Spin-lattices coupled to proximity-superconducting Rashba-surface states

Methods

In our work, we mainly employ the full ab-initio method based on DFT and implemented in the full-potential relativistic KKR method within the JuKKR code. This method, which we refer to as the Kohn-Sham BdG (KS-BdG) approach, is at the core of investigations carried out in this project. The electron and hole sectors of the KS-BdG Hamiltonian are coupled by the SC order parameter assumed to originate from the electron-phonon coupling, which can open a SC gap. The coupling is an input parameter, in the same spirit as done in the LDA+U approach when tackling electronic correlation effects. This approach is pragmatic and makes the KS-BdG method very flexible since SC can initially be imposed in a specific region of a material (part of the computational cell), which enables the self-consistent description of inhomogeneous systems. Therefore, the physics at interfaces of distinct materials (SC/non-SC) and proximity-induced effects at both sides of the interfaces can be efficiently explored and quantified. Besides the charge density, the anomalous density, which is the SC order parameter, becomes a crucial quantity in the formalism and allows to interpret the pairing symmetry. The code can address periodic materials or impurities, nanostructures and wires on surfaces including an arbitrary rotation of magnetic moments, within the embedding technique enabled by the KKR method.

Results

First, we simulated self-consistently the noble metal surfaces Ag(111) and BiAg${}_2$/Ag(111), and identified the effect that Rashba surface states have in their band structure. We then injected an electron phonon coupling$\lambda$ in the bottom Ag layer of the BiAg${}_2$Ag(111) film, to emulate the existence of the Nb s-wave superconductor, and through the proximity effect, were able to observe a superconducting gap open in the BiAg${}_2$ surface alloy. We proceeded with the simulation of a Fe adatom embedded in the hollow/bridge site of the Bi surface. The in-gap YSR state spectrum was calculated and compared with experimental measurements on the system. Next, we investigated the effect of magnetic moment rotation on the bound-state energy spectrum. A significant dependence on the moment orientation is observed, induced by the presence of strong SOC from the BiAg${}_2$ substrate. To further study the effects of Rashba surface states on the YSR spectrum, we constructed chains of Fe adatoms on the BiAg${}_2$ surface. Starting from a dimer and going up to an 8-atom chain, the Shiba spectrum was calculated and compared to experimental data. Using the the tensor of magnetic interactions and the magnetic anisotropy energy extracted from our DFT calculations, the magnetic ground state of each structure is evaluated and density of states calculations are performed for each and compared to results for a ferromagnetic arrangement of the atoms. There, we distinguished the effect the magnetic configuration of the system has on the Shiba spectrum, and derived the conclusion that in the proper moment misaligned order, zero-bias edge states can be observed. A qualitative match is established between our theoretical treatment and experimental measurements on the system.

Discussion

In our work, magnetic moment misalignment is identified as a key factor for the realization of zero-bias edge states in magnetic adatom chains, which are a cornerstone for the establishment of Majorana zero modes. An elaborate study of the effects of SOC in the BiAg${}_2$ superconducting heavy-metal surface leads to the conclusion that by modifying the spin chain construction and thus tuning the magnetic order of the system can result in a rich Shiba spectrum hybridization, which with the proper treatment can give rise to zero-bias states. Further investigation of these effects in chains comprised of different magnetic atoms, or in different magnetic setups are logical next steps. A confinement of both single adatom and chain systems in quantum corrals can also give insights into the spatial propagation of these features and the way they reshape the landscape of DOS in such surfaces.

Additional Project Information

DFG classification: 307-02 Theoretical Condensed Matter Physics
Software: JuKKR
Cluster: CLAIX