Prediction of superconducting transition temperatures of heterostructures based on first principles
In this paper we present material specific calculations of superconductor – normal metal heterostructures using density functional theory. In particular, we calculate the quasiparticle spectrum of different normal metal overlayers on a Nb(100) host. We find that the Andreev reflection leads to the formation of momentum dependent quasiparticle bands in the normal metal. As a consequence, the spectrum has a strongly momentum dependent induced gap. From the thickness dependence of the gap size we calculate the superconducting critical temperature of Au/Nb(100) heterostructures where we find very good agreement with experiments. Moreover, predictions are made for similar heterostructures of other compounds.
The experiments of Yamazaki et al. Yamazaki et al. (2006, 2010) poses a great challenge to the theory of superconductivity. In this experiment they studied the critical temperature of a very thin layer of gold grown on top of a thicker Nb host. While the thickness of the Nb sample (288 Å) was in the order of the superconducting coherence length (380 Å), the gold overlayer was only a couple of atomic layers thick. It was found that the superconducting transition temperature is significantly lowered in this composite compared to the bulk, and it decays monotonically as the thickness of the gold overlayer increases. While this appears to be a plausible consequence at first sight, it is surprising that the entire composite is superconducting. The explanation of these experimental findigs based on the standard theory of superconductivity is very difficult, because within the BCS (Bardeen-Cooper-Schriffer) theory Bardeen et al. (1957) it is not easy to describe inhomogeneous systems. However such systems can be treated efficiently in terms of the Bogoliubov-de Gennes (BdG) equations Bogoliubov (1958); de Gennes (1966), especially when they are reformulated within the framework of density functional theory Oliveira et al. (1988); Suvasini et al. (1993); Lüders et al. (2005); Marques et al. (2005); Csire et al. (2015) leading to the so called Kohn-Sham-Bogoliubov-de Gennes (KS-BdG) equations. In principle these equations are able to describe systems with inhomogeneities in the pairing potential.
In inhomogeneous systems, such as multilayers, it is expected that the Andreev reflection Andreev (1964); Blonder et al. (1982) leads to the formation of certain bound states, often referred to as Andreev bound states. Our main aim in this paper is to calculate the dispersion relation of Andreev states. We also verify that this effect enables the transport of supercurrents through non-superconducting materials, and is the key to predict the superconducting transition temperature in the whole system.
In what follows we numerically solve the KS-BdG equations, as it was described in Ref. Csire et al. (2015), for an Au/Nb(100) overlayer system. For varying Au thicknesses we calculate the quasiparticle electronic structure and obtain the superconducting gap for each thickness. Finally a method is developed to calculate the critical temperature from the gap and applied to several overlayer systems with different material constituents.
The first step down this road is to construct a realistic model of the interface lattice structure. Such a model can be built up from two-dimensional translational invariant atomic layers as follows. We divide the system into three regions: (i) a semi-infinite bulk (Nb); (ii) the interface region that – in our case – consists of six superconducting layers (Nb), various number of normal metal layers and three layers of empty spheres; (iii) and a semi-infinite vacuum. This division of space is typical in Screened KKR Szunyogh et al. (1994) calculations in overlayer systems. Although the thickness of the Nb layers in the experiment is not strictly semi-infinite, it is thick enough to be approximated by a semi-infinite bulk system. We use six Nb layers to mimic the transition between the semi-infinite bulk and the overlayer. The Nb has the body-centered cubic (bcc) crystal structure with a lattice parameter of Å. We assumed the realistic face-centered cubic (fcc) crystal structure for the gold overlayers, and the termination of the material layers were modeled by a semi-infinite vacuum. We also neglected imperfections at the interface, like interlayer relaxations and intermixing.
In such systems, where translational invariance is preserved only parallel to the interface, the quasiparticle spectrum is obtained as a function of a two dimensional momentum vector . To visualize this spectrum, it is customary to calculate the Bloch spectral function (BSF). In two dimensionally translational invariant systems the BSF for layer is defined as , where is the layer projected Green function (see Ref. Csire et al. (2015)). Since the BSF is equivalent to the quasiparticle spectrum, drawing a contour plot of the BSF as a function of energy along specified directions of is a powerful tool to visualize the quasiparticle states. Evidently, in a layered system this contour plot can be done for each layer.
In order to understand the quasiparticle spectrum in the superconducting state better, first we performed first principles calculations just in the normal state, by simply solving the Screend KKR equations Szunyogh et al. (1994) for the Au/Nb(001) overlayer system. In Fig. 1 we show the contour plot of the BSF for a layer that we considered to be in the ”middle” of the samples of various Au layer thicknesses. The plots are restricted in energy to the range of [-0.05 Ry, 0.05 Ry] (later we will choose , the superconducting gap to equal this value, and solve the BdG equations within this energy range). It can be seen from Fig. 1 that for energies where the DOS in the bulk Nb is low, the states in the Au are confined, as they can not scatter into the Nb, and on the other side the system is limited by vacuum. In regions where the DOS is high in the Nb, the states in the Au are smeared out, as here the electrons can scatter more easily into the other side of the interface. The confined states in the Au can be regarded as quantum-well (QW) states.
Now let us consider the Nb in the superconducting state and solve the first-principles BdG equations as described in Ref. Csire et al. (2015). In this formalism each layer is characterized by an additional, potential-like parameter, the superconducting gap . To solve the BdG equation we assumed that all Nb layers show the same superconducting gap Ry. This is a much larger value than the gap observed experimentally Pronin et al. (1998), in order to make it easier to visualize the quasiparticle spectrum. Later, when we will calculate the superconducting transition temperature, we will repeat the calculation for a more realistic . We also assumed, that the superconducting gap is zero in the Au layers, namely we set Ry, as Au is not superconducting at all in the bulk.
Based on the symmetry of the BdG equations de Gennes (1966) it is known, that in bulk superconductors the quasiparticle states can be obtained from the electronic states by mirroring the band structure of the normal state to the Fermi energy and opening up a gap. In the case of the Au/Nb(100) overlayer system the situation is entirely different, because the system is described by an inhomogeneous which, additionally, equals to zero in some layers. Therefore the main question is how the niobium host affects the quasiparticle spectrum of the gold overlayers. In Fig. 2 we plot the BSF summed up for all Au layers in the case of different total number of gold layers (right hand side of the figure). To illustrate the role of the crystal structure, on the left hand side of the figure we also show results of the same calculations performed by assuming bcc Au layers throughout the overlayer. This is not entirely unrealistic for very thin layers but most likely artificial for thicker ones, where the Au layers most certainly grow in an fcc structure. Unfortunately there is very little known of the crystal structures of thin Au layers on Nb.
By comparing Fig. 1/a,c,d and Fig. 2/a2,b2,c2, we can immediately conclude that the proximity of a superconductor in the studied heterostructures induces the expected mirroring of the electronic bands within the energy range of the Nb gap, and additionally opens up a smaller gap at each band crossing. Although it is quite trivial, it is much easier to see the effect first in the case of the bcc Au, fully described in Csire et al. (2015). One can also see from the figure that the size of the induced gaps in the gold is strongly -dependent, however - around the Fermi energy - it is the same for all layers, and it decreases as a function of the overall thickness of the gold overlayers. It should be noted as well that those regions of the spectrum which were more or less smeared out in the normal state (visualized in the figure by weaker lines) now sharpened up. This is the consequence of the opening of the superconducting gap in the Nb: the states where scattering into - on the other side of the interface - was allowed before, now disappeared because of the gap. We can also conclude from the figure that the quantum-well states, which we found to exist in the normal state band structure calculations, become bound Andreev states originating from the Andreev scattering at the interface. Finally, since Fig. 2 was obtained by summing up , and this sum does not show any significant broadening, it is clear that the quasiparticle spectrum is virtually identical in every interfacial layer. In Fig. 2 one can also observe changes in the quasiparticle spectrum owing to different lattice structures. In particular, more ’oscillations’ can be seen in fcc gold overlayers compared to the bcc ones, which is the consequence of more bands of the fcc gold in the normal state.
We are now ready to calculate the superconducting transition temperature . In the strong-coupling limit (e.g. in the case of Nb) the is given by the McMillan formula McMillan (1968); Allen and Dynes (1975)
where is the Debye temperature, is the dimensionless Coulomb pseudopotential which describes the effect of the Coulomb repulsion. For this two parameter we may assume the bulk values, justified by the fact that the niobium host is rather thick. The parameter is an effective electron-phonon coupling parameter for the whole system. Following the argument of de Gennes Yamazaki et al. (2010); De Gennes (1964) the simplest way to define such an effective electron-phonon interaction parameter in an inhomogeneous system is by an averaging process described in the following way. Let us interpret as a probability of finding an electron in the normal metal layers (Au) and in the superconducting Nb layers. Then the effective electron-phonon coupling can be defined as
where and are the thickness of the Nb and Au layers, respectively, and is the density of states at the Fermi level. While the density of states for the Nb can be approximated by the bulk value, the density of states in the gold should be defined as a layer average, , where is the number of the gold overlayers. Since the whole niobium – gold system is superconducting (a common obtained experimentally in Ref. Yamazaki et al. (2006, 2010)), Cooper pairs must exist throughout the whole system. Therefore, the existence of the superconducting gap in the Au layers can be viewed in a way that the proximity of the superconductor induces an effective electron-phonon coupling in the gold, which in turn can be obtained from the induced gap according to the BCS gap equation:
This induced gap can be read off from the quasiparticle spectrum, for which the whole calculation of the quasiparticle spectrum needs to be repeated with a more realistic value for , as we mentioned earlier. (For better visualization we used an artificially large in Fig. 2.) While the BCS theory is valid only for the weak-coupling limit, nevertheless we still argue that the BCS gap equation may be applied here to the gold layers, since the induced gap is much smaller than the gap in the Nb. To obtain the induced gap, we always take the maximum of the induced gap around the Fermi energy (as a function of ) in the 2D Brillouin zone because it is the most characteristic of the strength of the electron-phonon coupling. In summary, the induced gap can be read off from the calculated quasiparticle spectrum as the function of the thickness of the gold overlayers (see Fig. 3 inset), and the critical temperature can be calculated from the McMillan-formula given by Eq. (1). Figure 3 shows the critical temperature as a function of the thickness of the Au overlayers obtained from the procedure described above together with the experimental results taken from the papers Yamazaki et al. (2006, 2010). One can clearly see that our theoretical results are in very good agreement with the experimental findings, all points are within the experimental errors. Moreover, the slow decrease of the critical temperature as function of the Au overlayer thickness is well reproduced. We should emphasize that our theory contains only one parameter, namely the critical temperature for bulk niobium. While in principle this quantity can be calculated Klein and Papaconstantopoulos (1974), here for simplicity we take its experimental value.
The theory presented in this paper so far can be applied without changes to other overlayer systems like NbAg, NbIr, NbAl and NbMo. Nevertheless, it should be mentioned, that many of the properties of the quasiparticle spectrum we studied above are connected to the fact that QW states form in the Au in the normal state. The formation of QW states however, can not be regarded as a universal feature of every overlayer on Nb. This is illustrated in Fig. 4, where we show the normal state band structure and also the corresponding quasiparticle spectrum of Al and Mo overlayers on the Nb host. While there may be some quantum confinement present in these materials in the normal state, it is not as clearly exhibited as in the case of Au overlayers. It is clearly seen, that quantum confinement is still present in the superconducting state. It is important to notice as well, that in these systems it is hard to recognize how the quasiparticle specrtum could be obtained from the normal state band structure, as we mentioned before, based on the symmetry properties of the BdG equations. The most important feature of these plots is that in the superconducting state the induced gap around the Fermi-level can still be observed, therefore, the superconducting transition temperature can be predicted similarly to the NbAu heterostructure. This is shown in Fig. 5 for several overlayer systems.
It can be seen that it is a general trend that the superconducting transition temperature decays as the normal metal thickness increases, only the rate of the decay depends on the material of the overlayer. This behavior is a consequence of the size of the induced gap in the normal metal and also the change of the density of states at the Fermi level which enters the McMillan formula through the effective electron-phonon coupling parameter. This is most pronounced for the case of iridium overlayers, where the decay of the transition temperature is the fastest, mostly due to the much higher density of states at the Fermi level.
In summary, we have investigated the quasiparticle spectrum, and the superconducting transition temperature of superconductor – normal metal heterostructures. We have developed a method to predict the transition temperature of such heterostructures based on the first-principles solution of the KS-BdG equations. In the case of the Nb/Au(100) system we obtained very good agreement with the experimental findings. The theory was also applied for different metallic overlayers on a Nb host to predict the superconducting transition temperature.
Acknowledgment — This work was supported by the Hungarian National, Research, Development and Innovation Office under the contracts No. K115632 and K108676.
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