By: Jonas Bekaert, Levie Bringmans and Milorad V Milošević

**Long-standing research interest**

The behavior of superconductors in applied magnetic fields is a long-standing area of research. Initially, the highly influential physicist Lev Landau postulated the possibility of an intermediate state composed of a mix of normal and superconducting domains. This situation was later generalized by Ginzburg and Landau to a full description of superconducting-normal interfaces in the presence of an applied magnetic field, giving rise to the famed Ginzburg-Landau theory in 1950. To this end, they considered the energy associated with such an interface, called the surface energy. They demonstrated that it is fully parameterized by a single materials parameter, the Ginzburg-Landau parameter κ.

**New challenges**

The highly successful Ginzburg-Landau theory has undergone generalizations in recent years, to account for new superconducting materials with unconventional properties, such as multiband superconductors. In these materials, superconductivity is hosted by multiple dissimilar electronic bands. One notable example is magnesium diboride (MgB₂), which has attracted a lot of research interest because of its strong multiband features, and sustained superconductivity up to relatively high temperatures. A schematic picture of the interface between a normal region with an applied magnetic field and a superconducting region, composed of MgB₂, is shown in Figure 1.

*Figure 1: Model for a normal-superconducting interface, with an applied magnetic field (B) in the normal region and the layered multiband superconductor MgB₂ in the superconducting region. MgB₂ consists of alternating two-dimensional boron and magnesium networks.*

**Pen and paper at the start**

To account for these novel multiband effects, the first step we took was to elaborate an analytical model based on a generalized Ginzburg-Landau description. This model was applied to the superconducting-normal interface, yielding analytical expressions for key properties of this system. These include the surface energy and the critical magnetic field – the highest magnetic field strength below which a material remains uniformly superconducting. A final important result was the precise relation between the properties of the electrons in the material and the Ginzburg-Landau parameter κ. Once κ is calculated, the surface energy of multiband superconductors (denoted by δ) is entirely determined, as depicted in Figure 2*. *Importantly, we can determine if the multiband superconductor has a positive or negative surface energy. This has important consequences on whether the transition from superconducting to normal in an increasing magnetic field occurs abruptly (type-I superconductivity) or passes through an intermediate state where quantized magnetic flux tubes, called vortices, penetrate into the superconductor (type-II superconductivity).

*Figure 2: The surface energy as a function of the GL parameter 𝜿, encompassing multiband effects. The insets show the spatial dependence of the magnetic field (B) and the superconducting order parameter (ψ) around the superconducting-normal interface for different 𝜿values. *

**High-performance computing simulations for real materials**

Hence, to determine the surface energy of real superconducting materials, their microscopic parameters need to be determined. They can be calculated using quantum mechanical *ab initio* simulations, within the framework of Density Functional Theory (DFT). However, this does not come cheap. To this end we need to compute the complete interaction between the electrons and the lattice vibrations of the material, called the electron-phonon coupling, which mediates superconductivity. To perform these advanced calculations, the VSC infrastructure has proven indispensable. It has enabled us to determine the electron-phonon coupling and other microscopic parameters for several prime examples of multiband superconductors, such as bulk and monolayer MgB₂, to characterize their behavior in applied magnetic fields. This work showcases how theoretical work and high-performance computing can be highly complementary in solving advanced problems in the field of condensed matter physics.

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