Kondo effect in p-wave superconducting junctions

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Physica E 22 (2004) 542 – 545 www.elsevier.com/locate/physe

Kondo eect in p-wave superconducting junctions Tomosuke Aonoa;∗ , Anatoly Goluba , Yshai Avishaia; b a Department b Ilse

of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel Katz Center, Ben-Gurion University, Beer-Sheva 84105, Israel

Abstract We investigate electron transport through quantum dot in the Kondo regime with p-wave superconducting leads when TK ¿ . The current and the shot-noise power are calculated as functions of the applied voltage in the sub-gap region eV ¡ . In addition, the Josephson current is calculated as function of temperature and phase. We point out the qualitative dierence between junctions with s-wave and p-wave superconducting leads. ? 2003 Elsevier B.V. All rights reserved. PACS: 74.70.Kn; 72.15.Gd; 71.10.Hf; 74.20.Mn Keywords: Quantum dot; Kondo eect; p-wave superconductor; Andreev re8ection; Josephson eect

1. Introduction The Kondo eect has been observed in electron transport through semiconductor quantum dots [1]. Localized spin of electrons in the dot acts as magnetic impurity, interacting with electrons in leads. When the temperature T is lower than the Kondo temperature TK , electrons in the leads and the dot forms a singlet state, and consequently, the local density of states shows a sharp peak structure at the Fermi energy EF in the leads (the Abrikosov–Suhl resonance). This resonance result in an enhancement of the conductance G, approaching the unitary limit, G = 2e2 =h as T decreases below TK [2]. When one or both electrodes are superconductor (S) instead of normal metal (N ), a novel transport mechanism, the Andreev re8ection, plays a dominant role

∗

Corresponding author. E-mail address: [email protected] (T. Aono).

both in and out of equilibrium. In the former case it is responsible for occurrence of direct Josephson current for two superconducting leads while in the latter case it is the cause of dissipative currents at sub-gap voltages. In such systems, the interplay between the Andreev re8ection and the Kondo eect leads to a rich physical picture of electron transport. A natural candidate for this system is that in which the role of the quantum dot is played by a carbon nano-tube (CNT), where TK is rather high (TK ¿ 1 K) [3,4] and competes with the superconducting gap . Indeed, in a recent experiment, a superconducting junction through CNT in the Kondo regime has been realized with TK ¿  [4]. In this work, we investigate electron transport through quantum dot in the Kondo regime (K) with superconducting leads, namely SKN and SKS junctions, especially when TK ¿ , where the interplay between Andreev re8ection and the Kondo eect is prominent. We further elucidate the distinction between (conventional) s-wave and (unconventional) p-wave superconductor, the latter has recently been

1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.065

T. Aono et al. / Physica E 22 (2004) 542 – 545

discovered in Sr 2 RuO4 [5]. For p-wave superconductor, the pairing potential is anisotropic and also reverses its sign: () = −( − ), where  is the azimuthal angle. This property induces mid-gap zero-energy states (ZES) [6] near the interface between the lead and the dot while no such states appear for s-wave superconductor. This dierence becomes important when the Kondo eect takes place, because ZES strongly interact with the Kondo resonant state which appears also at EF = 0. 2. Model and method We adopt the Anderson model with one or two superconducting leads as the model Hamiltonian of SKN or SKS junctions H = HL + HR + Hd + Ht + Hc ;

(1)

in which Hj (j = L; R) are the Hamiltonians of the electrodes which depend on the electron Meld operators j (r; t) at r = (x; y) with the spin  = ±. For a superconducting lead we adopt the BCS Hamiltonian, including the chemical potential which depends on the bias voltage V , and the Cooper pairing potential . For s-wave superconductor,  is isotropic while for p-wave superconductor,  is anisotropic and depends on the two-dimensional momentum vector: () = ||(kx + iky )=|k| = || exp(i) with the azimuthal angle  = arc tan(ky =kx ). We choose  = 0 because the quantum dot is almost point-like. The quantum dot has a single energy level 0 ¡ 0 with the Coulomb interaction which is strong enough to exclude double occupancy of electrons in the dot. In the slave boson formalism; the annihilation operator d of electron in the dot is written as d = b† c with the slave boson operator b and the pseudo-fermion
operator the constraint term  † c and † of Hc : Hc =  c c + b b − 1 ; where  is a    Lagrange multiplier [7]. The corresponding dot and tunnelingHamiltonians, Hd and  Ht are expressed as: Hd = 0  c† c , and Ht = j Tj c† b j (0; t) + hc with the tunneling amplitude Tj . Since electron Meld in the dot couples only with j (0; t), (referred to as surface states), electron transport is determined by these states. Integrating out electron Melds inside the lead [8], we obtain the density of states () of surface states of p-value superconductor with  = 0:

√ () =

 2 − 2 (|| − ) + N() ||

543

(2)

with the unit-step function (x) and the Dirac function (x). This includes mid-gap zero-energy states (ZES) [6], while for the s-wave superconductor, () includes no ZES, having the well-known BCS form. Electron transport is calculated via the dynamical “partition function” [9]  Z ∼ D[F] exp(iS); (3) where the path integral is carried out over all Melds [F] and the action S is obtained by integrating the Lagrangian pertaining to the Hamiltonian (1) along the Keldysh contour. We treat the Hamiltonian (1) within the mean Meld approximation; b and  are c-numbers, determined self-consistently by the extremum conditions, Z=b2 = 0 and Z= = 0. These self-consistent equations depend on the two-dimensionless parameters, tK ≡ TK = and eV=. Electron transport observables such as the current I and the current shot noise S, can be calculated with the combination of the Keldysh Green functions [8,9]. 3. SKN junctions In Fig. 1(a), the dierential conductance G =dI=dV is plotted as a function of V in SKN junctions for both s-wave and p-wave superconducting leads for tK = TK = = 100; 5; 3; and 2 with $=TK = 200. When tK  1 [the upper curves in Fig. 1(a)], the G–V curves show no dierence between s-wave and p-wave superconductors; G =4e2 =h when  eV ¡  and   2 2 2 G=4e =h(eV )= eV + (eV ) −  when eV ¿ . In this limit, the Kondo resonance reaches the unitary limit, and consequently, this SKN junction reduces to an SN junction with pure ballistic contact, which has already been analyzed [10]. For lower values of tK the distinction between s-wave and p-wave leads becomes prominent. For s-wave superconductor, G is suppressed as tK decreases. For tK = 5 the G–V curve noticeably deviates from the one of tK = 100, re8ecting the suppression of the Kondo eect due to both  and V . For tK = 2 the competition between superconducting gap and the Kondo eect is much more prominent, leading to further decrease of the conductance.

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T. Aono et al. / Physica E 22 (2004) 542 – 545 4

G (e2∆ /h)

3

2

tK

1

0.0

1.0

1.5

eV=∆

K (e2∆ / h)

(a)

0.5

(b)

In Fig. 1(b), K is plotted as function of V with the same parameters as in Fig. 1(a); it is clearly correlated with the results of the G–V curve. When tK  1, the results for both s- and p-wave superconductors are consistent with those obtained for purely ballistic junctions, exhibiting strong suppression of K in the sub-gap region. At lower tK the physics is distinct. For s-wave superconductor, when tK = 5, the noise spectrum still shows features typical for a junction with relatively high transparency, while the result for tK =2 resembles the one for a low transparency junction. For p-wave superconductor, the noise spectrum is almost unchanged even when tK is small. This dierence is again explained by the existence of ZES; they turn the Kondo resonance to be less vulnerable to the impact of superconductivity and the junction remains close to the unitary limit. 4. Josephson current

eV/∆

Fig. 1. (a) The conductance G (in units of e2 =h) versus the bias V (in units of =e) for an s-wave (dash curves) and p-wave (solid lines) junctions with $=TK = 200. The parameter tK = TK = takes values 2,3,5,100 (tK decreases from top to down, which is indicated by the arrow). The upper line corresponding to tK =100 coincides for s- and p-wave superconducting leads. (b) The shot-noise power K(0) (in units of e2 =h) as a function of V (in units of =e) for an SKN junction and for several values of tK (it decreases from bottom to top, which is indicated by the arrow). The curves with tK = 100 for s- and p-wave superconductors coincide.

For p-wave superconductor, the G–V curves are less in8uenced by variations of tK in contrast to s-wave superconductor, indicating superconductivity plays a minor role in the suppression of the Kondo resonance and G is almost 4e2 =h when eV ¡  for smaller values of tK . This result shows that the ZES support the formation of a Kondo singlet for lower values of tK . The shot-noise power is given by K(t1 ; t2 ) = ˝[Iˆ(t1 )Iˆ(t2 ) − Iˆ 2 ];

(4)

with the current operator Iˆ. The Fourier transform of K(t1 ; t2 ) gives the shot-noise power spectrum K(!). Here we concentrate on the zero frequency shot-noise K ≡ K(0).

Next we investigate Josephson eect for SKS system in which both electrodes are p-wave superconductors, as well as s-wave superconductors for comparison. The Josephson current for p-wave superconductor is given by I= e ˝

!

×

!2 (1

('(!))2 sin  ; + (!))2 + ˜2 + ('(!))2 cos2 =2 (5)

where  is the phase dierence between the two superconductors, and the summation is taken over the Matsubara frequencies. When the junction is asymmetric, $L=R = $(1 ± p) (0 ¡ p ¡ 1), we should replace cos2 (=2) → (cos2 (=2) + p2 sin2 (=2)) and I ˙ sin  → I ˙ (1 − p2 )sin . In the unitary limit tK  1, Eq. (5) gives 2e (1 − p2 ) tanh('Nf()=2) J= sin  (6) ˝ 4 f() with f()= 1 + cos2 (=2) + p2 sin2 (=2), which is dierent from the one for s-wave [11]. In Fig. 2(a), the Josephson current is calculated as a function of the phase dierence  for both s- and p-wave superconducting leads with a small anisotropy parameter p = 0:1 at T → 0. For s-wave superconductor, I – curves (broken lines) have the maximum

T. Aono et al. / Physica E 22 (2004) 542 – 545

I (2e∆=ប)

0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

δ /


0.6

0.8

1.0

I (T)/I(0)

(a)

(b)

545

(dashed lines) and p-wave (solid lines) superconducting leads. For p-wave superconductor, the values of I (T ) keep relatively large values near T = TC while for s-wave junctions, they decrease more rapidly as T increases. The distinction becomes clear when tK = 2 (compare thin lines). These results are explained by the same reason above, the existence of ZES; the temperature dependence is similar to the usual SIS junction for the p-wave case, while for s-wave superconductors it is the one in SNS junctions. In conclusion, we have investigated G–V characteristics and the shot-noise power spectrum of SKN junctions, as well as the Josephson current, especially for p-wave superconductors. We Mnd qualitative dierences in electron transport between the conventional s-wave superconductors. Acknowledgements

T / TC

Fig. 2. (a) Josephson current I versus phase dierence = at T → 0 for s-wave (dashed lines) and p-wave (solid lines) superconductors. Thick and thin lines correspond to the unitary limit (tK  1) and tK = 2, respectively. (b) Temperature dependence of the maximum Josephson current. The current is normalized by I (T = 0); the temperature by the critical temperature TC . The notation is the same as in Fig. 2(a).

around  and are far from being sinusoidal; this result is peculiar to SNS junctions. On the other hand, for p-wave superconductor, I – curves (solid lines) have the maximal values around =2 and are almost sinusoidal; it is very similar to the one of SIS junctions. This result is interpreted as follows: The mid-gap states act as if they increase the eective normal region of the junction and therefore, the junction becomes more resistive, which makes the contact of the junction less transparent. Indeed, the amplitude of the Josephson current is reduced compared with its value for the s-wave junction. When tK decreases, I also decreases for both s- and p-wave (compare thick and solid lines). In Fig. 2(b), the maximum Josephson current is plotted as function of temperature T for s-wave

This research is supported by DIP German Israel Cooperation Project Quantum electronics in low dimensions by the Israeli Science Foundation Grant Many-Body eects in non-linear tunneling and by the US-Israel BSF Grant Dynamical instabilities in quantum dots. One of us (TA) is supported by JSPS Research Fellowships for Young Scientists. References [1] D. Goldhaber-Gordon, et al., Nature 391 (1998) 156; S.M. Cronenwett, et al., Science 281 (1998) 540; J. Schmid, et al., Physica B 256 –258 (1998) 182; P. Simmel, et al., Phys. Rev. Lett. 83 (1999) 804. [2] W.G. van der Wiel, et al., Science 289 (2000) 2105. [3] J. Nygard, D.H. Cobben, D.E. Lindelof, Nature 408 (2000) 342. [4] M.R. Buitelaar, T. Nussbaumer, C. SchUonenberger, Phys. Rev. Lett. 89 (2002) 256801. [5] Y. Maeno, et al., Nature 372 (1994) 532. [6] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641. [7] P. Coleman, Phys. Rev. B 29 (1984) 3035. [8] Y. Avishai, A. Golub, A.D. Zaikin, Phys. Rev. B 63 (2001) 134515; Y. Avishai, A. Golub, A.D. Zaikin, Europhys. Lett. 54 (2001) 640. [9] Y. Avishai, A. Golub, A.D. Zaikin, Phys. Rev. B 67 (2003) 041301. [10] T.M. Klapwijk, G.E. Blonder, M. Tinkham, Physica B 109+110 (1982) 1157. [11] L. Glazman, K.A. Matveev, Pis’ma Zh. Eksp. Teor. Fiz. 49 (1989) 570; L. Glazman, K.A. Matveev, JETP Lett. 49 (1989) 659.