Evolution from $B_{2g}$ nematics to $B_{1g}$ nematics in heavily hole-doped iron-based superconductors

TL;DR

This paper explains the transition from $B_{2g}$ to $B_{1g}$ nematic order in heavily hole-doped iron-based superconductors by analyzing magnetic ground state evolution and fluctuations using density functional theory.

Contribution

It reveals that the change in magnetic ground state from single to double stripes causes the nematic symmetry shift, highlighting the role of magnetic fluctuations in this process.

Findings

Nematic order shifts from $B_{2g}$ to $B_{1g}$ with hole doping.

Magnetic ground state evolves from single to double stripe configurations.

Magnetic fluctuations are crucial for stabilizing the double-stripe state.

Abstract

Recent experiments reported an unusual nematic behavior of heavily hole-doped pnictides $A$Fe$_{2}$As$_{2}$, with alkali $A$ = Rb, Cs. In contrast to the $B_{2g}$ nematic order of the parent $Ae$Fe$_{2}$As$_{2}$ compounds (with alkaline earth $Ae$ = Sr, Ba), characterized by unequal nearest-neighbor Fe-Fe bonds, in the hole-doped systems nematic order is observed in the $B_{1g}$ channel, characterized by unequal next-nearest-neighbor Fe-Fe (diagonal Fe-As-Fe) bonds. In this work, using density functional theory, we attribute this behavior to the evolution of the magnetic ground state along the series $Ae_{1-x}A_{x}$Fe$_{2}$As$_{2}$, from single stripes for small $x$ to double stripes for large $x$. Our simulations using the reduced Stoner theory show that fluctuations of Fe moments are essential for the stability of the double-stripe configuration. We propose that the change in the nature of the magnetic ground state is responsible for the change in the symmetry of the vestigial nematic order that it supports.

Equations

$$\begin{array}[]{rl}\hat{a}_{\mathrm{int}}&=\hat{a}_{1}+x\cdot(\hat{a}_{2}-\hat{a}_{1})\\[7.0pt] \mathbf{r}_{\mathrm{int}}&=\mathbf{r}_{1}+x\cdot(\hat{a}_{\mathrm{int}})^{-1}\hat{a}_{2}(\mathbf{r}_{2}-\mathbf{r}_{1})\end{array}$$

$$V_{\uparrow }^{\prime }(\mathbf{r})-V_{\downarrow }^{\prime }(\mathbf{r})=s(V_{\uparrow }(\mathbf{r})-V_{\downarrow }(\mathbf{r}))$$

Full text

Introduction. Unconventional superconductivity in iron-based superconductors is believed to be mediated by spin fluctuations and, therefore, closely related to the nature of the magnetic interactions Hirschfeld11 ; Chubukov11 . The most common type of magnetic order – single-stripe phase – is observed is many parent compounds, such as AeFe2As2 (Ae = Ca, Sr, Ba) review1 ; review21 ; review3 . Stripe magnetic order is characterized by an ordering vector $(\pi,0)$ or $(0,\pi)$ in the Fe-only square lattice. As such, it breaks the tetragonal symmetry of the lattice by making the nearest-neighbor Fe-Fe bonds inequivalent. External pressure or chemical substitution suppress this magnetic order and induce superconductivity (SC), suggesting a close link between the two phases. Besides possibly providing the glue that binds Cooper pairs together, magnetic fluctuations are also believed to be responsible for the nematic phase of the AeFe2As2 compounds Fernandes14 . This is corroborated by the experimental observation that the magnetic and nematic transition lines follow each other closely, and that the elastic and magnetic fluctuations obey a scaling relationship Fernandes13 ; Bohmer16 . One of the proposed mechanisms by which nematicity arises before the onset of single-stripe magnetic order is that magnetic fluctuations form a composite order parameter that condenses above the magnetic transition. In this scenario, nematic order is a vestigial phase of the magnetically ordered phase, breaking a subset of the symmetries broken by the latter (i.e. tetragonal symmetry) while preserving other symmetries (i.e. time-reversal symmetry) Sachdev08 ; Kivelson08 ; Fernandes12 ; Dagotto13 .

Very recently, a different type of nematic order was found in the stoichiometric RbFe2As2 and CsFe2As2 compounds Li2016 ; Liu2018 ; Shibauchi18 . Because these materials have an $n_{d}=5.5$ filling of the Fe $3d$ orbitals, in contrast to the $n_{d}=6$ filling of the AeFe2As2  compounds, they are heavily hole-doped versions of the latter. The nematic order displayed by RbFe2As2 lowers the tetragonal symmetry by breaking the equivalence between the next-nearest-neighbor Fe-Fe bonds, which correspond to the diagonals of the Fe-only square lattice. In group-theory language, this corresponds to a $B_{1g}$ nematic state, whereas the nematic state of AeFe2As2 is in the $B_{2g}$ channel (see insets in Fig. 1 and 4). Here, $B_{1g}$ and $B_{2g}$ refer to the irreducible representations of the tetragonal point group associated with the actual crystallographic unit cell, which contains two Fe atoms and is rotated $45^{\circ}$ with respect to the Fe-only square unit cell. An important question that remains unaddressed is the origin of this $B_{1g}$ nematic state. A recent theoretical studyOnari2018 proposes that this new type of nematicity is driven by incommensurate spin fluctuations on $d_{xy}$ orbitals, and is related to the Lifshitz transition as the system goes from $n_{d}=6$ to $n_{d}=5.5$ filling. On the other hand, structural differences between these systems can, in principle, produce significant changes in the magnetic interactions too, similarly to the well-known case of FeSexTe1-x.Moon2010 Moreover, electronic correlations are believed to be much stronger in RbFe2As2 and CsFe2As2 as compared to AeFe2As2, as manifested by the larger Sommerfeld coefficients of the former Meingast16 and suggested by theoretical studies based on the LDA+DMFT method.Yin2011 ; Medici2014 ; Diehl2014 ; Backes2015

In this paper, we address the change of nematic order from $B_{2g}$ to $B_{1g}$ as a function of hole-doping in a specific 122 family Sr1-xRbxFe2As2. In particular, we use first-principles density functional theory (DFT) calculations to analyze the competition between different magnetic orders as the iron 3d orbital filling evolves from $6$ to $5.5$ for increasing $x$. These DFT calculations include the effects of i) hole doping, ii) structural changes and, iii) Fe moment reduction by spin fluctuations within the recently proposed reduced Stoner theory.Ortenzi2012 ; Ortenzi2015 We find that the favored magnetic ground state changes from single-stripe to double-stripe, which is characterized by an ordering vector of $\left(\pi/2,\,\pi/2\right)$ in the Fe-only square Brillouin zone. The main results are shown in the phase diagram of Fig. 1. The stabilization of the double-stripe order in RbFe2As2, as compared to the single-stripe order in SrFe2As2, arises primarily from the changes in the electron count, rather than the changes in the lattice structure, and seems to be unrelated to Fermi surface nesting. Since the $B_{2g}$ nematic order is a vestigial phase of the single-stripe phase, whereas the $B_{1g}$ nematic order is a vestigial phase of the double-stripe phase, our results provide a unified picture in which the change in the nematic state as function of hole-doping reflects a change in the dominant magnetic interactions.

First-principle results: magnetic order. We focus on the series Sr1-xRbxFe2As2, since, among the three AFe2As2 stoichiometric compounds, $A=\mathrm{Rb}$ is the one that displays the strongest evidence for $B_{1g}$ nematic order,Liu2018 ; Shibauchi18 as compared to $A=\mathrm{Cs}$ and $A=\mathrm{K}$. The choice of Sr is to minimize steric effects related to the different sizes of the cations. To perform our DFT calculations, we used for the stoichiometric compounds RbFe2As2 and SrFe2As2 the experimental room-temperature structures, both belonging to the I4/mmm space group (Rb: $a=3.87198\,\mathrm{\AA}$, $c=14.46\,\mathrm{\AA}$, $z_{\mathrm{As}}=-0.1525$; Sr: $a=3.9243\,\mathrm{\AA}$, $c=12.3644\,\mathrm{\AA}$, $z_{\mathrm{As}}=-0.14$).Tegel2008 ; Eilers2016 The structural parameters of the series Sr1-xRbxFe2As2 were obtained by linear interpolation of the lattice parameters and all bond lengths following the equations:

[TABLE]

Here, $\hat{a}$ is the matrix of the lattice vectors and $\mathbf{r}$ is a vector of fractional coordinates of a given atom in the supercell. The indices $1$, $2$ and “int” refer to SrFe2As2, RbFe2As2 and Sr1-xRbxFe2As2, respectively. For Sr1-xRbxFe2As2, the spacer cation is replaced by an artificial atomic species with a fractional nuclear charge of $Z=38-x$, which simulates a random mixture of Sr and Rb using the virtual crystal approximation.

The competition between the non-magnetic state and three different types of magnetic states (single-stripe, N el, and double-stripe) in this compound series was studied using the FLAPW method as available in the Elk code.elk Fully relativistic magnetic energies were calculated using the PBE parameterizationPerdew1996 of the GGA density functional. The number of empty states was set to 24 per atom and spin, whereas the smearing was set to 0.002 Ha$\,\approx\,$50 meV and the k-mesh dimensions to $(10\times 10\times 10)$. The aforementioned magnetic configurations – single-stripe, N el, and double-stripe – are the usual competitors in Fe-based superconductors and have different spatial periodicities corresponding to the ordering vectors (in the coordinate system of the Fe-only square lattice) $\mathbf{Q}_{1}=(\pi,0)$, $\mathbf{Q}_{2}=(\pi,\pi)$ and $\mathbf{Q}_{3}=(\pi/2,\pi/2)$, respectively. In this work, these states are simulated using the minimal 8-Fe supercells of different shapes (see insets in Figs. 1 and 2). Each supercell includes a second Fe layer where all spins point in the opposite directions, which is commonly observed in iron pnictides.

One important ingredient of the physics in the iron-based superconductors that is often not captured by standard DFT approaches is the role of spin fluctuations. They are essential for instance to explain the reduced magnetic moment of the Fe atoms, which is usually overstimated in DFT.Ortenzi2015 Here, spin fluctuations are modeled using the reduced Stoner theoryOrtenzi2012 with a single adjustable parameter $s$ that scales down the spin splitting of the self-consistent potential $V(\mathbf{r})$ which enters then the DFT equations as:

[TABLE]

The role of $s$ is essentially to enhance the amplitude of the magnetic fluctuations, which in turn results in a suppression of the Fe magnetic moments. For the family of doped CaFe2As2 compounds, it has been found that values around $s=0.85$ provide a consistent and precise description of the Fe magnetic moments as a function of temperature and doping.Ortenzi2015 For this reason, hereafter we set $s=0.85$ in our calculations. It turns out that correctly capturing these enhanced spin fluctuations is crucial for the magnetic ground state, as indicated by the calculated magnetic energies for RbFe2As2 versus the $s$ parameter (Fig. 2). For the values around $s=0.85$, the double-stripe is the dominating magnetic configuration showing the correct order of magnitude of Fe moments $\sim\,0.5\,\mu_{\mathrm{B}}$.111We note that, in our simulations, the single-stripe order remains the ground state of SrFe2As2 even for smaller values of $s$. At the same time, the single-stripe and Néel orders strongly compete with this double-stripe ground state. In contrast, $s=1$ leads to overestimated Fe moments of the order of $2\mu_{B}$, commonly observed in previous DFT simulations of iron-based systems,Opahle2009 and the nature of the magnetic ground state of RbFe2As2 changes dramatically. Nevertheless, our results are robust for a reasonably wide range of values of $s$ corresponding to small Fe moments usually measured in experiments.Cruz2008

In the next step, we study the evolution of magnetic energies for Sr1-xRbxFe2As2 with simultaneous hole-doping $x$ and lattice deformation (Fig. 3a). As shown in Fig. 3a, the single-stripe order of the parent SrFe2As2 compound is the ground state up to a Rb concentration of about 75%, where the magnetic ground state switches to the double-stripe. At this point, the Fe occupation is $n_{d}=5.625$, compared to $n_{d}=6$ for pure SrFe2As2. We emphasize here again that the Néel state is a relatively close competitor of the double-stripe phase in the case of RbFe2As2 (Fig. 2).

In these simulations the effects of the electron count and the lattice parameters are both present. In order to disentangle them and determine which one plays the dominant role in driving the instability towards the double-stripe order in RbFe2As2, we compute the magnetic energies when the lattice parameters are tuned between those of SrFe2As2 and RbFe2As2, according to Eq. (1), and the Fe electron count is fixed at $n_{d}=5.5$. The main change in the lattice parameters is a stretch of the c-axis corresponding to a strain of ${\sim}17\%$. Importantly, such a stretch also affects various bond lengths. As shown in Fig. 3b, for all strain values investigated, the energy of the single-stripe magnetic state is always higher than that of the double-stripe state. While the energy difference is visibly affected by the $c$-axis lattice parameter, it is clear that the main contribution to the stabilization of the double-stripe order in RbFe2As2 arises from the change in the electron count. This is further illustrated in Fig. 1, where both parameters – electron count $n_{d}$ and strain – are varied independently. While we observe that strain alone can change the nature of the magnetic ground state for a fixed (smaller) hole-doping level, there is a clear trend that large hole-doping favors a double-stripe order. one. These results indicate that while bond lengths play a significant role in choosing the magnetic ground state of RbFe2As2, the electron filling of the Fe 3d orbitals appears to be the dominant factor determining the type of magnetic order.

Vestigial nematic orders. So far our DFT simulations show a change in the magnetic ground state of Sr1-xRbxFe2As2 from single-stripe, for small and moderate $x$, to double-stripe for large $x$. As we now argue, this evolution of the magnetic phase naturally accounts for the change in the nematic state from $B_{2g}$ (small and moderate $x$) to $B_{1g}$ (large $x$) observed experimentally. The key point is that both the single-stripe and the double-stripe states support symmetry-breaking composite order parameters that can condense even in the paramagnetic phase, forming so-called vestigial phases.Fernandes18

The case of the single-stripe phase has been widely studied Kivelson08 ; Sachdev08 ; Fernandes12 ; Dagotto13 . The Ising-like composite spin order parameter $\Phi_{B_{2g}}=\left\langle\mathbf{S}_{\mathbf{R}_{i}}\cdot\mathbf{S}_{\mathbf{R}_{i}+\mathbf{x}}\right\rangle-\left\langle\mathbf{S}_{\mathbf{R}_{i}}\cdot\mathbf{S}_{\mathbf{R}_{i}+\mathbf{y}}\right\rangle$ is clearly non-zero and uniform in the single-stripe phase, since spins are parallel to each other along one axis and anti-parallel to each other along the other axis (see Fig. 4a). As shown by a variety of different theoretical methods Kivelson08 ; Sachdev08 ; Fernandes12 ; Dagotto13 , $\Phi_{B_{2g}}$ can remain non-zero even when there is no long-range magnetic order, $\left\langle\mathbf{S}_{\mathbf{R}_{i}}\right\rangle=0$, because $\Phi_{B_{2g}}$ is a two-spin correlation function. Now, since $\Phi_{B_{2g}}$ breaks the equivalence between the $x$ and $y$ axis, its onset breaks tetragonal symmetry, while preserving translational symmetry. The state with $\Phi_{B_{2g}}\neq 0$ but $\left\langle\mathbf{S}_{\mathbf{R}_{i}}\right\rangle=0$ is a $B_{2g}$ nematic state. It is called a vestigial phase of the magnetically ordered state because it breaks a subset of the symmetries broken by the latter, as explained in the Introduction.

Analogously, inside the double-stripe magnetic phase, the Ising-like composite order parameter $\Phi_{B_{1g}}=\left\langle\mathbf{S}_{\mathbf{R}_{i}}\cdot\mathbf{S}_{\mathbf{R}_{i}+\mathbf{x}+\mathbf{y}}\right\rangle-\left\langle\mathbf{S}_{\mathbf{R}_{i}}\cdot\mathbf{S}_{\mathbf{R}_{i}-\mathbf{x}+\mathbf{y}}\right\rangle$ is non-zero and uniform Paul11 , since spins are parallel along one diagonal and anti-parallel along the other diagonal (see Fig. 4b). In this case, a vestigial $B_{1g}$ nematic phase, in which diagonals are not equivalent, can appear even in the absence of long-range magnetic order Fernandes17 ; Dagotto17 . Thus, the change in the magnetic ground state from single-stripe to double-stripe should be accompanied by a change in the vestigial nematic order from $B_{2g}$ to $B_{1g}$.

An interesting question that remains to be addressed is why RbFe2As2 shows nematic order but no apparent sign of long-range magnetic order, whereas SrFe2As2 displays both magnetic and nematic orders.Wu2014 The situation seems to be similar to the case of the Ti-based oxypnictides Fernandes17 , where a $B_{1g}$ vestigial nematic phase driven by double-stripe magnetic fluctuations has been proposed, but long-range magnetic order has not been observed. First, it is important to recall that the nematic transition, which must always happen above or simultaneously to the magnetic transition, is observed at lower temperatures in RbFe2As2 ($T_{\mathrm{nem}}\sim 40$ K)Liu2018 as compared to SrFe2As2 ($T_{\mathrm{nem}}\sim 200$ K).Wu2014 Second, the existence of a closely competing Néel state in the case of RbFe2As2 (Fig. 2) provides a possible mechanism for the suppression of the magnetic transition temperature, similarly to what has been proposed to explain the absence of magnetic order in FeSe.Glasbrenner15

Finally, the experimentalEilers2016 ; Meingast16 and theoretical indicationsHardy2013 ; Medici2014 ; Backes2015 that electronic correlations are stronger in hole-doped AFe2As2 iron pnictides as compared to AeFe2As2 could also be an aspect contributing to the change in nature of nematicity in these systems. In the present work, we concentrated on the underlying magnetism and found strong hints of its role in determining the various nematic phases.

*Conclusions. *In summary, our first-principles calculations of the magnetic ground states of 122-type iron-based compounds show that the dominant magnetic instability in these systems is strongly affected by the nature of the spacer cation substitution. For RbFe2As2, we found that the double-stripe state becomes the leading instability, in contrast to the single-stripe order in SrFe2As2. Analysis of the various contributions involved in the calculations showed that the magnetic order is more sensitive to the electron count variation due to the Sr$\to$Rb substitution than to the related structural changes. Based on these results, we proposed that the evolution of the nematic state from $B_{2g}$ for SrFe2As2 to $B_{1g}$ for RbFe2As2 is a consequence of the change in the vestigial ordered states supported by the single-stripe and double-stripe orders.

Our results have important experimental consequences that can in principle be verified. First, even though RbFe2As2 is not magnetic, it should have strong magnetic fluctuations peaked at the $\mathbf{Q}_{3}=\left(\pi/2,\pi/2\right)$ wave-vector. To the best of our knowledge, there are no inelastic neutron scattering measurements available for RbFe2As2. Neutron scattering in the related $n_{d}=5.5$ KFe2As2 compound, which shows no evidence of $B_{1g}$ nematic order, reveal incommensurate spin fluctuations peaked at $(\pi(1\pm 2\delta),0)$ with $\delta\approx 0.16$.Lee2011 Also, our phase diagram in Fig. 1 shows that uniaxial strain along the $c$-axis can tune the magnetic ground state for compounds with intermediate levels of hole doping. This offers the possibility to tune the nematic state from $B_{2g}$ to $B_{1g}$ as a function of strain or pressure.

Acknowledgments. We thank T. Shibauchi and J. Schmalian for fruitful discussions. VB and RV were financially supported by DFG Sonderforschungsbereich TRR 49. The computer time was allotted by the centre for supercomputing (CSC) in Frankfurt and by the computer center of Goethe University. RMF was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award number DE-SC0012336. Parts of some figures have been produced with VESTA3 (Ref. Vesta, ).