Observation of Leggett’s Collective Mode in a Multiband ${\mathrm{MgB}}_2$ Superconductor

We report observation of Leggett’s collective mode in a multiband ${\mathrm{MgB}}_2$ superconductor with $T_c=39\mathrm{K}$ arising from the fluctuations in the relative phase between two superconducting condensates. The novel mode is observed by Raman spectroscopy at 9.4 meV in the fully symmetric scattering channel. The observed mode frequency is consistent with theoretical considerations based on first-principles computations.

Figure 1

The Raman response spectra of an ${\mathrm{MgB}}_2$ crystal in the normal (red) and SC (blue) states for the $E_{2g}$ (top row) and $A_{1g}$ (bottom row) scattering channels. The $E_{2g}$ channel is accessed by RL (a)–(c) or VH (d) polarization and the $A_{1g}$ channel by RR (e)–(h) polarization. The low temperature data are acquired at 5–8 K. The normal state has been achieved either by increasing the crystal temperature to 40 K (d) or by applying a 5 T magnetic field parallel to the $c$-axis [(a)–(c), (e)–(h)] [32]. The columns are arranged in the order of increasing excitation energy ${\Omega }_{\mathrm{ex}}$. Solid lines are fits to the data points. The normal state continuum is fitted with $\omega /\sqrt{a+b{\omega }^2}$ function. The data in the SC state is decomposed into a sum of a gapped normal state continuum with temperature broadened $2{\Delta }_0=4.6\mathrm{meV}$ gap cutoff, the SC coherence peak at $2{\Delta }_l=13.5\mathrm{meV}$ (shaded in violet), and the collective modes at ${\omega }_{\mathrm{LR}}=9.4\mathrm{meV}$ and ${\omega }_{\mathrm{LR}2}=13.2\mathrm{meV}$ (shaded in dark and light green). The solid hairline is the sum of both modes. To fit the observed shapes the theoretical BCS coherence peak singularity ${\chi }^{\prime \prime }\sim 4{\Delta }_l^2/(\omega \sqrt{{\omega }^2-4{\Delta }_l^2})$ is broadened by convolution with a Lorentzian with $\mathrm{HWHM}=5\%–12\%$ of $2{\Delta }_l$ [22]. The collective mode ${\omega }_{\mathrm{LR}}$ is fitted with the response function shown in Fig. 3. Panels (d) and (h) also show the high energy part of spectra for respective symmetries. The broad $E_{2g}$ band at 79 meV is the boron stretching mode, the only phonon that exhibits renormalization below $T_C$ [25]. For the $A_{1g}$ channel the spectra are dominated by two-phonon scattering.

Figure 2

The comparisons of the $ab$-plane optical conductivity ${\sigma }_{ab}$ (solid line) [20] to the integrated spectral weight under SC coherence peaks as a function of excitation energy: $2{\Delta }_l$ in the $E_{2g}$ (circles) and Leggett’s collective modes ${\omega }_{\mathrm{LR}}$ (squares) and ${\omega }_{\mathrm{LR}2}$ (diamonds) in the $A_{1g}$ channel. All dashed lines are guides for the eye.

Figure 3

$\mathrm{Im}{\chi }_{A_{1g}}(\omega )$ given by Eq. (6) using the interaction matrix by Liu et al. [9]. Inset: An illustration of the ${\mathrm{MgB}}_2$ FS in the first Brillouin zone adapted from Ref. 33. A nearly cylindrical sheet of the FS around the $\Gamma -A$ line results from the $\sigma$-band. The $\pi$-band forms a FS of planar honeycomb tubular networks. For clarity only a single FS for each $\sigma$- and $\pi$-band pair is shown [9]. In the SC state the $\sigma$-band Cooper pairs are bound stronger than the $\pi$-band pairs, at the binding energies $2{\Delta }_{\sigma }$ and $2{\Delta }_{\pi }$, correspondingly. Leggett’s collective mode originates from dynamic scattering of the $\sigma$-band pairs of electrons (illustrated in red) with momentum ($k$, $-k$) into the $\pi$-band electron pairs (yellow) with momentum ($k^{\prime }$, $-k^{\prime }$).