Simultaneous loss of interlayer coherence and long-range magnetism in quasi-two-dimensional PdCrO2

In many layered metals, coherent propagation of electronic excitations is often confined to the highly conducting planes. While strong electron correlations and/or proximity to an ordered phase are believed to be the drivers of this electron confinement, it is still not known what triggers the loss of interlayer coherence in a number of layered systems with strong magnetic fluctuations, such as cuprates. Here, we show that a definitive signature of interlayer coherence in the metallic-layered triangular antiferromagnet PdCrO2 vanishes at the Néel transition temperature. Comparison with the relevant energy scales and with the isostructural non-magnetic PdCoO2 reveals that the interlayer incoherence is driven by the growth of short-range magnetic fluctuations. This establishes a connection between long-range order and interlayer coherence in PdCrO2 and suggests that in many other low-dimensional conductors, incoherent interlayer transport also arises from the strong interaction between the (tunnelling) electrons and fluctuations of some underlying order.

A previous interlayer magnetoresistance study in PdCoO 2 by Takatsu et al. [1] showed a striking angle dependence upon azimuthal rotation of the magnetic field within the conducting plane that was attributed to high mobility of the conduction electrons and fine details of the hexagonal Fermi surface of PdCrO 2 . We note however that the data reported in the Takatsu paper were taken with the magnetic field aligned approximately 3 o away from the conducting plane. As shown in Figure  2d of the main manuscript, the Hanasaki peak is suppressed completely once the field is rotated 2°away from the conducting plane. Thus, the sharp peaks in the interlayer magnetoresistance reported in Ref. [1] are not associated with the Hanasaki peaks discussed in the main paper.
Additionally, it should be pointed out that while the polar ADMR are extremely sensitive to details of the Fermi surface topology-as can be seen by inspection of the polar ADMR curves for PdCrO 2 (Figure 2b of the main manuscript) and PdCoO 2 (Figure 5a of [2])-the Hanasaki peak itself is similar in form in both systems. Indeed, to be visible, the Hanasaki peak requires only the existence of a three-dimensional Fermi surface (provided that ω c τ is large enough) and its width is determined uniquely by the ratio between k F and t ⊥ . It is therefore only weakly dependent on other details of the band structure. T e m p e r a t u r e ( K ) Supplementary Figure 1. | The c-axis resistance of different samples. The highest quality sample, with residual resistivity ratio of 108, was chosen for our angular-dependence measurements.

SUPPLEMENTARY NOTE 2. BACKGROUND SUBTRACTION
For analysis of the coherence peak sharpness, the resistivity values were normalized to a common scale as given by The broad quasi-sinusoidal background was then removed by subtracting the ρ N (T, θ, H) curve at 44 K, which has no AMRO nor coherence peak, via where the multiplicative factor α broadly corrects for the change in ω c τ with temperature and 'N.B.' stands for 'no background'. The factor α was found by empirically scaling the 44 K data until ρ N (44 K, θ, H) matched ρ N (T, θ, H) in the featureless low-angle region (0-30°). This results in a ρ N.B. (T, θ, H) that is approximately zero at low angles, with a series of peaks at higher angles. Thus one has a complete removal of the quasi-sinusoidal background, leaving only the AMRO features and the Hanasaki coherence peak that are observed at higher angles. As an example, see Supplementary Figure 3.
Using the 44 K curve to subtract the background in this way does not affect the analysis of the sharpness of the coherence peak. This is shown in the plot of d 2 ρ/ dθ 2 in Supplementary Figure 3b of the main manuscript, where d 2 ρ/ dθ 2 at θ = 90°is essentially zero for all four data points above 37.5 K. This confirms that the 44 K curve does not have a coherence peak, and the implies successful subtraction of the quasi-sinusoidal background in a way that does not influence our analysis of the temperature dependence of the Hanasaki peak.

SUPPLEMENTARY NOTE 3. ESTIMATE OFh/τ AND ωcτ
The mean free path can be estimated from the Drude formulae for a simple cylindrical Fermi surface: where n is electronic density, m * is the effective mass, d = 6.03Å is the interplanar distance, e is the electron charge and k f is the Fermi wave-vector. Thus given that ρ ab (37.5 K) = 0.7(1) µΩ cm [3,4], and assuming that the electronic transport is dominated by the biggest non-breakdown orbit (γ) for which k f = 0.57 (3)    Finally, it should be noted that in contrast to quantum oscillations measured by the de Haas-van Alphen effect, where the amplitude of ω c τ can be suppressed by both small-and large-angle scattering events, angledependent magnetoresistance (ADMR), being a transport property, is not affected or degraded by small-angle scattering. This is best illustrated in Ref. [5] where the polar ADMR in the interplane magnetoresistance of an overdoped cuprate could be fitted by precisely the same ω c τ value that is obtained from in-plane Hall effect measurements. Thus, the ω c τ product obtained from the interlayer magnetoresistance is, in principle, identical to that estimated from zero-field resistivity measurements.

SUPPLEMENTARY NOTE 4. FIELD-DEPENDENCE OF THE HANASAKI PEAK
It is interesting to investigate the field-dependence of the amplitude of the Hanasaki peak since any strong variation in the scattering rate with field, e.g. due to a change in the spin fluctuation spectrum, could be reflected in a departure from the expected quadratic dependence of the peak amplitude with field. In this section, we thus compare the field-dependence of the Hanasaki peak in PdCrO 2 with that measured in non-magnetic PdCoO 2 . The corresponding amplitudes are plotted as a function of field in Supplementary Figure 4. We find that the field-dependence in both systems is quadratic in field to within our experimental uncertainty. This finding sug-gests that the antiferromagnetic fluctuation rate does not vary appreciably at least within the field range of our experiments. With regards the cuprates, where antiferromagnetic fluctuations are clearly present, their influence on the magnetotransport, vis-á-vis the field-dependence of the magnetoresistance, for example, is also found to be negligible (see, e.g. Ref. [6]) despite the fact that it may have a significant impact on its temperature dependence [6].