Raman response modelling in unconventional multiband superconductors

Brief description

My research focuses on the theoretical understanding of Raman scattering in superconductors (SCs), paying special attention to non-electron-phonon systems, via analytical and computational methods.

What: Most specifically, in collaboration with my supervisor, Dr Maiti, I am developing a self-consistent framework of non-resonant Raman scattering in SCs using quantum many-body physics techniques. Our framework allows the investigation of the line spectra in SCs as a function of crystal lattice and fermionic interactions properties, in contrast with other models that consider isotropic systems and are insensitive to material-specific parameters. Also, the model is applicable to systems with any number of bands and for any polarisation configuration. Both pair-breaking features and collective modes, if realisable, are resolved in our model and contained in a fixed set of equations. The same set of equations is used to study the response with different ground states, interactions or polarisation configurations – only the input parameters need to be changed.

Why: By further developing the theory of Raman scattering for non-electron-phonon systems to study the behaviour of spectral features as a function of the fermionic interactions and the topology of the Fermi surface, one can use our model to gain insight into the interactions that led to the formation of the SC state in two-dimensional systems. Therefore, by giving constraints for the fermionic interaction, our work could help elucidate the pairing mechanism in non-electron-phonon SCs, which, as others have put it, “is among the most vexing problems of contemporary condensed matter physics”.

Full descriptionwith references

Superconductors (SCs) are materials that at a transition temperature Tc undergo a phase transition into a state in which the electrical resistivity is reduced to zero[1] and any magnetic field is expelled from its bulk, the Meissner–Ochsenfeld effect[2]. These characteristics allow SCs to be used in lossless transport of electric power and to create powerful electromagnets, among other technological applications. At the microscopic level, on the onset of superconductivity, pairs of fermions form an effective bound state and an energy gap centred around the Fermi energy appears in the two-particle energy dispersion[3]. This gap is defined as twice the magnitude of the order parameter Δ, which characterises the SC state, a macroscopic quantum state formed by phase-coherent pairs of fermions.

Conventional superconductors

In electron-phonon SCs, also known as conventional SCs, the order parameter Δ is constant in momentum and we can successfully describe their SC state microscopically[4–7]. They were the first SCs to be discovered and usually have very low Tc, with an upper limit of 40 K[8] under regular pressure conditions which requires refrigeration with liquid helium, a very expensive cooling agent. Hence, applications of superconductors in the delivery of electric power without losses, which would revolutionalise the electricity distribution landscape, remains restricted to specific and usually experimental scenarios. Their use to generate strong magnetic fields, by exploiting the Meissner–Ochsenfeld effect, is more explored both in research and commercially, vide MRIs and maglevs, as the environment can be more controlled.

Unconventional superconductors and their properties

More recently, another class of SCs were discovered: the unconventional SCs, which exhibit higher transition temperatures and distinct order parameters Δ[9–11]. They have a greater potential for technological applications as their Tc can go above the boiling point of nitrogen, a cheap refrigerant, and they are also capable of generating even stronger magnetic fields. The two most important families of such SCs are the cuprates, discovered in 1986[12], and the iron-based superconductors (FeSCs), discovered in 2006[13]. They are considered the most promising materials for applications requiring higher transition temperatures because of their Tc records of 138 K[14] and 103 K[15], respectively.

In these systems, experimental studies show, both s-wave, Δs = constant, and d-wave symmetric, Δd ∝ cos (2θk⃗), order parameters can occur in these systems, in contrast to the exclusive s-wave states of conventional SCs. Moreover, while the latter are usually single-band systems, several unconventional SCs are multiband[16], which can lead to new phenomena associated to the difference in phase of the order parameter across bands[17,18]. Not only that, but in some FeSCs systems, the ground-state order parameter Δs can compete with a sub-leading one having a different symmetry[19–21]. However, despite decades of efforts, unconventional SCs are still not described by a universally accepted microscopic theory[22,23] that describes the fermionic pairing mechanism or predicts the symmetry of Δ(θk⃗) in these unconventional systems[11,24–26].

Furthermore, by changing the temperature or doping or also by subjecting these materials to pressure or magnetic fields, in addition to the SC phase, one can access distinct lattice phases, electronic nematic phases, charge-density- or spin-density-wave phases, antiferromagnetic phases and other phases depending on the specific compound[11]. The interplay between these different phases in unconventional SCs could lead to unforeseen applications in quantum devices[26] and even the enhancement of the SC state[27]. All of this adds to the complexity of unconventional SCs, whose underlying excitations can provide us with a wealth of information about the system, but whose pairing mechanism remains elusive.

Smekal–Raman spectroscopy in the study of superconductors

To study SCs, both conventional and unconventional ones, we can use electronic Smekal–Raman spectroscopy, that relies on the inelastic scattering of light by the pairs of fermions, which creates or annihilates an excitation in a system. From the resulting spectra, we can study the temperature evolution and the symmetry of the order parameter Δ = Δ(θk⃗,T).

Moreover, by changing the light polarisation configuration, we can also verify the existence of sub-leading interactions by looking for signatures of collective excitations, usually not visible in other probes and which usually measure only ground-state properties[28]. Therefore, the presence of collective modes in the Raman spectra inform us about the interactions that led to the formation of the superconducting state[29,30]. This demonstrates the crucial importance of studying collective modes in SCs.

Furthermore, since collective modes are bosonic in nature, having no restriction in their occupation number, they will dominate the response when present, with single-particle features only being found in very special circumstances. This strengthens the case for a better understanding of the Raman response of SCs, which could help unveil the pairing mechanism of unconventional SCs. However, although Raman studies of SCs have a long history[31–34], the current models still only qualitatively agree with experiments[29,30,34,35] and are unable to explain features found in the responses of some unconventional systems[36–38].

My contribution

In this context, I am developing a framework to calculate the Raman response of two-dimensional SCs applicable to multiband unconventional systems, accounting for the momentum structure of the fermionic pairing interaction. The model is based on linear response theory and quantum many-body techniques and it would capture all the collective excitations of the system, identifying them with the polarisation geometry of the experiment and properties of the system.

The current theoretical models[29,30,35,39] are non-sensitive to lattice properties unless additional fitting parameters are added. They are also restricted to single or two-band systems and applicable only to specific scattering polarisations. Hence, the challenge is to extend the existing toy models by connecting the microscopic details of materials, such as effects of the lattice, the number of bands and electronic interactions, to the Raman response with minimal or self-consistent parameters.

Furthermore, the framework is expected to be applicable to SCs with any number of bands, with the order parameter Δ(θk⃗) belonging to any symmetry channel and for any polarisation configuration. With it, one could better interpret the features found in the experimental Raman response of unconventional SCs.

Finally, once the connection between features found in Raman experiments and their origins using our framework is done, one could use this link to verify if a given pairing mechanism model for unconventional SCs is indeed the appropriate one: the correct prediction of collective modes would be a test of validity for such models. And our framework would be the tool used to verify which collective modes are present in the system, helping infer qualitative and quantitative aspects of the pairing interaction. This method of verification of pairing mechanism models would lead to a better understanding of the electronic interactions taking part in the formation of the SC state in cuprates, iron-based SCs and other systems. This, in turn, could help to solve one of the biggest open questions in condensed matter physics: what is the pairing mechanism behind superconductivity in non-electron-phonon superconductors?

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