Observation of quantum interference conductance fluctuations in metal rings with strong spin-orbit coupling

Conductive materials with strong spin-orbit coupling are at the central stage of spintronics research. The mutual interaction between the charge and spin currents by the spin Hall effect (SHE) and its Onsager reciprocal: the inverse spin Hall effect (ISHE), has vastly expanded the field; enabling the investigation of spin-information transport,Kajiwara et al. (2010); Uchida et al. (2010a, b); Wu et al. (2015); Seki et al. (2015); Wu et al. (2016); Ramos et al. (2018); Lebrun et al. (2018); Oyanagi et al. (2019); Ramos et al. (2019) spin-orientation readout Nakayama et al. (2013); Hoogeboom et al. (2017); Hou et al. (2017); Fischer et al. (2018); Baldrati et al. (2018); Oyanagi et al. (2020) and electrical manipulation of the spin state of magnetic insulators.Avci et al. (2017); Moriyama et al. (2018); Chen et al. (2018) These phenomena are characterized by lengthscales in the nanometer range, therefore investigation of the fundamental transport properties at the nanoscale level can further expand the field, merging spin dependent transport phenomena and mesoscopic physics, and possibly leading to new functionalities of potential interest for future quantum technologies.Nitta et al. (1999); Shen et al. (2004); Souma and Nikolić (2005)

One interesting possibility is the study of quantum phase interference effects on the magnetotransport. These phenomena typically manifests in different ways: (1) a magnetoresistance background due to weak (anti-) localization effectsHikami et al. (1980); Gerd (1984) in conductive samples, (2) the appearance of a randomly oscillating conductance pattern, known as universal conductance fluctuations (UCF),Umbach et al. (1984); Lee and Stone (1985); Lee et al. (1987) characteristic of the microscopy impurity configuration and resulting from a magnetic flux piercing the body of the sample in laterally confined structures (nanowires) and (3) the periodic Aharonov–Bohm (AB) oscillations Aharonov and Bohm (1959) superimposed to the aperiodic UCF background in nanoring structures with characteristic dimensions of the order of the electron phase coherence length.

The magnetoconductance oscillations have been previously studied in non-magnetic metal rings with coherence lengths in the micrometer range,Webb et al. (1985); Washburn and Webb (1986); Milliken et al. (1987); Häussler et al. (2001) or metallic ferromagnets with relatively large coherence lengths.Kasai et al. (2002) The investigation of quantum phase interference effects in strong spin-orbit metals can further expand the capabilities of these materials for spintronics and possibly lead to new functionalities, as in the previously proposed spin interference devices without magnetic elements.Nitta et al. (1999); Shen et al. (2004); Souma and Nikolić (2005) However, the observation of conductance oscillation effects in metals with strong spin-orbit coupling and short phase coherence length has remained elusive, due to the difficulty to fabricate suitable nanostructures and the expected suppression of the electron phase coherence by the strong spin-orbit interaction.
Here we report the observation of magnetoconductance oscillations in Pt, a material with strong spin-orbit interaction and short phase coherence length. The observation of this effect in Pt opens the possibility to explore the mutual interaction between effects governed by the electron phase coherence and the spin degree of freedom, such as possible interactions between conductance oscillations and SHE-driven spin accumulation.
The samples investigated in the present study were fabricated on a 10 nm Pt film deposited at room temperature on a SiO₂/Si substrate by dc-magnetron sputtering (ULVAC QAM-4 STS). Two types of lateral nanostructures were fabricated: wires and rings, by combination of electron beam lithography and argon milling. In this work we investigate one nanowire and three nanorings of different diameters, with nominal dimensions: lateral width ($w$ = 40 nm), distance between current/voltage leads ($L$ = 1.5 $\mu$m) and diameters ($d$) of 250, 350 and 500 nm. Figure 1 shows a typical scanning electron microscopy (SEM) image of one of the fabricated nanorings. The magnetotransport properties of the samples at mK temperatures were investigated in a ³He–⁴He dilution refrigerator (KelvinoxMX200). The magnetoresistance was measured by a four probe method (13.1 Hz, 100 nA) with an ac resistance bridge (Lake Shore 372). The sweep rate of the magnetic field was kept at 12 mT/min to minimize heating effects by eddy currents of the sample holder.
Figure 2 shows the longitudinal resistance of a Pt ring with an average diameter of 250 nm measured at 0.05 K. The data shows the typical positive magnetoresistance at low field due to weak anti-localization (WAL), characteristic of materials with a strong spin-orbit coupling and in agreement with previous reports in Pt wires and thin films.Niimi et al. (2013); Shiomi et al. (2014); Ryu et al. (2016) To determine the phase coherence length ($L_{\phi}$) and spin-orbit length ($L_{\text{SO}}$) we employ the Hikami–Larkin–Nagaoka formula for a 1D wire, valid when the width of the wires forming the ring structures is small compared to the electron coherence length:Akkermans and Montambaux (2007); Niimi et al. (2013)

where $\Delta R_{\text{WAL}}=R_{\text{WAL}}-R_{\infty}$, $R_{\infty}$, $L$ and $w$ are the weak localization correction factor, the resistance at high enough field, the length between contacts and width of the Pt nanostructures, and $e$, $\hbar$ and $l_{\text{B}}=(\hbar/eB)^{0.5}$ are the electron charge, the reduced Planck constant and the magnetic length, respectively. From the WAL fitting we can estimate the electron coherence and spin-orbit lengths, obtaining $L_{\phi}$ = 150 nm and $L_{\text{SO}}$ = 27 nm at 0.05 K. The magnetoconductance ($\Delta G$) estimated after subtraction of the WAL magnetoresistance background, and expressed in units of the conductance quantum ($e^{2}/h$) is shown in Figure 2b, we can see a clear fluctuation of the sample conductance characteristic of the electron phase interference effects. Before calculating the Fourier transform to estimate the spectrum of the oscillations frequencies, we calculated the autocorrelation function of $\Delta G$ to further reduce the random noise. Moreover, following this procedure we can directly obtain the power spectrum of the measured signal from the Fourier transform of the autocorrelation function.Landau et al. (2015) The obtained Fourier spectrum comprises two regions: a low frequency region (see inset of Fig. 2c) originated from the aperiodic fluctuations due to UCFLee and Stone (1985); Lee et al. (1987), arising from interference effects within the body of the Pt and resulting in an aperiodic fluctuation pattern dependent on the defect distribution within the sample. At higher frequencies (Fig. 2c), we can distinguish the presence of Aharonov–Bohm oscillations due to the interference between electrons traveling clockwise and counterclockwise through the arms of the ring, which can be confirmed by the presence of peaks in the Fourier spectrum in the frequency range from 6 to 18 T^-1. These correspond to conductance oscillations due to a quantum of magnetic flux ($\phi_{0}=h/e$) being enclosed within the area of the Pt rings (the frequency range for the AB oscillation in the Fourier spectrum is a result of the finite width of the Pt wires forming the ring structure). The presence of AB oscillations in Pt ring is quite surprising, given the short value of the estimated coherence length from the WAL fitting, being much shorter than the half-perimeter of the ring (390 nm). From the amplitude of the AB oscillations we can also obtain an estimate of the phase coherence length. For a fully coherent sample, the expected amplitude of the AB oscillations is 0.3 $e^{2}/h$. Whereas for a sample with a length of the ring arms larger than $L_{\phi}$, the amplitude of the oscillations has an exponential suppression factor of (2$L_{\phi}/U)\exp(-U/L_{\phi})$, where U is the circunference of the ring.Häussler et al. (2001) Considering this expression and the amplitude of the AB oscillations in our measurement ($\sim$ 0.02 e²/h at 0.05 K, Fig. 2b) we can estimate a coherence length of $L_{\phi}\sim$ 300 nm, within the same order of magnitude as the one obtained from the fit to the WAL expression. We can also obtain an estimate of the phase coherence length from the UCF measurement of a Pt wire, using the expression $L_{\phi}=\phi_{0}/(wB_{\text{C, UCF}})$Häussler et al. (2001), where $B_{\text{C,UCF}}$ is the correlation field of the UCF, obtained from the conductance autocorrelation function.Lee et al. (1987); Alagha et al. (2010); Meng et al. (2019) Using this expression, we can estimate for a Pt nanowire a value of $B_{\text{C,UCF}}$ = 0.4 T (see Supplementary for estimation of $B_{\text{C, UCF}}$) and a phase coherence length of $L_{\phi}$ = 260 nm, in reasonable agreement with the previously obtained values from the WAL fit and AB oscillation amplitude. The temperature dependence of the conductance oscillations is depicted in Fig. 2d, which shows the Fourier spectrum obtained for three different temperatures, we can observe that the amplitude of the AB oscillations decreases upon a temperature increase, as expected.Umbach et al. (1984); Webb et al. (1985); Washburn and Webb (1986) Our result suggests that we can observe magnetoconductance oscillations due to electon phase interference effects, despite the much shorter phase-coherence length of Pt compared to previously studied materials.Webb et al. (1985); Washburn and Webb (1986); Milliken et al. (1987); Häussler et al. (2001); Kasai et al. (2002)