The scaling function at strong coupling from the quantum string Bethe equations

Stimulated by the initial proposal that molecules could be used as the functional building blocks in electronic devices 1, researchers around the world have been probing transport phenomena at the single-molecule level both experimentally and theoretically 2–11. Recent experimental advances include the demonstration of conductance switching 12–16, rectification 17–21, and illustrations on how quantum interference effects 22–26 play a critical role in the electronic properties of single metal–molecule–metal junctions. The focus of these experiments has been to both provide a fundamental understanding of transport phenomena in nanoscale devices as well as to demonstrate the engineering of functionality from rational chemical design in single-molecule junctions. Although so far there are no ‘molecular electronics’ devices manufactured commercially, basic research in this area has advanced significantly. Specifically, the drive to create functional molecular devices has pushed the frontiers of both measurement capabilities and our fundamental understanding of varied physi-cal phenomena at the single-molecule level, including mechan-ics, thermoelectrics, optoelectronics and spintronics in addition to electronic transport characterizations. Metal–molecule–metal junctions thus represent a powerful template for understanding and controlling these physical and chemical properties at the atomic- and molecular-length scales. I n this realm, molecular devices have atomically defined precision that is beyond what is achievable at present with quantum dots. Combined with the vast toolkit afforded by rational molecular design 27, these techniques hold a significant promise towards the development of actual devices that can transduce a variety of physical stimuli, beyond their proposed utility as electronic elements 28.n this Review we discuss recent measurements of physi-cal properties of single metal–molecule–metal junctions that go beyond electronic transport characterizations (Fig. 1). We present insights into experimental investigations of single-molecule junc-tions under different stimuli: mechanical force, optical illumina-tion and thermal gradients. We then review recent progress in spin- and quantum interference-based phenomena in molecular devices. I n what follows, we discuss the emerging experimentalSingle-molecule junctions beyond electronic transportSriharsha V. Aradhya and Latha Venkataraman*The id ea of using ind ivid ual molecules as active electronic components provid ed the impetus to d evelop a variety of experimental platforms to probe their electronic transport properties. Among these, single-molecule junctions in a metal–molecule–metal motif have contributed significantly to our fundamental understanding of the principles required to realize molecular-scale electronic components from resistive wires to reversible switches. The success of these techniques and the growing interest of other disciplines in single-molecule-level characterization are prompting new approaches to investigate metal–molecule–metal junctions with multiple probes. Going beyond electronic transport characterization, these new studies are highlighting both the fundamental and applied aspects of mechanical, optical and thermoelectric properties at the atomic and molecular scales. Furthermore, experimental demonstrations of quantum interference and manipulation of electronic and nuclear spins in single-molecule circuits are heralding new device concepts with no classical analogues. In this Review, we present the emerging methods being used to interrogate multiple properties in single molecule-based devices, detail how these measurements have advanced our understanding of the structure–function relationships in molecular junctions, and discuss the potential for future research and applications.methods, focusing on the scientific significance of investigations enabled by these methods, and their potential for future scientific and technological progress. The details and comparisons of the dif-ferent experimental platforms used for electronic transport char-acterization of single-molecule junctions can be found in ref. 29. Together, these varied investigations underscore the importance of single-molecule junctions in current and future research aimed at understanding and controlling a variety of physical interactions at the atomic- and molecular-length scale.Structure–function correlations using mechanicsMeasurements of electronic properties of nanoscale and molecu-lar junctions do not, in general, provide direct structural informa-tion about the junction. Direct imaging with atomic resolution as demonstrated by Ohnishi et al.30 for monoatomic Au wires can be used to correlate structure with electronic properties, however this has not proved feasible for investigating metal–molecule–metal junctions in which carbon-based organic molecules are used. Simultaneous mechanical and electronic measurements provide an alternate method to address questions relating to the struc-ture of atomic-size junctions 31. Specifically, the measurements of forces across single metal–molecule–metal junctions and of metal point contacts provide independent mechanical information, which can be used to: (1) relate junction structure to conduct-ance, (2) quantify bonding at the molecular scale, and (3) provide a mechanical ‘knob’ that can be used to control transport through nanoscale devices. The first simultaneous measurements of force and conductance in nanoscale junctions were carried out for Au point contacts by Rubio et al.32, where it was shown that the force data was unambiguously correlated to the quantized changes in conductance. Using a conducting atomic force microscope (AFM) set-up, Tao and coworkers 33 demonstrated simultaneous force and conductance measurements on Au metal–molecule–metal junc-tions; these experiments were performed at room temperature in a solution of molecules, analogous to the scanning tunnelling microscope (STM)-based break-junction scheme 8 that has now been widely adopted to perform conductance measurements.Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA. *e-mail: lv2117@DOI: 10.1038/NNANO.2013.91These initial experiments relied on the so-called static mode of AFM-based force spectroscopy, where the force on the canti-lever is monitored as a function of junction elongation. I n this method the deflection of the AFM cantilever is directly related to the force on the junction by Hooke’s law (force = cantilever stiff-ness × cantilever deflection). Concurrently, advances in dynamic force spectroscopy — particularly the introduction of the ‘q-Plus’ configuration 34 that utilizes a very stiff tuning fork as a force sen-sor — are enabling high-resolution measurements of atomic-size junctions. In this technique, the frequency shift of an AFM cantilever under forced near-resonance oscillation is measuredas a function of junction elongation. This frequency shift can be related to the gradient of the tip–sample force. The underlying advantage of this approach is that frequency-domain measure-ments of high-Q resonators is significantly easier to carry out with high precision. However, in contrast to the static mode, recover-ing the junction force from frequency shifts — especially in the presence of dissipation and dynamic structural changes during junction elongation experiments — is non-trivial and a detailed understanding remains to be developed 35.The most basic information that can be determined throughsimultaneous measurement of force and conductance in metalThermoelectricsSpintronics andMechanicsOptoelectronicsHotColdFigure 1 | Probing multiple properties of single-molecule junctions. phenomena in addition to demonstrations of quantum mechanical spin- and interference-dependent transport concepts for which there are no analogues in conventional electronics.contacts is the relation between the measured current and force. An experimental study by Ternes et al.36 attempted to resolve a long-standing theoretical prediction 37 that indicated that both the tunnelling current and force between two atomic-scale metal contacts scale similarly with distance (recently revisited by Jelinek et al.38). Using the dynamic force microscopy technique, Ternes et al. effectively probed the interplay between short-range forces and conductance under ultrahigh-vacuum conditions at liquid helium temperatures. As illustrated in Fig. 2a, the tunnel-ling current through the gap between the metallic AFM probe and the substrate, and the force on the cantilever were recorded, and both were found to decay exponentially with increasing distance with nearly the same decay constant. Although an exponential decay in current with distance is easily explained by considering an orbital overlap of the tip and sample wavefunctions through a tunnel barrier using Simmons’ model 39, the exponential decay in the short-range forces indicated that perhaps the same orbital controlled the interatomic short-range forces (Fig. 2b).Using such dynamic force microscopy techniques, research-ers have also studied, under ultrahigh-vacuum conditions, forces and conductance across junctions with diatomic adsorbates such as CO (refs 40,41) and more recently with fullerenes 42, address-ing the interplay between electronic transport, binding ener-getics and structural evolution. I n one such experiment, Tautz and coworkers 43 have demonstrated simultaneous conduct-ance and stiffness measurements during the lifting of a PTCDA (3,4,9,10-perylene-tetracarboxylicacid-dianhydride) molecule from a Ag(111) substrate using the dynamic mode method with an Ag-covered tungsten AFM tip. The authors were able to follow the lifting process (Fig. 2c,d) monitoring the junction stiffness as the molecule was peeled off the surface to yield a vertically bound molecule, which could also be characterized electronically to determine the conductance through the vertical metal–molecule–metal junction with an idealized geometry. These measurements were supported by force field-based model calculations (Fig. 2c and dashed black line in Fig. 2d), presenting a way to correlate local geometry to the electronic transport.Extending the work from metal point contacts, ambient meas-urements of force and conductance across single-molecule junc-tions have been carried out using the static AFM mode 33. These measurements allow correlation of the bond rupture forces with the chemistry of the linker group and molecular backbone. Single-molecule junctions are formed between a Au-metal sub-strate and a Au-coated cantilever in an environment of molecules. Measurements of current through the junction under an applied bias determine conductance, while simultaneous measurements of cantilever deflection relate to the force applied across the junction as shown in Fig. 2e. Although measurements of current throughzF zyxCantileverIVabConductance G (G 0)1 2 3Tip–sample distance d (Å)S h o r t -r a n g e f o r c e F z (n N )10−310−210−11110−110−210−3e10−410−210C o n d u c t a n c e (G 0)Displacement86420Force (nN)0.5 nm420−2F o r c e (n N )−0.4−0.200.20.4Displacement (nm)SSfIncreasing rupture forcegc(iv)(i)(iii)(ii)Low HighCounts d9630−3d F /d z (n N n m −1)(i)(iv)(iii)(ii)A p p r o a chL i ft i n g110−210−4G (2e 2/h )2051510z (Å)H 2NNH 2H 2NNH 2NNFigure 2 | Simultaneous measurements of electronic transport and mechanics. a , A conducting AFM set-up with a stiff probe (shown schematically) enabled the atomic-resolution imaging of a Pt adsorbate on a Pt(111) surface (tan colour topography), before the simultaneous measurement of interatomic forces and currents. F z , short-range force. b , Semilogarithmic plot of tunnelling conductance and F z measured over the Pt atom. A similar decay constant for current and force as a function of interatomic distance is seen. The blue dashed lines are exponential fits to the data. c , Structural snapshots showing a molecular mechanics simulation of a PTCDA molecule held between a Ag substrate and tip (read right to left). It shows the evolution of the Ag–PTCDA–Ag molecular junction as a function of tip–surface distance. d , Upper panel shows experimental stiffness (d F /d z ) measurements during the lifting process performed with a conducting AFM. The calculated values from the simulation are overlaid (dashed black line). Lower panel shows simultaneously measured conductance (G ). e , Simultaneously measured conductance (red) and force (blue) measurements showing evolution of a molecular junction as a function of junction elongation. A Au point contact is first formed, followed by the formation of a single-molecule junction, which then ruptures on further elongation. f , A two-dimensional histogram of thousands of single-molecule junctionrupture events (for 1,4-bis(methyl sulphide) butane; inset), constructed by redefining the rupture location as the zero displacement point. The most frequently measured rupture force is the drop in force (shown by the double-headed arrow) at the rupture location in the statistically averaged force trace (overlaid black curve). g , Beyond the expected dependence on the terminal group, the rupture force is also sensitive to the molecular backbone, highlighting the interplay between chemical structure and mechanics. In the case of nitrogen-terminated molecules, rupture force increases fromaromatic amines to aliphatic amines and the highest rupture force is for molecules with pyridyl moieties. Figure reproduced with permission from: a ,b , ref. 36, © 2011 APS; c ,d , ref. 43, © 2011 APS.DOI: 10.1038/NNANO.2013.91such junctions are easily accomplished using standard instru-mentation, measurements of forces with high resolution are not straightforward. This is because a rather stiff cantilever (with a typical spring constant of ~50 N m−1) is typically required to break the Au point contact that is first formed between the tip and sub-strate, before the molecular junctions are created. The force reso-lution is then limited by the smallest deflection of the cantilever that can be measured. With a custom-designed system24 our group has achieved a cantilever displacement resolution of ~2 pm (com-pare with Au atomic diameter of ~280 pm) using an optical detec-tion scheme, allowing the force noise floor of the AFM set-up to be as low as 0.1 nN even with these stiff cantilevers (Fig. 2e). With this system, and a novel analysis technique using two-dimensional force–displacement histograms as illustrated in Fig. 2f, we have been able to systematically probe the influence of the chemical linker group44,45 and the molecular backbone46 on single-molecule junction rupture force as illustrated in Fig. 2g.Significant future opportunities with force measurements exist for investigations that go beyond characterizations of the junc-tion rupture force. In two independent reports, one by our group47 and another by Wagner et al.48, force measurements were used to quantitatively measure the contribution of van der Waals interac-tions at the single-molecule level. Wagner et al. used the stiffness data from the lifting of PTCDA molecules on a Au(111) surface, and fitted it to the stiffness calculated from model potentials to estimate the contribution of the various interactions between the molecule and the surface48. By measuring force and conductance across single 4,4’-bipyridine molecules attached to Au electrodes, we were able to directly quantify the contribution of van der Waals interactions to single-molecule-junction stiffness and rupture force47. These experimental measurements can help benchmark the several theoretical frameworks currently under development aiming to reliably capture van der Waals interactions at metal/ organic interfaces due to their importance in diverse areas includ-ing catalysis, electronic devices and self-assembly.In most of the experiments mentioned thus far, the measured forces were typically used as a secondary probe of junction prop-erties, instead relying on the junction conductance as the primary signature for the formation of the junction. However, as is the case in large biological molecules49, forces measured across single-mol-ecule junctions can also provide the primary signature, thereby making it possible to characterize non-conducting molecules that nonetheless do form junctions. Furthermore, molecules pos-sess many internal degrees of motion (including vibrations and rotations) that can directly influence the electronic transport50, and the measurement of forces with such molecules can open up new avenues for mechanochemistry51. This potential of using force measurements to elucidate the fundamentals of electronic transport and binding interactions at the single-molecule level is prompting new activity in this area of research52–54. Optoelectronics and optical spectroscopyAddressing optical properties and understanding their influence on electronic transport in individual molecular-scale devices, col-lectively referred to as ‘molecular optoelectronics’, is an area with potentially important applications55. However, the fundamental mismatch between the optical (typically, approximately at the micrometre scale) and molecular-length scales has historically presented a barrier to experimental investigations. The motiva-tions for single-molecule optoelectronic studies are twofold: first, optical spectroscopies (especially Raman spectroscopy) could lead to a significantly better characterization of the local junction structure. The nanostructured metallic electrodes used to real-ize single-molecule junctions are coincidentally some of the best candidates for local field enhancement due to plasmons (coupled excitations of surface electrons and incident photons). This there-fore provides an excellent opportunity for understanding the interaction of plasmons with molecules at the nanoscale. Second, controlling the electronic transport properties using light as an external stimulus has long been sought as an attractive alternative to a molecular-scale field-effect transistor.Two independent groups have recently demonstrated simulta-neous optical and electrical measurements on molecular junctions with the aim of providing structural information using an optical probe. First, Ward et al.56 used Au nanogaps formed by electromi-gration57 to create molecular junctions with a few molecules. They then irradiated these junctions with a laser operating at a wavelength that is close to the plasmon resonance of these Au nanogaps to observe a Raman signal attributable to the molecules58 (Fig. 3a). As shown in Fig. 3b, they observed correlations between the intensity of the Raman features and magnitude of the junction conductance, providing direct evidence that Raman signatures could be used to identify junction structures. They later extended this experimental approach to estimate vibrational and electronic heating in molecu-lar junctions59. For this work, they measured the ratio of the Raman Stokes and anti-Stokes intensities, which were then related to the junction temperature as a function of the applied bias voltage. They found that the anti-Stokes intensity changed with bias voltage while the Stokes intensity remained constant, indicating that the effective temperature of the Raman-active mode was affected by passing cur-rent through the junction60. Interestingly, Ward et al. found that the vibrational mode temperatures exceeded several hundred kelvin, whereas earlier work by Tao and co-workers, who used models for junction rupture derived from biomolecule research, had indicated a much smaller value (~10 K) for electronic heating61. Whether this high temperature determined from the ratio of the anti-Stokes to Stokes intensities indicates that the electronic temperature is also similarly elevated is still being debated55, however, one can definitely conclude that such measurements under a high bias (few hundred millivolts) are clearly in a non-equilibrium transport regime, and much more research needs to be performed to understand the details of electronic heating.Concurrently, Liu et al.62 used the STM-based break-junction technique8 and combined this with Raman spectroscopy to per-form simultaneous conductance and Raman measurements on single-molecule junctions formed between a Au STM tip and a Au(111) substrate. They coupled a laser to a molecular junction as shown in Fig. 3c with a 4,4’-bipyridine molecule bridging the STM tip (top) and the substrate (bottom). Pyridines show clear surface-enhanced Raman signatures on metal58, and 4,4’-bipy-ridine is known to form single-molecule junctions in the STM break-junction set-up8,15. Similar to the study of Ward et al.56, Liu et al.62 found that conducting molecular junctions had a Raman signature that was distinct from the broken molecu-lar junctions. Furthermore, the authors studied the spectra of 4,4’-bipyridine at different bias voltages, ranging from 10 to 800 mV, and reported a reversible splitting of the 1,609 cm–1 peak (Fig. 3d). Because this Raman signature is due to a ring-stretching mode, they interpreted this splitting as arising from the break-ing of the degeneracy between the rings connected to the source and drain electrodes at high biases (Fig. 3c). Innovative experi-ments such as these have demonstrated that there is new physics to be learned through optical probing of molecular junctions, and are initiating further interest in understanding the effect of local structure and vibrational effects on electronic transport63. Experiments that probe electroluminescence — photon emis-sion induced by a tunnelling current — in these types of molec-ular junction can also offer insight into structure–conductance correlations. Ho and co-workers have demonstrated simultaneous measurement of differential conductance and photon emissionDOI: 10.1038/NNANO.2013.91from individual molecules at a submolecular-length scale using an STM 64,65. Instead of depositing molecules directly on a metal sur-face, they used an insulating layer to decouple the molecule from the metal 64,65 (Fig. 3e). This critical factor, combined with the vac-uum gap with the STM tip, ensures that the metal electrodes do not quench the radiated photons, and therefore the emitted photons carry molecular fingerprints. Indeed, the experimental observation of molecular electroluminescence of C 60 monolayers on Au(110) by Berndt et al.66 was later attributed to plasmon-mediated emission of the metallic electrodes, indirectly modulated by the molecule 67. The challenge of finding the correct insulator–molecule combination and performing the experiments at low temperature makes electro-luminescence relatively uncommon compared with the numerous Raman studies; however, progress is being made on both theoretical and experimental fronts to understand and exploit emission pro-cesses in single-molecule junctions 68.Beyond measurements of the Raman spectra of molecular junctions, light could be used to control transport in junctions formed with photochromic molecular backbones that occur in two (or more) stable and optically accessible states. Some common examples include azobenzene derivatives, which occur in a cis or trans form, as well as diarylene compounds that can be switched between a conducting conjugated form and a non-conducting cross-conjugated form 69. Experiments probing the conductance changes in molecular devices formed with such compounds have been reviewed in depth elsewhere 70,71. However, in the single-mol-ecule context, there are relatively few examples of optical modula-tion of conductance. To a large extent, this is due to the fact that although many molecular systems are known to switch reliably in solution, contact to metallic electrodes can dramatically alter switching properties, presenting a significant challenge to experi-ments at the single-molecule level.Two recent experiments have attempted to overcome this chal-lenge and have probed conductance changes in single-molecule junctions while simultaneously illuminating the junctions with visible light 72,73. Battacharyya et al.72 used a porphyrin-C 60 ‘dyad’ molecule deposited on an indium tin oxide (I TO) substrate to demonstrate the light-induced creation of an excited-state mol-ecule with a different conductance. The unconventional transpar-ent ITO electrode was chosen to provide optical access while also acting as a conducting electrode. The porphyrin segment of the molecule was the chromophore, whereas the C 60 segment served as the electron acceptor. The authors found, surprisingly, that the charge-separated molecule had a much longer lifetime on ITO than in solution. I n the break-junction experiments, the illuminated junctions showed a conductance feature that was absent without1 μm Raman shift (cm –1)1,609 cm –1(–)Source 1,609 cm–1Drain (+)Low voltage High voltageMgPNiAl(110)STM tip (Ag)VacuumThin alumina 1.4 1.5 1.6 1.701020 3040200400Photon energy (eV)3.00 V 2.90 V 2.80 V 2.70 V 2.60 V2.55 V 2.50 VP h o t o n c o u n t s (a .u .)888 829 777731Wavelength (nm)Oxideacebd f Raman intensity (CCD counts)1,5001,00050000.40.30.20.10.01,590 cm −11,498 cm −1d I /d V (μA V –1)1,609 cm –11,631 cm–11 μm1 μmTime (s)Figure 3 | Simultaneous studies of optical effects and transport. a , A scanning electron micrograph (left) of an electromigrated Au junction (light contrast) lithographically defined on a Si substrate (darker contrast). The nanoscale gap results in a ‘hot spot’ where Raman signals are enhanced, as seen in the optical image (right). b , Simultaneously measured differential conductance (black, bottom) and amplitudes of two molecular Raman features (blue traces, middle and top) as a function of time in a p-mercaptoaniline junction. c , Schematic representation of a bipyridine junction formed between a Au STM tip and a Au(111) substrate, where the tip enhancement from the atomically sharp STM tip results in a large enhancement of the Raman signal. d , The measured Raman spectra as a function of applied bias indicate breaking of symmetry in the bound molecule. e , Schematic representation of a Mg-porphyrin (MgP) molecule sandwiched between a Ag STM tip and a NiAl(110) substrate. A subnanometre alumina insulating layer is a key factor in measuring the molecular electroluminescence, which would otherwise be overshadowed by the metallic substrate. f , Emission spectra of a single Mg-porphyrin molecule as a function of bias voltage (data is vertically offset for clarity). At high biases, individual vibronic peaks become apparent. The spectra from a bare oxide layer (grey) is shown for reference. Figure reproduced with permission from: a ,b , ref. 56, © 2008 ACS; c ,d , ref. 62, © 2011 NPG; e ,f , ref. 65, © 2008 APS.DOI: 10.1038/NNANO.2013.91light, which the authors assigned to the charge-separated state. In another approach, Lara-Avila et al.73 have reported investigations of a dihydroazulene (DHA)/vinylheptafulvene (VHF) molecule switch, utilizing nanofabricated gaps to perform measurements of Au–DHA–Au single-molecule junctions. Based on the early work by Daub et al.74, DHA was known to switch to VHF under illumina-tion by 353-nm light and switch back to DHA thermally. In three of four devices, the authors observed a conductance increase after irradiating for a period of 10–20 min. In one of those three devices, they also reported reversible switching after a few hours. Although much more detailed studies are needed to establish the reliability of optical single-molecule switches, these experiments provide new platforms to perform in situ investigations of single-molecule con-ductance under illumination.We conclude this section by briefly pointing to the rapid pro-gress occurring in the development of optical probes at the single-molecule scale, which is also motivated by the tremendous interest in plasmonics and nano-optics. As mentioned previously, light can be coupled into nanoscale gaps, overcoming experimental chal-lenges such as local heating. Banerjee et al.75 have exploited these concepts to demonstrate plasmon-induced electrical conduction in a network of Au nanoparticles that form metal–molecule–metal junctions between them (Fig. 3f). Although not a single-molecule measurement, the control of molecular conductance through plas-monic coupling can benefit tremendously from the diverse set of new concepts under development in this area, such as nanofabri-cated transmission lines 76, adiabatic focusing of surface plasmons, electrical excitation of surface plasmons and nanoparticle optical antennas. The convergence of plasmonics and electronics at the fundamental atomic- and molecular-length scales can be expected to provide significant opportunities for new studies of light–mat-ter interaction 77–79.Thermoelectric characterization of single-molecule junctions Understanding the electronic response to heating in a single-mole-cule junction is not only of basic scientific interest; it can have a tech-nological impact by improving our ability to convert wasted heat into usable electricity through the thermoelectric effect, where a temper-ature difference between two sides of a device induces a voltage drop across it. The efficiency of such a device depends on its thermopower (S ; also known as the Seebeck coefficient), its electric and thermal conductivity 80. Strategies for increasing the efficiency of thermoelec-tric devices turned to nanoscale devices a decade ago 81, where one could, in principle, increase the electronic conductivity and ther-mopower while independently minimizing the thermal conductiv-ity 82. This has motivated the need for a fundamental understandingof thermoelectrics at the single-molecule level 83, and in particular, the measurement of the Seebeck coefficient in such junctions. The Seebeck coefficient, S = −(ΔV /ΔT )|I = 0, determines the magnitude of the voltage developed across the junction when a temperature dif-ference ΔT is applied, as illustrated in Fig. 4a; this definition holds both for bulk devices and for single-molecule junctions. If an addi-tional external voltage ΔV exists across the junction, then the cur-rent I through the junction is given by I = G ΔV + GS ΔT where G is the junction conductance 83. Transport through molecular junctions is typically in the coherent regime where conductance, which is pro-portional to the electronic transmission probability, is given by the Landauer formula 84. The Seebeck coefficient at zero applied voltage is then related to the derivative of the transmission probability at the metal Fermi energy (in the off-resonance limit), with, S = −∂E ∂ln( (E ))π2k 2B T E 3ewhere k B is the Boltzmann constant, e is the charge of the electron, T (E ) is the energy-dependent transmission function and E F is the Fermi energy. When the transmission function for the junction takes on a simple Lorentzian form 85, and transport is in the off-resonance limit, the sign of S can be used to deduce the nature of charge carriers in molecular junctions. In such cases, a positive S results from hole transport through the highest occupied molecu-lar orbital (HOMO) whereas a negative S indicates electron trans-port through the lowest unoccupied molecular orbital (LUMO). Much work has been performed on investigating the low-bias con-ductance of molecular junctions using a variety of chemical linker groups 86–89, which, in principle, can change the nature of charge carriers through the junction. Molecular junction thermopower measurements can thus be used to determine the nature of charge carriers, correlating the backbone and linker chemistry with elec-tronic aspects of conduction.Experimental measurements of S and conductance were first reported by Ludoph and Ruitenbeek 90 in Au point contacts at liquid helium temperatures. This work provided a method to carry out thermoelectric measurements on molecular junctions. Reddy et al.91 implemented a similar technique in the STM geome-try to measure S of molecular junctions, although due to electronic limitations, they could not simultaneously measure conductance. They used thiol-terminated oligophenyls with 1-3-benzene units and found a positive S that increased with increasing molecular length (Fig. 4b). These pioneering experiments allowed the iden-tification of hole transport through thiol-terminated molecular junctions, while also introducing a method to quantify S from statistically significant datasets. Following this work, our group measured the thermoelectric current through a molecular junction held under zero external bias voltage to determine S and the con-ductance through the same junction at a finite bias to determine G (ref. 92). Our measurements showed that amine-terminated mol-ecules conduct through the HOMO whereas pyridine-terminatedmolecules conduct through the LUMO (Fig. 4b) in good agree-ment with calculations.S has now been measured on a variety of molecular junctionsdemonstrating both hole and electron transport 91–95. Although the magnitude of S measured for molecular junctions is small, the fact that it can be tuned by changing the molecule makes these experiments interesting from a scientific perspective. Future work on the measurements of the thermal conductance at the molecu-lar level can be expected to establish a relation between chemical structure and the figure of merit, which defines the thermoelec-tric efficiencies of such devices and determines their viability for practical applications.SpintronicsWhereas most of the explorations of metal–molecule–metal junc-tions have been motivated by the quest for the ultimate minia-turization of electronic components, the quantum-mechanical aspects that are inherent to single-molecule junctions are inspir-ing entirely new device concepts with no classical analogues. In this section, we review recent experiments that demonstrate the capability of controlling spin (both electronic and nuclear) in single-molecule devices 96. The early experiments by the groups of McEuen and Ralph 97, and Park 98 in 2002 explored spin-depend-ent transport and the Kondo effect in single-molecule devices, and this topic has recently been reviewed in detail by Scott and Natelson 99. Here, we focus on new types of experiment that are attempting to control the spin state of a molecule or of the elec-trons flowing through the molecular junction. These studies aremotivated by the appeal of miniaturization and coherent trans-port afforded by molecular electronics, combined with the great potential of spintronics to create devices for data storage and quan-tum computation 100. The experimental platforms for conducting DOI: 10.1038/NNANO.2013.91。

a r X i v :0705.3936v 2 [n u c l -t h ] 11 N o v 2007Λ∗-hypernuclei in phenomenological nuclear forcesA.Arai,M.Oka and S.YasuiDepartment of Physics,Tokyo Institute of Technology,Tokyo 152-8551,JapanFebruary 1,2008Abstract The Λ∗-hypernuclei,which are bound states of Λ(1405)and nuclei,are discussed as a possible interpretation of the ¯K -nuclei.The Bonn and Nijmegen potentials are extended and used as a phenomenological potential between Λ∗and N .The K -exchange potential is also considered in the Λ∗and N interaction.The two-body (Λ∗N )and three-body (Λ∗NN )systems are solved by a variational method.It is shown that the spin and isospin of the ground states are assigned as Λ∗N (S =1,I =1/2)and Λ∗NN (S =3/2,I =0),respectively.The binding energies of the Λ∗-hypernuclei are discussed in comparison with experiment.1Introduction Possibility of kaon-nuclear bound states is of great interest in hadron and nuclear physics.A deeply bound kaonic nuclear state with a relatively small decay width was recently predicted and further studied in literatures.[1,2,3,4]Possible high density matter caused by the strong kaon attractive force is also a subject of heated discussion.[1,2,3,4]In order to find such exotic states,several experimental searches have been carried out.[5,6]FINUDA collaboration has reported an observation of bound ppK −state at the binding energy,115MeV.[6]Interpretation of the peak found in this experiment is yet under discussion,while some advanced calculations,such as a 3-body calculation `a la Fadeev equation with ¯KN−πΣchannel coupling,have been performed.[7,8]Other approaches include an interpretation as a nine-quark state studied in the MIT bag model [9],and a kaon absorption process between two nucleons [10].While the coming J-PARC facility will certainly answer to the question whether such deeply bound kaonic nuclear states exist,we need to study such a system in more extensive views.The purpose of this paper is to give a new interpretation to the “kaon-nucleus”states.We consider Λ∗-hypernuclear states.Λ∗is the lowest negative parity baryon with mass around 1405MeV and strangeness −1.This baryon is in many senses unique.For instance,it is below any other non-strange baryons with negative parity and is isolated1not forming an octet in SU(3).If it is assumed to be a p-wave baryon with spin1/2,then it requires a large spin-orbit splitting.Its uniqueness has attracted a lot of attention and various exotic views ofΛ∗have been proposed.We here do not consider specific composition ofΛ∗,but simply assume that it belongs to aflavor singlet representation,and thus isolated.Our claim is that the so-called kaon-nuclear bound states can be interpreted as a bound state ofΛ∗in a nucleus,or Λ∗-hypernucleus.Then the FINUDA observation is regarded as a two-body bound state ofΛ∗and N,whose binding energy is rger systems,such as strange tribaryon can be aΛ∗NN system and so on.In this paper,we construct a model ofΛ∗N interaction according to the one-boson exchange model and consider the two-body(Λ∗N)and three-body(Λ∗NN)systems.As the NN interaction,we choose the same one-boson exchange model,the Bonn potential [11]and the Nijmegen soft-core potential models[12,13].We extend these models to the Λ∗N systems and the K-exchange is also included in the same context betweenΛ∗and N.We then solve the two-body and three-body Schr¨o dinger equations using a variational method.The content of the paper is as follows.In Section2,we construct the potential model forΛ∗N system.The features of the model in the context of possible quantum num-bers of theΛ∗-nuclear states are also given.In Section3,the numerical results for the binding energies of two-body and three-body bound states are shown.In Section4,some discussion on the present results are given.The conclusion is given in Section5.2ModelThe phenomenological nuclear force is quite successful in studying nuclear systems.In order to setup the model for theΛ∗-hypernuclei,we discuss the phenomenologicalΛ∗N interaction.The nuclear forces among hyperons are not yet established,although a great amount of studies have been performed experimentally and theoretically so far.In general, it is considered that the interaction between baryons is described by exchange of mesons supplemented by phenomenological short-range repulsion.In the present study,based on the one-boson exchange picture,we extend the phenomenological nuclear forces between the NN pair to theΛ∗N pair interaction.In the Bonn[11]and Nijmegen potentials[12, 13],the scalar(σ,a0),pseudoscalar(π,η)and vector(ω,ρ)bosons are exchanged between the NN pair.The explicit equations and the parameter sets for the Bonn potential are shown in Appendix.By considering the SU(3)symmetry,we assume that these potentials are also applied to theΛ∗N pair.Here the isovector mesons(a0,π,ρ)are irrelevant to theΛ∗N interaction,since theΛ∗is isosinglet.Instead the K(and¯K)meson comes into the game in the exchange processΛ∗N→NΛ∗.Then,theΛ∗N potential is given by the sumVΛ∗N=Vσ+Vη+Vω+V K.(1) Theσ-exchange potential,Vσ,is the most attractive force,and theω-exchange potential, Vω,plays an essential role for the spin determination,as we see below.In the following,2we investigate several possibilities of theΛ∗N potential.In the extended phenomenological nuclear forces in theΛ∗N pair,we have some unknown parameters;the coupling constants gΛ∗NM(for M meson)and the momen-tum cutoffs,which appear in the form factors.Among them,theΛ∗N¯K vertex con-stant,gΛ∗N¯K,is determined by the SU(3)relation from the observed decay width of the Λ∗→Σπchannel.TheΛ∗Σπcoupling g2Λ∗Σπ/4π=0.064is obtained from the decay widthΓ(Λ∗→Σπ)=50.0±2.0MeV.Assuming thatΛ∗is a purely SU(3)singlet state, the SU(3)isoscalar factor,(Λ∗)=(N¯KΣπΛηΞK)=18(23−1−2)1/2,(2)predicts the coupling constant g2Λ∗NK/4π=g2Λ∗Σπ/4π=0.064.It should be noted that this coupling constant is much smaller than the NNσcoupling constant g2NNσ/4π=7.78. Therefore,the K meson plays only a minor role in theΛ∗N bound state as seen in the numerical calculation later.Here,the K-exchange has to be treated carefully.BecauseΛ∗lies close to the¯KN threshold,the kaon may have four momentum,qµ,near the on-mass-shell values.Then the kaon propagator can be enhanced strongly.In order to include this effect,we replace the kaon mass by an effective mass.Assuming that the baryons are static,we obtain the finite energy transfer,q0≃MΛ∗−M N,and the effective mass,˜m K≡ m2K−(MΛ∗−M N)2=171MeV/c2.(3) This replacement indeed enhances theΛ∗N potential more strongly than the case ofΛN orΣN potential.This is because that theΛ∗N¯K coupling is a scalar coupling and the effect of˜m K comes only in the range of the potential.In contrast,a p-wave coupling gives rise to an extra˜m2K factor,which significantly suppresses its effect on theΛN orΣN potential.As a result,the K-exchange potential forΛ∗N pair is given by the coupling constant g2Λ∗NK/4π=0.064and the effective mass˜m K in Eq.(10)in Appendix.Yet the other parameters in theΛ∗N interaction are notfixed due to lack of experi-mental information.In the present study,we treat the gΛ∗Λ∗σas a free parameter,since the results happen to be most sensitive to theσ-exchange.We further assume that the other parameters are the same as the NN interaction for simplicity.Here it is important for the later discussions to prospect the properties of theΛ∗N interaction.Let us see the spin-dependence of theΛ∗N interaction.The spin-spin inter-action is induced by theη,K andωmesons.We consider theω-exchange force,since theωmeson coupling is much stronger than theηand K meson couplings.The static ω-exchange potential is written in the momentum space asVω( k2)=−C( σ1· σ2) k2k2+m2ω( σ1· σ2),(4)3with a positive constant C,and theωmeson mass mω.In the real space,thefirst term is a delta function type,while the second term is the standard Yukawa potential.With a form factor introduced,the delta type potential becomes afinite range potential.For the Yukawa type potential,the spin singlet is more attractive than the spin triplet.On the other hand,for the delta type potential,the spin triplet is more attractive than the spin ually the delta type potential at short distance plays a minor role due to strong repulsive core of the NN force,hence the Yukawa potential is much more important.However,this is not the case for theΛ∗N potential.There,at short distances, the delta function is overwhelming to the Yukawa type potential.Consequently,in the Λ∗N potential,the spin triplet potential is more attractive at short distance than the spin singlet one.Now we look at the possible quantum states of theΛ∗-hypernuclei,which are obtained in the spin and isospin combinations.It is assumed that all the particles occupy the lowest-energy s-orbit.SinceΛ∗has spin S=1/2and isospin I=0,theΛ∗N system has two states of spin and isospin;(S=0,I=1/2)and(S=1,I=1/2).On the other hand,theΛ∗NN system takes the isosinglet and isotriplet states.The isosinglet state has the spin S=1/2and S=3/2states,where the NN pair is isosinglet and spin triplet.The isotriplet state has spin S=1/2,where the NN pair is isotriplet and spin singlet.Therefore the possible quantum numbers of the s-waveΛ∗NN system are (S=1/2,I=0),(S=3/2,I=0)and(S=1/2,I=1).Here,we prospect the quantum numbers of the ground states of theΛ∗-hypernuclei by using the spin dependence of theω-exchange at short distance.It is expected that the ground state of the two body system(Λ∗N)is spin triplet.In order to understand the ground state of the three body system(Λ∗NN),the three possible states,(S=1/2,I=0), (S=3/2,I=0)and(S=1/2,I=1),are expanded in terms of theΛ∗N pair;|Λ∗(NN)S=1;S=1/2,I=0 =1332|(Λ∗N)S=0N ,(7) for the isotriplet state.Among these states,the spin triplet interaction is contained most strongly in the(S=3/2,I=0)state due to the largest value of coefficients of the (Λ∗N)S=1component.Therefore,it is expected that the(S=3/2,I=0)is the ground state for theΛ∗NN state.In the next section,it will be shown that the above analysis is consistent with the numerical results.The above conclusion of the spin of the two-body and three-body states is in strong contrast with the kaon bound state approach.There,the spin dependence is induced by the isospin dependence of¯KN interaction.For instance,in the¯KNN system,the (¯KN)I=0interaction is the driving force for the bound state.It is easy to see that the ¯K(NN)state contains more(¯KN)I=0component than¯K(NN)I=0,S=1state.Thus, I=1,S=0the ground state of¯KNN is expected to have S=0.4-400-2002000 0.2 0.4 0.60.8 1 1.2 1.4V (r ) [M e V ]r [fm]Figure 1:The Bonn potentials applied for Λ∗N pair in the S =0(dashed line)and S =1(solid line)channels for the coupling constants g Λ∗Λ∗σ/g NNσ=0.39.3Numerical ResultsIn this section,we discuss the numerical results for the two-body (Λ∗N )and three-body (Λ∗NN )systems by solving the Schr¨o dinger equation with the Bonn [11]and Nijmegen (SC89[12]and ESC04[13])potentials for the Λ∗N and NN interactions.For simplicity,derivative terms are not considered.We mainly show the results for the Bonn potential.The results do not differ qualitatively for the Nijmegen potentials with SC89and ESC04.3.1Two-body systemThe two-body system (Λ∗N )has the spin singlet (S =0,I =1/2)and triplet (S =1,I =1/2)states.In Fig.1,the Bonn potentials are plotted as functions of the relative distance r between Λ∗and N in the S =0and S =1channels for the coupling constant g Λ∗Λ∗σ/g NNσ=0.39.The S =1potential is strongly attractive at short distance (r <∼0.4fm),while the S =0one is repulsive.This difference is caused by the delta type interaction in the ωpotential (4),which is attractive for S =1and repulsive for S =0.It should be noted that the delta function is smeared to a finite range potential due to the form factor in the phenomenological nuclear forces.In order to see the contributions from individual mesons in the S =1Λ∗N potential,the components from the σ,ωand η-exchange potentials and the additional K -exchange potentials are plotted in Fig.2.The σ-exchange potential is attractive at medium range (r >∼0.4fm),while the ωpotential is repulsive.However,the ωpotential is strongly attractive at shorter distances (r <∼0.4fm)due to the delta function in (4).Hence,the sum of them employs a deeply attractive potential at short distance,which is one of the characteristic properties in the Λ∗N potential.The contributions from the ηand K mesons are relatively weak.Here we investigate the on-shell effect of the K -exchange.As we have already dis-5-400-2002000 0.2 0.4 0.60.8 1 1.2 1.4V (r ) [M e V ]r [fm]ωηKFigure 2:The components from the σ,ω,ηand K mesons in the Bonn potential between Λ∗N pair for S =1for g Λ∗Λ∗σ/g NNσ=0.39.The Bonn potential is indicated by the solid line.cussed,the K meson propagating between the Λ∗and N may have large energy and thus can be close to the on-mass-shell kinematics.Such K -exchange might be largely enhanced.We treat this effect as an effective mass ˜m K defined in Eq.(3).In Fig.3,the K -exchange potential is plotted for the bare mass m K =495MeV (solid line)and for the reduced mass ˜m K =171MeV (dashed line).As the effective K meson mass gets smaller,both the potential strength and the range increase.However,this enhancement has little significance to the binding energy of the Λ∗N ,since the K -exchange potential is still weaker than the σand ω-exchange potentials.Solving the Schr¨o dinger equation for the Λ∗N with the Bonn potential,we obtain the binding energy and the wave function of the Λ∗N bound states.As a set of trial wave functions,we use a linear combination of Gaussian functions;ψ(r )=n i =1a i e −αi r 2,(8)where a i are the normalization constants,αi the width parameters,and r the relative distance between Λ∗and N .As a result,the binding energy strongly depends on the coupling constant g Λ∗Λ∗σ/g NNσas shown in Fig.4.The σpotential plays an essential role for the formation of the bound state.Indeed,without the Λ∗Λ∗σcoupling,the bound state does not exist.For S =1,the Λ∗N is bound for the coupling constant g Λ∗Λ∗σ/g NNσ>∼0.37.At g Λ∗Λ∗σ/g NNσ=0.39,the binding energy reaches 88MeV.The observed binding energy 115MeV of ppK −state reported by the FINUDA collaboration [6]is interpreted as 88MeV for the binding energy of the Λ∗N state.For S =0,a stronger coupling g Λ∗Λ∗σ/g NNσ>∼0.98is required for the bound state.The binding energy 88MeV is obtained at g Λ∗Λ∗σ/g NNσ=1.01.In the current experimental status,the quantum number of the bound state is not yet known.Our result suggests an S =1bound state rather than S =0.The probability density of the Λ∗N with S =1is plotted in Fig.5.The wave function is very compact as compared with the nucleon size,suggesting that the60 510152025 303540450 0.5 11.5 2V (r ) [M e V ]r [fm]m K =495MeV m K =171MeV Figure 3:The K -exchange component in the Λ∗N potential.The solid line is for the bare mass m K =495MeV and the dashed line is for the reduced mass ˜m K =171MeV.See the text.obtained bound state is produced mainly by the short-range ωattraction assisted by the medium-range σ-exchange that cancels the medium range ωrepulsion.So far,we have discussed the results of the Bonn potential.In order to check the model dependence,we also apply different types of the nuclear force.In particular,as the short-range part of the interaction is important,the mechanisms of short-range NN repulsion is in question.The Bonn potential acquires the short-range repulsion mainly from the ω-exchange potential,while the other models,such as the Nijmegen potential,introduce a new component for the repulsion.The pomeron exchange is represented in the Nijmegen potential by a gaussian potential of short range.In the case of NN interaction,it is strong enough to expel the wave functions away from the center.When we apply the same repulsion to Λ∗N system,we find that the short-range attraction from ω-exchange may still make a bound state,if the σ-exchange attraction at the medium range is strong enough.However,compared with the Bonn potential,the extra pomeronic repulsion for Λ∗N makes the system less bound.That means that the required g Λ∗Λ∗σcoupling constant becomes larger.We employ the Nijmegen potential of the versions SC89and ESC04.It is found that SC89requires the minimal coupling constants g Λ∗Λ∗σ/g NNσ=0.77and g Λ∗Λ∗σ/g NNσ=0.98to form a Λ∗N bound state for S =1and S =0,respectively.The binding energy (88MeV)reported by the FINUDA group is obtained by setting g Λ∗Λ∗σ/g NNσ=0.832for S =1and g Λ∗Λ∗σ/g NNσ=1.128for S =0.The probability density is pushed out from the center as compared with the Bonn potential.This is because the range of the Λ∗N potential is longer than that from the Bonn potential.The result of ESC04is qualitatively the same as that of SC89.Only difference is that absolute values of the σ,ωand pomeron exchange potentials in ESC04are smaller than those in SC89.Then,the minimum coupling constants of Λ∗Λ∗σbecome larger,and the experimental binding energy is reproduced at g Λ∗Λ∗σ/g NNσ=1.118for S =1and g Λ∗Λ∗σ/g NNσ=1.398for S =0.70.360.370.380.390.4g Λ∗Λ∗σ/g NN σ020406080100120B i n d i n g E n e r g y [M e V ](S=1, I=1/2)1 1.1 1.2g Λ∗Λ∗σ/g NN σ(S=0, I=1/2)Figure 4:The binding energy of the Λ∗N as functions of the coupling constant g Λ∗Λ∗σ/g NNσfor the Bonn potential.The solid line is for S =1and the dashed line for S =0.0 0.2 0.4 0.60.8 1 1.2 1.4D i s t r i b u t i o nr [fm]Figure 5:The probability density of the bound Λ∗N system with S =1and g Λ∗Λ∗σ/g NNσ=0.39.The binding energy is 88MeV.The horizontal axis is the relative distance between Λ∗and N .8Table1:The ratios of the coupling constants gΛ∗Λ∗σ/g NNσrequired to obtain the binding energy,88MeV,of theΛ∗N two-body system.Λ∗N Nijmegen(SC89) B.E.[MeV]0.39 1.1181.1 1.398The results above discussed are summarized in Table1.The ratios of coupling con-stants gΛ∗Λ∗σ/g NNσthat correspond to the experimental observation in the FINUDA collaboration are listed.The smaller value indicates that the system can form a bound state more easily.Wefind that in all cases that the coupling constant in S=1bound state is smaller than that in S=0one in the Bonn and Nijmegen potentials.Therefore, we expect that the lowest-energy bound state is the S=1state.3.2Three-body systemNow we discuss the three-body system(Λ∗NN).The binding energies and wave functions are obtained variationally by solving the Schr¨o dinger equation by using the phenomeno-logical NN andΛ∗N potentials.The trial wave function is given byψ(r,r′)= i,j a ij e−αi r2e−βi r′2(9)with a ij are the normalization constants,αi andβi the width parameters,and r and r′the Jacobi coordinates forΛ∗NN.Then we obtain the binding energies as func-tions of the coupling constant gΛ∗Λ∗σ/g NNσfor the(S=3/2,I=0),(S=1/2,I=1) and(S=1/2,I=0)states indicated by solid,long-dashed,short-dashed lines,respec-tively,in Fig.6.Wefind that a bound(S=3/2,I=0)state can be formed only for gΛ∗Λ∗σ/g NNσ>∼0.406,while the(S=1/2,I=1)and(S=1/2,I=0)bound states re-quire stronger coupling constant.Therefore our analysis shows the(S=3/2,I=0)state as the ground state of theΛ∗NN system.Note that forΛ∗N pair in the S=1channel is more attractive than the S=0channel.From theΛ∗NN states written as combinations of(Λ∗N)S=1N and(Λ∗N)S=0N as Eq.(6)and(7),one sees that the(S=3/2,I=0) state contains more the(Λ∗N)S=1component.Through the discussions given above,it is clear that the short-distance behavior of the nuclear potential plays an important role.We recall that at short distance the smeared delta interaction in theω-exchange inΛ∗N interaction overwhelms the Yukawa type potential,and it induces a strong attractive potential for S=1.This is contrary to the long range behavior of theω-exchange potential;repulsive for S=1and attractive for S=0.The strong attractive potential at short distance is also seen in the three-body system.It is directly observed from the probability densities ofΛ∗N andΛ∗NN,which concentrate at short distances.90.40.410.420.43g Λ∗Λ∗σ/g NN σ020406080100120140160B i n d i n g E n e r g y [M e V ](S=3/2, I=0)0.640.650.660.67 g Λ∗Λ∗σ/g NN σ(S=1/2, I=1)0.90.920.94g Λ∗Λ∗σ/g NN σ(S=1/2, I=0)Figure 6:The binding energy of the Λ∗NN as functions of the coupling constant g Λ∗Λ∗σ/g NNσfor the Bonn potential.The solid,long-dashed and short-dashed lines indicate the (S =3/2,I =0),(S =1/2,I =1)and (S =1/2,I =0)states,respectively.Table 2:The ratios of the coupling constants g Λ∗Λ∗σ/g NNσrequired to obtain the binding energy,167MeV,of the Λ∗NN three-body system.Λ∗NNNijmegen (SC89) B.E.[MeV]0.4241.2170.64 1.325104DiscussionWe here compare our results with some other studies for the same system.In the original Akaishi-Yamazaki picture[1,2,3],the binding energy of ppnK−is shown to be larger than ppK−.Furthermore,it is indicated that¯K-nuclei can be stable for larger baryon numbers.However,this is not the case for theΛ∗-hypernuclei.Let us compare the minimum coupling constants for the two-body(Λ∗N)and three-body(Λ∗NN)bound systems.For example,for the Bonn potential,the coupling gΛ∗Λ∗σ/g NNσ=0.37gives a bound state ofΛ∗N with(S=1,I=1/2),while gΛ∗Λ∗σ/g NNσ=0.406is required for Λ∗NN with(S=3/2,I=0).Therefore,the present framework leads to the conclusion that theΛ∗N two-body state is bound more easily than theΛ∗NN three-body state.This is qualitatively the same for the Nijmegen potentials with SC89and ESC04.Furthermore, it may be expected that the theΛ∗-hypernuclei becomes unbound as the baryon number increases.This picture is in opposite direction as compared with the¯K-nucleus bound states.[1,2,3]A recent proposal[15]of interpreting the binding of¯K in terms of“migration”of¯K is in fact quite similar to the K-exchange part of our picture.The difference between their analysis and ours is mainly in the coupling strengths.We determine the coupling constant so as to reproduce the relatively narrow width ofΛ∗,while Ref.[15]employs the coupling to reproduce the binding energy ofΛ∗as a¯KN bound state.A further analysis on this difference will be needed.A comment is in order for the dependence of the result on the coupling constant in the phenomenological nuclear potentials forΛ∗N.The coupling constant gΛ∗Λ∗σis taken as a free parameter,and the couplings for the other mesons arefixed to the same value as the NN interaction.Theσpotential is the most attractive force among them,and theωpotential is the second strongest force.Therefore,as a further step,it is interesting to alter the coupling strength ofΛ∗Λ∗ω.However,concerning the spin of the bound state,the result will not be changed as long as theω-exchange potential is attractive at short-range.This is because that the spin of theΛ∗N pair is almost determined by the ω-exchange potential,not by the other mesons.Lastly,we note that in the present framework only the meson exchange potential has been considered as interaction betweenΛ∗and N.However,the resulting bound states are compact where the substructure of the baryons should be important.In the present analysis,we have introduced the form factors representing the baryon structure.It may be important to consider the quark structure of baryons explicitly so that the short range baryon-baryon interactions are correctly taken into account.[14]This subject is left for a future work.5ConclusionThe possibility of theΛ∗-hypernuclei is discussed by considering theΛ∗as a compound state in the kaonic-nuclei.The two-body(Λ∗N)and three-body(Λ∗NN)systems are investigated by using phenomenological nuclear forces.Based on the one-boson exchange11picture,the Bonn and Nijmegen(SC89and ESC04)potentials are extended to theΛ∗N interaction.The K-exchange is also included in the extended nuclear forces.The gΛ∗N¯K coupling is determined from the experimental value of the decay width of theΛ∗.Only Λ∗Λ∗σcoupling constant is left as a free parameter with the other parametersfixed.As a result,for the two-body system(Λ∗N),an appropriateΛ∗Λ∗σcoupling constant reproduces the binding energy of the ppK−which is comparable to the value reported by the FINUDA collaboration.The most stable state in theΛ∗N bound state is the S=1state.For the three-body system(Λ∗NN),the(S=3/2,I=0)state is the most stable state.These results are understood by the fact that theω-exchange potential is strongly attractive at short distances for S=1channel rather than in S=0.There the K-exchange plays only a minor pared with theΛ∗N andΛ∗NN,the minimum coupling constant gΛ∗Λ∗σin the former is smaller than that of the latter.Therefore,the Λ∗N state is more easily produced in comparison with theΛ∗NN state.Our conclusion is qualitatively the same for the Bonn and Nijmegen(SC89and ESC04)potentials.In the present study,Λ∗is considered as a stable particle.In reality,however,multi-channel decays are open forΛ∗.The decay toΣπis the main source ofΛ∗free decay, while the in-medium decaysΛ∗N→ΛN,ΣN are new and interesting.In our discussion, the boundΛ∗N andΛ∗NN states are very compact objects.Therefore,the conversion width toΛN orΣN may be modified.These subjects are closely related to experimental researches in DAΦNE and J-PARC.Further studied are left for future works. AcknowledgementThis work is supported by a Grant-in-Aid for Scientific Research for Priority Areas,MEXT (Ministry of Education,Culture,Sports,Science and Technology)with No.17070002. AThe Bonn potential used in the present paper is explicitly shown in a coordinate formalism [11].In the followings,m is the mass of the meson,g ij and f ij are coupling constants in the reaction process,1+2→3+4with the baryon mass M1and M2.The parameter set is summarized in Table3.The pseudoscalar type potential is given byV ps(m,r)=g13g244M1M2m 14π 1−m24π g13g24 1+m24MM1+f13g24m216M2M1M2 φ(mr) 12Table3:The parameter set of the Bonn potential.(The superscript a and b denote the NN pairs with isospin T=1and T=0,respectively.)π138.0314.9 1.3η548.83 1.5ρ7690.95 5.8 1.3ω782.620 1.5a0983 2.6713 2.0σ550a7.7823a 2.0715b16.2061b 2.04M1M2 g13g24+g13f24M2M+f13f24M1M28M1M22x,(13)χ(x)= m x+3r2− σ1· σ2,(15) with the propagating meson mass m.When the exchanged meson has isospin,τ1·τ2is multiplied.The form factor is introduced for each meson by replacing asVα(m,r)→Vα(m,r)−Λ22−m2Λ22−Λ12Vα(Λ2,r),(16)whereΛ1=Λ+ǫ,Λ2=Λ−ǫ,(17) withǫ/Λ≪1,such asǫ≈10MeV.References[1]Y.Akaishi and T.Yamazaki,Phys.Rev.C65(2002),044005.[2]A.Dot´e,H.Horiuchi,Y.Akaishi and T.Yamazaki,Phys.Lett.B590(2004),51.13[3]A.Dot´e,H.Horiuchi,Y.Akaishi and T.Yamazaki,Phys.Rev.C70(2004),044313.[4]A.Dot´e and W.Weise,arXiv:nucl-th/0701050.[5]T.Suzuki et al.,Phys.Lett.B597(2004),263.[6]M.Agnello et al.(FINUDA Collaboration),Phys.Rev.Lett.94(2005),212303.[7]N.V.Shevchenko,A.Gal and J.Mareˇs,Phys.Rev.Lett.98(2007),082301.[8]Y.Ikeda and T.Sato,arXiv:nucl-th/0701001and arXiv:0704.1978[nucl-th].[9]Y.Maezawa,T.Hatsuda and S.Sasaki,Prog.Theor.Phys.114(2005),317.[10]V.K.Magas,E.Oset,A.Ramos and H.Toki,Phys.Rev.C74(2006),025206.[11]R.Machleidt,K.Holinde and Ch.Elster,Phys.Rep.149(1987),1.[12]M.M.Nagels,T.A.Rijken and J.J.de Swart,Phys.Rev.D17(1978),768.[13]Th.A.Rijken,Phys.Rev.C73(2006),044007.[14]M.Oka and K.Yazaki,Phys.Lett.B90,41(1980).[15]T.Yamazaki and Y.Akaishi,Phys.Rev.C76(2007),045201.14。