Recent Research on Theoretical Superconductivity

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1 Recent Research on Theoretical Superconductivity Sungkit Yip Institute of Physics, Academia Sinica, Nankang 115, Taipei, Taiwan Sungkit Yip Although Taiwan is only a small country, there are many groups actively performing research on theoretical superconductivity. Below is a snapshot of these activities summarized roughly according to group leaders, though the assignment of some collaborative work to a particular group is somewhat arbitrary. Many physical quantities calculated in variational theory [3] have been shown in good agreement with experiments. 3. Strong correlation must be a primary reason [4, 5] of the incoherent spectral weights seen by ARPES and asymmetric di/dv observed by STM/STS. The AFMM and AFMM+SC states have been observed recently in multilayered cuprates. By examining various physical quantities, it was found that AFMM state has a stronger antiferromagnetic order but much weaker superconductivity than AFMM+SC one (not shown). The phase diagram is shown in Fig. 1 below. T. K. Lee (Academia Sinica, with some work in collaboration with C. T. Shih at Tung-Hai University, C. Y. Mou at Tsinghua University and C. M. Ho at Tamkang University) has been researching on high temperature superconductors using numerical techniques. The discovery of high-temperature superconductivity in cuprates has brought about an era of great excitement for condensed matter physics. Theoretically, it is well known that the t-j model captures the essence of the low-lying electronic excitations of the cuprates. In their recent work, they have investigated with the variational Monte Carlo method the extended t-j models. Based on this approach, several important results in high-t C cuprates have been obtained: 4. The presence of cluster glassy states with a random distribution of shortranged modulation is a natural outcome in the extended t-j models [6]. First of all, the AFMM, coexisting antiferromagnetic and superconducting orders (AFMM+SC), and d-wave RVB wave functions describing the ground states of the t-j-type models at different hole dopings were discussed [1, 2]. Second, a systematic comparison between the d-wave RVB theory and a broad spectrum of experimental data of low-lying physical properties in cuprates was made [3]. The doping dependence of both Fermi surface area and Fermi momentum along the nodal direction have been examined using the fitted quasi-hole dispersion [1]. The results are rather consistent with the same quantities observed in various high-t C cuprates, as shown in Fig The antiferromagnetic metallic (AFMM) wave function [1, 2] has been confirmed to be a good candidate for a metallic phase with longrange antiferromagnetic order observed by recent NMR experiments. 2. Low-energy excitations described by the d-wave resonating-valencebond (RVB) wave function can be well realized within a renormalized Bogoliubov quasi-particle picture. Fig. 1: Ground-state phase diagram for the t-t -t -J model with (t, t, J)/t=(-0.2, 0.1, 0.3). Dashed (Dashed-dotted) line represents schematic phase boundary of the AFMM and AFMM+SC (AFMM+SC and d-wave RVB) states. <M> represents the staggered magnetization and P d-wave for long-range pair-pair correlation for R max =2. The results are obtained on the lattice of size AAPPS Bulletin June 2008, Vol. 18, No. 3

2 Recent Research on Theoretical Superconductivity Fig. 2: Doping dependence of Fermi surface area x FS obtained from the fitted quasi-hole dispersion on the lattice system. Dashed line represents the relation x FS =x. Solid circles and triangles are numerical results for (t, t, J)/t=(-0.3, 0.2, 0.3) and (-0.1, 0.05, 0.3), respectively. Empty symbols represent experimental data extracted from ARPES, with Na-CCOC and LSCO. Inset: The doping dependence of the k F position at the nodal point. Fig. 3: The doping dependence of the average spectral weights Z ave - = (Σk Z kσ - )/N for removing an electron of d-wave RVB state obtained numerically for the lattice and by RMFT. Solid symbols are optimized results for (t, t, J)/t=(-0.3, 0.2, 0.3). Dashed line without data points represents results by RMFT. They have also found a qualitatively good agreement between the theory and experiments on the antinodal quasi-hole energy. The agreement on the doping dependence of these properties is remarkable [3]. Third, they have studied the spectral weights for adding and removing an electron for a d-wave RVB state [4, 5]. The average spectral weight for removing an electron deviates clearly from results obtained by renormalized mean-field theory (RMFT) in the low doping regime as shown in Fig. 3. More specifically, at this doping level the conductance-related quantity of d-wave RVB state computed exactly is particle-hole asymmetric below the gap energy, consistent qualitatively with what is seen in recent STM experiments. Finally, they propose an inhomogeneous state with randomly-distributed clusters of electronic density, spin density and pairing, which is an inherent nature of the extended t-j models [6]. This state has variational energy close to the uniform d-wave RVB state when one considers a repulsive hole-hole correlation. For the t /t less than -0.1, the energy difference between the random stripe and d-wave RVB states is rather small less than 0.3%. Due to this degeneracy, random distribution of defects in a system may likely stabilize the inhomogeneous state. The state also has nodes and linear density of states near the nodal region. Near the antinodal region, the spectral weight is almost completely suppressed [6]. Other theoretical aspects of high T C cuprates were also studied in the group of C.-Y. Mou (National Tsing-hua University). The spin-1 excitation (the so-called called 41 mev excitations) in high T C materials was one of foci. With T.-K. Lee (and visitor J.-X. Li), he has investigated data from angle resolved photo emission (ARPES) using the slave-boson approach. By including the resonance caused by the spin-1 excitation through random phase approximation, the peak-dip-hump structure observed in ARPES was reproduced. Furthermore, their numerical results showed excellent agreement with experiments with positions of peak-diphump showing the same quantitative trend versus doping as observed by experiments [7]. The effects of spin-1 excitation was also first considered on the tunneling spectroscopy of high T C superconductors [8] using the Keldysh formalism. Their results extended the conventional BTK approach and showed that the Andreev peak can coexist with the dip-hump structure. The tunneling spectroscopy with hybrid superconductor junctions was another focus of studies. With Shin-Tza Wu (now at National Chung-Cheng University), they developed a generalized image method that allows one to include the lattice effect of superconductors systematically [9, 10]. The merit of this new method lies in its simplicity and provides a useful criterion to judge when the so-called zero-bias conductance peak would arise. In addition, he has investigated the newly discovered superconductor hydrate cobaltate Na 0.35 CoO 2 yh 2 O. The experimentally found kink structure in the upper critical field H c2 was shown to be best fitted by assuming a phase transition from s-wave to d+id wave [11]. The result provides an important clue to the understanding of the superconductivity in this system. Ying-Jer Kao (National Taiwan University) has been investigating the physics of nematic phases in strongly correlated electronic systems. These systems are of relevance to a wide-range of materials, such as high temperature superconductor cuprates, the stripe phases in magnites with colossal magnetoresistance. The underlying mechanism of the formation of phases are still not clear, and whether they are competing or enhancing the effects of superconductivity or other orders AAPPS Bulletin June 2008, Vol. 18, No. 3 13

3 of interests are still under heated debate. In collaboration with Prof. Hae Young Kee at University of Toronto in Canada, he studied the spin and charge correlations in the nematic phase. The spin correlation result agrees very well with the neutron experiments in untwinned YBa 2 Cu 3 O 6+x [12]. In addition, they study the effects of Fermi surface fluctuations on the singleparticle lifetime, focusing on the non- Fermi liquid behavior near the diagonal nematic to Fermi liquid transition. The singular longitudinal fluctuations of the order parameter near this quantum critical point leads to a non-fermi liquid behavior over the whole Fermi surface except along the kx- and ky-directions [13]. W. C. Wu (National Taiwan Normal University) have theoretically studied the possible coexistence of an AF SDW order and the d-wave superconducting order parameter in high-t C electron-doped cuprate superconductors. Through the calculations of doping-dependent point-contact spectroscopy [14], inelastic neutron scattering [15], Raman scattering [16], and London penetration depth [17], consistent pictures are obtained when experimental results are compared. The results highly suggest that AF SDW order coexists with the d- wave pairing in electron-doped cuprate superconductors. The coexistence is more important in optimally doped samples, but less immaterial in the underdoped samples. This opens up a new question why the coexistence is suppressed in the underdoped regime. Unconventional superconductivity in ruthenate Sr 2 RuO 4 was explored by looking into the transport properties [18, 19]. Recent experiments showed that the order parameter of the unconventional superconductor Sr 2 RuO 4 has likely a spin-triplet f-wave symmetry. They studied ultrasonic absorption and thermal conductivity of superconducting Sr 2 RuO 4 and fit to the experimental data for various f-wave candidates. It was shown that only f x2 -y2-wave symmetry can account qualitatively for the transport data [19]. Baruch Rosenstein (National Chiao- Tung University) has been studying vortex matter in superconductors, in particular high T C cuprates. When an applied magnetic field exceeding the lower critical field H c1 (which is very small in high T C cuprates), it necessarily creates an inhomogeneity in form of an array of the Abrikosov flux lines (vortices) in the superconductor. Understanding of magnetic, thermodynamic and transport properties of the vortex matter of type II superconductors is an important long standing problem in physics of strongly correlated electrons for both pure science and applications. Thermal fluctuations are particularly important in these superconductors due to high T C and strong anisotropy. Their work started with the quantitative theory of the Abrikosov vortex lattice. Perturbation theory of the thermal fluctuations was abandoned in the seventies due to infrared divergencies, which sometimes were interpreted as non-existence of the lattice in the thermodynamic sense. Clear experiments in the nineties however showed otherwise. Rosenstein proved that all the divergencies cancel exactly [20] and this allowed him to develop a theory of lattice melting. He, in collaboration with postdoc D. P. Li (now faculty in Beijing University, China), constructed a quantitative theory of the vortex liquid state using several sophisticated field theoretical methods, including optimized perturbation theory and renormalization group [21]. It was applied to a number of systems like YBCO, BSCCO, and even good low T C materials (with Ph.D. student P. J. Lin) [22]. This work is by now generally accepted. In the last three years he concentrated on the thermodynamic, magnetic and transport properties of the vortex matter in type II superconductors subject to both the quenched disorder and thermal fluctuations. Disorder in these superconductors may in addition give rise to a vortex glass phase in this system. In this vortex glass phase, dynamics is irreversible and vortices are pinned. In collaboration with D. P. Li, they explained the unified order disorder line (vortex lattice melting line mainly due to thermal fluctuations and the second peak line due mainly to disorder) and the Kauzmann point [21]. Later (with participation of postdoc V. Zhuravlev) they found the glass line or irreversibility line in both the amorphous and crystalline phases and made a considerable progress in applying the theory to experiments [23]. In framework of the Gaussian approximation they have obtained analytical expressions for two-particle correlation functions and for singular part of fourparticle vertex functions. The first one provides us the equation for transition line from vortex liquid state to glass state as in static case as well as in presence of an electric field, whereas the second one determines the dependence of the resistivity at glass transition. Relying on theoretical results for melting and glass transitions they were able to quantitatively describe the phase diagram near mean field superconducting transition which is composed of three superconducting phases: vortex liquid (not pinned), homogeneous vortex glass (pinned) and crystalline Bragg glass (pinned). After a years long effort, Rosenstein and D. Li summarized the Ginzburg Landau theory of vortex matter in a review article (Rev. Mod. Phys.) and are writing a monograph. In addition, the group of Rosenstein also actively pursued several other topics in vortex matter physics. We mention here the theory of spontaneous vortex state in p-wave superconductors and structural transformations in the vortex lattice of the low and high T C superconductors. In p-wave superconductors the vortex state can exist even without applied magnetic field. The topic was motivated in part by experiments at NTHU and was studied with postdoc A. Knigavko [24]. He was also working on understanding the transitions between different vortex lattice structures for some time. Yeshurun s group recently measured a second peak line in high T C superconductor LaSCO, which was interpreted by Rosenstein et al. as a transition from the square lattice at high 14 AAPPS Bulletin June 2008, Vol. 18, No. 3

4 Recent Research on Theoretical Superconductivity fields to a rhombic lattice at low field. They also resolved recently [25] the long standing controversy initiated by the Eskildsen neutron scattering experiment on borocarbides by incorporating the disorder effect and proving that thermal fluctuations have nothing to do with it. The group of Yip (Academia Sinica) has been studying non-centrosymmetric superconductors. These superconductors lack inversion center so that the superconducting pairing states are automatically a combination of singlets and triplets, provided spin-orbit coupling exists. He has been investigating the unique properties of these superconductors expected due to this mixing, for example the magneto-electric effects where an external Zeeman field can induce a supercurrent (in addition to the usual vector potential) [26]. As a result, peculiar magnetic field and magnetization distributions were expected in these superconductors, both under the Meissner geometry as well as in the vortex phase. This has been studied for C4v symmetry (appropriate to CePt 3 Si) [27] as well as for O symmetry (appropriate to Li 2 (Pd 1-x Pt x ) 3 B) [28]. In addition, he (with S. T. Wu) has studied the properties of a point contact between two asymmetric superconductors. The differences from the case where the two superconductors have identical energy gaps, arising from the different bound-state spectrum in the two cases, were emphasized [29, 30]. REFERENCES [1] C. P. Chou, T. K. Lee, and C.-M. Ho, Low-energy spectra of the t-jtype models studied by variational approach, J. Mag. Mag. Mat. 310, 474 (2007). [2] C. T. Shih, J. J. Wu, Y. C. Chen, C. Y. Mou, C. P. Chou, R. Eder, and T. K. Lee, Antiferromagnetism and Superconductivity of the Two-Dimensional Extended t-j Model, Low Temp. Phys. 31, 757 (2005). [3] K.-Y. Yang, C. T. Shih, C. P. Chou, S. M. Huang, T. K. Lee, T. Xiang, and F. C. Zhang, Low energy physical properties of high-t C superconducting Cu oxides: A comparison between the resonating valence bond and experiments, Phys. Rev. B 73, (2006). [4] C. P. Chou, T. K. Lee, and C.-M. Ho, Spectral weights, d-wave pairing amplitudes, and particle-hole tunneling asymmetry of a strongly correlated superconductor, Phys. Rev. B 74, (2006). [5] C. P. Chou, T. K. Lee, and C.-M. Ho, J. Phys. Chem. Solids, in press. [6] C. P. Chou, T. K. Lee, and N. Fukushima, unpublished. [7] J. X. Li, C.-Y. Mou, and T. K. Lee, Consistent picture for resonanceneutron-peak and angle-resolved photoemission spectra in high-t C superconductors, Phys. Rev. B 62, 640 (2000). [8] C. L. Wu, C. -Y. Mou, and C. Chang, The effects of spin fluctuations on the tunneling spectroscopy in high- T C superconductors, Phys. Rev. B 63, (2001). [9] S. T. Wu and Chung-Yu Mou, Generalized Method of Image and the Tunneling Spectroscopy in High-T C Superconductors, Phys. Rev. 66, (2002). [10] S. T. Wu and Chung-Yu Mou, Zerobias conductance peak in tunneling spectroscopy of hybrid superconductor junctions, Phys. Rev. B 67, (2003). [11] Z. T. Kao, J. Y. Lin, and C. -Y. Mou, Pairing Symmetry and Upward Curvature of Upper Critical Field in Superconducting Na 0.35 CoO 2 yh 2 O, Phys. Rev. B 75, (2007). [12] Y.-J. Kao and H. Y. Kee, Anisotropic spin and charge excitations in superconductors: Signature of electronic nematic order, Phys. Rev. B 72, (2005). [13] Y.-J. Kao and H. Y. Kee, Theory of non-fermi liquid near a diagonal electronic nematic state on a square lattice, Phys. Rev. B 76, (2007). [14] C.-S. Liu and W. C. Wu, Theory of point-contact spectroscopy in electron-doped cuprate superconductors, Phys. Rev. B 76, R (2007). [15] C.-S. Liu and W. C. Wu, Gap-function symmetry and spin dynamics in electron-doped cuprate superconductors, Phys. Rev. B 76, (2007). [16] C.-S. Liu, H. G. Lou, W. C. Wu, and T. Xiang, Two-band model of Raman scattering on electron-doped high-t C superconductor, Phys. Rev. B 73, (2006). [17] K.-K. Voo and W. C. Wu, An alternative interpretation of the magnetic penetration depth data on Pr 2-x Ce x CuO 4-y and La 2-x Ce x CuO 4-y, Physica C 417, 103 (2005). [18] P. Lou, M. C. Chang, and W. C. Wu, Evidence for the coupling between gamm-band carriers and the incommensurate spin fluctuations in Sr 2 RuO 4, Phys. Rev. B 68, (2003). [19] W. C. Wu and R. Joynt, Transport and the Order Parameter of Superconducting Sr 2 RuO 4, Phys. Rev. B 64, R (2001). [20] B. Rosenstein, First principles theory of fluctuations in vortex liquids and solids, Phys. Rev. B60, 4762 (1999). [21] D. P. Li and B. Rosenstein, Precision calculation of magnetization and specific heat of vortex liquids and solids in type II superconductors, Phys. Rev. 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5 E. H. Brandt, and T. Tamegai, Equilibrium First-Order Melting and Second-Order Glass Transitions of the Vortex Matter in Bi 2 Sr 2 CaCu 2 O 8, Phys. Rev. Lett. 95, (2005); R. Lortz, F. Lin, N. Musolino, Y. Wang, A. Junod, B. Rosenstein, and N. Toyota, Thermal fluctuations and vortex melting in the Nb 3 Sn superconductor from high resolution specific heat measurements, Phys. Rev. B74, (2006). [23] H. Beidenkopf, T. Verdene, Y. Myasoedov, H. Shtrikman, E. Zeldov, B. Rosenstein, D. Li, and T. Tamegai, Interplay of Anisotropy and Disorder in the Doping-Dependent Melting and Glass Transitions of Vortices in Bi 2 Sr 2 CaCu 2 O 8+δ, Phys. Rev. Lett. 98, (2007); B. Rosenstein and V. Zhuravlev, Quantitative theory of transport in vortex matter of the type II superconductors in the presence of random pinning, Phys. Rev. B76, (2007). [24] A. Knigavko and B. Rosenstein, Spontaneous Vortex State and Ferromagnetic Behaviour of Type II p-wave Superconductor, Phys. Rev. B58, 9354 (1999); B. Rosenstein and A. Knigavko, Magnetic skyrmion lattices in heavy fermion superconductor UPt3, Phys. Rev. Lett. 82, 1261 (1999). [25] B. Rosenstein and A. Knigavko, Anisotropic peak effects due to structural phase transition in the vortex lattice, Phys. Rev. Let. 83, 844 (1999); D. Chang, C. Y. Mou, B. Rosenstein and C. L. Wu, An Interpretation of the Neutron Scattering Data on Flux Lattices of Superconductor, Phys. Rev. Let. 80, 145 (1998); B. Rosenstein, B. Ya. Shapiro, I. Shapiro, Y. Bruckental, A. Shaulov, and Y. Yeshurun, Peak effect and square-to-rhombic vortex lattice transition in La 2-x Sr x CuO 4, Phys. Rev. B72, (2005); D. P. Li, P.-J. Lin, B. Rosenstein, B. Ya. Shapiro, I. Shapiro, Influence of quenched disorder on the square-to-rhombohedral structural transformation of the vortex lattice of type-ii superconductors, Phys. Rev. B74, (2006). [26] S. K. Yip, Two Dimensional Superconductivity with Strong Spin-Orbit Interaction, Phys. Rev. B 65, (2002). [27] S. K. Yip, Magnetic Properties of a Superconductor with no Inversion Symmetry, J. Low Temp. Phys. 140, 67 (2005). [28] Chi-Ken Lu and Sungkit Yip, Signature of superconducting states in cubic crystal without inversion symmetry, Phys. Rev. B 77, (2008). [29] S. K. Yip, Supercurrent and Noise in Point Contacts between Two Different Superconductors, Phys. Rev. B 68, (2003). [30] S. T. Wu and S. K. Yip, AC Josephson effect in asymmetric superconducting point contact, Phys. Rev. B 70, (2004). 16 AAPPS Bulletin June 2008, Vol. 18, No. 3