{'fraction_non_alphanumeric': 0.03936494127881688, 'fraction_numerical': 0.014245324053936494, 'mean_word_length': 4.744534665833854, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Interference Effects Due to Commensurate Electron Trajectories and Topological Crossovers in (TMTSF)2ClO4

We report angle-dependent magnetoresistance measurements on (TMTSF)2ClO4 that provide strong support for a new macroscopic quantum phenomenon, the interference commensurate (IC) effect, in quasi-one dimensional metals. In addition to observing rich magnetoresistance oscillations, and fitting them with one-electron calculations, we observe a clear demarcation of field-dependent behavior at local resistance minima and maxima (versus field angle). Anticipated by a theoretical treatment of the IC effect in terms of Bragg reflections in the extended Brillouin zone, this behavior results from 1D-2D topological crossovers of electron wave functions as a function of field orientation.

In recent years, several new types of angular magnetoresistance oscillation (AMRO) phenomena have been discovered in quasi-one dimensional metals such as the organic conductors (TMTSF) 2 X [1]. Such materials can be envisioned by considering one's hand. The fingers represent highly conducting chains along x that are weakly coupled within the hand along y, forming a conducting layer with in-plane anisotropy. The hands are themselves coupled even more weakly along z, forming a highly anisotropic, 3D system. The AMRO effects occur when a magnetic field is rotated within different planes in this system.
For example, y-z plane field rotations in (TMTSF) 2 ClO 4 [2,3,4,5] and (TMTSF) 2 PF 6 [6,7] reveal the Lebed [8] "magic angle" effect, wherein the interlayer (interhand) resistivity develops pronounced minima at characteristic orientations given by ( ) , where N is an integer, b//y and c//z are interchain (interfinger) and interplane (interhand) crystal spacings, respectively, and θ is the tilt angle from b to c. Resonant oscillations up to ~10 th order in N have been observed to date. As first shown by Danner, Kang and Chaikin [9], rotating in the x-z plane results in the appearance of a single resistivity peak centered about a small tilt angle from B//x (from the fingers). Finally, rotations from x to y (within the plane of the hand) result in the "third angular effect" (TAE) which is manifest as a resistance minimum near B//x [10,11,12].
There have been many proposed explanations of the y-z plane magic angle effect, [8,13,14,15,16] including some involving interactions between electrons [7,8,15]. While these theories share many common aspects, a consensus is still lacking. There even exists experimental evidence [7] of a non-Fermi liquid origin to some magic angle effects. The x-z plane effect is simpler to understand, being tied directly to details of the warped open Fermi surface. Through this effect, the transfer integral ratio t b /t a may be measured [9]. Since t a ~ E F , the Fermi energy, and can be independently determined, this effect provides a means to measure t b , the interchain coupling, or the degree to which the Fermi sheets are warped in the k b direction. The x-y plane TAE is ascribed to the velocity-preserving nature of "effective" electrons, via their proximity to geometrical inflection points on the Fermi surface [17]. As such, this can be used to quantify the interplane coupling t c as well as t b . These latter two AMRO effects result, to varying degrees, from the semiclassical behavior of electrons on a warped, open Fermi surface in a magnetic field, and can be explained by solving the Boltzmann kinetic equation in one Brillouin zone [9,17].
When one rotates a magnetic field through something other than one of these principal planes, the simplest expectation is for some trivial combination of the above three AMRO effects.
In fact, in the first such experiment [12] on (TMTSF) 2 PF 6 , x-y plane rotations with a finite field component along z, a rich AMRO spectrum was observed. These so-called "LN oscillations" were first interpreted in the simple manner above, with resistivity minima ascribed to projections of commensurate directions according to ( ) φ θ sin tan = c b N (N = integer and φ is the tilt angle from a to b) [12]. However, it has recently been suggested via analytical theory that the origin of these new complex oscillations is instead related to interference effects due to commensurate electron trajectories in a tilted magnetic field in the extended Brillouin zone [18,19]. Moreover, as shown in Ref. [19], the interference commensurate (IC) effect leads to topological 1D → 2D crossovers at commensurate directions of a magnetic field. In this Letter, we provide evidence for this IC interpretation in the sister compound (TMTSF) 2 ClO 4 via angle-dependent magnetoresistance measurements. We use the IC theory to accurately calculate the angle dependence, and we report a new manifestation of the IC effect in fixed-angle experiments, where the magnetoresistance displays qualitatively different behavior at field orientations when the resistance is at a local maximum versus a minimum, corresponding to 1D and 2D transport, respectively.
3 The (TMTSF) 2 X (X= ClO 4 , PF 6 , etc.) family of organic conductors can be modeled more quantitatively by a tight-binding band approximation, with electron band energy given by With the strongest overlap of π-molecular orbitals along the a-direction (x, chain, finger), the electron energy E is mostly dominated by k k a and is nearly independent of k b and k c . Therefore, the Fermi surface consists of a pair of warped sheets, defined by the electron transfer energies t a , t b , and t c along the three crystal axes. These latter two are the inter-finger and inter-hand couplings, respectively, discussed above. Because of the open Fermi surface, traditional magnetic oscillations [20] cannot exist in these materials.
Nonetheless, the metallic phase of (TMTSF) 2 X exhibits the large number of unconventional magnetic oscillations directly related to this open Q1D Fermi surface discussed above.
The geometry of our experiment is shown in the inset to Figure 1. We show the two angles θ and φ, corresponding to magnetic field rotation directions out of and within the x-y (a-b) plane, respectively. Two (TMTSF) 2 ClO 4 crystals were mounted with 12-µm gold wires and graphite paste on an in situ θ −rotating stage of a dilution refrigerator, situated in a φ−rotatable split-coil superconducting magnet. The φ−dependence of the interlayer resistivity ρ zz for various angles θ was measured at fixed temperature and field. The two samples showed nearly identical behavior; results for one are shown here. We show in Fig. 1 to commensurate magnetic field directions [19] ( ) φ θ sin , where the coefficient 2 is due to the reduced size of the Brillouin zone (b→2b) in (TMTSF) 2 ClO 4 .
In the IC theory [18,19], the origin of the LN oscillations is related to special commensurate electron trajectories where the average electron velocity along z is non-zero. This analytical theory [18] well describes previous data [12] on (TMTSF) 2 PF 6 whereas, for the present case, a more complex approach is needed to account for the doubling of the unit cell along the baxis due to ordering of the ClO 4 anions. The details of this approach are published elsewhere [19]. The resulting conductivity, accounting for the anion ordering gap ∆ AO , is given by: Fig. 2, we compare the calculated conductivity using this formalism with the data of The key point of this work, which was not appreciated in our earlier Letter [18], is that the magnetoresistance oscillations in Figs. 1 and 2 can be interpreted in terms of unique 1 to 2−dimensional crossovers [19]. In the absence of Landau quantization for open Fermi surfaces, the "other" quantum effect in a magnetic field, Bragg reflections, results in a series of 1D to 2D crossovers at the LN oscillation minima. In other words, electron wave functions, localized on chains (i.e. 1D) at arbitrary field directions [8,15], become delocalized on planes (i.e. 2D) at the commensurate directions. The non-trivial physical origin of these 1D → 2D crossovers is related to interference effects between velocity components along the z-axis, k z , and electron motion along the y-direction [19]. These interference effects occur as electrons move along open FS sheets in the extended Brillouin zone and are qualitatively different from that responsible for the magic angle effect [16]. A discussion of how 1D 2D dimensional crossovers can lead to the appearance of oscillation minima in ρ → zz can be found in Ref. 19.
We can use this dimensional crossover notion and the IC model to further discuss electron motion in such quasi-1D metals in a strong magnetic field. For electrons localized on conducting x-chains, one expects, in the absence of impurities (1 0 / = τ ), the conductivity between chains zz σ to be zero. saturates at commensurate directions (minima), while at non-commensurate directions (maxima), it exhibits nonsaturating behavior. As shown in Fig. 3, there is rather good agreement between experiment and the calculated magnetoresistance using the conductivity equation above and the same parameters as were used for the fits in Fig. 2 We would like to acknowledge Harukazu Yoshino for helpful discussions.