# Condensed Matter Seminars2010 â€“ 2011

Where: HZ 116
When: Mondays at 4:00 p.m.

 September 13, 2010 COOPERATIVE RECOMBINATION (SUPERFLUORESCENCE) OF A DENSE ELECTRON-HOLE PLASMA IN A HIGH MAGNETIC FIELD Prof. Alexey Belyanin Texas A&M University  Superfluorescence is one of the very few quantum phenomena in which a macroscopic ensemble of initially incoherent quantum oscillators demonstrates cooperative, coherent behavior due to an efficient self-phasing. It places a fundamental upper limit on the rate of radiative decay in any excited system and has been observed in gases and crystals doped with active impurities. An analog of superfluorescence in semiconductors would be a cooperative recombination of non-equilibrium carriers. However, in semiconductors, ultrafast carrier scattering tends to destroy optical coherence. As a result, superfluorescence in semiconductors remained elusive for many years, despite a significant amount of experimental effort. Here, we present a theoretical study on the light emission properties of ultrahigh-density magnetized electron-hole plasmas in semiconductor quantum wells created via intense femtosecond excitation. We show that strong quantum confinement and carrier degeneracy in this system can suppress decoherence processes and enable superfluorescence. In recent experiments conducted at the National High Magnetic Field Lab, several key signatures of superfluorescence were observed, in agreement with our predictions. These include ultrafast depopulation of Landau levels accompanied by picosecond bursts of radiation and the transition in the band-edge photoluminescence from omnidirectional amplified spontaneous emission to a randomly-directed but highly-collimated output. Interestingly, the system maintains excitonic behavior up to extremely high carrier densities well above the Mott transition density. September 14, 2010 SU(N) Magnetism with Cold Atoms and Chiral Spin Liquids Prof. Victor Gurarie University of Colorado, Boulder  Certain cold atoms, namely the alkaline earth-like atoms whose electronic degrees of freedom are decoupled from their nuclear spin, can be thought of as quantum particles with an SU(N)-symmetric spin. These have recently been cooled to quantum degeneracy in the laboratories around the world, including one at Rice University. A new world of SU(N) physics has thus become accessible to experiment, including that described by the SU(N) Hubbard model in various dimensions as well as many others. We show that the Mott insulator of such cold atoms is a SU(N) symmetric antiferromagnet of the type not commonly studied in the literature. We further show that in 2 dimensions, this antiferromagnet is a chiral spin liquid, a long sought-after topological state of magnets, with fractional and non-Abelian excitations. September 27, 2010 Computational Studies of Models for Manganites and Pnictides Prof. Elbio Dagotto University of Tennessee and Oak Ridge National Lab  The study of strongly correlated electronic materials provides several examples where complex behaviors emerge from systems where the interaction rules, such as the model Hamiltonians, appear to be deceptively simple. In this presentation, the results of computational studies of models for the Mn-oxides with the colossal magnetoresistance (CMR), the manganites, and for the novel superconductors based on Fe, the pnictides, will be presented. In the manganite case, phase competition between a metal and an insulator, with spin, charge, and orbital order, is shown to induce CMR resistivity curves in good agreement with experiments. In the case of the pnictides, the multi-orbital Hubbard models studied at intermediate interaction strengths contain an interesting competition between several phases and pairing channels. Moreover, charge inhomogeneities also emerge in the pnictide systems within mean-field approximations. October 25, 2010 DIRAC POINT OF GRAPHENE: Parity Anomaly, Giant Flavor-Hall Effect and Correlated Electron States Dr. Dmitry Abanin Princeton University  Previous studies of non-relativistic two-dimensional electron systems have revealed a plethora of new phenomena, including quantum effects in transport, localization, and integer and fractional quantum Hall effects. Recently, the advent of graphene made it possible to study two-dimensional quasi-relativistic Dirac fermions in a condensed matter laboratory. The massless nature of quasiparticles alters, and in some cases completely changes fundamental physical properties, from tunneling to localization. I will discuss new transport properties and correlated states that arise near the Dirac point (low carrier density) as a result of the interplay between electron and hole states. In particular, I will show that the parity anomaly of Dirac fermions, known in high-energy physics, manifests itself in giant flavor (spin, valley) Hall effect that arises in low (non-quantizing) magnetic field. Experimentally, flavor-Hall effect (FHE) can be detected by measuring flavor accumulation. Another, all-electric way of probing neutral flavor currents arising from FHE is to employ nonlocal geometry, which allows one to filter out otherwise dominant Ohmic contribution to transport and distill more subtle flavor phenomena. The large nonlocal resistance of graphene has been recently observed by the Manchester group, indicating the presence of the long-range flavor currents resulting from FHE. At quantizing magnetic fields, the Dirac spectrum gives rise to a sequence of unusual Landau levels. The Coulomb interactions split these Landau levels into integer and fractional sub-levels with novel valley, spin, and charge properties. November 1, 2010 BROKEN ROTATIONAL SYMMETRY IN UNDERDOPED Fe-ARSENIDE SUPERCONDUCTORS Prof. Ian Fisher Stanford University  Common to the high Tc cuprates, superconductivity in the recently discovered Fe arsenides and related compounds is associated with suppression of an antiferromagnetic ground state. Evidence points to spin fluctuations associated with a partially nested Fermi surface playing an important role in the pairing mechanism. On the underdoped side of the phase diagram, in addition to the antiferromagnetic transition, the materials also suffer a phase transition that breaks the 4-fold rotational symmetry of the high-temperature crystal structure, this occurring at either the same or higher temperature then the Neel transition. Emerging evidence based on measurements of detwinned single crystals reveals a dramatic electronic anisotropy associated with this nematic transition. Aspects of this behavior are reminiscent of the pseudogap phase observed in the cuprates, suggestive of an additional commonality between the two classes of material. The distinct phase transition observed in the Fe pnictides allows a cleaner investigation of the origins, and consequences, of this effect. November 8, 2010 Tuning Magnetism in FeAs-Based Materials via Tetrahedral Structure Prof. Johnpierre Paglione University of Maryland  The discovery of high-temperature superconductivity in iron-based materials has motivated extensive studies of structural, magnetic and electronic properties of a wide variety of crystallographic systems with the common iron-arsenide building block. A common element of the FeAs-based intermetallic series of materials is the occurrence of a simultaneous structural and antiferromagnetic phase transition, which occurs at temperatures ranging between 130 K and 200 K in the Ba, Sr, and Ca-based "1-2-2" parent compounds. I will present a systematic study of the evolution of the magnetic and structural properties of solid solutions of these parent compounds, including Ba1-xSrxFe2As2 and Sr1-xCaxFe2As2, obtained through electrical transport, magnetic susceptibility, x-ray and neutron scattering measurements of single-crystalline samples. Our results reveal an intimate relationship to exist between the magnetic energy scale and the internal structure of the FeAs4 tetrahedra, even far above the magnetic transition temperatures. December 3, 2010 HIGH MAGNETIC FIELD EXPERIMENTS WITH MINI-COILS-APPLICATIONS FOR X-RAY AND NEUTRON DIFFRACTION Prof. Hiroyuki Nojiri Tohoku University, Japan  A high magnetic field is one of the most powerful parameters to control the magnetic states of matters. Possible experimental probes have been much limited, however, compared with those available at zero magnetic fields. One of the technical problems is a massive high magnetic field generator that requires complete re-arrangement of experimental setup. Recently, we have developed a compact high magnetic field generator for various types of experiments such as X-ray and neutron diffractions. Those experiments are usually performed in special facilities and the installations of high magnetic field generators are rather difficult. Moreover, the limited space of advanced spectrometer is often not compatible with those of standard superconducting and pulsed magnets. The concept of the mini-coil is to reduce the size and energy of coils by keeping the magnetic field intensity. In fact, a millisecond pulse magnetic fields of 30-50 T is easily achieved. In this talk, two recent examples of such applications are introduced. The first one is the determination of non-trivial magnetic structures in frustrated anti-ferromagnets by neutron diffractions. We show recent results on spinel compounds and on multiferroics. The second example is the X-ray magnetic circular dichroism to examine the orbital and element selective magnetic moment in the metamagnetic transitions of complex materials. Possible applications for table top optical spectrometer, adiabatic control of spins are also discussed. December 6, 2010 STRANDS OF SUPERCONDUCTIVITY AT THE NANOSCALE Prof. Paul Goldbart University of Illinois at Urbana-Champaign  Superconducting circuitry can now be fabricated at the nanoscale, e.g., by depositing suitable materials on to single molecules, such as DNA or carbon nanotubes. I shall discuss various themes that arise when superconductivity is explored in this new regime, including the thermal passage over and quantum tunneling through barriers by the superconducting condensate as a whole, as well as a strange, hormetic effect that magnetism can have on nanoscale superconductors. I shall describe nanoscale superconducting quantum interference devices, which are subtly sensitivity to magnetic fields and patterns of supercurrent -- features that hint at uses of superconducting nanocircuitry, e.g., in mapping quantum phase fields and testing for superconducting correlations in novel materials. I shall also mention settings in which superconducting nanosamples show a particular sensitivity to their geometry or topology, and shall conclude by touching on two emerging themes: the interplay between graphene and superconductivity, and what nanoprobes might be revealing about exotic forms of superconductivity. January 10, 2011 Magneto-optical and Magneto-electric Effects in Thin-film Topological Insulators Dr. Wang-Kong Tse University of Texas, Austin  Topological insulators are a novel class of materials theoretically predicted five years ago whose existence has subsequently been confirmed by experiments. These materials are insulating in the bulk but carry gapless surface states that are protected by time-reversal symmetry. Under broken time-reversal symmetry condition, the resulting half-quantized surface quantum Hall effect is a distinctive signature of the topological insulating state and can be probed via the associated magneto-optical response. In this talk, I will present a microscopic theory for the magneto-electric and magneto-optical effects of topological insulator thin films. When time-reversal symmetry is broken, our theory predicts that the low-frequency Faraday effect is quantized in integer multiples of the fine structure constant 1/137, whereas the Kerr effect exhibits a giant full-quarter rotation from the incident polarization plane. Possible experimental detection schemes for these effects will be discussed. January 24, 2011 Adventures in Crystal Growth of Intermetallics: Challenges and Opportunities Prof. Julia Y. Chan Louisiana State University  The central motivation of our research group has been and continues to be the discovery of novel, highly correlated electron systems. Ternary phases composed of lanthanides, transition metals, and main group elements present competing and/or complementary characteristics which show the potential for exciting physical properties and crystal chemistry. These systems are known to exhibit exotic properties such as: superconductivity, heavy fermion behavior, and unusual forms of magnetism. We employ flux-growth methods to discover and grow large single crystalline materials so that we may determine structure-physical property relationships. The ability to derive correlations between the crystal structure and physical properties such as magnetism and transport phenomena requires the growth of high quality single crystals. In this seminar, I will highly the synthesis, crystal structures, physical properties, and structure/property correlations of select group of interesting intermetallic compounds which were grown using Group 13, 14, or 15 fluxes. January 28, 2011 OBSERVATION OF SHOCK WAVES AND BEYOND LUTTINGER LIQUID PHYSICS IN COLD ATOMS Dr. Manas Kulkarni SUNY at Stony Brook  I will present our work with experimentalists at Duke, involving the study of collisions between two strongly interacting atomic Fermi gas clouds. They observed exotic nonlinear hydrodynamic behavior, distinguished by the formation of a very sharp and stable density peak as the clouds collide and subsequent evolution into a box-like shape. We model the nonlinear dynamics of these collisions using quasi-1D hydrodynamic equations. Our simulations of the time-dependent density profiles show near perfect agreement with the data and provide clear evidence of shock wave formation in this universal quantum hydrodynamic system. We argue that these experiments on strongly interacting Fermi gases form an ideal playground for studying out-of-equilibrium nonlinear hydrodynamics. I will then talk about nonlinear collective field theory for a two-component integrable model with inverse square interactions and spin-exchange. In this context, I will present several results such as spin-charge drag, gradient catastrophe, solitons all of which are hallmarks of physics beyond the luttinger liquid pradigm. January 31, 2011 NONCONVENTIONAL ODD DENOMINATOR FRACTIONAL QUANTUM HALL STATES IN THE SECOND LANDAU LEVEL Prof. Gabor Csathy Purdue University  The fractional quantum Hall states of the lowest Landau level can be accounted for by the elegant model of noninteracting composite fermions. Those of the second Landau level, however, appear to be different and continue to challenge our understanding. The most puzzling state is the one developed at the odd denominator Landau level filling factor 5/2, a state which is believed to arise from a p-type of pairing of the composite fermions described by the Pfaffian wavefunction and might support non-Abelian quasiparticles. An equally interesting and related problem is the origin of the odd denominator fractional quantum Hall states in the second Landau level. While at first sight these states could be part of the conventional composite fermion hierarchy, several recent theories suggest that they support generalized Pfaffian-like correlations instead. Recent progress in cooling electrons to about 5~mK allowed us to observe a new fractional state at filling factor 2+6/13. We find that the energy gap of this newly seen state and that of the 2+2/5 state deviate significantly from the predictions of the free composite fermion model. This observation constitutes a first evidence of the exotic origin of these states and is therefore an important milestone in our understanding of the perplexing physics of the second Landau level. February 7, 2011 Physics of Oxide Heterostructures Prof. Alex Demkov University of Texas, Austin  Multifunctional oxides (MO) are functional materials exhibiting more than one ferroic response. MO can store and release electrical, magnetic, and mechanical energy, which makes them useful as sensors, actuators, as well as memory elements. However, the coupling of even two ferroic responses i.e., switching of ferroelectric order leading to commensurate changes in magnetic order is typically very weak in bulk materials. On the other hand, multifunctional oxide structures, such as epitaxially grown heterojunctions and superlattices do provide a pathway to coupling, and enable ‘designer’ polarization, strain, and magnetization effects. When combined with monolithic integration with semiconductors this technology may lead to highly integrated oxide-based electronics. I will review our theoretical studies of several functional oxide structures. First I will discuss our calculations of the two-dimensional electron gas at the SrTiO3/LaAlO3 interface recently reported by Hwang and others (A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004)). We find that a complex balance of the crystal filed, Jahn-Teller effect, lattice dynamics and internal electric field results in the robust electrostatic doping for carefully chosen thickness of the polar oxide. I will then show how this electrostatic doping could be extended to a ferromagnetic oxide such as EuO. Because the conduction band in EuO is spin split, one can expect introducing the spin polarized charge at the interface of EuO with a polar oxide such as LaAlO3. To introduce more compex structures I will describe magnetoelectric coupling in tri-color superlattices comprised of a ferromagnetic metal, a normal metal and a ferroelectric. All calculations are done within density functional theory. To achieve correct band gap values for SrTiO3, EuO and YMnO3 we use the LDA+U approximation. For EuO we apply a Hubbard correction within the GGA (GGA+U) to the localized 4f states. If time permits I will discuss our experimental attempt to integrate LaCoO3 with Si and induce the ferromagnetic transition in LCO via strain induced by a buffer SrTiO3 layer. This work is supported by the Office of Naval Research under grants N000 14-06-1-0362 and N000 14-09-1-0908, the National Science Foundation under grant DMR-0548182, and Texas Advanced Computing Center. J.K. Lee and A.A. Demkov, “Charge origin and localization at the n-type SrTiO3/LaAlO3 interface”, Phys. Rev. B 78, 146839 (2008). T. Cai, Q. Niu, J.K. Lee, Na Sai, and A.A. Demkov, “Magnetoelectric Coupling and Electric Control of Magnetization in Ferromagnet-Ferroelectric-Metal Superlattices”, Phys. Rev. B 80, 140415(R) (2009). J.K. Lee, Na Sai, T. Cai, Q. Niu and A.A. Demkov, “Interfacial Magnetoelectric Coupling in Tri-component Superlattices”, Phys. Rev. B 81, 144425 (2010). J.K Lee, N. Sai, and A.A. Demkov, “Spin-polarized 2DEG through electrostatic doping in LaAlO3-EuO heterostructures”, Phys. Rev. B 82, 235305 (2010). A. Posadas, M. Berg, H. Seo, D.J. Smith, H. Celio, A.P. Kirk, D. Zhernokletov, R.M. Wallace, A. de Lozanne, and A.A. Demkov, “Strain-induced ferromagnetism in correlated oxide LaCoO3 epitaxially grown on Si (100)”, App. Phys. Lett. 98, 1 (2011). February 8, 2011 MAJORANA SPIN LIQUIDS ON THE TRIANGULAR LATTICE Dr. Rudro Biswas Harvard University  In this talk I will describe a new kind of spin liquid on the triangular lattice. This work is motivated by recent experiments on the dmit-131 compounds (Yamashita et al, Science, 2010) in which a spin liquid state has been observed which has a Fermi liquid-like thermal conductivity that increases linearly with temperature but exhibits no thermal Hall effect, unlike that expected from a theory of fermionic spinons. Our calculations show that the spin liquid state in our theory is characterized by spin one Majorana excitations with a Fermi surface composed of Fermi lines meeting at a Fermi point. At low temperatures the thermodynamic properties of our spin liquid vary as power laws of the temperature while the thermal transport varies linearly with it. This spin liquid state is also found to have zero thermal Hall effect. February 11, 2011 MAJORANA MODES AT THE ENDS OF SUPERCONDUCTOR VORTICES IN DOPED TOPOLOGICAL INSULATORS Dr. Pouyan Ghaemi University of California, Berkeley  Recent experiments have observed bulk superconductivity in doped topological insulators. In this talk I discuss that the vortex Majorana zero modes, previously predicted to occur when superconductivity is induced on the surface of topological insulators, survive even in the doped systems with metallic normal states. We find that Majorana zero modes indeed appear but only below a critical doping. The critical doping is associated with a topological phase transition of the vortex line, where it supports gapless excitations along its length. Generally, it is shown that the critical chemical potential depends only on the orientation of the vortex line, and a Berry phase property of the normal state Fermi surface. We use this criterion and available band structures to argue that n-doped Bi$_2$Te$_3$ under pressure supports vortex end Majorana modes, along with other materials candidates. Surprisingly, even topologically trivial band structures in spin orbit materials, when suitably doped, may lead to surface Majorana fermions. February 14, 2011 TRANSPORT IN NETWORKS OF JOSEPHSON JUNCTIONS Dr. Sergey Syzranov Ruhr University Bochum, Germany  We consider transport and superconductor-insulator transitions (SITs) in weakly disordered arrays of Josephson junctions and granulated superconductors. Their low-temperature properties are described by an effective Ginzburg-Landau-type action, which in addition to the superconductive order parameter contains a guage field that accounts for the intergranular Coulomb interactions. The RG analysis at $T=0$ show that the SIT is of the first order in all weakly disordered 3D and in realistic 2D arrays. The conductivity in the insulating phase comes from the motion of bosonic excitations, scattered by offset charges and irregularities in the array. At weak disorder we calculate the Drude-like conductivity and obtain weak localization corrections to it. At sufficiently low temperatures or strong disorder Anderson localization of Cooper pairs ensues, the conductivity is determined then by the variable-range hopping of bosons. At arbitrary disorder strength the conductivity in the insulating phase obeys the activation law $\propto\exp(-E_g/T)$ with the activation gap $E_g$ independent or nearly independent of the temperature. February 21, 2011 FERMIONIC SPINONS IN THE S=1/2 HEISENBERG ANTIFERROMAGNET MODEL ON KAGOME LATTICE Dr. Zhihao Hao Johns Hopkins University  Conventional magnetic systems display long range magnetic order in ground state. Their excitations are bosons with spin 1. Thanks to geometrical frustration and strong quantum fluctuations, S=1/2 antiferromagnet on kagome has no magnetic long range order in its ground state; the magnetic excitations are fermionic spinons with S=1/2. These spinons are topological defects and exchange-mediated attraction binds them into S=0 pairs. The motion of the bound states are strongly frustrated by the presence of valence bonds. A low energy S=1 excitation corresponds to breaking a bound state into a pair of spinons of total spin one at energy cost 0.06J. Experimental signatures of the bound states can be identified by inelastic neutron scattering on single crystal kagome compound ZnCu3(OH)6Cl2, the herbertsmithite. March 7, 2011 Interacting Electrons in 1D beyond the Luttinger-Liquid Limit - Transport and Relaxation Prof. Alex Levchenko Argonne National Lab  Over the past several decades the traditional framework for studying one-dimensional systems was provided by the Luttinger-liquid theory. It exploits an approximation of the linearized fermionic dispersion relation which makes this model exactly solvable. While being extremely fruitful in many cases an ideal Luttinger-liquid model, however, possesses certain deficiencies. For example, physically expected equilibration effects are completely absent in the Luttinger liquid. The effects of interactions show no sign in application to the dc transport coefficients. When relaxing on the linear spectrum approximation the Luttinger-liquid model provides new intriguing consequences of the interactions. The latter have direct implications on many experimental systems that include Quantum Hall Edge states driven out of equilibrium, or quantum wires where transport and themalization processes are probed by the momentum-resolved tunneling spectroscopy. In this seminar I would like to present recent results and progress in understanding the physics in one-dimension beyond the standard paradigm of the Luttinger-liquid model. March 8, 2011 TBA Dr. Jie Li Penn State University  TBA March 18, 2011 A SIGN OF CHANGE: Pinning Down the Order Parameter Symmetry of the Iron-based Superconductors Dr. Erez Berg Penn State University  The iron-based superconductors, discovered a couple of years ago, hold a great promise, from both a technological and a scientific point of view. With an unusually high maximum critical temperature exceeding 50K, second only to that of the so-called copper-oxide high-temperature superconductors, they are likely to belong to a class of "unconventional" superconductors, whose mechanism lies beyond the Bardeen-Cooper-Schrieffer theory of phonon-mediated pairing. A crucial step to unveil the mechanism of superconductivity in these materials is to understand the structure of their order parameter. In this talk, I will discuss theoretical investigations of the iron-based superconductors, which indicate an exotic order parameter with an internal sign change. Experimentally, detecting such a sign change poses a significant challenge. I will show how a recent intriguing experiment, which shows fractional flux quantization in a composite superconducting loop, can be intepreted as evidence for a sign-changing order parameter. March 28, 2011 ANISOTROPY OF NORMAL AND SUPERCONDUCTING STATES OF IRON ARSENIDE SUPERCONDUCTORS Prof. Makariy Tanatar Ames Lab  TBA April 4, 2011 Phase Transition and Crossover in Electron-Hole-Photon Systems Prof. Tetsuo Ogawa Osaka University, Japan  We introduce several aspects of exciton/electron-hole and electron-hole-photon physics, which have been an important field of optical condensed-matter physics. In particular, we stress optical responses and cooperative phenomena related to optically-excited states, called the photoinduced phase transitions (PIPTs). In other words, we explore coupled systems, where the fermionic (electronic, trionic) fields and the bosonic (photonic, phononic, excitonic, biexcitonic, polaritonic) fields are interacting with each other. We will show that the Coulomb interaction leads to the formation of various phases such as exciton gas, e-h plasma, e-h liquids, exciton Bose-Einstein condensate, and exciton-polariton condensate in a quasiequilibrium situation. These findings are useful not only for understanding composite many-body systems but also for designing new semiconductor lasers and/or new quantum light sources, leading to coherence-tunable light sources. April 18, 2011 VISUALIZATION OF NANOSCALE ELECTRONIC PHASES BY MICROWAVE MICROSCOPY Dr. Keji Lai Stanford University   The microscopic electronic properties of novel materials are of fundamental importance in nanoscale science and technology. Combining state-of-the-art micro-fabrication process, microwave engineering, and scanning probe platforms, we have developed a microwave microscope to resolve the local dielectric constant and conductivity at 1GHz down to the sub-micrometer length scale. Two experiments will be discussed here. First, narrow strips with either metallic or insulating screening properties are observed along edges of a 2D electron gas in the quantum Hall regime. The quantitative local conductivity information provides a complete microscopic description of the evolution through the bulk filling factor 2. Second, a mesoscopic glassy order is observed in a strained manganite thin film and the magnetic-field-induced metallic domains show a period of 100nm. This orientation-ordered percolating network indicates that the substrate-induced strain rather than the Coulomb interaction plays the dominant role in the phase separation. Improved design of the microscope is expected to further impact many areas of solid-state physics. April 25, 2011 QUANTUM PHASE TRANSITIONS IN HOLOGRAPHIC MODELS OF STRONGLY CORRELATED SYSTEMS Dr. Thomas Faulkner University of California at Santa Barbara   TBA May 2, 2011 EVOLUTION OF SUPERCONDUCTIVITY IN Fe-BASED SYSTEMS WITH DOPING Prof. Andrey Chubukov University of Wisconsin, Madison   We study the symmetry and the structure of the gap in Fe-based superconductors by decomposing the pairing interaction obtained in the RPA into s- and d-wave components and into contributions from scattering between different Fermi surfaces. We show that each interaction is well approximated by the lowest angular harmonics and use this simplification to analyze the origin of the attraction in s^\pm and d_{x^2-y^2} channels, the competition between s- and d-wave solutions, and the origin of superconductivity in heavily doped systems, when only electron or only hole pockets are present.

[ Back ]