Evolution of Magnetic Double Helix and Quantum Criticality near a Dome of Superconductivity in CrAs

At ambient pressure CrAs undergoes a first-order transition into a double-helical magnetic state at TN = 265 K, which is accompanied by a structural transition. The recent discovery of pressure-induced superconductivity in CrAs makes it important to clarify the nature of quantum phase transitions out of its coupled structural/helimagnetic order. Here we show, via neutron diffraction on the single-crystal CrAs under hydrostatic pressure (P), that the combined order is suppressed at Pc ~ 10 kbar, near which bulk superconductivity develops with a maximal transition temperature Tc ~ 2 K. We further show that the coupled order is also completely suppressed by phosphorus doping in CrAs1-xPx at a critical xc ~ 0.05, above which inelastic neutron scattering evidenced persistent antiferromagnetic correlations, providing a possible link between magnetism and superconductivity. In line with the presence of antiferromagnetic fluctuations near Pc (xc), the A coefficient of the quadratic temperature dependence of resistivity exhibits a dramatic enhancement as P (x) approaches Pc (xc), around which Res(T) has a non-Fermi-liquid form. Accordingly, the electronic specific-heat coefficient of CrAs1-xPx peaks out around xc. These properties provide clear evidences for quantum criticality, which we interpret as originating from a nearly second-order helimagnetic quantum phase transition that is concomitant with a first-order structural transition. Our findings in CrAs highlight the distinct characteristics of quantum criticality in bad metals, thereby bringing out new insights into the physics of unconventional superconductivity such as occurring in the high-Tc iron pnictides.

The study of unconventional superconductivity (SC) is one of the most vigorous research fields in condensed matter physics. [1] Among the families of unconventional superconductors are the celebrated cuprate [2] and iron-based [3] high-Tc superconductors as well as the heavyfermion [4] low-Tc ones. Although the underlying mechanisms responsible for the observed SC remains elusive, extensive investigations over the last decades have provided evidences for quantum criticality as a candidate mechanism for these diverse classes of superconductors. [5][6][7] The phase diagrams of these materials often feature a superconducting dome situated adjacent to a magnetically ordered state, with the optimal superconducting transition temperature (Tc) located near a quantum critical point (QCP). The close proximity of SC to magnetic order makes it important to elucidate the nature of magnetism. In the iron-based superconductors, for instance, the mechanism for SC might differ significantly depending on the description of magnetism, [8] which can arise from either the Fermi-surface nesting in the weak coupling limit or the incipient Mott physics when the electron correlations are strong. In addition, the fluctuations associated with the QCP provide a means to understand the non-Fermi-liquid behavior in the normal state.
The properties that deviate noticeably from the conventional Fermi-liquid forms include, [7,9] e.g. the resistivity   T n with 1 < n < 2. In this work, motivated by the recent discovery some of us made of pressure-induced SC in CrAs, [10] we explore the role of magnetic QCP in this system.
CrAs crystallizes in the orthorhombic MnP-type structure at ambient conditions. It is an antiferromagnetic metal and develops a double helical magnetic order below TN  265 K. [11,12] Interestingly, the antiferromagnetic order at TN was found to be accompanied by a first-order isostructural transition, which is characterized by a sudden expansion of b axis by ~4% and a slight (< 1%) reduction of a and c. [13] Neutron powder diffraction revealed a large ordered Cr moment of ~1.7 μB below TN. [12] The nature of the first-order antiferromagnetic transition in CrAs, in particular, the strong anisotropic lattice changes across TN, have not been well understood. Earlier studies have shown that the coupled structural/helical order can be readily suppressed by either the application of external pressure [14] or chemical substitutions, [15] e.g. 5% P for As in CrAs1-xPx, providing a model system for studying the quantum criticality associated with the destruction of a helimagnetic order. However, the electrical transport properties for CrAs and related compounds were rarely explored because these samples always develop many cracks or are even shattered into pieces when crossing TN due to a large volume change. Besides, all earlier studies were performed on the polycrystalline samples. These factors together make the plausible magnetic quantum phase transition unexplored. Work by some of us [16] obtained the first CrAs single crystal and characterized in detail its intrinsic resistivity and magnetic susceptibility. More recently, we found CrAs as the first superconductor among Cr-based compounds. [10] The transport measurements also indicated that the development of SC is accompanied by the reduction and eventual suppression of TN. The findings were later confirmed by Kotegawa et al. [17] and their subsequent 75 As-nuclear quadrupole resonance (NQR) study evidenced no coherence effect in the nuclear spin relaxation rate 1/T1, thus suggesting an unconventional nature for the observed SC in CrAs. [18] These observations make CrAs an ideal material for in-depth studies on how the coupled structural and helimagnetic orders are tuned to the critical point and how the magnetism is coupled to the unconventional SC developed nearby. In combinations of elastic and inelastic neutron scattering, resistivity, and specific-heat measurements on undoped CrAs and P-doped CrAs1-xPx single crystals and powders, we have studied systematically the evolutions of static helical order and dynamic spin correlations as well as the electronic properties of CrAs when its coupled structural/helical order is suppressed by either external pressure or the chemical substitutions. Our results show explicitly the suppression of structural and magnetic transitions by external or chemical pressure, and reveal the persistent antiferromagnetic correlations above TN and xc (Pc) where the long-range magnetic order is absent. In addition, our results provide clear evidences for magnetic quantum criticality around xc = 0.05 and Pc  10 kbar, near which the Tc exhibits a broad maximum. We argue that these results can be understood in terms of a proximity to a magnetic QCP [19] tuned by the degree of the incipient localization of the Cr-3d states, [20] possibly via an orbital-selective Mottness mechanism, [21,22] which have been proposed for the iron-based high-Tc superconductors.
CrAs and doped CrAs1-xPx single crystals used in this study were grown out of the Sn flux as described elsewhere. [16] Single-crystal neutron diffraction measurements were performed on a triple-axis spectrometer HB-1 installed at the High Flux Isotope Reactor (HFIR) of Oak Ridge National Laboratory (ORNL). The horizontal collimator sequence was 48'-opensample-open-240'. The fixed incident neutron energy was 13.5 meV with the energy resolution of ~ 1.5 meV. Since the crystals have needle-like shape with the longest dimension along the a axis, eight pieces of CrAs single crystals were co-aligned in the pressure cell so that the (0KL) scattering plane can be measured. The high pressures were generated with a self-clamped pistoncylinder cell made of a Zr-based amorphous alloy [23]. Fluorinert was chosen as the pressure transmitting medium (PTM). The pressure inside the piston-cylinder cell was determined at room temperature by measuring the lattice constant of a comounted NaCl crystal. Cryogenic temperatures down to 4 K were realized by attaching the pressure cell to a closed-cycle 4  CrAs crystals were aligned in the pressure cell with H//a. Since the magnetic signal from the pressure cell is in the same level as that from CrAs samples, we employed the background record/subtract function in SQUID, which was found to be essential in order to follow correctly the magnetic transition of CrAs under pressure. The resistivity of CrAs single crystal under pressure was measured with the standard four probe method; the pressures up to 21 kbar and 70 kbar were generated by employing a piston-cylinder-type cell [24] and a "Palm" cubic anvil apparatus [25], respectively. In both cases, glycerol was used as PTM. Resistivity and specific heat for the CrAs1-xPx single crystals at ambient pressure were measured by the Physical Property Measurement System (PPMS, Quantum Design).  Fig. 1(a, b), with increasing pressure the transition temperature decreases and the temperature range where the two phases coexist is enlarged. In addition, the temperature window of the thermal hysteresis also becomes wider with pressure ( Fig. S1). At 6.9 kbar, the high-temperature paramagnetic phase is terminated at 70 K upon cooling. At 9.1 kbar, the high-temperature phase is stabilized down to the lowest temperature. However, the lowtemperature magnetically ordered phase still exists; its volume fraction is estimated to be ~ 33% at 4 K from the relative intensity of the (020) nuclear Bragg peaks of these two phases (see Fig.   S2). Eventually, the structural transition is completely eliminated at 12 kbar. The pressure dependence of the transition temperature for CrAs determined from the present structural study agrees well with that based on the resistivity measurements as shown below.
A characteristic feature of the low-temperature phase below TN is that both lattice parameters b and c are nearly temperature independent, in contrast to the normal thermal expansion observed at T > TN. In addition, the expansion of b axis when cooling across TN becomes enhanced with pressure. In contrast for the c axis, the structural anomaly when cooling across TN changes from a sudden drop at 1 bar to a slight expansion at P  3.5 kbar, Fig. 1 Notably, this latter observation on the single-crystal sample is different from what observed on the polycrystalline samples that always show a sudden contraction across TN. [26] The chemical pressure applied via P substitution for As in CrAs1-xPx plays the same role as the external pressure in suppressing the structural transition of CrAs. As shown in Fig. 1 both the transition temperature and the temperature dependence of b axis for x = 0.035 are similar to those of CrAs under 3.5 kbar, except that the transition is sharper without having a two-phase coexistence region for the P-doped samples. However, the c axis response at TN differs for the external and chemical pressures: it is expanded slightly under P = 3.5 kbar, but contracted suddenly for x = 0.035. For the critical composition xc = 0.05, there is no transition down to the lowest temperature, and both b and c decrease gradually with temperature, which are similar to the behaviors of CrAs under 12 kbar. As noted above, the two-phase coexistence region observed under external pressures is absent in the P-doped samples, which exhibit a sharp transition as in the undoped CrAs at ambient pressure. This observation indicates that the external pressure is not applied homogeneously through the whole sample, which might explain the filamentary SC observed in the pressure range 3 < P < 7 kbar. [10] However, superconductivity has not been realized in these P-doped samples presumably because the substantial disorders introduced via chemical doping hinder the Cooper pairing, [10] which will be discussed in detail below.
In order to follow directly the evolution of helimagnetic order for CrAs, we examined a magnetic Bragg peak at (0, 0, 2-) as a function of temperature and pressure as well as the P doping. The data are summarized in Fig. 2. It has been well established that the magnetic structure of CrAs at ambient pressure is characterized by two coupled interpenetrating helical sublattices with the propagation vector (0, 0,  ). Instead of new fundamental reflections, such a double helical order of CrAs is featured by some double satellite reflections to the fundamental ones, such as (0, 0, 2). As shown in Fig. 2(a), our results at ambient pressure confirmed the double helical structure with  = 0.356 in reciprocal lattice unit (r.l.u.) at 4 K, and that  becomes larger with increasing temperature, in excellent agreement with the previous studies on powder samples. [12] With increasing pressure, the magnetic Bragg peak broadens and diminishes gradually with a slight decrease of . As shown in Fig. 2(b), no magnetic signal can be observed down to 4 K at P  9.1 kbar. All these results are in general consistent with our previous resistivity measurements under pressure, [10] but provide direct evidence for the disappearance of coupled structural/helical order.
Recently, Keller et al. [27] and Shen et al. [26] performed high-pressure neutron diffraction experiments on polycrystalline CrAs samples and found evidences in support of the possible connection between magnetism and SC. In particularly, Shen et al. [26] reported a pressure-induced spin reorientation from the ab to the ac plane with an abrupt reduction of  at around 6 kbar, above which  is well below 0.3. However, our results in Fig. 2(d) shows a  = 0.334 at 6.9 kbar, indicating the absence of such transition at least up to 6.9 kbar for the singlecrystal sample. Such discrepancy might arise from the different forms of samples studied, i.e. single crystal versus polycrystalline samples. As noted above, the lattice constant c indeed displays a distinct response to pressure for the single crystal and polycrystalline samples. In addition, a recent high-pressure NQR study on CrAs single crystals also found no clear signature of a change in the magnetic structure under pressure. [18] If we assume that the helical structure does not change other than  with pressure and normalize the magnetic signal by the (020) nuclear Bragg intensity, the ordered magnetic moment at 6.9 kbar, which is proportional to the square root of the magnetic intensity, is estimated to be ~ 65% of that at ambient pressure. This estimated moment is between those two polycrystalline results (~25% [26] and ~85% [27]).
In addition to reduce the transition temperature, the P doping also changes the modulation vector of the helical order in a similar manner as the external pressure.  respectively. The spectrum at 100 K shown in Fig. 3(a) can illustrate the low-energy magnetic excitations clearly. The shape of the magnetic excitation spectra does not show strong temperature dependence below TN. Only the intensity distribution in energy changes, which follows the Bose factor. Note that there exist phonon scattering above Q > 2.5 Å -1 and also in the energy ranges of 10  E  25 meV and 28  E  38 meV. We found that the magnetic excitations at low-Q are extended up to ~110 meV, as shown in Fig. 3(b). Above TN, very broad peaks remain at very low-Q and ~1.6 Å -1 , Fig. 3(c), indicating that the short-range antiferromagnetic correlations survive above TN, but the steep incommensurate magnetic excitations are broadened.
For the nominal CrAs0.94P0.06 without static magnetic order, a very broad magnetic excitation at Q ~1.0 Å -1 was observed. The temperature dependence of S(|Q|), which is integrated with respect to energy between 4.0 and 7.5 meV, is shown in Fig. 3(e). The broad peak at ~1.0 Å -1 gradually increases with increasing temperature, which follows the Bose factor. This broad peak feature at also close to pure CrAs. We thus tend to believe them originating from the phase separated CrAs.
In complementary to the above neutron diffraction study, we also tracked down the pressure dependence of bulk magnetic properties from dc magnetic susceptibility (T) under pressure. Except the (T) curve at 1 bar, all curves shown in Fig. 4 are shifted above for clarity.
In agreement with the previous measurement at ambient pressure, the (T) curves at P < 3 kbar increases with temperature and exhibit sudden jump at TN. For P > 3 kbar, the magnetic transition is manifested as a continuous cusp-like anomaly and broadens upwards with increasing pressure. For P > 6 kbar, it is hard to define TN and, more interestingly, the (T) display a Curie-Weiss-like temperature dependence, which suggests the presence of strong spin fluctuations when the long-range magnetic order gets ready to collapse, in agreement with the above INS results.
Based on the above elastic and inelastic neutron scattering as well as high-pressure magnetic susceptibility measurements, we have not only followed directly the coupled structural/helimagnetic order as a function of external/chemical pressure, but also evidenced persistent short-range antiferromagnetic correlations above TN or xc when the static order is absent, thus providing strong evidences in favor of unconventional magnetism mediated SC emerged near a helimagnetic QCP.
To search for further evidences of quantum criticality, we have performed high-pressure resistivity measurements on CrAs single crystal up to 70 kbar and the resistivity and specificheat measurements on CrAs1-xPx single crystals at ambient-pressure. The low-temperature resistivity data of CrAs under various pressures up to 70 kbar are shown in Fig. 5(a, b). The present high-pressure resistivity study considerably extends the pressure range of previous studies, [10] , [17] allowing us to construct a complete superconducting phase diagram shown in  Fig. 5(c, d) in the form of (=-0) versus T 2 . As can be seen, (T 2 ) under P < 3 kbar follow nicely the linear behavior below 10 K. A clear deviation from the linear dependence is evidenced for P > 3 kbar. The strongest deviations occur in the intermediate pressure range 5 < P < 13 kbar. The sublinear behavior in  versus T 2 plot indicated a resistivity exponent n < 2 in  ~ T n . With increasing P > 20 kbar, the linear dependence is gradually restored. We thus have fitted the resistivity data below 10 K with the general power-law formula: (T) = 0+ BT n , as shown by the broken lines in Fig. 5(a), and extracted the exponent n, which contains valuable information about the nature of the metallic state. The Landau Fermi-liquid (FL) theory predicts an n = 2 for correlated metals, while the observation of n < 2 has been taken as a common signature for non-Fermi-liquid (nFL) metals near a QCP. As shown in Fig. 5(e), the application of high pressure that suppresses TN also converted the behavior from FL to nFL; n deceases quickly to a minimum of 1.3(1) at ~ 6 kbar, slightly lower than Pc. For P > 6 kbar, the exponent n increases gradually and reaches ~1.82 at 70 kbar. Although the above analysis evidenced a dramatic enhancement of the B coefficient around Pc, the variation of exponent n prevents a quantitative comparison. We therefore perform a linear fit to the  versus T 2 plot in the low-temperature limit, Fig. 5(c, d), which yields the T-quadratic coefficient A defined by -0 = AT 2 . As shown in Fig. 5(g), the A coefficient displays a pronounced peak centered at Pc. Since the A coefficient of FL is related to the effective mass of charge carriers via A  (m * /m0) 2 , the significant enhancement of A, especially on the side of P < Pc, signifies a dramatic enhancement of the effective mass associated with the suppression of helimagnetic order. The observations of nFL behavior and dramatic enhancement of the A coefficient for CrAs near Pc thus provide important clues for the presence of magnetic quantum criticality associated with the helimagnetic order. [28,29] This point is further supported by the P-doped CrAs1-xPx samples. Fig. 6 summarizes the resistivity and specific-heat results on the CrAs1-xPx samples. As can be seen in Fig. 6(a), the first-order structural/helical order of CrAs manifested as a sudden change in resistivity is gradually suppressed to lower temperatures by P doping, and disappears completely in the x = 0.05 sample, in agreement with the structural data shown in Fig. 1(c, d). It should be noted that the resistivity curves for the 0 < x <0.05 samples exhibit a strong jump, instead of a drop as seen in CrAs, at TN upon cooling. As a result, all these P-doped samples are featured by a large residual resistivity 0 in the range of 40-70  cm, which should be attributed to presence of chemical disorders and/or strains induced by the large volume change at the structural transition. We have shown in our previous study [10] that the pressure-induced SC in CrAs is very sensitive to the residual resistivity 0, and bulk SC can only be achieved on highquality CrAs single crystals with 0  1-2  cm, in which the electron mean free path lmfp is large than the superconducting coherent length . Such a criterion, however, cannot be fulfilled for these P-doped samples having a very large 0, which can rationalize the absence of SC for these P-doped samples. Nonetheless, characteristics of QCP are observed near xc. As seen in Figs. 6(b, c, d), nFL behavior with a resistivity exponent n  1.5 and a dramatic enhancement of the Tquadratic coefficient A were clearly evidenced at the critical xc from the analysis of lowtemperature resistivity data. Moreover, the advantage of accessing the magnetic QCP at ambient pressure for these P-doped samples is that it allows us to probe directly the electronic properties near the QCP via the specific heat C(T) measurements. The low-temperature C(T) of CrAs1-xPx single crystals are plotted in Fig. 6(e) in the form of C/T vs T 2 . A linear fit has been applied on these data to extract the electronic specific-heat coefficient , which was plotted in Fig. 6(g) together with the A coefficient. As can be seen clearly, both quantities exhibit significant enhancement near xc, thus providing unequivocal proof for the presence of magnetic QCP in CrAs.
In order to understand how quantum criticality can arise from the coupled magnetic/structural transition, we carry out an analysis within the framework of Ref. [20]. CrAs has a large room-temperature resistivity, [16] which reaches the Ioffe-Regel limit. It is therefore a bad metal, just as the parent iron pnictides [30]. Consequently, its single-electronic spectrum can be divided into an incoherent part responsible for the quasi-localized moment and a coherent part near the Fermi energy for metallic conduction. [21] A magnetic quantum phase transition may then arise when the spectral weight of the coherent part increases to certain threshold at the expense of the incoherent part. [20] To make further progress, we model the exchange interactions in a Heisenberg form, = ∑ • , , where Jij contains the leading exchange couplings up to fourth nearest neighbors [31] denoted as Ja, Jb, Jc1, and Jc2, as illustrated in Fig.7(a). The ground-state phase diagram is shown in Fig. 7(b). The double-helical magnetic order is stabilized between the two hyperbola in the second and fourth quadrants. With the helical wavevector  = 0.353, the phase angle for the double-helical order  ~ /3 arises when Jc1/Ja ~ -0.52 and Jc2/Ja ~ 7.01, as shown in Fig. 7(c), indicating a strong bond overlap along the c axis. Because Jc2 is considerably stronger than Jc1 and Ja, the band maximum of the magnetic excitations at low-Q that are extended up to ~110 meV in the INS of CrAs shown in Fig. 3(b) could be roughly expressed as 2SJc2, where S is the spin value. Since the effective S ~ 0.85, Jc2 is estimated to be ~65 meV.
Both  and  vary slightly in the magnetic phase, which suggests that the exchange couplings are sensitive to lattice distortions. We model this through a Ginzburg-Landau free energy function: where m and Δ respectively refer to the helimagnetic order parameter and the unit-cell volume change. It follows from this function that the finite-temperature magnetic and structural transitions are either both first-order or both second-order. This leads to two possible types of the phase diagram for the pressure tuning of the coupled magnetic/structural transitions, as shown in Fig. 7(d). The zero-temperature magnetic transition is treated by a generalization of the Ginzburg-Landau function to an action, in which the order parameters depend on both wavevector and frequency. A damping term for the magnetic order parameter occurs in this action, reflecting the coupling of the local moments to the coherent itinerant electrons. This term makes it possible for the magnetic quantum phase transition to be nearly second order [20], thereby yielding a large dynamical range for magnetic quantum criticality. In CrAs, the nearly continuous nature of the magnetic transition at T = 0 is reflected in the gradual reduction of the jump in the dc magnetic susceptibility at TN as pressure is increased, Fig. 4. The structural transition, however, can remain to be first order. This is consistent with the prominent persistence under pressure of the jump in the lattice constant and the thermal hysteresis reported here, as well as the apparent first-order behavior observed in the NQR measurements. [18] Our analysis encodes the bad-metal characteristics of CrAs. To further highlight this, it is instructive to compare CrAs with Cr, a good metal with a room-temperature resistivity far below the Ioffe-Regel limit. None of the characteristics of the quantum criticality discussed above for CrAs --viz. the divergent tendency of both the resistivity A coefficient and the effective carrier mass, the nFL form of the resistivity, and the emergence of SC in its vicinity --occurs near the magnetic quantum phase transition in Cr under either hydrostatic or chemical pressure. [32][33][34] By contrast, these same features have been observed in P-doped iron arsenides [29,35], which are also bad metals with strong correlations. Our results in CrAs, therefore, highlight the distinct features of quantum criticality in bad metals, including its ability to drive SC. This provides new insights into the essential role the bad metal behavior plays in the emergence of unconventional SC, which appears also important in other correlated superconductors such as the iron pnictides. [36] The bad metal behavior in CrAs indicates that the electron correlations play a crucial role in this material. Given the multiorbital character of the compound, an important ingredient responsible for our results is the orbital-selective Mottness (OSM), which has been actively discussed recently in the iron pnictides. The OSM in the case of CrAs is characterized by three prominent features: (i) it remains metallic in the whole temperature range, (ii) it acquires below TN a considerably large localized magnetic moment 1.7 B, which is yet smaller than the fully localized value of 3B for Cr 3+ , (iii) its lattice constants undergo a strong anisotropic change at TN with the abrupt expansion of the b-axis upon cooling below TN. These observations can be rationalized by considering the distinct localized/itinerant characters of Cr-3d orbitals, which are split into doubly degenerate eg and triply degenerate t2g states in an octahedral crystal field. Since the eg orbitals are not filled for Cr 3+ , the t2g orbitals (dxy, dyz, and dzx) are mainly involved for the observed electrical and magnetic properties. As discussed previously, [11,14] dxy orbitals that are directed along the a axis with the shortest Cr-Cr nearest-neighbor distance should form a broad band below the Fermi level, EF, while the dyz and dzx orbitals form narrow bands near the Fermi level and their bandwidths and positions vary sensitively depending on change of lattice constants. First-principles calculations [37] on the series of CrX (X = P, As, Sb) indeed found a systematic reduction of bandwidth near EF (or enhancement of magnetism) with increasing the b axis (the size of X anion). This is also consistent with the observation of sudden expansion of b axis below TN that gives rise to a large localized moment of 1.7 B in CrAs. It is such an intimate correlation between structure and orbital-dependent electronic states that enables pressure to be a very effective knob to tune the system towards the electronic instability and in favor of superconductivity. Such an orbital dependent correlation effect is quite similar with the iron pnictides. [21,22] In summary, we have performed a comprehensive elastic and inelastic neutron scattering study on the CrAs under hydrostatic pressures up to 12 kbar and under chemical pressure via P substitutions for As. Our results provide evidence for quantum criticality associated with the double helical magnetic order, and its association with a diverging carrier mass and an emergent superconductivity.