MgB$_{2}$: Novel properties due to multibands*

About 40 years ago A.J. Leggett proposed a new collective mode arising from cross-tunneling of Cooper pairs residing on different Fermi surfaces of a multiband superconductor: Leggett's collective mode is caused by a counter flow of the interacting superfluids leading to small fluctuations of the relative phase of the condensates while the total electron density is locally conserved.\footnote{A.J. Leggett, Progr. Theor. Phys. \textbf{36}, 901 (1966).} Here we present direct spectroscopic observation of the Leggett's excitation in the MgB$_{2}$ superconductor containig two pairs of Fermi surfaces resulting from $\pi$- and $\sigma$-bands. Electronic Raman scattering studies have revealed three distinct superconducting (SC) features: (i) a clean threshold of Raman intensity at 4.6 meV consistent with the $\pi$-band SC gap; (ii) the SC pair breaking coherence peak at 13.5 meV consistent with excitations above the $\sigma$-band gap; and (iii) the SC collective mode at 9.4 meV which we assign to an excitation first discussed by Leggett.\footnote{G. Blumberg et al., Phys. Rev. Lett. \textbf{99} (2007); \urllink{arXiv:0710.2803}{http://arxiv.org/pdf/0710.2803}.} Our calculation of the Raman response function for MgB$_{2}$ superconductor based on multiband interaction matrices by first principle computations show good agreement with spectroscopic observations. The temperature and field dependencies for all three features (i) -- (iii) have been established;\footnote{G. Blumberg et al., Physica (Amsterdam) \textbf{456C}, 75 (2007).} the effects of magnetic field on the pair cross-tunneling in multiband system will be discussed. In addition, anharmonicity and superconductivity-induced self-energy effects for the E$_{2g}$ boron stretching phonon have been studied.\footnote{A. Mialitsin et al., Phys. Rev. B \textbf{75}, 020509(R) (2007).} We show that anharmonic two-phonon decay is mainly responsible for the unusually large linewidth of the E$_{2g}$ mode. We observe 2.5\% hardening of the E$_{2g}$ phonon frequency upon cooling into the SC state and estimate the electron-phonon coupling strength associated with this renormalization.