A. Bussmann-Holder; A. Bianconi (Universit`a di Roma, Italy) The model of two (σandπ) channel
supercon-ductivity known to be necessary to explain the superconductivity in MgB2has been applied to the Al1xMgxB2 diborides by tuning x from MgB2 to AlMgB4. The evolution of the inter-band coupling parameter (probing the strength of the interchannel pairing due to quantum in-terference effects) and the two gaps in theσand
π channel as a function of x have been calcu-lated. While in MgB2the quantum interference effects give an amplification of Tc by a factor of 1.5 in comparison with the dominant intra σ-band single channel pairing, in AlMgB4the am-plification is about 100, in comparison with the dominant intraπ-band single channel pairing.
Various recent experimental results for MgB2 show that the two-band model (TBM) is needed to explain both the normal and the supercon-ducting properties. Here the key ingredient that allows the physical realization of the process of two channel Tcamplification is the fact that the two gaps refer to two different parts in k-space (a large gap on theσFermi surface, a small gap on theπFermi surface) and are well separated in real space, one for the s holes in the boron layers, and the other forπ-electrons in the inter-stitial Mg layers. Therefore the system can be viewed as a multilayer structure of alternating metallic and superconducting planes.
Figure 5: (a) The superconducting transition tem-perature for Al1xMgxB2from x = 0.5 (AlMgB4) to x = 1 (MgB2). (b) The PDOS of theσ-band Nσand of theπ-band Nπas function of x.
The Al1xMgxB2 alloys show a continuous evolution through a complicated mixed phase from MgB2(x = 1) to the end member AlMgB4
(x = 0.5) where an ordered superlattice structure of boron layers intercalated by alternating lay-ers of Al and Mg is formed. Even though the alloys with intermediate x are rather disordered, their Tcis well defined and drops with decreas-ing x as shown in Fig. 5(a). Around x = 0.7, Tc shows a kink which is attributed to a dimen-sionality crossover of the Fermi surface and at x = 0.5, Tc= 3 K. The superconducting phase in the ordered phase AlMgB4 (x = 0.5) is highly interesting since the Fermi level is driven to the top of the σ-band, the PDOS in the σ-band is strongly reduced and the Fermi energy EF for theσ-holes is only 100–200 meV.
The two-gap scenario which has already been used for MgB2, has also been used for the alloys and the gaps and interband couplings have been calculated as function of Al doping.
Figure 6: The calculated superconducting transition temperature Tcfor two ideal systems made of only σ- andπ-electrons, respectively, are compared with the experimental data.
Going from x = 1 to x = 0.5 a dramatic increase in the E2gphonon mode energyωE2gtakes place increasing from 70 meV to 115 meV indicating a strong decrease in the electron-phonon inter-action and a reduction of the phonon damping.
Since this mode is known to couple strongly, it was first suggested that the large value of its coupling constant is sufficient to explain the high Tc within the single band approach. The optimum superconducting transition tempera-tures for two different ideal metals made of only σ- and π-electrons, respectively, without interband interactions are shown in Fig. 6. The corresponding screened effective couplings are plotted in Fig. 7. From Fig. 6 we can clearly see that the single-band model fails to predict the experimental Tcxdependence and is inca-pable to reproduce experimental data for vary-ing x. This motivated us to model also the alloys within the two-band scenario using the BCS approximation and the experimental data for the two intraband couplingsλ1xandλ2x (Fig. 7).
Figure 7: The screened effective couplings λ1x andλ2x for the σ- and π-electrons respectively.
The interband couplingλ12xhas been calculated by using the two-band model in such a way as to re-produce the experimental values of Tcfor each value of x.
Taking as input the experimental values of the phonon mode energiesωE2gxandωlnx, Tcx,λ1x andλ2x, the interband coupling λ12xand the intraband gaps∆1xand∆2x for the σ- and π-band, respectively, are ob-tained. The results are shown in Fig. 7 where
the effective electron-phonon couplings are pre-sented. The corresponding energy gaps at T = 0 K are shown in Fig. 8(a) and the gap to Tc ratios are depicted in Fig. 8(b).
Figure 8: (a) The energy gaps, at T = 0 K,∆1xand
∆2xfor theσ- and π-electrons, respectively, and (b) the corresponding gap to Tcratios as a function of x.
As is well known for the two-band model, both gap to Tcratios deviate substantially from BCS predictions – one being strongly enhanced, while the other is far below the predicted value.
The obtained values of ∆1(x = 1) and∆2(x = 1) are in very good agreement with the experimen-tal ones.
Figure 9: The predicted temperature dependence of the gaps for three different systems AlMgB4 (top panel), Al025Mg075B2 (middle panel) and MgB2 (lower panel).
Interestingly the interband coupling λ12x (Fig. 7) increases with decreasing x to reach a maximum around x0.6–0.7 where the
strength of the interchannel pairing due to quan-tum interference effects is optimum. Here also the Tcxcurve shows a kink signaling that the Fermi level has been tuned at the cross-over of the Fermi surface of theσ-band from 2D to 3D dimensionality. The related σ- and π-gaps as a function of x (Fig. 8) show the very interest-ing case of interchange of their dominance and a gap crossing takes place at x = 0.6 where the σ-band related gap becomes smaller than theπ related one. For AlMgB4 we have therefore a different physical situation for the two-gap sce-nario. In fact, in MgB2 the interchannel inter-ference effects push Tcup to the strong coupling regime (2∆1/Tc= 4.2) with an effective ampli-fication of Tc of the order of 1.5–2 increasing the strong-intermediate coupling regime of the dominant 2D σ-band. In AlMgB2 the π-band is the dominant one which is supported by the 3D σ-band with small intraband coupling λ1. While the intraband pairing alone yields a Tc of 1–10 mK, the actual Tc is 3 K, corre-sponding to an amplification of 100–1000. The consequence of the interchange of the driving band going through the ‘shape resonance’ at x = 0.6–0.7 is that the gap separation is strongly doping dependent, being large for MgB2, inter-mediate for x = 0.75 and reversed at x = 0.5. The temperature dependence of the gaps for these three cases is shown in Fig. 9 where substantial differences are predicted which can be tested by further experiments.
In conclusion, we have shown that also the al-loyed systems Al1xMgxB2 with 0.5x1.0 are best described within a two-band model where interband interactions are the dominant force that drives Tc to the experimentally ob-served values. Predictions for the ratio of the two gaps are made, where especially a reverse in gap magnitudes is obtained.