Quasilinear quantum magnetoresistance in pressure-induced nonsymmorphic superconductor chromium arsenide

Abstract

In conventional metals, modification of electron trajectories under magnetic field gives rise to a magnetoresistance that varies quadratically at low field, followed by a saturation at high field for closed orbits on the Fermi surface. Deviations from the conventional behaviour, for example, the observation of a linear magnetoresistance, or a non-saturating magnetoresistance, have been attributed to exotic electron scattering mechanisms. Recently, linear magnetoresistance has been observed in many Dirac materials, in which the electron-electron correlation is relatively weak. The strongly correlated helimagnet CrAs undergoes a quantum phase transition to a nonmagnetic superconductor under pressure. Here we observe, near the magnetic instability, a large and non-saturating quasilinear magnetoresistance from the upper critical field to 14 T at low temperatures. We show that the quasilinear magnetoresistance may arise from an intricate interplay between a nontrivial band crossing protected by nonsymmorphic crystal symmetry and strong magnetic fluctuations.

ARTICLE

Received 23 Sep 2016 | Accepted 15 Mar 2017 | Published 5 Jun 2017

Q. Niu1, W.C. Yu1, K.Y. Yip1, Z.L. Lim1, H. Kotegawa2, E. Matsuoka2, H. Sugawara2, H. Tou2, Y. Yanase3

& Swee K. Goh1

In conventional metals, modication of electron trajectories under magnetic eld gives rise to a magnetoresistance that varies quadratically at low eld, followed by a saturation at high eld for closed orbits on the Fermi surface. Deviations from the conventional behaviour, for example, the observation of a linear magnetoresistance, or a non-saturating magnetoresistance, have been attributed to exotic electron scattering mechanisms. Recently, linear magnetoresistance has been observed in many Dirac materials, in which the electronelectron correlation is relatively weak. The strongly correlated helimagnet CrAs undergoes a quantum phase transition to a nonmagnetic superconductor under pressure. Here we observe, near the magnetic instability, a large and non-saturating quasilinear magnetoresistance from the upper critical eld to 14 T at low temperatures. We show that the quasilinear magnetoresistance may arise from an intricate interplay between a nontrivial band crossing protected by nonsymmorphic crystal symmetry and strong magnetic uctuations.

DOI: 10.1038/ncomms15358 OPEN

Quasilinear quantum magnetoresistance in pressure-induced nonsymmorphic superconductor chromium arsenide

1 Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 999077, China. 2 Department of Physics, Kobe University, Kobe 658-8530, Japan. 3 Department of Physics, Kyoto University, Kyoto 606-8502, Japan. Correspondence and requests for materials should be addressed to Y.Y. (email: mailto:[email protected]

Web End [email protected] ) or to S.K.G. (email: mailto:[email protected]

Web End [email protected] ).

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Linear magnetoresistance (MR) is a fascinating material property that can be harnessed for magnetic-eld sensing owing to its simple eld-to-resistance conversion and its

equal sensitivity across the entire eld range. As such, the mechanisms underpining the linear MR continue to attract interest. Linear MR is most commonly observed in systems with a relatively weak electronelectron correlation and comparatively low carrier density, such as doped semiconductors13 and topological semimetals47. The observation of a linear MR in strongly correlated electron systems has been a rare occurence. A notable case of a striking linear MR is found in BaFe2(As1 xPx)2

near the antiferromagnetic quantum critical point8, where non-Fermi liquid behaviour in the temperature (T) dependence of the resistivity (r) has also been observed9, leading to a phenomenological model to describe the electrical transport with a scattering rate that is proportional to the quadrature sum of magnetic eld and temperature8. Signature of linear MR has also been found in some heavy fermion compounds such as UPt3 and CeCu6 (refs 10,11), although the conventional quadratic component or the negative MR was found to coexist with the linear component in these systems. To develop the concept further, it is important to study other strongly correlated electron systems near magnetic instability which exhibit clear linear MR to expand the material base for detailed investigations.

CrAs undergoes a strong rst-order transition at TN 265 K

into an antiferromagnetic state, in which the spins assume a double helical structure (winding axis || c axis)12,13. This antiferromagnetic state can be completely suppressed by applying 7 kbar (refs 14,15), or by substituting 5% of phosphorous for arsenic, that is, CrAs0.95P0.05 (ref. 16). The

magnetic behaviour of phosphorous-substituted samples is similar to that of CrAs under high pressure, and hence phosphorous acts as the chemical pressure analogous to other systems such as BaFe2(As1 xPx)2 (ref. 17). Concomitant with the

suppression of the helimagnetic state, superconductivity can be induced in CrAs by pressure. The superconducting transition temperature (Tc) takes a dome-shaped pressure dependence, with a maximum TcE2.17 K at 10 kbar (refs 14,15). The temperature pressure (Tp) phase diagram constructed is reminiscent of those constructed for many quantum critical systems, for example, refs 1822, except that the magnetic transition at ambient pressure is strongly rst order, although the rst-order signature is signicantly weakened at elevated pressures14,15. Nevertheless, nuclear quadrupole resonance detected substantial magnetic uctuations in the paramagnetic state23. In the phosphorous-substituted series, superconductivity has not been observed due to the strong disorder introduced by phosphorous substitution, which is consistent with the unconventional nature of superconductivity23.

We have measured the MR of CrAs down to 14 mK in representative parts of the Tp phase diagram. The space group of CrAs, Pnma, includes nonsymmorphic glide and screw symmetries, which involve half translation. We will show that the nonsymmorphic symmetry of the crystal structure plays an essential role on the realization of a particular band structure, leading to a nontrivial band crossing on the Brillouin zone face protected by nonsymmorphic symmetry. At 14 T, the MRs are positive, sizeable and non-saturating. Furthermore, distinct quasilinear MRs are observed in the vicinity of the magnetic instability, which we suggest is the result of an intricate interplay between having this particular band structure and the proximity to magnetic instability.

ResultsTp phase diagram and MR. Figure 1a shows the temperature pressure phase diagram of CrAs. Three crystals were used for this

study, namely CrAs at ambient pressure (pA), CrP at ambient pressure (pCrP) and CrAs under pressure (pis with i indicates the sequence of the measurement). Both CrAs and CrP crystallize in an orthorhombic structure with space group Pnma (inset to Fig. 1a)12. Using the x-dependence of the unit cell volume in CrAs1 xPx (ref. 24), we estimate that Dx 1 corresponds to

DpE145 kbar. Therefore, we place CrP at the location of p 145 kbar relative to CrAs. CrP has an excellent purity with

a residual resistance ratio (RRR) of 455, in stark contrast to members of phosphorous-substituted CrAs. For example, RRRE5 for CrAs0.95P0.05. On the other hand, the RRR of the

pressurized CrAs is far larger, for example, RRR 194 at p2,

provided that care is taken not to thermally cycle the crystal through the rst-order line (see below). Figure 1bd show the MR, dened as MR rH r0r0 100% , at the lowest attainable

temperature of each run in the helimagnetic state (pA),

nonmagnetic superconducting state (p1 9.2 kbar) and the

paramagnetic non-superconducting state (pCrP), respectively. All traces exhibit a large and non-saturating MR at 14 T. In the helimagnetic state (pA), the eld dependence of the MR is sublinear at low eld with a seemingly absence of the H2 region. At p1, the MR is quasilinear from the upper critical eld (Hc2) to14 T. Finally, at pCrP, the MR is conventional and follows a H2 variation from 0 to 14 T. The CrAs system thus provides a useful platform for accessing different MR behaviours by tuning the unit cell volume. All these traces are clearly an even function of H, indicating a negligible contamination from the Hall component. Therefore, the data presented here are truly the transverse MR and hence only the positive H portion of the data are plotted in the following.

Figure 2ad show the MR collected over a wide temperature range at pressures where the ground state is superconducting. At these pressures, the MR is quasilinear at the lowest temperature, and it reaches at least 70% at 14 T. For p1 and p2, the single crystal was directly pressurized to the required pressure values at room temperature before cooling. MR(14 T) at the lowest temperature for p1 and p2 are larger than 110%, and it decreases with increasing temperature. To investigate the possibility of a disorder-induced linear MR behaviour, we take advantage of the strong magnetostriction effect associated with the rst-order transition into the helimagnetic state12. Empirically, it is known that on warming up across the rst-order line, the crystal can crack. Therefore, we reduced the pressure to 5.2 kbar (p3) and

performed a thermal cycle, where the rst-order transition exists but much weaker than the ambient pressure case (pA)12,14,15, with the aim of inducing microcracks that are not completely detrimental to measurements. Indeed, at p4 and p5, which are the runs after p3, RRR decreases to 120 and 96, respectively. Interestingly, although MR(14 T) decreases signicantly, the curvature of the MR remains quasilinear. In Fig. 2e, we compare the MR at 3 and 10 K for p2 and p4, since these pressure values are very close. We nd that the MRs at p4 can be satisfactorily scaled to the curves at p2 using the ratio RRR(p2)/RRR(p4)D1.62. This exercise works extremely well for the data sets at 3 K. Therefore, the functional form of the MR is robust in this system. Although it is not possible to completely rule out linear MR mechanisms that assume sample inhomogeneity2,25, we note that our samples are high-quality single crystals: RRR and the residual resistivity at p2 are 194 and 0.83 mO cm, respectively (Supplementary Note 2). These observations suggest that disorder, particularly spatial inhomogeneity associated with the rst-order transition, is not expected to play a dominant role.

Band structure and the nonsymmorphic symmetry. Figure 3a shows the band structure of CrAs at 14.3 kbar, calculated using

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Figure 1 | Transverse magnetoresistance in pressure-tuned CrAs. (a) Schematic temperaturepressure phase diagram showing the helimagnetic region, the superconducting region (SC) and the paramagnetic region (PM), constructed using the data in ref. 14. The area bounded by dashed lines indicates the region with coexistence of magnetism and superconductivity. The open squares denote the experimental data of the current work, which are consistent with the previously published data14,15. For clarity, the values of Tc are multiplied by 30. The position of CrP is determined by taking into account the chemical pressure effect of P on CrAs. The crystal structure of CrAs, prepared using VESTA42, is shown in the inset. (bd) Magnetoresistance, dened as MR [r(H) r(0)]/r(0) 100%, from 14 to 14 T at pA, p1, and pCrP, collected at 25.7, 15.5 and 16.2 mK, respectively. For the data collected in the

superconducting state, the magnetoresistance is calculated using a zero-eld resistivity r(0) estimated from a smooth extrapolation of the data above Hc2.

(e) High-eld magnetoresistance in CrP, shown as the green rectangle in d, with steep background subtracted. The left and the right arrows are at 13 and14 T, respectively. Clear Shubnikovde Haas oscillations can be seen. (f) Fast Fourier transform of the oscillatory signal in e from 8 to 14 T.

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Figure 2 | Magnetoresistance of CrAs under pressure. (ad) Magnetoresistance of CrAs over a wide temperature range at p1, p2, p3 and p5. The spikes at some traces are experimental artefacts. (e) Magnetoresistance at p2 (solid lines) and p4 (dashed lines). The traces at p4 are multiplied by 1.62, which is the ratio RRR(p2)/RRR(p4). (f) The eld dependence of the resistance at p2 on logarithmic scales, showing the determination of the crossing over eld m0H*.

(g) The temperature dependence of m0H* (squares) at p2, which can be described by m0H* a(T b)3/2 (solid line).

the experimental lattice parameters24. At the Y point, a band crossing is found at B8.6 meV below EF along the ky direction, giving rise to a linear energymomentum dispersion with a slope :c. Along the YS direction, which is parallel to GX (Fig. 3c), the dispersion is quadratic, parametrized by m. The crossing point at

Y is thus analogous to the semi-Dirac point discussed in the literature26,27, except that the parabolic branches are degenerate in our case. Such a hybrid dispersion relation is a result of the

nonsymmorphic symmetry of the space group Pnma. The two bands crossing at the Y point are degenerate along the kx direction (YS line on the Brillouin zone face), as the nonsymmorphic symmetry protects the degeneracy. Thus, the band crossing has to occur on the YS line on which the linear term in kx is prohibited and the usual quadratic dispersion appears (see Supplementary Note 4 for theoretical proof). A similar band crossing at the Y point is obtained for CrP,

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1.0

0.5

1.0

p6pe c f g

3=2 EF Ec

kBT

3=2; 1

where Ec is the energy of the crossing point, and f(g) (1 g)2/3 g2/3 is a dimensionless quantity. Since this pocket has an

anisotropy akxF=kyF in the kxky plane, we get another constraint

m a2(EF Ec)/(2c2).

Following Narayanan et al.6, we plot logR against log(m0H) in Fig. 2f, which clearly reveals how the low-eld MR transforms into the linear region above a crossover eld. Given the absence of quantum oscillations in our high-quality sample, which we will discuss later, it is tempting to assign the crossover eld as m0H*.

That is to say, we speculate that the linear MR in pressurized CrAs arises from the Fermi surface pocket at the Y point, which reaches the extreme quantum limit above m0H*. The

T-dependence of m0H* at p2 13.1 kbar, which is the closest to

the pressure value of the calculated band structure, can be satisfactorily described by m0H* a(T b)3/2 (see Supplementary

Note 1 for other pressures). (EF Ec) can be instantly determined

from b to be 3.7 meV. If the band structure displayed in Fig. 3b is rigidly shifted so that (EF Ec) 3.7 meV, we obtain a 4.7.

From the tting parameter a, we can estimate that c ranges between 10.5 104 and 15.6 104 m s 1 when g goes from 0 to 1.

This is in satisfactory agreement with c 9.9 104 m s 1

extracted from the band structure calculation at 14.3 kbar. Note that the band shifting does not affect c, hence the comparison between experimental data and the calculated slope :c is particularly meaningful. Therefore, it is conceivable that the

MR in the pressurized CrAs is governed by the tiny Fermi pocket at Y, which reaches the extreme quantum limit above m0H*.

For a multiband system like CrAs, the applicability of Abrikosovs theory requires a careful justication. In fact, in his own paper30, Abrikosov provided a mechanism to show how the theory can be applied when a big Fermi surface coexists with a small Fermi pocket. Following the same line of thought, we propose that in the pressurized CrAs, the Fermi pocket around Y dominates the MR. Along the GY direction, the effective mass is essentially zero because of the linear dispersion. Along the YS direction, m ranges between 0.660.30 me for g between 0 and 1.

Therefore, the mobility is high and this pocket can dominate the MR. In the vicinity of the magnetic instability, magnetic uctuations are signicant, as indicated by the enhancement of A-coefcient which characterizes the weight of the T2 component in the electrical resistivity14,15, as well as the spin-lattice relaxation rate23. We speculate that the scattering lifetime is short in other Fermi surface sheets due to strong magnetic uctuations, and hence they do not contribute signicantly to MR. To accurately analyse a multiband system, an extension of Abrikosovs theory to properly include the multiband effect is desirable.

To further examine the relevance of the pocket at Y, we now move to CrP. In CrP, we can take (EF Ec)E47 meV (Fig. 3e). If

g remains the same in CrP, m0H*E171 T at 0 K. Therefore, with a magnetic eld of 14 T, it is not possible to have all carriers in the lowest Landau level in CrP. Furthermore, since CrP is far from the magnetic instability, multiple Fermi surface sheets with lower effective mass are contributing to the MR. Hence, the MR is quadratic.

Absence of Shubnikovde Haas oscillations in CrAs. We have not observed any trace of Shubnikovde Haas (SdH) oscillations in CrAs. This might be particularly puzzling for p1 and p2, given that extra care has been taken to completely avoid the magnetic

2m

Energy (eV)

Energy (eV)

0.5

1.0

Figure 3 | Comparison of the band structure of CrAs at 14.3 kbar and CrP. (a,b) Dispersion relation along several high-symmetry directions for CrAs at 14.3 kbar. An expanded view of the band structure near EF around Y (indicated by the circle) is shown, where a band crossing is found

B8.6 meV below the Fermi energy. The dispersion is linear along YG and parabolic along YS. (c) The rst Brillouin zone of CrAs/CrP with the relevant high-symmetry points labelled. (d,e) Dispersion relation for CrP. A similar band structure is found in CrP near Y, but with a crossing point located at B47 meV below EF instead.

with the crossing point located at B47 meV below EF, which is much deeper compared with CrAs at 14.3 kbar (Fig. 3e). Although a similar band crossing protected by another nonsymmorphic space group P4/nmm was reported in ZrSiS (ref. 28), the crossing point appears 500 meV below EF. This is to be contrasted with the present case where the crossing point is much closer to EF, and the separation between the crossing point and EF is tunable by applied pressure (compare Fig. 3b,e).

Landau levels associated with the hybrid band structure. The close proximity of the crossing point to EF implies a small Fermi pocket, which contains neither conventional electrons nor Dirac electrons. In the kxky plane, the dispersion relation around the Y point near EF can be written as e cky 2k2x=2m, where

(kx, ky) is the wavevector measured from Y. Combining with the quantization condition S(e) 2p(n g)em0H/: within the

semiclassical approach, the Landau level spacing is proportional to H2/3 when H || kz, and gA[0, 1] is a phase factor that can not be determined from the semiclassical approach. According to Abrikosovs theory, a Fermi surface sheet will contribute to a linear quantum MR if all carriers occupy the lowest Landau level29,30. This is the extreme quantum limit, and it requires the band extrema to be very near the Fermi energy (EF). It is a straightforward exercise to calculate the characteristic eld m0H*,

above which all carriers in this pocket are in the lowest Landau level:

m0H?

8

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100

10

1

transition, so that potential spatial inhomogeniety introduced by crossing the rst-order line can be ruled out. Indeed, the residual resistivity at p2 is 0.83 mO cm (Supplementary Note 2). However, the absence of SdH oscillations is consistent with our interpretation above, namely, that the MR is dominated by the Fermi pocket with carriers at the lowest Landau level. The SdH effect arises when successive Landau levels cross the Fermi pocket as the magnetic eld is varied. In our case, since the extreme quantum limit for the pocket at Y is reached at m0H? 4 T, magnetic eld

higher than m0H* will not give SdH oscillations. For magnetic eld below m0H*, SdH oscillations can in principle appear. However, the larger reduction at low eld due to Dingle factor, the appearance of superconductivity below m0Hc2, and the

extremely small SdH frequency all make the detection of SdH signals from this pocket extremely challenging. For other Fermi surface sheets, their short scattering lifetime due to signicant magnetic uctuations and hence their substantially weaker contributions to MR make SdH an insensitive probe to detect quantum oscillations from them.

Much can again be learnt by examining the high-pressure analogue CrP. Here the constraints stipulated above for CrAs are lifted, and SdH oscillations can readily be seen, as displayed in Fig. 1e,f. Multiple SdH frequencies are detected, with the largest frequency being B3.6 kT, signifying the multiband, high carrier density nature of CrP (and CrAs). In addition, the lowest frequency detected is B210 T (Fig. 1f). The Landau level index nEF/(m0H), where F is the quantum oscillation frequency.

Therefore, to enter the lowest Landau level (no1) the applied eld m0H* is around F. A satisfying agreement between the detected SdH frequency of 210 T and m0H*E171 T estimated in the preceeding section is noted.

HT scaling in CrAs under pressure. Recently, quasilinear MR has been observed in BaFe2(As1 xPx)2 in the vicinity of the

quantum critical concentration xcE0.3 (ref. 8). Near xc, a linear T-dependence of electrical resistivity was reported. Incidentally, the MR is also linear in eld, which led to the proposal that the resistivity would be proportional to a new energy scale that is a quadrature sum of the temperature and applied eld, that is,

rH; T r0; 0 /

akBT 2 bmBm0H

2

[ (H,T) (0,0)]/T (cm K1 )

0.1

0.01

0.01 0.1 1 100 1,000

10

0H/T (T K1)

Figure 4 | HT scaling in CrAs at 13.1 kbar. The magnetoresistance replotted as [r(H, T) r(0, 0)]/T against m0H/T. The data below 3.0 K can

all be described by a single curve A(1 Bx2)0.5 (thick solid curve), which

has the same functional form as equation (3). Note that the axes are on logarithmic scales, and the dashed line indicates a unity slope.

q

; 2

or equivalently,

rH; T r0; 0

=T /

s ; 3

where a, b and l are numerical parameters. Hence, the resistivity traces plotted as [r(H, T) r(0, 0)]/T versus m0H/T will all lie on

a universal curve. In Fig. 4, we replot our data at p2 13.1 kbar on

the axes of [r(H, T) r(0, 0)]/T and m0H/T. All the normal state

data below 3 K indeed fall onto the same curve, except for the datapoints just above Hc2. Note that both axes are on logarithmic scales. Therefore, an excellent agreement is found over ve orders of magnitude in m0H/T. Similar behaviour is found for p1 and p5 (Supplementary Note 2). On the contrary, a severe violation of Kohlers rule is found for the same data sets (Supplementary Note 3). These observations imply that the low temperature r(T)

in the zero-eld limit is linear in T over a wide pressure range, even though the system is tuned to a ground state that is far away from magnetism. This is reminiscent of the observation in overdoped cuprates31 and highlights the importance of the proximity to magnetic instability which renormalizes the effective masses of the massive bands, thereby allowing a clear dominance of high mobility fermions on magnetotransport properties. Nontrivial band crossings protected by nonsymmorphic

symmetry have been attracting attention recently28,32,33; in the case of CrAs, it gives rise to the small Fermi pocket that allows a clear manifestation of the physics at the extreme quantum limit. Such intriguing band structure may also give rise to exotic superconducting properties as recently illustrated in UPt3 (ref. 34). Future studies on the inuence of the nonsymmorphic symmetry will be an important theme.

ConclusionsThe subject of linear MR has a long history and it continues to attract fresh theoretical efforts (for example, refs 3537), it is therefore possible that an alternative explanation can be found for our data. In three-dimensional Dirac semimetal Cd3As2, linear MR was attributed to mobility uctuations in ref. 6, while alternative views were proposed for the same material4,35. In Cd3As2, Kohlers rule holds extremely well6, in stark contrast to pressurized CrAs. These highlight the intense interest in unraveling the underlying mechanisms giving rise to linear MR. Nevertheless, our high quality and transparent MR data, excellent HT scaling, as well as the violation of Kohlers scaling in pressurized CrAs will provide useful constraints for a critical examination of various models.

Methods

Single-crystal growth. Single crystals of CrAs were grown with the Sn-ux method from a starting composition of Cr:As:Sn 1:1:10 (ref. 14). Single crystals of

CrP were prepared by chemical vapour transport method38. The powdered polycrystalline sample of CrP was sealed in a silica tube with iodine under vacuum. The charge zone was kept at 900 C and the growth zone at 800 C for 2 weeks.

Electrical resistance measured under extreme conditions. The high-pressure environment was provided by a piston-cylinder clamp cell made of MP35N alloys. Glycerin was used as the pressure medium and the pressure values were estimated by the zero-eld Tc of a Pb manometer placed near the sample. The pressure cell was cooled down to about 15 mK in a dilution refrigerator (BlueFors) equipped with a 14 T superconducting magnet (American Magnetic Inc.) for all pressure points except p4 12.5 kbar, which was examined with a Physical Property

Measurement System (Quantum Design), with a base temperature of 1.8 K. The MR was measured with a conventional four-probe method with the magnetic eld parallel to the c axis and the current along a axis.

Band structure calculations. Band structure calculations were performed with the all-electron full-potential linearized augmented plane-wave code WIEN2k (ref. 39).

1 l

m0H

T

2

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The generalized gradient approximation of Perdew, Burke and Ernzerhof40 was employed for the exchange-correlation potential. Experimental lattice constants24,41 were used in the calculation and internal structure optimization was performed. The mufn-tin radii were set to 1.96 a.u. for the P atoms and2.24 a.u. for the Cr and As atoms. RminMTKmax 8 and a k-point mesh of 5,000 in the

rst Brillouin zone were used.

Data availability. The data used for supporting the ndings of this study are available from the corresponding authors on request.

References

1. Xu, R. et al. Large magnetoresistance in non-magnetic silver chalcogenides. Nature 390, 5760 (1997).

2. Parish, M. & Littlewood, P. Non-saturating magnetoresistance in heavily disordered semiconductors. Nature 426, 162165 (2003).

3. Hu, J. & Rosenbaum, T. Classical and quantum routes to linear magnetoresistance. Nat. Mater. 7, 697700 (2008).

4. Feng, J. et al. Large linear magnetoresistance in Dirac semimetal Cd3As2 with Fermi surfaces close to the Dirac points. Phys. Rev. B 92, 081306 (2015).5. Liang, T. et al. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2. Nat. Mater. 14, 280284 (2014).

6. Narayanan, A. et al. Linear magnetoresistance caused by mobility uctuations in n-doped Cd3As2. Phys. Rev. Lett. 114, 117201 (2015).

7. Novak, M., Sasaki, S., Segawa, K. & Ando, Y. Large linear magnetoresistance in the Dirac semimetal TlBiSSe. Phys. Rev. B 91, 041203 (2015).

8. Hayes, I. M. et al. Scaling between magnetic eld and temperature in the high-temperature superconductor BaFe2(As1 xPx)2. Nat. Phys. 12, 916919 (2016).

9. Kasahara, S. et al. Evolution from non-Fermi- to Fermi-liquid transport via isovalent doping in BaFe2(As1 xPx)2 superconductors. Phys. Rev. B 81, 184519

(2010).10. Lippman, T., Davis, J., Choi, H., Pollanen, J. & Halperin, W. Magnetoresistance of UPt3. J. Low Temp. Phys. 148, 863867 (2007).

11. Coleridge, P. Magnetoresistance and growth of the coherent state in CeCu6.

J. Phys. F 17, L79L85 (1987).12. Motizuki, K., Ido, H., Itoh, T. & Morifuji, M. Electronic Structure and Magnetism of 3d-Transition Metal Pnictides (Springer, 2009).

13. Watanabe, H., Kazama, N., Yamaguchi, Y. & Ohashi, M. Magnetic structure of CrAs and Mn-substituted CrAs. J. Appl. Phys. 40, 11281129 (1969).

14. Kotegawa, H., Nakahara, S., Tou, H. & Sugawara, H. Superconductivity of 2.2 K under pressure in helimagnet CrAs. J. Phys. Soc. Jpn 83, 093702 (2014).

15. Wu, W. et al. Superconductivity in the vicinity of antiferromagnetic order in CrAs. Nat. Commun. 5, 6508 (2014).

16. Kanaya, K., Abe, S., Yoshida, H., Kamigaki, K. & Kaneko, T. Magnetic and structural properties of pseudo-binary compounds CrAs1 xPx. J. Alloy. Comp.

383, 189194 (2004).17. Klintberg, L. E. et al. Chemical pressure and physical pressure in BaFe2(As1 xPx)2. J. Phys. Soc. Jpn 79, 123706 (2010).

18. Mathur, N. D. et al. Magnetically mediated superconductivity in heavy fermion compounds. Nature 394, 3943 (1998).

19. Gegenwart, P., Si, Q. & Steglich, F. Quantum criticality in heavy-fermion metals. Nat. Phys. 4, 186197 (2008).

20. Paglione, J. & Greene, R. L. High-temperature superconductivity in iron-based materials. Nat. Phys. 6, 645658 (2010).

21. Klintberg, L. E. et al. Pressure- and composition-induced structural quantum phase transition in the cubic superconductor (Sr,Ca)3Ir4Sn13. Phys. Rev. Lett.

109, 237008 (2012).22. Goh, S. K. et al. Ambient pressure structural quantum critical point in the phase diagram of (CaxSr1 x)3Rh4Sn13. Phys. Rev. Lett. 114, 097002 (2015).

23. Kotegawa, H. et al. Detection of an unconventional superconducting phase in the vicinity of the strong rst-order magnetic transition in CrAs using

75As-nuclear quadrupole resonance. Phys. Rev. Lett. 114, 117002 (2015).24. Yu, Z. et al. Anomalous anisotropic compression behavior of superconducting CrAs under high pressure. Proc. Natl Acad. Sci. USA 112, 1476614770 (2015).

25. Parish, M. M. & Littlewood, P. B. Classical magnetotransport of inhomogeneous conductors. Phys. Rev. B 72, 094417 (2005).

26. Dietl, P., Pichon, F. & Montambaux, G. New magnetic eld dependence of Landau levels in a graphenelike structure. Phys. Rev. Lett. 100, 236405 (2008).

27. Banerjee, S., Singh, R. R. P., Pardo, V. & Pickett, W. E. Tight-binding modeling and low-energy behavior of the semi-Dirac point. Phys. Rev. Lett. 103, 016402 (2009).

28. Schoop, L. M. et al. Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS. Nat. Commun. 7, 11696 (2016).

29. Abrikosov, A. A. Quantum magnetoresistance. Phys. Rev. B 58, 27882794 (1998).

30. Abrikosov, A. Quantum linear magnetoresistance. Europhys. Lett. 49, 789793 (2000).

31. Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2 xSrxCuO4. Science 323, 603607 (2009).

32. Young, S. M. & Kane, C. L. Dirac semimetals in two dimensions. Phys. Rev. Lett. 115, 126803 (2015).

33. Watanabe, H., Po, H. C., Vishwanath, A. & Zaletel, M. Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals. Proc. Natl Acad. Sci. USA 112, 1455114556 (2015).

34. Yanase, Y. Nonsymmorphic weyl superconductivity in UPt3 based on E2u representation. Phys. Rev. B 94, 174502 (2016).

35. Song, J. C. W., Refael, G. & Lee, P. A. Linear magnetoresistance in metals: guiding center diffusion in a smooth random potential. Phys. Rev. B 92, 180204 (2015).

36. Alekseev, P. S. et al. Magnetoresistance in two-component systems. Phys. Rev. Lett. 114, 156601 (2015).

37. Kisslinger, F., Ott, C. & Weber, H. B. Origin of nonsaturating linear magnetoresistivity. Phys. Rev. B 95, 024204 (2017).

38. Nozue, T., Kobayashi, H., Yamagami, H., Suzuki, T. & Kamimura, T.de Haas-van Alphen effect in CrP. J. Magn. Soc. Jpn 23, 430432 (1999).

39. Schwarz, K. & Blaha, P. Solid state calculations using WIEN2k. Comput. Mater. Sci. 28, 259273 (2003).

40. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 38653868 (1996).

41. Selte, K., Hjersing, H., Kjekshues, A., Andresen, A. F. & Fischer, P. Magnetic structures and properties of CrP1 xAsx. Acta Chem. Scand. A 29, 695698

(1975).42. Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 12721276 (2011).

Acknowledgements

We acknowledge stimulating discussion with Anthony Leggett, Xi Dai, Christos Panagopoulos, Takasada Shibauchi and Yasuyuki Nakajima. This work was supported by Research Grant Council of Hong Kong (ECS/24300214), CUHK Startup Grant (No. 4930048), Grant-in-Aids for Scientic Research (No. JP15H03689, JP15H05745, JP15K05885, JP15H05884, JP15K05164 and JP16H00991) from Japan Society for the Promotion of Science, and National Science Foundation China (No. 11504310).

Author contributions

S.K.G. conceived and designed the experiment. Q.N., K.Y.Y. and Z.L.L. performed the low-temperature measurements. H.K., E.M., H.S. and H.T. prepared the single crystals. W.C.Y. performed the band structure calculations. S.K.G. and Q.N. analysed the data. Y.Y. provided the theoretical support. S.K.G., Y.Y. and Q.N. wrote the manuscript.

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How to cite this article: Niu, Q. et al. Quasilinear quantum magnetoresistance in pressure-induced nonsymmorphic superconductor chromium arsenide. Nat. Commun. 8, 15358 doi: 10.1038/ncomms15358 (2017).

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