Small Hole Pockets Research Articles

Evidence of compensated semimetal with electronic correlations at charge neutrality of twisted double bilayer graphene

Recently, magic-angle twisted bilayer graphene (MATBLG) has emerged with various interaction-driven novel quantum phases at the commensurate fillings of the moiré superlattice, while the charge neutrality point (CNP) remains mostly a trivial insulator. Here, we show an emerging phase of compensated semimetallicity at the CNP of twisted double bilayer graphene (TDBLG), a close cousin of MATBLG, with signatures of electronic correlation. Using electrical and thermal transport, we find two orders of magnitude enhancement of the thermopower at magnetic fields much smaller than the extreme quantum limit, accompanied by large magnetoresistance ( ~ 2500%) at CNP, providing strong experimental evidence of compensated semimetallicity at CNP of TDBLG. Moreover, at low temperatures, we observe unusual sublinear temperature dependence of resistance. A recent theory1 predicts the formation of an excitonic metal near CNP, where small electron and hole pockets co-exist. We understand this sublinear temperature dependence in terms of critical fluctuations in this theory.

Enhancing thermoelectric properties in TiNiSi structure-type semimetal ZrNiSi by doping

The orthorhombic TiNiSi structure-type compounds show interesting electronic structures comprising in most cases a pseudogap in the density of states and several small electron and hole pockets at the Fermi energy. These features are promising and can be exploited to test their potential as thermoelectric materials for waste heat conversion. Here, we investigate the effect of electron doping in the semimetallic member $\mathrm{ZrNiSi}$ of this family. We show that by doping with Sb for Si in $\mathrm{ZrNiSi}, S$ and $\ensuremath{\sigma}$ can both be increased simultaneously for initial Sb doping defying the oppositely directed trend commonly observed in most materials. In the doped samples, $\ensuremath{\sigma}$ at $300\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ increases from $1000\phantom{\rule{0.28em}{0ex}}\mathrm{S}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ to as high as $2500\phantom{\rule{0.28em}{0ex}}\mathrm{S}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$; at the same time, the peak value of $S$, which is $\ensuremath{-}20\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{V}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ in $\mathrm{ZrNiSi}$, increases by more than a factor of two. The simultaneous enhancement of $\ensuremath{\sigma}$ and $S$ has been explained using the first-principles density functional theory based band structure calculations. The as-cast (i.e., unannealed) ${\mathrm{ZrNiSi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ samples show phase segregation due to a spinodal-type decomposition with two coexisting TiNiSi structure-type phases with different Sb/Si ratios. The thermal conductivity ($\ensuremath{\kappa}$) in the doped samples drops significantly from $12\phantom{\rule{0.28em}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ ($x=0$) to nearly $2\phantom{\rule{0.28em}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ ($x=0.2$) at 300 K. As a result, the peak thermoelectric figure of merit ($zT$) increases from 0.005 $(x=0)$ to 0.023 $(x=0.2)$. Further enhancement in $zT$ is obtained by codoping of Hf (Zr site) and Sb (Si site), which improves the phase stability and chemical homogeneity while keeping the thermal conductivity still very low due to Zr-Hf point mass fluctuation, resulting in a peak $zT$ value of 0.055, i.e., almost an order of magnitude higher value than for the pristine $\mathrm{ZrNiSi}$. We show that the thermoelectric properties of TiNiSi structure-type semimetals can be enhanced by aliovalent doping. This principle can be employed on other members of the TiNiSi to improve the $zT$ in this family of compounds further.

L -hole pockets of the Fermi surface of palladium revealed by positron annihilation spectroscopy

Using the combined Density Functional and Dynamical Mean Field theory we study relativistic corrections to the Fermi surface of palladium. We find indeed that relativistic corrections create a small hole pockets at the $L$-symmetry points. Furthermore we show that the computed two dimensional Angular Correlation of Electron Positron Annihilation Radiation (the so called $2D$-ACAR) clearly demonstrates the existence of these $L$-hole pockets, which remains robust against electronic correlations. A $2D$-ACAR experiment should therefore provide the "smoking-gun" proof for the existence of the $L$-hole pockets in the palladium Fermi surface.

Low thermal conductivity and semimetallic behavior in some TiNiSi structure-type compounds

Motivated by recent advances in half-Heusler based thermoelectric materials, we investigated the phase stability and thermoelectric properties of compounds ZrNiSi, ZrNiGe, HfNiSi, NbCoSi, and ZrNiSb, some of which were recently reported in literature as promising half-Heuslers for thermoelectric applications using the first-principles density functional theory based calculations. Here, we show that all the named compounds actually crystallize with the orthorhombic TiNiSi structure type, which remains stable above room temperature up to at least 1100 K. In ZrNiSb, $5%$ excess Zr is required to obtain the pure orthorhombic phase. Our first-principles electronic band structure calculations reveal that they are semimetals. In ZrNiSi, ZrNiGe, and HfNiSi, the Fermi surface consists of small electron and hole pockets with electrons as the majority charge carriers. In NbCoSi and ZrNiSb, the majority carriers are holes. A pseudogaplike feature is observed in the electronic density of states with Fermi energy (${E}_{F}$) located either slightly below (ZrNiSi, ZrNiGe, and HfNiSi) or above the pseudogap (NbCoSi). In ZrNiSb no pseudogap is observed; however, the density of states at ${E}_{F}$ is still small. The electrical conductivity ($\ensuremath{\sigma}$) near room temperature is of the order of ${10}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{S}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, which is intermediate between that of the degenerate semiconductors and metallic alloys. Near room temperature the thermopower is negative for $\mathrm{ZrNi}X$ ($X$ = Si, Ge) and HfNiSi, and positive for NbCoSi and ZrNiSb as predicted theoretically. The average value of Seebeck coefficient is small, of the order of $10\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{V}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$. Despite reasonably high electrical conductivity, the thermal conductivity ($\ensuremath{\kappa}$) of these compounds is found to be generally low ($<15\phantom{\rule{4pt}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ near $300\phantom{\rule{4pt}{0ex}}\mathrm{K}$). In ${\mathrm{Zr}}_{1.05}\mathrm{Ni}\mathrm{Sb}$, which has the highest electrical conductivity ($\ensuremath{\approx}4000\phantom{\rule{4pt}{0ex}}\mathrm{S}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$), $\ensuremath{\kappa}$ is as low as $\ensuremath{\approx}4\phantom{\rule{4pt}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ at $300\phantom{\rule{4pt}{0ex}}\mathrm{K}$, of which almost 70% is estimated to be due to the electronic contribution resulting in a lattice contribution which is $<1\phantom{\rule{4pt}{0ex}}\mathrm{W}\phantom{\rule{0.16em}{0ex}}{\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$. This uncommon combination of high electrical conductivity and low thermal conductivity is interesting and invites further attention.

Transport property of multi-band topological material PtBi_2 studied by maximum entropy mobility spectrum analysis (MEMSA)

Electrical transport of both longitudinal and transverse directions carries rich information. Mobility spectrum analysis (MSA) is capable of extracting charge information from conductivity tensor, including charge types, concentration and mobilities. Using a numerical method based on maximum entropy principle, i.e., maximum entropy mobility spectrum analysis (MEMSA), mobility spectrum for beta -type PtBi_2 is studied. Three hole-pockets and two electron-pockets were found, including a small hole pocket with very high mobility, which is very likely corresponding to Dirac Fermions. Benefiting from our high resolution result, we studied temperature dependence of carrier properties and explained the sign change phenomenon of Hall conductivity. We further compared the results with band structure obtained by our first principle calculation. The present results prove MEMSA is a useful tool of extracting carries’ information in recently discovered Iron-based superconductors, and topological materials.

Structural, elastic, electronic, bonding, and optical properties of topological CaSn3 semimetal

In recent days, topological semimetals have become an active branch of materials research. The topological Weyl semimetal CaSn3, belonging to the AuCu3 type cubic structure, is an important electronic system to investigate both from the point of view of fundamental physics and prospective applications. In this work, we have studied the structural, elastic, mechanical, electronic, bonding, Fermi surface and optical properties of CaSn3 in detail via first-principles method using the density functional theory. A comprehensive study of elastic constants and moduli shows that CaSn3 possesses low level of elastic anisotropy, reasonably good machinability, mixed bonding characteristics with ionic and covalent contributions, brittle nature and relatively high Vickers hardness with a low Debye temperature. The mechanical stability conditions are fulfilled. Analysis of bond population supports the bonding nature as indicated by the elastic parameters. The bulk electronic band structure reveals clear semimetallic features with signature Dirac cone-like dispersions near the Fermi level. A pseudogap in the electronic energy density of states at the Fermi level separating the bonding and the antibonding peaks points towards significant electronic stability of cubic CaSn3. The Fermi surface mostly consists of electron-like sheets with very few small hole pockets. The band structure is fairly isotropic in the k-space. The optical constants show interesting characteristics. The reflectivity spectra show almost non-selective behavior over a wide range of photon energy encompassing infrared to mid-ultraviolet regions. High reflectivity over wide spectral range makes CaSn3 a suitable material for reflecting coating. CaSn3 is an efficient absorber of ultraviolet radiation. The refractive index is very high in the infrared to visible range. All the energy dependent optical parameters exhibit clear metallic signatures and are in complete accord with the underlying bulk electronic density of states calculations.

Pressure-induced phase transitions and superconductivity in a quasi–1-dimensional topological crystalline insulator α-Bi4Br4

Great progress has been achieved in the research field of topological states of matter during the past decade. Recently, a quasi-1-dimensional bismuth bromide, Bi4Br4, has been predicted to be a rotational symmetry-protected topological crystalline insulator; it would also exhibit more exotic topological properties under pressure. Here, we report a thorough study of phase transitions and superconductivity in a quasihydrostatically pressurized α-Bi4Br4 crystal by performing detailed measurements of electrical resistance, alternating current magnetic susceptibility, and in situ high-pressure single-crystal X-ray diffraction together with first principles calculations. We find a pressure-induced insulator-metal transition between ∼3.0 and 3.8 GPa where valence and conduction bands cross the Fermi level to form a set of small pockets of holes and electrons. With further increase of pressure, 2 superconductive transitions emerge. One shows a sharp resistance drop to 0 near 6.8 K at 3.8 GPa; the transition temperature gradually lowers with increasing pressure and completely vanishes above 12.0 GPa. Another transition sets in around 9.0 K at 5.5 GPa and persists up to the highest pressure of 45.0 GPa studied in this work. Intriguingly, we find that the first superconducting phase might coexist with a nontrivial rotational symmetry-protected topology in the pressure range of ∼3.8 to 4.3 GPa; the second one is associated with a structural phase transition from monoclinic C2/m to triclinic P-1 symmetry.

Phase stiffness in an antiferromagnetic superconductor

We analyze the suppression of the phase stiffness in a superconductor by antiferromagnetic order. The analysis is based on a general expression for the phase stiffness in a mean-field state with coexisting spin-singlet superconductivity and spiral magnetism. Neel order is included as a special case. Close to half-filling, where the pairing gap is much smaller than the magnetic gap, a simple formula for the phase stiffness in terms of magnetic quasi-particle bands is derived. The phase stiffness is determined by charge carriers in small electron or hole pockets in this regime. The general analysis is complemented by a numerical calculation for the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping amplitudes at a moderate interaction strength. The resulting phase stiffness exhibits a striking electron-hole asymmetry. In the ground state, it is larger than the pairing gap on the hole-doped side, and smaller for electron doping. Hence, in the hole-doped regime near half-filling the ground state pairing gap sets the scale for the Kosterlitz-Thouless temperature T_c^{KT}, while in the slightly electron-doped regime T_c^{KT} is determined essentially by the ground state phase stiffness.

Effect of CuO2 Lattice Strain on the Electronic Structure and Properties of High-Tc Cuprate Family

Doping and strain dependences of the electronic structure of the CuO6-octahedra layer within LDA+GTB method in the frameworks of six-band p-d model are calculated. Band structure and Fermi surface of the quasiparticle excitations in the effective Hubbard model are characterized by inhomogeneous distribution of the k-dependent spectral weight. Doping results in reconstruction of the band structure, redistribution of the spectral weight over dispersion surface and reconstruction of Fermi surface from small hole pockets in the underdoped region to large hole contour in the overdoped region through two quantum phase transitions (QPT). Strain increasing leads to displacement of the valence and conductivity bands, bandwidths decreasing and shift of the concentrations corresponding to QPTs. Strain dependences of the antiferromagnetic exchange coupling and DOS at the Fermi level determining superconducting temperature Tc are obtained. Effective exchange coupling in the equation for Tc demonstrates monotonic strain dependence.

Electron spectral functions in a quantum dimer model for topological metals

We study single electron spectral functions in a quantum dimer model introduced by Punk, Allais and Sachdev (Ref. [1]). The Hilbert space of this model is spanned by hard-core coverings of the square lattice with two types of dimers: ordinary bosonic spin-singlets, as well as fermionic dimers carrying charge +e and spin 1/2, which can be viewed as bound-states of spinons and holons in a doped resonating valence bond (RVB) liquid. This model realizes a metallic phase with topological order and captures several properties of the pseudogap phase in hole-doped cuprates, such as a reconstructed Fermi surface with small hole-pockets and a highly anisotropic quasiparticle residue in the absence of any broken symmetries. Using a combination of exact diagonalization and analytical methods we compute electron spectral functions and show that this model indeed exhibits a sizeable antinodal pseudogap, with a momentum dependence deviating from a simple d-wave form, in accordance with experiments on underdoped cuprates.

Absence of Dirac states in BaZnBi2 induced by spin-orbit coupling

We report magnetotransport properties of ${\mathrm{BaZnBi}}_{2}$ single crystals. Whereas electronic structure features Dirac states, such states are removed from the Fermi level by spin-orbit coupling (SOC) and consequently electronic transport is dominated by the small hole and electron pockets. Our results are consistent with not only three-dimensional, but also with quasi-two-dimensional portions of the Fermi surface. The SOC-induced gap in Dirac states is much larger when compared to isostructural ${\mathrm{SrMnBi}}_{2}$. This suggests that not only long-range magnetic order, but also mass of the alkaline-earth atoms $A$ in ${ABX}_{2}\phantom{\rule{4pt}{0ex}}(A=$ alkaline-earth, $B=$ transition-metal, and $X=$ Bi/Sb) are important for the presence of low-energy states obeying the relativistic Dirac equation at the Fermi surface.

Fermi Surface Topology in the Case of Spontaneously Broken Rotational Symmetry

The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f(p, p') having a sharp peak at |p − p'| = q0. It is shown that, in the case of attraction and q0 = 2pF the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q0, the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ≃ π(pF/q0 − 1) small hole pockets at the distance pF − q0 from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L. At q0 → 0, when the model under study transforms to the two-dimensional Nozieres model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.

Comparison of band structure and superconductivity in FeSe0.5Te0.5 and FeS**Project supported by the National Natural Science Foundation of China (Grant Nos.11604303, 11604168, and 11574108).

The isovalent iron chalcogenides, FeSe0.5Te0.5 and FeS, share similar lattice structures but behave very differently in superconducting properties. We study the underlying mechanism theoretically. By first principle calculations and tight-binding fitting, we find the spectral weight of the orbital changes remarkably in these compounds. While there are both electron and hole pockets in FeSe0.5Te0.5 and FeS, a small hole pocket with a mainly character is absent in FeS. We find the spectral weights of orbital change remarkably, which contribute to electron and hole pockets in FeSe0.5Te0.5 but only to electron pockets in FeS. We then perform random-phase-approximation and unbiased singular-mode functional renormalization group calculations to investigate possible superconducting instabilities that may be triggered by electron-electron interactions on top of such bare band structures. For FeSe0.5Te0.5, we find a fully gapped -wave pairing that can be associated with spin fluctuations connecting electron and hole pockets. For FeS, however, a nodal dxy (or in an unfolded Broullin zone) is favorable and can be related to spin fluctuations connecting the electron pockets around the corner of the Brillouin zone. Apart from the difference in chacogenide elements, we propose the main source of the difference is from the orbital, which tunes the Fermi surface nesting vector and then influences the dominant pairing symmetry.

On the Origin of a Small Hole Pocket in the Fermi Surface of Underdoped YBa2Cu3O y

The comprehensive experimental studies of the quantum oscillations (QOs) in YBa2Cu3O y (YBCO) lead N. Doiron-Leyraud et al. (Nature Communications 6:6034, 2015) to detection of small hole pockets below the transition into the charge-ordered (CO) phase. Their interpretation of the observed energy spectrum imposes serious constraints for the possible origin of electron and hole pockets—both of them should be due to the Fermi surface (FS) reconstruction at charge ordering with biaxial order parameter. However, the results of recent X-ray experiments which reveal the uniaxial CO in the underdoped YBCO make such interpretation questionable. We show that uniaxial CO parameter in combination with preexisting electron pocket at the Γ-point of Brillouin zone suggests a plausible interpretation of QOs associated with small hole pockets in YBCO.

Manifestations of Quasi-Two-Dimensional Metallicity in a Layered Ternary Transition Metal Chalcogenide Ti2PTe2

Quasi-two-dimensional (2D) metallicity in a ternary layered transition metal chalcogenide of single crystalline Ti2PTe2 is studied by transport prop-erties measurements and electronic structure analyses. The logarithmic dependence of resistivity in the low temperature region appears as a charac-teristic feature of the quasi-2D transport in a weakly disordered metallic system. A large Sommerfeld coefficient value of 6.94 mJ/mol·K2 and negative temperature dependence of the Hall coefficient indicate that electron-electron interaction is dominant in a phase-breaking mechanism originating from bipolar behavior. Angle-resolved photoemission spectroscopy combined with theoretical calculations successfully proves that quasi-2D elec-tron pockets encompassing a small three-dimensional hole pocket mainly determine the quasi-2D metallic transport properties. We also adopt an orbital character calculation to verify that Ti character is dominant in valence bands, revealing that 2D Ti-networks form quasi-2D conducting ...

Electronic quasiparticles in the quantum dimer model: Density matrix renormalization group results

We study a recently proposed quantum dimer model for the pseudogap metal state of the cuprates. The model contains bosonic dimers, representing a spin-singlet valence bond between a pair of electrons, and fermionic dimers, representing a quasiparticle with spin-$1/2$ and charge $+e$. By density matrix renormalization group calculations on a long but finite cylinder, we obtain the ground-state density distribution of the fermionic dimers for a number of different total densities. From the Friedel oscillations at open boundaries, we deduce that the Fermi surface consists of small hole pockets near $(\pi/2, \pi/2)$, and this feature persists up to a doping density of $1/16$. We also compute the entanglement entropy and find that it closely matches the sum of the entanglement entropies of a critical boson and a low density of free fermions. Our results support the existence of a fractionalized Fermi liquid in this model.

Suppression of orbital ordering by chemical pressure inFeSe1−xSx

We report a high-resolution angle-resolved photo-emission spectroscopy study of the evolution of the electronic structure of FeSe1-xSx single crystals. Isovalent S substitution onto the Se site constitutes a chemical pressure which subtly modifies the electronic structure of FeSe at high temperatures and induces a suppression of the tetragonal-symmetry-breaking structural transition temperature from 87K to 58K for x=0.15. With increasing S substitution, we find smaller splitting between bands with dyz and dxz orbital character and weaker anisotropic distortions of the low temperature Fermi surfaces. These effects evolve systematically as a function of both S substitution and temperature, providing strong evidence that an orbital ordering is the underlying order parameter of the structural transition in FeSe1-xSx. Finally, we detect the small inner hole pocket for x=0.12, which is pushed below the Fermi level in the orbitally-ordered low temperature Fermi surface of FeSe.

Quantum dimer model for the pseudogap metal

We propose a quantum dimer model for the metallic state of the hole-doped cuprates at low hole density, p. The Hilbert space is spanned by spinless, neutral, bosonic dimers and spin S = 1/2, charge +e fermionic dimers. The model realizes a "fractionalized Fermi liquid" with no symmetry breaking and small hole pocket Fermi surfaces enclosing a total area determined by p. Exact diagonalization, on lattices of sizes up to 8 × 8, shows anisotropic quasiparticle residue around the pocket Fermi surfaces. We discuss the relationship to experiments.

Evidence for a small hole pocket in the Fermi surface of underdoped YBa2Cu3Oy.

In underdoped cuprate superconductors, the Fermi surface undergoes a reconstruction that produces a small electron pocket, but whether there is another, as yet, undetected portion to the Fermi surface is unknown. Establishing the complete topology of the Fermi surface is key to identifying the mechanism responsible for its reconstruction. Here we report evidence for a second Fermi pocket in underdoped YBa2Cu3Oy, detected as a small quantum oscillation frequency in the thermoelectric response and in the c-axis resistance. The field-angle dependence of the frequency shows that it is a distinct Fermi surface, and the normal-state thermopower requires it to be a hole pocket. A Fermi surface consisting of one electron pocket and two hole pockets with the measured areas and masses is consistent with a Fermi-surface reconstruction by the charge–density–wave order observed in YBa2Cu3Oy, provided other parts of the reconstructed Fermi surface are removed by a separate mechanism, possibly the pseudogap.

Large, non-saturating magnetoresistance in WTe2.

Magnetoresistance is the change in a material's electrical resistance in response to an applied magnetic field. Materials with large magnetoresistance have found use as magnetic sensors, in magnetic memory, and in hard drives at room temperature, and their rarity has motivated many fundamental studies in materials physics at low temperatures. Here we report the observation of an extremely large positive magnetoresistance at low temperatures in the non-magnetic layered transition-metal dichalcogenide WTe2: 452,700 per cent at 4.5 kelvins in a magnetic field of 14.7 teslas, and 13 million per cent at 0.53 kelvins in a magnetic field of 60 teslas. In contrast with other materials, there is no saturation of the magnetoresistance value even at very high applied fields. Determination of the origin and consequences of this effect, and the fabrication of thin films, nanostructures and devices based on the extremely large positive magnetoresistance of WTe2, will represent a significant new direction in the study of magnetoresistivity.