Electron tunneling spectroscopy on superconducting Al-doped ${\text{MgB}}_2$ thin films: $\pi$ energy gap and Eliashberg function

Superconducting thin films of composition ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ with $0\le x<0.5$ were prepared in situ by sublimation of Mg combined with B and Al magnetron sputtering. The critical temperature $T_{\text{c}}$ decreased linearly with $x$ up to 0.4. For $0.4<x<0.5$ the formation of a plateaulike feature at a $T_{\text{c}}\approx 12\text{K}$ was observed. This effect is supposed to be due to the incipient formation of the superstructure ${\text{MgAlB}}_4$ with ordered alternating Mg and Al planes separated by $B$ planes. To detailedly study the influence of Al doping on the electron-phonon coupling in the polycrystalline films with a preferred $c$-axis texture quasiparticle tunneling experiments were performed on planar tunnel junctions with natural thermal oxide or artificial aluminum oxide tunnel barriers. Differential conductance measurements at low-bias voltage and low temperature of superconductor-insulator-superconductor tunnel junctions allowed the direct determination of the energy gap of the Fermi surface $\pi$ sheet. The $\pi$ energy gap decreased linearly with decreasing $T_{\text{c}}$ of the films in agreement with the model of band filling. Whereas all the tunneling studies published so far mainly revealed features of the two energy gaps, the observation of phonon-induced structures in the differential conductance measurements at high bias voltage, i.e., in the phonon region, in this study enabled the important determination of the energy-dependent Eliashberg function ${\alpha }^{2}F$ of the Fermi surface $\pi$ sheet for various Al doping levels. Compared to the undoped ${\text{MgB}}_2$, significant changes in ${\alpha }^{2}F$ could be observed that were confirmed by first-principles calculations.

Figure 1

Exemplary RBS spectrum of an Al doped ${\text{MgB}}_2$ thin film on $r$-plane sapphire substrate. Such spectra were used to determine the Al doping level $x$, here $x=0.38\pm 0.02$ ($F=\text{film}$, $S=\text{substrate}$).

Figure 2

Set of inductively measured superconductive transitions with the Al doping level $x$ as a parameter. The transition curves are described by the characteristic temperatures onset, midpoint, and downset (see dashed horizontal lines).

Figure 3

Normalized conductance after elimination of the inelastic tunneling contribution versus bias voltage for various junctions A to I. Each characteristic is normalized to its minimum value. The characteristics are shifted in vertical direction for clarity. The inset shows the conductance increase due to inelastic tunneling for junctions E and G. The voltage polarity is that of the In counterelectrode. A, B, and C: Junctions with native oxide barrier. D to I: Junctions with artificial ${\text{Al}}_2{\text{O}}_3$ overlayer.

Figure 4

(a) Transition temperature $T_{\text{c}}$ of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ thin films (●, this work), polycrystals (○, Ref. 13, and ▲, Ref. 53) and single crystals (◻, Ref. 54, and $★$, Ref. 12) in dependence on the Al concentration $x$. The straight line is the result of a calculation by Kortus et al., Ref. 19. The dashed lines are guides to the eye. The horizontal and vertical bars indicate the absolute error $\pm \Delta x$ and the transition widths $\Delta T_{\text{c}}$, respectively. (b) Transition width $\Delta T_{\text{c}}$ of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ thin films (●, this work), single crystals (◼, Ref. 12), and polycrystals (△, Ref. 53, and ◇, Ref. 55) versus $x$.

Figure 5

Voltage dependence of the normalized conductance of three ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$/insulator/In junctions with $x=0.16$ (junction A), 0.38 (junction B), and 0.49 (junction C) measured at 1.2 K in zero magnetic field with a rms modulation voltage $\Delta V$ of $32\mu \text{V}$. Both electrodes are superconducting. The conductance is normalized to the constant value of each junction with normal electrodes. The gap ${\Delta }_{\pi }$ of the diboride films can be directly determined from the position of the conductance peaks. The modulation voltage broadening of $1.7\Delta \text{V}$ can be neglected compared with the width of the conductance peaks.

Figure 6

Enlargement of Fig. 5 by a factor of 10 between $-2$ and 2 mV. The inner-gap features of the junctions are clearly resolved and discussed in the text. For a better differentiation between curve B and C, curve C was slightly shifted upwards.

Figure 7

Differential conductance $dI/dV$ at zero bias, $V=0$, of a ${\text{Mg}}_{0.88}{\text{Al}}_{0.12}{\text{B}}_2$/insulator/In junction versus temperature. The conductance is normalized to the constant value of the junction with both electrodes in the normal state (straight line, left scale). The open circles are the result of a simulation with $2{\Delta }_0/k{T}_{\text{C}}=1.65$ as a fit parameter. The local gap $T_{\text{c}}$ measured on the junction area of $\approx {10}^{-3}{\text{cm}}^2$ is 29.1 K. The nonlocal $T_{\text{c}}$ of 28.9 K on a film area of $\approx 1{\text{cm}}^2$ is measured by the inductive transition curve (right scale). The good $T_{\text{c}}$ agreement is another manifestation of the film homogeneity.

Figure 8

${\Delta }_{\pi }$ close to zero temperature versus $T_{\text{c}}$ of diverse ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ samples; ●, this work (thin films, ${\text{SIS}}^{\prime }$ junctions at 1.2 K); ◻, Ref. 27 (sc: single crystals, STS: scanning tunneling spectroscopy); $+$, Ref. 24 (PCS: point contact spectroscopy); ◇, Ref. 26 (PCAR: point-contact Andreev-reflection); ∗, Ref. 25 (pc: polycrystals); △, Ref. 14 ($C_{\text{p}}$: specific heat measurements); $\times$, Ref. 28; –, Ref. 19 (band filling calculation); --, this study (band filling calculation). Vertical bars on full circles indicate the statistical error of ${\Delta }_{\pi }$ upon gap measurements on different junctions with the same $T_{\text{c}}$ of the ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ base electrode.

Figure 9

(a) Differential conductance $dI/dV$ versus bias voltage of a ${\text{Mg}}_{0.89}{\text{Al}}_{0.11}{\text{B}}_2/\text{Al-oxide}/\text{In}$ tunnel junction measured at 4.2 K (${\text{Mg}}_{0.89}{\text{Al}}_{0.11}{\text{B}}_2$ superconducting, In normalconducting; bold line) and at 30 K (both films normalconducting, thin line). (b) Negative second derivative of the $I\text{−}V$ characteristic versus voltage measured at 4.2 K. The normal-state background has been subtracted.

Figure 10

Reduced density of states (RDOS) versus energy measured on a ${\text{Mg}}_{0.89}{\text{Al}}_{0.11}{\text{B}}_2/\text{Al-oxide}/\text{In}$ junction (dashed-dotted line). The straight line is the result of a calculation using the PMMR-program with parameters $d/\text{l}=0.4$ and $R=0.002{\text{meV}}^{-1}$.

Figure 11

(a) to (c) Eliashberg functions ${\left({\alpha }^{2}F\right)}_{\pi -\pi }$, ${\left({\alpha }^{2}F\right)}_{\pi -\sigma }$, and ${\left({\alpha }^{2}F\right)}_{\sigma -\sigma }$, respectively, of the two-band superconductor ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ for $x=0$ (thin lines) and $x=0.39$ (thick lines) determined by first principles calculations in the generalized gradient approximation (GGA).

Figure 12

(a) to (d) ${\left({\alpha }^{2}F\right)}_{\text{eff}}^{\text{th}}$ of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ for $x=0$, 0.11, 0.31, and 0.39, respectively, as determined by the inversion of the calculated RDOS of the $\pi$ sheet using the single-band Eliashberg equations (open circles). The solid lines are the result of a superposition of ${\left({\alpha }^{2}F\right)}_{\pi -\pi }$ and ${\left({\alpha }^{2}F\right)}_{\pi -\sigma }$ (e) to (h) ${\left({\alpha }^{2}F\right)}_{\text{eff}}^{\text{ex}}$ of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ for $x=0$, 0.11, 0.31, and 0.39, respectively, as determined by the tunneling experiment.