- Superconductivity

__Electron cooling and refrigeration__

In collaboration with Dr. Procolo Lucignano at the University of Napoli (Italy), this activity investigates the thermoelectric properties of hybrid junctions realised proximitizing the surface states of three-dimensional topological insulators with conventional s-wave superconductors either by considering plane or wire geometry. We focus on ballistic devices and study the quasiparticle flow. We are looking at electric and thermal currents by adopting a scattering matrix approach based on conventional Blonder-Tinkham-Klapwijk formalism. We calculate the cooling efficiency of the junction as a function of the microscopic parameters of the normal region.

For the case of plane geometry, we have shown that the cooling power increases when moving from a regime of Andreev retro-reflection to a regime dominated by Andreev specular-reflection. Differently from the case of a conventional N/S interface, we have proven that we can achieve efficient cooling of the normal region without including any explicit impurity scattering at the interface to increase normal reflection. We are currently working to extend these results to the case of nanowires and disordered systems.

D. Bercioux & P. Lucignano, Eur. Phys. J. Spec. Top. **2**, 105 (2018).

__Transport in unconventional d-wave superconductors__

In collaboration with the experimental group of Javier Villegas (Thales-CNRS), we have investigated the transport properties of graphene when proximitized by an unconventional d-wave superconductor YBCO. The experimental set-up consisted of a hybrid junction containing two normal regions with a different chemical potential, one serving as a contact, and a superconducting one, serving as a contact as well. We have shown that two different ballistic charge carrier oscillations with respect to a back gate are present in the normal central region. The difference is observed when considering processes with energy below or above the induced superconductivity gap in graphene. The oscillations are standard Fabry-Pérot oscillations for energies above the gap, typical of graphene with different gating.

In contrast, the oscillations are due to coherent interference between electrons and holes below the gap. The two types of processes differ for the different gate dependence, and one is double the other. Our finding that shows a strong directional dependence related to the order parameter of the d-wave superconductivity can be of interest for the realization of exotic topological states.

D. Perconte *et al.*, Phys. Rev. Lett. **125**, 087002 (2020).

D. Perconte *et al.*, Annalen der Physik (2022).

**Electronic properties of low-dimensional carbon-based materials**

__Carbon nanotubes__

Since 2007, I have collaborated with the experimental group of Dr. Oliver Gröning at EMPA (Switzerland). We work together on the spectral properties of defected carbon nanotubes, and my role is to model their results for scanning tunnelling spectroscopy. We published together several works related to metallic and semiconducting tubes. In our most recent manuscript, we have proposed a THz detector based on splitting the metallic mode of a zig-zag nanotube. Additionally, we have generalized our findings to all the possible defected metallic tubes. We have theoretically shown that the underlying mechanism at the basis of our detector can work even at liquid nitrogen temperature, making it appealing for practical purposes.

G. Buch *et al.*, Phys. Rev. Lett. **102**, 245505 (2009).

D. Bercioux *et al.*, Phys. Rev B **83**, 165439 (2011).

G. Buhs, B. Bercioux , L. Mayrhofer & O. Gröning, Carbon **132**, 304 (2018).

G. Buchs *et al.*, Appl. Phys. Rev. 8, 021406 (2021).

__Graphene__

In collaboration with Dr. Alessandro de Martino at City, University of London (UK), we are interested in the transport properties of graphene in the presence of spin-orbit interaction. We published 2009 a paper in which we showed how to generalize the use of the transfer matrix technique for the case of graphene with the inclusion of spin-orbit interaction. We have been employing this technique to investigate the formation of spin-slit snake states in *pn*-junctions. We are currently studying how to include the effects of electron-electron interaction and consider the more relevant case of circular *pn*-junctions.

D. Bercioux & A. De Martino, Phys. Rev. B **81**, 165410 (2010).

D. Bercioux & A. De Martino, Phys. Rev. B **100**, 115407 (2019).

__Twisted-bilayer Graphene__

In collaboration with colleagues at the Phenikaa Institute for Advanced Studies & Phenikaa University (Vietnam), we investigate localised states’ propagation in twisted bilayer graphene. We employ the Kernel Polynomial Method for studying the spread of initially localised electronic states in one of the layers. The advantage of our approach is that we can investigate any twist-angle of the bilayer structure. We have a current project in which we are investigating the optical response of the system by using the Kubo formalism adapted to the kernel polynomial method.

H. Nam Do, H. Anh Le, & D. Bercioux, Phys. Rev. B **99**, 165127 (2019).

V. Nam Do, H. Anh Le, V. Duy Nguyen, D. Bercioux, Phys. Rev. Research 2, 043281 (2020).

**Analogue Quantum Simulators**

Analogue quantum simulators represent ideal testbed systems for validating the properties of more complex quantum systems that would require a more demanding effort to find the system properties, especially in the case of interacting systems.

__Case of electronic simulators__

This simulator is constituted by the quasi-free electrons on Cu's surface (111). The quasi-particles are enforced to stay in a lattice designed by placing CO molecules on the top of the Cu surface. Experimentally, the CO molecules are placed with atomic precision using a scanning tunnelling microscope tip. The tip is used successively for characterising the local density of states. On the theory side, we corroborate the experimental findings with two complementary effects: a muffin-tin and a tight-binding approach. In the former, we consider the standard Schrödinger equation complemented by a periodic potential defined by cylindrical potentials set ad-hoc for simulating the CO molecules. This technique allows obtaining the spectral properties for an infinite and a finite-size system.

Furthermore, it permits obtaining the optimal hopping parameters in the tight-binding description. We have analysed so far crystalline topological systems in one and two dimensions. Currently, we are working on interpreting our results within the framework of topological quantum chemistry. Furthermore, we have results for the signature of charge fractionalisation, but the experimental team is working on improving the substrate to have indisputable experimental evidence. We are further working to extend the results to the surface states of three-dimensional topological insulators. Last but not least, we are working to include the effects of spin-orbit and electron-phonon interaction.

S. N. Kempkes *et al.*, Nat. Mater. **18**, 1292 (2019).

D. Bercioux *et al.*, Eur. Phys. J. Plus **135**, 811 (2020).

Herrera *et al.*, Phys. Rev. B 105, 085411 (2022).

__Case of photonic simulators__

In collaboration with colleagues at the Donostia International Physics Center, we are investigating theoretically the topological properties of analogue quantum simulators based on photonic crystals. The photonic simulator is constituted by a periodic arrangement of rods with different shapes and modulated dielectric constant. We have been able to extend most of the concepts of topological quantum chemistry into the framework of photonic systems. We have also found a photonic system that exhibits a fragile topological phase. Additionally, in a recent preprint, we have given a general strategy for classifying high-order topological states in two-dimensional photonic systems. We are also working parallel on the quantum aspect of topological photonic systems. Specifically, we are investigating the back action of a topological photonic bath on the properties of one or more quantum emitters.

M. Blanco de Paz *et al.*, Phys. Rev. Research **1**, 032005(R)(2019).

M. Blanco de Paz *et al.*, Advanced Quantum Technology, **3** 1900117 (2020).

M. Proctor *et al.*, Phys. Rev. Research 2, 042038(R) (2020).

M. Proctor *et al.*, Appl. Phys. Lett. 118, 091105 (2021).

M. Blanco de Paz *et al.*, Journal of Physics: Condensed Matter **34**, 314002 (2022).

**Volkov-Pankratov states in topological matter**

Recently, we started a research activity on topological systems in the presence of a smooth modulation of the topological parameters. This activity is carried on within my group and collaborates with Prof. Reyes Calvo at the University of Alicante (Spain) and Dr. Alessandro De Martino at City, University of London (UK). A smooth transition in the system parameter leads to the coexistence of the topological edge states with massive edge states — these states are known as Volkov-Pankratov states.

A surface in a topological system is an interface between two regions characterized by a different value of a topological invariant. This invariant classifies the two regions differently: one trivial and one topological. An energy gap characterises both sides of the interface; thus, both are not conducting. The bulk-boundary correspondence corroborates the existence of a topological surface state living at the boundary between these two regions: this state is robust against external perturbation. Typical examples of topological systems are the quantum Hall and the quantum spin-Hall (QSH) states. This project will focus mainly on the newly discovered time-reversal invariant QSH case. Standard approaches to these interface regions consider an abrupt change of some key parameters defining the topological system. However, it is now better understood that a smooth transition in this parameter describes a real system more appropriately. A set of new unprotected boundary states emerges in this condition, aka Volkov-Pankratov (VP) states.

Most of the research focused so far on three-dimensional topological systems. We worked on extending these results for the case of two-dimensional topological insulators such as *graphene* and *HgTe quantum wells*.

T. L. Van den Berg, M.R. Calvo & D. Bercioux, Phys. Rev. Research **2**, 013171 (2020).

T. L. Van den Berg, A. De Martino, M.R. Calvo & D. Bercioux, Phys. Rev. Research **2**, 023373 (2020).