Pressure-driven phase transition from antiferromagnetic semiconductor to nonmagnetic metal in two-leg ladders $A$Fe$_2$$X$$_3$ ($A$=Ba or K, $X$=S or Se)

TL;DR

This study uses first principles calculations to explore pressure-induced phase transitions in 123-type iron chalcogenides, revealing a transition from antiferromagnetic semiconductor to nonmagnetic metal, which may relate to superconductivity.

Contribution

It provides the first detailed computational analysis of pressure-driven magnetic and electronic phase transitions in BaFe$_2$S$_3$, KFe$_2$S$_3$, and KFe$_2$Se$_3$, highlighting the role of self-doping and magnetic suppression.

Findings

BaFe$_2$S$_3$ undergoes a first-order transition to a nonmagnetic metal at high pressure.

KFe$_2$S$_3$ also shows a first-order transition, but KFe$_2$Se$_3$ exhibits a second-order or weakly first-order transition.

Higher pressures are needed to quench magnetism in KFe$_2$S$_3$ and KFe$_2$Se$_3$ compared to BaFe$_2$S$_3$.

Abstract

The recent discovery of superconductivity in BaFe$_2$S$_3$ [Takahashi {\it et al.}, Nat. Mater. {\bf 14}, 1008 (2015)] has stimulated considerable interest in 123-type iron chalcogenides. This material is the first reported iron-based two-leg ladder superconductor, as opposed to the prevailing two-dimensional layered structures of the iron superconductors family. Once the hydrostatic pressure exceeds $11$ GPa, BaFe$_2$S$_3$ changes from a semiconductor to a superconductor below $24$~K. Although previous calculations correctly explained its ground state magnetic state and electronic structure, the pressure induced phase transition was not successfully reproduced. In this work, our first principles calculations find that with increasing pressure the lattice constants as well as local magnetic moments are gradually suppressed, followed by a first-order magnetic transition at a critical pressure, with local magnetic moments dropping to zero suddenly. Our calculations suggests that the self-doping caused by electrons transferred from S to Fe may play a key role in this transition. The development of a nonmagnetic metallic phase at high pressure may pave the way to superconductivity. As extensions of this effort, two other 123-type iron chalcogenides, KFe$_2$S$_3$ and KFe$_2$Se$_3$, have also been investigated. KFe$_2$S$_3$ also displays a first-order transition with increasing pressure, but KFe$_2$Se$_3$ shows instead a second-order, or weakly first-order, transition. The required pressures for KFe$_2$S$_3$ and KFe$_2$Se$_3$ to quench the magnetism are higher than for BaFe$_2$S$_3$. Further experiments can confirm the predicted first-order nature of the transition in BaFe$_2$S$_3$ and KFe$_2$S$_3$, as well as the possible metallic/superconductivity state in other 123-type iron chalcogenides under high pressure.

Full text

I Introduction

Since the discovery of superconductivity in fluorine doped LaFeAsO Kamihara et al. (2008), the iron pnictides and chalcogenides have rapidly developed into one of the main branches of research in the field of unconventional superconductors Lumsden and Christianson (2010); Dai et al. (2012); Stewart (2011). Almost all previously reported iron-based superconductors have similar crystal structures, involving a slightly distorted two-dimensional Fe square lattice Johnston (2010); Dagotto (2013); Dai (2015); Si et al. (2016), where each Fe atom is caged in a tetrahedral structure. The magnetism of the non-superconducting parent state is primarily given by the collinear stripe-like order, namely the C-type antiferromagnetic (C-AFM) order, although some exceptions exist such as in FeTe, FeSe, KFe2Se2, and K2Fe4Se5 Bishop et al. (2016); Li et al. (2009); Moon and Choi (2010); Ma et al. (2009); Yu and Si (2015); Li et al. (2012a, b); Zhang et al. (2016); Bao et al. (2011); Ye et al. (2011); Yin et al. (2010). The structural and magnetic similarities of all these iron-based superconductors suggest common physical mechanisms leading to their magnetic and superconducting properties, despite the different ratios of the atomic elements involved.

However, recently iron chalcogenides with a different structure, the so-called 123-type $A$Fe${}_{2}X_{3}$ ($A$=K, Cs, Rb, or Ba; and $X$=S, Se, or Te), have drawn considerable attention Krzton-Maziopa et al. (2011); Saparov et al. (2011); Caron et al. (2011, 2012); Lei et al. (2011); Arita et al. (2015); Wang et al. (2016); Patel et al. (2016). These materials display unique quasi-one-dimensional two-leg ladder iron structures (see Fig. 1(a)) that are clearly qualitatively distinct from the other extensively studied iron pnictides/chalcogenides with iron layers. At least from the perspective of the electronic structure, the frequently mentioned Fermi surface nesting effect involving two pocket cylindrical Fermi surfaces (corresponding to the quasi-two-dimensional structure) in several iron pnictides/chalcogenides can not be relevant in these ladder systems Lumsden and Christianson (2010); Stewart (2011); Johnston (2010); Dagotto (2013); Dai (2015). Thus, the reduced dimensionality of the iron network makes $A$Fe${}_{2}X_{3}$ a physically fascinating material that deserves further experimental and theoretical scrutiny. In fact, the two-leg ladders made of Fe atoms remind us of the previously studied two-leg ladders superconducting cuprates Dagotto (1994, 1999). In the past, the study of cuprate ladders much illuminated the physics of Cu oxides, primarily because theoretical calculations involving model Hamiltonian can be carried out with good accuracy in chains and ladders and, thus, accurate theory-experiment comparisons can be done. A similar important impact of iron ladders on the field of iron superconductors is now to be expected.

In this family of ladder materials, BaFe2Se3 was the first compound reported to be superconducting at approximately $11$ K Krzton-Maziopa et al. (2011). However, others experiments claimed the material to be a semiconductor with the observed superconductivity probably induced by impurities Caron et al. (2011). Even under these circumstances, one of our previous studies predicted an interesting property, namely that its block-type antiferromagnetic (Block-AFM) state is multiferroic due to magnetostriction effects Dong et al. (2014, 2015). The theoretically predicted polar structure has indeed been verified by a subsequent neutron study Lovesey et al. (2016). In addition, doping of K in the Ba site changes the ground state to the so-called CX-type antiferromagnetism (CX-AFM) (see Fig. 1(b)) Caron et al. (2012). Such CX-AFM order was also predicted to be the ground state of BaFe2S3, which was later also confirmed using neutron techniques Takahashi et al. (2015). Under ambient conditions, BaFe2S3 has a semiconducting ground state with a very small gap about $0.06-0.07$ eV Gönen et al. (2000); Reiff et al. (1975). The most striking recent experimental discovery is that high pressure can drive BaFe2S3 to become superconducting when the hydrostatic pressure exceeds about $11$ GPa Takahashi et al. (2015) and its highest $T_{\rm c}$ can reach $24$ K at $13.5$ GPa Yamauchi et al. (2015).

A subsequent density functional theory (DFT) study by Suzuki et al. on BaFe2S3 confirmed the CX-AFM nature of the magnetic state (see also our earlier work Dong et al. (2014)) as well as its semiconducting behavior under ambient conditions Suzuki et al. (2015). However, the pressure induced magnetic to nonmagnetic transition (as phenomenologicall required by superconductivity) was not obvious from that DFT study, although the calculation indeed showed a semiconductor-metal transition at about $5$ GPa. In those previous DFT calculations, the magnetism persisted with only a small suppression upon increasing pressure. This problem might be due to the process followed to allow the relaxation of the crystal structure under pressure, since for simplicity only the positions of sulfur atoms were optimized in the $x$-$y$ plane in that previous study, which is sufficient for a qualitative analysis. However, in our study presented below all atomic positions were simultaneously relaxed, allowing us to be not only qualitative but also quantitative in our analysis.

In the present publication, the magnetic properties, electronic structure, and pressure effects corresponding to BaFe2S3 are revisited using first-principles DFT calculations. Our results can be divided in two classes. First, similar results as in previous DFT efforts and experiments for the ambient conditions have been obtained and confirmed. Second, a semiconductor-metal transition accompanying the quenching of magnetism has been observed, in good agreement with experimental observations. This phase transition is probably of first order because we observe a sudden drop of the iron’s magnetic moment and also anomalies in the crystal structure. More specifically, one of our most important results is that the underlying physics of this transition lies in the modifications by pressure of the effective electronic density at the iron network of relevance. Thus, the net effect of increasing pressure is equivalent to doping charge into the iron atoms, a nontrivial effect difficult to deduce from the existing experimental information. Following these arguments, two additional materials $A$Fe${}_{2}X$3 (KFe2S3 and KFe2Se3) have been also studied theoretically by a similar procedure. The quenching of magnetic moments and metallic phases have also been obtained both for the cases of KFe2S3 and KFe2Se3, although at higher pressures. The theoretical observation of these pressure induced transitions suggest possible pressure-induced superconducting states also in KFe2S3 and KFe2Se3 by analogy with what occurs in BaFe2S3, although we cannot explicitly prove these predictions due to the limitations of DFT techniques to address superconductivity. We expect that our results should motivate experimental efforts for their confirmation.

II Methods

The DFT calculations are performed based on the generalized gradient approximation (GGA) with the projector augmented wave (PAW) potentials, as implemented in the Vienna ab initio Simulation Package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller (1996); Blöchl et al. (1994); Perdew et al. (1996). The PBE exchange function has been adopted. The plane-wave cutoff is $500$ eV. The $k$-point mesh is $4\times$3$\times 6$ for the minimum crystalline unit cell, which is accordingly adapted for the various magnetic cells studied (e.g. $2\times$6$\times 2$ for CX-AFM ($\pi$, $\pi$, [math])). Starting from experimental values, both the lattice constants and atomic positions are fully relaxed until the force on each atom was below $0.01$ eV/Å.

To study the magnetic properties, various possible (in-ladder) magnetic structures are imposed on the Fe ladders, as shown in Fig. 1(b). Despite the in-ladder correlation, the magnetic correlation between ladders can also slightly affect the physical properties. Therefore, besides the simplest ([math], [math], [math]) order, the ($\pi$, $\pi$, [math]) order is also studied, as indicated in Fig. 1(c).

In addition to the standard DFT calculation, the maximally-localized Wannier functions (MLWFs) have also been employed to fit five Fe’s $3d$ bands and the Fermi surface, using the WANNIER90 packages Mostofi et al. (2007).

Several previous DFT studies have found that even the pure GGA (or LSDA) procedures overestimates the local magnetic moments in iron pnictides and chalcogenides Hansmann et al. (2010); Mazin et al. (2008); Mazin and Johannes (2009); Dong et al. (2009); Zhang et al. (2016). Thus, using GGA+$U$ with a positive $U$ will render this inconsistency even more serious. Thus, in some studies, a negative $U$ correction was adopted to better describe the 122-type Fe-based materials Ferber et al. (2010); Yi et al. (2010). Alternatively, the exchange and correlation kernels were rescaled by an appropriate factor Ortenzi et al. (2015). In the present study, both GGA+$U$ and pure GGA have been tested, and we found that the later provides a better description of $A$Fe2X3 (regarding its crystalline constants, magnetic moments, as well as band gaps). For this reason, only the pure GGA results will be presented in the rest of this publication.

III Results & discussion

III.1 Magnetism & electronic structure of BaFe2S3

Our main DFT results with regards to the magnetic and electronic properties of the BaFe2S3 ladder can be summarized in four statements as described below:

(1) Without the external pressure, both the lattice constants and atomic positions were fully optimized in the presence of magnetism. In the pure GGA calculation, the CX-AFM ($\pi$, $\pi$, [math]) state has the lowest energy among all candidate configurations investigated, in agreement with experiments. The CX-AFM ([math], [math], [math]) is only slightly higher in energy by $6.3$ meV/Fe, which is reasonable considering the similarity between these two CX-AFM configurations.

(2) Even the pure GGA result gives a magnetic moment ($2.08$ $\mu_{\rm B}$/Fe) that is slightly higher than the experimental one ($\sim 1.2$ $\mu_{\rm B}$/Fe at low temperature, as measured by neutron scattering Takahashi et al. (2015)). This overestimated local moment is quite common in DFT calculations of iron-based superconductors Hansmann et al. (2010); Mazin et al. (2008); Mazin and Johannes (2009), which may be due to the coexistence of localized Fe spins and itinerant electrons Tam et al. (2015). Note that this calculated value also depends on how large the Wigner-Seitz radius of the iron atom is set to be. Thus, the inconsistency described above may partially originate from the methodological difference between the VASP procedure and neutron experiments.

(3) For the experimentally relevant CX-AFM ($\pi$, $\pi$, [math]) state, the pure GGA calculation gives a small gap of $0.088$ eV, which agrees with the experimental value ($0.06-0.07$ meV) Gönen et al. (2000); Reiff et al. (1975) and previous DFT results Suzuki et al. (2015).

(4) According to the calculated density of states (DOS) (not shown), the bands near the Fermi levels are highly hybridized between Fe’s $3d$ orbitals and S’s $3p$ orbitals. Such hybridization is quite common in iron pnictides/chalcogenides.

III.2 Pressure induced transition in BaFe2S3

Although previous DFT calculations correctly reproduced the magnetic ground state and electronic structure of BaFe2S3, the pressure induced magnetic-nonmagnetic transition (presumably also associated to superconductivity, although beyond the DFT scope) was not reproduced Suzuki et al. (2015), as explained before. Below, these pressure effects will be revisited using DFT.

By increasing the hydrostatic pressure in the calculation, our energies and Fe’s magnetic moments are summarized in Fig. 3(a-b). For all magnetic orders, the local moments decrease with pressure but the CX-AFM ($\pi$, $\pi$, [math]) state always has the lowest energy. When the pressure reaches $10.8$ GPa (very close to the experimental critical pressure Takahashi et al. (2015); Yamauchi et al. (2015)), the static values of the local moments drop to zero. Meanwhile, the system becomes metallic. Such nonmagnetic metallic phase provides the conditions for superconductivity if there are still antiferromagnetic fluctuations present (although short-range magnetic quantum fluctuations also can not be captured in the DFT calculation). In this sense, without fine tuning parameters, our calculation correctly reproduces the antiferromagnetic-to-nonmagnetic transition. Furthermore, we found that the quenching of magnetic moments occurs as in a first-order transition, with a sudden drop from $0.83$ $\mu_{\rm B}$/Fe to zero at $10.8$ GPa (inset of Fig. 3(b)).

Figure 3(c) shows the compressibility, i.e. the lattice constants normalized to the ambient ones. The $b$-axis lattice is the softest while the $c$-axis lattice is the hardest, as in experiments Takahashi et al. (2015). Such anisotropic compressibility can be intuitively understood considering the loose space between ladders and the compact bonds along ladders. The first-order character of the transition at $10.8$ GPa can also be observed in the lattice structure (especially for the lattice constant along the $b$-axis), as emphasized in the inset of Fig. 3(c). This first-order characteristic was not reported previously and could be verified in future experiments.

According to the Bader charge analysis (Fig. 2(d)) Bader (1990); Tang et al. (2009); Henkelman et al. , there is a significant charge transfer ($\sim 0.15$ electron) from S to Fe by increasing pressure from [math] GPa to $12$ GPa. This tendency is equivalent to the effects of electron doping by, for example, chemical substitution, which is the standard procedure to generate superconductivity from a magnetic parent compound. In this sense, it is reasonable to suspect that the superconductivity observed in BaFe2S3 probably is induced by electron doping, in analogy with the superconductivity triggered by F doping in LaFeAsO. This is also quite similar to what happened in previous investigations of the Cu-ladders, where a transfer of charge from chains to ladders triggers superconductivity Dagotto and Rice (1996). In summary, our calculations suggest that the superconductivity of BaFe2S3 is probably caused by self-doping of electrons into the iron network. This effect can occur in addition to the previously proposed scenario based on the broadening of the electronic bandwidth by pressure Takahashi et al. (2015). Only further experimental work can clarify which of the two tendencies is more dominant to generate superconductivity.

A careful analysis of the DOS at the Fermi level just before the critical pressure finds a sharp peak, i.e. a van Hove singularity, which suddenly drops to become a valley around $11$ GPa, as shown in Fig. 3(a). The first-order magnetic-nonmagnetic transition leads to the sudden disappearance of this van Hove singularity since electronic bands are seriously reconstructed at the critical pressure. The Fermi surface at $11$ GPa is shown in Fig. 3(b), with four bands crossing the Fermi level. Two of them are nesting around the $\Gamma$ point while the other two are isolated.

III.3 Pressure induced transition in KFe2S3

Until now, BaFe2S3 is the only experimentally confirmed superconductor in the 123-type series of iron ladders. As a consequence, it is interesting to investigate whether there are other 123-type iron ladders that can also potentially become superconductors. KFe2S3 is a sister member of BaFe2S3, where the nominal valence of Fe is higher by $+0.5$. Experimentally, KFe2S3 has been synthesized Mitchell et al. (2004) but their detailed physical properties have not been reported, particularly under high pressure. Structurally, this K-based 123 ladder is similar to BaFe2S3, with the same $Cmcm$ group.

Here, DFT calculations have been performed also on KFe2S3. In this study we have found that the magnetic ground state is also of the CX-AFM type. Moreover, the local magnetic moment is about $2.3$ $\mu_{\rm B}$/Fe in the pure GGA calculations. Compared to BaFe2S3, the band gap of KFe2S3 ($\sim 0.51$ eV in pure GGA calculations) is slightly larger. According to the atomically-resolved DOS (not shown), the states near the Fermi level are also primarily contributed by the Fe’s $3d$ orbitals, which are also highly hybridized with the S’s $3p$ orbitals.

According to the Bader charge analysis, the electronic densities at the Fe and S sites in KFe2S3 are lower by $0.09$ and $0.16$ electrons than those in BaFe2S3, respectively. Therefore, the replacement of Ba2+ by K1+ does not really dope the iron sites by the nominal amount of $0.5$ holes, but those holes mostly go to the S’s sites due to the partially covalent Fe-S bonds.

Since the electronic density of Fe is slightly lower in KFe2S3, the critical pressure should be higher according to the Bader charge analysis. To verify this expectation, the calculated crystal constants and magnetism (with pure GGA) are presented in Fig. 4 as a function of pressure. As shown in Fig. 4(a), the CX-AFM ($\pi$, $\pi$, [math]) state is always the ground state if it is magnetically ordered. The suppression of the magnetic moment by pressure is shown in Fig. 4(b). The first-order character of the transition is similar to that observed in BaFe2S3. The critical pressure ($\sim 23$ GPa) is indeed larger, as expected. The band gap of the CX-AFM phase reduces to zero at $17$ GPa, inducing a semiconductor to metal transition. The metallic nonmagnetic phase above $23$ GPa may be superconducting, according to the previous experience with BaFe2S3. Of course, this reasoning is merely by analogy between similar materials because DFT cannot address superconductivity directly. The lattice constants under pressure, normalized to their ambient values, are shown in Fig. 4(c); they are also very similar to those reported for BaFe2S3. The Bader charge analysis applied to KFe2S3 under pressure leads to the same behavior as in BaFe2S3, namely pressure enhances the local electronic density at the Fe sites, as shown in Fig. 4(d). There is also significant charge transfer ($\sim 0.24$ electron per Fe) from S to Fe by increasing pressure from [math] GPa to $25$ GPa. Interestingly, the Bader charge densities at the critical pressures for magnetic quenching are almost identical (error bar $\delta<0.005e$ in our calculations) for KFe2S3 and BaFe2S3, suggesting a similar physical mechanism for both compounds.

To summarize this subsection, our calculations here predict that KFe2S3 should be similar to BaFe2S3, regarding its magnetic ground state as well as its behavior upon pressure. Thus, superconductivity is possible under higher pressure. By increasing pressure, the transfer of electrons from S to Fe occurs, namely a self-doping process takes place that eventually could lead to superconductivity as in the canonical layered iron superconductors.

III.4 Pressure induced transition in KFe2Se3

Since reducing the electronic density at the Fe atoms is a disadvantage to suppress magnetism with increasing pressure, as demonstrated in KFe2S3 where a higher pressure than for BaFe2S3 was needed to induce the metallic phase, it is natural to expect the opposite tendency in other $A$Fe${}_{2}X_{3}$ compounds that naturally have higher electronic density at the Fe atoms.

According to our Bader charge analysis, at ambient conditions the electronic density at the Fe atoms in KFe2Se3 (experimentally confirmed to display the CX-AFM order Caron et al. (2012)) is higher than in the case of KFe2S3 by $0.12$ electrons, and even higher than that in BaFe2S3 by $0.03$ electrons, due to the weak electronegativity of Se. Then, a natural speculation arises: could it be that by increasing pressure KFe2Se3 is closer to metallicity, and thus perhaps also superconductivity, than BaFe2S3 is?

Despite the experimental studies by Caron et al. Caron et al. (2012), DFT calculations on KFe2Se3 have not been performed to our knowledge. To remedy this problem, here the pure GGA calculation has been carried out for KFe2Se3. At ambient conditions, the CX-AFM state is indeed the ground state (Fig. 5(a)). The local moment of Fe is large, reaching the value $2.65$ $\mu_{\rm B}$/Fe (slightly higher than the experimental one $2.1$ $\mu_{\rm B}$/Fe Caron et al. (2012)) at ambient conditions (Fig. 5(b)), which is a negative signal for metallicity, and thus for potential superconductivity.

Upon pressure, the quenching of the magnetic moment and semiconductor-metal transition indeed occurs. The gap of KFe2Se3 is about $0.56$ eV at ambient condition, which is gradually closed by increasing pressure to $25$ GPa. The required critical pressure for magnetic quenching reaches $29$ GPa, even higher than that of KFe2S3. And a different feature is that this magnetic phase transition seems to be more gradual, probably of second order or weak first order, rather than occurring by a sudden jump as observed in BaFe2S3 and KFe2S3. The Bader charge analysis is shown in Fig. 5(d) as a function of pressure. Furthermore, for the higher magnetic moment in KFe2Se3, the critical pressure should be higher to suppress the magnetism. Then, in this case additional charge transfer from Se to Fe may be required to suppress the magnetism.

In summary of this subsection, our DFT calculations have confirmed the CX-AFM magnetic ground state for KFe2Se3. Although KFe2Se3 owns a relative high electronic density at the Fe atoms, its large gap and large moment make it even more difficult to induce a nonmagnetic metallic phase upon pressure. And this magnetic-nonmagnetic transition may be of the second order, or weak first order. These different features may arise from the Se atoms, which are larger in size and weaker in their electronegativity.

IV Conclusion

In this work, the magnetic and electronic properties of BaFe2S3, KFe2S3, and KFe2Se3 have been analyzed using first-principles calculations. The CX-AFM magnetic order is confirmed to be the common magnetic ground state for all these materials. The pressure-driven semiconductor to metal transition, as well as the antiferromagnetic-to-nonmagnetic transition, has been properly reproduced. Although the DFT technique can not directly address a superconducting state, our study can provide helpful information to understand the superconducting transition of BaFe2S3 at high pressure ($11$ GPa), which is predicted to be a first-order transition. Our main conclusion is that the electron transfer from S to Fe, i.e. a self-doping process, may play a key role to tune the magnetism in BaFe2S3 and eventually induce metallicity and potentially superconductivity.

A similar first-order transition has also been predicted for KFe2S3, although the required critical pressure is higher (about $23$ GPa). By contrast, although the magnetism can also be quenched in KFe2Se3, the required pressure (about $29$ GPa) is even higher and the transition seems to be of the second order, or weak first order. Further experiments are encouraged to verify our predictions as well as the possible existence of metallicity, and perhaps superconductivity, in KFe2S3 and KFe2Se3.

Acknowledgements.

We acknowledge valuable discussions with Rong Yu, Hongyan Lu, and Songxue Chi. This work was primarily supported by the National Natural Science Foundation of China (Grant No. 11674055). E.D. was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. Most calculations were performed on Tianhe-2 at the National Supercomputer Center in Guangzhou (NSCC-GZ).