- Email: [email protected]
Theory of the nonequilibrium Kondo effect in a quantum dot
Available online at www.sciencedirect.com
Physica E 22 (2004) 498 – 501 www.elsevier.com/locate/physe
Theory of the nonequilibrium Kondo e&ect in a quantum dot Tatsuya Fujiia;∗ , Kazuo Uedaa; b a Institute
b Advanced
for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Science Research Center, Japan Atomic Energy Research Institute, Tokai 319-1195, Japan
Abstract Based on the recent approach to the Kondo e&ect in a dot with a -nite bias voltage by using the perturbation theory, we discuss the new peak found in the di&erential conductance, in addition to the zero-bias peak. The new peak originate from the splitting of the Kondo resonance under a -nite voltage which may be measured by the recent proposed three-terminal quantum dot. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.63.Kv; 72.15.Qm Keywords: Nonequilibrium Kondo e&ect; Quantum dot
Recently, it has been established that the Kondo e&ect is relevant to the transport property through quantum dots [1–5]. Clearly, the new feature of the Kondo transport compared with the usual Kondo effect of magnetic impurities is the nonequilibrium nature, since the current is measured with a -nite bias voltage. One possible method to study the nonequilibrium Kondo e&ect is the perturbation theory with respect to the Coulomb interaction U in the dot [6] based on the Keldysh formalism [7]. According to the results obtained by the second-order perturbation theory, the Kondo resonance is simply suppressed and does not show any particular structure in the nonequilibrium situation. Concerning the equilibrium Kondo problem it is well known that the second-order perturbation theory gives remarkably good results [8]. Indeed, the exact solution shows a rapid convergence ∗
Corresponding author. E-mail address: [email protected] (T. Fujii).
of the perturbation series [9]. However, it is not clear whether the second-order perturbation theory works well in the nonequilibrium conditions. Actually, the noncrossing approximation (NCA) has predicted the double peak structure at chemical potentials of both leads [10]. This result was obtained also by other approaches, the equation of motion [10], a real-time diagrammatic formulation [11] and scaling method [12]. However it remains unclear how the e&ects of the double peak structure, if it is true, appear in the di&erential conductance. Thus we have developed a theory of quantum transport through a dot under -nite voltage by using the perturbation theory up to the fourth order of U based on the Keldysh formalism [13]. We have shown that a single Kondo peak splits into double peaks when the voltage exceeds the Kondo temperature, eV ¿ kB TK , which leads to the appearance of the new peak in conductance G(V ) in addition to the zero-bias peak. Moreover, we have discussed the possible relevance of the present study to the 0.7 conductance anomaly in QPC [14–16]. It has been
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.054
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
suggested that Kondo e&ect plays signi-cant role as the origin of the 0.7 structure in QPC because the structure is found when the conductance takes a value close to the unitary limit 2e2 =h [16] with decreasing temperature. In its experimental report [16], it is clearly stated that the separatrix between the region of the conductance of the unitary limit and the 0.8 plateau region follows the Kondo temperature, which is precisely our prediction of the appearance of the new peak. However, theoretically, it is necessary to discuss the origin of the new peak in more detail because the other theoretical treatments including Refs. [10–12] do not show this additional peak. In this paper, -rst we brieFy review our recent results of the nonequilibrium Kondo e&ect at a dot [13]. We show that the new peak is originated in the behavior of @ (!)=@eV which appears in the di&erential conductance expressed as G(V ) ˙ (eV=2) + eV=2 @ (!)=@eV d!. Here (!) is the density of −eV=2 states (DOS) of the electron with spin at a dot. Recently, it was suggested that the DOS can be measured by using a three-terminal con-guration. In order to consider a single-level quantum dot attached to two leads, we employ the impurity Anderson model H=
k
† jk ck ck
+
k
+
† (Vk ck d
d n + Un↑ n↓
+ h:c:);
First we calculate the density of states at the dot 1 (!) = − Im G r (!);
(1)
where = L, R and ckL (ckR ) annihilates an electron in the left(right)-lead, d annihilates an electron with spin in the dot and n = d† d . The coupling constants, Vk describe the tunneling matrix elements between the dot and leads. Nonequilibrium steady state is realized by a -nite di&erence between L and R which are chemical potentials of the leads in both sides. We concentrate on the symmetric Anderson model without loss of generality, where the symmetric conditions are stated as L = R ≡ , L = −R = eV=2 and −d = d + U . Here L; R represent resonance width at the chemical potentials, L; R (!) = 2 k |VkL; R |2 (! − jk ). In this paper we restrict ourselves to the ground state, T = 0.
(2)
where G r (!) is the retarded Green function. Then the current through the dot is expressed by L R e ∞ d! (!) I= ˝ −∞ L + R × (fL (!) − fR (!));
where fL; R (!) = 1=(1 + e (!−L·R ) ). From the current the di&erential conductance is obtained by G(V ) =
@I : @V
(3)
In order to estimate G r (!), we employ the perturbation theory based on the Keldysh formalism [7]. The Dyson equations for the Keldysh formalism can be written in a matrix form G (!) = g (!) + g (!)
(!)G (!);
(4)
where g −+ (!) = −i2Im g r (!)fe& (!);
(5)
g +− (!) = +i2Im g r (!)(1 − fe& (!));
(6)
g − − (!) =
499
1 − fe& (!) fe& (!) + ; ! + i ! − i
g ++ (!) = −g − − (!)∗
(7) (8)
with fe& (!) = (L fL (!) + R fR (!))=(L + R ) and g r (!) = (! + i)−1 . fe& (!) is the weighted average of the Fermi functions of leads. So far the analysis of the perturbation theory of U has been restricted up to the lowest second order [6]. In order to estimate higher order contributions the four-point vertex which is obtained by extending the method used by Keldysh for the electron–phonon vertex [7] is very convenient. This procedure is to insert vertices, − + + (0)1 2 ; 3 4 = U ( − 1 2 3 4 − 1 2 3 4 );
(9)
in each of diagrams, which may be called as the Keldysh vertex. Here 31 2 = 1 2 z 2 3 , with z 2 3 being the third Pauli matrix.
500
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
1
1
ρσ(ω) (1/πΓ)
G(V) (2e2/h)
eV/Γ = 0
0.8
eV/Γ = 0.5 eV/Γ = 1.0 eV/Γ = 2.0
0.6 0.4
eV/Γ = 8.0
-10
-5
0 ω (Γ)
5
0.4 0.2 0
1
10
Fig. 1. The density of states in units of 1= for various eV= where U= = 6.
All diagrams up to the fourth order are shown in Ref. [13] for the symmetric case where the third-order terms vanish in the same way as the equilibrium case. We can write down the analytic expressions of 1 2 up to the fourth order described by these diagrams by using Feynman rules where the insertion of the Keldysh vertex is considered additionally. The imaginary part of r is de-ned by using the o&-diagonal components of the self-energy matrix 1 2 as
r
(!)
0.6
-0.2
0.2
Im
G1(V) G(V) G2(V)
0.8
= 12 (i
+−
(!)
−i
−+
(!)):
(10)
Then the real part of r is obtained by the Kramers– Kronig relation, ∞ Im r (! ) 1 (11) Re r (!) = P d! : −! ! −∞ The Dyson equation for the retarded component is derived from Eq. (4) as G r (!) = g r (!) + g r (!) r (!)G r (!). Thus the DOS de-ned by Eq. (2) can be calculated. The density of states is shown in Fig. 1 in units of 1=. In the equilibrium case, the sharp Kondo peak reaches to the unitary limit value 1= at != = 0. Under a -nite voltage of eV= = 0:5, the Kondo peak is broadened with keeping a single-peak because the real transitions between dot and leads are slightly enhanced. This behavior is con-rmed by the analysis based on the Ward identities [17]. With further increasing the voltage, the Kondo peak splits into double peaks at ! ∼ ±eV=2, which locate near
2
3
4 5 eV (Γ)
6
7
8
Fig. 2. The di&erential conductance for U= = 6, G(V ), is given by the sum of G1 (V ) and G2 (V ). The new peak is originated in the behavior of G2 (V ).
the chemical potentials of the two leads. This behavior is qualitatively consistent with the results obtained by the NCA [10]. This change occurs when the potential drop, eV , exceeds kB TK de-ned as the full-width at half-maximum of the Kondo peak. For U= = 6, kB TK = is estimated to be about 0.6. Let us discuss the e&ects of the DOS to G(V ). As shown in Fig. 2 the new peak appears in G(V ) for U= = 6 between the zero-bias peak and the broad peak at eV ∼ U . To consider the new peak, we rewrite the expression of G(V ) in Eq. (3) as eV=2 2e2 @ (!) G(V ) = (eV=2) + d! h @eV −eV=2 ≡ G1 (V ) + G2 (V ):
(12)
As shown in Fig. 2, the -rst term, G1 (V ), decreases monotonically as V is increased. Concerning the second term, G2 (V ), -rst we note that if the integrand is the entire frequency space it should vanish due to the sum-rule of the spectral weight. For eV ¡ kB TK the spectral weight shifts from the integrand to higher frequencies, giving a negative contribution. When eV ∼ kB TK a single Kondo peak splits to double peaks. Then in a considerable part of the frequency space, @ (!)=@eV becomes positive, which is shown in Fig. 3. A signi-cant part of the positive contributions enter in the integrand of G2 (V ), thus giving less negative contribution. As a result we may conclude that the new peak appears when the bias voltage exceeds the Kondo energy.
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
501
values when eV ∼ kB TK , giving the appearance of the new peak in the di&erential conductance. The authors would like to thank K. Kobayashi and A. Oguri for helpful discussions.
References
Fig. 3. @ (!)=@eV in units of 1= of U= = 6, where eV= = 0:5 is near by the Kondo temperature kB TK = 0:6.
It is not so easy to measure the DOS experimentally because its full information cannot be obtained only by measuring G(V ) at a dot with two leads. However recently a theoretical prediction was made to measure the DOS at a dot with two leads by introducing an additional probe source lead which is rather weakly coupled with a dot [18,19]. It is found that when the chemical potential of an additional lead is changed with keeping a certain bias between the source and drain, the di&erential conductance of the current into the third lead follows the shape of the DOS at a dot with two leads. Moreover in Ref. [19] it has been shown that this behavior is not much a&ected even its coupling is -nite, not necessary very small. This method is eMcient to explore the DOS at a dot with two leads. Actually, the splitting of the Kondo peak is successfully observed [20] based on this prediction. In summary, we have overviewed our recent results on the nonequilibrium Kondo e&ect under a -nite bias voltage. It is shown that @ (!)=@eV has positive
[1] T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768. [2] L.I. Glazman, M.E. Raikh, JETP Lett. 47 (1988) 452. [3] D. Goldhaber-Gordon, et al., Nature (London) 391 (1998) 156; D. Goldhaber-Gordon, et al., Phys. Rev. Lett. 81 (1998) 5225. [4] S.M. Cronenwett, et al., Science 281 (1998) 540. [5] W.G. van der Wiel, et al., Science 289 (2000) 2105. [6] S. Hersh-eld, et al., Phys. Rev. Lett. 67 (1991) 3720; S. Hersh-eld, et al., Phys. Rev. B. 46 (1992) 7046. [7] L.V. Keldysh, Sov. Phys. JETP 20 (1965) 1018. [8] K. Yosida, K. Yamada, Progr. Theor. Phys. 46 (Suppl.) (1970) 244; K. Yosida, K. Yamada, Progr. Theor. Phys. 53 (1975) 1286; K. Yamada, Progr. Theor. Phys. 53 (1975) 970; K. Yamada, Progr. Theor. Phys. 54 (1975) 316. [9] V. ZlatiQc, et al., Phys. Rev. B 28 (1983) 6904; K. Ueda, et al., J. Phys. C 16 (1983) L849. [10] Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601; N.S. Wingreen, Y. Meir, Phys. Rev. B. 49 (1994) 11040. [11] J. KRonig, et al., Phys. Rev. B 54 (1996) 16820. [12] A. Rosch, et al., Phys. Rev. Lett. 87 (2001) 156802; A. Rosch, et al., cond-mat/0202404. [13] T. Fujii, K. Ueda, Phys. Rev. B 68 (2003) 155310. [14] K.J. Thomas, et al., Phys. Rev. Lett. 77 (1996) 135. [15] A. Kristensen, et al., Phys. Rev. B 62 (2000) 10950. [16] S.M. Cronenwett, et al., Phys. Rev. Lett. 88 (2002) 226805. [17] A. Oguri, Phys. Rev. B. 64 (2001) 153305. [18] Q.-F. Sun, et al., Phys. Rev. B 64 (2001) 153306. [19] E. Lebanon, et al., Phys. Rev. B 65 (2002) 035308. [20] S. De Franceschi, et al., Phys. Rev. Lett. 89 (2002) 156801.
Physica E 22 (2004) 498 – 501 www.elsevier.com/locate/physe
Theory of the nonequilibrium Kondo e&ect in a quantum dot Tatsuya Fujiia;∗ , Kazuo Uedaa; b a Institute
b Advanced
for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Science Research Center, Japan Atomic Energy Research Institute, Tokai 319-1195, Japan
Abstract Based on the recent approach to the Kondo e&ect in a dot with a -nite bias voltage by using the perturbation theory, we discuss the new peak found in the di&erential conductance, in addition to the zero-bias peak. The new peak originate from the splitting of the Kondo resonance under a -nite voltage which may be measured by the recent proposed three-terminal quantum dot. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.63.Kv; 72.15.Qm Keywords: Nonequilibrium Kondo e&ect; Quantum dot
Recently, it has been established that the Kondo e&ect is relevant to the transport property through quantum dots [1–5]. Clearly, the new feature of the Kondo transport compared with the usual Kondo effect of magnetic impurities is the nonequilibrium nature, since the current is measured with a -nite bias voltage. One possible method to study the nonequilibrium Kondo e&ect is the perturbation theory with respect to the Coulomb interaction U in the dot [6] based on the Keldysh formalism [7]. According to the results obtained by the second-order perturbation theory, the Kondo resonance is simply suppressed and does not show any particular structure in the nonequilibrium situation. Concerning the equilibrium Kondo problem it is well known that the second-order perturbation theory gives remarkably good results [8]. Indeed, the exact solution shows a rapid convergence ∗
Corresponding author. E-mail address: [email protected] (T. Fujii).
of the perturbation series [9]. However, it is not clear whether the second-order perturbation theory works well in the nonequilibrium conditions. Actually, the noncrossing approximation (NCA) has predicted the double peak structure at chemical potentials of both leads [10]. This result was obtained also by other approaches, the equation of motion [10], a real-time diagrammatic formulation [11] and scaling method [12]. However it remains unclear how the e&ects of the double peak structure, if it is true, appear in the di&erential conductance. Thus we have developed a theory of quantum transport through a dot under -nite voltage by using the perturbation theory up to the fourth order of U based on the Keldysh formalism [13]. We have shown that a single Kondo peak splits into double peaks when the voltage exceeds the Kondo temperature, eV ¿ kB TK , which leads to the appearance of the new peak in conductance G(V ) in addition to the zero-bias peak. Moreover, we have discussed the possible relevance of the present study to the 0.7 conductance anomaly in QPC [14–16]. It has been
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.054
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
suggested that Kondo e&ect plays signi-cant role as the origin of the 0.7 structure in QPC because the structure is found when the conductance takes a value close to the unitary limit 2e2 =h [16] with decreasing temperature. In its experimental report [16], it is clearly stated that the separatrix between the region of the conductance of the unitary limit and the 0.8 plateau region follows the Kondo temperature, which is precisely our prediction of the appearance of the new peak. However, theoretically, it is necessary to discuss the origin of the new peak in more detail because the other theoretical treatments including Refs. [10–12] do not show this additional peak. In this paper, -rst we brieFy review our recent results of the nonequilibrium Kondo e&ect at a dot [13]. We show that the new peak is originated in the behavior of @ (!)=@eV which appears in the di&erential conductance expressed as G(V ) ˙ (eV=2) + eV=2 @ (!)=@eV d!. Here (!) is the density of −eV=2 states (DOS) of the electron with spin at a dot. Recently, it was suggested that the DOS can be measured by using a three-terminal con-guration. In order to consider a single-level quantum dot attached to two leads, we employ the impurity Anderson model H=
k
† jk ck ck
+
k
+
† (Vk ck d
d n + Un↑ n↓
+ h:c:);
First we calculate the density of states at the dot 1 (!) = − Im G r (!);
(1)
where = L, R and ckL (ckR ) annihilates an electron in the left(right)-lead, d annihilates an electron with spin in the dot and n = d† d . The coupling constants, Vk describe the tunneling matrix elements between the dot and leads. Nonequilibrium steady state is realized by a -nite di&erence between L and R which are chemical potentials of the leads in both sides. We concentrate on the symmetric Anderson model without loss of generality, where the symmetric conditions are stated as L = R ≡ , L = −R = eV=2 and −d = d + U . Here L; R represent resonance width at the chemical potentials, L; R (!) = 2 k |VkL; R |2 (! − jk ). In this paper we restrict ourselves to the ground state, T = 0.
(2)
where G r (!) is the retarded Green function. Then the current through the dot is expressed by L R e ∞ d! (!) I= ˝ −∞ L + R × (fL (!) − fR (!));
where fL; R (!) = 1=(1 + e (!−L·R ) ). From the current the di&erential conductance is obtained by G(V ) =
@I : @V
(3)
In order to estimate G r (!), we employ the perturbation theory based on the Keldysh formalism [7]. The Dyson equations for the Keldysh formalism can be written in a matrix form G (!) = g (!) + g (!)
(!)G (!);
(4)
where g −+ (!) = −i2Im g r (!)fe& (!);
(5)
g +− (!) = +i2Im g r (!)(1 − fe& (!));
(6)
g − − (!) =
499
1 − fe& (!) fe& (!) + ; ! + i ! − i
g ++ (!) = −g − − (!)∗
(7) (8)
with fe& (!) = (L fL (!) + R fR (!))=(L + R ) and g r (!) = (! + i)−1 . fe& (!) is the weighted average of the Fermi functions of leads. So far the analysis of the perturbation theory of U has been restricted up to the lowest second order [6]. In order to estimate higher order contributions the four-point vertex which is obtained by extending the method used by Keldysh for the electron–phonon vertex [7] is very convenient. This procedure is to insert vertices, − + + (0)1 2 ; 3 4 = U ( − 1 2 3 4 − 1 2 3 4 );
(9)
in each of diagrams, which may be called as the Keldysh vertex. Here 31 2 = 1 2 z 2 3 , with z 2 3 being the third Pauli matrix.
500
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
1
1
ρσ(ω) (1/πΓ)
G(V) (2e2/h)
eV/Γ = 0
0.8
eV/Γ = 0.5 eV/Γ = 1.0 eV/Γ = 2.0
0.6 0.4
eV/Γ = 8.0
-10
-5
0 ω (Γ)
5
0.4 0.2 0
1
10
Fig. 1. The density of states in units of 1= for various eV= where U= = 6.
All diagrams up to the fourth order are shown in Ref. [13] for the symmetric case where the third-order terms vanish in the same way as the equilibrium case. We can write down the analytic expressions of 1 2 up to the fourth order described by these diagrams by using Feynman rules where the insertion of the Keldysh vertex is considered additionally. The imaginary part of r is de-ned by using the o&-diagonal components of the self-energy matrix 1 2 as
r
(!)
0.6
-0.2
0.2
Im
G1(V) G(V) G2(V)
0.8
= 12 (i
+−
(!)
−i
−+
(!)):
(10)
Then the real part of r is obtained by the Kramers– Kronig relation, ∞ Im r (! ) 1 (11) Re r (!) = P d! : −! ! −∞ The Dyson equation for the retarded component is derived from Eq. (4) as G r (!) = g r (!) + g r (!) r (!)G r (!). Thus the DOS de-ned by Eq. (2) can be calculated. The density of states is shown in Fig. 1 in units of 1=. In the equilibrium case, the sharp Kondo peak reaches to the unitary limit value 1= at != = 0. Under a -nite voltage of eV= = 0:5, the Kondo peak is broadened with keeping a single-peak because the real transitions between dot and leads are slightly enhanced. This behavior is con-rmed by the analysis based on the Ward identities [17]. With further increasing the voltage, the Kondo peak splits into double peaks at ! ∼ ±eV=2, which locate near
2
3
4 5 eV (Γ)
6
7
8
Fig. 2. The di&erential conductance for U= = 6, G(V ), is given by the sum of G1 (V ) and G2 (V ). The new peak is originated in the behavior of G2 (V ).
the chemical potentials of the two leads. This behavior is qualitatively consistent with the results obtained by the NCA [10]. This change occurs when the potential drop, eV , exceeds kB TK de-ned as the full-width at half-maximum of the Kondo peak. For U= = 6, kB TK = is estimated to be about 0.6. Let us discuss the e&ects of the DOS to G(V ). As shown in Fig. 2 the new peak appears in G(V ) for U= = 6 between the zero-bias peak and the broad peak at eV ∼ U . To consider the new peak, we rewrite the expression of G(V ) in Eq. (3) as eV=2 2e2 @ (!) G(V ) = (eV=2) + d! h @eV −eV=2 ≡ G1 (V ) + G2 (V ):
(12)
As shown in Fig. 2, the -rst term, G1 (V ), decreases monotonically as V is increased. Concerning the second term, G2 (V ), -rst we note that if the integrand is the entire frequency space it should vanish due to the sum-rule of the spectral weight. For eV ¡ kB TK the spectral weight shifts from the integrand to higher frequencies, giving a negative contribution. When eV ∼ kB TK a single Kondo peak splits to double peaks. Then in a considerable part of the frequency space, @ (!)=@eV becomes positive, which is shown in Fig. 3. A signi-cant part of the positive contributions enter in the integrand of G2 (V ), thus giving less negative contribution. As a result we may conclude that the new peak appears when the bias voltage exceeds the Kondo energy.
T. Fujii, K. Ueda / Physica E 22 (2004) 498 – 501
501
values when eV ∼ kB TK , giving the appearance of the new peak in the di&erential conductance. The authors would like to thank K. Kobayashi and A. Oguri for helpful discussions.
References
Fig. 3. @ (!)=@eV in units of 1= of U= = 6, where eV= = 0:5 is near by the Kondo temperature kB TK = 0:6.
It is not so easy to measure the DOS experimentally because its full information cannot be obtained only by measuring G(V ) at a dot with two leads. However recently a theoretical prediction was made to measure the DOS at a dot with two leads by introducing an additional probe source lead which is rather weakly coupled with a dot [18,19]. It is found that when the chemical potential of an additional lead is changed with keeping a certain bias between the source and drain, the di&erential conductance of the current into the third lead follows the shape of the DOS at a dot with two leads. Moreover in Ref. [19] it has been shown that this behavior is not much a&ected even its coupling is -nite, not necessary very small. This method is eMcient to explore the DOS at a dot with two leads. Actually, the splitting of the Kondo peak is successfully observed [20] based on this prediction. In summary, we have overviewed our recent results on the nonequilibrium Kondo e&ect under a -nite bias voltage. It is shown that @ (!)=@eV has positive
[1] T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768. [2] L.I. Glazman, M.E. Raikh, JETP Lett. 47 (1988) 452. [3] D. Goldhaber-Gordon, et al., Nature (London) 391 (1998) 156; D. Goldhaber-Gordon, et al., Phys. Rev. Lett. 81 (1998) 5225. [4] S.M. Cronenwett, et al., Science 281 (1998) 540. [5] W.G. van der Wiel, et al., Science 289 (2000) 2105. [6] S. Hersh-eld, et al., Phys. Rev. Lett. 67 (1991) 3720; S. Hersh-eld, et al., Phys. Rev. B. 46 (1992) 7046. [7] L.V. Keldysh, Sov. Phys. JETP 20 (1965) 1018. [8] K. Yosida, K. Yamada, Progr. Theor. Phys. 46 (Suppl.) (1970) 244; K. Yosida, K. Yamada, Progr. Theor. Phys. 53 (1975) 1286; K. Yamada, Progr. Theor. Phys. 53 (1975) 970; K. Yamada, Progr. Theor. Phys. 54 (1975) 316. [9] V. ZlatiQc, et al., Phys. Rev. B 28 (1983) 6904; K. Ueda, et al., J. Phys. C 16 (1983) L849. [10] Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601; N.S. Wingreen, Y. Meir, Phys. Rev. B. 49 (1994) 11040. [11] J. KRonig, et al., Phys. Rev. B 54 (1996) 16820. [12] A. Rosch, et al., Phys. Rev. Lett. 87 (2001) 156802; A. Rosch, et al., cond-mat/0202404. [13] T. Fujii, K. Ueda, Phys. Rev. B 68 (2003) 155310. [14] K.J. Thomas, et al., Phys. Rev. Lett. 77 (1996) 135. [15] A. Kristensen, et al., Phys. Rev. B 62 (2000) 10950. [16] S.M. Cronenwett, et al., Phys. Rev. Lett. 88 (2002) 226805. [17] A. Oguri, Phys. Rev. B. 64 (2001) 153305. [18] Q.-F. Sun, et al., Phys. Rev. B 64 (2001) 153306. [19] E. Lebanon, et al., Phys. Rev. B 65 (2002) 035308. [20] S. De Franceschi, et al., Phys. Rev. Lett. 89 (2002) 156801.