Transverse-Field $\mu$SR (TF-$\mu$SR)

In a transverse-field muon spin rotation experiment (Fig. 2.3), a magnetic field $\vec{H}$ is applied perpendicular to the initial muon-spin polarization direction. A muon stopping in the sample Larmor precesses about the local magnetic field $\vec{B}$ (which in general is different than the external field) at an angular frequency $\omega_{\mu}$ = $\gamma_{\mu}B$, where $\gamma_{\mu}/2\pi$ = $135.5342$ MHzT$^{-1}$ is the muon gyromagnetic ratio.

Fig. 2.3: Schematic diagram of a TF-$\mu$SR experiment. The initial muon spin polarization $\vec{P}_{0}$, indicated by the open arrow at $\mu^+$, is rotated by 90$^{\circ}$ so that it is perpendicular to the direction of the external magnetic field $\vec{H}$. The open arrows at the sample S illustrate the Larmor precession of the muon spin.

In a superconductor the muons stop at well defined sites in the crystal lattice. However, in the vortex state of a type-II superconductor, the muons stop randomly on the length scale of the vortex lattice (which is typically two to three orders of magnitude larger than that of the crystal lattice). Consequently, $\mu$SR is an effective local probe of the spatial variation of internal magnetic fields due to the periodic arrangement of vortices. For the case where the external field is directed along ${\hat z}$, the $x$ component of the muon-spin polarization function is

$$P_{x}(t) = \int^{\infty}_{0} n(B)\cos(\gamma_{\mu}Bt+\phi)\,dB\, ,$$ (3.6)

where $n(B)$ is the probability that a muon sees a local magnetic field between $B$ and $B$+$dB$, and $\phi$ is the initial phase. There are several sources of the local magnetic field sensed by the muons. First there is the magnetic field inhomogeneity associated with the vortex lattice. For a type-II superconductor, the spatial profile of the magnetic field in the ${\hat a}$-${\hat b}$ plane due to an applied field along the ${\hat c}$ axis (${\hat z}$ direction) is reasonably described by the phenomenological model [19]

$${\vec B}({\vec r}) = B_{0}(1-b^4)\sum_{\vec K}\frac{e^{-i{\vec K}\cdot{\vec r}}uK_{1}(u)}{\lambda_{ab}^{2}K^2}{\hat z}\, ,$$ (3.7)

where $\vec{K}$ is a reciprocal lattice vector, $b$ = $B/B_{c_{2}}$ is the reduced field, $K_{1}(u)$ is a modified Bessel function and $u^2$ = $2\xi_{ab}K^2(1+b^4)[1-2b(1-b)^2]$. Equation (2.7), which is derived from Ginzburg-Landau theory, assumes that $\lambda_{ab}^{2}K_{min}^{2} \gg 1$, where $K_{min}^{2}$ is the magnitude of the smallest non-zero reciprocal-lattice vector in the summation. This condition is satisfied for an extreme type-II superconductor like the high-$T_c$ cuprates, where $\lambda_{ab} \gg \xi_{ab}$. This self-consistent analytic function, agrees extremely well with the exact numerical solutions of the Ginzburg-Landau equations at low reduced fields $b$. Random vortex pinning and thermal fluctuations modify the field distribution associated with Eq. (2.7), which assumes a perfect periodic arrangement of vortices.

The muon is also sensitive to both nuclear and electronic dipole moments. The nuclear moments are randomly oriented at temperatures reachable in a $\mu$SR experiment (i.e. $T \geq 15$ mK), whereas electronic magnetic moments may order. In general, the magnetic moments that are static on the $\mu$SR time scale broaden the measured internal magnetic field distribution. To account for the effects of this additional source of field inhomomgeneity on the muon-spin polarization function, Eq. (2.6) can be multiplied by a depolarization function $G(t)$ such that

$$P_{x}(t) = G(t)\int^{\infty}_{0} n(B)\cos(\gamma_{\mu}Bt+\phi)\,dB\, .$$ (3.8)

The precise functional form of $G(t)$ depends on the nature of the additional sources of magnetic field at the muon site. For example, a Gaussian function $G(t)$ = $\exp{(-\sigma_{\rm dip}^2t^2/2)}$ with a depolarization rate $\sigma_{\rm dip}$, characterizes the damping of the muon-spin precession signal due to dense static dipole moments [20].