=-2.0Δ
1
EQD=-3.0Δ
0
-1
(6)
here rQD;s ðoÞ ¼ ð1=pÞ ImðG sQD ðoÞÞ is the density of states (DOS) for the electrons with spin s in the QD, nF ðoÞ is the Fermi–Dirac distribution function and D ¼ pV 2 rc ðoÞ ¼ pV 2 =2D is the mixing width between the QD and the leads. rc ¼ ð1=pÞ ImðG sc ðoÞÞ ¼ 12 D is the DOS for the conduction band, connected to the leads, considering that these are ballistic channels, described by the GF G sc ðoÞ ¼ ð1=2DÞ ln½ðo þ DÞ=ðo DÞ, where o ¼ o þ iZ, with Z ! 0þ . Using the X-boson method, the QD GF is given by [7]
1.0
-1.0
-2 10-2
10-1
101
100 T/Δ
0.9 0.8 T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
0.7 0.6 κ/Δ (1/h)
L12
2.0
S (1/e)
The amplitude V a is responsible for the tunneling between the QD and the lead a. For simplicity, we assume symmetric junctions (i.e. V L ¼ V R ¼ V Þ and identical leads. Dong et al. [1] derived the particle current and thermal flux formulas, through an interacting QD connected to leads, within the framework of the Keldish non-equilibrium Greens functions (GF). The electric and thermoelectric transport coefficients were obtained in the presence of the chemical potential and temperature gradients with the Onsager relation in the linear regime automatically satisfied. We can calculate the linear thermopower S ¼ ð1=eTÞðL12 =L11 Þ, the thermal conductance k ¼ ð1=T 2 ÞðL22 L212 =L11 Þ and the electrical conductance G ¼ ðe2 =TÞL11 ; with the transport coefficients given by Z pT X qnF D L11 ¼ ðoÞ do, (4) r h qo QD;s s
T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
3.0
a¼L;R k;s
Z pT 2 X qnF D ¼ rQD;s ðoÞ do, h qT s
e243
0.5 0.4 0.3 0.2 0.1 0.0 -8
-6
-4
-2
0 EQD/Δ
2
4
6
8
Fig. 1. Linear thermopower (S) (a) vs. the QD energy ðE QD Þ for different temperatures, (b) vs. temperature for different values of the QD energy and (c) thermal conductance k as a function of the QD energy ðE QD Þ for different temperatures.
oscillatory behavior characteristic of the Kondo regime, as indicated by recent experimental reports of thermovoltage ðV T / SÞ [3], especially for the few-electron QD case [3, PRB 75 (2007) 041301]. The change in the sign of S is
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
5 T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
ZT=S2GT/κ
4
3
2
1
0 -8
-6
-4
-2
0 EQD/Δ
4
2
6
8
2.0 1.8
EQD=2.0Δ EQD=Δ
1.6
EQD=0.0Δ
1.4 ZT=S2GT/κ
associated with the QD DOS at the chemical potential rQD;s ðm ¼ 0Þ [1,2,4], which presents a maximum in the Kondo region (the Kondo resonance). This change is associated with the different regimes of the AIM: when the dot is in the Kondo regime ðE QD o DÞ the thermopower is mainly positive, in the intermediate valence (IV) region ðDoE QD oDÞ, S changes the sign and finally in the crossover from the IV to the empty dot region the thermopower becomes negative. In Fig. 1(b) we represent the linear thermopower (S) as a function of the temperature, for different values of the QD energy ðE QD Þ. The change in the thermopower sign is evident in this figure and if we fix one particular temperature and change the gate voltage, from the empty dot to the Kondo regime, the thermopower goes continuously from negative to positive values. In Fig. 1(c) we present the thermal conductance k as a function of the QD energy ðE QD Þ, for different temperatures. We clearly identify in this figure, two relevant energy scales: a low temperature scale dominated by the Kondo effect and characterized by the Kondo temperature T K (the k maximum occurs in the Kondo region for T ¼ 0:1DÞ and a TD temperature scale characterized by charge fluctuations (the k maximum occurs in the crossover from the IV to the empty dot region, for T ¼ 0:5D–2:0DÞ. The shape of the curve agrees qualitatively with that expected by the Wiedemann–Franz law ðk ¼ ðp2 =3e2 ÞTGÞ, for a fixed temperature [1,2,4]. The thermoelectric figure of merit Z ¼ S2 G=k [2,4] is a measure of the usefulness of materials or devices for thermopower generators or cooling systems. Since Z for simple systems varies inversely proportional to the temperature T, it is convenient to plot ZT, which indicates the system performance. In Fig. 2(a) we show the ZT, at several temperatures, vs. the QD energy E QD . It is evident that the low temperature contribution to the ZT is completely negligible; the values assumed by ZT are much smaller than the unity for all regimes of the system what means a very limited practical applicability to develop thermal nanoscopic devices in the Kondo region; the same conclusion was achieved by Sakano et al. [4]. But the situation changes completely for temperatures around T ’ D, in the crossover from the IV to empty dot region where ZT assumes higher values. In Fig. 2(b) we present the ZT vs. temperature, for several QD energy E QD values. The ZT presents a maximum around the T ’ D temperature scale; this peak is associated with charge fluctuations, characteristic of the IV regime. This regime should be considered interesting to explore practical applications, because the ZT achieve values greater than the unity. Finally in Fig. 2(c) we present the Wiedemann–Franz ratio vs. temperature. This law describes transport in Fermi liquid bulk metals and in general is not obeyed in QD systems where the transport takes place through a small
EQD=-Δ
1.2
EQD=-1.75Δ
1.0 0.8 0.6 0.4 0.2 0.0 10-2
100
10-1
101
T/Δ 1.4 EQD=2.0Δ
1.2
EQD=Δ EQD=0.0Δ
1.0 3e2κ/ (π2TG)
e244
EQD=-Δ
0.8
EQD=-2.0Δ
0.6 0.4 0.2 0.0 10-2
100
10-1
101
T/Δ Fig. 2. Thermoelectric figure of merit Z times temperature T, ZT (a) vs. energy of the QD, for different temperature values (b) vs. temperature for different values of the E QD and (c) the Wiedemann–Franz ratio vs. temperature for the different regimes of the system.
confined region, however, at very low temperatures, the Wiedemann–Franz law is recovered. Our results are in qualitative agreement with the reported by other authors [1,2].
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
4. Conclusions We study thermoelectric properties in a embedded QD connected to paramagnetic conducting leads. We describe the system employing the Anderson impurity model within the X-boson treatment [7]. We study all the characteristic regimes of the AIM: Kondo, IV and empty dot regimes. The system present two relevant energy scales: a low temperature scale dominated by the Kondo effect and a TD temperature scale characterized by charge fluctuations, where the ZT assumes values higher than the unity, what should indicates possible practical applications of the nanoscopic systems at temperatures characterized by the scale T ’ D, that is in qualitative agreement with a recent experimental report [5] (R. Scheibner, et al., cond-mat 0703514). Acknowledgments We thank the financial support of DINAIN: 20601003550, DIB:8003060 and COLCIENCIAS:1101-333-
e245
18707 (Colombia); CNPq and FAPERJ-‘‘Primeiros Projetos’’ (Brazil). References [1] B. Dong, et al., J. Phys. Condens. Matter 14 (2002) 11747. [2] M. Krawiec, et al., Phys. Rev. B 73 (2006) 075307; Phys. Rev. B 75 (2007) 155330 [3] R. Scheibner, et al., Phys. Rev. Lett. 95 (2005) 176602; R. Scheibner, et al., Phys. Rev. B. 75 (2007) 041301. [4] R. Sakano, et al., J. Phys. Soc. Jpn 76 (2007) 074709; J. Magn. Magn. Mater 310 (2007) 1136 [5] R. Scheibner, et al., cond-mat 0703514, 2007; C.W. Chang, et al., Science 314 (2006) 1121. [6] Olli-Pentti Saira, et al., Phys. Rev. Lett 99 (2007) 027203. [7] R. Franco, et al., Phys. Rev. B 73 (2006) 195305; R. Franco, et al., Phys. Rev. B 67 (2003) 155301; R. Franco, et al., Braz. J. Phys 36 (2006) 925; R. Franco, et al., Phys. Rev. B 66 (2002) 045112; R. Franco, et al., J. Magn. Magn. Mater 226–230 (2001) 194; R. Franco, et al., Physica A 308 (2002) 245. [8] T. Lobo, et al., Nanotechnology 17 (2006) 6016; T. Lobo, et al., Physica B 384 (2006) 113; T. Lobo, et al., Braz. J. Phys. 36 (2006) 397; T. Lobo, et al., Braz. J. Phys. 36 (2006) 401.
Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245 www.elsevier.com/locate/jmmm
Thermopower and thermal conductance for a Kondo correlated quantum dot R. Francoa,, J. Silva-Valenciaa, M.S. Figueirab a
Departamento de Fı´sica, Universidad Nacional de Colombia, A. A. 5997 Bogota´, Colombia Instituto de Fı´sica, Universidade Federal Fluminense (UFF), Avenida litoraˆnea s/n, CEP: 24210-346, Nitero´i, Rio de Janeiro, Brazil
b
Available online 21 February 2008
Abstract We study the thermopower and thermal conductivity of a gate-defined quantum dot, with a very strong Coulomb repulsion inside the dot, employing the X-boson approach for the impurity Anderson model. Our results show a change in the sign of the thermopower as function of the energy level of the quantum dot (gate voltage), which is associated with an oscillatory behavior and a suppression of the thermopower magnitude at low temperatures. We identify two relevant energy scales: a low temperature scale dominated by the Kondo effect and a TD temperature scale characterized by charge fluctuations. We also discuss the Wiedemann–Franz relation and the thermoelectric figure of merit. Our results are in qualitative agreement with recent experimental reports and other theoretical treatments. r 2008 Elsevier B.V. All rights reserved. PACS: 72.15.Jf; 73.21.La; 73.63.Kv; 73.50.Lw; 73.23 Hk Keywords: Quantum dot; Kondo effect; Transport; X-boson; Thermoelectric and thermomagnetic effects
1. Introduction
2. Model and theory
The thermoelectrical effects in nanostructures have gained renewed interest in recent years [1–5]. It happens due to the possibility of using nanoscopic systems to enhance the efficiency of macroscopic devices through the control of the energy transport on a microscopic scale. Recently, it was reported the experimental realization of nanoscopic thermal rectifiers [5], as well as the Coulombian control over a refrigeration process in a mesoscopic system [6]. In this work, we study the thermopower (S), thermal conductance ðkÞ and the thermoelectric figure of merit (Z), in a single quantum dot (QD) as a function of the gate voltage. We employ the X-boson treatment for the impurity Anderson model (AIM), considering very strong Coulombian repulsion inside the QD [7].
We employ the model Hamiltonian H ¼ H L þ H R þ H D þ H T to describe a embedded QD, connected to a paramagnetic leads system. The Hamiltonian for the left (L) and right (R) leads are given by X Ha ¼ E k;s cyk;a;s ck;a;s ða ¼ L; RÞ, (1)
Corresponding author.
E-mail address: [email protected] (R. Franco). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.02.056
k;a;s
where cyk;a;s ðck;a;s Þ is a creation (destruction) operator of an electron with energy E k;s , momentum k and spin s on the lead a. The interacting QD is described by X E d;s X d;ss , (2) HD ¼ s
where we employ the AIM characterized by a localized d level E d;s , associated with the QD, in the representation of the Hubbard operators, that are convenient for working with local state linked with the QD and are defined in general by X d;ab ¼ jd; aihd; bj [7,8]. The tunneling
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
Hamiltonian H T is X X ðV a X yd;0s ck;a;s þ H:c:Þ. HT ¼
4.0
(3)
L22 ¼
pT 2 X D h s
Z qnF orQD;s ðoÞ do, qT
(5)
G sQD ðoÞ ¼
Ds , o E~d þ V 2 Ds Gsc ðoÞ
(7)
with the quantity Ds ¼ hX d;00 i þ nQD;s being responsible for the correlation in the cumulant X-boson approach and E~d ¼ E d þ L, where the parameter L renormalizes the localized level in such a way, that the completeness in the QD occupation numbers is fulfilled. The solution is obtained by the minimization of the Helmholtz free energy. 3. Results and discussion In our calculations we choose the following parameters D ¼ pV 2 =2D ¼ 1, D ¼ 100D, with all the energies expressed in D units. In Fig. 1(a) we represent the linear thermopower (S) vs. the gate voltage of the QD, here represented by the QD energy ðE QD Þ, for different temperatures. The linear thermopower result shows the
S (1/e)
0.0
-2.0 -3.0 -10
-8
-6
-2
-4
0 EQD/Δ
2
4
8
6
10
3 EQD=2.0Δ EQD=Δ
2
EQD=0.0Δ EQD=-Δ EQD=-2.0Δ
1
EQD=-3.0Δ
0
-1
(6)
here rQD;s ðoÞ ¼ ð1=pÞ ImðG sQD ðoÞÞ is the density of states (DOS) for the electrons with spin s in the QD, nF ðoÞ is the Fermi–Dirac distribution function and D ¼ pV 2 rc ðoÞ ¼ pV 2 =2D is the mixing width between the QD and the leads. rc ¼ ð1=pÞ ImðG sc ðoÞÞ ¼ 12 D is the DOS for the conduction band, connected to the leads, considering that these are ballistic channels, described by the GF G sc ðoÞ ¼ ð1=2DÞ ln½ðo þ DÞ=ðo DÞ, where o ¼ o þ iZ, with Z ! 0þ . Using the X-boson method, the QD GF is given by [7]
1.0
-1.0
-2 10-2
10-1
101
100 T/Δ
0.9 0.8 T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
0.7 0.6 κ/Δ (1/h)
L12
2.0
S (1/e)
The amplitude V a is responsible for the tunneling between the QD and the lead a. For simplicity, we assume symmetric junctions (i.e. V L ¼ V R ¼ V Þ and identical leads. Dong et al. [1] derived the particle current and thermal flux formulas, through an interacting QD connected to leads, within the framework of the Keldish non-equilibrium Greens functions (GF). The electric and thermoelectric transport coefficients were obtained in the presence of the chemical potential and temperature gradients with the Onsager relation in the linear regime automatically satisfied. We can calculate the linear thermopower S ¼ ð1=eTÞðL12 =L11 Þ, the thermal conductance k ¼ ð1=T 2 ÞðL22 L212 =L11 Þ and the electrical conductance G ¼ ðe2 =TÞL11 ; with the transport coefficients given by Z pT X qnF D L11 ¼ ðoÞ do, (4) r h qo QD;s s
T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
3.0
a¼L;R k;s
Z pT 2 X qnF D ¼ rQD;s ðoÞ do, h qT s
e243
0.5 0.4 0.3 0.2 0.1 0.0 -8
-6
-4
-2
0 EQD/Δ
2
4
6
8
Fig. 1. Linear thermopower (S) (a) vs. the QD energy ðE QD Þ for different temperatures, (b) vs. temperature for different values of the QD energy and (c) thermal conductance k as a function of the QD energy ðE QD Þ for different temperatures.
oscillatory behavior characteristic of the Kondo regime, as indicated by recent experimental reports of thermovoltage ðV T / SÞ [3], especially for the few-electron QD case [3, PRB 75 (2007) 041301]. The change in the sign of S is
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
5 T=0.1Δ T=0.2Δ T=0.5Δ T=Δ T=2.0Δ
ZT=S2GT/κ
4
3
2
1
0 -8
-6
-4
-2
0 EQD/Δ
4
2
6
8
2.0 1.8
EQD=2.0Δ EQD=Δ
1.6
EQD=0.0Δ
1.4 ZT=S2GT/κ
associated with the QD DOS at the chemical potential rQD;s ðm ¼ 0Þ [1,2,4], which presents a maximum in the Kondo region (the Kondo resonance). This change is associated with the different regimes of the AIM: when the dot is in the Kondo regime ðE QD o DÞ the thermopower is mainly positive, in the intermediate valence (IV) region ðDoE QD oDÞ, S changes the sign and finally in the crossover from the IV to the empty dot region the thermopower becomes negative. In Fig. 1(b) we represent the linear thermopower (S) as a function of the temperature, for different values of the QD energy ðE QD Þ. The change in the thermopower sign is evident in this figure and if we fix one particular temperature and change the gate voltage, from the empty dot to the Kondo regime, the thermopower goes continuously from negative to positive values. In Fig. 1(c) we present the thermal conductance k as a function of the QD energy ðE QD Þ, for different temperatures. We clearly identify in this figure, two relevant energy scales: a low temperature scale dominated by the Kondo effect and characterized by the Kondo temperature T K (the k maximum occurs in the Kondo region for T ¼ 0:1DÞ and a TD temperature scale characterized by charge fluctuations (the k maximum occurs in the crossover from the IV to the empty dot region, for T ¼ 0:5D–2:0DÞ. The shape of the curve agrees qualitatively with that expected by the Wiedemann–Franz law ðk ¼ ðp2 =3e2 ÞTGÞ, for a fixed temperature [1,2,4]. The thermoelectric figure of merit Z ¼ S2 G=k [2,4] is a measure of the usefulness of materials or devices for thermopower generators or cooling systems. Since Z for simple systems varies inversely proportional to the temperature T, it is convenient to plot ZT, which indicates the system performance. In Fig. 2(a) we show the ZT, at several temperatures, vs. the QD energy E QD . It is evident that the low temperature contribution to the ZT is completely negligible; the values assumed by ZT are much smaller than the unity for all regimes of the system what means a very limited practical applicability to develop thermal nanoscopic devices in the Kondo region; the same conclusion was achieved by Sakano et al. [4]. But the situation changes completely for temperatures around T ’ D, in the crossover from the IV to empty dot region where ZT assumes higher values. In Fig. 2(b) we present the ZT vs. temperature, for several QD energy E QD values. The ZT presents a maximum around the T ’ D temperature scale; this peak is associated with charge fluctuations, characteristic of the IV regime. This regime should be considered interesting to explore practical applications, because the ZT achieve values greater than the unity. Finally in Fig. 2(c) we present the Wiedemann–Franz ratio vs. temperature. This law describes transport in Fermi liquid bulk metals and in general is not obeyed in QD systems where the transport takes place through a small
EQD=-Δ
1.2
EQD=-1.75Δ
1.0 0.8 0.6 0.4 0.2 0.0 10-2
100
10-1
101
T/Δ 1.4 EQD=2.0Δ
1.2
EQD=Δ EQD=0.0Δ
1.0 3e2κ/ (π2TG)
e244
EQD=-Δ
0.8
EQD=-2.0Δ
0.6 0.4 0.2 0.0 10-2
100
10-1
101
T/Δ Fig. 2. Thermoelectric figure of merit Z times temperature T, ZT (a) vs. energy of the QD, for different temperature values (b) vs. temperature for different values of the E QD and (c) the Wiedemann–Franz ratio vs. temperature for the different regimes of the system.
confined region, however, at very low temperatures, the Wiedemann–Franz law is recovered. Our results are in qualitative agreement with the reported by other authors [1,2].
ARTICLE IN PRESS R. Franco et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e242–e245
4. Conclusions We study thermoelectric properties in a embedded QD connected to paramagnetic conducting leads. We describe the system employing the Anderson impurity model within the X-boson treatment [7]. We study all the characteristic regimes of the AIM: Kondo, IV and empty dot regimes. The system present two relevant energy scales: a low temperature scale dominated by the Kondo effect and a TD temperature scale characterized by charge fluctuations, where the ZT assumes values higher than the unity, what should indicates possible practical applications of the nanoscopic systems at temperatures characterized by the scale T ’ D, that is in qualitative agreement with a recent experimental report [5] (R. Scheibner, et al., cond-mat 0703514). Acknowledgments We thank the financial support of DINAIN: 20601003550, DIB:8003060 and COLCIENCIAS:1101-333-
e245
18707 (Colombia); CNPq and FAPERJ-‘‘Primeiros Projetos’’ (Brazil). References [1] B. Dong, et al., J. Phys. Condens. Matter 14 (2002) 11747. [2] M. Krawiec, et al., Phys. Rev. B 73 (2006) 075307; Phys. Rev. B 75 (2007) 155330 [3] R. Scheibner, et al., Phys. Rev. Lett. 95 (2005) 176602; R. Scheibner, et al., Phys. Rev. B. 75 (2007) 041301. [4] R. Sakano, et al., J. Phys. Soc. Jpn 76 (2007) 074709; J. Magn. Magn. Mater 310 (2007) 1136 [5] R. Scheibner, et al., cond-mat 0703514, 2007; C.W. Chang, et al., Science 314 (2006) 1121. [6] Olli-Pentti Saira, et al., Phys. Rev. Lett 99 (2007) 027203. [7] R. Franco, et al., Phys. Rev. B 73 (2006) 195305; R. Franco, et al., Phys. Rev. B 67 (2003) 155301; R. Franco, et al., Braz. J. Phys 36 (2006) 925; R. Franco, et al., Phys. Rev. B 66 (2002) 045112; R. Franco, et al., J. Magn. Magn. Mater 226–230 (2001) 194; R. Franco, et al., Physica A 308 (2002) 245. [8] T. Lobo, et al., Nanotechnology 17 (2006) 6016; T. Lobo, et al., Physica B 384 (2006) 113; T. Lobo, et al., Braz. J. Phys. 36 (2006) 397; T. Lobo, et al., Braz. J. Phys. 36 (2006) 401.