Tuning the distance to a possible ferromagnetic quantum critical point in A2Cr3As3

TL;DR

This study investigates how changing alkali elements in A2Cr3As3 compounds tunes the system toward a ferromagnetic quantum critical point, affecting superconductivity and revealing point nodes in the gap function.

Contribution

It demonstrates the tuning of A2Cr3As3 compounds toward a ferromagnetic QCP and links this to changes in superconducting properties, a novel insight in Cr-based superconductors.

Findings

Approaching FM QCP correlates with increased Tsc in Na2Cr3As3.

Superconducting gap exhibits point nodes, indicated by T^5 dependence of 1/T1.

Superconductivity is enhanced away from the QCP, contrasting AFM systems.

Abstract

Although superconductivity in the vicinity of antiferromagnetic (AFM) instability has been extensively explored in the last three decades or so, superconductivity in compounds with a background of ferromagnetic (FM) spin fluctuations is still rare. We report 75As nuclear quadrupole resonance measurements on the A2Cr3As3 family, which is the first group of Cr-based superconductors at ambient pressure, with A being alkali elements. From the temperature dependence of the spin-lattice relaxation rate (1/T1), we find that by changing A in the order of A=Na, Na0.75K0.25, K, and Rb, the system is tuned to approach a FM quantum critical point (QCP). This may be ascribed to the Cr2-As2-Cr2 bond angle that decreases towards 90 degrees, which enhances the FM interaction via the Cr2-As2-Cr2 path. Upon moving away from the QCP, the superconducting transition temperature Tsc increases progressively up to 8.0 K in Na2Cr3As3, which is in sharp contrast to the AFM case where Tsc usually shows a maximum around a QCP. The 1/T1 decreases rapidly below Tsc with no Hebel-Slichter peak, and ubiquitously follows a T5 variation below a characteristic temperature T*=0.6 Tsc, which indicates the existence of point nodes in the superconducting gap function commonly in the family. These results suggest that the A2Cr3As3 family is a possible solid-state analog of superfluid 3He.

Equations

$$\frac{(T_1{)}_{T_{\mathrm{sc}}}}{T_1}=\frac{2}{k_{B}T}\int N_{\mathrm{s}}(E{)}^2(1+\frac{{△}^2}{E^2})f(E)(1-f(E))dE,$$

Full text

The interplay between magnetism and superconductivity is a key topic in condensed matter physics. In the past thirty years or so, a large amount of superconductors in proximity to an antiferromagnetic (AFM) ordered phase have been found. In heavy fermionsheavyfermion ; QCP , cupratescuprate , and iron pnictides ironbased superconductivity appears on the verge of antiferromagnetic instability, and the critical temperature $T_{\rm sc}$ usually takes a maximum at the AFM quantum critical point (QCP). However, superconductivity in the vicinity of a ferromagnetic ordered phase is still rare. UGe2 is a ferromagnet, but becomes superconducting under high pressure when the Curie temperature $T_{\rm C}$ is reduced inside the ferromagnetic phase UGe2 . However, superconductivity has not been found when the ferromagnetic order is completely destroyed.

Ferromagnetic interactions can also promote quantum states other than superconductivity. For example, superfluidity in 3He emerges in the background of ferromagnetic spin fluctuations. There are two phases, namely A phase and B phase, in 3He He3 ; ABM1 ; ABM2 ; BW . The A phase is a $p$-wave ABM (Anderson-Brinkman-Morel) state with equal spin pairing (\uparrow$$\uparrow and \downarrow$$\downarrow ) ABM1 ; ABM2 . The B phase is a $p$-wave BW (Balian-Werthamer) state with an additional component 1/$\sqrt{2}$(\uparrow$$\downarrow+\downarrow$$\uparrow) BW . There are point nodes in the gap function in ABM state but the gap is isotropic in BW stateleggett . It has been pointed out that the spin triplet pairing in 3He is induced by FM spin fluctuations ABM2 . Notably, 3He phases are topologically non-trivial, which has received new and intensive interests in the past few years QiandZhang . The B phase of 3He belongs to the so-called DIII topological class class , and the A phase bares similarities to topological Weyl semi-metals. Therefore, searching for a solid state analog of 3He serves to bridge three large research areas: strong correlations, unconventional superconductivity, and topological quantum phenomena.

Recently, a 3$d$-electron system, chromium-based superconductors A2Cr3As3(A = Na, K, Rb, Cs), has been discovered Na2Cr3As3 ; K2Cr3As3 ; Rb2Cr3As3 ; Cs2Cr3As3 . In this family, superconductivity emerges from a paramagnetic state. K2Cr3As3 and Rb2Cr3As3 have a Tsc=6.1 and 4.8 K K2Cr3As3 ; Rb2Cr3As3 , respectively, and Na2Cr3As3 has the highest Tsc = 8.0 K. Resistivity measurement suggests that Cs2Cr3As3 superconducts below $T$=2.2 K Cs2Cr3As3 . Although anomaly at $T$=2.2 K was not found by nuclear quadrupole resonance (NQR) imaiCsCrAs ; Luojununpublished , magnetic susceptibility measurement does confirm bulk superconductivity below $T$=1.2 K Sumiyama . Density function theory (DFT) calculations show that there are three bands across the Fermi level, namely, two quasi one-dimensional (1D) band $\alpha$ and $\beta$, and one three-dimensional (3D) band $\gamma$ ferromagneticfluctuation1 ; magnetism . The $\gamma$ band makes the main contribution to the density of states (DOS).

NQR, penetration depth, muon spin rotation ($\mu$SR), upper critical field Hc2 and specific heat measurements show signatures of unconventional superconductivity imai ; yang ; specific ; penetration ; muSRK2Cr3As3 ; muSRRb2Cr3As3 ; Hc2 ; angularHc2 . Theoretically, spin-triplet superconducting state has been proposed fwave ; pwave ; spintriplet . In the normal state, ferromagnetic spin fluctuations have been found from the Knight shift and spin-lattice relaxation rate (1/T1) measurements in Rb2Cr3As3 yang . Neutron scattering measurements also suggest short-range magnetic order in K2Cr3As3 magneticins . However, how the spin fluctuations evolve with changing A is unknown. In addition, the gap symmetry is still controversial. For example, 1/T1 $\propto$ $T^{4}$ was reported in K2Cr3As3 imai , but 1/T1 $\propto$ $T^{5}$ was found in Rb2Cr3As3 yang . The latter result suggests point nodes in the gap function yang . On the other hand, line nodes were claimed by penetration depth and specific heat measurements specific ; penetration .

In this work, we systematically study the normal and superconducting states of the family A2Cr3As3 by NQR. We find that 1/T1$T$ in the normal state for all compounds can be fitted by Moriya’s theory for 3D FM spin fluctuations SCRtheory . Our results show that on going from A = Na to Na0.75K0.25, K, Rb, the system progressively approaches a possible FM QCP, which may be ascribed to the Cr2-As2-Cr2 bond angle that decreases toward 90∘ in the same order and enhances the FM interaction. In the superconducting state, 1/T1 for all compounds show no Hebel-Slichter peak, and follows a T5 variation below a characteristic temperature T* $\approx$ 0.6Tsc, indicating a common formation of point nodes in the gap function. Our results indicate that A2Cr3As3 shows some properties similar to superfluid 3He.

Polycrystal sample of Na2Cr3As3 was prepared by a low temperature ion-exchange method Na2Cr3As3 , and the others were prepared by solid state reaction method K2Cr3As3 . We crush the samples into powders or cut them into pieces to avoid skin depth problem in the NQR measurements. To protect the samples against air and water vapor, we seal the samples into an epoxy (stycast 1266) tube in an Ar-filled glove box. The T1 was measured by using the saturation-recovery method, and obtained by a good fitting yang of the nuclear magnetization to $1-M(t)/M_{0}=\exp(-3t/T_{1})$, where $M(t)$ is the nuclear magnetization at time $t$ after the single saturation pulse and $M_{0}$ is the nuclear magnetization at thermal equilibrium.

Figure 1(a) and (b) show the crystal structure of the system and the NQR spectra, respectively. Similar to K2Cr3As3imai and Rb2Cr3As3yang , the NQR spectra of (Na0.75K0.25)2Cr3As3 and Na2Cr3As3 also have two peaks originating from two inequivalent As sites. Since the temperature dependence of 1/$T_{1}$ for two As sites is the same imai ; yang , we measured 1/$T_{1}$ for A=Na, Na0.75K0.25 and K at the stronger peak.

As can been seen in Fig. 2, 1/$T_{1}T$ is a constant above $T\sim$150 K and increases with decreasing temperature down to $T_{\rm sc}$ for all samples. For a conventional non-interacting metal, 1/$T_{1}T$ is a constant. Therefore, the results indicate electron correlations. Previous Knight shift measurements in Rb2Cr3As3 found that spin susceptibility increases with decreasing temperature, which indicates that the electron correlation is ferromagnetic in character. DFT calculations also show that the interaction within each Cr sublattice is ferromagnetic ferromagneticfluctuation1 ; magnetism ; pwave . Below we apply the 3D ferromagnetic spin fluctuations theory of Moriya to characterize the spin fluctuations SCRtheory . The 1/$T_{1}T$ can be expressed as 1/$T_{1}T$ = (1/$T_{1}T$)SF + (1/$T_{1}T$)0. The first part originates from spin fluctuations of 3$d$ electrons, and the second part is due to non-interacting electrons. For 3D ferromagnetic spin fluctuations SCRtheory , (1/$T_{1}T$)SF follows a relation of $C/(T+\theta)$, with the parameter $\theta$ describing a distance to FM QCP. The obtained $\theta$ decreases in the order of A=Na, Na0.75K0.25, K, and Rb. The parameter $\theta$=4$\pm$1.5 K for Rb2Cr3As3 and $\theta$= 57$\pm$7 K for Na2Cr3As3.

Figure  3 shows the phase diagram of the family, where $\theta$ and $T_{\rm sc}$ are plotted as a function of the ionic radius of alkali element A. As the ionic radius increases, the parameter $\theta$ decreases, indicating that the system is tuned closer to a possible FM QCP. Concomitantly, $T_{\rm sc}$ decreases. Below we show that the transverse axis of Fig. 3 correlates with ferromagnetic interaction strength. The larger the ionic radius is, the stronger the ferromagnetic interaction is.

As shown in Fig. 4, analysis of the available data shows that the Cr2-As2-Cr2 bond angle $\alpha$ has a linear relationship with the ionic radius, while the Cr1-As1-Cr1 angle $\beta$ is almost a constant. Notably, as the ionic radius increases, $\alpha$ decreases towards 90∘. It was pointed out that the Cr-Cr coupling would be antiferromagnetic according to Goodenough-Kanamori-Anderson rule pwave . However, since Cr is in an averaged valence of 2.3, double exchange interaction through Cr-As-Cr path is possible pwave , which is ferromagnetic. At $\alpha$=90∘, the As-4$p_{x}$ and As-4$p_{y}$ orbitals become degenerated with respect to Cr-3$d$ orbitals, which will maximize the double exchange interaction between the two Cr2 along the $c$-axis via the As-4$p_{x}$ and As-4$p_{y}$ orbitals. Therefore, on going from A = Na to Na0.75K0.25, K, and Rb, an increase in the ferromagnetic interaction can be expected, which drives the system towards a FM QCP. In order to directly access such QCP, we propose to replace A with Ca, Sr or Ba, thereby a long-range ordered phase can hopefully be obtained.

In Fig. 3, one sees that Tsc decreases upon approaching the FM QCP. This is in sharp contrast to the AFM case where Tsc forms a peak around a QCP. Superconductivity near a FM QCP in paramagnetic side was discussed by Fay and Appel in their seminal work appel , but to our knowledge, had not been confirmed thus far. In the AFM case, the pairing interaction is enhanced due to increased quantum fluctuations Monthoux . In the FM case, when it is approached from the paramagnetic side, increased quantum fluctuations also enhances pairing strength appel ; Monthoux . However, Fay and Appel found, based on a random phase approximation, that mass enhancement due to FM spin fluctuations will kill a spin-triplet superconducting state so that Tsc is zero right at ferromagnetic QCP appel . Later on, Monthoux and Lonzarich Monthoux and Wang et al WangZQ calculated Tsc in the strong-coupling limit. They pointed out that mass enhancement and a finite quasiparticle life time act as pair breaking, but found a non-zero Tsc at ferromagnetic QCP. In UGe2, however, no Tsc was found in the paramagnetic side. In a related compound UCoGe UCoGe , although superconductivity survives after ferromagnetic order is suppressed, the ferromagnetic-paramagnetic transition is a first-order phase transition Ishida and thus cannot be directly compared to theories. Our results experimentally demonstrate Supple , for the first time, the evolution of Tsc in paramagnetic side predicted by theories appel ; Monthoux . Looking forward, it would be interesting to experimentally probe the evolution of effective electron mass by London penetration depth at the zero-temperature limit WangCG , for example.

Next, we discuss the properties of superconducting state. In Fig 5(a) are shown the 1/T1 data for the four compounds. There is no coherence peak for all cases. In contrast to the previous report for a K2Cr3As3 sample with a lower $T_{sc}$=5.7 K than our case (6.1 K) imai ; Supple , our data show that K2Cr3As3 exhibits a temperature variation very similar to Rb2Cr3As3. In Fig 5(b), we show the data with reduced scales for the axes. As can be seen there, 1/T1 commonly shows a $T^{5}$ behavior below a characteristic $T$*/Tsc $\approx$ 0.6. This suggests that the gap symmetry is the same for all members of this family.

The 1/T1 in superconducting state can be expressed as

[TABLE]

where $N_{\rm s}(E)$ is the DOS below Tsc, $f$(E) is Fermi distribution function, $\Delta$ is the gap function. When there exist point nodes in the gap function, such as in the ABM model ABM1 ; ABM2 with $\Delta=\Delta_{0}sin\theta e^{i\phi}$, $N_{\rm s}(E)\propto E^{2}$. This results in 1/T1 $\propto$ $T^{5}$ at low temperatures. Previously, Katayama $et$ $al$ found such a $T^{5}$ variation in filled skutterudite superconductor PrOs4Sb12 under pressure T5 . Notably, 1/T1 does not deviate from $T^{5}$ in the lowest temperature even for (Na0.75K0.25)2Cr3As3 where partial substitution of K for Na would induce disorder. Generally, disorders or impurities cause finite DOS at the Fermi level in the case of nodal gap, which will result in a deviation of 1/T1 , which has indeed been observed in cuprates ingapstate . The current result therefore indicates that the superconducting state is not affected by the disorder out of [Cr3As3]∞ tube. Finally, the existence of $T$* is unclear at the moment. A possible reason is multiple-bands superconductivity, as seen in iron-based superconductors Matano . Another possibility is multiple phases arising from internal freedoms associated with spin-triplet pairing as seen in UPt3 multiphase . Clearly, more work is needed in this regard.

In summary, we have performed 75As-NQR measurements on the A2Cr3As3 family. we find that by changing A in the order of A=Na, Na0.75K0.25, K, and Rb, the system is tuned to approach a possible ferromagnetic QCP. We propose that the Cr2-As2-Cr2 bond angle ($\alpha$) that decreases towards 90 degree is responsible for the increase of ferromagnetic interaction. Upon approaching the QCP, the superconducting critical temperature $T_{\rm sc}$ decreases, which is in sharp contrast to the AFM case where $T_{\rm sc}$ usually forms a broad peak around a QCP. In the superconducting state, 1/$T_{1}$ decreases with no Hebel-Slichter peak just below $T_{\rm sc}$, and ubiquitously follows a $T$5 variation at low temperatures, which indicates the existence of point nodes in the gap function commonly in the whole family. Our results indicate that the A2Cr3As3 family is a possible solid-state analog of superfluid 3He. Therefore, further investigations on this family promise to enrich the physics across multiple research areas of electron correlations, unconventional superconductivity and topological quantum phenomena.

We thank G.H. Cao, Y. Haga, J.P. Hu, H. Kontani, K. Miyake, A. Sumiyama, Z.Q. Wang, H.M. Weng, G.M. Zhang and Y. Zhou for helpful discussions, and C.G. Wang for technical assistance. This work was partially supported by NSFC grants (Nos. 11674377, 11634015 and 11774399), MOST grants (Nos. 2016YFA0300502, 2017YFA0302901 and 2017YFA0302904), as well as JSPS/MEXT Grants (No. JP15H05852 and JP19H00657). J. Y. also acknowledges support by the Youth Innovation Promotion Association of CAS.