6. Análisis y hallazgos
6.1 Prácticas comunicativas
6.1.4 Prácticas comunicativas académicas
Figure 3.24 shows AC susceptibility measurements against tem- perature at a series of compressive strains. When the sample is subject to either uniaxial compression or tension the superconduct-
68 The Physics of Sr2RuO4 Approaching a Van Hove Singularity
Fig. 3.24: Susceptibility against temperature. Real part of the sus- ceptibility χ for sample 3 against
temperature. AStrains below the peakTc,Babove the peak. No nor-
malisation or offsets are applied to the curves. 0.00 % εxx: −0.20 % −0.30 % −0.38 % −0.44 % −0.50 % −0.52 % −0.54 % −0.56 %
A
χ
0(a.u.)
−0.56 % −0.62 % −0.63 % εxx: −0.66 % −0.70 % −0.74 % −0.78 % −0.83 % −0.87 % −0.92 %B
Temperature (K)
χ
0(a.u.)
1
2
3
4
ing transition temperature is enhanced, corroborating the results of Hicks et al. [33]. As can be seen from the figure the transitions become somewhat broader as the transition temperature moves to higher temperature with increasing strain. The amount and shape of this broadening varies from sample to sample, see figures B.1 and B.2 in appendix B for comparison, so it is likely this effect is extrinsic and is most probably due to differing strain homogeneity. As described in detail in section 2.5, imperfections in mounting can cause the sample to bend, imposing a strain gradient across the thickness of the sample when it is strained. The presence of dislocations or ruthenium inclusions in Sr2RuO4 could also present some local strain disorder producing variations between samples. However it is clearly apparent that the transition temperature reaches a maximum with applied strain and that as the maximum
3.4Results and discussions 69 Tc(transition midpoint) 20-80 % width
ε
xx(%)
Temp
erature
(K)
−0.55
±0.06
0.00
±0.04
1.5
2.0
2.5
3.0
3.5
Sample 1 Sample 2 Sample 3Fig. 3.25: Tc against strain for all three samples. The points are the mid points (50% levels) of the
transitions shown in figures B.1, B.2 and 3.24. The 20 and 80% levels of the transitions are shown as lines to give a measure of the transition width. Here the strain scales have been normalised so the peaks inTc coincide at their average value of (−0.55±0.06) %.
mumTc is enhanced by a factor of∼2.3 over the unstrained value.
Compressing beyond this maximum causes a rapid suppression ofTc, causing it to fall below even its zero strain value, and the
transitions broaden substantially once more. For broadening as the result of strain inhomogeneity the width of a particular transition at strain should be related to the slope of theTc vs. strain curve
at that strain. Qualitatively this is in agreement with the observed broadening, suggesting the dominant cause of the broadening is indeed strain inhomogeneity.
The response to applied strain was fully elastic. In fact the curves in figure 3.24 are not from one single sequence with increas- ing strain but rather are only a small subset of the total number of measurements where the strain was cycled four times over the maximum and back to zero, measuring both while increasing and releasing the strain, and each time reproducing the same results. This is in contrast to measurements by Taniguchiet al. [145] where there is strong evidence for plastic deformation in samples pres- surised at room temperature in traditional uniaxial pressure cells. Dislocations are known to induce local higher-Tc superconductiv-
ity [146] so by applying strain at low temperatures we reduced the risk of plastic deformation. The stringent requirements on sample preparation for traditional uniaxial pressure cells, as de-
70 The Physics of Sr2RuO4 Approaching a Van Hove Singularity
ε
xxDOS
at
E
F(e
V
− 1u.c.
− 1)
γ
α
β
γ α β Zero strain M Γ X Van Hove point εxx=εVHS kx ky Highεxx compressionε
VHS0
0
5
10
15
−π a π a −π b π bFig. 3.26:Density of states and Fermi surface calculations. The density of states at the Fermi energy as a function of applied anisotropic strain as calculated from a tight-binding model including strain dependent hopping terms, see main text for details. Three representative Fermi surfaces show the effects of applied strain on the band structure and highlight the Lifshitz transition as theγ-band reaches the
Van Hove point.
scribed in section 2.4, may also be playing a role in the Taniguchi measurements.
The nominal strain at which the peak inTc was observed varied
slightly between the three samples, see table 3.1, but within our uncertainty in determining the strain scale. The profile ofTcagainst
strain for all three samples is plotted in figure 3.25. Here the strain scales have been normalised so that the peaks inTc all coincide at
their average value ofεxx=(−0.55±0.06) %.
The combined strain data from these three new strained samples of Sr2RuO4, and from measurements on Sr3Ru2O7 presented in the next chapter, suggest that the strains determined by Hicks
et al. in Ref. [33] are ∼30 % too low. This is most probably
due to the technique used to measure the strain. In Ref. [33] a resistive strain gauge was used to monitor the displacement of the device but it now seems likely that this may have imposed some mechanical resistance on the motion of the device. Additionally, a temperature dependence of the gauge coefficient could have skewed results. The capacitive sensor used in this new strain rig is less affected by temperature and imposes no mechanical resistance so more confidence can be placed in the results presented here.
3.4Results and discussions 71
drive theγ-band towards the Van Hove singularity and increase
the density of states at the Fermi level. This can also be seen in a simple tight-binding model which incorporates the effects of anisotropic strain through strain dependent hopping parameters. To first approximation at low strains the hopping parameters can be taken to change linearly with applied strain. Using the tight- binding parametrisation by Bergemannet al. [6] from fits to the experimentally determined band structure and the correct band renormalization from Shenet al. [80] as a starting point, the effects of anisotropic strain can be included by scaling all hoppings along the pressurised direction by (1−αεxx) and along the transverse
direction by (1+ανxyεxx). νxyis the in-plane Poisson’s ratio which
has been experimentally determined by Paglioneet al. [68] andα
is an adjustable parameter to scale the effect of applied strain. In this model the chemical potential must also be adjusted slightly with strain to keep the total electron count constant. A plot of the density of states at the Fermi level against strain in this model is shown in figure 3.26 along with the Fermi surfaces predicted as the
γ-band reaches the Van Hove point and at a much higher strain
beyond the Van Hove point. The density of states diverges for the
γ-band as the Van Hove singularity is approached and the band
changes character from a closed electron pocket to an open orbit running alongky in the Brillouin zone at the Lifshitz transition.
In BCS theory the BCS gap,|∆|, grows with increasing density
of states at the Fermi level. Tc is related to the size of the BCS
gap and in a material with ak-dependent gap it is proportional to
thek-space average of|∆(k)|. The experimental observation of an
enhancedTc is therefore qualitatively expected as the density of
states grows with applied strain. However, for the widely favoured
p-wave pairing symmetry the gap must change phase byπunder
inversion. This leads to frustration as theγ-band approaches the
Van Hove point which is inversion invariant, and therefore the gap must locally be zero at the Van Hove point. In contrast, an even-parity pairing symmetry is not subject to the same frustration constraints and one might expect a stronger enhancement ofTc for
an even-parity state as the density of states is increased by the Van Hove point.10
10BCS estimates are expected to
be accurate for a single band metal with a small gap but in a multiband system like Sr2RuO4 these estimates refer only to the average gap which can already be a big approximation when, like in Sr2RuO4, there are large changes around the Fermi surface. This analysis also completely overlooks the potential for interband coupling, so it should only be taken as a guide.
Bearing this in mind, the peak in Tc may be a result of the
peaking density of states and coincide with the Lifshitz transition or frustration may take over before the Van Hove singularity is reached and the enhancement could be cut off at a lower strain. Weak- coupling calculations by T. Scaffidi [147] on a similar tight-binding parametrisation of the strained Fermi surface suggest that, even in the presence of this frustration, odd-parity order is still enhanced andTc peaks as the Van Hove singularity is surpassed. The slow
72 The Physics of Sr2RuO4 Approaching a Van Hove Singularity
Hove singularity are seen to contribute to superconductivity on the αandβ bands through inter-orbital interaction terms. Further
evidence supporting the coincidence of the peak in Tc and the Van
Hove singularity comes from normal state resistivity measurements which will be presented in the next part of this section. One other alternative proposed by Liu et al. [148] is competition with a spin density wave state that is predicted to be stabilized with compressive strain, and whose formation cuts off the increase in
Tc. However, the transport data in the next section seems to
be inconsistent with this and together with the resistivity results about to be presented it seems most likely that the peak inTc does
coincide with the Van Hove singularity.