Unconventional Large Linear Magnetoresistance in Cu$_{2-x}$Te

We report a large linear magnetoresistance in Cu$_{2-x}$Te, reaching $\Delta\rho/\rho(0)$ = 250\% at 2 K in a 9 T field. This is observed for samples with $x$ in the range 0.13 to 0.22, and the results are comparable to the effects observed in Ag$_2 X$ materials, although in this case the results appear for a much wider range of bulk carrier density. Examining the magnitude vs. crossover field from low-field quadratic to high-field linear behavior, we show that models based on classical transport behavior best explain the observed results. The effects are traced to misdirected currents due to topologically inverted behavior in this system, such that stable surface states provide the high mobility transport channels. The resistivity also crosses over to a $T^2$ dependence in the temperature range where the large linear MR appears, an indicator of electron-electron interaction effects within the surface states. Thus this is an example of a system in which these interactions dominate the low-temperature behavior of the surface states.

Considerable attention has recently been devoted to systems exhibiting linear magnetoresistance (MR), starting with the effects observed in Ag 2 Te and Ag 2 Se 1,2 . In these systems linear MR extends to very large applied fields, in contrast to conventional conductors exhibiting quadratic MR which eventually saturates with increasing field. The origin of this effect can be traced to high mobility topological surface states in these systems [3][4][5][6] , and meanwhile many related systems have been discovered to exhibit such behavior, including Dirac semimetals [7][8][9] .
Models proposed to explain the Ag 2 X effects include the quantum mechanism of Abrikosov 10 , with low mass carriers generated through disorder-induced band contacts leading to orbital quantization in relatively small fields. Weak anti-localization 11,12 can also be important for magneto-transport of topological surface states, and the interplay of these effects with linear MR has been examined, for example, in Bi 2 Te 3 -based topological insulators 13,14 . On the other hand Parish and Littlewood 15,16 showed that a classical mechanism will give linear MR extending over a wide range of fields in the case of a distribution of carrier mobilities significantly exceeding the mean value (∆µ ≫ µ ). Herring 17 had earlier shown that linear MR may occur in weakly inhomogeneous systems, for fields where the cyclotron orbit period exceeds the scattering time, equivalent to B µ > 1. A common feature of these models is low mass/high mobility states, although 2 dimensional surface states are not specifically required.
With the Ag 2 Se linear MR observed to track with mobility 18 , a classical model would appear to apply. Nevertheless the origin remains unclear because of the small fields needed to limit massless surface states to the lowest Landau level 3 , thus suggesting a quantum origin. It has further been proposed 19 that more conventional processes involving compensating charge carriers may dominate in Ag 2 Te, and a mechanism based on spin splitting of surface states has also been advanced for topological insulators 20 . Weak antilocalization as a bulk rather than surface effect 12 may also occur in these layered systems in the presence of spin-orbit coupling. Furthermore it was recently demonstrated 9 that even very weak disorder may lead to such effects in 3D high-mobility systems such as the Dirac semimetals.
Cu 2 Te has been of significant interest for potential applications including thermoelectric and solar energy conversion, as well as a variety of nano-devices [21][22][23][24] , and it has been connected to a topologically nontrivial band configuration 25,26 . Synthesized materials in bulk have a Cu 2−x Te stoichiometry, with the Cu deficit corresponding to vacancies which lead to p-type semiconducting behavior. Here we present magnetotransport properties of materials in the range x = 0.13 to 0.22, exhibiting a large linear MR which can be traced to surface states, reinforcing the topological insulator nature of this system, and occurring in a regime of high carrier density and with strong electron interactions distinct from what has been observed in other systems.
The three polycrystalline Cu 2−x Te samples were obtained by solid state reaction and vacuum annealing. Their properties have been described in more detail in Ref. 27 bridge. Transport measurements were performed on barshaped samples cut from the polycrystalline ingots, with magnetoresistance measured in the geometry with the field perpendicular to the current direction. Figure 1 exhibits resistivities below 60 K. The behavior is quadratic in the low-temperature limit, particularly for the lowest-vacancy composition: fitting to ρ 0 + AT n below 30 K yielded n = 2.02, 2.13 and 2.34 for increasing x. Extended to room temperature, the residual resistivity ratios for the three samples are 32, 31, and 14 for Cu 1.87 Te, Cu 1.82 Te, and Cu 1.78 Te respectively. The inset of Fig. 1 also shows carrier densities derived from the Hall resistances, shown vs. temperature for Cu 1.87 Te and Cu 1.82 Te. These show a low temperature downturn where the T 2 resistivity sets in, apparently a result of parallel conduction paths rather than a change in the bulk carrier density, since as noted below it is in this range that the surface states are believed to influence the transport properties.
Fixing the low-temperature exponent to n = 2, the fitted resistivity pre-factors are A = 0.019, 0.009, and 0.010 µΩcm/K 2 for the three samples with increasing x. These compare to the lower end of the range for Fermi liquid behavior in heavy-Fermion materials 29 , although with the low-temperature behavior attributed to high-mobility threading states, this implies considerably smaller effective A values for these states, comparable for example to elemental transition metals. Mobilities derived from the resistivities and room-temperature Hall-effect carrier densities are µ = 10 cm 2 /Vs or less at room temperature, increasing to 170, 280, and 55 cm 2 /Vs for Cu 1.87 Te, Cu 1.82 Te, and Cu 1.78 Te respectively at low temperature. These are not unexpected for semiconductors with large vacancy densities and large hole band mass 25 close to 0.5 m e , although the temperature dependences are large for such a case, apparently due to threading states. the resistivity measured in applied field B. The 2 K magnitudes reach 250%, 180%, and 37% at 9 T for Cu 1.87 Te, Cu 1.82 Te, and Cu 1.78 Te, respectively. The largest of these are comparable to effects observed 1 in Ag 2 Te, although differing in that the Ag 2 Te results are observed in a much narrower composition window for carrier densities near zero, and decrease more slowly vs. temperature.
The insets of Fig. 2 also display Kohler plots, often used to understand whether a single scattering process controls the magnetoresistance 30 . The curves deviate from a common line at 30 K and below, showing that there are parallel scattering processes corresponding to the low-temperature conduction mechanism. Fig. 3(a) shows crossover fields (B c ), where the MR changes from quadratic to linear. B c was obtained by fitting the resistivity in the low-field limit to a B 2 dependence, and linear at higher fields, and extracting the fields where these curves cross 8 . The highest-temperature curves were excluded since the small response makes the fits unreliable.
In the Parish and Littlewood (PL) classical transport model, B c corresponds to a condition B µ = 1. In Fig. 3(a) values of the average mobility are given on the right axis according to this condition. For the two lowestx samples these are close to 7500 cm 2 /Vs at 2 K, considerably larger than the Hall mobilities extracted for these samples. This confirms that there must be high mobility regions within the overall low mobility material, reinforcing the evidence for threading conduction due to topological surface states, as was noted above.
For carriers of one sign, in the PL model the magnitude of the linear MR should also scale with average mobility, and in some cases a direct proportionality has been observed for linear MR vs. measured mobility. As a measure of the linear MR we plotted the 9 T values (MR Max) vs. T in Fig. 3(b). In the inset these are plotted vs. the normalized inverse resistivities measured at the corresponding temperatures, a measure of the mobilities for the case of constant carrier densities as we have here. However the plots are not linear, reflective of the discontinuous mobility distribution in these samples, and the crossover in resistivity mechanisms at low temperatures.
A more direct measure of the connection between the crossover field and the high mobility carriers responsible for the linear MR is shown in the plot of 1/B c vs. inverse resisitivity, Fig. 3(c). There is a linear relation below 30 K for the two lowest-x samples, which exhibit the largest MR and low-T resistivities closest to T 2 behavior. Thus there is strong evidence that the high mobility threading carriers for these samples can be traced directly to the measured resistivity in the low temperature limit, and these carriers apparently dominate the resistivity in this limit. The 1/(a + bT 2 ) curves in Fig. 3(a) and (b) are also drawn to correspond to this relation, with both 1/B c and MR Max connected to the inverse of the mobility for low temperatures.
A plot of 1/B c vs. MR max is also given in Fig. 4, and we see that there is a universal scaling between these quantities for all samples. A straight-line relationship is expected for a classical transport model, but with zero intercept in the PL model. This result is also similar to the universal scaling identified in Ref. 31 for nanoparticle films, although again in the present case there is a large offset. Note however that other systems have been observed to exhibit such an offset, for example in results for Ag 2 Se films 18 one can see that the corresponding offset has a much larger value of about -23 T −1 . The reasons are not clear, although in the present materials the source of the linear MR appears more likely to be misdirected currents 32,33 rather than a broad distribution of mobilities, with the threading nature of the high mobility currents a distinguishing feature relative to more continuous models. Aside from a surface state mechanism, because of its layered nature 2D weak anti-localization from spin-orbitsplit bulk states could play a role in Cu 2 Te. Such effects can be difficult to separate from classical linear MR, and the corresponding magnetoconductance 11 indeed works reasonably well as an alternative fitting model for the present data (not shown). These fits, with addition of a large quadratic classical magnetoconductance term, yield maximum dephasing lengths 11 at 2 K of between 19 and 30 nm, not unreasonable values. For comparison, in the analog material Cu 2 Se weak anti-localization effects were also identified at low temperatures 34 , although with a rather different amplitude and field dependence than what is observed here. Thus though it seems possible that some of the observed response in Cu 2 Te is due to such effects, the scaling with resistivity established here points to a classical model based on high mobility surface states as a more reasonable model to explain these observations.
It is also possible to attribute the results to the existence of bulk Dirac-like electronic states, such as for the guiding center mechanism recently introduced to account for inhomogeneous transport in such systems 9 . However in such cases a generally high mobility would be expected 8,[35][36][37] , as opposed to the situation here. Alternatively, a compensation-based mechanism due to multiple carrier pockets 19 might also explain the present results, however it has been shown 27 that hole pockets alone account well for the bulk transport and NMR behavior in Cu 2−x Te, and the observation of linear MR in compositions with different carrier concentrations appears inconsistent with the balance of carrier occupations required for such a mechanism.
Another consideration would be whether magnetic quantization conditions are reached here, such that a quantum MR 10 model is appropriate. Given the effective mass obtained for Cu 2 Te and the carrier densities present in these samples, we expect 25 that the Fermi energy in the bulk corresponds to several tenths of eV. With a corresponding Fermi level for Dirac-like surface states having the Fermi velocity of graphene, a field of 100 T or more would be required to occupy only the lowest Landau level 38 . Thus under the conditions used here we expect that many Landau levels will be occupied, a situation far from the quantum limit.
In a classical treatment, it is also clear that a model based on uniformly distributed weak disorder 17 cannot account for the present behavior, since the crossover fields are well out of range of the nearly-uniform high mobility needed for this to work. Indeed, as noted above it appears more reasonable to treat this system as discontinuous, with carrier behavior closer to bimodal, consisting of that of the bulk and of the polycrystal interfaces. There have been a number of works analyzing such situations, including through models based on resistor networks 32 as well as in effective medium theories 39,40 . However we are not aware of specific predictions related to the behavior of Fig. 4, in which the behavior appears to go smoothly from an inhomogeneous transport-based liner MR to a purely quadratic classic behavior as the temperature and carrier density increases.
Returning to the observed T 2 resistivities, there has been considerable interest in understanding the electron-electron interaction behavior of topological surface states 41,42 . In some cases these are predicted to have T 2 behavior analogous with that of ordinary Fermi liquids, through processes that should be strongly dependent on umklapp scattering, and correspondingly on the symmetry and curvature of the Fermi surface of the 2D topological states 43,44 . An alternative mechanism for the observed results would be scattering between bulk and surface electronic states, although with a scattering rate proportional to the density of states 45 , this mechanism would not be expected to produce the observed T 2 dependence. However it seems possible that such a mechanism is responsible for the much more rapid drop in linear MR with increasing bulk carrier density as observed 1 in Ag 2 Te vs. what is seen here. Cu 2 Te presents a case where a clear T 2 behavior sets in at low temperatures dominated by the surface states, a situation for which there are few experimental examples.
In conclusion, we observe a large linear magnetoresistance in samples of Cu 2−x Te with increasing carrier densities, with the magnitude reaching ∆ρ/ρ(0) = 250% at 2 K in a 9 T field, comparable to the effects observed in Ag 2 Se and Ag 2 Te. Examining the magnitude of the effect vs. the crossover field where the behavior changes from low-field quadratic to high-field linear behavior, we demonstrated that models based on classical transport behavior best explain the observed results. We also identified a universal scaling between the MR magnitude and the crossover field independent of carrier density. The effects are traced to previously identified indications of topologically inverted behavior in this system, such that topologically stable surface states provide the high mobility transport channels. There is a crossover to a T 2 resistivity behavior at low temperatures where the large linear MR appears, which we connected to electron-electron interaction effects within the surface states, so this system also provides an experimental example of such strongly interacting surface states.

ACKNOWLEDGMENTS
This work was supported by the Robert A. Welch Foundation, Grant No. A-1526. Work at the University of Michigan were supported as part of the Center for Solar and Thermal Energy Conversion, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences under Award DE-SC-0000957.