4.1 The Field Distribution of the Vortex Lattice

Figure 4.1 shows a typical muon-spin precession signal in the normal and vortex states of YBa₂Cu₃O_6.95 obtained by applying a magnetic field parallel to the $\hat{c}$-axis. For convenience these signals are displayed in a reference frame rotating at about 3 MHz below the average muon Larmor precession frequency in the vortex lattice. A damped signal results from the inhomogeneous distribution of magnetic field in the sample. The undamped signals arising from individual muons precessing in different local static fields combine to give a signal which decays over time. Above T_c where flux penetrates the sample uniformly, there is only a slight damping of the signal which is attributed mainly to the random local fields of nuclear dipolar moments. On the other hand, below T_c the strongly damped signal is primarily due to the inhomogeneous field distribution of the vortex lattice.

 

Figure 4.2 shows the finite Fourier transforms of the time spectra in Fig. 4.1. The real amplitude of the Fourier transform represents a good approximation to the internal field distribution. Above T_c the $\mu$SR line shape is symmetric with some broadening due to the nuclear dipolar moments (see top panel of Fig. 4.2). Below T_c the observed line shape is primarily due to the vortex lattice. The sharp peak at 67.3 MHz is attributed to the residual background signal from muons which miss the sample.

 

Figure 4.3 shows a general theoretical field distribution corresponding to a triangular vortex lattice. The sharp cutoff at low fields is due to the minimum in the field distribution which occurs at the center of the triangle formed from three adjacent vortices. The peak is due to the saddle point midway between two adjacent vortices. The long tail is due to the region around the vortex core, and the high-field cutoff is due to the maximum field at the center of the core. As shown in the bottom panel of Fig. 4.2, the sharp features expected from the vortex lattice are smeared in the Fourier transform of the measured muon precession signal. This is primarily due to the broadening effects associated with the Fourier transform (which were discussed in the previous chapter). The measured line shape also contains broadening effects due to the nuclear dipolar moments, fluctuations in the temperature and magnetic field, demagnetization effects associated with the sample geometry and disorder in the vortex lattice caused by pinning.