Figure 5.5 presents the interpolated resistivity data derived from a large number of field sweeps, each at slightly different angle, on a crystal oriented
with its c-axis close to the magnet HZ axis. Magnetic field provides a 3D
coordinate space, which complicates the presentation of data. Here, I aim to show data in a way which is convenient both when demonstrating the physical
limits of the magnet, but also when investigating the inversion symmetry which must exist around the c-axis. The choice of presentation is then to
cut the data set at constant field modulus, leaving rotation angles alongHX
and HY as free variables ΘX and ΘY (always expressed in units of degrees).
A large data set is needed to perform the interpolations effectively, making this technique less useful if even more parameters, such as temperature, are introduced.
The modulus of the field vector is identical in the magnet and crystal frames of reference. Any difference in orientation between the two frames
may be expressed as a difference in positions on the ΘX-ΘY plane. The
orientation of the c-axis can be found at the point of high symmetry in the
ΘX-ΘY plane. In figure 5.5 every grid has this high-symmetry point around
{-2◦,-2◦}, which is especially clear from the plot at|H|= 7.82T. We conclude that {-2◦,-2◦} is the misalignment of the crystal c-axis with respect to HZ.
The convenience of a completed grid derives from the fact that it allows for an extremely precise measurement of the orientation of the c-axis as well as the orientation of the current direction, which turns out to be crucial for later measurements. In addition, the 3D data set can be intersected in different ways, by which many measurements, such as the ‘circles’ described in the section 5.5, may be verified independently. An underlying assumption
is that Sr3Ru2O7 has no history dependence in its anisotropic phase. This
point will be further investigated in section 5.7.
The field window available for grids is easily understood in terms ofθlim,
as defined in appendix E. Because ΘX and ΘY are defined in the magnet
reference frame, they are limited by θlim as
p
Θ2
X + Θ2Y ≤θlim = 6.4
◦. The
actual maximum field angle depends on the field modulus, which is why the grids at high fields cover a smaller range of angles. The grids in figure 5.5 were measured right to the limit of the magnet specifications.
The interpolated grids reveal several features of the anisotropic phase
which were not previously understood. BelowHC1, the grids show very little
dependence of the resistivity on θ. Not only is there no anisotropy in the
‘normal metallic’ region, but there is also very little susceptibility to Hab
at all. As the field modulus is gradually increased, the field along c-axis is the last orientation at which the phase is finally entered. Before that, the transition shows up on the grid as a circle centred around c-axis, as shown in figure 5.5b. This may be understood by considering the phase transitions mapped in figure 5.4. Constant field modulus grids sample data along a
horizontal line on the |H| versus θ plot. At |H| = 7.82T the phase may be
entered as a function ofθ aroundθ = 3◦. At these small angles, the in-plane
field angle dependence of HC1 is minimal, so the transition shows up as a
Figure 5.5: Interpolated resistivity maps for the needle sample from 88 in- dependent fixed angle field sweeps, evaluated at progressively higher values
of |H|[a)→f)]. All data were taken atT = 50mK. Black markers indicate
the orientations of the field sweeps from which the surfaces are interpolated. The total measurement time amounted to around 45 hours.
When the phase has been entered for all field angles (figure 5.5c), the resistivity shows up as a ridge aligned with the current direction. In the figure
this direction is approximately along ΘY. The shape of this ridge is two-
fold symmetric, reflecting the broken four-fold symmetry in transport which exists phase. We observe that an 8 degree in-plane field perpendicular to the
current direction (‘easy’, corresponding to {6◦,-2◦} on the grid) suppresses
about half the resistive peak, consistent with the results in figure 5.3. At even higher fields the grids describe the shoulder feature (figures 5.3d- f), where the symmetry remains twofold even though the amplitude of the
resistive peak along ‘hard’ gradually disappears. The orientation of the
anisotropy is the same as inside the phase, but more variation in its ampli-
tude is measured as a function ofθ. At 8.1T there appears to be a maximum
in resistivity around 7 degrees away from thec-axis along hard, with a saddle
point at the c-axis itself. At 8.4T, the highest field for which this range of
tilting angles can be interpolated, the grid looks almost four-fold symmet- ric again. From fixed angle field sweeps up to higher fields, such as those presented in figure 5.2, we know four-fold symmetry is restored around 8.5T. Some care must be taken in analysing data at fixed field modulus, as the
critical fields change withθ. It is therefore possible to see regions above and
below HC1 on a single grid (figure 5.3b), orρ decreasing asθ is increased in
the hard direction (figure 5.3c). This decrease simply indicates that HC2 is
being approached, rather than uncovering strange fundamental physics about the properties of the phase.
It is possible to rescale the field scale as a function of θ in such a way
that HC1 (but not HC2) becomes θ-independent. However, the difficulty in
interpreting grids, and especially making direct comparisons with raw data, becomes much greater. Finally, for a comparison of grids taken with all four current directions for octagon 2, I refer to appendix D.