Coexistence of Superconductivity and Density Waves in Quasi-Two-Dimensional Metals

Transcription

1 Coexistence of Superconductivity and Density Waves in Quasi-Two-Dimensional Metals D I S S E R T A T I O N zur Erlangung des Grades eines Doktors der Naturwissenschaften in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum von Jan-Peter Ismer aus Berlin - Schöneberg Bochum 2

2 . Gutachter: Prof. Dr. Ilya Eremin 2. Gutachter: PD Dr. Dirk Manske Datum der Disputation:

3 Name: Jan-Peter Ismer Versicherung gemäß 7 Abs. 2 Nr. 5 PromO 987 Hiermit versichere ich, dass ich meine Dissertation selbstständig angefertigt und verfasst und keine anderen als die angegebenen Hilfsmittel und Hilfen benutzt habe. Meine Dissertation habe ich in dieser oder ähnlicher Form noch bei keiner anderen Fakultät der Ruhr-Universität Bochum oder bei einer anderen Hochschule eingereicht. Bochum, den Unterschrift

4 Previous Publications of partial results of this Thesis: J.-P. Ismer, I. Eremin, and D.K. Morr, Dynamical spin susceptibility and the resonance peak in the pseudogap region of the underdoped cuprate superconductors, Physical Review B 73, J.-P. Ismer, I. Eremin, and D.K. Morr, Dynamical Spin Susceptibility in the underdoped cuprate superconductors: DDW state and influence of orthorhombicity, in Electron Correlation in New Materials and Nanosystems, Pages 87-97, K. Scharnberg and S. Kruchinin eds., 27 Springer J.-P. Ismer, I. Eremin, and D.K. Morr, Resonant spin excitations in high-tc cuprates: Influence of the pseudogap, Physica C J.-P. Ismer, I. Eremin, E. Rossi, and D.K. Morr, Magnetic Resonance in the Spin Excitation Spectrum of Electron-Doped Cuprate Superconductors, Physical Review Letters 99, D. Reznik, J.-P. Ismer, I. Eremin, L. Pintschovius, T. Wolf, M. Arai, Y. Endoh, T. Masui, and S. Tajima, Local-moment fluctuations in the optimally doped high-t c superconductor YBa 2 Cu 3 O 6.95, Physical Review B 78, J.-P. Ismer, I. Eremin, E. Rossi, D.K. Morr, and G. Blumberg, Theory of Multiband Superconductivity in Spin-Density-Wave Metals, Physical Review Letters 5, 373 2

5 Contents Introduction. Superconductivity Unconventional Superconductors Density Waves High-T c Cuprates Coexistence of Superconductivity and Density Waves Spin Susceptibility and Inelastic Neutron Scattering Overview Basic electronic states 4 2. Basic electronic structure Superconducting state Spin-Density Wave state Charge-Density Wave state D-Density Wave state Superconductivity on SDW background 4 3. Derivation of the gap equations s-wave superconductivity d-wave superconductivity Phase Diagrams Superconductivity on CDW backgrounds Superconductivity on a CDW background Derivation of the gap equations s-wave superconductivity d-wave superconductivity Phase Diagrams Superconductivity on a DDW background Derivation of the gap equations s-wave superconductivity i

6 CONTENTS ii d-wave superconductivity Phase Diagrams Spin Susceptibility in superconductors 9 5. Susceptibility in hole-doped cuprates Susceptibility in electron-doped cuprates Temperature evolution towards a low-temperature SDW phase Influence of a magnetic field Spin Susceptibility of DDW states 8 6. Pure DDW order DDW order coexisting with superconductivity Summary and Outlook 22 Appendix 27 A Superconductor in a Magnetic Field 28 A. Hamiltonian A.2 Bare Susceptibility A.3 RPA Susceptibility B Spin Susceptibility of DDW states 35 B. Pure DDW state B.2 DDW order coexisting with superconductivity C Summation of the Matsubara frequencies 44 Bibliography 46 Curriculum Vitae 56 Acknowledgements 57

7 Chapter Introduction. Superconductivity About years ago Heike Kamerlingh Onnes found during his seminal lowtemperature experiments surprisingly that mercury suddenly lost all of its electrical resistance when cooled below a certain critical temperature T c [6]: Superconductivity was discovered. In the following years it was established that superconductivity is a very common feature at low temperatures of the order of K. As seen in Fig.. most elements have been found to become superconducting at low temperatures. However, frequently it is necessary to apply pressure. Superconductors are characterized not only by the lack of electrical resistance. They were furthermore found to exhibit certain unusual properties [22]. Most notably among those is the Meissner-Ochsenfeld-effect [9]: A superconductor is a perfect diamagnet, i. e. a magnetic field is completely expelled from its bulk. However, for one kind of superconductors, the so-called type-i superconductors, there is a critical field H c above which the superconductivity is destroyed and the magnetic field can penetrate the bulk. In type-ii superconductors on the other hand there exists a lower critical field H c above which the magnetic flux lines can penetrate the bulk of the superconductor without destroying the bulk superconductivity and an upper critical field H c2 above which the superconductivity is destroyed []. Most elemental superconductors show type-i behavior, while most superconducting alloys and compounds but also some elements as vanadium exhibit type-ii behavior. Another striking property of superconductors is the opening of an energy gap of size 2 in the elementary excitation spectrum which can be observed directly and indirectly in countless ways, e. g. [42]. Furthermore, the transi-

8 CHAPTER. INTRODUCTION 2 Figure.: Elements known to have a superconducting phase [7]. Blue color indicates that superconductivity occurs at ambient pressure, while green elements have to be subject to high pressure in order to become superconductive. Carbon is yellow because ordinary bulk carbon is not superconducting, but carbon nanotubes can be [33]. tion temperature T c was found to depend on the isotope and thus the mass of the superconducting element [8]. Though later superconductors showing weak to no isotope effect were found [33]. From the beginning superconductivity attracted large efforts from theoretical physicists. But the first successful step on the way to a microscopic understanding of superconductivity was not made before 95 when Fröhlich realized [44] that quantized lattice vibrations, phonons, can lead to an attractive effective interaction between electrons in a metal. In 957, Bardeen, Cooper and Schrieffer developed their so-called BCS theory of superconductivity [3]. Together with the extensions made by Eliashberg [34] this theory could describe all superconductors known at that time. The BCS theory is based on the idea that electrons in a shell of width of the Debey frequency around the Fermi surface can overcome the Coulomb repulsion via the exchange of virtual phonons leading to an effective attractive interaction. Due to this interaction two electrons with opposite spin and momentum form bound so-called Cooper pairs which then build up

9 CHAPTER. INTRODUCTION 3 a superconducting phase-coherent condensate. This new state of matter is characterized by the gap function or order parameter. is non-zero in the vicinity of the Fermi surface only, where, within the original BCS theory, it is isotropic and finite..2 Unconventional Superconductors Soon after publication of the BCS theory, there was speculation about generalizations of the BCS theory. These generalizations concern, among others, the kind of Fermions forming the Cooper pairs, the pairing interaction, the spin structure of the Cooper pair, the momentum structure of the gap function, the number of bands crossing the Fermi surface and the coexistence of superconductivity with other forms of order [5, 4, 2, 3, 2, 26, 27]. A simple classification of the superconducting state can be derived from the wave-function of the Cooper pair. In the state proposed by BCS the mates of the pair are in a singlet state and thus have opposite spins. Therefore the spin part of the wave-function is antisymmetric with respect to particle exchange. This requires the momentum dependent part of the wave-function to be even, which is realized if the relative angular momentum of the mates is even. In the original BCS proposal the relative angular momentum was zero. In general, as long as the unit cell is symmetric under inversion, the Cooper pairs form either singlets with even relative angular momentum, i. e. s, d, g,... states, or triplets with odd relative angular momentum, i. e. p, f,... states. Within this thesis every superconducting state different from nodeless s-wave with one band crossing the Fermi surface will be referred to as unconventional superconductivity. Results of first experiments suggesting that at least some of these generalizations may in fact be realized in nature were published in 972 when superfluidity in 3 He was discovered [8]. Rapidly consensus was reached that in 3 He p-wave triplet pairing of the fermionic 3 He atoms is realized due to the exchange of spin fluctuations [79]. However, due to the charge-neutrality of its atoms 3 He is not able to transport a supercurrent and thus does not show perfect diamagnetism. In 979, one discovery stimulated new theoretical and experimental efforts on superconductivity. The compound CeCu 2 Si 2 was found to become superconducting below T c.5k [3]. This was the first member of a new class of materials, the so-called heavy-fermion superconductors, which have very diversified properties [5]. The common ground of the heavy-fermion superconductors is that they contain rare-earth or actinide atoms. Their f- electrons are strongly correlated and determine the properties of the quasi-

10 CHAPTER. INTRODUCTION 4 particles at the Fermi level. This leads to very large effective masses of the quasi-particles of the order of m e. The properties of the heavy-fermion superconductors include ferromagnetism, antiferromagnetism, unconventional superconductivity and even coexistence of magnetism and superconductivity. As an example for this variety of phases present in the heavy-fermion superconductors the pressure-temperature phase diagram of CeRhIn 5 is shown in the left panel of Fig..2. By now theoretically and to some extent even experimentally there is only little understanding of the heavy-fermion superconductors. Remarkably, recently heavy-fermion superconductors have been discovered which have non-centrosymmetric unit cells, like for example CePt 3 Si [5] and UIr [2]. In this case mixing of singlet and triplet pairing is expected to take place and the standard classification scheme is not applicable anymore [92]. In 98, superconductivity was found for the first time in an organic compound [65]. Later one found besides TMTSF 2 PF 6 additional other organic superconductors. They show a giant variety in their properties including reduced dimensionality, magnetism and unconventional superconductivity [77]. The superconductivity in the cuprates, which are the materials this thesis is focussed on, was found in 986. MgB 2 was found to become superconducting below 39 K in 2 [97]. It is regarded as a phonon-mediated s-wave superconductor with two-bands crossing the Fermi level. Due to the interband coupling the, for phonon mediated superconductivity, exceptionally high T c is made possible [23]. In 28, a new class of superconductors has been discovered with the alloy LaFeAsO.89 F. being the first known member [67]. It attracted giant attention to this new family of superconductors. These so-called iron pnictides are quasi-two-dimensional layered materials containing FePn Pn = pnictogen atom layers. They exhibit a rich phase diagram, containing phases with spin density wave antiferromagnetism and superconductivity as well as the coexistence of both [63, ]. As an example the phase diagram of BaFe x Co x 2 As 2 is shown in the right panel of Fig..2. It should be noted that there are also pnictide superconductors containing no iron. Remarkably, one of them was very recently found to show coexistence of charge density wave order and superconductivity [76]..3 Density Waves The concept of a Charge-Density Wave CDW was developed by Peierls [4]. In 93, he pointed out that a one-dimensional electron system coupled to the lattice is unstable with respect to the formation of a periodic

11 CHAPTER. INTRODUCTION 5 Figure.2: Left panel: Pressure-Temperature phase diagram of the heavyfermion superconductor CeRhIn 5 [7]. Right panel: Doping-Temperature phase diagram of the iron pnictide superconductor BaFe x Co x 2 As 2 [6]. modulation of the electron-density and an accompanying periodic distortion of the lattice. Interestingly, Fröhlich [45] and to some extent Bardeen [4] developed theories of superconductivity due to Charge-Density Wave formation. Though these theories were not confirmed when Charge-Density Wave phases were studied, it was found that Charge-Density Waves can transport electrical current by a movement of the Charge-Density Wave as a whole [93]. In general due to an enhanced tendency of Fermi surface nesting Charge- Density Waves form preferably in low dimensional materials. Examples for quasi one-dimensional materials known to have a Charge Density Wave phase are NbSe 3 and the so-called blue bronze K.3 MoO 3 [54]. A Spin-Density Wave SDW is conceptually similar to a CDW [9]. In this case the modulated Density is the Spin-Density, which leads to an antiferromagnetic behavior of such a state [4]. The most prominent example of a SDW material is probably Cr [27]. Theoretically there might exist more exotic forms of Density Waves DW [] as for example the id-wave Charge Density Wave DDW [2], where a pattern of circulating currents is formed. However, experimentally there is no unambiguous indication that a DDW exists in nature. It should be mentioned that mathematically Density Wave states can be regarded as a generalization of the superconducting state []. While in a standard s-wave superconductor electrons with opposite momentum and spin pair, in a Density Wave the pairing takes place between electrons and holes with momentum differing by the fixed momentum Q. In singlet Density Waves, which can be viewed as generalizations of the CDW, the electrons and holes participating in the pairing have the same spin, while in triplet Density Waves, which can be viewed as generalizations of the SDW, the

12 CHAPTER. INTRODUCTION 6 spin structure of the pairs can be more complex. The momentum Q is the momentum of the density modulation. If this momentum is related to the reciprocal lattice vectors by a simple rational number the Density Wave is referred to as commensurate, otherwise as incommensurate. Throughout this thesis only commensurate DW will be discussed..4 High-T c Cuprates In 986, Bednorz and Müller [6] while studying doped layered copper oxides found superconductivity in the La 2 x Ba x CuO 4 system with T c 3K. Within only a few months this by then highest known T c was not only confirmed by other groups, but even tripled when superconductivity with T c 9K was discovered in the YBa 2 Cu 3 O 6+y YBCO system [38]. For potential technological applications this was a giant breakthrough since it was now possible to use relative cheap liquid nitrogen as coolant. By now the highest known T c of 38 K is realized in Hg.8 Tl.2 Ba 2 Ca 2 Cu 3 O 8.33 [3] at ambient pressure and 64 K in HgBa 2 Ca 2 Cu 3 O 8+x under high pressure [48]. The vast majority of the known cuprate superconductors, including all aforementioned, are hole-doped. However, in 989 superconductivity was found in electron-doped cuprates, too [32]. Nd 2 x Ce x CuO 4 is the most commonly examined example of these n-type cuprates. The cuprate superconductors are based on ceramic perovskites and have a complex layered structure. It is illustrated in Fig..3 considering the n- type family Re 2 x Ce x CuO 4 where RE = Nd, Pr, Sm, Eu and the p-type cuprate La 2 x Sr x CuO 4 as examples. Here, CuO 2 layers alternate with MO layers, where M = Nd, Pr, Sm, Eu, Ce for the n-type example and M = La, Sr for La 2 x Sr x CuO 4. The most notable difference between the shown compounds is that the p-type example has the T crystal strucure, while the n-type compound exhibits T structure characterized by the missing apical oxygen atoms in the n-type compound. Most importantly, cuprates show a quasi two-dimensional character. Interestingly, the critical temperature appears to be increasing with increasing number of CuO 2 layers per unit cell. The important structural element common to all cuprates is the CuO 2 - layers. They are believed to be responsible for the formation of superconductivity, while the rest of the material merely acts as a charge reservoir controlling the carrier concentration in the CuO 2 -layers. Therefore, in theoretical treatments of the cuprates usually only models of the two-dimensional CuO 2 -layers are considered. The properties of all cuprates are extremely sensitive to doping. Upon

13 CHAPTER. INTRODUCTION 7 Figure.3: Examples for the structure of electron- left and hole-doped right cuprates. Figure was taken from [7]. replacing La 3+ ions in La 2 CuO 4 by Sr 2+ for example one hole per Sr atom is created in the CuO 2 -layers. In a similar, but not always quite obvious fashion the carrier concentration in the CuO 2 -layers can be controlled in the other cuprates via doping with some kind of atoms. In Fig..4 is shown what is believed to be the generic temperature-doping phase diagram of all cuprates. There one can see that without doping both types of cuprates are antiferromagnetic Mott insulators with Néel temperatures of the order of a few K. Upon hole-doping the Néel temperature is rapidly reduced and the antiferromagnetic phase vanishes completely at about 5 percent of hole doping. Roughly symmetrical around a hole concentration of about.6 an unconventional superconducting phase is found. The maximal T c is reached at a doping level of.6, therefore this doping level is referred to as optimal while smaller and higher hole-doping levels are called under-doped and overdoped, respectively. It is now well established that the superconducting phase on the holedoped side of the phase diagram is unconventional. In particular, the superconducting order parameter has d x 2 y2-wave symmetry which was confirmed by several experimental techniques including NMR [7], ARPES [25], phasesensitive measurements [37], penetration-depth measurements [58], and polarization dependent Raman-scattering [72]. Such a gap function changes sign within the first Brillouin Zone and has two lines of nodes. Another possible indication of unconventional superconductivity is the value of the isotope ef-

14 CHAPTER. INTRODUCTION 8 Figure.4: Schematic phase diagram of n- and p-type cuprates. Taken from [7] and modified. The bright green colored area shows the extension of the antiferromagnetic phase in the case of no coexistence of SC and AF. The grey colored are shows the region with coexistence in the case that there is coexistence of SC and AF in the cuprates. Which of the two scenarios is realized is unclear. fect coefficient α.5 at optimal doping [], which is much smaller than the BCS value of α =.5. This indicates a non-phonon-mediated pairing mechanism. Besides the antiferromagnetic and superconducting phases one finds also two different regimes. In the over-doped region one finds ordinary Fermi liquid behavior. At smaller dopings below a temperature T, however, exists a regime which exhibits some unusual properties. Most notable among those is the opening of a so-called pseudo-gap, i. e. a gap on parts of the Fermi surface [86]. Like the superconducting gap the pseudo-gap appears to have d x 2 y2-wave symmetry. However, more recently there has been indication that in the so-called pseudo-gap region the Fermi surface is not simply gapped but folded back and forms a number of disconnected, but closed Fermi pockets []. Such a behavior is expected to be realized in the presence of a Density Wave. Due to these experiments, Density Waves are one of the most favored explanations of the pseudo-gap, but by far not the only suggested explanation [49, 35, 68]. Nevertheless, many questions regarding the nature of the pseudogap remain unresolved both experimentally and theoretically. The presence of the pseudo-gap region above the under-doped side of the superconducting dome raises the question whether the order responsible for the pseudo-gap behavior persists in the superconducting phase. This

15 CHAPTER. INTRODUCTION 9 would imply that in the under-doped part of the dome superconductivity coexists with another order parameter, maybe some kind of Density Wave [2]. In this case the T -line in Fig..4 does not terminate at T c, but extends into the superconducting phase and terminates at zero temperature at a doping level of about.9. By now, however, one has not found an unambiguous indication of such a coexistence, though the under-doped part of the superconducting dome shows in some ways behavior different from what is found on the over-doped side [56, 78]. Motivated by this possible indication of coexistence of superconductivity and Density Waves, this thesis analyzes some aspects of such a phase. As mentioned before there are not only p- but also n-type cuprates. Their phase diagram is shown in the right half of Fig..4. Similar to the holedoped side at very low doping there is an antiferromagnetic phase. However, in the n-type compounds the antiferromagnetism is much more stable against doping than in the p-type and extends to more than percent electron doping. As in the hole-doped case there is a superconducting phase around a doping level of about.6, but the maximal T c and the doping width of this phase are significantly smaller. Until now, there is no general consensus reached on the symmetry of the gap function. There is growing experimental evidence suggesting d x 2 y2-wave symmetry as in the hole-doped cuprates but with a peculiar non-monotonic behavior. But it is still not completely ruled out that within over-doped parts of the superconducting phase the gap function has s-wave symmetry [7]. Furthermore, since the T c and T N lines cross at finite temperatures there has been speculation about a coexistence of antiferromagnetism and superconductivity in the n-type cuprates. In the last years there has been growing evidence of superconductivity coexisting with a different order parameter, probably a Spin-Density Wave [7, 87, 8, 4, 26, 35, 69, 3]. Therefore, in this thesis the electron-doped cuprates also are analyzed. However, several aspects of the theoretical results presented here do not depend strongly on the parameter values which are used to model the cuprates, but are rather general and might be relevant for some of the other quasi-two dimensional superconductors mentioned before. It should be noted that the experimental situation in the electron-doped cuprates is very difficult and therefore several questions remain unanswered [7]. The experimental problems are due to the small number of known electron-doped compounds, but also due to the quite difficult sample production. For example electron-doped cuprates become superconducting only after an oxygen-depletion process which is in large parts not yet understood.

16 CHAPTER. INTRODUCTION.5 Coexistence of Superconductivity and Density Waves As pointed out before, experimentally the coexistence of superconductivity and density waves is a well established phenomenon. Theoretically this problem has been studied for several decades from a large number of different perspectives, e. g. [, 26, 27, 29, 53, 98, 99, 2, 95, 9, 22, 9, 39, 62, 75, 52, 8,, 5, 5]. However, no unified and generally accepted framework for this problem could be established. It is a well known fact that Density Waves lead to a reconstruction of the band structure [55, 73]. Depending on the underlying band structure the formation of Density Wave order can change the number of bands crossing the Fermi level. Therefore, even in one-band systems the formation of DW order can result in two bands crossing the Fermi level. Thus, in general one will have to deal with multi-band superconductivity [3, 4]. Generally, it quickly has been realized that due to the presence of the Density Wave several different kinds of pairing become possible and thus the superconducting state is no longer characterized by a single superconducting gap, e. g. [98]. Not only the number of the superconducting gaps can be influenced by the Density Wave, but also its momentum dependence and nodal structure [84, 26, 27, 73]. Using a different approach than the one employed here, it was shown in [26, 27] that CDW order does not introduce additional nodes in a superconducting s-wave gap, while SDW does. This is in agreement with the results derived here. Especially in the context of the pnictide superconductors the influence of SDW on the symmetry, number and position of nodes and relative phases of the superconducting gaps has been addressed recently [3]. To some extent a consensus on the stability of superconductivity in the presence of DW order has been reached. It is clear that the DW order finally destroys superconductivity. The reason for that is that the Fermi surface becomes increasingly gapped when the DW order parameter is turned on. So, in the end there are just no states left at the Fermi level which could form Cooper pairs. However, the exact way in which the DW order destroys the superconductivity is a rather complicated issue and depends on the types of superconductivity as well as Density Wave present. In [53] it was found using a different approach than here that singlet superconductivity is suppressed only weakly by CDW order, but exponentially by SDW order. Triplet superconductivity, however, was found to be suppressed weakly by SDW order. These results are compatible with the results derived here. In this thesis the approach introduced in [23, 24] will be extended.

17 CHAPTER. INTRODUCTION It is based upon the idea that the DW ordering temperature T DW is much larger than the superconducting T c. Then at T c there are DW Bogoliubov quasi-particles present which can form Cooper pairs. Therefore the superconducting pairing interaction needs to be rewritten in terms of the DW Bogoliubov quasi-particles. This will be done considering all DW bands and also retaining inter-band contributions to the pairing. However, due to the substantial Fermi surface mismatch inter-band pairing can and will be neglected. Using this approach the coexistence of SDW, CDW and DDW order with s- and d-wave superconductivity is studied. In [23, 24] there was used an oversimplified band structure which allowed to take only one DW band into account. Furthermore, only SDW order coexisting with s-wave and triplet p-wave superconductivity were considered. Here, in particular, the coupled set of three gap equations is derived, the T c W curves are calculated, the momentum dependence and phasing of the superconducting gaps are discussed and possible phase diagrams are presented..6 Spin Susceptibility and Inelastic Neutron Scattering Apart from the question of the stability of superconductivity coexisting with Density Waves, it is interesting to know how to recognize such a phase experimentally. Therefore in this thesis the dynamical spin susceptibility will be calculated and analyzed for several different phases. The dynamical spin susceptibility is of interest in this context because its imaginary part can be extracted from Inelastic Neutron Scattering INS experiments [83]. The INS spectra in the superconducting state of the cuprates are dominated by one feature: the so-called resonance peak. While in the normal state a broad peak exists at the antiferromagnetic momentum Q = π, π, the spectra in the superconducting state are strongly changed. In particular, the intensity at small frequencies is removed completely while the peak is sharpened and shifted towards higher frequencies. This behavior is illustrated in Fig..5. Furthermore, the frequency ω res at which the resonance peak is observed was found to scale with the SC transition temperature T c in hole-doped cuprates and very recently also in electron-doped cuprates [36]. Obviously the resonance peak is intimately related to superconductivity. Among other scenarios [38], the resonance peak can be understood as a sign of a spin exciton, a bound particle-hole state below the spin gap [36]. Essential to this scenario is that the gap function changes sign between momenta differing by the antiferromagnetic momentum Q, which is the case

18 CHAPTER. INTRODUCTION 2 Figure.5: Measured INS count as a function of frequency at fixed momentum Q = π, π in the normal at 2 K and superconducting state at 2 K. Experiment was performed on YBCO [43]. for d x 2 y2-wave symmetry, but not for s-wave superconductivity. This picture is, with some limitations, known to work for the p-type cuprates, where the disagreement with experimental data increases with increasing under-doping. For the n-type cuprates there has been some theoretical work based on the spin exciton scenario suggesting the absence of a resonance peak [7], in contrast to experiment. The applicability of the spin exciton scenario to the n- and p-type cuprates will be discussed in this thesis. In particular, the limitations of the spin exciton scenario in the case of optimal hole-doping will be presented in detail. Furthermore, the recent experiments on electrondoped cuprates from [36] will be addressed extensively. Since it was suggested that the pseudo-gap region of the under-doped p-type cuprates is an id-wave Charge Density Wave phase [2], in this thesis the susceptibility of such a phase will be discussed. At the moment it is unclear whether the order responsible for the pseudo-gap does coexist with superconductivity in the under-doped p-type cuprates or not. Therefore, the susceptibility will also be calculated for the coexistence of DDW with superconductivity. In [34] this problem has been addressed previously. However, the band structure used there allowed for an oversimplified discussion. Therefore, the full analytic expressions which are presented here have not been derived previously and the discussion missed some points essential to a realistic description of the cuprates. Partial results have also been obtained in [64].

19 CHAPTER. INTRODUCTION 3.7 Overview This thesis is organized as follows: In chapter 2 the model Hamiltonian is motivated and introduced. Solutions with only one mean-field order are discussed. In chapter 3 the coexistence of SDW order with superconductivity is discussed. In chapter 4 the coexistence of both CDW and DDW order with superconductivity is discussed. In chapter 5 the spin susceptibility in superconductors is discussed. The focus is on the resonance peak observed by INS in both the electron and hole doped cuprates. In chapter 6 the spin susceptibility in a DDW state as well as a DDW state coexisting with d-wave superconductivity are discussed. The focus is again on the formation of a resonance peak. In chapter 7 the most important results are summarized. In appendix A the analytic expression for the spin susceptibility in the superconducting state in the presence of a Zeeman magnetic field is derived. The RPA expression for the spin susceptibility in that case is derived. In appendix B the analytic expressions for the spin susceptibility in the DDW state and in a state of DDW coexisting with superconductivity are derived. In appendix C the summation over the Matsubara frequencies for the cases needed in the previous appendices is executed.

20 Chapter 2 Basic electronic states 2. Basic electronic structure Motivated by their layered structure it was quickly concluded that the cuprates are quasi-two-dimensional materials. As pointed out originally by Anderson [6] the defining feature of all cuprates is the copper oxide layers. It is now generally agreed that they contain the essential physics of the cuprates. The electronic structure of such a CuO 2 -plane is shown in Fig. 2.. The formal valences of the ions in the layers are Cu 2+ and O 2, so that without doping the highest occupied orbitals are the copper 3d, which are occupied by 9 electrons, and the filled oxygen 2p orbitals. Because of the octahedral crystal field due to the oxygen ions the degeneracy of the copper 3d orbitals is removed. In particular, the five 3d levels are split into three degenerate t 2g -levels and two degenerate e g. However, there is an additional Jahn-Teller-effect which splits the degeneracy of the e g -levels. This leaves the d x 2 y2-orbital as the highest in energy. Since the configuration is d 9, the d x 2 y2-orbital is then occupied by a single hole, which results in a half-filled band. Upon hole-doping electrons are removed from the oxygen p-orbitals. However, as shown by Zhang and Rice [42] the doped hole hybridizes with the hole at the copper ion forming a so-called Zhang-Rice singlet which behaves as if the hole was doped into the copper 3d x 2 y 2-orbital. In the case of electron-doping on the other hand, the doped electron is located in the copper 3d x 2 y2 orbital. This suggests that the cuprates can be described by an one-band model on the two-dimensional square lattice with the copper ions being the lattice sites. Furthermore, the phase diagram of the cuprates should exhibit electron-hole symmetry. As is obvious from Fig..4 cuprates are electron-hole asymmetric. How this asymmetry can be captured by an one-band model is still under debate 4

21 CHAPTER 2. BASIC ELECTRONIC STATES 5 Figure 2.: Schematic illustration of the electronic structure of a CuO 2 -layer. [7]. One possible way is that the parameters entering the one-band model have to be different for electron- and hole-doping. This is reasonable because the Zhang-Rice singlet certainly requires different parameters than an electron doped directly into the copper 3d-orbital. Nevertheless, in this thesis the parameters used to describe the doped holes and electrons will differ only in the chemical potential which controls the doping level. The model most commonly discussed in the context of the high-t c cuprates is the Hubbard model [6]: H = t ij c iσ c jσ + h.c. + U n i n i 2. i,j,σ i where c iσ creates a fermion with spin σ at site i of the two-dimensional square lattice, n iσ = c iσ c iσ, t i,j is the hopping amplitude for hopping from site j to site i and U is the on-site Coulomb repulsion. Frequently only hopping to nearest neighbors with an amplitude t is considered. Note that in principle U and t i,j have to be derived from models containing the full electronic structure of the cuprates. Sometimes also models derived from the Hubbard model or extended versions of this simple model are discussed. However, since the Hubbard model obviously is a highly oversimplified model, there is little doubt that it is too simple to describe real materials quantitatively. Nev-

22 CHAPTER 2. BASIC ELECTRONIC STATES 6 ertheless, it is a very important theoretical model exhibiting various phases [85], probably including those observed in the cuprates [28]. There exists no solution for the Hubbard model valid in the full parameter space [85]. In the large U limit the half-filled Hubbard model describes a Mott insulator, i. e. a material which according to band theory should be metallic but due to strong electron-electron repulsion is an insulator. Furthermore, this state is antiferromagnetic since by the antiparallel alignment of neighboring spins the exchange energy J given by J = 4t2 U 2.2 can be gained. This state might actually correspond to the zero doping state in the cuprates. A popular route to a theory of the cuprates is to start from this state and to dope holes or electrons into it [78]. If U is small compared to t the system behaves metallic. This might correspond to the highly doped regimes of the cuprates phase diagram. Probably this limit is applicable to the n-type cuprates in a somewhat larger doping range than in the p-type materials. To start from this limit and decrease doping, and with it increase the importance and value of U, is the second very popular route towards a theory of the cuprates [24]. This thesis follows the second approach and assumes that the small U limit is applicable for every doping level where the ground state is metallic. Both of these scenarios appear to work in and close to their starting limits. Unfortunately, they start to fail when one leaves the vicinity of their starting limiting cases. Therefore, none of them is able to describe the full doping range [78, 24]. This is especially unfortunate since the most interesting physics, i. e. the superconductivity, appears to be located in the region where both approaches start to fail. However, it should be stressed that it is absolutely not clear whether the Hubbard model can yield an appropriate qualitative description of the cuprates. In this respect experimental efforts to study the Hubbard model using optical lattices are extremely interesting [89]. Though the extended Hubbard model and its derivatives might contain DW phases, superconductivity and even coexistence of both [23, 2, 28, 5], we do not discuss how these phases could originate from the Hubbard model. Instead we consider the following phenomenological model Hamiltonian: H = k,σ + ε k c k,σ c k,σ k,p,q,α,β,γ,δ V DW αβγδk, p, qc k+qα c kβc pγc p qδ 2.3 k,k,q,α,α,β,β V SC αα ββ k k c k α c k +qβ c k+qβ c kα

23 CHAPTER 2. BASIC ELECTRONIC STATES 7 where ε k is the energy dispersion, the interaction V DW gives rise to different kinds of Density Wave order depending on the spin-momentum dependence and V SC gives rise to different kinds of superconductivity depending on the spin-momentum dependence. This Hamiltonian can be used to study the coexistence of Density Waves and different kinds of superconductivity. The discussion of possible origins of the interactions is beyond the scope of this thesis. Let us for the moment assume V DW V SC. In this case we are left with the first term in equation 2.3 only, which describes a non-interacting system. For ε k we use a tight-binding dispersion on the two-dimensional square lattice. Here it is assumed that it is sufficient to include hopping to the three nearest shells of neighboring sites, so that the dispersion is given by the lattice constant is set to unity ε k = µ + 2 t cos k x + cos k y + t 2 cos k x cos k y t 3 cos 2k x + cos 2k y where µ is the chemical potential which is adjusted in order to control the doping level, t = ev, t 2 =.4eV and t 3 =.ev are the hopping amplitudes for hopping to the nearest, next-nearest and next-next-nearest neighbors, respectively. The parameters were chosen such that the agreement of the experimental and numerical results presented in chapter 5. is optimal. In particular, the dispersion of the resonance mode away from Q was used as the essential fitting criterion. However, the resulting Fermi surface agrees for appropriate values of the chemical potential well with the Fermi surface observed by angle-resolved photoemmission spectroscopy ARPES in the normal state of both the hole- and electron-doped cuprates [32, 7]. The resulting Fermi surface is shown in the left panel of Fig It consists of almost circular hole-pockets centered around the momentum points ±π, ±π. When the number of holes doped into the system is raised the volume of the hole-pockets increases while the overall form of the pockets remains unchanged. Similarly, reducing the number of holes in the system leads to a shrinking of the hole-pockets, again leaving the shape of the pockets mainly unchanged. The corresponding density of states as a function of energy is shown in the right panel of Fig The only prominent feature is the van-hovesingularity at about 2 mev below the Fermi level, which is typical for low-dimensional systems. Changing the doping level corresponds to shifting the frequency axis. The van-hove-singularity crosses the Fermi level at a hole-doping level of about.5.

24 CHAPTER 2. BASIC ELECTRONIC STATES 8 k y [π] k x [π] DoS [/ev] ω [ev] Figure 2.2: Left panel: Fermi surface for µ =.29eV corresponding to a holedoping level of.5. Upon decreasing hole-doping the hole-pockets centered around ±π, ±π shrink. Right panel: Density of states as a function of energy for the same doping level. 2.2 Superconducting state We now discuss the Hamiltonian in equation 2.3 with V DW and V SC. Following [23] the following three types of pairing potentials are considered: V k k δ α αδ β β charge fluctuation, phonons Vαα SC ββ k k = V k k σα 3 α σ3 β β amplitude spin fluctuation V k k σ + α α σ β β orientational spin fluctuation 2.5 This thesis deals with singlet superconductivity and therefore even angular momentum states of the Cooper pairs only. Therefore α = β is required additionally. This notation means that if α =, then β = and vice versa. Furthermore, pairs are allowed to have total momentum of zero only, which means that q from equation 2.3 has to be zero for now. With these restrictions the Hamiltonians for the different types of potentials can be written as H c = ε k c kσ c kσ + V c k k c k α c k ᾱ c kᾱc kα 2.6 kσ k,k,α H z = ε k c kσ c kσ V z k k c k α c k ᾱ c kᾱc kα kσ k,k,α H + = ε k c kσ c kσ + V + k k c k α c k ᾱ c kαc kᾱ kσ k,k,α where in H + the Hermitian conjugate has already been added to make the Hamiltonian hermitian. Apparently, if one incorporates the overall sign into

25 CHAPTER 2. BASIC ELECTRONIC STATES 9 the V and brings the creation and annihilation operators in the same order, all three Hamiltonians can be written as H pair = ε k c kσ c kσ + V k k c k α c k ᾱ c kᾱc kα 2.7 kσ k,k,α In order to treat this Hamiltonian we will employ a mean-field theory. The idea of a mean-field theory is that it is a good approximation to replace a product of two fermion operators in the four fermion operator interaction by its average value. This gives an effective one-particle Hamiltonian, which can be treated easily. However, the approximation is only good if the fluctuations around the average value are small. In general, one expects mean-field theories to work well in high-dimensional systems and for long-ranged interactions [7]. Let us introduce the following mean-field k = q V q c k q, c k+q, 2.8 The mean-field k is usually referred to as the gap function, which in general can be complex. However, here it is taken to be real. With this mean-field we arrive at H = ε k c kσ c kσ k c k, c k, + c k, c k, 2.9 kσ k = k ψ khkψk where we have introduced ψ k = Hk = c k, c k εk k k ε k The eigenvalues of the matrix H k can be calculated easily and one finds E k = E 2k = E k = ε 2 k + 2 k 2.2 Apparently, a simple two-dimensional rotation can diagonalize the mean-field Hamiltonian. Such a rotation is equivalent to introducing new operators, which are certain linear combinations of the old ones, where the fermionic anticommutation relations remain valid. This kind of rotation is known as

26 CHAPTER 2. BASIC ELECTRONIC STATES 2 canonical or Bogoliubov transformation [8]. Let us now express the c k - operators through new operators according to c k = u k α k + v kβ k 2.3 c k = v k α k u k β k 2.4 The new operators are mixtures of creation and annihilation operators with opposite spin and momentum. The amplitude of the creation and annihilation operator in the mixture for a certain momentum point is governed by the so-called coherence factors u k and v k, which need to satisfy the relations u 2 k, vk 2 = ± ε k E k u k v k = k 2E k 2.6 in order to bring the mean-field Hamiltonian into the following diagonal form: H = E k α k α k + β k β k 2.7 k From the Hamiltonian in equation 2.9 the Greensfunction can be calculated easily using Gk, iω = Hk iω I 2.8 where I is the 2 2 unit-matrix. One finds: u Gk, iω = E k iω 2 k u k v k u k v k vk 2 v + E k iω 2 k u k v k u k v k u 2 k 2.9 The Greensfunction has poles at ±E k, which means we are dealing with two non-degenerate bands now. For a gap function with k > this implies that there is no Fermi surface at all because E k k >. From the left panel of Fig. 2.3 also the reason why the gap function is called such becomes apparent: 2 is the gap between the two bands. This implies that 2 is the minimal energy necessary in order to create excited quasi-particles. However, there is also the possibility that the gap function becomes zero for certain momentum points, i. e. has nodes. If these nodes and the normal state Fermi surface intersect, the intersection points form the Fermi surface of the superconducting state. In the high-t c cuprates the gap function is

27 CHAPTER 2. BASIC ELECTRONIC STATES E [ev] 2 k y [π].5 2, π,π π,, k x,k y.5.5 k x [π] Figure 2.3: Left panel: Band structure along the path, π, π π,, in the normal state black and in a superconducting state with a momentum independent gap function blue and red. Right panel: Normal state Fermi surface red together with lines of nodes of a d x 2 y 2-wave gap function black and the resulting Fermi surface in the superconducting state grey which consists of four points. believed to have d x 2 y2-wave symmetry and thus have lines of nodes on the diagonals of the BZ. These lines of nodes intersect the Fermi surface and lead to Fermi points in the superconducting state. The position of the nodes for a d x 2 y2-wave superconductor, the Fermi surface of the cuprates and the resulting Fermi points of the superconducting state are shown in the right panel of Fig Accordingly, the density of states can be obtained from gω = u 2 k + vk 2 δ ω Ek + u 2 k + vk 2 δ ω + Ek 2.2 k The result is shown in Fig. 2.4 for two examples: In the left panel for the case of a momentum independent gap and in the left panel for a gap with d x 2 y2-wave symmetry. Such a gap is usually modelled as k = 2 cos k x cos k y = φ k 2.2 which is the momentum dependence of the gap observed in p-type cuprates. For both cases the main difference in comparison to the normal state result is that the density of states has become symmetric around the Fermi level. The density of states can be viewed as the sum of two modified normal state densities of states, one for usual frequencies and one for an inverted frequency axis. However, the way the normal state density of states has to be modified depends on the kind of gap function present. In the case of a

28 CHAPTER 2. BASIC ELECTRONIC STATES normal ssc.7.6 normal dsc DoS [/ev] DoS [/ev] ω [ev] ω [ev] Figure 2.4: Left panel: Density of states in superconducting state with a momentum independent s-wave gap ssc as a function of energy blue. For comparison the normal state densisty of states is shown, too red. Right panel: Density of states in a superconducting state with a d x 2 y2-wave gap dsc as a function of energy blue. For comparison the normal state density of states is shown, too red. momentum independent s-wave gap the density of states increases sharply when the Fermi level is approached, forming the so-called coherence peaks and then suddenly drops to zero at ω = ±. For a gap with nodes the overall picture is similar, here exemplified by a d x 2 y2-wave gap. However, the coherence peaks are less pronounced and, more importantly, the density of states does not suddenly drop to zero. Instead it decreases power-law like, reaching zero at the Fermi level [28]. Using the Bogoliubov transformation the expectation value in equation 2.8 can be calculated, yielding the following so-called gap-equation k = V p k p Ep tanh E p p 2k B T If there is a non-zero solution for k, a superconducting state forms. The highest temperature for which a non-trivial solution is possible, is the phasetransition temperature T c. In general, this equation has to be solved numerically using a self-consistent approach. However, if one assumes a spherical Fermi surface with a constant density of states of N F around the Fermi level and a interaction that is constant and attractive with value V for states within a shell of width ω D around the Fermi level and zero elsewhere, one

29 CHAPTER 2. BASIC ELECTRONIC STATES 23 can get some analytical results for the gap and the critical temperature T c : 2 = 4ω D exp 2.23 N F V k B T c =.4ω D exp 2.24 N F V 2 = k B T c It should be noted that the superconducting state forms for any arbitrarily weak attraction as long as a Fermi surface exists, though the transition temperature might be also arbitrarily small. Furthermore, the gap function is increasing with increasing density of states at the Fermi level N F and increasing pairing strength V. Also, the gap is proportional to the cut-off frequency ω D, which determines the number of particles participating in the pairing. It must be stressed that these results remain valid qualitatively for any type of interaction. It is worth noting that within this approximation the ratio of gap and critical temperature does not depend on any material dependent parameter. Usually, one is most interested in calculating the transition temperature T c and the momentum dependence of the gap function for a given bandstructure and interaction. If the pairing interaction V p k in equation 2.22 is such that it can be written as V fkgp the sum becomes independent of k and therefore the momentum dependence of k is given by fk. If the pairing interaction does not factorize, it is helpful to realize that the gap equation is an eigenvalue equation. The momentum dependence of the gap is given by its eigenvectors. Unfortunately, the matrix whose eigenvectors we are interested in depends again on the gap through E k. Close to T c, where k vanishes, it is certainly justified to neglect the 2 k in E k. This gives the so-called linearized gap equation. The eigenvector belonging to the largest eigenvalue of the linearized gap equation gives the momentum dependence of the gap. This procedure gives by the way also the transition temperature: The temperature where the largest eigenvalue equals one is T c. However, it is also possible to solve the gap equation by simply guessing any gap function and calculate a corresponding new gap function. The new gap function is inserted back into the gap equation. The procedure is stopped when convergence has been reached. One problem with this approach is, that the speed of the convergence is not necessarily sufficient. Furthermore, one has to be careful whether the gap function does at all converge towards the correct result. Certain solutions might not be reachable by this approach at all.

30 CHAPTER 2. BASIC ELECTRONIC STATES 24 Throughout this thesis we will discuss two kinds of pairing interactions, always assuming them to be non-zero only in a shell of width ω D around the Fermi level. Any details considerations on the origins of these interactions are beyond the scope of this thesis. The first one is a constant attraction, i. e. negative V, which gives rise to a momentum independent s-wave gap. This is the same interaction that was originally used by BCS. A typical origin for this interaction type is the electron-phonon coupling. The second is given by V p k = V 2 cosp x k x + cosp y k y 2.26 = V 4 {cos p x cos p y cos k x cos k y + cos p x + cos p y cos k x + cos k y + sin p x sin p y sin k x sin k y + sin p x + sin p y sin k x + sin k y } with V <. This interaction is of the nearest neighbor type and might be due to charge or spin-fluctuations. Apparently, this interaction can be written as the sum of four terms which each factorize. However, the spin structure of the pairing we introduced is a singlet, which excludes the last two terms with the sin-functions, as they would give rise to p-wave triplet superconductivity. Furthermore, the cos + cos interaction is zero or very small in the vicinity of the Fermi surface and can therefore be neglected with respect to the cos cos-part. Note, that for other Fermi surfaces than the ones considered here, this might not be the case. The cos cos-part leads to a gap function with d x 2 y2-wave symmetry. The momentum dependence of such a gap function is shown in equation 2.2. Here a remark is in order. It should be noted, that the d x 2 y2-wave gap as well as the pairing interaction change sign when shifted by Q = π, π: k = k+q and V q = V q + Q 2.27 The pairing interaction is repulsive for some momenta. In fact superconductivity with lines of nodes can form due to purely repulsive interactions. The repulsion can be overcome if the gap function changes sign when shifted by the momentum for which the repulsion is maximal. However, in general for comparable interaction strengths T c decreases with increasing number of gap-nodes on the Fermi surface. If there are multiple bands crossing the Fermi level the situation becomes more complex. In the easiest case with two bands crossing the Fermi level,

31 CHAPTER 2. BASIC ELECTRONIC STATES 25 a Hamiltonian that can give rise to pair formation in each of the bands can be written as [3] H = kσ ε kc c kσ c kσ + kσ ε kd d kσ d kσ k,k V cc c k c k c k c k 2.28 k,k V dd d k d k d k d k k,k V cd c k c k d k d k + d k d k c k c k where c kσ c kσ creates annihilates an electron in the c-band, while d kσ d kσ creates annihilates an electron in the d-band. The momentum dependencies of the interactions V cc, V dd, V cd have been neglected for the sake of simplicity. V cc and V dd describe the c-intra-band and d-intra-band pairing strengths respectively, while V cd describes the strength of the inter-band contribution to the intra-band pairing. Apparently, it is necessary to introduce two gap functions here: c k = k V cc c k, c k, q d k = k V dd d k, d k, q V cd d k, d k, 2.29 V cd c k, c k, 2.3 After introduction of these mean-fields the two-bands are decoupled inside the Hamiltonian, but through the definitions of the gap functions the bands remain coupled. Now two independent Bogoliubov transformations of the kind used earlier in this section diagonalize the resulting mean-field Hamiltonian. Application of the two Bogoliubov transformations yields the two coupled gap equations: c k = k V cc c k 2E c k tanh Ec k 2k B T q d k = k V dd d k 2E d k tanh Ed k 2k B T q V cd d k 2E d k tanh Ed k 2k B T V cd c k 2E c k tanh Ec k 2k B T For such a set of gap equations it is well known that there are non-trivial solutions [3]. However, it is not immediately clear how the gaps are related to each other. Especially, it is not clear if both gaps have the same or opposite sign. If one takes V cc, V dd and V cd to be negative, i. e. attractive, apparently both gaps have the same sign. If, on the other hand, one has negative V cc

32 CHAPTER 2. BASIC ELECTRONIC STATES 26 and V dd, but positive, i. e. repulsive, V cd, one can see immediately that it is favorable for the gaps to have opposite sign, i. e. to be phase-shifted by π. The transition temperatures in both cases are equal, if only the sign of V cd is changed and the absolute values of all interactions are left unaltered. The first situation is widely believed to be realized in MgB 2 [23], while the latter is currently discussed in the context of the iron-based superconductors [4]. Clearly, the relative phase of the gap functions is of interest in multi-band superconductors, since it can give information on the nature of the pairing interactions and thus the mechanism of superconductivity. This discussion can straightforwardly be generalized to an arbitrary number of bands at the Fermi level and pairing interactions that are momentum dependent. Note, that within this thesis sometimes it will be stated that T c vanishes. As long as there are a non-zero pairing interaction and a Fermi surface, this statement is not true. In this context, it means that the transition temperature is strongly suppressed and extremely small, i. e. smaller than numerical resolution. 2.3 Spin-Density Wave state It will be shown here that the model from equation 2.3 can yield a Spin- Density Wave phase. To this end take V SC. In order to induce Spin- Density Wave order with ordering momentum Q = π, π and polarization along the z-direction the interaction V DW has to have the form V DW αβγδk, p, q = U δ q Q σ z αβσ z γδ 2.33 with U >. We introduce now the following mean-field, which we will call in analogy to the superconducting state SDW gap, W = U c pγ σγδc z p+qδ 2.34 p,γ,δ This leads to the mean-field Hamiltonian: H = k,σ ε k c kσ c kσ + k,α,β W c k+qα σz αβc kβ 2.35 = k,σ ε k c kσ c kσ + k,σ W sgnσc k+qσ c kσ = k,σ ψ k, σhk, σψk, σ

33 CHAPTER 2. BASIC ELECTRONIC STATES 27 with sgn = + and sgn = and where we have introduced ψ k, σ = c kσ, c k+qσ ε Hk, σ = k sgnσw sgnσw ε k+q and the at the summation sign indicates that the momentum summation is restricted to the reduced or antiferromagnetic Brillouin zone RBZ defined by RBZ = {k x, k y k x + k y π} 2.38 The reduced Brillouin zone is shown in both panels of Fig It is the area inside the black diamond. The eigenvalues of Hk, σ are easily found to be with E α,β k = ε + k ± ε k 2 + W ε ± k = ε k ± ε k+q The mean-field Hamiltonian can be diagonalized by a Bogoliubov transformation similar to the superconducting case. However, the new operators have a different form than for the superconductor. Here, the c-operators need to be replaced according to c kσ = u k α kσ + v kβ kσ 2.4 c k+qσ = sgnσ v k α kσ u kβ kσ 2.42 The new operators are mixtures of operators with momentum inside the first and second reduced Brillouin zone, respectively. The amplitude of the contribution from the first and second reduced Brillouin zone is governed by the coherence factors. They need to satisfy u 2 k, vk 2 = ± ε k with u k v k = W 2 E k E k 2.44 E k = Eα k Eβ k

34 CHAPTER 2. BASIC ELECTRONIC STATES k y [π] k y [π] k x [π].5.5 k x [π] Figure 2.5: Fermi surface in a Spin-Density Wave state for hole- left and electron- right doping. The Ek α-fermi surface is red, the Eβ k blue. For comparison the normal state Fermi surface is shown green. The black diamond indicates the border of the RBZ. The SDW gap used here is W =.ev. in order to yield the following diagonal Hamiltonian: H = k,σ Ek α α kσ α kσ + E β k β kσ β kσ 2.46 From Hk, σ the Greensfunction can be obtained easily and one finds: Gk, iω, σ = Ek α iω u 2 k sgnσ u k v k sgnσ u k v k vk E β k iω vk 2 sgnσ u k v k sgnσ u k v k u 2 k The Greensfunction has poles at Ek α and Eβ k. So, here we are dealing with a four-band system consisting of two two-fold degenerate bands. However, unlike in a superconductor where the bands are gapped or at most touching, here the two bands can overlap, touch or be gapped depending on the normal state band-structure and the value of the SDW gap. Only if t 2 = t 3 = µ =, i. e. if the normal state Fermi surface is perfectly nested, the two bands are gapped for any non-zero W. The resulting Fermi surfaces are shown in Fig. 2.5 for an intermediate value of W. The Fermi surface in the Spin-Density Wave consists no longer of hole-pockets around ±π, ±π as in the normal state, but becomes folded back when crossing the border of the reduced Brillouin zone. This results in the formation of electron-pockets around ±π, and, ±π due to the

35 CHAPTER 2. BASIC ELECTRONIC STATES 29 E [ev] DoS [/ev] SDW normal α band β band.5, π,π, k x,k y π, ω [ev] Figure 2.6: Left panel: Band structure along the path, π, π π,, in the normal state black and in a Spin-Density Wave state blue and red. The border of the Reduced Brillouin Zone is marked by the vertical dashed lines. Right panel: Density of states in a Spin-Density Wave state as a function of energy black. For comparison the contributions from the Ek α red and Eβ k blue are shown separately as well as the normal state density of states magenta. Both figures are for W =.ev. Ek α-band and hole-pockets around ±π/2, ±π/2 due to the Eβ k -band. Both kinds of pockets shrink upon increasing W and eventually disappear one after another. For hole-doping the E α -band Fermi surface is smaller than the one belonging to the E β -band and disappears first, while for electron-doping the E β -band Fermi surface is the smaller one and is destroyed first. The corresponding band structure is shown for an intermediate W in the left panel of Fig There, one can see again that both bands cross the Fermi level and no gap forms. At the marked points the bands cross the border of the first reduced Brillouin zone and one can nicely observe how the bands are folded back. Away from these points, however, one of the bands is always almost identical with the normal state band. The influence of increasing W on the band structure is shifting the α- β-band towards higher lower energies, similar to a chemical potential. Unlike a usual chemical potential W does not have the same influence at every momentum point as it enters in squared form under the square root together with ε k. The density of states can be obtained from gω = k u 2 k δ ω E α k + v 2 k δ ω E β k 2.48 Note, that the sum covers the full first Brillouin zone here. For the same value of W as before the result is shown in the right panel of Fig There, it becomes obvious that the Spin-Density Wave order has a much less striking

36 CHAPTER 2. BASIC ELECTRONIC STATES 3 influence on the band structure than the superconducting order. There are only two regions where there are visible differences between the densities of states in the normal and Spin-Density Wave state. Both are located at the borders of the frequency region where the two SDW-bands overlap. Each of them consists of a drop below the normal state level right outside the overlap region and an increase above the normal state level further outside. So, we can see a gap and coherence peaks forming similar to the behavior in the superconducting state. When W increases above a critical value the density of states becomes zero for a certain frequency between the two bands. However, the density of states will remain asymmetric with respect to the Fermi level and notably a full gap, i. e. a zero density of states, will in general not begin to form at the Fermi level. The exact position depends on the band structure. For a band structure with only µ and t the full gap begins to form at ω = µ for W =. With further increasing W the fully gapped region becomes larger and eventually the Fermi level becomes gapped, too. The Fermi surface of n-type cuprates has been measured as a function of doping [8]. It was found that for low doping the Fermi surface consists of pockets at ±π, and, ±π. At larger doping levels, around the same doping level where superconductivity forms, a Fermi pocket begins to form around ±π/2, ±π/2. With further doping finally a large Fermi surface centered around ±π, ±π is formed by connecting both kinds of pockets. This kind of overall evolution with doping is what one would expect for a Spin-Density Wave phase, if the SDW gap decreases with increasing electrondoping and finally reaches zero. In the p-type cuprates there is also indication of Fermi pockets in the underdoped regime []. Therefore, it is reasonable to analyze the possibility that in both kinds of cuprates a Spin-Density Wave phase coexists with superconductivity. Using the SDW Bogoliubov transformation we are now able to calculate the expectation value in the SDW gap equation One rapidly finds W = U U = k W E α E tanh k tanh Eβ k k 2k B T 2k B T E α tanh k tanh Eβ k 2k B T 2k B T k E k 2.49 The β-band usually lies mainly below the Fermi level, while the α-band is located mostly above. Therefore, in order to find a solution U has to be positive, as was already stated above. In principle, one should calculate the SDW gap W from the gap equation for a given interaction U. However, due

37 CHAPTER 2. BASIC ELECTRONIC STATES W [ev] U [ev] Figure 2.7: SDW gap W as a function of the interaction strength U at zero temperature for optimal hole doping. to the second line, in fact it is easier to put some W into the sum and obtain the corresponding U directly from the reciprocal of the sum. If one wants to start from U one needs to self-consistently solve the first line of equation In Fig. 2.7 we show the SDW gap as a function of the interaction U. There is a minimal interaction strength necessary to form a finite SDW gap. Upon increasing the interaction strength the SDW gap increases as well. Note, that there are two kinks in the W U curve at the W values where the Fermi surface of one of the bands disappears. It is easily seen that the relation k c k c k+q = k c k c k+q 2.5 holds in the Spin-Density Wave state. Via Fourier transformation one rapidly finds constant prefactors are dropped c k c k+q = c i c i e ir iq 2.5 k i = c i c i e ir iq i where the sum covers all lattice sites i at positions r i. Apparently, as e i n π is + for even and for odd n, this puts the constraint on the spin densities ϱ iσ = c iσ c iσ to be large at the even and small at the odd lattice sites for spin up and the other way round for spin down. This leads to a magnetization of the even and an opposite magnetization of the odd sublattice, while there is no net magnetization: An antiferromagnetic state with ordering momentum Q forms. This antiferromagnetic ordering is observed in both kinds of cuprates in certain doping ranges.

38 CHAPTER 2. BASIC ELECTRONIC STATES Charge-Density Wave state A Charge-Density Wave state is mathematically rather similar to the Spin- Density Wave state from the previous section. It can originate from the model in equation 2.3, too. To show this set V SC. Furthermore, we want to obtain a Charge-Density Wave with ordering momentum Q = π, π and thus take V DW αβγδk, p, q = U δ q Q δ αβ δ γδ 2.52 with U >. We now introduce the following mean-field, the Charge-Density Wave gap: W = U c pγ δ γδ c p+qδ 2.53 p,γ,δ Using this CDW gap one arrives at the mean-field Hamiltonian H = k,σ ε k c kσ c kσ + k,σ W c k+qσ c kσ 2.54 = k,σ ψ k, σhkψk, σ where we have introduced ψ k, σ = Hk = c kσ, c k+qσ εk W W ε k+q and the at the summation sign again indicates that the summation covers the reduced Brillouin zone only. The eigenvalues of Hk are the same as in the Spin-Density Wave case: E α,β k = ε + k ± ε k 2 + W ε ± k = ε k ± ε k+q Since the CDW Hamiltonian is quite similar to the SDW Hamiltonian, it is diagonalized by a similar Bogoliubov transformation. Let us replace the c-operators according to c kσ = u k α kσ + v kβ kσ 2.59 c k+qσ = v k α kσ u kβ kσ 2.6

39 CHAPTER 2. BASIC ELECTRONIC STATES 33 If the coherence factors u k and v k obey the same relations as in the previous section, this transformation brings the Hamiltonian into the same diagonal form as shown for the SDW case. From Hk the Greensfunction can be obtained easily and one finds: u Gk, iω = Ek α iω 2 k u k v k u k v k vk v + E β k iω 2 k u k v k u k v k u 2 k Remember that Hk and Gk, iω are in fact 4 4-matrices. The full matrices are obtained by using the shown 2 2-matrices as diagonal elements and putting 2 2-zero-matrices as off-diagonal elements. The Greensfunctions in the CDW and SDW case are identical if one replaces sgnσ +. The position of the poles in the Greensfunctions, the band structure, Fermi surface and density of states of the Charge-Density Wave state are exactly the same as in the Spin-Density Wave state. After evaluation of the expectation value in equation 2.53 using the Bogoliubov transformation we find the Charge-Density Wave gap equation. The gap equation is identical to the SDW case. Similarly to the Spin-Density Wave case one finds that in a Charge- Density Wave state k c k c k+q = k c k c k+q 2.62 holds. Using Fourier transformation to real space one can write this as constant prefactors are dropped c kσ c k+qσ = c iσ c iσ e ir iq 2.63 k i This means that the Charge-Density is modulated with an ordering momentum Q. In contrast to the Spin-Density Wave state, here both Spin-Densities are equal at every site and not shifted by one lattice site. Therefore, no lattice or sublattice exhibits any magnetization. The modulation of the Charge- Density is observable. In electron-doped cuprates there is no indication of any charge ordering. In the hole-doped cuprates, however, there is indication of charge order [57, 78]. Until now it remains unclear what the exact nature of this charge order is. The charge order considered here is one which gives rise to a Fermi surface which is similar to the one observed.

40 CHAPTER 2. BASIC ELECTRONIC STATES D-Density Wave state As the last kind of a Density Wave now the formation of an id-charge-density Wave or D-Density Wave DDW state is considered. Mathematically, this can be viewed as a generalization of the Charge-Density Wave state from the preceding section. In the Charge-Density Wave state the CDW gap is momentum independent. Therefore, one might refer to this state as s-wave Charge-Density Wave. Here, the Density Wave gap is momentum dependent and has d x 2 y2-wave symmetry. This requires the Density Wave gap to be purely imaginary in order to yield a hermitian Hamiltonian. Note, that the DDW state can also be regarded as a generalization of the inhomogeneous d-wave Fulde-Ferrel-Larkin-Ovchinnikov superconducting state [46] to the particle-hole channel [2]. In a DDW state there is a non-zero order parameter which is proportional to φ k c k Qσ c kσ = 2 cos k x cos k y c k Qσ c kσ 2.64 k k In real space the momentum sum reads overall constant prefactors were dropped c j+,σ c jσ + c j,σ c jσ c j+,σ c jσ c j,σ c jσ e iqr j j This means that in a DDW state a certain pattern of currents exists, which is shown in Fig These currents circulate around the inter-lattice positions a + /2, b + /2 where a, b is any site of the square-lattice and the lattice constant is set to unity. The direction of the circulation is different for the even and odd inter-lattice sites. Therefore, the magnetic moments, which exist due to the circulating currents, order antiferromagnetically. So, in fact in a DDW state there is no modulation of the Charge-Density. Instead a DDW state is characterized by circulating currents which lead to orbital antiferromagnetism. As before, our starting point is equation 2.3 with V SC. To obtain a DDW state we take V DW to be V DW αβγδk, p, q = U φ k φ p δ q Q δ αβ δ γδ 2.65 Introduce the DDW gap W k = i U φ k φ p c pγ δ γδ c p+qδ pγδ 2.66

41 CHAPTER 2. BASIC ELECTRONIC STATES 35 Figure 2.8: Pattern of circulating currents which forms in a DDW state. The diamonds indicate the sites of the square-lattice and the arrows show in which direction the quasi-particles move. This leads us to the following mean-field Hamiltonian H = k,σ ε k c kσ c kσ + k,σ iw k c k+qσ c kσ 2.67 = k,σ ψ k, σhkψk, σ where we have introduced ψ k, σ = c kσ, c k+qσ εk iw Hk = k+q iw k ε k+q εk iw = k iw k ε k+q The corresponding energy eigenvalues are again given by E α,β k = ε + k ± ε k 2 + W 2 k 2.7 ε ± k = ε k ± ε k+q The mean-field Hamiltonian can again be diagonalized by a Bogoliubov trans-

42 CHAPTER 2. BASIC ELECTRONIC STATES 36 formation. Here, the c-operators need to be replaced according to c k = u kα k + v kβ k 2.72 c k+q = vkα k + u kβ k 2.73 c k = u k α k v kβ k 2.74 c k+q = v k α k + u kβ k 2.75 The coherence factors u k and v k are required to obey u k 2, v k 2 = ± ε k 2 E k u kv k = iw k 2 E k with E k = Eα k Eβ k 2 in order to bring the mean-field Hamiltonian into the form 2.78 H = k,σ Ek α α kσ α kσ + E β k β kσ β kσ 2.79 From Hk the Greensfunction can be found to be Gk, iω = Ek α iω uk 2 u k v k u k v k v k 2 + E β k iω vk 2 u k v k u k v k u k Again, remember that Hk and Gk, iω are due to the spin actually 4 4- matrices. The band structure of the DDW state is still very similar to the SDW and CDW states. However, though the expressions look identical, they are not. One has to remember that the DDW gap W k is not constant, but strongly momentum dependent. This has several effects on the band structure and the density of states. The momentum dependence of W k is given by φ k. Since φ k is maximal at ±π, and, ±π, while it is zero on the lines defined by k x = ±k y, it follows that the DDW gap is large for the α-band which is centered around

43 CHAPTER 2. BASIC ELECTRONIC STATES k y [π] k y [π] k x [π].5.5 k x [π] Figure 2.9: Fermi surface in a D-Density Wave state for hole- left and electron- right doping. The Ek α-fermi surface is red, the Eβ k blue. For comparison the normal state Fermi surface is shown green. The black diamond indicates the border of the RBZ. The DDW gap used here is W =.ev. φ k s maxima, while it is small for the β-band which is centered around φ k s node lines. The most important result of this is that no matter how large W is, the β-band will always have at least Fermi points on the diagonal of the Brillouin zone. The resulting Fermi surfaces are shown in Fig For hole-doping left panel there is no visible difference to the Spin-Density Wave Fermi surface with W SDW = W DDW. That is because the Fermi surface of the β-band is large, so that it is located where W k is already close to W. For electrondoping right panel on the other hand the β-band has a small Fermi surface. Due to the momentum dependent DDW gap that Fermi surface becomes prolonged along the border of the reduced Brillouin zone. On the border of the reduced Brillouin zone ε k ε k+q = and thus W k has maximal influence on that line. Compared with the SDW Fermi surface one can notice that the volume of the β-band pocket is almost doubled. A similar prolongation occurs for the α-band Fermi surface. However, in this case the effect is less pronounced since W k is at the relevant momenta only slightly smaller than W. The band structure along the same path as was used in the SDW section is shown in the left panel of Fig. 2.. There is only one notable difference to the band structure in the SDW case: The two bands touch at π/2, π/2 and are not gapped there. This is again a combined result of ε k ε k+q = on the border of the reduced Brillouin zone and φ k having lines of nodes for k x = ±k y.

44 CHAPTER 2. BASIC ELECTRONIC STATES 38 E [ev] , π,π π,, k x,k y DoS [/ev] ω [ev] DDW normal α band β band Figure 2.: Left panel: Band structure along the path, π, π π,, in the normal state black and in a D-Density Wave state blue and red. Right panel: Density of states in a D-Density Wave state as a function of energy black. For comparison the contributions from the Ek α red and E β k blue are shown separately as well as the normal state density of states magenta. Both figures are for W =.ev. The density of states can be obtained from gω = u k 2 δ ω Ek α + v k 2 δ ω E β k k 2.8 where the summation covers the full Brillouin zone. The result is shown in the right panel of Fig. 2.. The overall picture is again very similar to the SDW case. However, the feature at higher frequencies here is less pronounced. This again happens because the β-band is not as strongly affected by the DDW gap as it was by the SDW gap. From equation 2.66 one rapidly finds using the Bogoliubov transformation W k = U φ k W φ k = U φ k U = p φ 2 p E p p p tanh W p φ p E p W φ 2 p E p E α p tanh tanh 2k B T E α p tanh tanh 2k B T tanh Eβ p 2k B T E α p 2k B T Eβ p 2k B T Eβ p 2k B T 2.82 Apart from the momentum dependence of the interaction and the DDW gap given by φ k this is the same result as for the Spin-Density Wave. Therefore one might expect that the plot of W U looks similar to the result for SDW

45 CHAPTER 2. BASIC ELECTRONIC STATES W [ev] U [ev] Figure 2.: DDW gap W as a function of the interaction strength U at zero temperature for optimal hole doping. order shown in Fig However, this is not the case. The actual result is shown in Fig. 2.. As in the SDW case a minimal interaction strength exists that is necessary in order to obtain a non-zero W. Unlike in the previous cases this minimal interaction strength does not yield infinitely small W but exactly the W value at which the Fermi surface of the Ek α -band vanishes. For larger U two solutions for W exist until an interaction strength U cr is reached and the smaller W solution has reached W =. At this U cr this W branch ends. This behavior can be easily understood from the gap equation The DDW gap enters the gap equation via Ep and via the energy dispersions inside the tanh-functions. Approximately, Ep W holds. This would lead to U = k W with some constant k. This is exactly the behavior of the upper W branch. In the limit of zero temperature, the effect of the tanh-functions is to give the sign of the energy dispersion. For the upper W branch the Ek α- band is completely above the Fermi level and the only remaining points of the E β k -band above the Fermi level are located close to the nodes of φ2 p. Therefore the tanh-functions are not changing for the upper W branch. For the lower W branch on the other hand the situation is different. Here the E β k -band still remains unimportant. But the Ek α-band is shifted with decreasing W below the Fermi level. The points that are shifted below the Fermi level are located at the maximum of φ 2 p and thus are weighted strongly. These points are strongly antagonistic to the DDW formation. Thus, even though W decreases, the corresponding U has to increase. The very peculiar behavior of W U is therefore a direct consequence of the momentum dependence of the order parameter. It is reasonable to believe that at least at zero temperature the DDW state with the larger DDW gap W is the most stable one. Therefore, a sta-

46 CHAPTER 2. BASIC ELECTRONIC STATES 4 ble DDW state can form only when the E α k -Fermi surface is gone. However, this property of the DDW gap equation has never before been perceived. The question which DDW state, if any, is stable at zero and especially finite temperatures requires further study. However, this behavior may be in agreement with measurements on the underdoped cuprates [, 32].

47 Chapter 3 Superconductivity on SDW background 3. Derivation of the gap equations As shown before there is indication of Density Waves coexisting with superconductivity in a large variety of materials. In some of these materials, including p- and n-type cuprates, there might exist a regime where T DW T c, with T DW being the onset temperature of the Density Wave order and T c the superconducting transition temperature. In that case the normal state i. e. the regime above T DW c-quasi-particles are not the relevant quasi-particles close to T c, but the Density Wave quasi-particles α and β are. Therefore, the Cooper pairs forming should not be c k c k but α k α k and β k β k. Since the DW gap is large here, Cooper pairs of the form α k β k are not supposed to form, because the corresponding two Fermi surfaces show a substantial mismatch. In this chapter the formation of a superconducting phase on a Spin- Density Wave background is analyzed in some detail. The main focus is on the stability of the superconductivity as a function of the Spin-Density Wave gap and the influence of the Spin-Density Wave on the symmetry and phase of the superconducting gap. The following chapter is devoted to the same problem, but there with a background of different Charge-Density Wave orders. Starting point is as in the preceding chapter the model Hamiltonian 2.3: H = k,σ + ε k c k,σ c k,σ k,p,α,β,γ,δ V DW αβγδk, p, qc k+qα c kβc pγc p qδ 3. k,k,q,α,α,β,β V SC αα ββ k k c k α c k +qβ c k+qβ c kα 4

48 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 42 In the previous chapter it was shown that V DW can give rise to Density Wave order while V SC can yield superconductivity. Assume for the moment that a well-developed Spin-Density Wave is already present. In this case, the c- operators need to be replaced by the α- and β-operators introduced in the preceding chapter. The introduction of the Spin-Density Wave gap and of the SDW Bogoliubov quasi-particles diagonalizes the first two terms in the model Hamiltonian, as shown in the previous chapter. In order to be able to apply the Bogoliubov transformation to the SC pairing term, it must be rewritten with k and k limited to the first RBZ: H pair = = k,k,q,α,α,β,β V SC αα ββ k k c k α c k +qβ c k+qβ c kα 3.2 k,k q,α,α,β,β V SC αα ββ k k c k α c k +qβ c k+qβ c kα +V SC αα ββ k k + Qc k +Qα c k +q+qβ c k+qβ c kα +Vαα SC ββ k k + Qc k α c k +qβ c k+q+qβ c k+qα +Vαα SC ββ k k c k +Qα c k +q+qβ c k+q+qβ c k+qα where advantage of the fact that 2Q is a reciprocal lattice vector has been taken. In order to count all pairing contributions q has to be q = or q = Q. This yields: H pair = k,k α,α,β,β V SC αα ββ k k c k α c k β c kβ c kα 3.3 +V SC αα ββ k k + Qc k +Qα c k +Qβ c kβ c kα +V SC αα ββ k k + Qc k α c k β c k+qβ c k+qα +V SC αα ββ k k c k +Qα c k +Qβ c k+qβ c k+qα + V SC αα ββ k k c k α c k +Qβ c k+qβ c kα +V SC αα ββ k k + Qc k +Qα c k β c k+qβ c kα +Vαα SC ββ k k + Qc k α c k +Qβ c kβ c k+qα +Vαα SC ββ k k c k +Qα c k β c kβ c k+qα

49 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 43 After reordering the terms one has H pair = k,k α,α,β,β V SC αα ββ k k c k α c k β c kβ c kα V SC αα ββ k k c k α c k +Qβ c k+qβ c kα +V SC αα ββ k k c k +Qα c k β c kβ c k+qα +V SC αα ββ k k c k +Qα c k +Qβ c k+qβ c k+qα +V SC αα ββ k k + Qc k +Qα c k +Qβ c kβ c kα +V SC αα ββ k k + Qc k α c k β c k+qβ c k+qα +Vαα SC ββ k k + Qc k +Qα c k β c k+qβ c kα +Vαα SC ββ k k + Qc k α c k +Qβ c kβ c k+qα Now the calculation is restricted to the singlet pairing channel, i. e. it is required that α = β and α = β. For pairing potentials of the charge and amplitude spin fluctuation form, the pairing part of the Hamiltonian can then be written as H pair = k,k α V SC k k c k α c k ᾱ c kᾱc kα V SC k k c k α c k +Qᾱ c k+qᾱc kα +V SC k k c k +Qα c k ᾱ c kᾱc k+qα +V SC k k c k +Qα c k +Qᾱ c k+qᾱc k+qα +V SC k k + Qc k +Qα c k +Qᾱ c kᾱc kα +V SC k k + Qc k α c k ᾱ c k+qᾱc k+qα +V SC k k + Qc k +Qα c k ᾱ c k+qᾱc kα +V SC k k + Qc k α c k +Qᾱ c kᾱc k+qα where V SC for the charge fluctuation channel is different from the interaction in the spin amplitude fluctuation channel. For a pairing potential of the spin orientation fluctuation type the result reads, after adding the complex

50 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 44 conjugate: H pair = k,k α V SC k k c k α c k ᾱ c kαc kᾱ V SC k k c k α c k +Qᾱ c k+qαc kᾱ +V SC k k c k +Qα c k ᾱ c kαc k+qᾱ +V SC k k c k +Qα c k +Qᾱ c k+qαc k+qᾱ +V SC k k + Qc k +Qα c k +Qᾱ c kαc kᾱ +V SC k k + Qc k α c k ᾱ c k+qαc k+qᾱ +V SC k k + Qc k +Qα c k ᾱ c k+qαc kᾱ +V SC k k + Qc k α c k +Qᾱ c kαc k+qᾱ The rest of the chapter will focus on the spin orientation fluctuation mediated pairing. The reason for that is that the other two pairing potentials, essentially the charge and zz-component of the spin susceptibility respectively, are rather weak in a SDW state. The spin orientation fluctuation pairing potential, essentially the + -component of the spin susceptibility, on the other hand is strong, because of the Goldstone mode forming in the SDW state [23]. Nevertheless, the result for the charge and spin amplitude fluctuation channel will also be given and discussed qualitatively. Now the SDW Bogoliubov transformation is applied to the pairing Hamil-

51 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 45 tonian in equation 3.6: H pair = k,k σ [ V SC k k u k α k σ + v k β k σ u k α k σ + v k β k σ u k α kσ + v k β kσ u k α k σ + v k β k σ V SC k k u k α k σ + v k β k σ v k α k σ u k β k σ v k α kσ u k β kσ u k α k σ + v k β k σ V SC k k v k α k σ u k β k σ u k α k σ + v k β k σ u k α kσ + v k β kσ v k α k σ u k β k σ +V SC k k v k α k σ u k β k σ v k α k σ u k β k σ v k α kσ u k β kσ v k α k σ u k β k σ V SC k k + Q v k α k σ u k β k σ v k α k σ u k β k σ u k α kσ + v k β kσ u k α k σ + v k β k σ V SC k k + Q u k α k σ + v k β k σ u k α k σ + v k β k σ v k α kσ u k β kσ v k α k σ u k β k σ +V SC k k + Q v k α k σ u k β k σ u k α k σ + v k β k σ v k α kσ u k β kσ u k α k σ + v k β k σ +V SC k k + Q u k α k σ + v k β k σ v k α k σ u k β k σ u k α kσ + v k β kσ v k α k σ u k β k σ ] 3.7 The formation of inter-band pairs can be neglected for sufficiently large W, because of the substantial Fermi surface mismatch. Therefore, all terms that do not have the form γ k σ γ k σ δ kσδ k σ with γ = α, β and δ = α, β 3.8

52 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 46 are dropped and one arrives after some rearrangements at H pair = k,k σ α k σ α k σ {[ V SC k k {u } 2 k u2 k 2u k v k u k v k + vk 2 v2 k V SC k k + Q {u 2 k v2 k 2u k v k u k v k + vk k}] 2 u2 α kσ α k σ + [ V SC k k {u } 2 k v2 k + 2u k v k u k v k + vk 2 u2 k V SC k k + Q {u }] } 2 k u2 k + 2u k v k u k v k + vk 2 v2 k β kσ β k σ +β k σ β k σ {[ V SC k k {u } 2 k v2 k + 2u k v k u k v k + vk 2 u2 k V SC k k + Q {u 2 k u2 k + 2u k v k u k v k + vk k}] 2 v2 α kσ α k σ + [ V SC k k {u } 2 k u2 k 2u k v k u k v k + vk 2 v2 k 3.9 V SC k k + Q {u }] } 2 k v2 k 2u k v k u k v k + vk 2 u2 k β kσ β k σ Introduce now the following SC mean-fields here the gaps are assumed to be real: [ u γ k = V SC 2 p k ku 2 p 2u k v k u p v p + vkvp 2 2 γ p γ p p p + u 2 kv 2 p + 2u k v k u p v p + v 2 ku 2 p γ p γ p [ u + V SC 2 p k + Q kvp 2 2u k v k u p v p + vkup 2 2 γ p γ p + u 2 ku 2 p + 2u k v k u p v p + v 2 kv 2 p γ p γ p ] ] 3. with γ = α, β and ᾱ = β and vice versa. With these mean-fields, and the earlier introduced SDW order, the model Hamiltonian from equation 2.3 is reduced to the following mean-field form: H = kσ Ek α α k,σ α k,σ + E β k β k,σ β k,σ ] ] α k [α k α k + α k α k + β k [β k β k + β k β k k 3. where the E γ k are the SDW eigenenergies. Apparently, there is no coupling between the α- and β-bands left in the Hamiltonian. However, there is an indirect coupling between the bands through the two SC gaps, which each contain contributions from both bands. The remaining two sub-hamiltonians

53 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 47 each are of the same form as the superconducting mean-field Hamiltonian we discussed in the preceding chapter. Therefore, this Hamiltonian can be diagonalized easily by application of two superconducting Bogoliubov transformations, one for each band: α k = w k d k + x kf k 3.2 α k = x k d k w k f k β k = y k g k + z kh k β k = z k g k y k h k where the coherence factors are required to obey with w 2 k, x 2 k, y 2 k, z 2 k = 2 w k x k, y k z k = α,β k 2Ω α,β k ± Eα,β k Ω α,β k Ω γ k = E γ 2 + γ Using this transformation the Hamiltonian can be written as H = k [ ] Ω α k d k d k + f k f k + Ω β k g k g k + h k h k 3.6 With this transformation one can now evaluate the expectation values in the definition of the two superconducting gaps given above and one obtains [ u γ γ k = V SC 2 p k ku 2 p 2u k v k u p v p + vkv 2 p 2 p p 2Ω γ p + γ u 2 kvp 2 + 2u k v k u p v p + vku 2 2 p Ω γ p p 2Ω γ tanh p 2k B T [ u γ + V SC 2 p + Q k kvp 2 2u k v k u p v p + vku 2 2 p p p + u 2 ku 2 p + 2u k v k u p v p + v 2 kv 2 p γ p 2Ω γ p Ω γ p tanh 2k B T 2Ω γ p Ω γ p tanh 2k B T ] Ω γ p tanh 2k B T ] 3.7 The structure of these gap equations is similar to the structure of the original SC gap equation 2.22: The gap is given by a momentum-integral of the

54 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 48 interaction multiplied by the SC gap divided by twice the energy and a final tanh-factor which determines the temperature dependence. However, here two important additions occur. First of all, the gaps are now given by the sum of two times two contributions. For every gap there are contributions from both bands. Furthermore, as the integration covers only the first RBZ, there are contributions to the gaps in the first RBZ from the first and the second RBZ. These two effects lead to a total of four terms. The second addition is that the pairing interaction is modified to yield an effective interaction which is different for each term. Due to the presence of SDW order the pairing interaction V SC q needs to be multiplied by certain combinations of SDW coherence factors. Since in each gap equation there are contributions from both bands, it is necessary to solve both equations simultaneously. It should also be noted that due to the coupling of the two gap equations there will be only one transition temperature. The gap equations for a pairing potential of the charge or spin amplitude fluctuation type can be derived in exactly the same way. The result reads γ k = p p V SC c/zzp k [ u γ 2 ku 2 p + 2u k v k u p v p + vkv 2 p 2 p 2Ω γ p + γ u 2 kvp 2 2u k v k u p v p + vku 2 2 p Ω γ p p 2Ω γ tanh p 2k B T [ u γ + Vc/zzp SC 2 + Q k kvp 2 + 2u k v k u p v p + vku 2 2 p p + u 2 ku 2 p 2u k v k u p v p + v 2 kv 2 p γ p 2Ω γ p Ω γ p tanh 2k B T ] 2Ω γ p Ω γ p tanh 2k B T ] Ω γ p tanh 2k B T The only change is the inverted signs of the u k v k u p v p factors. By now, we have considered the influence of the SDW order on superconductivity only. But it might be also necessary to discuss the effect of superconductivity on the Spin-Density Wave. To this end, we need to inspect the Spin-Density Wave gap equation In order to calculate the expectation value it is now no longer sufficient to just apply the SDW Bogoliubov transformation. If superconductivity coexists with a Spin-Density Wave, also the SC Bogoliubov transformation needs to be applied. By subsequent application of the SDW and SC Bogoliubov transformations one finds [ ] W Ek α Ω α W = U 2 E tanh k Eβ k Ω β k tanh 3.9 k Ω α k 2k B T Ω β 2k k B T k Apparently, there are only minor differences to the original SDW gap equation 2.49: First, in the tanh-parts the SDW eigenenergies are replaced by the 3.8

55 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 49 coexistence eigenenergies, which is expected to have only small impact on the SDW order as long as W γ, as we assumed in the beginning. Secondly, the tanh-parts are now modulated by the ratio of the SDW and the corresponding coexistence eigenenergies. This mainly makes sure that the tanh-parts are counted with the same sign as the SDW eigenenergy would have yielded. In the limit of γ the original SDW gap equation is recovered. In total the coexistence phase of SDW and SC orders is described by a set of three coupled gap equations. In general, the three equations need to be solved simultaneously. However, here the focus is on the case where W γ and T DW T c. In this case, to a good approximation the equation determining the SDW gap can be replaced by the equation 2.49 which is valid in the pure SDW phase. This decouples the SDW from the SC gap equations and greatly simplifies the solution. Note, that for T T c the decoupling of the SC and SDW gap equations becomes exact. Since not U, but W enters the SC gap equations, W is used as the parameter describing the strength of the SDW order. The two superconducting gaps have a remarkable property independent of the actual interaction. As can be seen from their definition, the SDW coherence factors obey the following relations: u 2 k = v 2 k+q 3.2 u k v k = u k+q v k+q 3.2 Using these relations one easily finds that the SC gaps change sign when shifted by Q: γ k = γ k+q 3.22 This means that in the presence of SDW order no superconducting state without lines of nodes is possible. But it should be noted that the SDW order does not necessarily induce additional lines of nodes. If the superconducting state without SDW background, already obeys k = k+q, the SDW order does not have to change the symmetry of the gap function. However, in general one can expect that the presence of the coherence factors leads to a modification of the momentum-dependence of the gap function. In the derivation of the gap equations we made the assumption that T SDW T c. Therefore it is not clear, whether the two SC gap equations contain the superconducting gap equation for a normal metal background in the limit of W. One finds that in fact they contain the correct limiting case, if one makes the following identification: { β k = k for k RBZ 3.23 for k / RBZ α k

56 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 5 We will use this as a motivation to assume that the gap equations are valid for any value of W. We now proceed with the discussion using two kinds of pairing interactions. First, we take a momentum independent interaction that usually gives rise to a momentum independent s-wave gap. After that we will discuss an interaction that usually leads to d x 2 y2-wave superconductivity. 3.2 s-wave superconductivity In this section the pairing interaction V SC q is taken to be V SC q = V 3.24 From the standard SC gap equation 2.22 one sees immediately that this interaction leads to a momentum independent gap. However, as shown above, this is not possible here, for the gap is required to change sign under shift by Q. To find the gap function it is instructive to simplify the gap equation for the actual interaction: α k = V u 2 k vk 2 p u 2 p v 2 p α p 2Ω α p V u 2 k vk 2 u 2 p vp β 2 p p 2Ω β p Ω α p tanh 2k B T Ω β p tanh 2k B T = β k 3.25 Apparently, both gaps have the same magnitude, but opposite sign, i. e. their phases differ by π. Their momentum dependence is given by u 2 k v2 k which depends strongly on the value of the SDW gap W. For W this function is sgnt in the first and sgnt in the second RBZ. At the border of the RBZ it changes step-like between these two values. The effect of an increasing W is to widen the transition between the values in the center of the first and second RBZ. However, at the border of the RBZ it remains zero for all W. Therefore, both gaps acquire a line of nodes due to the presence of the SDW order. The superconductivity becomes unconventional. In the case of a superconducting gap that is the sum of several contributions, the possibility of a momentum dependence that changes as a function of temperature arises. If the contributions add different momentum dependencies to the gap and at the same time their magnitudes behave differently as a function of temperature, the momentum dependence of the gap needs

57 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 5 to change with temperature. However, here both contributions add the same momentum dependence to the gap and thus the momentum dependence of the gap is temperature independent. Furthermore, one can get from the form of the gap equations some hint on the stability of s-wave superconductivity in the presence of SDW order. Since β k = α k, the contributions from both bands enhance the gap. However, the pairing interaction is dressed by u 2 k v2 k multiplied by the same with k replaced by p. The fraction of the first RBZ where u 2 k v2 k deviates significantly from sgnt and is close to increases with increasing W. Thus the effective pairing interaction is reduced with increasing W. The physical reason for this can be seen from equation 3.7: Contributions to the gap in the first RBZ from the second RBZ are pair-breaking. Therefore, as long as the density of states at the Fermi level does not increase with increasing W, the SDW order suppresses rapidly the s-wave superconductivity. To get a more quantitative insight into the above equation we now solve it numerically. Therefore, we need to select a certain energy dispersion and thus material. In order to model both slightly underdoped p- and n-type cuprates we use the t i defined earlier and take µ =.795eV, 27225eV for electron- hole-doped cuprates, corresponding to a doping level of about.2. Here, we need to remember that we took the interaction to be non-zero in a shell of width ω D around the respective Fermi surfaces only. Therefore, both gaps can be non-zero only in the same shell of width ω D and α k = β k is not true for every k. We neglected ω D up to here, since it made the discussion of general properties of the gap equations much simpler. For the numerics we take ω D =.ev. To obtain numerical results we begin by fixing the temperature and SDW gap W. Now we split the RBZ into a mesh of 5 5 points all results have been checked to not change for larger meshes. Then we choose start values for the gap function. Here we used small constant gaps with opposite sign but equal magnitude. Using the start values as input we calculate new gaps from the gap equations. The new gaps are put into the gap equations to give a second iteration of the gaps. This procedure is repeated until the gaps do no longer change. By varying temperature and W the complete parameter space can be covered. The results for the superconducting gaps are shown in Fig. 3.. In the left panel the SDW Fermi surfaces together with the borders of the first RBZ are shown. The + and close to the Fermi surfaces indicate the sign of the respective gap on that part of the Fermi surface. From this panel one can see two important results. First, if one follows the parts of the Fermi surface which for W connect to form the normal state Fermi surface, that is β- band inside first RBZ and α-band inside second RBZ, the gaps do not change

58 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 52 k y [π] ϕ k x [π] + α/β [ ] max.5.5 α β ϕ [ ] Figure 3.: Left panel: Fermi surfaces in the SDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the SDW order as a function of the Fermi surface angle. α is red β is blue. Both panels are for n-type cuprates with W =.. sign. Second, if one follows the Fermi surface of one of the bands and crosses the border of the RBZ, the corresponding gap changes sign, again confirming that a momentum independent pairing interaction in the presence of SDW order leads to superconducting gaps with lines of nodes. In the right panel of Fig. 3. the two gaps are shown as a function of the Fermi surface angle φ defined in the left panel. The gaps remain almost constant in most of their existence region. Close to the border of the RBZ, however, both gaps rapidly approach zero and finally vanish when the border is reached. The region of φ where the gaps deviate from their maximal values is determined by the SDW gap W. For W the gaps go step-like to zero, while with increasing W the range of φ where the gaps are not maximal increases. The results shown here are for n-type cuprates with an SDW gap W =.ev. The overall symmetry of the gaps is the same for p-type cuprates and the φ-dependence is similar. It should be noted, that a very similar behavior has been found in [73]. However, there the explicit multi-gap character of the coexistence phase was not considered. Let us now calculate the superconducting transition temperature T c. At the transition temperature the gaps vanish. Therefore, close to T c it is certainly a good approximation to neglect the squared gap in the Ω γ k, that is we

59 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 53.8 n-type a.8 p-type b T c [T c W=] W cr W cr2 T c [T c W=] W cr W cr2 [arb. u.] α/β V intra/inter α V intra β V intra V inter c [arb. u.] α/β V intra/inter 5 5 α V intra β V intra V inter d DoSE F [/ev] W cr total α band β band W cr2 e DoSE F [/ev] total α band β band W cr W cr2 f W [ev] W [ev] Figure 3.2: Normalized s-wave superconducting transition temperature T c as a function of SDW gap for electron- a and hole-doping b. The corresponding effective interactions and densities of states at the Fermi level are shown in panels c and e n-type and d and f p-type. The SDW gaps W cr and W cr2 where the Fermi surfaces of the first and second band, respectively, disappear are marked by the dashed lines. linearize the gap equations. In that case the gap equation can be written as α = V Dp α u 2 p vp 2 2 α E α 2 p p E α p tanh 2k B T V Dp β u 2 p vp 2 2 β E β p tanh p 2 2k B T E β p = β = 3.26

60 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 54 where D γ p is in a shell of width ω D around the Fermi surface of the γ = α, β band and elsewhere. This can be transformed to give = + V +V p p D β p Dp α u 2 p vp 2 2 E α 2 p E α p tanh 2k B T u 2 p vp 2 2 E β p 2 Ep β tanh 2k B T 3.27 The temperature for which this equation holds is the transition temperature. The numerical results for both p- and n-type cuprates are shown in Fig The transition temperature is determined by two contributions. The first is the number of quasi-particles participating in the pairing. It is obtained from the integral over the SDW state density of states in the shell of width ω D around the Fermi surfaces. This can be approximated by the density of states at the Fermi level multiplied by twice ω D. The density of states at the Fermi level is shown in panels e and f of Fig. 3.2 as a function of the SDW gap W. There one can see that the density remains almost constant up to a critical SDW gap W cr, where it suddenly drops to about one half of its initial value. This occurs since the first Fermi surface, i. e. the Fermi surface of the β-band α-band for n-type p-type cuprates, disappears there. The density again remains constant up to W cr2, where it suddenly vanishes, since there the second Fermi surface is destroyed. If the effective interactions were independent of W, the T c curves would look similar to the plots of the densities of states. However, due to ω D the jumps would be significantly broadened. The second contribution determining T c is the strength of the effective interactions. They can be measured by V γ intra W V RBZ V inter W V D γ k D α k u 2 k vk 2 2 d 2 k RBZ u 2 k vk 2 2 d 2 k D γ p D β p u 2 p vp 2 2 d 2 p 3.28 u 2 p vp 2 2 d 2 p RBZ RBZ again with γ = α, β. It can be seen in panels c and d of Fig. 3.2 that the emerging SDW order rapidly suppresses the effective interactions. This is a combined effect of the Fermi surfaces approaching the border of the RBZ, where u 2 k v2 k = and the region where u2 k v2 k increasing for larger W. Upon comparing the panels c, e d, f with the panel a b of Fig. 3.2, where the superconducting transition temperature T c is shown as a function of W,

61 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 55 DoS [/ev] SDW+sSC SDW+dSC DoS [/ev] SDW+sSC SDW+dSC ω [ev] ω [ev] Figure 3.3: Density of states as function of the frequency around the Fermi level in a SDW state with SDW gap W =.ev left and with W =.ev right coexisting with superconductivity induced by a momentum independent pairing interaction blue. For comparison the result for the same SDW state coexisting with a superconductor with a single d x 2 y 2-wave gap is shown red. one finds that it is the effective interactions which completely determine the evolution of T c. In fact, for both types of cuprates the reduction of the T c happens so quickly that almost no differences can be made out and T c becomes invisibly small for W W cr. The transition temperature actually decreases much faster than the effective interactions. This is a result of T c depending non-linearly on the effective interactions. Here it should be stressed that T c does not vanish as long as a Fermi surface and attractive interaction exist. So, until W = W cr2 there always exists a SC phase for some temperature, though T c can be arbitrarily small. As long as the Fermi surface in the normal state is located in the vicinity of the border of the RBZ, we expect to observe similar behavior. However, if the normal state Fermi surface were located in the vicinity of, or ±π, ±π the influence of the SDW order on the effective interaction would be much smaller. In such a case the superconductivity would be much less affected from the coexisting SDW order. Unfortunately, such a material is not expected to show SDW order with ordering momentum π, π, since it lacks the necessary nesting of the Fermi surface at this wave-vector. The density of states of the coexistence phase can be obtained from gω = k [ w 2 k + xk 2 δ ω Ω α k + ] yk 2 + zk 2 δ ω Ω β k 3.29 The result is shown in Fig. 3.3 for W =.ev and W =.ev in the

62 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 56 vicinity of the Fermi level. For comparison the density of states of a state with SDW order coexisting with superconductivity with a single d x 2 y 2-wave gap function is shown. This can be obtained from the same equation, simply using α k = β k = φ k. Due to the superconductivity the density of states is symmetric with respect to the Fermi level. In all cases the density of states exhibits a depletion around the Fermi level. Remarkably, due to the lines of nodes in the gap functions the density of states of the unconventional s-wave superconducting state goes in all cases linearly to zero at ω =. However, for the W =.ev case the slope is extremely small. Therefore, it is probably difficult to distinguish this state experimentally from a standard s-wave superconducting state. For W =.ev the influence of the SDW order becomes more visible. Most notable is the splitting of the SC coherence peaks into two on each side of the gap due to the splitting of the two SDW bands in which the gap functions reach different maximal values. Due to the large W here the unconventional s-wave gap deviates in a substantial region of the Fermi surfaces from its maximal value. This results in a density of states that strongly resembles the result for the d-wave superconductor. The resemblance is largest for frequencies close to zero, where the slopes of the densities of states differ by a factor of about two only. Therefore, this state will in several experimental quantities look like a d-wave state. But it should be noted that for W =.ev the unconventional s-wave superconductivity is very strongly suppressed which renders it unlikely that such a state can be observed. Note, that for a pairing potential in the charge or spin amplitude channel all results from this section remain valid. If the pairing potential obeys V q = V q+q, the u k v k u p v p terms cancel out. Therefore, for all three types of pairing potential the gap equations are equal, as long as V q = V q + Q is valid. 3.3 d-wave superconductivity Here the pairing interaction V p k is taken to be V p k = V 4 cos p x cos p y cos k x cos k y = V φ k φ p 3.3 It was already mentioned that this interaction yields a superconducting state with a gap function having d x 2 y2-wave symmetry. It should be noted that this interaction satisfies V p k = V p + Q k. Therefore the contributions to the gaps in the first RBZ from the second RBZ are no longer

63 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 57 pair-breaking, which makes it possible that the superconductivity is in this case not as rapidly destroyed by the SDW order as it was in the case of a momentum independent interaction. Taking advantage of this property of the interaction one can write the gap equations as [ γ k = V p k 4u k v k u p v p γ p p + + 4u k v k u p v p γ p 2Ω γ p with ᾱ = β. Here the interactions are dressed by 2Ω γ p Ω γ p tanh 2k B T Ω γ p tanh 2k B T ] 3.3 ± 4u k v k u p v p 3.32 with for V eff eff intra and + for Vinter. Excluding the boundary of the RBZ this is simply for W. With increasing W, however, V eff inter increases, while V eff intra is reduced. The sum of the dressing factors remains constant. The dressing factor for the intra-band component is at the border of the RBZ, while it is two for the inter-band component. Therefore, the gap is non-zero at the border of the RBZ. In fact, the dressing factors have the same sign inside the entire BZ and thus the SDW background does not introduce any additional nodes here, i. e. there is no phase shift between the two gaps. It should also be noted, that the intra- and inter-band contributions add, for non-zero W, different momentum dependencies to the gaps. Therefore, if the contributions show different temperature dependencies, the momentum dependence of the gaps changes with temperature. The gap equations are now solved numerically. This is done following the same procedure as for the momentum independent interaction, only the starting values for the gaps are different. Here in the beginning both gaps have the same magnitude and a momentum dependence given by φ k. The resulting superconducting gaps are shown in Fig In the left panel the SDW Fermi surfaces together with the borders of the first RBZ are shown. The + and - close to the Fermi surfaces indicate the sign of the respective gap function on that part of the Fermi surface. The grey points mark the nodes of the gap-functions. Apparently, the position of the nodes is the same as in the normal d x 2 y2-wave superconductor, i. e. they are located on the diagonal of the BZ. The symmetry of the gaps is not affected by the SDW background. In the right panel of Fig. 3.4 the superconducting gaps are shown as a function of the Fermi surface angle. To make a detailed comparison with the standard d x 2 y2-wave gap function possible, for each gap there is also a

64 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 58 k y [π] ϕ k x [π] α/β [ ] max ϕ [ ] α β α d wave fit β d wave fit Figure 3.4: Left panel: Fermi surfaces in the SDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the SDW order as a function of the Fermi surface angle. For comparison d-wave fits are shown. α is red β is blue, corresponding fits are dashed. Both panels are for n-type cuprates with W =.. d x 2 y2-wave fit shown. From that picture one can identify two results. First, close to the border of the RBZ, i. e. for the intermediate values of φ, the gaps deviate from the d x 2 y2-wave fits. The α-gap drops below the fit, while the β-gap is larger than the fit. For the α-band the difference reaches about 25 %. The deviation for the β-band is significantly smaller, which partly occurs because the fit was made using the gap value at about 37. Secondly, the magnitude of the β-gap is significantly larger, by almost 5 %, than for the α-band. This is a combined result of the inter-band dressing factor being larger than the intra-band dressing and the bare interaction being maximal close to the α-band. The result for the p-type cuprates is similar, though there the difference in the magnitudes of the two gaps is even stronger since the β-band has a density of states at the Fermi level larger than that of the α-band. However, in general one can only say that the SDW background leads to deviations from exact d x 2 y2-wave form of the gaps. The particular nature of the deviations is very sensitive to the details of the band structure. Also here it should be noted, that a very similar behavior has been found in [73]. However, there the explicit multi-gap character of the coexistence phase was not considered. Concerning the possible change of the momentum dependence of the gaps with temperature, these changes where found to be negligible. Numerically

65 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 59 it was found that an increase of temperature from T = K to T = T c induces changes in the normalized gaps which are below numerical accuracy. The resulting density of states not shown is similar to the density of states of a SDW state coexisting with superconductivity with a single d x 2 y 2- wave gap as shown in Fig However, there are some subtle differences. Most important, the d-wave fits of the gaps in the α- and the β-band have different amplitude, with the β being the larger one. This effect increases with increasing W. Therefore, the splitting of the superconducting coherence peaks due to the splitting of the two SDW bands is hardly, if at all, observable. Now the superconducting transition temperature of the system is calculated. Since the inter- and intra-band dressing factors have different momentum dependencies, the linearization of the gap equations does not yield an useful equation to calculate the transition temperature. If one neglects the deviations of the gap from the d x 2 y2-wave momentum dependence, one arrives via linearization at a simple expression from which the transition temperature can be calculated. However, we want to keep the exact momentum dependence of the gap. A possible route to calculate the transition temperature is to treat the linearized gap equations as an eigenvalue equation and to find the temperature for which the largest eigenvalue equals one. Unfortunately, in order to achieve a stable numerical solution one needs a momentum mesh that is very large and one needs a lot of calculation time. Therefore here, we use another procedure: For a given temperature and SDW gap W we start from very small gap values with d x 2 y2-wave momentum dependence and calculate in iterations new gaps. The magnitude of the new gaps is compared with the starting magnitudes. If at least one of the new magnitudes is larger than the starting magnitude, we increase the temperature and repeat the procedure. For a given W the highest temperature, which gives at least one increasing gap magnitude is the transition temperature. This procedure is not able to capture superconducting phases with extremely small gap magnitudes, corresponding to extremely small transition temperatures, and suffers from the discretization of temperature necessary to perform the numerics. For SDW gaps W and pairing strengths V where the superconducting transition temperature is sufficiently large, i. e. larger than K, the procedure gives reliable results. In Fig. 3.5 the results for the superconducting transition temperature are shown for hole- and electron-doped cuprates. As before the transition temperature is determined by two contributions. The first one is the number of quasi-particles experiencing the pairing interaction, which can be approximately measured by the SDW state density of states at the Fermi level. This quantity was shown before, for the sake of completeness it is shown again as a

66 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 6 T c [T c W = ] W cr n-type W cr2 a T c [T c W = ] p-type W cr W cr2 b [arb. u.] α/β V intra/inter α V intra β V intra V inter c [arb. u.] α/β V intra/inter α V intra β V intra V inter d DoSE F [/ev].5..5 W cr total α band β band W cr2 e DoSE F [/ev] total α band β band W cr W cr2 f W [ev] W [ev] Figure 3.5: Normalized d-wave superconducting transition temperature T c as a function of SDW gap for electron- a and hole-doping b. The corresponding effective interactions and densities of states at the Fermi level are shown in panels c and e n-type and d and f p-type. The SDW gaps W cr and W cr2 where the Fermi surfaces of the first and second band, respectively, disappear are marked by the dashed lines. function of the SDW gap W in panels e for electron- and f for hole-doping of Fig Here we take a closer look at the result for hole-doping in panel f. Though up to W cr the total density of states remains almost constant, the densities of the α- and β-band do change. The density of states of the β- band decreases, while it increases for the α-band by approximately the same amount. This is important as φ 2 k, which tells us how strong the bare pairing interaction is for a given k, is small in the vicinity of ±π/2, ±π/2 where the β-band is located, while it is maximal close to ±π, and, ±π where

67 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 6 the α-band is located. Therefore, while the number of quasi-particles close to the Fermi level is not changing, the average strength of the bare pairing interaction experienced by the quasi-particles grows. In the n-type cuprates on the other hand the densities of states remain constant for each band up to W cr and W cr2, respectively. The second ingredient determining T c is the strength of the effective interactions. Here, a measure of their strength can be obtained from V γ intra W V D γ k Dγ p 4u k v k u p v p φ 2 kφ 2 pd 2 kd 2 p 3.33 RBZ V inter W V RBZ D α k D β p + 4u k v k u p v p φ 2 kφ 2 pd 2 kd 2 p 3.34 Note, that the use of these effective interaction strengths corresponds to neglecting deviations of the gaps from the exact d x 2 y2-wave symmetry. The magnitudes of the effective interaction strengths as a function of W are shown in panels c and d of Fig In both cases the intra-band interaction strengths decrease rapidly, while the inter-band interaction becomes stronger for small W. For larger W also the inter-band interaction weakens and finally vanishes because the bands are shifted out of the range of ±ω D around the Fermi level. The total interaction not shown initially remains almost constant due to the increase of the inter-band interaction. For larger W the total interaction decreases and in the end vanishes as well. In total this explains the shape of the T c curves shown in panels a and b of Fig. 3.5: Initially, T c remains almost constant. When W becomes larger the critical temperature decreases more rapidly and finally vanishes together with the first SDW band. In the hole-doped cuprates superconductivity is somewhat more stable. That occurs because as mentioned before spectral weight from the β-band is transferred to the α-band with increasing W and thus a larger fraction of the quasi-particles is subjected to the stronger pairing interaction of the α-band. This can compensate for some time for the weakening total effective interaction. But here too superconductivity vanishes at W cr. As we saw, the exact shape of T c W depends on the details of the band structure. However, in general it is clear that due to the increase of the interband interaction the d-wave superconductivity remains stable over a wide range of W. But finally superconductivity is destroyed by the SDW order, when the first band vanishes and the inter-band pairing stops working. In principle, superconductivity can exist between W cr and W cr2 but this would require an extremely strong pairing interaction. For the case of a charge or spin amplitude channel pairing interaction the expressions for the inter- and intra-band dressing factors are interchanged.

68 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 62 T AFI AFM e FS AFM e+h FS dsc ssc W W cr2 W cr doping Figure 3.6: Schematic phase diagram for electron-doped cuprates. The system is an antiferromagnetic insulator AFI for W > W cr2, an antiferromagnetic metal with a single, here electron-like, Fermi surface AFM e FS for W cr2 > W > W cr and an antiferromagnetic metal with one electron- and one hole-like Fermi surface AFM e+h FS for W < W cr. For W < W cr there might be superconducting phases of different symmetry coexisting with the SDW order. Since then the intra-band pairing interaction increases with increasing W, one can expect that T c remains approximately constant or even increases. However, at W = W cr2 superconductivity has to vanish, since at that point both Fermi surfaces are destroyed. But note again that there should exist no sizeable pairing interaction in the charge or spin amplitude channel in the SDW state. 3.4 Phase Diagrams Here possible phase diagrams for the coexistence of superconductivity and Spin-Density Wave order are analyzed shortly. A set of possible phase diagrams is shown schematically in Fig As x-axis the SDW gap is used, which is supposed to be directly related to the doping level in cuprates and iron pnictides or pressure in other systems. The doping arrow corresponds to electron-doping cuprates. For large W the system is an antiferromagnetic insulator. Let us now reduce W. When W reaches W cr2 the first Fermi surface the electron-like α-band FS for n-type cuprates forms and we have an antiferromagnetic metal. In principle, now a superconducting phase could form. However, for both types of interactions due to the influence of the SDW order this would require an extremely strong pairing interaction and

69 CHAPTER 3. SUPERCONDUCTIVITY ON SDW BACKGROUND 63 hence is highly unlikely. When W is further reduced it reaches W cr and the second Fermi surface the hole-like β-band FS for n-type cuprates forms. Now, as we have seen before, d-wave superconductivity can emerge on the SDW background if a suitable pairing interaction is present. With further reducing the SDW gap the d-wave T c increases. If there exists a momentum independent pairing interaction in the system, there can be an unconventional s-wave superconducting phase for small SDW gaps. However, the W range, where the s-wave superconductivity coexists with the Spin-Density Wave can only be very narrow, since the Spin-Density Wave rapidly destroys s-wave superconductivity. It is also possible that both types of pairing interactions exist in the system. In that case one of the two following scenarios can be realized. First, if the momentum independent pairing interaction leads in the absence of SDW order to a smaller T c than the d-wave pairing interaction, only the d-wave superconductivity is realized. If, on the other hand, the momentum independent pairing interaction yields a T c larger than the d-wave interaction, the following might happen: In the phase diagram region where the SDW order is destroyed, the s-wave superconductivity forms. But with decreasing doping the SDW order forms and W increases. The SDW order introduces nodes in the superconducting gaps and rapidly suppresses the unconventional s-wave superconductivity, while it leaves the d-wave superconductivity almost unaffected. Therefore, at a certain doping and W level the s-wave T c becomes smaller than the d-wave T c and the symmetry of the superconducting order parameter changes. It should be noted that the phase diagram with the solid lines appears to be compatible with the available experimental data on the n-type cuprates. It should be mentioned that the problem of coexistence of Spin-Density Waves and superconductivity has attracted the attention of theoretical physicists for decades. Therefore, a lot of literature exists on the subject based on very different approaches to the problem and applied to very different materials, e. g. [26, 53, 98, 99, 2, 95, 9, 22, 9, 39, 62, 75, 52]. Overall, in the literature there is a consensus that Spin-Density Waves suppress superconductivity in agreement with the present results. However, the strength of the suppression is not clear by now. As seen above and also found by others the strength of the suppression differs for the various kinds of superconductivity from weak to exponential. The influence of the SDW order on the momentum dependence of the superconducting gap was also considered before. Results similar to the ones obtained here were found in [3, 73]. The new aspect of the theory presented here, is the explicit inclusion of both the multi-band and multi-gap aspects, with both inter- and intra-band pairing contributions, of superconductivity coexisting with SDW order.

70 Chapter 4 Superconductivity on CDW backgrounds In this chapter superconductivity on two different kinds of Charge-Density Wave backgrounds is discussed. First the situation with a momentum independent CDW gap is analyzed. Then the same is repeated for a CDW gap with d x 2 y2-wave symmetry. To keep notation short the first kind of CDW background is referred to as Charge-Density Wave, while the latter is called D-Density Wave. 4. Superconductivity on a CDW background 4.. Derivation of the gap equations As mentioned earlier there is indication of Charge-Density Waves coexisting with superconductivity in a variety of materials. Therefore, in this section we will focus on the gap-symmetry and stability of a superconducting phase on a Charge-Density Wave background with ordering momentum Q = π, π and momentum independent CDW gap. As before we start from the Hamiltonian in equation 2.3. Following exactly the same procedure as in the SDW case but with the CDW Bogoliubov transformation introduced earlier in section 2.4, we arrive at the same mean-field Hamiltonian as in the case of superconductivity on the SDW background. However, here the γ k are defined by slightly modified expressions. Again, the two DW-bands decouple in the Hamiltonian, while some indirect coupling via the two SC gap equations remains. The Hamiltonian can be diagonalized using the same two SC Bogoliubov transformations from the preceding chapter yielding the same diagonal Hamiltonian. Application of 64

71 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 65 the Bogoliubov transformations to the definitions of the CDW superconducting gaps gives [ u γ γ k = V SC 2 p k ku 2 p + 2u k v k u p v p + vkv 2 p 2 p p 2Ω γ p + γ u 2 kvp 2 2u k v k u p v p + vku 2 2 p Ω γ p p 2Ω γ tanh p 2k B T [ u γ V SC 2 p + Q k kvp 2 + 2u k v k u p v p + vku 2 2 p p p + u 2 ku 2 p 2u k v k u p v p + v 2 kv 2 p γ p 2Ω γ p Ω γ p tanh 2k B T 2Ω γ p Ω γ p tanh 2k B T ] Ω γ p tanh 2k B T where the u k, v k are the CDW coherence factors and γ {α, β}. Note, that these gap equations are valid for any of the three types of SC pairing potentials, since the CDW Bogoliubov transformation is spin-independent. Furthermore, these gap equations have the same structure as the ones for the coexistence of Spin-Density Wave with superconductivity. However, here the contributions to the gaps in the first RBZ from the second RBZ do not act pair-breaking. Also, the sign of the u k v k contributions is exactly opposite to the result for the spin orientational channel from the preceding chapter. In the same way as in the preceding chapter we find that the Charge- Density Wave gap can be obtained from W = U k W 2E k [ E α k Ω α k Ω α tanh k Eβ k 2k B T Ω β k tanh ] Ω β k 2k B T ] 4.2 which is just identical to the result for SDW coexisting with superconductivity. Therefore, we again do not use this equation but the one for the pure CDW state. This decouples the set of three coupled gap equations and allows to solve the two superconducting gap equations with W just entering as a parameter. Does the CDW background induce additional lines of nodes in the SC gaps? From the gap equations one easily finds that γ k = γ k+q 4.3 This means that the gaps need to have an even number of nodes between two momenta connected by Q. Therefore, any superconducting state with an odd number of nodes, as a d-wave state, acquires an additional line of nodes in the presence of CDW order, while states with an even number of nodes, 4.

72 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 66 as an s-wave state, are expected to experience no changes in the symmetry of the gaps, but only some changes in the momentum dependence of the gap functions. If one makes the identification k = { β k α k for k RBZ for k / RBZ 4.4 the two gap equations yield the normal state result in the limit of W. Therefore, it is assumed here that the two SC gap equations are valid for any W and not only for T CDW T c. We proceed now by choosing certain pairing interactions. We begin with a momentum independent interaction giving rise to s-wave superconductivity, followed by an interaction usually giving rise to d x 2 y2-wave superconductivity s-wave superconductivity Here the pairing interaction V SC q is taken to be V SC q = V 4.5 In the absence of CDW order this yields a momentum independent gap as seen from equation For this interaction V SC q = V SC q + Q holds. Therefore the contributions to the gaps from both RBZs enhance the pairing. This implies that superconductivity does not need to be destroyed rapidly by an increasing CDW order. For such an interaction the gap equations reduce to [ γ k = V SC p k + 4u k v k u p v p γ p p + 4u k v k u p v p γ p 2Ω γ p 2Ω γ p Ω γ p tanh 2k B T Ω γ p tanh 2k B T ] 4.6 This is similar to the equation for SDW coexisting with d-wave superconductivity. The only difference is that the inter- and intra-band interaction dressing factors are permuted. Here, in the limit of W both dressing factors are just one. Upon increasing W the intra-band dressing factor gets enhanced, while the inter-band dressing factor diminishes by the same amount. These dressing factors do not induce nodes in the gap functions. Let us note that though in principle the gaps could have a momentum dependence changing with temperature, here again no significant changes are

73 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 67 k y [π] ϕ k x [π] α/β [ ] max α β ϕ [ ] Figure 4.: Left panel: Fermi surfaces in the CDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The superconducting gaps have everywhere the same sign and there are no nodes of the gap functions. Right panel: Superconducting gaps in the presence of the CDW order as a function of the Fermi surface angle. α is red β is blue. Both panels are for n-type cuprates with W =.. expected due to the similarity with the SDW + dsc case which was confirmed numerically. As a next step the gap equations are solved numerically. This is done as before. The results for the gap functions are displayed in Fig. 4.. In the left panel the Fermi surfaces are shown together with the nodes and signs of the gap functions. However, in the present case the gaps have no nodes and have both the same sign throughout the BZ. In the right panel the gaps are shown as a function of the Fermi surface angle φ defined in the left panel. Apparently, both gaps have the same sign and almost the same magnitude. Furthermore, they both display only very weak momentum dependence and are constant within a few percent. They deviate from their constant value only close to the border of the RBZ, i. e. for intermediate angles, since there the dressing factors of the pairing interaction deviate maximally from one. However, since the sum of the dressing factors remains constant and both bands have approximately the same density of states at the Fermi level, this does not induce any significant changes. Due to the larger imbalance in the density of states at the Fermi level, the changes are slightly larger in the p-type cuprates, but still remain on the order of a few percent. With decreasing W the momentum dependence of the gaps vanishes and both gaps have the same magnitude. When W increases the momentum dependence of the gaps gets stronger, but remains small for all values of the CDW gap W. Since the gaps show only weak momentum dependence and have similar

74 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 68 magnitude, the density of states looks simply like the result for CDW order coexisting with a superconducting phase with one momentum independent gap. Such a density of states can easily be understood as combination of the CDW density of states and the effects of a standard s-wave superconductor. The resultant density of states would look similar to the SDW+sSC curve shown in the left panel of Fig However, the CDW+sSC result does not approach linearly zero for ω but is exactly zero for ω < min due to the lack of nodes in the gap function. Here min is the minimal absolute value of the two gap functions. n-type a p-type W cr W cr2 b T c [T c W=] [arb. u.] α/β V intra/inter W cr α V intra β V intra V inter W cr2 c T c [T c W=] [arb. u.] α/β V intra/inter α V intra β V intra V inter d DoSE F [/ev].5..5 W cr total α band β band W cr2 e DoSE F [/ev] total α band β band W cr W cr2 f W [ev] W [ev] Figure 4.2: Normalized s-wave superconducting transition temperature T c as a function of the CDW gap for electron- a and hole-doping b. The corresponding effective interactions and densities of states at the Fermi level are shown in panels c and e n-type and d and f p-type. The CDW gaps W cr and W cr2 where the Fermi surfaces of the first and second band, respectively, disappear are marked by the dashed lines.

75 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 69 Now the transition temperature is calculated. Here it is again necessary to calculate the gaps for a given pair W, T with very small start values for the gaps and check whether the resultant gaps are smaller or larger than the starting gaps. The critical temperature is defined as the highest temperature where one of the gaps is larger than the start value. The results for T c in n- and p-type cuprates are shown as a function of W in panels a and b of Fig These curves can be understood as a combination of the effective interactions strengths, shown in panels c and d and the density of states at the Fermi level in the pure CDW state, shown in panels e and f, which gives a measure for the number of quasi-particles participating in the pairing. The density of states is the same as in the SDW state. A measure for the strength of the dressed interactions can be obtained from V γ intra W V D γ k Dγ p + 4u k v k u p v p d 2 kd 2 p 4.7 RBZ V inter W V RBZ D α k D β p 4u k v k u p v p d 2 kd 2 p 4.8 Apparently, the intra-band interaction strengths increase with growing W, while the inter-band interaction strength decreases. As soon as the bands are shifted out of the vicinity of the Fermi level, for W W cr, W cr2, the intraband interactions do no longer grow and start to decrease rapidly and finally vanish. The inter-band interaction strength becomes zero for W = W cr +ω D. In the electron-doped case all three interaction strengths are for W approximately the same, since the bands have the same density of states at the Fermi level. Start to turn on W now. For small W the decrease in the inter-band pairing strength can be compensated by the increase in the intra-band pairing. This happens because the effect of the dressing factors is simply to transfer weight from the inter- to the intra-band pairing, or in other words: to decouple the two bands. In total, however, the interaction strength remains constant and so does the critical temperature T c. When W = W cr ω D the β-band begins to loose quasi-particles which can participate in the pairing and thus the effective intra-β-band as well as the interband interaction suddenly start to weaken. This loss cannot be compensated for and therefore T c decreases. When the β-band has vanished, there is no inter-band pairing anymore and the two bands are decoupled. In contrast to the situation in the case of coexistence of SDW and superconductivity, here the intra-band pairing for the α-band is enhanced due to the CDW order. Therefore, superconductivity survives as one-band superconductivity. In fact the critical temperature reaches a local minimum but increases again when

76 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 7 the increase in the intra-band pairing for the α-band is larger than the decrease in the inter-band pairing with increasing W. The T c reaches a global maximum when W = W cr2 ω D. At this W the intra-band dressing factor is essentially everywhere two. The slight increase of T c above its W value is made possible due to a slight increase of the density of states at the Fermi level above its W value. With further increasing W the α-band is slowly destroyed and superconductivity vanishes around W W cr2. In the hole-doped cuprates the situation is similar. However, here a few differences need to be noted: First, the roles of the α- and β-bands are interchanged in the sense that the α-band Fermi surface vanishes first. Second, the β-band has a much larger density of states at the Fermi level than the α-band and is therefore the band driving the superconductivity. Third, the density of states of the β-band decreases so strongly with increasing W that also the total density of states at the Fermi level decreases. This leads in total to two differences in the T c curve with respect to the result for n-type cuprates. First, for small W the T c W is not constant but decreases slowly and secondly, the global maximum of T cw is at W. However, the overall structure with a local minimum at W cr and a local maximum in between W cr and W cr2 is preserved. It should be noted again that for W W cr + ω D there is only one-band superconductivity, due to the α- or β-band in the n- or p-type cuprates, respectively. In total we have seen that for a momentum independent pairing interaction superconductivity is very stable against Charge-Density Wave formation. In fact, the maximal transition temperature can even be reached for CDW gaps W so large that only the Fermi surface of one band is left. In this case superconductivity is entirely due to one band and the pairing takes place inside this band only. Furthermore, the Charge-Density Wave order has only weak impact on the momentum dependence of the gap functions. Most importantly, the CDW background does not induce any nodes in the gaps and does not lead to a phase-shift in between the two gaps d-wave superconductivity We proceed by analyzing the situation for a pairing interaction given by V SC p k = V 4 cos p x cos p y cos k x cos k y = V φ k φ p 4.9 As we saw, in the absence of Charge-Density Wave order this interaction gives rise to a superconducting state with a gap of d x 2 y2-wave symmetry. This

77 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 7 pairing interaction obeys V k = V k + Q. Making use of that relation the gap equations can be simplified, giving α α k = V φ k u 2 k vk 2 φ p u 2 p vp 2 p 2Ω α p p β V φ k u 2 k vk 2 φ p u 2 p vp 2 p = β k = φ k u 2 k vk 2 p 2Ω β p Ω α p tanh 2k B T tanh Ω β p 2k B T 4. Notably the dressing factors arising due to the CDW order are the same as in the case of a SDW background and a momentum independent pairing interaction. Due to the CDW background the gaps of the two bands acquire a phase-shift of π, but the gaps have the same magnitude. This phase shift is necessary to make sure that the inter-band pairing contributions do not act pair-breaking. Furthermore, we saw that due to the CDW order there needs to be an even number of nodes in each gap between the momentum points k and k + Q. The momentum dependence of the gaps is given by φ k u 2 k v2 k. This function has lines of nodes at the border of the RBZ due to the CDW dressing factor and at the diagonals of the BZ due to φ k. Therefore, in total there are four nodes on each Fermi pocket. Since the gap equations are very similar to the momentum independent pairing on the SDW background, one can expect a similar behavior of the critical temperature, i. e. superconductivity is rapidly suppressed when the CDW order is turned on. That is because here the contributions to the gaps in the first RBZ from the second RBZ are pair-breaking. In order to get more insight into these gap equations they are solved numerically. This is done following the same procedure as before. The results for the superconducting gaps are shown in Fig In the left panel the CDW state Fermi surfaces are shown with the signs close to the different parts of it indicating the sign of the superconducting gap on that specific part of the Fermi surface. One can clearly see that the gaps show four nodes on each Fermi pocket. However, if one follows the parts of the CDW Fermi surfaces that for W reconnect to form the normal state Fermi surface, the gaps exhibit standard d x 2 y2-wave behavior. In the right panel the gaps are shown as a function of the Fermi surface angle φ. For comparison also a single d x 2 y2-fit with additional nodes at the border of the RBZ is shown. From that Figure one can again clearly see that the α and β gaps have opposite sign but the same magnitude. Because of the CDW dressing factors the d x 2 y2-fit is not perfect. For some angles the gaps deviate substantially

78 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 72 k y [π] ϕ k x [π] α/β [ ] max ϕ [ ] α β d wave fit Figure 4.3: Left panel: Fermi surfaces in the CDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the CDW order as a function of the Fermi surface angle. α is red β is blue. The black line is a d-wave fit. Both panels are for n-type cuprates with W =.. from the d-wave fit with additional nodes. The deviation gets stronger for larger W. As a next step we calculate the critical temperature. Here one can use the linearized gap equations again. In the same way as earlier one finds that the critical temperature is the temperature for which Dp α φ 2 p u 2 p v 2 2 p E α = + V 2 p p E α p tanh 2k B T Dp β φ 2 p u 2 p v 2 2 p E β +V p tanh 4. p 2 2k B T E β p is satisfied. The numerical results are shown in Fig. 4.4 for electron- and hole-doped cuprates. Again the critical temperature is determined by two contributions. The first one is the number of quasi-particles participating in the pairing, which can approximately be measured by the density of states at the Fermi level, which is shown in panels e and f of Fig The densities of states at the Fermi level are the same as before. The second contribution is the strength of the effective interactions. A measure for them

79 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 73 T c [T c W=] W cr n-type W cr2 a T c [T c W=] p-type W cr W cr2 b [arb. u.] α/β V intra/inter α V intra β V intra V inter c [arb. u.] α/β V intra/inter.5.5 α V intra β V intra V inter d DoSE F [/ev] W cr total α band β band W cr2 e DoSE F [/ev] total α band β band W cr W cr2 f W [ev] W [ev] Figure 4.4: Normalized d-wave superconducting transition temperature T c as a function of CDW gap for electron- a and hole-doping b. The corresponding effective interactions and densities of states at the Fermi level are shown in panels c and e n-type and d and f p-type. The CDW gaps W cr and W cr2 where the Fermi surfaces of the first and second band, respectively, disappear are marked by the dashed lines. can be obtained from V γ intra W V V inter W V RBZ RBZ D γ k φ2 k u 2 k v 2 k 2 d 2 k RBZ D α k φ 2 k u 2 k v 2 k 2 d 2 k RBZ D γ pφ 2 p u 2 p v 2 p 2 d 2 p D β p φ 2 p u 2 p v 2 p 2 d 2 p 4.2 They are shown in panels c and d of Fig As expected all interactions

80 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 74 weaken rapidly when W is turned on. The intra-band interaction belonging to the α-band β survives longest for the n-type p-type materials, since the corresponding Fermi surface exists longest. This leads to the T c curves in the panels a and b, where the critical temperature is very rapidly suppressed and vanishes long before the first Fermi surface is destroyed. Since the effective interaction decreases so rapidly for both kinds of materials there are no visible differences in the T c -curves. The density of states of the coexistence state can be calculated as was done before. However there is no noteworthy influence of the additional nodes in the gaps. The density of states therefore looks almost exactly like the d-wave result shown in Fig. 3.3 and is not shown separately. In summary, for a pairing interaction usually leading to d-wave superconductivity the superconductivity on the background of Charge-Density Wave order is very rapidly suppressed when the CDW gap grows. Furthermore, the Charge-Density Wave order has strong impact on the momentum dependence of the gap functions. Most importantly the CDW background induces additional lines of nodes at the border of the RBZ, so that the gaps have four nodes on each Fermi pocket. The two gaps have opposite sign, i. e. their phases differ by π Phase Diagrams Let us shortly discuss the different possible phase diagrams that can arise in systems with Charge-Density Wave order. The possible phase diagrams are schematically shown in Fig The x-axis is the W -axis. In the cuprates that corresponds directly to the doping. In other materials that might correspond to the doping, pressure, or any other parameter controlling the CDW order. For W > W cr2 there is no Fermi surface and the material is an insulator. Upon decreasing W at W cr2 the Fermi surface of one band appears. If in the system there is a momentum independent pairing interaction, possibly due to phonons, now s-wave superconductivity can emerge. The gap in that superconducting phase is not quite isotropic but shows weak momentum dependence due to the CDW background. When the CDW gap is further reduced the Fermi surface of the second band forms at W cr. The T c shows a very peculiar behavior: It exhibits a local, or maybe even global maximum somewhere between W cr2 and W cr and a local minimum around W cr. Note that this special shape of T c W might not occur if the band structure is such that the density of states at the Fermi level is varying so strongly with W that this feature cannot be resolved. When W < W cr and there is a d-wave pairing interaction present, there can form a superconducting phase with d-wave symmetry of the gaps, addi-

81 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 75 T I M FS M e+h FS ssc dsc W W cr2 W cr pressure, doping,... Figure 4.5: Schematic phase diagram for superconductivity on a Charge- Density Wave background. The system is an insulator I for W > W cr2, a metal with a single Fermi surface M FS for W cr2 > W > W cr and a metal with one electron- and one hole-like Fermi surface M e+h FS for W < W cr. For W < W cr there might be superconducting phases of different symmetry coexisting with the CDW order, while for W cr < W < W cr2 there can only be superconductivity with s-wave symmetry. tional lines of nodes at the RBZ and a π-phase shift between the two gap functions. However, this phase is rapidly suppressed by the Charge-Density Wave, therefore the coexistence regime of CDW and dsc needs to be rather narrow. In the case that both momentum independent and d-wave pairing interactions are present in the system, one of the two following scenarios may be realized. If the d-wave pairing interaction leads for W = to a T c smaller than the s-wave pairing interaction, only an s-wave superconducting phase exists on the CDW background. If on the other hand the d-wave pairing interaction yields for W = a critical temperature larger than the s-wave pairing interaction, the following happens: In the absence of a Charge-Density Wave there exists d-wave superconductivity. While W is small there exists phase-shifted d-wave superconductivity with additional lines of nodes. When W reaches a certain value the d-wave superconductivity is so strongly suppressed that Tc s wave > Tc d wave and the symmetry of the superconducting gaps changes from phase-shifted d-wave with additional nodes to s-wave. However, at the moment there is no system known that exhibits such a behavior. The possibility of CDW coexisting with superconductivity has been discussed theoretically several times, e. g. [53, 75, 27, 29, 8]. The approaches

82 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 76 and considered materials are versatile and so are the results obtained. Here it was pointed out that the symmetry of the superconducting gap function in absence of a DW background is determining the stability of the superconductivity against CDW formation, even more than in the SDW case, which to some extent might explain the variety of the results obtained in the literature. The influence of CDW on the momentum dependence of the superconducting gaps was not considered in all approaches. Results similar to some of the ones presented here were obtained e. g. in [27, 29]. 4.2 Superconductivity on a DDW background Now the same discussion is repeated for a Charge-Density Wave background with a d x 2 y2-wave CDW gap, i. e. a DDW background Derivation of the gap equations One of the most intensively discussed models for the pseudo-gap region in the hole-doped cuprates identifies this region of the phase diagram with an id-charge-density Wave DDW phase. The under-doped part of the superconducting dome is in this scenario a phase of DDW order coexisting with d x 2 y2-wave superconductivity [2]. Within this scenario in the extremely under-doped part of the superconducting dome the SC transition temperature T c is much smaller than the DDW transition temperature T DDW. Here we will assume that T c T DDW holds and analyze the coexistence of DDW and superconductivity. Though there is no indication that a DDW phase is present in n-type cuprates, numerical results for the n-type materials are presented to indicate which features are stable against weak changes in the band structure. Following the same procedure as in the previous chapter and in the previous section but now with the Bogoliubov transformation from section 2.5, the superconducting gap equations in the presence of a DDW background

83 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 77 are found to be γ k = V p k [ u k 2 u p 2 2u kv k u pv p + v k 2 v p 2 p γ p Ω γ p 2Ω γ tanh p 2k B T u k 2 v p 2 + 2u kv k u pv p + v k 2 u p 2 γ p 2Ω γ p ] Ω γ p tanh 2k B T V p + Q k [ u k 2 v p 2 + 2u kv k u pv p + v k 2 u p 2 p γ p Ω γ p 2Ω γ tanh p 2k B T + u k 2 u p 2 2u kv k u pv p + v k 2 v p 2 γ p 2Ω γ tanh p Ω γ p 2k B T ] 4.3 where the u k, v k are the DDW coherence factors. As for the superconductivity on the CDW background, these gap equations are valid for all types of SC pairing potentials considered in this thesis. The structure of the gap equations is the same as in the two preceding chapters. However, here the intra-band contributions to the gaps in the first RBZ from the second RBZ are pair-breaking, while the inter-band contributions from the first RBZ are pair-breaking. Also the signs of the u k v k -contributions are different from the previous two cases. The feedback of the superconductivity on the DDW order needs to be taken into account. That leads the DDW gap equations to become W k = U φ k p φ p W p 2E p [ E α p Ω α p tanh Ω α p 2k B T Eβ p Ω β p tanh Ω β p 2k B T ] 4.4 In the same way as in the SDW and CDW cases the influence of superconductivity is expected to be small, at least in the considered limit of T c T DDW. Therefore, we do not use this equation but simply use W as a parameter entering the two superconducting gap equations. As can be seen from their definition the DDW coherence factors obey the following relations: u k 2 = v k+q u kv k = u k+qv k+q 4.6

84 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 78 Thus from the two superconducting gap equations one finds easily that the gap functions obey γ k = γ k+q 4.7 This means that between two momenta differing by Q each gap is required to have an odd number of nodes. This is automatically satisfied by the d x 2 y2-wave gap, while it is not satisfied by the momentum independent gap. Therefore, in certain cases like s-wave superconductivity the DDW background has to induce additional lines of nodes in the gap functions, while it may affect other states like d x 2 y2-wave superconductivity only weakly. If one makes the identification { β k = k for k RBZ 4.8 for k / RBZ α k the two gap equations yield the normal state result in the limit of W. Note the minus sign in front of the α k. This means that there is no continuous evolution between the equations for T DDW T c and T DDW. This is not exactly surprising. As was shown previously, there appear to be no solutions of the DDW gap equation for small W. Therefore, it may be that only the case T DDW T c can be realized. Nevertheless, we assume here for the moment that W can be tuned continuously from zero to any finite value and that the two SC gap equations are valid for any W. We will proceed now by choosing certain pairing interactions. We begin with a momentum independent interaction giving rise to s-wave superconductivity, followed by an interaction usually giving rise to d x 2 y2-wave superconductivity s-wave superconductivity The pairing interaction V SC q is assumed to be given by V SC q = V 4.9 Usually this gives rise to superconductivity with a momentum independent gap function. But we have seen earlier that this is not possible here. That interaction obeys V SC q = V SC q + Q. Exploiting this relation the gap equations can be written as γ k = + p p = γ k [ Vk p SC uk 2 v k 2 u p 2 v p 2 ] γ 4u kv k u p pv p 2Ω γ tanh p Ω γ p 2k B T [ Vk p SC uk 2 v k 2 u p 2 v p 2 ] γ 4u kv k u p Ω γ p pv p 2Ω γ tanh p 2k B T 4.2

85 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 79 which implies that both gaps have the same sign. The dressing factor of the pairing potential can be written as ε k E k ε p E p + W k E k Wp E p 4.2 It must be noted that the dressing factor is the sum of two terms which have different symmetries. The first summand has nodes for k or p on the border of the RBZ, while the second one has nodes for k or p on the diagonal of the BZ. In fact the two parts are orthogonal. Therefore, each term leads to two gap functions with the respective symmetry. The first term leads to unconventional s-wave symmetry similar to the result in the case of a momentum independent pairing interaction on an SDW background. Here the gaps have the same sign, contrasting with the phase shift of π between the two gaps induced by the SDW background. The second term leads to d x 2 y 2- wave symmetry of the superconducting gaps, but the momentum dependence of the gap is not exactly given by φ k. Since the second term gets stronger while the first one gets weaker with growing W, the possibility arises that as a function of W the symmetry of the gap function changes. From the experience with the SDW background we can expect that the unconventional s-wave superconductivity is suppressed rapidly when the DDW background is turned on. On the other hand, for sufficiently large W there might be a phase of d x 2 y2-wave superconductivity forming. This is remarkable since the pairing interaction is momentum independent. Each gap-symmetry has its own transition temperature independent of the other gap-symmetry. The gap-symmetry with the larger transition temperature is the one being realized. In order to get a more quantitative insight into the above equations they are solved numerically. This is done following the same procedure as in the preceding section and chapters. The results are shown for n-type cuprates in Figs. 4.6 for W =.ev which leads to unconventional s-wave superconductivity and 4.7 for W =.35 which leads to d x 2 y2-wave superconductivity. In the left panels the Fermi surfaces are shown together with the signs and nodes of the gaps on the respective part of the Fermi surface. In both cases one sees that both gaps have the same sign and the nodes are at the expected positions, i. e. at the boundary of the RBZ for W =.ev and at the diagonals of the BZ for W =.35eV. In the right panels of Figs. 4.6 and 4.7 the gaps are shown as a function of the Fermi surface angle φ defined in the left panels. As expected the result for W =.ev looks very similar to the SDW result shown in the right panel of Fig. 3.. The main difference is the missing phase-shift between the gaps.

86 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 8 k y [π] ϕ k x [π] α/β [ ] max α β ϕ [ ] Figure 4.6: Left panel: Fermi surfaces in the DDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the DDW order as a function of the Fermi surface angle. α is red β is blue. Both panels are for n-type cuprates with W =.. However, there are also some minor differences to the SDW picture. Those are due to the momentum dependence of the DDW gap. The result for W =.35eV exhibits d x 2 y2-wave symmetry. Since the momentum dependence of the gap is given by the k-dependent part of the second dressing factor there are significant deviations from the φ k behavior, which is shown for comparison. These deviations occur because the dressing factor is ± on the border of the RBZ which leads to the gaps having maxima for intermediate angles φ. Such a behavior of the superconducting gaps can be modelled by taking into account higher-harmonics of the φ k function. Now the transition temperature is calculated. Since the two dressing factors are orthogonal the transition temperatures can be calculated using two sets of decoupled linearized gap equations. The transition temperature for the unconventional s-wave superconductivity can be obtained from [ Dp α up 2 v p 2 2 E α = + V 2 p p E α p tanh 2k B T Dp β up 2 v p 2 2 E β + p p 2 Ep β tanh k B T which is formally the same result as in the case of momentum independent

87 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 8 k y [π] ϕ k x [π] α/β [ max ] α.2 β d wave fit ϕ [ ] Figure 4.7: Left panel: Fermi surfaces in the DDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the DDW order as a function of the Fermi surface angle. α is red β is blue. For comparison a d x 2 y2-wave fit is shown black. Both panels are for n-type cuprates with W =.35. pairing on an SDW background. The only difference is the momentum dependence of the DDW gap. The transition temperature for d x 2 y2-wave superconductivity is found from [ Dp α 2u 2 p v p E α = V 2 p p E α p tanh 2k B T Dp β 2u 2 p v p E β + p p 2 Ep β tanh k B T In both cases for a given W the temperature T for which the equation holds is the transition temperature. The resulting superconducting transition temperatures are shown as a function of the DDW gap W in panels a and b of Fig. 4.8 for electron- and hole-doped cuprates, respectively. As before the transition temperatures are governed by the combination of the dressed interaction strengths, shown in panels c and d, and the density of states at the Fermi level, shown in panels e and f of Fig The density of states at the Fermi level of the α-band behaves very similar to the CDW and SDW states, which happens because for the α-band Fermi surface the momentum dependence of the DDW gap is weak and gets even weaker with increasing W. The β-band density of states on the other hand behaves differently than

88 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 82 [arb. u.] T c [T c W=] α/β V intra/inter DoSE F [/ev] n-type s-wave d-wave α V intra β V intra V inter.5 total α band W cr β band W [ev] W cr a c e T c [T c W=] [arb. u.] α/β V intra/inter DoSE F [/ev] p-type W cr.5 total W cr α band β band W [ev] b s-wave d-wave d α V intra β V intra V inter f Figure 4.8: Normalized s- red and d-wave black superconducting transition temperatures T c as a function of the DDW gap for electron- a and hole-doping b. The corresponding effective interactions and densities of states at the Fermi level are shown in panels c and e n-type and d and f p-type. The effective interactions for s-wave superconductivity are solid, for d-wave dashed. The DDW gap W cr where the Fermi surface of the α-band disappears is marked by the dashed lines. in the previous cases: It monotonously decreases, but does not reach zero for any finite W. This is a result of the momentum dependence of W. In total the density of states first increases, reaches a maximal value right before W cr, drops when the α-band Fermi surface vanishes and then follows the monotonous decrease of the β-band density of states.

89 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 83 The strengths of the dressed interactions can be obtained from V γ intra W V u 2 k vk 2 2 d 2 k Dp γ u 2 p vp 2 2 d 2 p 4.24 RBZ V inter W V D γ k D α k u 2 k vk 2 2 d 2 k RBZ D β p u 2 p vp 2 2 d 2 p 4.25 RBZ RBZ for the unconventional s-wave superconductivity solid lines in Fig. 4.8 c and d and from V γ intra W V D γ k 2u kv k 2 d 2 k Dp γ 2u 2 p v p d 2 p 4.26 RBZ V inter W V D α k 2u kv k 2 d 2 k RBZ D β p 2u p v p 2 d 2 p 4.27 RBZ RBZ for the d-wave superconductivity dashed lines in Fig. 4.8 c and d. The effective interactions for the unconventional s-wave superconductivity rapidly decrease. The α-intra and inter-band components vanish around W cr together with the α-band Fermi surface. However, the β-band intracomponent is only strongly dampened but never actually eliminated. Therefore for W > W cr and a sufficiently large V one-band superconductivity may be observed. This evolution of T c is reflected by the results in panels a and b of Fig When the DDW gap is turned on T c is strongly suppressed below numerical resolution, but in fact never vanishes. The behavior of the d-wave effective interactions is completely different. For W there is no effective interaction. When W is turned on, all effective interactions increase with the α-intra-band component being largest and β-intra-band smallest. The α-intra-band component and the inter-band pairing reach a maximum at W W cr ω D and then decrease due to the vanishing of the α-band Fermi surface. The weak β-intra-band component increases further. The d-wave T c -curves follow this overall behavior. For the electron-doped cuprates there is a large-w regime where the d-wave T c is much larger than the s-wave T c and thus there is a phase with d-wave superconductivity. Since the d-wave superconductivity is mostly driven by the α-band which vanishes at W cr the d-wave superconductivity vanishes there as well. For both kinds of materials there is a region of intermediate W where the d- and s-wave T c s are similar. However, for small W the unconventional s-wave T c is always much larger than the d-wave T c. Here it should be noted that there can be no state with a mixture of s- and d-wave gaps. The non-linearized gap equations cannot be decoupled

90 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 84 into one set for each symmetry. The coupling through Ω γ k has the effect that even for parameter where both symmetries have similar T c s, always only one symmetry is realized and no mixture of both gaps exists. The density of states of the possible superconducting states on a DDW background is similar to cases discussed earlier. The result for unconventional s-wave superconductivity is almost identical to the result for unconventional s-wave superconductivity on an SDW background. There are only very small differences due to the momentum dependence of the DDW gap. Close to the Fermi level the density of states is dominated by superconductivity and thus there are no differences between the SDW and DDW backgrounds. The d- wave superconductivity on the DDW background has a density of states very similar to the d-wave result shown in Fig However, here the increase of the density of states close to the Fermi level is initially steeper. In both cases the density of states goes linearly to zero towards the Fermi level, but in each case with a different slope, depending on W and the kind of superconductivity present. Here it should be stressed again that there are no solutions of the DDW gap equation with W < W cr. For W > W cr only a rather small β-band Fermi pocket exists and therefore no sizeable T c can be expected. But in both types of cuprates one has T c W cr T c W = /2, which in actual materials might still be observable d-wave superconductivity We proceed by analyzing the situation for a pairing interaction given by V SC p k = V 4 cos p x cos p y cos k x cos k y = V φ k φ p 4.28 Taking advantage of V SC k = V SC k + Q we find γ k = V SC p k γ p Ω γ p 2Ω γ tanh p 2k B T p γ + V SC p p k p 2Ω γ p Ω γ p tanh 2k B T = γ k 4.29 Apparently, there is no dressing of the interaction present in this equation. The only influences of the DDW background are the two-band character of

91 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 85 k y [π] ϕ k x [π] α/β [ max ] ϕ [ ] α β Figure 4.9: Left panel: Fermi surfaces in the DDW α-band red, β-band blue and normal state green together with the border of the RBZ black and the definition of the Fermi surface angle. The + and signs indicate the phase of the superconducting gaps on the next part of the Fermi surface. The grey points mark the positions of the nodes of the gap functions. Right panel: Superconducting gaps in the presence of the DDW order as a function of the Fermi surface angle. α is red β is blue. Both panels are for n-type cuprates with W =.. the superconductivity and a phase-shift of π between the two gaps. Furthermore, due to the specific interaction chosen here the superconductivity will have d x 2 y2-wave symmetry leading to an odd number of nodes between points connected by Q, as required due to the DDW background. After inclusion of the cut-off factors the gap equations can be solved. The results for the momentum dependence of the gaps are shown in Fig In the left panel the DDW Fermi surfaces are shown with signs indicating the phase and gray points indicating the nodes of the superconducting gaps. The gaps exhibit standard d x 2 y2-wave behavior with a π-phase shift between the gaps. The values of the gaps on the DDW Fermi surface as a function of the Fermi surface angle φ are shown in the left panel. The resulting density of states is very similar to the d-wave results shown in Fig. 3.3 and therefore not shown. There are only minor differences outside the gapped region around the Fermi level due to the momentum dependence of the DDW gap. The transition temperature can be calculated from the linearized gap

92 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 86 T c [T c W=] DoSE F [/ev] n-type.5 total α band β band W cr W [ev] W cr a c T c [T c W=] DoSE F [/ev] p-type W cr total.5 α band W cr β band W [ev] b d Figure 4.: Normalized d-wave superconducting transition temperature T c as a function of the DDW gap for electron- a and hole-doping b. The corresponding densities of states at the Fermi level are shown in panels c n-type and d p-type. The DDW gap W cr where the Fermi surfaces of the α-band disappears is marked by the dashed line. equations. One easily finds that at T c = + V +V p p D α p φ 2 p 2 E α p tanh Dp β φ 2 p 2 Ep β tanh E α p 2k B T E β p 2k B T 4.3 is satisfied. The resulting transition temperature is shown in Fig. 4. for electron- a and hole-doped b cuprates. This time the pairing interaction is not dressed and thus independent of the DDW gap W. Therefore the evolution of T c with W is entirely determined by the evolution of the density of states at the Fermi level with W, shown in panels c and d, and the corresponding shift of the Fermi surface. Because of the increasing density of states T c grows when W is turned on. The transition temperature reaches a maximum around W W cr ω D and then decreases. The increase of T c is much larger than the increase of the density of states since T c depends

93 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 87 approximately exponentially on the density of states at the Fermi level. Furthermore, with increasing W the α-band Fermi surface approaches, ±π and ±π, which results in a larger average pairing interaction experienced per quasi-particle. When the α-band Fermi surface has vanished, superconductivity vanishes. Of course there can be superconductivity in the β-band alone, but the pairing interaction vanishes for quasi-particles close to the diagonals of the BZ, where the β-band Fermi surface is located, and thus the resulting T c is extremely small. Again, it must be stressed that from the pure DDW gap equation follows, that W must be larger than W cr. For such a DDW gap no sizeable T c is possible Phase Diagrams Here the results from the preceding section will be summarized by discussing possible phase diagrams with superconductivity on a D-Density Wave background. The possible phase diagrams are shown in Fig. 4.. The x-axis is the D-Density Wave gap W which in actual systems like the cuprates corresponds to doping. But any parameter that controls the DDW gap is possible. For large W there is only the β-band Fermi surface in the system and the system is a metal M h FS. In this regime there is no superconductivity with a sizeable transition temperature possible. For W < W cr there are two Fermi surfaces. If there is a d-wave pairing interaction in the system then in this regime there exists a superconducting phase with d-wave symmetry and a phase shift of π between the two superconducting gaps, as indicated by the solid blue line in Fig. 4.. Due to the increase of the density of states at the Fermi level with W and the shift of the α-band Fermi surface towards points where the pairing potential is larger, the transition temperature Tc d phase can increase with growing W. When the α-band Fermi surface begins to vanish, the phase-shifted d-wave superconductivity vanishes. This results in a maximal Tc d phase at approximately W W cr ω D. However, the exact shape of Tc d phase depends on the details of the band structure, but the phaseshifted d-wave superconductivity is certainly rather stable against the DDW background until the first band vanishes. This phase diagram is overall compatible with the observed phase diagram of the hole-doped cuprates if one identifies the pseudo-gap phase with the DDW phase. But it should be noted that to current knowledge the pseudo-gap phase is no actual thermodynamic phase [78], which contradicts the interpretation of the pseudo-gap phase as the DDW phase. Furthermore, in the cuprates superconductivity appears to be suppressed by the pseudo-gap, while here the T c is first enhanced and then reduced. However, this intermediate increase might be removed due

94 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 88 T M h FS M e+h FS d-phase-sc d-wave-sc u-ssc W W cr pressure, doping,... Figure 4.: Schematic phase diagram for superconductivity on a D-Density Wave background. The system is never an insulator. For W > W cr the system is a metal with a single hole-like Fermi surface M h FS and a metal with one electron- and one hole-like Fermi surface M e+h FS for W < W cr. For W > W cr there is no superconducting phase with an observable T c coexisting with the DDW order, while for W < W cr there can be superconductivity with d-wave symmetry and a π-phase shift between the two gaps or superconductivity with unconventional s-wave symmetry and no phase shift between the gaps. For W close to W cr there can also be d-wave superconductivity without phase shift. to the doping dependence of or influences of the DDW order on the pairing interaction not considered here. If in the system a momentum independent pairing interaction exists, there is an unconventional s-wave superconducting phase with transition temperature Tc u ssc solid red for small W. When W is turned on the s-wave superconductivity is rapidly suppressed. The momentum independent pairing interaction gets dressed by the DDW background and for large W can give rise to d-wave superconductivity without phase shift between the two gaps with transition temperature Tc d wave solid black. d-wave superconductivity does not coexist with s-wave superconductivity. Only the gap-symmetry with the larger T c forms. Depending on the band structure, for W close to W cr the transition temperature Tc d wave can become much larger than the Tc u ssc, so that a change of the symmetry of the order parameter can be observed. However, it should be noted that in this process only the position of the nodes changes, not the number of nodes. If both types of pairing interactions are present there can be one of three

95 CHAPTER 4. SUPERCONDUCTIVITY ON CDW BACKGROUNDS 89 cases. If the d-wave pairing interaction has the largest T c at W it will remain larger than the other transition temperatures for all W not shown. If the s-wave Tc u ssc is much larger than the Tc d phase at W, this will be the case for all W not shown. If Tc u ssc is only somewhat larger than Tc d phase, with increasing W the Tc u ssc will be suppressed while Tc d phase increases and the symmetry of the superconducting gaps changes, as shown in Fig. 4.. For certain band structures there can even be the possibility that Tc d wave for W close to W cr is larger than Tc d phase not shown. In that case another phase transition occurs and the symmetry of the superconductivity changes from phase shifted d-wave to d-wave without phase shift. But note that this second change of symmetry is only possible for a rather special choice of parameters. Finally, it should be noted that the region in the phase diagram with < W < W cr is not allowed, according to the DDW gap equation. Therefore, one can conclude that within a mean-field approach there is no superconductivity on a DDW background with a sizeable SC T c. The coexistence of DDW and superconductivity is much less studied than the coexistence of SDW/CDW and superconductivity, probably because DDW order has not been identified unambiguously in a single material. Until now only the coexistence of DDW with d x 2 y2-wave superconductivity has been discussed, e. g. [, 5, 5]. The approaches used were different from the present one and yielded different results. The d x 2 y2-wave superconductivity was there found to be less stable against DDW order than here. Furthermore, in contrast to the results presented here no phase-shift of π between the two SC gaps was found. Both differences are results of the different approaches to the problem.

96 Chapter 5 Spin Susceptibility in superconductors The imaginary part of the dynamical spin susceptibility is accessible by inelastic neutron scattering measurements [83, 47]. This makes the spin susceptibility an interesting quantity for comparison of theory and experiment. Here the analytical expression for the dynamical spin susceptibility in the superconducting state will be given. This expression will be evaluated numerically for both hole- and electron-doped cuprates and compared to the available experimental data. The momentum and frequency dependent susceptibility tensor is defined by [85, 3] χ ij q, q, iω n = β dτe iω nτ T τ S i qs j q 5. where the S i q are the spin-operators defined in equation A.. For the most common systems the susceptibility is non-zero for q = q only. Here, it should be noted that in the normal and superconducting states spin-rotation symmetry holds and thus [23] χ zz = 2 χ+ 5.2 Thus, as long as one is not interested in the absolute magnitude of the susceptibility, the superscripts can be dropped. Therefore, for now the notation χ q, ω is used, with the subscript indicating the bare susceptibility, which has to be dressed by residual interactions. Note, that in the DDW state also q = q + Q gives non-zero χ and in the presence of a magnetic field the 9

97 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 9 spin-rotation symmetry is broken. Hence, later the notation will have to be adjusted. The Spin Susceptibility of a superconductor, defined by the Hamiltonian in Equation 2.9, is well known. It is given by, e. g. [9, 22, 4] χ BCS q, ω = k { c n k, q fe k+q fe k ω E k+q E k + iγ + c ak, q fe k+q fe k 2 ω + E k+q + E k + iγ + c ak, q fe } k+q + fe k 2 ω E k+q + E k + iγ 5.3 where the normal and anomalous coherence factors c n and c a are given by c n k, q = 2 + ε k ε k+q + k k+q 5.4 E k E k+q c a k, q = 2 ε k ε k+q + k k+q 5.5 E k E k+q and Γ can be used to model lifetime effects and experimental resolution. Typical values for Γ are of the order of mev. In the normal state, i. e. for k =, the susceptibility becomes χ q, ω = fε k+q fε k 5.6 ω ε k+q ε k + iγ k In order to be able to account for the experimental observations it is necessary to include residual effective interactions. To this end the random phase approximation RPA is employed [25, 85]. The dressed susceptibility within the RPA reads χ q, ω χ RP A q, ω = 5.7 Uq, ωχ q, ω The function Uq, ω depends on the specific model. A common approximation is Uq, ω = U J 2 cos q x + cos q y 5.8 which might originate from a Coulomb on-site repulsion U and an antiferromagnetic nearest-neighbor interaction J. The coefficients U and J are adjusted to yield optimal agreement with experiment. Note, the theory does not contain any free parameter since UQ is determined by the observed position of the resonance. The ratio of U and J is determined by the observed dispersion of the resonance mode, while the energy dispersion and the superconducting gap can be extracted from other experiments as ARPES.

98 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS Susceptibility in hole-doped cuprates The equations for the susceptibility in both the superconducting and normal states are evaluated numerically now. To this end the doping level has to be fixed. The chemical potential is chosen to be µ =.296eV corresponding to an approximately optimal hole doping level of x.6. The numerical results for the imaginary and real parts of the bare and RPA susceptibility in the normal dashed lines and two different superconducting solid lines states are shown in Fig. 5. for momentum fixed at q = Q as a function of frequency ω. In order to understand the shape of the curves it is helpful to note that the real and imaginary part of the bare normal state susceptibility can be written as Reχ q, ω = k Imχ q, ω = k π k ω ε k+q ε k fε k+q fε k ω ε k+q ε k 2 + Γ Γ fε k+q fε k ω ε k+q ε k 2 + Γ 2 5. fε k+q fε k δ ω ε k+q ε k 5. Note, the approximation becomes exact for Γ +. The Fermi function factors make sure that only processes connecting a free with an occupied state are counted. The magnitude of the imaginary part of the susceptibility is then determined by the density of states of the function ε k+q ε k. As a result the imaginary part of the bare susceptibility is zero at ω = and increases approximately linearly with increasing ω. For larger frequencies this behavior changes, but larger frequencies are not considered here, since they are less interesting with respect to comparison of theory and experiment. Correspondingly, since real and imaginary parts are connected by Kramers- Kronig relation [25], the real part of the bare susceptibility is approximately constant for the frequencies considered here. The imaginary part of the RPA susceptibility is given by Imχ RP A q, ω = Imχ q, ω U q Reχ q, ω 2 + U 2 q Imχq, ω The combination of the linear increase of the imaginary and the almost constant real part of the bare susceptibility in the normal state at Q leads to a broad small peak at some finite frequency in the imaginary part of the RPA susceptibility, as shown in Fig. 5.. For momenta close to Q the susceptibility is similar to the result at Q. However, for momenta deviating more

99 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 93 Re χ Q,ω [arb. u.] Im χ Q,ω [arb. u.] U- normal dsc ssc normal dsc ssc ω res.5..5 ω [ev] a b Re χ RPA Q,ω [arb. u.] Im χ RPA Q,ω [arb. u.] ω res normal dsc ssc.5..5 ω [ev] c normal dsc ssc d Figure 5.: Imaginary b and d and real a and c parts of the susceptibility in three different states, normal blue, d x 2 y 2 SC red, s SC black, as a function of frequency for q = Q. Left the bare susceptibilities are shown, RPA susceptibilities are in the right panels. The dashed lines show UQ used in the RPA and the resulting position of ω res. Gap values used here are d = 4meV and s = 33meV, Γ = 2meV and temperature K. strongly from Q, i. e. q = η π, π with η.7, the real part of the bare susceptibility is strongly reduced and thus the imaginary part of the RPA susceptibility does not exhibit any noticeable intensity any more, as shown in Fig. 5.3.d. In the superconducting state the bare susceptibility gets strongly reshaped due to the coherence factors in the susceptibility and due to the opening of the gap k in the elementary excitation spectrum. Because of k the eigenenergies now obey E k which results in a negligible first term of the bare susceptibility. The second and third term both give non-zero contributions even at T. The real and imaginary parts of the third term

100 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 94 can be written as Reχ q, ω = k Imχ q, ω = k π k c a k, q ω E k+q + E k fe k+q + fe k ω E k+q + E k 2 + Γ 2 c a k, q Γ fe k+q + fe k ω E k+q + E k 2 + Γ 2 c a k, q fe k+q + fe k δ ω E k+q + E k 5.3 Again, the last line becomes exact for Γ +. The expressions for the second term look similar. However, since E k the second term does not contribute to the imaginary part and does add only a structureless background to the real part of the bare susceptibility at positive frequencies. For the present discussion it is thus sufficient to regard the third contribution as the total bare susceptibility in the superconducting state. In the limit of small temperatures the imaginary part of the susceptibility is determined by the δ-function factor and the anomalous coherence factor. For a given q the δ-function makes sure that Imχ is zero up to ω cr = min k E k + E k+q = min k,k+q F S E k + E k+q 5.4 The minimum value of this function is located at k and k + q on the underlying normal state Fermi surface. For a given q the critical frequency up to which Imχ vanishes is thus given by the sum of the absolute values of the superconducting gaps at the points of the normal state Fermi surface that are connected by q. The behavior at frequencies ω ω cr depends strongly on the symmetry of the superconducting order parameter. The anomalous coherence factor at the Fermi surface for s- and d x 2 y2-wave symmetry and q = Q is: c s wave a = 2 c d wave a = k k+q k k+q = 5.5 = 2 k = k Because of the sign-change of a d x 2 y2-wave gap function under shift by Q, the anomalous coherence factor has the value in contrast to for a momentum independent s-wave gap. Therefore the d x 2 y2-wave superconducting susceptibility makes a discontinuous jump from zero to a finite value at ω = ω cr, while for an s-wave superconductor the imaginary part of the susceptibility

101 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 95 starts to increase linearly from zero at ω = ω cr as shown in Fig. 5..b. There the discontinuous jump has been broadened due to finite temperature and Γ. The real part of the susceptibility shows correspondingly a very high peak at ω cr in the d x 2 y2-wave case, while it jumps to a larger constant value in the s-wave case. The discontinuous jump in the imaginary part of the susceptibility of the d x 2 y2-wave superconductor at zero temperature and in the absence of damping would correspond not only to a high peak, but a logarithmic divergence in the real part. It must be noted that the real part of the susceptibility in the s-wave superconductor is overall reduced with respect to the d x 2 y2-wave superconductor and the normal metal, which results from the anomalous coherence factor as well. What happens for momenta away from Q? In the s-wave case the gap does not depend on the momentum and thus the susceptibility remains mostly unchanged, in particular ω cr does not depend on q. In the d x 2 y2-wave superconductor, however, there are actually three ω cr, which are degenerate at q = Q. This degeneracy is removed for momenta away from Q. The three different processes responsible for the three ω cr are indicated in Fig. 5.2.a. The processes 2 and 3 are Umklapp processes. The dispersion of the critical frequencies is shown in Fig. 5.2.b. Away from q = Q the imaginary part of the susceptibility is zero up to the smallest ω cr i, where it makes a discontinuous jump. For larger frequencies Imχ increases linearly and exhibits, depending on the momentum, up to two additional discontinuous jumps at the other ω cr i. All these jumps have corresponding peaks in Reχ. Here, it should be noted that the real part of the susceptibility shows an overall decrease for smaller momenta, similar to the normal state. The imaginary part of the RPA susceptibility can diverge under certain circumstances, forming a so-called resonance peak, which entirely dominates the susceptibility. It can be seen from the expression given in equation 5.2 that two conditions need to be fulfilled in order to give rise to a resonance peak. First, the imaginary part of the bare susceptibility needs to be very small. Second, the inverse of the residual effective interaction has to equal the real part of the bare susceptibility: Imχ q, ω res 5.7 Reχ q, ω res = U q 5.8 The first condition is satisfied in a superconductor for ω < ω cr. The second condition can be easily satisfied if the real part of the bare susceptibility exhibits a sufficiently large peak around ω cr, which is the case in the d x 2 y 2- wave superconductor. In fact, in the limit of zero temperature and Γ a resonance peak forms for any U below a critical U cr, since the peak in Reχ

102 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 96 k y [π] 2 a k x [π] 2 Critical frequency/.5.5 b ω cr q[π,π] ω 2 cr ω 3 cr Figure 5.2: Panel a: Normal state Fermi surface red lines together with the value of the d x 2 y2-wave gap function. The arrows indicate the connected Fermi surface points responsible for the three ω cr i. Panel b: Dispersion of the ω cr i for momenta along the diagonal of the Brillouin zone. becomes a logarithmic divergence. The second resonance condition cannot be satisfied in the s-wave superconductor and therefore there the formation of a resonance is impossible. The reason for this is that Reχ s wave q, ω = < Reχ normal q, ω =. This means that the resonance condition could be satisfied only for U such that the normal state were instable. Therefore the resonance peak is a direct reflection of the symmetry of the superconducting gap. It can only form if k = k+q is fulfilled. The formation absence of the resonance peak in the imaginary part of the RPA susceptibility and its reflection in the real part can be seen nicely in the right panels of Fig. 5. for a d x 2 y2- s-wave superconductor. In the remainder of this chapter only d x 2 y2-wave superconductors are considered. The resonance forms always at ω < ω cr q, since otherwise Imχ would not be small. Therefore, the structure of the RPA susceptibility is determined in the superconducting state by the smallest ω cr q. Furthermore, it should be noted that the critical frequency ω cr vanishes at the momentum q node bridging the Fermi surface at the diagonal of the BZ where the superconducting gap has its node. Therefore, the resonance peak has to vanish. For momenta further away from Q the real part of the bare susceptibility strongly decreases. This has the effect that the resonance peak is observed only close to Q and may reappear only weakly after vanishing at q node. The resulting Imχ RP A is shown in Fig. 5.3.c. There, one can clearly see how the resonance peak follows the shape of ω cr. Apart from the resonance mode

103 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 97 there is no other major structure. In Fig. 5.3 experimental measurements and calculations of the imaginary part of the susceptibility are displayed. The left panels show the difference of the susceptibilities at K and 34 K. Panel b shows the measured difference of the susceptibilities at K and 34 K, panel d shows the susceptibility in the normal state. At K the sample was in the superconducting state, while at K and 34 K it was in the normal state. From panel b, where some U-shaped intensity can be observed, one can see that the imaginary part of the susceptibility changes substantially as a function of temperature in the normal state. Theory failed entirely to capture this behavior, since the calculated imaginary part of the susceptibility in the normal state is almost completely temperature independent. A possible explanation for that might be that the sample was actually not in the normal state at K and maybe even already at 34 K, but in a pseudo-gap state or another non-normal state. It is also possible that the U-shaped intensity is a signature of magnetic collective modes due to correlated local moments. In panel a one can observe an X-shaped dispersion of the intensity. The upwards dispersing branch can probably be attributed to changes in the non-superconducting state, since this part of the intensity is mostly already present in panel b. The downward dispersing branch is probably entirely due to superconductivity. Upon comparison of panel a with panel c, which shows the calculated results, one can clearly see that the downward dispersing branch is captured well by this theory. The only difference is that the experimental result disperses down to smaller frequencies than the calculated. This might be a result of a different distance of the different Fermi surface branches along the diagonal of the BZ in the theoretical and the real Fermi surface. In summary, the genuine features of the superconducting state in the holedoped cuprates are captured rather well with the presented theory. However, the underlying normal state employed here appears to be a completely inadequate description of the hole-doped cuprates above T c. If the underlying normal state is not the correct description for the cuprates, this raises the question if a superconducting state based on this can be a correct description of the superconducting state. 5.2 Susceptibility in electron-doped cuprates The resonance peak is a well-established feature of hole-doped cuprates [38]. Unfortunately, for a long time it was not possible to perform conclusive INS experiments on the electron-doped cuprates. Therefore it remained unclear

104 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 98 a b c d Figure 5.3: Experimental and theoretical results for Imχ as a function of frequency and momentum along the diagonal of the BZ for different temperatures. Panel a: Difference of the experimental results in the normal state at 34 K and in the superconducting state at K. Panel b: Difference of the experimental results in the normal state at 34 K and K. Panel c: Difference of the theoretical results for the normal state at 34 K and the superconducting state at K. This is almost just the superconducting state result, since the normal state result is much smaller. Panel d: Theoretical result for the normal state. This is almost temperature independent. Therefore the difference of the K and 34 K results is not shown. whether the resonance peak is present in all kinds of cuprates. In 26 finally the resonance peak was observed in the first electron-doped cuprate compound [36]. The experimental results from this work are addressed in this section. In order to model the observed susceptibility in superconducting electrondoped cuprates the chemical potential needs to be adjusted. It is taken to be µ =.5eV corresponding to an electron-doping level of x =.5. The resulting Fermi surface is very similar to the one in the previous section. However, here the Fermi surface circles centered around ±π, ±π are smaller and thus the distance of the different Fermi surface sheets along the diagonal of the zone is larger. Moreover, in experiments on electron-doped cuprates there is indication that the d x 2 y2-wave gap is extremal at the hot spots of the Fermi surface, i. e. the points connected by Q, e. g. [88]. In order to account for this behavior the superconducting gap is here assumed to have

105 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS e doped h doped k y [π].5 Q [ hs ] k x [π] ϕ Figure 5.4: Left panel: Fermi surface in the electron-doped red and holedoped blue cuprates. The hot spots are indicated in both cases by the wave-vector Q, which connects these Fermi surface points. The dashed lines mark the nodes of the SC gap. Right panel: Superconducting gaps in the e- red and h-doped blue cuprates as a function of the Fermi surface angle normalized to the value at the hot spots hs. the form k = 2 cos k x cos k y + 2 cos 2k x cos 2k y 5.9 = =.ev 5.2 Here it should be noted that similar results can be obtained using different higher harmonic functions than the one used here, see e. g. [9]. The resulting behavior of the gap as a function of the Fermi surface angle is shown in Fig The gap has at the hot spots the absolute value hs = 5meV. Based on the results presented in [88] this is the right order of magnitude, though the exact gap value of the sample used in the experiments from [36] remains unknown. The bare susceptibility of such a superconducting state looks very similar to the one presented in the previous section. Of course one has to adjust the frequency scale because of the reduced magnitude of the gap. Moreover, the different chemical potential and momentum dependence of the gap lead to changes of the ω cr i q. Most notably, ωcr reaches zero much earlier than before, at about.9π, π.

106 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS Im χ Q,ω [arb. u.] RPA a b ω cr ω [ hs ] c Im χ Q,ω [arb. u.] RPA q [π,π] Figure 5.5: a and b: Imχ RP A at q = Q as a function of frequency in the normal black dashed and superconducting state red solid for two different values of Γ: Γ =.5meV a and Γ = 3meV b. All other parameter remain the same. Note the shift of the resonance. c: Intensity map of Imχ RP A as a function of frequency and momentum along the diagonal. For comparison ω cr q is shown white dashed. Γ =.5meV was used. For a suitable value of UQ again a resonance peak in the imaginary part of the RPA susceptibility emerges. The resonance peak has to be confined to a rather narrow region around q = Q, because of the changes in the dispersion of the ω cr i. In experiment the resonance is observed at ω res = mev. For the chosen value of the gap, this is.ω cr Q, which poses a challenge to the theory from the preceding section. However, since the exact value of the gap is unknown, the resonance might also be located below ω cr Q. In Fig. 5.5 the Imχ RP A Q, ω is shown for the normal and superconducting state. Panel a is for a small value of Γ, while b is for a larger value of Γ. All other parameters are the same. Apparently, the effect of increasing Γ is to shift the position of the resonance from below ω cr to above ω cr. The reason for that is that Imχ is larger above ω cr than below. In panel b of Fig. 5.5 the broadening of the resonance due to Γ is so large that the larger Imχ at higher frequencies can shift the peak position of the resonance to above ω cr. Therefore, a resonance peak located above ω cr is not impossible. However, a large Γ has also the effect to substantially broaden the resonance, which somehow makes the term "resonance" inadequate. Note, that this effect is similar to the results obtained in [96]. In panel b of Fig. 5.5 one can clearly see that due to the large Γ the susceptibility of the superconducting state somewhat resembles the normal state result, while the difference between the two states is much more pronounced in panel a. Which one of the two values of Γ is correct is unclear. Neverthe-

107 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS Intensity 2K-3K [arb. u.] Im χ dsc -Im χ normal [arb. u.] RPA RPA ω [mev] Figure 5.6: Comparison of the experimental and theoretical results. The blue squares and error bars are the difference in the measured intensity between T = 2K and T = 3K data taken from [36], blue line is a guide to the eyes, while the red curve shows the difference between the calculated Imχ RP A in the superconducting and normal states. The figure is for q = Q. In the calculation Γ = 2meV was used. less there is one important hint toward a larger Γ. As can be seen from Fig. 5.6, where the difference of the imaginary part of the normal and superconducting susceptibility with large Γ and the difference of the measured INS intensity in the normal and superconducting state is shown, the agreement between theory and experiment seems to be rather good for the large value of Γ. However, since the experimental curve is not the susceptibility extracted from the measurements, the agreement might be coincidental and misleading. Moreover, unfortunately from that agreement one cannot decide whether the large Γ accounts for experimental resolution or for intrinsic effects like for example short-lived quasi-particles. Clearly, the resonance in the n-type materials is located much closer to ω cr than in the p-type ones. Since the damping due to non-zero Imχ increases when the resonance approaches ω cr, the intensity of the resonance decreases. Therefore, the resonance peak is here expected to be a much less dominating, but still very large, feature than in the hole-doped materials. Experimentally, the momentum dependence of the resonance peak in the p- and n-type cuprates appears to be very different. While in the holedoped cuprates the resonance peak exhibits pronounced dispersion, as seen in the previous section and e. g. [2], there is no observable dispersion in the electron-doped cuprates [36]. The resonance peak in the electron-doped

108 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 2 cuprates appears to consist of a single peak at q = Q with a limited frequency and momentum resolution. Generally, a reduced momentum width of the resonance mode in electron-doped cuprates can be understood as a result of an enlarged distance of the different Fermi surface sheets along the diagonal of the Brillouin Zone. However, this effect alone cannot explain the missing dispersion. Furthermore, one needs to adjust the momentum dependence of the effective interaction U q accordingly. If one takes U to vanish and only J as non-zero, U q is maximal at q = Q. Now the combination of the resonance at Q being located very close to ω cr and the decreasing of both U q and Reχ q, ω away from Q, results in the resonance condition being satisfied only in the immediate vicinity of Q, where ω cr q can be regarded as dispersionless. This results in the intensity map of Imχ RP A q, ω shown in Fig There, one can clearly see that the resonance consists of a peak centered at Q, ω res and exhibits a finite momentum and frequency resolution but no dispersion in agreement with the measurements of [36]. Here, it should be pointed out that the presented interpretation of the numerical results in the electron-doped cuprates is not generally accepted [74, 7]. There are two major concerns. First, it is argued that the chosen value of hs is too large for the material used in the measurements from [36]. For example in [88] on a different material with an even larger T c than the material from [36], the measured gap was hs = 2.5meV, which is by a factor of 2 smaller than the value used here. Such a small value of hs would indeed render the description of the observed data by the presented theory impossible. However, the exact value of hs is unknown and therefore this debate can only be ended by a conclusive measurement in the sample used in [36]. Second, the chosen form of U q with U = and J nonzero is criticized. However, in order to capture the phenomenology correctly U q is required to be strongly momentum dependent. The origin of this strong momentum dependence and the weak on-site repulsion U is unclear and further analysis is required, but beyond the scope of this thesis Temperature evolution towards a low-temperature SDW phase In electron-doped cuprates there might exist a doping range where T N < T c, i. e. for high temperatures there is a pure superconducting and at low temperature there is a SDW phase, either pure or coexisting with superconductivity. In the pure superconducting phase a resonance peak forms, while the susceptibility in the presence of SDW order is dominated by the forma-

109 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 3 tion of a Goldstone mode, which is located at q = Q and zero frequency [23]. A resonance peak at higher frequencies cannot form since Imχ cannot be satisfied there in the presence of a Goldstone mode. Thus, naively one can expect that the resonance peak of the superconducting state transforms into the Goldstone mode of the SDW state when the temperature is lowered. Within the presented theory this requires a temperature dependent J. In order to obtain the lowering of the resonance frequency with increasing temperature, J needs to increase. In order to be able to numerically model the described temperature evolution of the resonance peak, the onset temperature of the SDW order T N has to be fixed. To be concrete T N = T c /6 is taken. Furthermore, a concrete temperature dependence of U Q has to be chosen. For the sake of simplicity, J varies linearly between values chosen such that the resonance peak at Q is located at T =.75T c at ω res.92ω cr and for T = T N at ω res =. However, the actual temperature dependence of J has no impact on the qualitative features of the temperature evolution. The mean-field temperature evolution of the superconducting gap is taken into account. The resulting temperature evolution of Imχ RP A q, ω is shown in Fig There, one can see that initially panels a and b the resonance mode keeps its flat dispersion and the resonance frequency is affected only weakly. However, when temperature is decreased even further, the resonance mode starts to exhibit a V-shaped upward dispersion centered at Q and the resonance frequency at q = Q decreases panel c. Finally, when the Néel temperature is reached, the resonance mode transforms into the Goldstone mode of the SDW state at zero frequency and q = Q with an almost linear V-shaped dispersion. Until now no INS experiments have been performed on electron-doped cuprates with T c > T N Influence of a magnetic field The question of the effect of a magnetic field on the resonance has been considered both experimentally [2] and theoretically [66, 38, 39]. In [38, 39] the magnetic field was aligned along the c-axis of the material. In this case Abrikosov vortices form and complicate the calculations. On the other hand, [66] considered a magnetic field oriented parallel to the a-b-plane. In this case there are no vortices and the magnetic field has only two effects. First, the energy band becomes split into two due to the Zeeman effect and second, the superconducting gap gets reduced. Here a calculation similar to [66] is presented. However, the results obtained are somewhat different, since in [66] only χ + was considered. In the presence of a magnetic field H aligned along the a-b-plane, the

110 CHAPTER 5. SPIN SUSCEPTIBILITY IN SUPERCONDUCTORS 4 ω [ hs ] a T=.75T C b T=.6T C c T=.25T d T=T =T /6 C x 3 x 9 N C ω [ hs ] q [π,π] q [π,π] Figure 5.7: Contour plots of Imχ RP A in arb. u. for Γ =.5meV at four different temperatures. Here JT =.8784eV.4eV/K T was employed. The white dashed line in d indicates the dispersion of the Goldstone mode. energy band in the normal state gets split into two ξ ± k = ε k ± H 5.22 Moreover, the superconducting gap closes when a magnetic field is turned on. However, the upper critical fields H c2 for field orientation parallel to the a-b-plane exceed 5 T [82]. Thus for measurements at magnetic fields of, say, 8 T, the closing of the superconducting gap can be neglected and one is left with the Zeeman splitting nevertheless in the numerical calculations the gap closing was considered, modelled by a standard behavior. In the superconducting state the eigenenergies in the presence of a magnetic field are given by Ξ ± k = ε 2 k + 2 k ± H 5.23 The magnetic field does not enter the square-root. Therefore, the superconducting eigenenergies are simply split as in the normal state. Furthermore, the presence of the magnetic field lifts the degeneracy of the transverse and longitudinal components of χ, which are now given by for details of the