Charge and spin effects in mesoscopic Josephson junctions (Review Article)

We consider the charge and spin effects in low dimensional superconducting weak links. The first part of the review deals with the effects of electron—electron interaction in Superconductor/ Luttinger liquid/Superconductor junctions. The experimental realization of this mesoscopic hybrid system c...

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Дата:	2004
Автори:	Krive, I.V., Kulinich, S.I., Shekhter, R.I., Jonson, M.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2004
Назва видання:	Физика низких температур
Теми:	Сверхпроводимость и мезоскопические структуры
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Цитувати:	Charge and spin effects in mesoscopic Josephson junctions (Review Article) / I.V. Krive, S.I. Kulinich, R.I. Shekhter, M. Jonson // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 738-755. — Бібліогр.: 83 назв. — англ.

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spelling	irk-123456789-1198302017-06-11T03:05:09Z Charge and spin effects in mesoscopic Josephson junctions (Review Article) Krive, I.V. Kulinich, S.I. Shekhter, R.I. Jonson, M. Сверхпроводимость и мезоскопические структуры We consider the charge and spin effects in low dimensional superconducting weak links. The first part of the review deals with the effects of electron—electron interaction in Superconductor/ Luttinger liquid/Superconductor junctions. The experimental realization of this mesoscopic hybrid system can be the individual single wall carbon nanotube that bridges the gap between two bulk superconductors. The dc Josephson current through a Luttinger liquid is evaluated in the limits of perfectly and poorly transmitting junctions. The relationship between the Josephson effect in a long SNS junction and the Casimir effect is discussed. In the second part of the paper we review the recent results concerning the influence of the Zeeman and Rashba interactions on the thermodynamical properties of ballistic S–QW–S junction fabricated in two dimensional electron gas. It is shown that in magnetically controlled junction there are conditions for resonant Cooper pair transition which results in giant supercurrent through a tunnel junction and a giant magnetic response of a multichannel SNS junction. The supercurrent induced by the joint action of the Zeeman and Rashba interactions in 1D quantum wires connected to bulk superconductors is predicted. 2004 Article Charge and spin effects in mesoscopic Josephson junctions (Review Article) / I.V. Krive, S.I. Kulinich, R.I. Shekhter, M. Jonson // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 738-755. — Бібліогр.: 83 назв. — англ. 0132-6414 PACS: 71.10.Pm, 72.15.Nj, 73.21.Hb, 73.23.–b, 74.50.+r http://dspace.nbuv.gov.ua/handle/123456789/119830 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Сверхпроводимость и мезоскопические структуры Сверхпроводимость и мезоскопические структуры
spellingShingle	Сверхпроводимость и мезоскопические структуры Сверхпроводимость и мезоскопические структуры Krive, I.V. Kulinich, S.I. Shekhter, R.I. Jonson, M. Charge and spin effects in mesoscopic Josephson junctions (Review Article) Физика низких температур
description	We consider the charge and spin effects in low dimensional superconducting weak links. The first part of the review deals with the effects of electron—electron interaction in Superconductor/ Luttinger liquid/Superconductor junctions. The experimental realization of this mesoscopic hybrid system can be the individual single wall carbon nanotube that bridges the gap between two bulk superconductors. The dc Josephson current through a Luttinger liquid is evaluated in the limits of perfectly and poorly transmitting junctions. The relationship between the Josephson effect in a long SNS junction and the Casimir effect is discussed. In the second part of the paper we review the recent results concerning the influence of the Zeeman and Rashba interactions on the thermodynamical properties of ballistic S–QW–S junction fabricated in two dimensional electron gas. It is shown that in magnetically controlled junction there are conditions for resonant Cooper pair transition which results in giant supercurrent through a tunnel junction and a giant magnetic response of a multichannel SNS junction. The supercurrent induced by the joint action of the Zeeman and Rashba interactions in 1D quantum wires connected to bulk superconductors is predicted.
format	Article
author	Krive, I.V. Kulinich, S.I. Shekhter, R.I. Jonson, M.
author_facet	Krive, I.V. Kulinich, S.I. Shekhter, R.I. Jonson, M.
author_sort	Krive, I.V.
title	Charge and spin effects in mesoscopic Josephson junctions (Review Article)
title_short	Charge and spin effects in mesoscopic Josephson junctions (Review Article)
title_full	Charge and spin effects in mesoscopic Josephson junctions (Review Article)
title_fullStr	Charge and spin effects in mesoscopic Josephson junctions (Review Article)
title_full_unstemmed	Charge and spin effects in mesoscopic Josephson junctions (Review Article)
title_sort	charge and spin effects in mesoscopic josephson junctions (review article)
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2004
topic_facet	Сверхпроводимость и мезоскопические структуры
url	http://dspace.nbuv.gov.ua/handle/123456789/119830
citation_txt	Charge and spin effects in mesoscopic Josephson junctions (Review Article) / I.V. Krive, S.I. Kulinich, R.I. Shekhter, M. Jonson // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 738-755. — Бібліогр.: 83 назв. — англ.
series	Физика низких температур
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first_indexed	2025-07-08T16:44:43Z
last_indexed	2025-07-08T16:44:43Z
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fulltext	Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8, p. 738–755 Charge and spin effects in mesoscopic Josephson junctions (Review Article) Ilya V. Krive1,2, Sergei I. Kulinich1,2, Robert I. Shekhter1, and Mats Jonson1 1Department of Applied Physics, Chalmers University of Technology Göteborg University, SE-412 96 Göteborg, Sweden 2B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: Shekhter@fy.chalmers.se Received January 30, 2004 We consider the charge and spin effects in low dimensional superconducting weak links. The first part of the review deals with the effects of electron—electron interaction in Superconduc- tor/Luttinger liquid/Superconductor junctions. The experimental realization of this mesoscopic hy- brid system can be the individual single wall carbon nanotube that bridges the gap between two bulk superconductors. The dc Josephson current through a Luttinger liquid is evaluated in the limits of per- fectly and poorly transmitting junctions. The relationship between the Josephson effect in a long SNS junction and the Casimir effect is discussed. In the second part of the paper we review the recent re- sults concerning the influence of the Zeeman and Rashba interactions on the thermodynamical proper- ties of ballistic S–QW–S junction fabricated in two dimensional electron gas. It is shown that in mag- netically controlled junction there are conditions for resonant Cooper pair transition which results in giant supercurrent through a tunnel junction and a giant magnetic response of a multichannel SNS junction. The supercurrent induced by the joint action of the Zeeman and Rashba interactions in 1D quantum wires connected to bulk superconductors is predicted. PACS: 71.10.Pm, 72.15.Nj, 73.21.Hb, 73.23.–b, 74.50.+r Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 739 2. Josephson current in S–QW–S junction . . . . . . . . . . 740 2.1. Quantization of Josephson current in a short ballistic junction . . . . . . . . . . . . . . . . . . . . . . 741 2.2. Luttinger liquid wire coupled to superconductors . . . . 741 2.2.1. Tunnel junction . . . . . . . . . . . . . . . . . . . 743 2.2.2. Transparent junction . . . . . . . . . . . . . . . . . 745 2.3. Josephson current and the Casimir effect . . . . . . . . 746 3. The effects of Zeeman splitting and spin—orbit interaction in SNS junctions . . . . . . . . . . . . . . . . . . . . . 747 3.1. Giant critical current in a magnetically controlled tunnel junction . . . . . . . . . . . . . . . . . . . . . . 748 3.2. Giant magnetic response of a multichannel quantum wire coupled to superconductors . . . . . . . . . . . . . . 749 3.3. Rashba effect and chiral electrons in quantum wires . . . 750 3.4. Zeeman splitting induced supercurrent . . . . . . . . . 751 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 753 References . . . . . . . . . . . . . . . . . . . . . . . 753 © Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson, 2004 1. Introduction Since the discovery of superconductivity in 1911 this amazing macroscopic quantum phenomena has influ- enced modern solid state physics more then any other fundamental discovery in the 20th century. The mere fact that five Nobel prizes already have been awarded for discoveries directly connected to superconductivity indicates the worldwide recognition of the exceptional role superconductivity plays in physics. Both at the early stages of the field development and later on, research in basic superconductivity brought surprises. One of the most fundamental discoveries made in superconductivity was the Josephson effect [1]. In 1962 Josephson predicted that when two super- conductors are put into contact via an insulating layer (SIS junction) then (i) a dc supercurrent J Jc� sin� (Jc is the critical current, � is the superconducting phase difference) flows through the junction in equilib- rium (dc Josephson effect) and (ii) an alternating cur- rent ( , ,� � �� J Jt eV/2 � where V is the bias volt- age) appears when a voltage is applied across the junction (ac Josephson effect). A year latter both the dc and the ac Josephson effect were observed in experi- ments [2,3]. An important contribution to the experi- mental proof of the Josephson effect has been made by Yanson, Svistunov, and Dmitrenko [4], who were the first to observe rf-radiation from the voltage biased contact and who measured the temperature dependence of the critical Josephson current J Tc( ). As a matter of fact the discovery of the Josephson effect gave birth to a new and unexpected direction in superconductivity, namely, the superconductivity of weak links (weak superconductivity, see, e.g., Ref. 5). It soon became clear that any normal metal layer be- tween superconductors (say, an SNS junction) will support a supercurrent as long as the phase coherence in the normal part of the device is preserved. Using the modern physical language one can say that the physics of superconducting weak links turned out to be part of mesoscopic physics. During the last decade the field of mesoscopic physics has been the subject of an extraordinary growth and development. This was mainly caused by the recent advances in fabrication technology and by the discovery of principally new types of mesoscopic systems such as carbon nanotubes (see, e.g., Ref. 6). For our purposes metallic single wall carbon nanotubes (SWNT) are of primary interest since they are strictly one-dimensional conductors. It was experi- mentally demonstrated [7–9] (see also Ref. 10) that electron transport along metallic individual SWNT at the low bias voltage regime is ballistic. At first glance this observation looks surprising. For a long time it was known (see Ref. 11) that 1D metals are unstable with respect to the Peierls phase transition, which opens up a gap in the electron spectrum at the Fermi level. In car- bon nanotubes the electron—phonon coupling for con- ducting electrons is very weak while the Coulomb corre- lations are strong. The theory of metallic carbon nanotubes [12,13] shows that at temperatures outside the mK-range the individual SWNT has to demonstrate the properties of a two channel, spin-1/2 Luttinger liq- uid (LL). This theoretical prediction was soon con- firmed by transport measurements on metal-SWNT and SWNT–SWNT junctions [14,15] (see also Ref. 16, where the photoemission measurements on a SWNT were interpreted as a direct observation of LL state in carbon nanotubes). Both theory and experiments re- vealed strong electron—electron correlations in SWNTs. Undoped individual SWNT is not intrinsically a superconducting material. Intrinsic superconductivity was observed only in ropes of SWNT (see Refs. 17 and 18). Here we consider the proximity-induced super- conductivity in a LL wire coupled to superconductors (SLLS). The experimental realization of SLLS junc- tion could be an individual SWNT, which bridges the gap between two bulk superconductors [19,20]. The dc Josephson current through a LL junction was evaluated for the first time in Ref. 21. In this pa- per a tunnel junction was considered in the geometry (see subsection 2.2.), which is very suitable for theo- retical calculations but probably difficult to realize in an experiment. It was shown that the Coulomb corre- lations in a LL wire strongly suppress the critical Josephson current. The opposite limit — a perfectly transmitting SLLS junction was studied in Ref. 22, where it was demonstrated by a direct calculation of the dc Josephson current that the interaction does not renormalize the supercurrent in a fully transparent (D � 1, D is the junction transparency) junction. In subsection 2.2. we re-derive and explain these results using the boundary Hamiltonian method [23]. The physics of quantum wires is not reduced to the investigations of SWNTs. Quantum wires can be fab- ricated in a two-dimensional electron gas (2DEG) by using various experimental methods. Some of them (e.g., the split-gate technique) originate from the end of 80’s when the first transport experiments with a quantum point contact (QPC) revealed unexpected properties of quantized electron ballistic transport (see, e.g., Ref. 24). In subsection 2.1. we briefly re- view the results concerning the quantization of the critical supercurrent in a QPC. In quantum wires formed in a 2DEG the elec- tron—electron interaction is less pronounced [25] than in SWNTs (presumably due to the screening ef- fects of nearby bulk metallic electrodes). The electron transport in these systems can in many cases be Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 739 successfully described by Fermi liquid theory. For noninteracting quasiparticles the supercurrent in a SNS ballistic junction is carried by Andreev levels. For a long (L v / LF�� 0 � �, is the junction length, � is the superconducting energy gap) perfectly transmitting junction the Andreev–Kulik spectrum [26] for quasiparticle energies E �� is a set of equi- distant levels. In subsection 2.3. we show that this spectrum corresponds to twisted periodic boundary conditions for chiral ( right- and left-moving) electron fields and calculate the thermodynamic potential of an SNS junction using field theoretical methods. In this approach there is a close connection between the Josephson effect and the Casimir effect. In Section 3 of our review we consider the spin effects in ballistic Josephson junctions. As is well-known, the electron spin does not influence the physics of standard SIS or SNS junctions. Spin effects become significant for SFS junctions (here «F» denotes a magnetic material) or when spin-dependent scattering on magnetic impuri- ties is considered. As a rule, magnetic impurities tend to suppress the critical current in Josephson junction by inducing spin-flip processes [27,28]. Another system where spin effects play an important role is a quantum dot (QD). Intriguing new physics appears in normal and superconducting charge transport through a QD at very low temperatures when the Kondo physics starts to play a crucial role in the electron dynamics. Last year a vast literature was devoted to these problems. Here we discuss the spin effects in a ballistic SNS junction in the presence of: (i) the Zeeman splitting due to a local magnetic field acting only on the normal part of the junction, and (ii) strong spin-orbit interaction, which is known to exist in quantum heterostructures due to the asymmetry of the electrical confining potential [29]. It is shown in subsection 3.1. that in magnetically controlled single barrier junction there are conditions when superconductivity in the leads strongly enhances electron transport, so that a giant crit- ical Josephson current appears J Dc ~ . The effect is due to resonant electron transport through de Gen- nes–Saint–James energy levels split by tunneling. The joint action of Zeeman splitting and supercon- ductivity (see subsection 3.2.) results in yet another unexpected effect — a giant magnetic response, M N B~ � , (M is the magnetization, N� is the num- ber of transverse channels of the wire, B is the Bohr magneton) of a multichannel quantum wire coupled to superconductors [30]. This effect can be understood in terms of the Andreev level structure which gives rise to an additional (superconductivity-induced) contri- bution to the magnetization of the junction. The magnetization peaks at special values of the supercon- ducting phase difference when the Andreev energy levels at E Z � � , (� Z is the Zeeman energy split- ting) become 2N� -fold degenerate. The last two subsections of Sec. 3 deal with the in- fluence of the Rashba effect on the transport proper- ties of quasi-1D quantum wires. Strong spin—orbit (SO) interaction experienced by 2D electrons in hete- rostructures in the presence of additional lateral con- finement results in a dispersion asymmetry of the elec- tron spectrum in a quantum wire and in a strong correlation between the direction of electron motion along the wire (right/left) and the electron spin projection [31,32]. The chiral properties of electrons in a quantum wire cause nontrivial effects when the wire is coupled to bulk superconductors. In particular, in subsection 3.4. we show that the Zeeman splitting in a S–QW–S junction induces an anomalous supercurrent, that is a Josephson current that persists even at zero phase dif- ference between the superconducting banks. In Conclusion we once more emphasize the new fea- tures of the Josephson current in ballistic mesoscopic structures and briefly discuss the novel effects, which could appear in an ac Josephson current through an ul- tra-small superconducting quantum dot. 2. Josephson current through a superconductor—quantum wire—superconductor junction In this chapter we consider the Josephson current in a quantum wire coupled to bulk superconductors. One could expect that the conducting properties of this sys- tem strongly depend on the quality of the electrical con- tacts between the QW and the superconductors. The normal conductance of a QW coupled to electron reser- voirs in Fermi liquid theory is determined by the trans- mission properties of the wire (see, e.g., Ref. 33). For the ballistic case the transmission coefficient of the sys- tem in the general situation of nonresonant electron transport depends only on the transparencies of the po- tential barriers which characterize the electrical contacts and does not depend on the length L of the wire. As al- ready was mentioned in the Introduction, the Coulomb interaction in a long 1D (or few transverse channel) QW is strong enough to convert the conduction elec- trons in the wire into a Luttinger liquid. Then the barri- ers at the interfaces between QW and electron reservoirs are strongly renormalized by electron—electron interac- tion and the conductance of the N–QW–N junction at low temperature strongly depends on the wire length [34]. For a long junction and repulsive electron—elec- tron interaction the current through the system is strongly suppressed. The only exception is the case of perfect (adiabatic) contacts when the backscattering of electrons at the interfaces is negligibly (exponentially) 740 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson small. In the absence of electron backscattering the con- ductance G is not renormalized by interaction [35] and coincides with the conductance quantumG e /h� 2 2 (per channel). From the theory of Luttinger liquids it is also known [36] that for a strong repulsive interaction the resonant transition of electrons through a double-barrier structure is absent even for symmetric barriers. The well-known results for the transport properties of 1D Luttinger liquid listed above (see, e.g., Rev. 37) allows us to consider two cases when studing bal- listic S–QW–S junctions: (i) a transparent junction ( )D � 1 , and (ii) a tunnel junction ( )D �� 1 . These two limiting cases are sufficient to describe the most significant physical effects in S–QW–S junctions. 2.1. Quantization of the Josephson current in a short ballistic junction At first we consider a short L �� 0; ballistic S–QW–S junction. One of the realizations of this mesoscopic device is a quantum point contact (QPC) in a 2DEG (see Fig. 1,a). For a QPC the screening of the Coulomb interaction is qualitatively the same as in a pure 2D geometry and one can evaluate the Josephson current through the constriction in a noninteracting electron model. Then due to Andreev backscattering of quasiparticles at the SN interfaces, a set of Andreev levels is formed in the normal part of the junction [26]. In a single mode short junction the spectrum of bound states takes the form [38] (L/�0 0� ) E D / � �� 1 22sin ) � (1) where � is the superconducting phase difference. This spectrum does not depend on the Fermi velocity and therefore the Andreev levels, Eq.(1), in a junction with N� transverse channels are 2N� degenerate (the factor 2 is due to spin degeneracy). It is well known (see, e.g., Refs. 39 and 40) that the continuum spectrum in the limit L/�0 0� does not contribute to the Josephson current, J e � � � � �� , (2) where � is the thermodynamic potential. It is evident from Eqs. (1) and (2) that the Josephson current through a QPC (D � 1) is quantized [39]. At low tem- peratures (T �� ) we have [39] J N e � � � � sin � 2 . (3) This effect� is the analog of the famous conductance quantization in OPCs (see Ref. 41). Now let us imagine that the geometry of the con- striction allows one to treat the QPC as a 1D quantum wire of finite length L smoothly connected to bulk su- perconductors (Fig. 1,b). The 1D wire is still much shorter that the coherence length �0. How does the weakly screened Coulomb interaction in a 1D QW in- fluence the Josephson current in a fully transmitting (D � 1) junction? Notice that the charge is freely transported through the junction since the real elec- trons are not backscattered by the adiabatic constric- tion [42]. So, it is reasonable to assume that the Cou- lomb interaction in this case does not influence the Josephson current at all. We will prove this assump- tion for the case of a long junction in the next section. If the QW is separated from the leads by potential barriers (quite a natural situation in a real experi- ment) the charging effects have to be taken into ac- count. As a rule the Coulomb correlations, which tend to keep the number of electrons in the normal region (quantum dot in our case) constant, suppress the criti- cal supercurrent due to the Coulomb blockade effect (see, e.g., Ref. 43, where a consistent theory of the Coulomb blockade of Josephson tunneling was devel- oped). They can also change the �-dependence of the Josephson current. One possible scenario for how charging effects influence the Josephson current in a short SNS junction is considered in Ref. 44. 2.2. Luttinger liquid wire coupled to superconductors A consistent theory of electron—electron interac- tions effects in weak superconductivity has been devel- oped for a long 1D or quasi-1D SNS junction, when the normal region can be modelled by a Luttinger liquid (LL). The standard approach to this problem (see, e.g., Ref. 23) is to use for the description of electron trans- port through the normal region the LL Hamiltonian Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 741 a b S S S SN N Fig. 1. A schematic display of a superconducting point con- tact (a). Quantum wire adiabatically connected to bulk superconductors (b). � It was recently observed in: T. Bauch et al., Supercurrent and Conductance Quantization in Superconducting Quantum Point Contact, cond-mat/0405205, May 11, 2004. with boundary conditions which take into account the Andreev [45] and normal backscattering of quasiparticles at the NS interfaces. The LL Hamiltonian HLL expressed in terms of charge density operators ~ ,�R/L /� � of right/left mo- ving electrons with up/down spin projection takes the form (see, e.g., Ref. 46) H dx uLL R L R L � � � � �� [ (~ ~ ~ ~ )2 2 2 2 � � � � �� V R R L L R L R L R L 0 � � � � � � � � � � � � (~ ~ ~ ~ ~ ~ ~ ~ ~ ~ � � � ~ ~ )] ,� �R L (4) whereV0 is the strength of electron—electron interac- tion (V e0 2~ ) and the velocity u v V /F� � 0 2��. The charge density operators of the chiral ( )R/L fields obey anomalous Kac–Moody commutation relations (see, e.g., Ref. 46) [~ ( ), ~ ( )]( ) ( )� �R L j R L kx x� � � � � � � � � � � � � jk i x x x j k 2 ( ), , , . The Hamiltonian (4) is quadratic and can easily be diagonalized by a Bogoliubov transformation H dx v vLL d R L R L ( ) [ ( ) ( )],� � � �� 2 2 2 2 (5) where v� �( ) are the velocities of noninteracting bosonic modes (plasmons), v v /gF� � � �( ) ( )� , and g V v g F / � �� 1 2 10 1 2 � , . (6) Here g� and g� are the correlation parameters of a spin-1/2 LL in the charge ( )� and spin ( )� sectors. Notice that g� �� 1 for a strongly interacting ( )V vF0 �� electron system. The Andreev and normal backscattering of quasi- particles at the NS boundaries (x � 0 and x L� ) can be represented by the effective boundary Hamiltonian H H HB B A B N� �( ) ( ) HB A B l R L R L ( ) ( )[ ( ) ( ) ( ) ( )]� � �� 0 0 0 0 � � �� B r R L R LL L L L( )[ ( ) ( ) ( ) ( )] .,h. c (7) H VB N B l j j j ( ) ( ) † , ( ) ( )� � � �� 0 0 � V L LB r j j j ( ) † , ( ) ( )� �� , (8) where j L R� � � �( , ), ( , )� . Here � B l r( , ) is the effective boundary pairing potential at the left (right) NS interface and VB l r( , ) is the effective boundary scatter- ing potential. The values of these potentials are re- lated to the phase of the superconducting order pa- rameters in the banks and to the normal scattering properties at the left and right interfaces. They can be considered either as input parameters (see, e.g., Ref. 47) or they can be calculated by using some particular model of the interfaces [23]. In what follows we will consider two limiting cases: (i) poorly transmitting interfaces VB l r( , ) � ! (tunnel junction) and (ii) per- fectly transmitting interfaces VB l r( , ) � 0. At first we relate the effective boundary pairing poten- tials � B l r( , ) to the amplitudes rA l r( , ) of the Andreev back- scattering process [48,49]. Let us consider for example the Andreev backscattering of an electron at the left in- terface. This process can be described as the annihilation of two electrons with opposite momenta and spin projections at x � 0. The corresponding Hamiltonian is h r a aA A l p p~ ( ) , , " � � � , or equivalently in the coordinate representation h rA A l R L~ ( ) ( ) ( )" � �� 0 0 . Here rA is the amplitude of Andreev backscattering at the left interface, r t i / t r A l l l l l ( ) ( ) ( ) ( ) \| \| exp[ ( )] \| \| \| \| ,� � � 2 4 2 2 4 � � (9) 742 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson Cooper pair Electron Hole Fig. 2. A schematic picture of Andreev reflection. t l( ) is the transmission amplitude (\| \| \| \|( ) ( )t rl l2 2 1� � ) and � l is the phase of superconducting order parame- ter at the left bank. An analogous expression holds for the right interface. Notice that for a tunnel junc- tion \| \|( , )t l r �� 1 the amplitude of Andreev backscatte- ring is small — it is proportional to the transparency D tl r l r , ( , )\| \|# ��2 1 of the barrier at the right (left) interface. So in our model the effective boundary pairing potential is � �B l F A l B r F A rC v r C v r( ) ( ) ( ) ( ), ,� � �" " � � (10) where C is a numerical factor which will be specified later. 2.2.1. Tunnel junction. For poorly transmitting in- terfaces Dr l, �� 1 the amplitude of Andreev backscat- tering is small and we can use perturbation theory when evaluating the phase dependent part of the ground state energy. In second order perturbation the- ory the ground state energy takes the form � �E j H E E B A jj ( ) ( ) ( )2 2 0 0 � � � � � � � � ! � 1 0 0 0 0 � d H HB A B A$ $( )† ( )( ) ( ) . (11) Here HB A( )( )$ is the boundary Hamiltonian (7) in the imaginary time Heisenberg representation. After sub- stituting Eq. (7) into Eq. (11) we get the following formula for �E( )2 expressed in terms of electron corre- lation functions � �E C v r rF A l A r( ) ( ) ( )( ) Re(2 24� � " � d L LR L L R $ $ $ 0 0 0 0 0 ! � � � �� % � �[ ( , ) ( , ) ( , ) ( , ) ]† †� � � � . (12) We will calculate the electron correlation function by making use of the bosonization technique. The standard bosonisation formula reads � &' � '( � � ' �, ( , ) exp[ ( , )] ,x t a i x t� 1 2 4 (13) where a is the cutoff parameter ( )~a F) , ' � � # �( , ) ( , ),R L 1 1 � � � � # �( , ) ( , )1 1 . The chiral bosonic fields in Eq. (13) are represented as follows (see, e.g., Ref. 51) & ' � ' � � ' �� ' �, , ,( , ) � � ( , ).x t x vt L x t� � � � 1 2 (14) Here the zero mode operators � , � ,�' � � obey the stan- dard commutation relations for «coordinate» and «mo- mentum» [� , � ], ,� '�' � � � �* � �� i . They are introduced for a finite length LL to restore correct canonical com- mutation relations for bosonic fields [50,51]. Notice that the topological modes associated with these opera- tors fully determine the Josephson current in a transpar- ent (D � 1) SLLS junction [22]. The nontopological components �' �, ( , )x t of the chiral scalar fields are rep- resented by the series � '' �, ( , ) {exp[ ( )] � }x t qL iq x vt b q q� � � 1 2 h.c. , (15) where � ( � )†b bq q are the standard bosonic annihilation (creation) operator; L is the length of the junction, v is the velocity. It is convenient here to introduce [46] the charge ( )� and spin (�) bosonic fields � +� �, , which are related to above defined chiral fields�' �, by simple linear equation � + � � � �� 1 2 ( )R L R L� (16) (the upper sign corresponds to �� and the lower sign denotes +�). After straightforward transformations Eq. (12) takes the form � � � $ $ $ , E C v D dF ( ) (( ) cos [ ( ) )]2 24� � ! � �� * * , (17) where D D Dl r� �� 1 is the junction transparency and * �� ( ) ( ) exp { [ ( , )$ � � � $ �� 2 22 2a L � �� + $ + + $ �� ( , ) ( , )L L �� $ + $� �( , ) ]} ( )L Q . (18) Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 743 L S S Fig. 3. A schematic picture of a SLLS junction formed by an effectively infinite Luttinger liquid coupled to bulk su- perconductors by side electrodes. Here � � + +� � � �# #( , ), ( , )0 0 0 0 and the double brac- kets �� denote the subtraction of the correspond- ing vacuum average at the points ( , ) ( , )$ x � 0 0 . Note that the superconducting properties of a LL are deter- mined by the correlators of +� and �� bosonic fields unlike the normal conducting properties where the fields +� and �� play a dominant role. The factors Q ( )$ originate from the contribution of zero modes, Q iv L vF � � � �� - . / 0 1 2 ( ) exp [ � � ( � � )] exp($ � $ � 2 2* * * * F /L$ ). (19) With the help of a Bogoliubov transformation the chiral bosonic fields in Eq. (16) can be expressed in terms of noninteracting plasmonic modes with known propagators (see, e.g., Ref. 46). Two different geome- tries of SLLS junction have been considered in the lit- erature, viz, an effectively infinite LL connected by the side electrodes to bulk superconductors [21] (see Fig. 3) and a finite LL wire coupled via tunnel barri- ers to superconductors [47,52]. Notice, that both model geometries can be related to realistic contacts of a single wall carbon nanotube with metals (see, e.g., Ref. 53 and references therein). The geometry of Fig. 3 could model the junction when electron beam lithography is first used to define the leads and then ropes of SWSN are deposited on top of the leads. A tunnel junction of the type schematically shown in Fig. 4 is produced when the contacts are applied over the nanotube rope. The topological excitations for an effectively infi- nite LL ( )L� ! play no role and the corresponding contributions can be omitted in Eqs. (15) and (18), Q #( ) .$ 1 The propagators of noninteracting chiral bosonic fields are (see, e.g., [46]) �� R/L j R/L k jk kt x a x s t a, ,( , ) ln , 4 � (20) where j k, ,� 12 and the plasmonic velocities s v1 � �, s v vF2 � �� (see Eq.(6)). Finally the expression for the Josephson current through a «side-contacted» LL (Fig. 3) takes the form [21] J J R gLL i c i ( ) ( ) ( )sin� 0 � �, (21) where J Dev / L C/c F ( ) ( )( )0 4� � is the critical Josephson current for noninteracting electrons, R gi ( )� is the inter- action induced renormalization factor ( ( ) )R gi � � �1 1 R g g / g / / g F g gi ( ( ) ( ) , ; ;)� � � � � � � � � � � 3 3 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2� � � � � � � � � � � � � � � �a L g� 1 1 . (22) Here g� is the correlation parameter of a spin-1/2 LL in the charge sector Eq. (6), 3( )x is the gamma func- tion and F z( , ; ; )4 5 6 is the hypergeometric function (see, e.g., Ref. 54). For the first time the expression for R gi ( )� in the integral form was derived in Ref. 21. In the limit of strong interaction V / vF0 1� �� the renormalization factor is small R g v V a Li F / V / vF ( ) ,� �� 1 2 1 0 3 2 2 0 � � � (23) and the Josephson current through the SLLS junction is strongly suppressed. This is nothing but a manifestation of the Kane–Fisher effect [34] in the Josephson current. To evaluate the correlation function, Eq. (18), for a LL wire of finite length coupled to bulk superconductors via tunnel barriers, (Fig. 4), we at first have to formu- late boundary conditions for the electron wave function � � �� ( ) exp( ) ( ) exp( ) ( ), ,x ik x x ik x xF R F L� � � , � � � � (24) at the interfaces x L� 0, . To zeroth order of perturba- tion theory in the barrier transparencies the electrons are confined to the normal region. So the particle cur- rent J i x� � �~ Re( )� �" � through the interfaces is 744 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson S S L D1 LL D r Fig. 4. A Luttinger liquid wire of length L coupled to bulk superconductors via tunnel barriers with transparencies Dl r( ). zero. For a single mode LL this requirement is equiva- lent to the following boundary condition for the chiral fermionic fields [50,52] � � � �R R x L L L x Lx x x x, , , , , ,( ) ( )\| ( ) ( )\|� � � � " � " ��0 0 . (25) These boundary conditions (LL with open ends) result in zero eigenvalues of the «momentum»-like zero mode operator �*� and in the quantization of nontopological modes on a ring with circumference 2L (see Ref. 50). In this case the plasmon propagators take the form �� L j RL kt x, ,( , ) � � � � �� jk ki x s t ia a/L4 1 ln exp[ ( )] . (26) Using Eqs. (2), (17)–(19), and (26) one readily gets the expression for the Josephson current analogous to Eq. (21) J J R gLL f c f ( ) ( ) ( )sin� 0 � �, where now the crit- ical Josephson current of noninteracting electron is J Dev / L C/c F ( ) ( )( )0 4� � and the renormalization factor (R gf ( )� � �1 1) reads R g g g F g g g g g a Lf ( ) ; ; ,� � � � � � � � � � � � � � � � � � � � � � � 2 2 2 2 2 1 1 2 2 � � � � � � � �2 11( ) . g� (27) Comparing Jc ( )0 with the well known formula for the critical Josephson current in a low transparency SINIS junction (see, e.g., Ref. 40) we find the nu- merical constant C � �. In the limit g� �� 1 of strong interaction Eq. (27) is reduced to the simple formula R g v V a Lf F V / vF ( ) .� �� 1 2 1 0 2 2 0 � � � (28) The dependence of the renormalization factor given by Eqs. (22), (27) on the strength of the elect- ron—electron interaction V / vF0 � is shown in Fig. 5. The behavior of the Josephson current as a function of the interaction strength is similar for the two consid- ered geometries. However we see that the interaction influences the supercurrent more strongly for the case of «end-coupled» LL wire. 2.2.2. Transparent junction. The case of perfectly transmitting interfaces in terms of the boundary Hamiltonians (7), (8) which formally correspond to the limit VB � 0 and not small � B . It cannot be treated perturbatively. Physically it means that charge is freely transported through the junction and only pure Andreev reflection takes place at the NS boundaries. It is well known that at energies much smaller than the supercon- ducting gap (E �� ) the scattering amplitude of quasiparticles becomes energy independent (see Eq. (9)). This enable one to represent the Andreev scatter- ing process as a boundary condition for a real space fermion operator. It was shown in Ref. 22 that the corre- sponding boundary condition for chiral fermion fields takes the form of a twisted periodic boundary condition over the interval 2L, � �L/R L/Rx L t i x t, ,( , ) exp( ) ( , ) � � �72 (29) (the upper sign corresponds to the left-moving fer- mions, lower sign — to right moving particles), where 7 � �� , � is the superconducting phase difference and the phase � is acquired due to the Andreev reflection on two interfaces (see, e.g., Eq. (9)). So the problem can be mapped [22] to the one for the persistent current of chiral fermions on a ring of circumference 2L. It is well known [51,55] (see also the Rev. 56) that the persistent current in a per- fect ring (without impurities) in the continuum model does not depend on the electron—electron in- teraction due to the translational invariance of the problem. This «no-renormalization» theorem allows us to conclude that the Josephson current in a per- fectly transmitting SLLS junction coincides with the supercurrent in a one-dimensional long SNS ballistic junction [26,57] Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 745 1.0 0.8 0.6 0.4 0.2 0 2 4 6 8 10 R i,f 1 2 v / v0 Fh Fig. 5. Dependence of the renormalization factor Ri f( ) on the dimensionless electron—electron interaction strength V / vF0 � . Curve 1 corresponds to the case of «side-coupled» LL wire (i), curve 2 to an «end-coupled» LL wire (f). J J eT k kT/LL k k L � � � � � ! nonint 4 1 2 1 1 � ( ) sin sinh( ) , � � � (30) where T is the temperature and � L Fv /L� � . The for- mal proof of this statement [22] consist in evaluating the partition function of the LL with the twisted boundary conditions, Eq. (29), supplemented by a connection between the �R,� and �L,� fields that fol- lows from the chiral symmetry. The superconducting phase difference � couples only to zero modes of the charge current field +� . In a Galilelian invariant sys- tem zero modes are not renormalized by the interaction and the partition function for a SLLS junction exactly coincides with the one for a long SNS junction. We notice here that Eq. (30) holds not only for per- fectly transmitting interfaces. It also describes asym- ptotically at T �� the Josephson current through a tunnel junction when the interaction in the wire is as- sumed to be attractive. We have seen already in the previous subsection that the electron—electron interac- tion renormalizies the bare transparency of the junction due to the Kane–Fisher effect. The renormalization is known to suppress the electron current for a repulsive interaction and to enhance it for an attractive forces [34]. So one could expect that for an attractive inter- action the electron interface scattering will be re- normalized at low temperatures to perfect Andreev scattering [23]. 2.3. Josephson current and the Casimir effect More then fifty years ago Casimir predicted [58] the existence of small quantum forces between grounded metallic plates in vacuum. This force (a kind of van der Waals force between neutral objects) arises due to a change of the vacuum energy (zero-point fluctuations) induced by the boundary conditions imposed by the metallic plates on the fluctuating electromagnetic fields (see Refs. 59 and 60). This force has been mea- sured (see, e.g., one of the recent experiments [61] and the references therein) and in quantum field theory the Casimir effect is considered as the most spectacular manifestation of zero-point energy. In a general situa- tion the shift of the vacuum energy of fluctuating fields in a constrained volume is usually called the Casimir energy EC. For a field with zero rest mass di- mensional considerations result in a simple behavior of the Casimir energy as a function of geometrical size. In 1D, E v/LC ~ � where v is the velocity. Now we will show that the Josephson current in a long SNS junction from a field-theoretical point of view can be considered as a manifestation of the Casimir effect. Namely, the Andreev boundary condition changes the energy of the «Fermi sea» of quasiparticles in the nor- mal region. This results in the appearance of: (i) an additional cohesive force between the superconduct- ing banks [30], and (ii) a supercurrent induced by the superconducting phase difference. As a simple example we evaluate the Josephson cur- rent in a long transparent 1D SNS junction by using a field theoretical approach. Andreev scattering at the NS interfaces results in twisted periodic boundary conditions, Eq. (26), for the chiral fermion fields [51]. So the problem is reduced to the evaluation of the Casimir energy for chiral fermions on an S1 mani- fold of circumference 2L with «flux» 7. Notice that the left- and right-moving quasiparticles feel opposite (in sign) «flux» (see Eq. (29)). The energy spectrum takes the form (�L Fv /L� � ) E L nn L, ( , ) ,' � � ' � � � � �� 1 2 2 n � � 0 1 2 1, , ,..., ,' (31) and coincides (as it should be) with the electron and hole energies calculated by matching the quasi- particle wave functions at the NS boundaries [26]. The Casimir energy is defined as the shift of the vacuum energy induced by the boundary conditions E L E L E LC n n n n ( , ) ( , ) ( ), , , , � �' ' ' ' � �� ! 8 9 : : ; 2 1 2 < = = . (32) Notice, that the factor (–1/2) in Eq. (32) is due to the zero-point energy of chiral fermions, the addi- tional factor of 2 is due to spin degeneracy. Both sums in Eq. (32) diverge and one needs a certain regular- ization procedure to manipulate them. One of the most efficient regularization methods in the calcula- tion of vacuum energies is the so-called generalized zeta-function regularization [62]. For the simple en- ergy spectrum, Eq. (31), this procedure is reduced to the analytical continuation of the infinite sum over n in Eq. (32) to the complex plane, E n aC L s s n ( ) lim ( ) , � � ' ' � � � � �� ! � ! � 1 1 � � � � � � � � � > > ' ' ' '� L a a a[ ( , ) ( , ) ], 1 1 1 (33) where >( , )s a is the generalized Riemann >-function [54] and a /' � '� �� ( ) 2 . Using an expression for >( , )�n a in terms of Bernoulli polynomials that is well-known from textbooks (see Ref. 54) one gets the desired formula for the Casimir energy of a 1D SNS junction as 746 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson E v LC F� � � � � � � � 8 9 : : ; < = = ?2 2 1 12 2 � � � � � � , \| \| . (34) The Casimir force FC and the Josephson current J at T � 0 are F E E L J e E ev LC C C C F� � � � � � � � � ? � � � � � �, , \| \| � . (35) the expression for the Josephson current coincides with the zero-temperature limit of Eq. (30). The gen- eralization of the calculation method to finite temper- atures is straightforward. The additional cohesive force between two bulk metals induced by supercon- ductivity is discussed in Ref. 30. In this paper it was shown that for a multichannel SNS junction this force can be measured in modified AFM-STM experiments, where force oscillations in nanowires were observed. The calculation of the Casimir energy for a system of interacting electrons is a much more sophisticated problem. In Ref. 47 this energy and the corresponding Josephson current were analytically calculated for a special exactly solvable case of double-boundary LL. Unfortunately the considered case corresponds to the attractive regime of LLs (g� � 2 in our notation, see Eq. (6)) and the interesting results obtained in Ref. 47 can not be applied for electron transport in quantum wires fabricated in 2DEG or in individual SWNTs were the electron—electron interaction is known to be repulsive. 3. The effects of Zeeman splitting and spin—orbit interaction in SNS junctions In the previous Section we considered the influence of electron—electron interactions on the Josephson current in a S–QW–S junction. Although all calcula- tions were performed for a spin-1/2 Luttinger liquid model, it is readily seen that the spin degrees of free- dom in the absence of a magnetic field are trivially in- volved in the quantum dynamics of our system. In es- sence, they do not change the results obtained for spinless particles. For noninteracting electrons spin only leads to an additional statistical factor 2 (spin de- generacy) in the thermodynamic quantities. At the first glance spin effects could manifest themselves in SLLS junctions since it is known that in LL the phe- nomena of spin-charge separation takes a place [46]. One could naively expect some manifestations of this nontrivial spin dynamics in the Josephson current. Spin effects for interacting electrons are indeed not re- duced to the appearance of statistical factor. How- ever, as we have seen already in the previous sections, the dependence of the critical Josephson current on the interaction strength is qualitatively the same for spin-1/2 and spinless Luttinger liquids. So it is for ease of calculations a common practice to investigate weak superconductivity in the model of spinless Luttinger liquid [47]. Spin effects in the Josephson current become impor- tant in the presence of a magnetic field, spin—orbit in- teractions or spin—dependent scattering on impurities. At first we consider the effects induced by a magnetic field. Generally speaking a magnetic field influences both the normal part of the junction and the supercon- ducting banks. It is the last impact that determines the critical Josephson current in short and wide junctions. The corresponding problem was solved many years ago and one can find the analytical results for a short and wide junction in a magnetic field parallel to the NS in- terface (e.g., in Refs. 63 and 64). In this review we are interested in the supercon- ducting properties of junctions formed by a long bal- listic quantum wire coupled to bulk superconductors. We will assume that a magnetic field is applied lo- cally, i.e., only to the normal part of the junction (such an experiment could be realized for instance with the help of a magnetic tip and a scanning tunnel- ing microscope). In this case the only influence of the magnetic field on the electron dynamics in a single channel (or few channel) QW is due to the Zeeman in- teraction. For noninteracting electrons the Zeeman splitting lifts the double degeneracy of Andreev levels in an SNS junction and results in a periodic depend- ence of the critical Josephson current on magnetic field [65]. Interaction effects can easily be taken into account for a 1D SLLS junction in a magnetic field by using bosonization techniques. The term in the Hamiltonian �HZ, which describes the interaction of the magnetic field B with the electron spin S( )x is in bosonized form (see, e.g., Ref. 46) � ( ), ( )H g B dx S x S xZ f B z z z x� � � �� 1 2 ,(36) where gf is the g-factor, B is the Bohr magneton and the scalar field �� is defined in Eq. (16). As is easy to see, this interaction can be transformed away in the LL Hamiltonian by a coordinate-dependent shift of the spin bosonic field � � �� @ � � z Fx/ v� 2 , � z f Bg B� is the Zeeman splitting. So the Zeeman splitting intro- duces an extra x-dependent phase factor in the chiral components of the fermion fields and thus the Zeeman interaction can be readily taken into account [66] by a slight change of the bosonization formula (13) A A' � ' � ' �, ( ) , ,( , ) exp( ) ( , ),Z x t iK x x t� Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 747 K v z F ' � '� ' �, , , .� � � 4 1 � (37) The phase factor appearing in Eq. (37) results in a peri- odic dependence of the Josephson current on magnetic field. In the presence of Zeeman splitting the critical current, say, for a tunnel SLLS junction, Eq. (21), ac- quires an additional harmonic factor cos( )� �Z L/ , the same as for noninteracting particles. 3.1. Giant critical current in a magnetically controlled tunnel junction Interesting physics for low-transparency junctions appears when resonant electron tunneling occurs. In this subsection we consider the special situation when the conditions for resonant tunneling through a junc- tion are induced by superconductivity. The device we have in mind is an SNINS ballistic junction formed in a 2DEG with a tunable tunnel barrier («I») and a tunable Zeeman splitting which can be provided for instance with the help of a magnetic tip and a scan- ning tunneling microscope (STM). In quantum wires fabricated in 2DEG the effects of electron—electron interactions are not pronounced and we will neglect them in what follows. Resonant electron tunneling through a double bar- rier mesoscopic structure is a well studied quantum phenomenon, which has numerous applications in solid state physics. Recently a manifestation of reso- nant tunneling in the persistent current both in super- conducting [67] and in normal systems [68] was stud- ied. In these papers a double-barrier system was formed by the two tunnel barriers at the NS interfaces [67] or in a normal metal ring [68]. It was shown that for resonance conditions (realized for a special set of junction lengths [67] or interbarrier distances [68]) a giant persistent current appears which is of the same order of magnitude as the persistent current in a sys- tem with only a single barrier. In the case of the SINIS junction considered in Ref. 67 the critical supercurrent was found to be proportional to D. Notice that the normal transmission coefficient for a symmetric dou- ble-barrier structure (i.e., the structure with normal leads) at resonance conditions does not depend on the barrier transparency at all. It means that for the hy- brid structure considered in Ref. 67 the superconduc- tivity actually suppresses electron transport. Now we show [69] that in a magnetically con- trolled single barrier SFIFS junction («F» denotes the region with nonzero Zeeman splitting) there are con- ditions when superconductivity in the leads strongly enhances electron transport. Namely, the proposed hy- brid SFIFS structure is characterized by a giant criti- cal current J Dc ~ . While the normal conductanceG is proportional to D. For a single barrier SFIFS junction of length L, where the barrier is located at a distance l L�� mea- sured from the left bank, the spectrum of Andreev lev- els is determined from the transcendental equation [69] cos cos cos , 2 2 0 2 E R E DZ L Z L l � � � � � � � � � (38) where � x Fv /x� � and D R� � 1, � Z is the Zeeman splitting. In the limit � Z � 0 Eq. (38) is reduced to a well-known spectral equation for Andreev levels in a long ballistic SNS junction with a single barrier [40,70]. At first we consider the symmetric single-barrier junction, i.e., the case when the scattering barrier is situated in the middle of the normal region l L/� 2. Then the second cosine term in the spectral equation is equal to one and Eq. (38) is reduced to a much simpler equation which is easily solved analytically. The eval- uation of the Josephson current shows [69] that for D �� 1 and for a discrete set of Zeeman splittings, � �Z k Lk k� � ��( ) , , , , ... ,2 1 0 1 2 (39) the resonance Josephson current (of order D) is de- veloped. At T � 0 it takes the form J ev L D / r F( ) sin \| sin( )\| .� � � � 2 (40) This expression has the typical form of a resonant Josephson current associated with the contribution of a single Andreev level [40]. One can interpret this result as follows. Let us assume for a moment that the potential barrier in a symmetric SNINS junction is in- finite. Then the system breaks up into two identical INS-hybrid structures. In each of the two systems de Gennes–Saint-James energy levels with spacing 2�� L are formed [71]. For a finite barrier these levels are split due to tunneling with characteristic splitting en- ergy � ~ D L� . The split levels being localized al- ready on the whole length L between the two su- perconductors are nothing but the Andreev–Kulik energy levels, i.e., they depend on the superconduct- ing phase difference. Although the partial current of a single level is large (~ D) (see Refs. 40 and 67), the current carried by a pair of split levels is small (~ D) due to a partial cancellation. At T � 0 all levels above the Fermi energy are empty and all levels below EF are filled. So in a system without Zeeman splitting the partial cancellation of currents carried by pairs of tunnel—split energy levels results in a small critical current (~ D). The Zeeman splitting � Z of order � L (see Eq. (39)) shifts two sets («spin-up» and «spin-down») of Andreev levels so that the Fermi en- 748 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson ergy lies in between the split levels. Now at T � 0 only the lower state is occupied and this results in an uncompensated large (~ D) Josephson current. Since the quantized electron-hole spectrum is formed by Andreev scattering at the NS interfaces, the resonance structure for a single barrier junction disappears when the leads are in the normal (non- superconducting) state. So, the electron transport through the normal region is enhanced by supercon- ductivity. Electron spin effects (Zeeman splitting) are crucial for the generation of a giant Josephson current in a single barrier junction. The described resonant transport can occur not only in symmetric junction. For a given value of Zeeman splitting � Z k( ) from Eq. (39) there is a set of points [69] (determined by their coordinates xm k( ) counted from the middle of the junction) x m k Lm k( ) � �2 1 (41) (m is the integer in the interval 0 1 2? ? �m k / ), where a barrier still supports resonant transport. The temperature dependence of the giant Josephson cur- rent is determined by the energy scale � ~ D L� and therefore at temperatures T ~ �, which are much lower then � L, all resonance effects are washed out. 3.2. Giant magnetic response of a quantum wire coupled to superconductors It is known that the proximity effect produced in a wire by superconducting electrodes strongly enhances the normal conductance of the wire for certain value of the superconducting phase difference (giant con- ductance oscillations [72]). For ballistic electron transport this effect has a simple physical explanation [73] in terms of Andreev levels. Consider a multichan- nel ballistic wire perfectly (without normal electron backscattering) coupled to bulk superconductors. The wire is assumed to be connected to normal leads via tunnel contacts. In the first approximation one can ne- glect the electron leakage through the contacts and then the normal part of the considered Andreev interferometer is described by a set of Andreev levels produced by superconducting mirrors . When the dis- tance L between the mirrors is much longer then the superconducting coherence length L v /F�� 0 � � (� is the superconducting gap), the spectrum takes a simple form [26] E v L n nn j F j , ( ) ( ) [ ( ) ], , , , ... , � � � � 2 2 1 0 1 2� � (42) where vF j( ) is the Fermi velocity of the jth transverse channel (j N� �1 2, , ..., ). It is evident from Eq. (42) that at special values of phase difference � n � � �� ( )2 1n energy levels belonging to different trans- verse channels j, collapse to a single multi-degenerate (N� ) level exactly at the Fermi energy. So resonant normal electron transport through a multichannel wire (the situation which is possible for symmetric barriers in the normal contacts) will be strongly enhanced at � �� n . The finite transparency of the barriers results in a broadening and a shift of the Andreev levels. These effects lead to a broadening of the resonance peaks in giant conductance oscillations at low temperatures [73]. Magnetic properties of a quantum wire coupled to superconductors can also demonstrate a behavior analo- gous to the giant conductance oscillations. We consider a long perfectly transmitting SNS junction in a local (applied only to the normal region) magnetic field. In this case the only influence of the magnetic field on the Andreev level structure is through the Zeeman cou- pling. The thermodynamic potential � A B( , )� calcu- lated for Zeeman-split Andreev levels is [30] � � A k j L j kj B T k k k kT/ ( , ) ( ) cos cos sinh( )( ) { � � B � � � � ! 4 1 21} . N� (43) Here B j Z L j Z B/ g B� �� ( ), is the Zeeman ener- gy splitting, � L j F jv /L( ) ( )� � and vF j( ) is the Fermi ve- locity in the jth transverse channel, { }j is the set of transverse quantum numbers. In Ref. 30 the normal part of the SNS junction was modelled by a cylinder of length L and cross-section area S V/L� . Hard-wall boundary conditions for the electron wave function on the cylinder surface were assumed. Then the set { }j is determined by the quantum numbers (l n, ) that label the zeroes 6 l n, of the Bessel function Jl l n( ),6 � 0 and the velocity vF l n( , ) takes the form v m L mVF l n F ( , ) .� � � � � � � � � � 2 2 2 2 C 6 � ln � (44) It is evident from Eq. (43) that the superconductiv- ity-induced magnetization M B BA A� � � � � ( , )� (45) at high temperatures (T L�� ) is exponentially small and does not contribute to the total magnetiza- tion of the junction. At low temperatures T the mag- netization peaks at M N gA B~ � where the super- conducting phase difference is an odd multiples of � (see Fig. 6 which is adapted from Ref. 30). The quali- tative explanation of this resonance behavior of the magnetization is as follows. It is known [74] that for � � �� # �n n( )2 1 (n is the integer) the two Andreev levels E /A Z ( ) � � 2 become 2N� -fold degenerate. Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 749 At T � 0 the filled state EA ( )� dominates in the mag- netization at � �� n since at other values of supercon- ducting phase the sets of Andreev levels corresponding to different transverse channels contribute to magneti- zation Eqs. (43), (45) with different periods in «mag- netic phase» B j (i.e., in general, incoherently) and their contributions partially cancel each other. Notice also that for a fixed volume V, the number of trans- verse channels N� has a step-like dependence on the wire diameter. So at resonance values of the phase dif- ference � �� n one can expect a step-like behavior of the magnetization as a function of wire diameter [30]. This effect is a magnetic analog of the Josephson cur- rent quantization in a short SNS junction [39] consid- ered in Sec. 2.1. 3.3. Rashba effect and chiral electrons in quantum wires Another type of system where spin is nontrivially involved in the quantum dynamics of electrons are conducting structures with strong spin—orbit interac- tion. It has been known for a long time [29] that the SO interaction in the 2DEG formed in a GaAs/AlGaAs in- version layer is strong due to the structural inversion asymmetry of the heterostructure. The appearance in quantum heterostructures of an SO coupling linear in electron momentum is now called the Rashba effect. The Rashba interaction is described by the Hamiltonian H i x yso SO y x R( ) ,� � � � � � � � �� 4 � � (46) where � x y( ) are the Pauli matrices. The strength of the spin—orbit interaction is determined by the cou- pling constant 4 so , which ranges in a wide interval (1–10) D �10 10 eV D cm for different systems (see, e.g., Ref. 31 and references therein). Recently it was ex- perimentally shown [75–77] that the strength of the Rashba interaction can be controlled by a gate volt- age 4 so GV( ). This observation makes the Rashba ef- fect a very attractive and useful tool in spintronics. The best known proposal based on the Rashba effect is the spin-modulator device of Datta and Das [78]. The spin—orbit interaction lifts the spin degener- acy of the 2DEG energy bands at p E 0 (p is the elec- tron momentum). The Rashba interaction, Eq. (46) produces two separate branches for «spin-up» and «spin-down» electron states C 4 ( )p p \| p \|� 2 2m SO � . (47) Notice that under the conditions of the Rashba effect the electron spin lies in a 2D plane and is always per- pendicular to the electron momentum. By the terms «spin-up» («spin-down») we imply two opposite spin projections at a given momentum. The spectrum (47) does not violate left-right symmetry, that is the elec- trons with opposite momenta ( p) have the same en- ergy. Actually, the time reversal symmetry of the spin—orbit interaction, Eq. (46), imposes less strict limitations on the electron energy spectrum, namely, C C� ��( ) ( )� �p p and thus, the Rashba interaction can in principle break the chiral symmetry. In [31] it was shown that in quasi-1D quantum wires formed in a 2DEG by a laterally confining potential the electron spectrum is characterized by a dispersion asymmetry C C� �( ) ( )� Ep p . It means that the electron spectrum linearized near the Fermi energy is characterized by two different Fermi velocities v F1 2( ) and, what is more important, electrons with large (Fermi) momenta behave as chiral particles in the sense that in each subband (characterized by Fermi velocity vF ( )1 or vF ( )2 ) the direction of the electron motion is corre- lated with the spin projection [31,79] (see Fig. 7). It is natural in this case to characterize the spectrum by the asymmetry parameter ) a F F F F v v v v � � � 1 2 1 2 , (48) which depends on the strength of Rashba interaction ) 4a SO( )� �0 0. The asymmetry parameter grows with the increase of 4 SO and can be considered in this model as the effective dimensional strength of the Rashba interaction in a 1D quantum wire [31]. Notice that the spectrum proposed in [31,79] (Fig. 7, solid lines for spin projections) does not hold for strong SO interactions, when ) a is not small. Spin is not con- served in the presence of the SO interaction and the prevailing spin projection of electron states in quasi 1D wires has to be independently calculated. It was shown in Ref. 32 by a direct calculation of the average 750 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson 0.4 0.6 0.8 1.0 1.2 1.4 1.6 � �/ 50 40 30 20 10 0 –10 T = 0.8 K T = 3.2 K T = 0.2 K M , b Fig. 6. Dependence of the magnetization M of an SNS junction on the superconducting phase difference for dif- ferent temperatures. electron spin projection that for energies close to CF the electron spin projection for strong Rashba interac- tion (comparable with the band splitting in the confin- ing potential) is strongly correlated with the direction of the electron motion. Namely, the right (R)-and the left (L)-moving electrons always have opposite spin projections regardless of their velocities (see Fig. 7, where the parentheses indicate the spin projection for strong Rashba interaction). For our choice of Rashba SO Hamiltonian, Eq. (46), R-electrons (kx � 0) will be «down-polarized» ( � � � �� y 1) and L-electrons ( )kx � 0 will be «up-polarized» ( )� � � �� y 1 to minimize the main part of electron energy ~ ( )� 2 2/ m � k m /x y SO� �� 4 � 2 in the presence of strong spin—orbit interaction [32]. Chiral electrons in 1D quantum wire result in such interesting predictions as «spin accumulation» in normal wires [32] or Zeeman splitting induced supercurrent in S–QW–S junction [69]. 3.4. Zeeman splitting induced supercurrent It was shown in the previous subsection that under the conditions of the Rashba effect in 1D quantum wires the spin degree of freedom is strongly correlated with the electron momentum. This observation opens the pos- sibility to magnetically control an electric current. It is well known that in ring-shaped conductors the current can be induced by magnetic flux due to the momentum dependent interaction of the electromagnetic potential A with a charged particle H e/mcint ( )� pA. Chiral prop- erties of electrons in quasi-1D quantum wires allow one to induce a persistent current via pure spin (momentum independent) interaction H g B� SH. Below we con- sider the Josephson current in a ballistic S—QW—S junction in the presence of Rashba spin—orbit interac- tion and Zeeman splitting. We will assume at first that SO interactions exist both in the normal part of the junction and in the superconducting leads, so that one can neglect the spin rotation accompanied by electron backscattering induced by SO interactions at the NS in- terfaces. In other words the contacts are assumed to be fully adiabatic. This model can be justified at least for a weak SO interaction. The energy spectrum of electrons in a quantum wire is shown in Fig. 7 and the effect of the SO interaction in this approach is characterized by the dispersion asymmetry parameter ) a , Eq. (48). For a perfectly transparent junction (D � 1) the two subbands 1 and 2 (see Fig. 7) contribute inde- pendently to the Andreev spectrum which is described by two sets of levels [69] E nn L, ( ) ( ) ,' � ' � B � 1 1 11 2 2 � � � �� (49) E mm L, ( ) ( ) ,' � ' � B � 2 2 21 2 2 � � � �� where the integers n m, , , ,� 0 1 2 � are ordinary quantum numbers which label the equidistant Andreev levels in a long SNS junction [26], ' � 1, � L j jFv /L j( ) ( , )� �� 1 2 and � is the superconducting phase difference. The magnetic phases B j Z L j/� � �( ) characterize the shift of Andreev energy levels in- duced by Zeeman interaction. Notice that the relative sign between the superconducting phase � and the magnetic phase B j is different for channels 1 and 2. This is a direct consequence of the chiral properties of the electrons in our model. In the absence of a disper- sion asymmetry (v v vF F F1 2� # ) the two sets of levels in Eq. (49) describe the ordinary spectrum of Andreev levels in a long transparent SFS junction («F» stands for the normal region with Zeeman splitting) E nn L, , , ,' � � ' � � � B � ' �� 1 2 2 2 1. (50) Knowing explicitly the energy spectrum, Eq. (49), it is straightforward to evaluate the Josephson current. It takes the form [69] J T eT k kT/ Z k k L ( , , ) ( ) sin ( ) sinh( ( � � B � � � � � �� ! 2 1 2 1 1 1 1� ) ( )) sin ( ) sinh( ) .� �8 9 : : ; < = = k kT/ L � B � 2 22 � (51) Charge and spin effects in mesoscopic Josephson junctions Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 751 C C F p 1 2 (p) 1 2 Fig. 7. Schematic energy spectrum of 1D electrons with dispersion asymmetry. Particles with energies close to the Fermi energy CF have an almost linear dependence on momentum and are classified by their Fermi velocities (v F1 -subband 1, v F2 -subband 2). Solid line for spin pro- jections correspond to the case of weak SO interaction; spin in parentheses indicates the spin projections in sub- band 1 for strong Rashba interaction. Here T is the temperature. The formal structure of Eq. (51) is obvious. The two sums in Eq. (51) corre- spond to the contributions of magnetically shifted sets of levels 1 and 2 in Eq. (49). In the absence of any SO interaction the Zeeman splitting results only in an additional cos( )k /Z L� � factor in the standard formula for the supercurrent through a perfectly transmitting long SNS junction [57]. The most strik- ing consequence of Eq. (51) is the appearance of an anomalous Josephson current J Jan # �( )� 0 , when both the Zeeman splitting (� Z) and dispersion asymmetry () a) are nonzero. At high temperatures T L jF �( ) the anomalous supercurrent is exponentially small. In the low temperature regime T L j�� ( ) it is a piece-wise constant function of the Zeeman energy splitting � Z, J e L k v k van Z k k F Z L F( ) ( ) sin ( ) � � � � � � � � � � � � � � � � ! � 1 1 1 1 1 2 sin . ( ) k Z L � � 2 � � � � � � � � 8 9 : : ; < = = (52) For rational values v /v p/qF F1 2 � (p q? are the integers) Jan is a periodic function of the Zeeman en- ergy splitting with period � �� Z Lq� 2 1( ), otherwise it is a quasiperiodic function. The dependence of the normalized supercurrent J /Jan 0 (here J ev /LF0 � , v v v /F F F� �( )1 2 2) on the dimensionless Zeeman splitting B # � �Z L/ for ) a � 01. and for different temperatures is shown in Fig. 8. We see that at T � 0 the Zeeman splitting induced supercurrent appears abruptly at finite values of � Z of the order of the Andreev level spacing. Let us imagine now the situation when the Zeeman splitting arises due to a local magnetic field (acting only on the normal part of the junction) in the 2D plane applied normal to the quantum wire. Then the vector product of this magnetic field and the electric field (normal to the plane), which induces the Rashba interaction determines the direction of the anomalous supercurrent. In other words the change of the sign of the SO interaction in Eq. (46) or the sign of � Z makes the supercurrent Eq. (52) change sign as well. Now we briefly discuss the case of a strong Rashba interaction (the characteristic momentum kSO � � m/ VSO g�4 ( ) is of the order of the Fermi momen- tum). The electrons in a quantum wire with strong Rashba coupling are chiral particles, that is the right- and left-moving particles have opposite spin projec- tions [32]. There is no reason to assume a strong SO interaction in 3D superconducting leads. We will fol- low the approach taken in [32,80], where the system was modelled by a quantum wire (4 SO E 0) attached to semi-infinite leads with 4 SO � 0. In this model the SN interface acts as a special strong scatterer where backscattering is accompanied by spin-flip process. For a general nonresonant situation the dispersion asymmetry is not important in the limit of strong Rashba interaction and we can put v v vF F F1 2G G . Then the Josephson current at T � 0 up to numerical factor takes the form J D ev LZ SO F Z L ( , ) ( ) sin .� 4 �� G � � � �� eff (53) Here D SOeff ( )4 �� 1 is the effective transparency of the junction. It can be calculated by solving the transition problem for the corresponding normal junc- tion [32]. Anyway, in the considered model for NS interfaces (nonadiabatic switching on the Rashba interaction) even in the limit of strong Rashba inter- action the anomalous supercurrent J Jan Z� �( , )� 0 � is small because of smallness of the effective transpar- ency of the junction. One could expect large current only for special case of resonant transition. This prob- lem has not yet been solved. 752 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Ilya V. Krive, Sergei I. Kulinich, Robert I. Shekhter, and Mats Jonson 1 3 2 0 2 4 6 8 10 12 14 16 18 20 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 J/ J 0 � �/Z L Fig. 8. Dependence of the normalized anomalous Josephson current J /Jan 0 ( )J ev /LF0 � on the dimen- sionless Zeeman splitting � �z L/ ( )�L Fv /L� � for asym- metry parameter )a � 01. . The different plots correspond to different temperatures T: 0.1T"( )1 ; 1.5T"( )2 ; 3.5 T"( )3 , where T /L " � � 2�. 4. Conclusion The objective of our review was to discuss the quali- tatively new features of the Josephson effect that ap- pear in S–QW–S hybrid structures. Quantum wires are characterized by a 1D or quasi-1D character of the elec- tron conductivity. Electron transport along QWs is ballistic and due to the weak screening of the Coulomb interaction in 1D it is described by a Luttinger liquid theory. So the first question we would like to answer was — what is the Josephson effect in SLLS junction? It was shown that although electrons do not propagate in a LL weak link the supercurrent in a perfectly trans- mitting SLLS junction exactly coincides with the one in an SNS junction [22]. This «no renormalization» theorem is analogous to the result known for a LL adia- batically coupled to nonsuperconducting leads [35]. For a tunnel SILLIS junction the dc Josephson current is described by the famous Josephson current-phase relation, however now the effective transparency Deff �� 1 defined as J J D� 0 eff sin� (where J0 � � ev /LF ) strongly depends on the aspect ratio of the LL wire d/L (d F~ ) is the width of the nanowire), temperature and electron—electron interaction strength. This result [21] is a manifestation of the Kane–Fisher ef- fect [34] in mesoscopic superconductivity. It was also in- teresting for us (and we hope for the readers as well) to find a close connection, rooted in the Andreev boundary conditions, between the physics of a long SNS junction and the Casimir effect (see Sec. 2.3.). Qualitatively new behavior of the proximity in- duced supercurrent in nanowires is predicted for sys- tems with strong spin—orbit interactions. The Rashba effect in nanowires results in the appearance of chiral electrons [31,32] for which the direction of particle motion along the wire (right or left) is strongly corre- lated with the electron spin projection. For chiral electrons the supercurrent can be magnetically in- duced via Zeeman splitting. The interplay of Zeeman, Rashba interactions and proximity effects in quantum wires leads to effects that are qualitatively different from those predicted for 2D junctions [81]. It is worth-while to mention here another impor- tant trend in mesoscopic superconductivity, namely, the fabrication and investigation of superconductiv- ity-based qubits. Among different suggestions and projects in this rapidly developing field, the creation of a so-called single–Cooper–box (SCPB) was a remarkable event [82]. The SCPB consist of an ultrasmall superconducting dot in tunneling contact with a bulk superconductor. A gate electrode, by lift- ing the Coulomb blockade of Cooper-pair tunneling, allows the delocalization of a single Cooper pair be- tween the two superconductors. For a nanoscale grain the quantum fluctuations of the charge on the island are suppressed due to the strong charging energy associated with a small grain capacitance. By appro- priately biasing the gate electrode it is possible to make the two states on the dot, differing by one Coo- per pair, have the same energy. This twofold degener- acy of the ground state brings about the opportunity to create a long-lived coherent mixture of two ground states (qubit). The superconducting weak link which includes a SCPB as a tunnel element could be very sensitive to external ac fields. This problem was studied in [83], where the resonant microwave properties of a voltage biased single–Cooper–pair transistor were considered. It was shown that the quantum dynamics of the sys- tem is strongly affected by interference between mul- tiple microwave-induced inter-level transitions. As a result the magnitude and the direction of the dc Josephson current are extremely sensitive to small variations of the bias voltage and to changes in the fre- quency of the microwave field. This picture, which differs qualitatively from the famous Shapiro effect [3], is a direct manifestation of the role the strong Coulomb correlations play in the nonequilibrium superconducting dynamics of mesoscopic weak links. Acknowledgment The authors thank E. Bezuglyi, L. Gorelik, A. Kadigrobov, and V. Shumeiko for numerous fruitful discussions. Ilya V. Krive and Sergei I. Kulinich acknowledge the hospitality of the Department of Applied Physics at Chalmers University of Tech- nology and Göteborg University. Financial support from the Royal Swedish Academy of sciences (Sergei I. Kulinich), the Swedish Science Research Council (Robert I. 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Charge and spin effects in mesoscopic Josephson junctions (Review Article)

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