Resonant Inelastic X-ray Scattering as a Probe of the Phase and Excitations of the Order Parameter of Superconductors
The capability to probe the dispersion of elementary spin, charge, orbital, and lattice excitations has positioned resonant inelastic s-ray scattering (RIXS) at the forefront of photon science. Here we develop the scattering theory for RIXS on superconductors, calculating its momentum-dependent scattering amplitude. Considering superconductors with different pairing symmetries we show that the low-energy scattering is strongly affected by the superconducting gap and coherence factors. This establishes RIXS as a tool to disentangle pairing symmetries and to probe the elementary excitations of unconventional superconductors.
pacs:78.70.Ck, 74.72.-h, 74.20.Rp
In the past decade, resonant inelastic x-ray scattering (RIXS) Schuelke2007 (); Ament2011 () has made remarkable progress as a spectroscopic technique, establishing itself as an experimental probe of elementary spin Braicovich2009a (); Braicovich2009b (); Hill2008 (); Tacon2011 (); Kim2012b (); Dean2012 (), orbital Ulrich2009 (); Schlappa2012 (), and lattice excitations Yavas2010 (). In quite a number of cases, theoretical considerations have preceded and stimulated these experimental advances, prominent examples being the theoretical demonstration of the presence of strong single-magnon scattering channels in cuprates Groot1998 (); Ament2009a () and iridates Ament2010b (). Being a photon-in/photon-out spectroscopy, both the energy and the momentum change of the scattered photon are measured. As the energy and momentum lost by the photon are transferred to intrinsic excitations of the material under study, direct information on the dispersion of those excitations becomes available. The resonant character of the technique is due to the energy of the incident photon being chosen such that it coincides, and hence resonates, with an x-ray absorption edge of the system Schuelke2007 (); Ament2011 (). This year the energy resolution of RIXS has reached 30 meV in the hard x-ray regime Kim2012b (), will reach 50 meV in the soft x-ray regime by building on present instrumentation PrivComm () and is designed to reach 11 meV at the Cu -edges at the NSLS-II presently under construction NSLS-II_design_report (). This brings the RIXS energy resolution well into the regime of the energy gap of cuprate superconductors, which stretches out to 119 meV for mercury-based high systems Yu2009 (). Consequently the fundamental question arises of how the superconducting (SC) state leaves its fingerprints in RIXS spectra — in particular whether and how RIXS is sensitive to the phase and amplitude of the SC gap and to quasiparticle excitations.
Probing the order parameter in unconventional superconductors is generally the first step for an investigation of the pairing mechanism and of the character of the SC state. Compared to the available spectroscopic methods, such as scanning-tunneling spectroscopy (STS), photo-emission spectroscopy, optical spectroscopy or inelastic neutron scattering, RIXS uniquely combines the advantages of bulk-sensitivity and availability of momentum resolution while at the same time requiring only small sample volumes. Here we show how the sensitivity of the RIXS process to the dynamical structure factor (DSF) of the electron spin and density in the SC state enables the investigation of SC quasiparticle excitations. In particular, we find that the momentum dependence of RIXS spectra is intrinsically determined by the pairing symmetry, being sensitive not only to the magnitude of the SC gap and to the presence of nodes on the Fermi surface but also to the phase of the order parameter. This phase sensitivity is due to the appearance of SC coherence factors which, for instance, in STS determine to large extent the quasiparticle interference in the presence of impurities Hoffman2002 (); McElroy2003 (); Hanaguri2007 (); Kohsaka2008 (); Hanaguri2009 (); Hanke12 (); Sykora11 ().
Dynamical structure factors in RIXS
In what follows we concentrate on the so-called direct RIXS process at the transition metal (TM) ion edges, in which the incoming photon resonantly excites the core shell 2 electron into the 3 shell which consequently decays into an outgoing photon and a charge, spin, or orbital excitation in the electronic system Ament2011 (). In this case the RIXS cross section can be written as Ament2009a (); Ament2011 (); Haverkort2010 (); Marra2012 ()
where () is the initial (final) state of RIXS process with energy () and () is the transferred photon energy (momentum). Furthermore, in the fast collision approximation (FCA) Ament2009a (); Ament2011 (); Haverkort2010 (); Marra2012 () the Fourier transformed RIXS transition operator can be written as , where the operators and are the density and spin of conduction electrons Haverkort2010 (); Marra2012 (); Note1 (). The so-called RIXS form factors and depend on the TM ion, the specific geometry of the experiment, and on the polarization of the incoming and outgoing photon — their precise dependencies are provided in Refs. Ament2009a, ; Haverkort2010, ; Marra2012, . In terms of these form factors the RIXS cross section is
where is the charge and the spin dynamical structure factor (DSF), where one assumes that the spin DSF has the same momentum and energy dependence for all the three different components of the spin operator — as is the case of unconventional superconductors Andersen2005 (). As is clear from the above, the amplitude of the RIXS form factors and can be tuned by properly adjusting the experimental conditions in RIXS. This implies that RIXS at -edges can measure either spin or charge DSF depending on the chosen polarization, which is a unique feature of RIXS spectroscopy. Note that the FCA has been successfully used to describe low energy excitations in RIXS, e.g., at the Cu edge of both undoped and doped cuprates (cf. theoretical calculations Ament2009a (); Haverkort2010 (); Marra2012 () and experimental results with their interpretation fully based on the FCA Braicovich2010 (); Tacon2011 (); Dean2012 (); Schlappa2012 ()), at the Fe edge of the iron arsenides Kaneshita2011 (), and at the Ir edge in iridates Kim2012a (). It has also been shown theoretically that when the incoming photon energy in RIXS is tuned to a TM ion resonant edge in a TM oxide, the FCA describes the RIXS spectrum well Haverkort2010 ().
In the following we concentrate on determining the properties of DSF for different types of singlet-pairing superconductors, and to be even more specific below we consider the case of Cu ions in lattices with tetragonal symmetry, i.e., the one which has direct relevance to the high superconductors. Our main aim in this context is to establish how a variation of the phase of the SC order parameter is reflected in the spin and charge DSF. Following the most direct theoretical inroad and avoiding model-specific technical details, we performed calculations for a singlet-pairing superconductor with a SC order parameter varying along the Fermi surface. Even if by this electron correlations are not fully taken into account, this approach is commonly used — and is very successful to calculate quasiparticle interference in cuprates Hanaguri2007 (); Hanaguri2009 (); Kohsaka2008 (); Fischer2007 (). Besides this, in Appendix IIIIII we show that it is actually possible to introduce strong correlations between electrons into the calculations and that such does not affect the main results on the SC electronic system presented below.
DSF for a superconductor
Quasiparticle excitations in a single band superconductor with singlet-pairing are described by the Hamiltonian , where is the quasiparticle energy dispersion depending on the SC gap function and on the dispersion of the bare electrons . The Bogoliubov quasiparticle operators are related to the electron operators via and with and . The Bogoliubov transformation allows the evaluation of the DSF for a SC by evaluating the transition amplitudes and between the ground state and any excited state of the Hamiltonian. At zero temperature, the excited states which contribute to DSF have the form with a transition energy of . Using the Bogoliubov transformation one then finds that the DSF for a superconductor reads
where sign is for the charge (spin) DSF Kee1998 (); Kee1999 (); Voo2000 (). Thus, the DSF is a sum over all momenta within the Brillouin zone, where the transition amplitudes are strongly influenced by the character of the SC state. Although quasiparticle interactions substantially affect the DSF, they do not alter its intrinsic sensitivity to the character of the SC state (in particular to the symmetry of its gap function), as can be seen, e.g., at the random-phase approximation level Kao2005 (), or by considering a strongly correlated system with Hubbard interactions, see Appendix IIIIII.
To determine in detail how the RIXS spectra of unconventional superconductors reflect the phase of the order parameter, we consider (i) a -wave pairing with and (ii) an anisotropic -wave pairing with , i.e., two pairing symmetries which differ from each other only in the SC order parameter phase. Maps of the considered gap functions and the Fermi surface are shown in Fig. 1. In the wave case, the DSF , due to the ’’ sign in Eq. (DSF for a superconductor), is non-zero all over the Brillouin zone, while in the wave case excitations combining momenta with opposite phases of the order parameter, i.e., sign reversing processes, are suppressed. Note that for the spin DSF this situation is inverted.
The calculated quasiparticle spectra for the above two pairing symmetries are shown in Fig. 2(a) for a fixed and relatively large momentum transfer , where is the Fermi radius. A detailed expansion of the DSF for is provided in Appendix II. Note that direct RIXS at the -edge in 2D cuprates allows momentum transfers Braicovich2010 () and therefore one is able to access momentum transfers of the order of . Such momentum transfers correspond to momentum vectors combining two anti-nodal points on the Fermi surface [see inset of Fig. 2(a)] with the same gap value . For such an excitation, the sign of the order parameter in the wave case is preserved whereas in the wave case it is reversed. Clearly the spectral weight at in the charge DSF is enhanced in the wave case. On the other hand, for the wave case the sign reversal leads immediately to a suppression of the DSF according to Eq. (DSF for a superconductor). Therefore, the spectral weight at is expected to be very small which is confirmed by the dotted curve in Fig. 2(a). On the other hand, spin DSF is suppressed in the wave case, while the sign reversal enhances the spectral weight at . Note that due to the gap being equal in magnitude for both cases, the obtained effect is entirely due to phase changes of the SC order parameter along the Fermi-surface.
In unconventional superconductors, where the pairing is generally considered as mediated by antiferromagnetic spin fluctuations Schrieffer1989 (), the SC gap function is expected to exhibit a sign reversal between the Fermi momenta connected by characteristic wave vector of the spin fluctuations Kuroki2001 (). The conduction electrons of such superconductors show a tendency to Fermi-surface nesting with a typical nesting vector of . In Fig. 2(b) the scattering intensity as function of energy for the two pairing symmetries considered above are shown again, but now for a perfectly nested cuprate-like Fermi surface (see inset), with a transferred momentum equal to the nesting vector. It is clear that, in comparison to the case of an isotropic Fermi surface, the coherence peak in the anisotropic wave pairing case appears now strongly enhanced due to the nesting effect. As in the case of an isotropic Fermi surface, sign reversing excitations occurring in the wave case in the charge DSF are strongly suppressed, as well as sign preserving excitations in the anisotropic wave case in the spin DSF.
To highlight its strong dependence on the order parameter phase, we show in Fig. 3 the DSF for a fixed energy as a function of momentum in the entire Brillouin zone, both for the anisotropic wave and the wave pairing, for a perfectly nested Fermi surface. Due to the nesting effect, coherence peaks are clearly visible in the charge (spin) DSF in the anisotropic wave ( wave) case for any of the nesting vector , while they are strongly suppressed in the wave (anisotropic wave) case (Fig. 3). The case of an isotropic Fermi surface is presented in Appendix IIII. Clearly the symmetry of the order parameter is reflected by the symmetry of the charge and spin DSF spectrum. Since the charge and spin DSF are complementary with respect to the spectral suppression of the sign reversing and sign preserving excitations, the phase sensitivity is enhanced when these two components are fully disentangled. This can be done by tuning the polarization dependence in the form factors and in Eq. (2).
Origin of the phase sensitivity
It occurs that a strong dependence on the SC order parameter phase in DSF is found for transferred momenta and when the transferred energy is close to twice the energy of the SC gap . This sensitivity to the SC order parameter phase can be better understood if we further confine our discussion to the case where not only the transferred momentum is rather large, i.e., , but also the momentum is on the Fermi surface. The main contributions to the DSF correspond in fact to those excitations close to the Fermi surface which fulfill the conditions and . Assuming an unconventional superconductor with a pairing governed by a phase dependent order parameter the DSF in Eq. (DSF for a superconductor) for excitations near the Fermi surface () becomes
Thus, the momentum-dependent intensity distribution of the low energy DSF mainly represents the variation of the SC order parameter phase along the Fermi surface.
Comparison with other spectroscopies
In principle also other two-particle spectroscopies (see, e.g., Ref. Ament2011, for an overview) can be directly sensitive to the DSF of superconductors. Even if none can match RIXS in measuring both spin and charge DSF of superconductors, already probing either of the two is in general challenging as in, e.g., electron energy loss spectroscopy (EELS) one cannot reliably measure spectra with high momentum transfers, inelastic neutron scattering (INS) does not directly probe the charge DSF and non-resonant inelastic x-ray scattering (NIXS) is extremely photon-hungry. Nevertheless, transition amplitudes of the same type as in Eqs. (DSF for a superconductor,Origin of the phase sensitivity) are also encountered when determining the scattering rate of conduction electrons in the presence of impurities, as observed in the surface-sensitive STS Hoffman2002 (); McElroy2003 (); Hanaguri2007 (); Kohsaka2008 (); Hanaguri2009 (); Hanke12 (); Sykora11 (). This is because Eqs. (DSF for a superconductor,Origin of the phase sensitivity) have a similar structure as the ones which are known to govern the quasiparticle interference [in which case the transition amplitudes, whose sum over the momentum contribute to the DSF in Eq. (DSF for a superconductor), are termed ‘coherence factors’] in the presence of impurities. Since the quasiparticle interference patterns explored by STS have turned out to be very successful in uncover the pairing symmetries of the unconventional SC Hoffman2002 (); McElroy2003 (); Hanaguri2007 (); Kohsaka2008 (); Hanaguri2009 (); Hanke12 (), this gauges the potential of RIXS to observe and unravel symmetries of SC pairing and pairing-mediators. Compared to STS, however, RIXS has a succinct conceptual advantage. Whereas the theoretical interpretation of STS in the framework of quasiparticle interference relies crucially on the form of the underlying impurity system showing various components of scattering Sykora11 (), in the case of RIXS the interpretation of spectroscopic features does neither rely on the presence of impurities in the superconductor nor on the modeling thereof.
We have shown that RIXS, in contrast to other well-known two-particle spectroscopies, is directly sensitive to the spin and to the charge dynamical structure factor (DSF) of a superconductor. In particular we have shown that the DSF of a superconductor observed in RIXS is very sensitive to the symmetry of the order parameter. This is rooted in the quasiparticle spectra reflecting sign-reversing excitations at large transferred momenta which arise for order parameters with a phase that varies over the Fermi surface. This, together with the recent experimental successes of RIXS, including in particular the major enhancements in resolution and pioneering study of hole doped cuprates Tacon2011 (), establishes the potential of RIXS as a versatile and practical spectroscopic technique to investigate the fundamental properties of superconducting materials.
The authors thank K. W. Becker for fruitful discussions related to this work. K. W. acknowledges support from the Alexander von Humboldt Foundation. This research benefited from the RIXS collaboration supported by the Computational Materials Science Network (CMSN) program of the Division of Materials Science and Engineering, U.S. Department of Energy, Grant No. DE-FG02-08ER46540.
I I - Dynamical structure factor for small () scattering
In what follows we discuss the effect of the momentum dependence of the SC gap magnitude on the RIXS spectrum in the case transferred momentum . In this regime the spectral intensity is rather insensitive with respect to points where the sign of the SC order parameter changes. In the normal state () the dynamical structure factor (DSF) is non-zero only for particle-hole excitations close to the Fermi surface leading to a coherence peak in the spectrum which disperses according to the Fermi velocity. In Fig. 4 the DSF is shown for a conventional wave (a) and for an unconventional wave (b) superconductor, calculated using Eq. (3) in the main text of the paper for an isotropic Fermi surface as a function of the transferred momentum close to (0,0). Clearly seen is the absence of spectral weight for in the conventional case (a), whereas at higher energies the coherence peak follows the dispersion of the bare electrons (dotted line). Instead, for a -wave superconductor (b) two types of excitations contribute to the low energy spectrum, related to two different regimes of momenta contributing to the sum of Eq. (3) in the main text of the paper. If the momentum is close to the nodal lines, i.e., the Brillouin zone diagonals shown in Fig. 1 of the main text of the paper, where , the excitations are particle-hole like with an energy .
The second type of excitations is provided by momenta close to the anti-nodal points where is the maximum gap value. Here, for small values of the transition energy is approximately
where is the electron group velocity. Thus, as it is seen in Fig. 4(b) for we obtain a coherence peak showing the known dispersion of Fig. 4(a), but with less spectral weight since the excitations are restricted to the vicinity of the anti-nodal points. The -dependence of the excitation energy given by Eq. (5) determines the behavior of the coherence peak in a typical RIXS spectrum in the SC state. An additional coherent excitation showing a linear dispersion down to for indicates the presence of nodes in the SC order parameter.
Ii II - Dynamical structure factor for an isotropic Fermi surface superconductor
The spin and charge DSF spectra are shown in Fig. 5, for an isotropic Fermi surface and for different symmetries of the order parameter. As one can see, wave (anisotropic wave) pairing strongly suppresses the spectral weight in the charge (spin) DSF for transferred momenta that correspond to sign reversing (sign preserving) excitations, while no suppression occurs in the anisotropic wave ( wave) case.
Iii III - Dynamical structure factor for strongly correlated electron system
The main aim of this part of the appendix is to show that our main result for the phase dependence of the dynamical structure factor (DSF) holds also in the presence of strong Coulomb repulsion of the conduction electrons. To simplify the presentation of our arguments we will focus on the limit of infinite strong correlations where double occupancy is forbidden Trugman1990 (); Becca2000 (). As is well-known using a Schrieffer-Wolf transformation the Hubbard model can then be replaced with a - model where the superexchange , cf. Chao1978 (). Such regime of very strong correlations may be still regarded as rather realistic in describing many basic properties of the strongly transition metal oxides such as cuprates in the overdoped limit Putikka1992 (); Tandon1999 (); Cosentini1998 (). The superconductor is well described by the following Hamiltonian
The operators , and are Hubbard creation and annihilation operators which enter since doubly occupancies of local sites are strictly forbidden due to the presence of strong electronic correlations. They obey unusual anticommutator relations. For instance, , with .
For simplicity let us consider the spin DSF which is defined by the following correlation function
Here, the time dependence and the expectation value are formed with Hamiltonian (6). To calculate the expectation value and the dynamical behavior, we diagonalize the Hamiltonian using new approximate quasiparticle operators , which are related to the original correlated electron operators via and . In the case of a sufficiently large hole concentration the operator can approximately be replaced by its expectation value and the Bogoliubov quasiparticle operators fulfill the following relations Sykora2009 (): , where and . Replacing all operators by the quasiparticle operators , the time dependence can easily be evaluated. Each of the remaining expectation values contains a product of four quasiparticle operators. A final factorization leads to the following expectation values with two quasiparticle operators
where and and are defined by and . They are evaluated using the Gutzwiller approximation (cf. Ref. Fazekas, ), as and , where and is the Fermi function at .
Finally, one obtains the spin and charge DSF in the strongly correlated case
where the coherence factors and are defined by
To summarize, the main effect of correlations in the case is to rescale the magnitude of the order parameter. In fact, up to a renormalized gap function , the coherence factors in Eq. (10) have the same form as in the uncorrelated case [cf. Eq. (3) of the letter]. Due to this renormalization, the quasiparticle excitations gap is lowered in energy by a factor . Moreover, the phase sensitivity of the DSF is reduced by the presence of the third term of Eq. (III). However, when the hole doping is rather large (, ), the order parameter decrease is negligible, while the main contributions to the quasiparticle spectrum are given by the first term in Eq. (III). As a consequence, the phase sensitivity of the DSF is not affected. In Fig. 6 we compare the spin and charge DSF of a strongly correlated hole doped system with those of an uncorrelated one (cf. Fig. 3 of the letter), for different order parameter symmetries ( wave and anisotropic wave). As one can see, the presence of electronic correlations does not change RIXS spectra qualitatively. Hence the dynamical structure factor in a strongly correlated electron system is governed, as well as in an uncorrelated one, by coherence factors which are responsible for the sensitivity of RIXS spectra to the order parameter phase.
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