Material design from first principles

Balázs Újfalussy

István Balogh

Gábor Csire

Éva Fazakas

Ádám Huszár

Krisztina Kádas

Annamária Kiss

György Kovács

Zsolt Maksa

Balázs Nagyfalusi

Karlo Penc

Levente Rózsa

István Tüttő

Lajos Károly Varga

Levente Vitos

(* ***2016**)

**Superconductivity in layered heterostructures. **— During the previous years, a novel and unique computer code was developed which allows us to study the nature of the Andreev bound states related to the proximity effect in normal metal–superconductor heterostructures based on the first-principles Bogoliubov–deGennes (BdG) equations. For the first time, we succeeded in applying the SKKR method for solving the Kohn–Sham–Bogoliubov–deGennes (KSBdG) equation which allowed us to investigate the quasiparticle spectrum of superconducting heterostructures. This year, a fully relativistic generalization of the BdG equations within Multiple Scattering Theory has been derived. The method allows the solution of the first-principles Dirac–Bogoliubov–de Gennes equations combined with a semi-phenomenological parametrization of the exchange-correlation functional. The major difficulty during the development was to derive simple conditions for the case when the right-hand-side and left-hand-side solutions must be treated separately while setting up the corresponding Green function. As an application of the theory, we calculated the superconducting order parameter in Nb/Fe and Nb/Au/Fe systems. We found Fulde–Ferrell–Larkin–Ovchinnikov like oscillations in the iron layers, but more interestingly an oscillatory behaviour is observed in the gold layers as well.

**Thin Film Magnetism.** — Non-collinear magnetic structures have been investigated in ultrathin films by combining *ab initio* electronic structure calculations with numerical spin model simulations. The experimentally observed significant increase in the spin spiral period as a function of temperature of three-atomic-layer thick Fe films on Ir(111) has been explained. *Ab initio* calculations revealed how the addition of hydrogen to a two-atomic-layer thick Fe film on Ir(111) leads to the formation of magnetic skyrmions in place of the spin spiral ground state of the pristine system, in agreement with scanning tunneling microscopy measurements. Theoretical predictions have been made on the characterization of skyrmionic structures with various topological charges.

**High Entropy Alloys.** — Looking for high-strength and high-temperature-resistant high-entropy alloys (HEAs) new refractory HEA compositions have been predicted theoretically bycombining a refractory CrMoW alloy with late transition metals (LTM = Ni, Co, Fe, and Mn). Ab initio calculations revealed that the LTM additions increase the ductility, but reduce the strength of these CrMoW based alloys with single-phase BCC structure.

The magnetization components of permeability spectra for annealed nanocrystalline (Finemet) core have been studied and four contributions have been revealed for the first time in the literature: i) eddy current; ii) Debye relaxation of magnetization rotation, iii) Debye relaxation of damped domain wall motion and iv) resonant type DW motion. Although the relative weight of these contributions changes with the frequency and exciting field amplitude, the role of eddy current cannot be neglected even for the smallest applied field. These components can be found in the powder cores of soft magnetic composites as well.

**Observation of spin-quadrupolar excitations in Sr _{2}CoGe_{2}O_{7} by high-field electron spin resonance.** — When we think of a spin, usually we imagine an arrow pointing somewhere (representing the expectation values of the components of the spin operator), and with the arrow we associate a magnetic moment. Upon time reversal, the arrow reverses its direction. This is a reasonable picture for the spin 1/2 of the electron, but for larger spins this does not exhaust all the possibilities. For example, the dimension of the Hilbert space is 3 for the spin 1, and we can construct spin states for which the expectation values of all the three spin operators vanish — the state does not point anywhere, it cannot be represented by an arrow. The simplest example is the 0 eigenstate of the S

Similarly, the long-range-ordered states of interacting spins are usually time-reversal-breaking states, with a configuration of “arrows” that repeats itself on the lattice. However, under favorable conditions, interacting spins can produce ordered states where the order parameter is of spin-quadrupolar character which does not break the time reversal symmetry. Theoretically, such phases have been established in spin-one Heisenberg models extended with higher-order spin interactions. Even more interestingly, time-reversal invariant ordered states can also be realized in spin-1/2 systems, where the quadrupole-like order parameter is distributed between two spins on a bond, leading to a so-called nematic ordering.

These theoretical developments have inspired the quest to nematic and quadrupole phases in real materials. However, when relying on standard experimental methods, such phases usually remain hidden. Most of the experimental probes detect spin-dipolar (ΔS=1) transitions, and they do not interact with the spin-quadrupoles, as their detection requires ΔS=2 transitions.

In a collaboration with experimental researchers from Osaka University, we found an unambiguous experimental observation of spin-quadrupolar excitations in the layered Sr_{2}CoGe_{2}O_{7} multiferroic compound. In this compound, the Co ions are in the centers of tetrahedra formed by the four surrounding O ions (Fig. 1). Since the inversion symmetry is absent, the relativistic spin-orbit coupling allows the coupling of the electric polarizations to the spin-quadrupolar operators. Due to this magnetoelectric coupling present in the Sr_{2}CoGe_{2}O_{7}, the non-magnetic, purely spin-quadrupolar excitation becomes electrically active and detectable by electromagnetic waves, like the electron spin resonance spectroscopy.

**Figure 1.** The schematic crystal structure of the Sr_{2}CoGe_{2}O_{7} multiferroic compound projected onto the ab plane. The green spheres represent the magnetic Co^{2+} ions with S = 3/2 surrounded by four O^{2−} ions (red) in an alternating tetrahedral environment.

In the electron spin resonance spectra of Sr_{2}CoGe_{2}O_{7 }above the saturation field of 20T, a mode with twice the g-factor of the usual modes is observed (Fig. 2). This indicates the absorption of two magnons, just what is needed for the creation of a quadrupole wave. Indeed, we could explaine the features of the experimental spectra taken in different geometries by a simple theoretical model of the spin-quadrupolar wave providing not only a qualitative description, but also a quantitative agreement.

**Figure 2.** Frequency-field diagrams of the ESR resonance fields of Sr_{2}CoGe_{2}O_{7} for magnetic fields parallel to the [100] direction of the external magnetic field. The solid lines represent the dipolar resonance modes from the multiboson spin-wave theory. The red dashed line indicates a resonance mode with a slope twice larger than the others, corresponding to a two-magnon excitation — the quadrupolar mode.

The most significant point of our finding is the first observation of non-magnetic spin-quadrupolar excitation in an antiferromagnetic material (Fig. 3). Such quadrupolar degrees of freedom become inherent in systems with larger than S=1/2 magnetic moments, regardless of the presence of magneto-electric coupling. Upon condensing such multipolar excitations, magnetically disordered exotic quantum phases may arise. The experimental identification of quadrupole excitations with vanishing gap gives us a possibility to identify long-sought nematic phases, which stand without any usual magnetic fingerprint and are almost impossible to tell apart from other non-magnetic phases. Furthermore, our work will stimulate the application of the magnetoelectric effect as a spectroscopy tool.

**Figure 3.** Schematic plot of (a) the Q1 quadrupolar mode for H∥[100] and (b) the dipolar modes for H∥[110], as seen from the direction of the magnetic field. In both cases the oscillating component of the uniform electric polarization P (shown by orange ellipse) is perpendicular to the external magnetic field H, therefore they are active in the Faraday configuration. The green ellipse represents the rotating quadrupolar moments, while the green arrows the precessing dipolar spins on the two sublattices. The red arrows show the electric polarization vectors which are excited by the oscillating electric field.

**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