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Tunable quantum critical point and detached superconductivity in Al-doped CrAs

The origin of unconventional superconductivity and its relationship to a T=0 K continuous quantum phase transition (a quantum critical point, QCP), which is hidden inside the dome of a superconducting state, have long been an outstanding puzzle in correlated superconductors. The observation and tuning of the hidden QCP, which is critical in resolving the mystery, however, has been rarely reported due to lack of ideal systems. The helical antiferromagnet CrAs provides an example in which a dome of superconductivity appears at a pressure where its magnetic transition goes to zero temperature. Here we report the tuning of a projected critical point in CrAs via Al chemical doping (Al-CrAs) and separation of the magnetic critical point from the pressure-induced superconducting phase. When CrAs is doped with Al, its AFM ordering temperature TN increases from 260 K to 270 K. With applied pressure, TN decreases and extrapolates to zero Kelvin near 4.5 kbar, which is shifted from 8 kbar for undoped CrAs. A funnel of anomalously enhanced electron scattering and a non-Fermi liquid resistivity underscore an AFM QCP near 4.5 kbar in Al-CrAs. Pressure-induced superconductivity, in contrast, is almost independent of Al doping and forms a dome with essentially the identical maximum Tc and same optimal pressure as in pure CrAs. The clear separation between the tuned AFM QCP and Tc maximum in Al-CrAs suggests that superconductivity is independent of the AFM QCP, illustrating subtleties in the interplay between superconductivity and quantum criticality in correlated electron systems.

Unconventional superconductivity commonly emerges in proximity to a magnetically ordered phase, raising the possibility that critical spin fluctuations may mediate the formation of superconducting (SC) Cooper pairs [1][2][3][4][5][6][7][8][9][10]. In the vast majority of these cases, as sketched in Fig.   1(a), the zero-temperature limit of a continuous magnetic phase boundary (a quantum critical point, QCP), is veiled by a dome of superconductivity, making it difficult to prove the interplay between unconventional superconductivity and fluctuations arising from the presumed QCP.
Nevertheless, unusual normal state properties of these materials above their dome of superconductivity imply that a magnetic QCP may remain a viable concept even below T c in various classes of unconventional superconductors, such as those based on Fe and Cu as well as heavy fermion compounds, in which there is an intricate interplay among intertwined order parameters [11][12][13][14]. More definitive evidence for the connection between quantum criticality and unconventional superconductivity may come from experiments showing that a superconducting phase is pinned to or detached from tunable QCPs [15]. When a QCP is moved, as sketched in Fig 1(b), the superconducting phase will be pinned to the tuned QCP if Cooper pairing is produced by the critical quantum fluctuations, while it can detach from the QCP if the two phenomena are independent of each other, therefore providing a stringent test to resolve the potential relationship between QCP and unconventional superconducting state in strongly correlated systems.
CrAs, with MnP orthorhombic structure, orders in a non-collinear helimagnetic structure below T N =260 K, which is accompanied by a discontinuous lattice expansion along the crystalline baxis and contraction along a-and c-axes [16 -18]. The crystal structure remains unchanged through the transition but its cell volume, dominated by expansion along b, is larger below 260 K.
With initial applied pressure, the coupled magnetic and structural transitions move to lower temperatures. Though weak diamagnetism and zero resistance appear already at pressures near 3 kbar, the highest T c occurs near the critical pressure of 8 kbar where the coupled magnetic/structural transition is projected to zero Kelvin and electrical resistivity deviates from the Landau-Fermi liquid T 2 dependence, indicating a helical AFM QCP hidden below the dome of pressure-induced superconductivity in CrAs [19][20][21][22]. Recent neutron scattering suggest that the non-Fermi liquid behavior arises from a nearly second-order helical magnetic phase transition that is accompanied by a first-order isostructural transition [22]. Even though these experiments provide circumstantial evidence for a close relationship between quantum criticality and superconductivity, the complexity of simultaneous magnetic and isostructural transitions and the possibility of electronic phase separation in polycrystalline CrAs [23] cloud a straightforward connection between criticality and superconductivity.
Here we show that a projected critical point in CrAs is successfully shifted by Al chemical substitution and the pressure-induced superconducting phase is detached from the tuned magnetic critical point. Slight Al-doping increases T N from 260 K to 270 K but pressure rapidly suppresses T N of Al-CrAs to zero Kelvin near 4.5 kbar (=P C ), giving a suppression rate that is nearly two times faster than that of pure CrAs. The residual resistivity as well as the temperature coefficient of resistivity peak near P C shows that the projected critical point is shifted from 8 kbar for pure CrAs to 4.5 kbar by Al doping. Contrary to the tunable critical point, the maximum T c of pressure-induced SC state remains near 8 kbar, showing that the SC dome is detached from the shifted QCP. These discoveries evidence that superconductivity in CrAs is produced in spite of the QCP, not because of it. The unambiguous demonstration of detached superconductivity from the QCP in Al-CrAs illustrates that tuning via non-thermal control parameters can provide an alternative route to probe the intricate relationship between a hidden critical point and surrounding superconductivity. for pure and Al-doped CrAs, respectively, which arises from the coincidence of AFM and isostructural volume expansion transitions [19,20]. Aluminum substitution not only increases T N by 10 K, but also increases ρ at 290 K from 169 to 197 µΩ⋅cm, due in part to the increase in disorder and higher transition temperature. Likewise, the residual resistivity ρ 0 , estimated by extrapolating ρ(T) from base temperature to 0 K, increases from 1.4 to 6.0 µΩ⋅cm and the residual resistivity ratio (RRR) decreases from 120 to 33, again signifying that disorder from Al substitution contributes significantly to the electron scattering. Concomitant with the resistivity results, as shown in Fig. 2(d), magnetic susceptibility measurements find that T N increases from 260 K for pure CrAs to 270 K for Al-CrAs, demonstrating that Al substitution is of bulk nature.

Results and discussion
Though the larger cell volume of Al-CrAs is consistent with its higher magnetic/structural transition temperature, Al-doping is not a simple negative chemical pressure effect. Figure 3(a) and (b) comparatively show the pressure dependence of the magnetic and superconducting phase transition temperatures of Al-doped and pure CrAs, respectively. T N of Al-CrAs decreases gradually with initial pressure but drops rapidly for pressures higher than 4 kbar, similar to what happens in pure CrAs near 7 kbar. If Al were acting solely as a negative chemical pressure, the crossover to a steep decrease in T N should occur at a higher pressure in Al-CrAs. Nevertheless, a smooth extrapolation of T N (P) indicates that the magnetic transition reaches T= 0 near 4.5 kbar (= P C ), which is nearly half the critical pressure of 8 kbar for CrAs, even though its T N is 10 K lower.
The increase in T N from 260 to 270 K with Al-doping might reasonably be expected from a negative pressure effect because of the larger cell volume and particularly expanded b-axis.
Since the Cr-3d states are more localized due to reduction in p-d mixing between Cr 3d and the anion p states, the already sizeable ordered moment (1.73 µ B ) in CrAs should increase with Al doping as should T N [24]. The substantially lower P C , however, indicates that the effect of Aldoping cannot be explained simply by a reduction in p-d mixing. As with itinerant antiferromagnetism in V-doped Cr, pressure and chemical doping play very different roles due both to impurity scattering and to changes in the electronic structure [25,26]. This is likely as well to be the case in Al-CrAs, where Al introduces both disorder and additional carriers as V does in Cr. Interestingly, a few atomic percent V in Cr also substantially decreases the critical pressure of antiferromagnetic order and induces a very rapid drop in T N (P) as the critical pressure is approached [26]. Clearly, experimental and theoretical studies of the pressure-dependent electronic structure will be important to understand the microscopic role of Al-doping in CrAs.
Color contour plots of the temperature and pressure variation in ρ of Al-doped and pure CrAs are respectively depicted in Fig. 3  , respectively. At lower pressures (P < P C ), n is close to 2, as expected for Landau-Fermi liquid behavior. With increasing pressure, the exponent n of Al-CrAs sharply drops close to 1.5 at 4.5 kbar and gradually increases to 1.83 at 24.1 kbar. The non-Fermi liquid behavior can be ascribed to scattering by critical fluctuations associated with the AFM QCP at P C . Underpinning the presence of the QCP, ρ 0 shows a sharp peak and A abruptly increases by a factor of 30 at the critical pressure P C (=4.5 kbar), as shown in Fig. 4(b) and (c). In pure CrAs, analysis of the low-T resistivity by power-law fits also shows that the non-Fermi liquid behavior appears near 8 kbar, the pressure across which there occurs a change in the slope of resistivity, and the critical region extends over a wide pressure range (P > 8 kbar) -see Fig. S3 in the SI. The fact that non-Fermi liquid behavior is observed near the tuned QCP of Al-CrAs underscores that the strange metallic behavior near the optimal pressure (P c ) in CrAs is originated from the critical magnetic fluctuations associated with the AFM QCP veiled by the dome of SC phase.
The SC transition temperature T c , defined by the point of zero resistance in Al-doped CrAs, starts to appear for pressures above the critical pressure P C (=4.5 kbar), as marked by the triangles in Fig. 3(a). With further increasing pressure, T c reaches a maximum near 10 kbar, and gradually decreases to 1.27 K at 24.1 kbar. When compared with pure CrAs (see Fig. 3 where the zero-resistance state starts to appear at 3 kbar, the pressure required to induce an initial zero-resistance state is shifted to a higher pressure in Al-CrAs. However, the pressure-dependent dome of T c (P) is similar to that of pure CrAs, and a broad maximum appears near 10 kbar for both compounds. We note that the zero-resistance state starts to appear deep in the AFM state for pure CrAs, while it exists only after the AFM phase is completely suppressed for Al-CrAs.
The superconductivity detached from a quantum critical point in Al-CrAs is in stark contrast with those unconventional superconductors where non-Fermi liquid behaviors appear above the optimal doping or pressure at which the highest T c appears. Superconductivity in pure CrAs has been proposed to be mediated by magnetic fluctuations because of its proximity to a possible AFM QCP [19][20][21]. The absence of a coherence peak in the spin-lattice relaxation rate T 1 and a T 3 dependence of 1/T 1 below T c are consistent with unconventional superconductivity in CrAs [21]. The observations of both six-fold and two-fold symmetric components in the field-angle dependent upper critical field of CrAs support this conclusion and further argue for odd-parity spin-triplet pairing [27]. When CrAs is doped with Al, however, the SC dome as well as the maximum T c at the optimal pressure near 10 kbar are independent of the disorder even though the residual resistivity increases from 1.4 to 6.0 µΩ⋅cm with Al doping. This robustness of the superconductivity against introduction of non-magnetic impurities seems at odds with simple triplet superconductivity that is easily destroyed by any type of impurities [28]. Further, muonspin rotation measurements of CrAs under pressure find scaling of the superfluid density, n s ∝T c 3.2 , which is consistent with conventional phonon pairing [23]. The peculiar band structure of CrAs, where its possible non-trivial band crossing is protected by the non-symmorphic crystal structure, may be important in unraveling the mechanism and nature of superconductivity in this fascinating material [23,29]. Additional study that can give direct information on the SC gap, such as point contact spectroscopy and field-directional specific heat measurements under pressure, will be important to resolve these contradicting results. Al-CrAs. The representative raw data that Fig. 5 is based on are plotted in Fig. S4 in the SI. In the high-T regime, strong electron scattering (red) is observed mainly outside the magnetic phase boundary due to thermally induced critical fluctuations for both compounds. In the low-T regime of CrAs, as shown in Fig. 5(b), the area of enhanced scattering extends over the high-pressure regime (P>P c ), which is consistent with recent neutron scattering and NQR results that showed abundant magnetic fluctuations in the normal state near the T c maximum pressure [21,22]. There completely disappears at 6.9 kbar [22]. This discrepancy on the spin reorientation transition was then ascribed to the usage of single crystals in their study. As shown in Fig. 5(b), enhanced resistivity extends across the critical pressure of 8 kbar, but it is difficult to find any clear signature that may indicate the presence of an additional phase transition near 9.4 kbar -see also

Material and methods
Single crystals of pure CrAs and CrAs doped with 0.7% Al were grown out of a Sn-flux, as described elsewhere [30]. Powder X-ray diffraction was measured with a Rigaku miniflex-600 (Cu K-α source, λ ~ 1.5406 Å) and the data were refined in the Fullprof program to determine the lattice constants at room temperature. A conventional four-probe technique was applied to measure electrical resistivity of needle-shaped CrAs with current flow in the needle along its elongated crystalline a-axis. At ambient pressure, the residual resistivity ratio (RRR) of Al-CrAs is approximately 33, which is lower than that of pure CrAs. Pressure measurements to 24.1 kbar were performed in a hybrid Be-Cu/NiCrAl clamp-type pressure cell with silicone oil as the pressure medium. The pressure-dependent superconducting transition temperature of a Pb manometer was used to determine the pressure [31]. Resistivity under pressure was measured in CCR (closed cycle refrigerator) and 3 He refrigerators for relatively high-(2.8~305 K) and low-        In this supplement, we present additional data that support results in the main text.