Újfalussy Balázs

Balogh István

Csire Gábor

Fazakas Éva

Huszár Ádám

Kádas Krisztina

Kiss Annamária

Kovács György

Maksa Zsolt

Nagyfalusi Balázs

Penc Karlo

Rózsa Levente

Tütt%F5 István

Varga Lajos Károly

Vitos Levente

**Superconductivity in layered heterostructures**. — The physics of superconductor/normal metal heterostructures has become a very intensively studied research field since modern deposition techniques allow to create very high-quality thin films and overlayers. In such systems, superconducting correlations are introduced in the normal metal by the so-called Andreev scattering, when an electron, with energy lying in the superconducting gap, arriving from the normal metal to the superconductor/normal metal (S/N) interface is retro-reflected as a hole and a Cooper pair is formed in the superconductor. This effect controls the transport properties of such systems and allows the understanding of the proximity-effect on a microscopic scale. It is also known that the Andreev reflection is the key effect behind the formation of Andreev bound states. While a great many theoretical works were dedicated to study the Andreev reflection and the Andreev bound states, it was done on model systems only, their material specific dispersion, their ``band structure'' has never been calculated (nor observed experimentally) to date. While the theory of Bardeen, Cooper, and Schrieffer (BCS) successfully describes the universal properties of conventional (s-wave) superconductors, it can not be applied easily to inhomogeneous systems where the wave number is not a good quantum number.

We developed a new method, which allows the quantitative and material-specific description of superconductivity-related phenomena. Density functional theory (DFT) has already been generalized for the superconducting state (Kohn-Sham-Bogoliubov-de Gennes, KSBdG, equations) and applied successfully in bulk superconducting systems. At present, this is the most accurate theory which allows the first-principles calculation of the superconducting transition temperature.

The superconductor/normal metal hetereostructures can also be well described by these equations. By the generalization of the screened Korringa-Kohn-Rostoker (Green function) method to the superconducting state via the KSBdG equations, it is possible to calculate the dispersion relation, charge densities, density of states, bound-state energies, the superconducting order parameter and many other physical properties of the superconducting system with arbitrary (e.g., semi-infinite) geometry. A fully ab-initio approach can also be constructed by taking into account the electron-phonon coupling within a simple approximation for the exchange functional.

The new KSBdG-SKKR method was applied to Nb/Au heterostructures where the superconductor's thickness is in the range of the coherence length, i.e., thick superconductor.

**Simplified treatment of the electron-phonon interaction.** — To calculate the superconducting properties at the interface, a simple step function was used to model the changes of the pairing potential at the interface (assuming the experimental value of the bulk gap in the superconductor). We showed that the quantum-well states, which we found to exist in the normal-state band structure, become bound Andreev states due to Andreev scattering. We found that the proximity of a superconductor in the studied heterostructures induces the mirroring of the electronic bands, and opens up a gap at each band crossing. For those materials where no quantum-well states are present, this simple picture is not applicable for the quasiparticle spectrum. It was obtained that the induced gap observed in the normal metal remains constant for each layer for a given Au thickness; however, the size of the gap decays as a function of the Au thickness, and the superconducting order parameter extends well into the normal metal and, interestingly, follows a 1/L decay. Nevertheless, the anomalous charge per layer (which is related to the superconducting order parameter) shows the usual layer-dependent property of the proximity effect as it follows a 1/L decay in the normal metal, which agrees with one-dimensional model calculations in the literature.

Based on the properties of the Andreev spectrum, a simple phenomenological method was developed to predict the transition temperature of such heterostructures which give very good agreements with the experiments (see Figure 1) in the case of the Nb/Au system. The theory was also applied to several different metal overlayers on a Nb host to predict the superconducting transition temperature.

**Figure 1.** Dependence of the superconducting transition temperature on the overlayer thickness in a Au/Nb(110) sample. The red dots with errorbars are taken from experiments of Yamazaki et al., Phys. Rev. B 81, 094503 (2010). The inset shows the induced gap in the Andreev spectrum

If the superconductor is also ultrathin, the calculation of the electron-phonon coupling is necessary, which makes the theory fully first-principles.Therefore, the McMillan-Gaspari-Győrffy theory was extended to slabs and heterostructures and then it was connected to the exchange functional. The McMillan-Hopfield parameter was obtained from the Gaspari-Győrffy formula (using the SKKR method). KSBdG equations were solved self-consistently and the critical temperatue was obtained. This method was applied to Nb/Au thin films where the inverse proximity could be observed. The critical temperature grows if we add only one gold layer to the ultrathin niobium. It was shown that this effect is a consequence of the induced changes in the effective electron-phonon coupling.

**High-Entropy Alloys.** — The equimolar NiCoFeCr is a face-centered cubic single-phase high-entropy alloy (HEA). Four different sp elements were added in equimolar ratios: NiCoFeCrAl, NiCoFeCrGa, NiCoFeCrGe and NiCoFeCrSn. The initially non-magnetic and single-phase structure turned into multiphase magnetic alloys. Investigations done using first-principles calculations and key experimental measurements revealed that the equimolar FeCrCoNiGe system is decomposed into a mixture of face-centered cubic and body-centered cubic solid solution phases. The increased stability of the ferromagnetic order in the as-cast FeCrCoNiGe composite, with measured Curie temperature of 640 K, is explained using the exchange interactions.

Continuing the structural investigations of these sp-element doped HEAs, X-ray diffraction and scanning electron microscopy (SEM) measurements were performed. The nanoindentation test revealed a ‘fingerprint” of the two-phase structure. The Young’s and shear moduli of the investigated HEAs were also determined using ultrasound methods. The correlation between these two moduli suggests a general relationship for metallic alloys.

**Figure 2.** Comparison of the ground-state energy per site between the exact diagonalization of NS-site clusters (open circles) and the variational energy (full lines) based on Gutzwiller-projected fermionic wave functions with flux π/N per triangular plaquette. The θ measures the strength of the 3-site ring exchange. The chiral spin liquid is realised for intermediate values of θ/π, approximately between 0.1 and 0.2, where the two energies agree.

**Chiral spin liquids in triangular-lattice SU(N) fermionic Mott insulators with artificial gauge fields **— The competition of different interactions in frustrated spin systems may lead to entangled quantum mechanical states, where some kind of local order parameter is formed. Correlated phases without local order are more exciting, such as the chiral spin liquid, proposed by Kalmeyer and Laughlin in 1987 as a variational ground state for spin-1/2 Mott insulators. It can be viewed as a bosonic analogue of the fractional quantum Hall effect, with universal properties such as ground-state degeneracy which depends on the boundary conditions and topologically protected edge excitations. Most interestingly, its excitations are believed to be anyons - excitations that are neither fermionic nor bosonic in nature. These anyonic particles can be braided, allowing for a topological quantum computers, making them relevant also for technological applications. The chiral spin liquid state appears to be fragile, it was found only recently in some spin-1/2 models.

Mott insulating states of ultracold atomic gases in optical lattices promise an alternative way to study quantum states of matter. Alkaline rare earths allow one to realize SU(N) symmetric Mott phases with N as large as 10, in contrast to the SU(2) symmetry of the spin-1/2 models. In this case the quantum fluctuations are enhanced due to increased number N of local degrees of freedom.

Using a variety of numerical probes, including exact diagonalization and variational Monte Carlo calculations, we have shown that, in the presence of an artificial gauge field leading to ring exchange, Mott insulating phases of ultracold fermions with one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by a ground space of N low-lying singlets for periodic boundary conditions, and by chiral edge states described by the SU(N)_{1} Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. (Fig. 2)

**Skyrmions in multilayer systems ** — Skyrmions are non-collinear magnetic structures which are stabilised by interactions that are beyond the reach of a Heisenberg model. The most important of these is the Dzyaloshinsky–Moriya (DM) interaction. We were able to show that frustration in the isotropic exchange interaction also lead to the formation of skyrmions; however, these have very different properties compared to skyrmions stabilised by the DM interaction. Using first-principles calculations, we showed that in (Pt_{1‑x}Ir_{x})/Fe/Pd(111) ultrathin magnetic layers both frustrated Heisenberg couplings and DM interactions are present, and the properties of the skyrmions are primarily determined by the former interaction. As a consequence, it became possible to arrange skyrmions in regular patterns, which is usually prohibited by the repulsive interaction between them. We have also shown that beside skyrmion formation, the frustrated interaction may also lead to the formation of other localised magnetic structures, with non-cylindrical symmetry (Fig. 3). This result has been recently verified by STM experiments in the literature.

**Figure 3.** Attractive (left) and repulsive (right) skyrmions at T = 4.7K