Shiba band structure and topological edge states of artificial spin chains
In recent years there is an intense race to detect Majorana Zero Modes (MZMs), that could provide a unique platform for fault-tolerant quantum computing,the real breakthrough - that has created a great number of routes to such platforms - was the realization that one can create topological superconductivity based on artificial heterostructures. However, MZMs in superconducting heterostructures are still elusive because it is very difficult to uniquely identify them experimentally. Several promising STM experiments have been performed on various systems, which show peaks in the differential conductivity at zero energy in the superconducting gap of the host. However, this does not impose a strict evidence that the observed states at the end of the chain are indeed the long sought MZMs and further information about the nature of these peaks are difficult to obtain. At the same time, most theoretical models which claim to calculate Majoranas are able to do so only if unrealistic parameters are used, which, for example, gives rise to an enormous superconducting gap. To address this problem we developed a first-principles based, hence parameter less computational approach by solving of the Kohn-Sham-Dirac-Bogoliubov de Gennes (KSDBdG) equations. Such an approach can not only reproduce many aspects of the STM experiments, as we demonstrated earlier, but it also allows the calculation of other quantities, like spin-polarization and the superconducting order parameter (OP), which are important to understand the nature of these states, but are not easily accessible to experiments. Furthermore, some of their properties can be further explored and tested by computational experiments which are also beyond the capabilities of conventional experimental techniques. Therefore we performed first principles calculations in the superconducting state for Fe chains on Nb(110) hosts with a single Au overlayer, where relativity, superconductivity and the complex geometry and electronic structure is treated on the same level. Additionally within our Green’s function based approach it can be done without the introduction of a supercell. On Fig. 1 it can be seen that in the magnetic chain the Yu-Shiba-Rusinov (YSR) states of the single Fe impurity hybridize within the superconducting gap of the host, as it was seen in the experiments, and the hybridized states occupy almost the entire energy range of the gap, including the vicinity of zero energy. Most interestingly however, when repeating the calculation for a 90° Néel type spin spiral, the LDOS plotted in Fig.1d shows the opening of an internal gap of Δ=0.22 meV around zero energy within the hybridized YSR states. Moreover, one peak remains seemingly untouched right in the middle of this mini gap, exactly at zero energy - that is, at the Fermi energy - on the atoms at both ends of the chain. In connection to STM experiments, these peaks at zero energy are often referred to as Zero Bias Peaks (ZBPs). It is expected, that that MZMs appear at the two ends of the chains.
We visualize our results for a wide range of spirals on in Fig, 2 by plotting the DOS (LDOS summed over all atoms in a chain) obtained for many different spirals as a function of the spiraling rotation angle.
One can easily see on Fig. 2a, that there is no meaningful gap present for the FM case. However as the spiraling angle is increasing, a minigap opens and starts to increase in size from about 20°, and keeps increasing until around 110°, where it reaches its maximum value of 0.25 meV which is 16.5 % of the full Nb gap. For larger spiraling angles the minigap starts to decrease and it collapses at around 150° and then reopens again. Probably, the most interesting feature of Fig. 2a is the existence of a state at zero energy. This state is present even in the FM state - just there is no gap around it - and remains undisturbed while the minigap opens all the way until the minigap closes again at 150° and disappears as the gap reopens for even larger spiraling angles. In order to investigate the dependence of both the minigap and the ZBP on Spin-Orbit Coupling (SOC), we repeated our calculations with SOC scaled out from the KSDBdG equations. The results can be analyzed by comparing Fig. 2a and d. Calculations behind these figures are completely identical otherwise. Probably, the most prominent effect is, that without SOC the ferromagnetic state is gapped without a ZBP in it. On Fig, 2b, c, e, and f, one may observe the spatial extent of the Majorana state. The frequently assumed physical picture is that the larger coherence length (smaller gap sizes) will be much more likely to cause larger localization length and thus hybridization of MZMs. The results we obtained here significantly changes this picture emphasizing the importance of spin-orbit effects and the necessity of a material specific treatment.
One may assign band inversion as a key signature of a topological system, which is than typically observed in the band structure of infinite systems.
At the same time, in addition to the electron (and hole) densities, the KSDBdG equations also provides us with a recipe to calculate the singlet OPs. We found that in the superconducting state, band inversion may be indicated by the energy-resolved singlet OP, due to its antisymmetric property with respect to the Fermi level, even in the case of a finite chain. This can be seen in Fig. 3, where the singlet OP is plotted along with the LDOS in the presence and absence of an MZM. It is easily observable that while in the case of a 170° spiral, where no ZBP is present, the sign of the singlet OP is the same at the positive (negative) energy edge of the minigap and the positive (negative) energy edge of the superconducting gap. However, for a 100° spiral this changes around so, that the sign of the OP is opposite at the minigap edges than at the coherence peaks at superconducting gap. This we found to be an indication of band inversion, and, consequently, a topological minigap.
References:
[1] DOI: https://doi.org/10.1038/s42005-023-01196-y
[2] DOI: https://doi.org/10.1103/PhysRevB.108.134512
[3] DOI: https://doi.org/10.1103/PhysRevB.108.134513
Scanning tunneling microscopy simulations
Scanning tunneling microscopy and spectroscopy (STM/STS) are widely used experimental methods to study the atomic and electronic structures at the surfaces of materials on the atomic scale. The interpretation of experimental STM/STS results is often not straightforward. Therefore, based on a combined method involving density functional theory (DFT) and various models of tunneling electron transport theories, we perform simulations of high-resolution STM/STS of various material surfaces. Going beyond that, we are constantly developing new theories of tunneling electron charge and spin transport and implementing them into computer codes. Our theoretical and simulation works considerably contribute to the better understanding of physical and chemical phenomena on material surfaces at the atomic scale.
Magnetic nanostructures
The interaction between localized atomic magnetic moments gives rise to magnetic ordering. At very short length scales the atomic spins are often not parallel to each other, but form stable configurations where they point along various directions. These configurations include magnetic domain walls, vortices, skyrmions and hopfions. The equilibrium properties and the excitations of these spin structures are widely studied because of their potential applications in small and energy-efficient computational devices. We study the stabilization mechanisms of these structures by calculating the magnetic interactions between the spins from first- principles methods. Using the Hamiltonian parametrized by this method, we perform atomistic spin dynamics simulations to determine the equilibrium structure and lifetime of these objects. We also investigate the magnon excitations in these structures using analytical and numerical methods.
Magnetism of strongly correlated systems
• Study of the topological properties of nanoscale quantum magnets
• Description of the optical properties of multiferroic materials
• Investigation of the entangled states and the topological properties of Mott insulators, including frustrated spin and ultracold atomic systems
• Description of spin and charge fluctuations, numerical investigation of the static and dynamical properties of rare-earth compounds using quantum Monte Carlo methods
Mechanical properties of metals and alloys
• Characterization of the soft magnetic properties of nanocrystalline materials
• Ab initio simulation of the bulk and surface properties of multicomponent disordered alloys, particularly of high-entropy alloys grown and experimentally characterized in the laboratory of the group