Electronic and dynamical properties of CeRh_{2}As_{2}: Role of Rh_{2}As_{2} layers and expected hidden orbital order
Andrzej Ptok, Konrad J. Kapcia, Pawe? T. Jochym, Jan ?ażewski, Andrzej M. Ole?, Przemys?aw Piekarz
EElectronic and dynamical properties of CeRh As :Role of Rh As layers and expected hidden orbital order Andrzej Ptok, ∗ Konrad J. Kapcia, Pawe(cid:32)l T. Jochym, Jan (cid:32)La˙zewski, Andrzej M. Ole´s,
3, 4 and Przemys(cid:32)law Piekarz † Institute of Nuclear Physics, Polish Academy of Sciences, W. E. Radzikowskiego 152, PL-31342 Krak´ow, Poland Faculty of Physics, Adam Mickiewicz University in Pozna´n, Uniwersytetu Pozna´nskiego 2, PL-61614 Pozna´n, Poland Institute of Theoretical Physics, Jagiellonian University, Profesora Stanis(cid:32)lawa (cid:32)Lojasiewicza 11, PL-30348 Krak´ow, Poland Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Dated: February 5, 2021)Recently discovered heavy fermion CeRh As compound crystallizes in the nonsymmorphic P4/nmm symmetry, which allows unexpected behavior associated with topological protection. Ex-perimental results show that this material exhibits unusual behavior, which is manifested by theappearance of two superconducting phases. In this work, we uncover and discuss a role of Rh As layers and their impact on the electronic and dynamical properties of the system. The location of Ceatoms between two non-equivalent layers allows for the realization of hidden orbital order. We pointout that the electronic band structure around the Fermi level is associated with d orbital electronsand suggest the occurrence of the magnetic Lifshitz transition induced by the external magneticfield. In contrast, the Ce 4 f electrons only weakly influence the system properties. Introduction. — Recently discovered CeRh As su-perconductor [1] is one of rare examples of a heavyfermion systems crystallizing in P4/nmm space group. Incontrast to isostructural SrPt As [2] or R Pt Si (where R =Y, La, Nd, and Lu) [3], it does not exhibit the coex-istence of superconductivity and charge density waves.In the case of CeRh As , two anomalies in the specificheat are observed below 1 K [1]. First anomaly (at lowertemperature) is associated with the phase transition fromthe normal state to the superconducting (SC) phase (con-nected with a diamagnetic drop of the ac-susceptibilityand the specific heat jump of the same order of magni-tude as the Bardeen–Cooper–Schrieffer value). Secondanomaly (at higher temperature) is not associated withsuperconductivity, but likely signals some other kind oforder (its T c increases with the in-plane magnetic field).In the presence of the magnetic field perpendicular to theRh As layers, the system exhibits a phase transition in-side the SC state. It is suggested that at the transitionthe parity of superconductivity changes from even to oddone. This leads to H - T phase diagram in a characteris-tic form [1], which can be treated as a generic one for arealization of spin-singlet and spin-triplet SC phases [4].Here, we would like to emphasize that this behavior isalso observed in other systems, where a coexistence oftrivial and topological SC phases can occur [5–7] due tofinite spin-orbit coupling (SOC).The symmetry of the system described by P4/nmm space group is nonsymmorphic with multiple symmetriesprotecting the Dirac points [8]. The similar situation hasbeen recently reported for Dirac semimetals, crystalizingin the same symmetry [9–15]. However, in contrary tothese materials, where a dominant role is played by thesquare nets [16], in the CeRh As compound, there existtwo different types of Rh As layers (Fig. 1). What ismore important, these layers can be treated as planes of the glide symmetry. The purpose of this Letter is tohighlight important role played by the Rh As layers inpossible realization of the hidden orbital order as a resultof two distinguishable of Ce atom positions with respectto the neighboring Rh As layers.The density functional theory (DFT) calculations wereperformed using the Vasp code [17–19] and the
Quan-tum Espresso software [20, 21]. Phonon calculationswere conducted by
Alamode [22] for the thermal distri-bution of multi-displacement of atoms at T = 50 K, gen-erated within the hecss procedure [23]. More details ofnumerical calculation can be found in Refs. [24–27] andin the Supplemental Material (SM) [28]. Crystal structure. — CeRh As crystallizes in theCaBe Ge [29] tetragonal structure ( P4/nmm , spacegroup no. 129). The stacking sequence along c -axis isCe-Rh As -Ce-Rh As -Ce, in which Rh As layers arearranged in two non-equivalent forms: square array ofRh atoms sandwiched between two checkerboard layersof As atoms (one below and one above the Rh layer, al-ternately) and vice versa (square As layer decorated byRh atoms, alternately, from top and bottom). This cor- a bc AsRhAsCeRh AsRhAsRh FIG. 1. The unit cell of the CeRh As crystal structure. Thesystem is composed of two Rh As layers, which separatelyplay a role of the mirrors of the glide symmetry. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b AsRhCeAsRhAsRhCeCe
FIG. 2. Differences of (a) electron localization functionand (b) partial charge density (for states ± . As compound and Rh As layers (Ce atoms removed). responds to upper and lower Rh As layers in Fig. 1.The unit cell of the studied system consists of two for-mula units. From the ab initio calculations, we find thelattice constants as a = 4 . c = 9 . f electrons treated as valence elec-trons, the ground state (GS) of the system is found to benonmagnetic (cf. the SM [28]). Glide symmetry. — CeRh As crystallizes in tetrag-onal structure described with (unusual for heavy fermionsystems) nonsymmorphic space group P4/nmm [30, 31].It supports realization of the unconventional SC gap inthe electronic structure [32], which is protected by thespace group symmetry [33]. This space group exhibitsthe glide symmetry, which is here realized by the Rhand As square nets inside Rh As layers. Therefore, or-bitals of Ce atoms should also exhibit the glide symmetrywith respect to these planes. Simultaneously, the Ce elec-tron orbitals lose the mirror symmetry (along c -direction)due to different environments from “top” and “bottom”sides (cf. Fig. 1). Indeed, this behavior is clearly mani-fested indirectly via modifications of the electron localiza-tion function [34–36] [Fig. 2(a)] or partial charge density[Fig. 2(b)]. In the first case, we calculated a differencebetween localization functions of CeRh As and artificialmaterial Rh As (without Ce atoms), giving us informa-tion about localization of electrons around Ce atoms. Asone can see in Fig. 2(a), localization of electrons aroundCe atom (at center of the system) does not exhibit sym-metry with respect to the ab plane (marked by dashedblue line). Similar property is observed in the case of par-tial charge density coming from states around the Fermilevel (we take range of energies around ± . As lay-ers and effectively realized pseudo-orbitals.As a result, one can expect the occurrence of a hiddenorbital order due to broken reflection symmetry at Ceatoms. Similarly, the hidden order breaks local verticaland diagonal reflection symmetries at the uranium sites AsCeRh
FIG. 3. (a) Phonon dispersion curves along high-symmetrypoints path. (b) Total and partial phonon DOSs, shown bygray and color (as labeled) solid lines, respectively. Resultsfor Ce 4 f electrons treated as valence electrons. in URu Si resulting in the crystal-field states [37]. On-set of the hidden order in CeRh As associated with thesymmetry breaking at the Ce sites could lower the sym-metry from P4/nmm (space group no. 129) to
P4mm (space group no. 99). The same effect can be achievedby including antiferromagnetic order on the sublattice ofCe atoms. Such hidden orbital order can play importantrole in the realization of a non-trivial topological phase.
Lattice dynamics. — The phonon dispersion curvesand DOSs are shown in Fig. 3. As one can see, CeRh As is stable dynamically, i.e., the soft modes (imaginary fre-quencies) are not observed. The irreducible representa-tions at the the Γ point are: E u + A u for acoustic modes,and 4 E u + 4 A u + 5 E g + 3 A g + 2 B g for optic modes.Partial DOSs clearly show that the modes associatedto Ce atoms are located mostly at lower frequencies ina range of 2 ÷ As layers(cf. Fig. 1). Indeed, similar behavior is observed e.g. inthe case of KFe As [39] with I4/mmm symmetry. Theremodes involving K atom located between Fe Se layersare observed only at lower frequencies [Fig. 3(b)]. Suchsituation leads to emergence of nearly flat phonon bandswith weak contribution to lattice thermal conductivity,due to small group velocity and short phonon life-time.Moreover, this physical behavior can be examined ex-perimentally by the phonon life-time measurements [40].One finds that modes mixing vibrations of Rh and Asatoms are of particular interest and are visible in thewhole range of phonon spectrum.The effect of hidden orbital order on phonon spectrais noticable by comparing the results for the P4/nmm and
P4mm space groups (without and with orbital or-der, respectively). By reducing symmetry, we find smallshift of energies and splitting of phonon branches result-ing from slightly different charge distribution on two sites
FIG. 4. Electronic band structures of CeRh As in thepresence of the spin-orbit interaction. Solid blue and graylines correspond to different treatments of the Ce 4 f electrons(as core and valence electrons, respectively). of Ce atoms (Fig. S1 in the SM [28]). This effect couldbe verified in the future by the inelastic scattering mea-surements [41–43]. Electronic band structure. — Calculated electronicband structure in the presence of the SOC is presented inFig. 4. The absence of magnetic order in CeRh As leadsto spin degeneracy of electronic bands in the P4/nmm symmetry. For instance, at k z = 0 the degeneracy of thebands is preserved at the X and M points. Additionally,along the Γ–Z direction the bands cross below the Fermilevel (marked by the green arrow in Fig. 4). Similar be-havior is observed in the nonmagnetic Dirac semimetals(with the P4/nmm symmetry), like CeSbTe [13]. How-ever, note that the Dirac point and non-trivial topologycan be expected even in canonical heavy fermion systemslike CeCoIn [44] with the P4/mmm symmetry.CeRh As can be compared with SrPt As [45, 46] andLaPt Si [46], both crystallizing in the same P4/nmm symmetry. The SOC has weaker impact on the bandstructure of CeRh As than for SrPt As [45, 46] andis comparable with the impact on the band structure ofLaPt Si [46] (cf. Fig. S2 in the SM [28]). These changescan be a consequence of modifications of chemical com-position, due to the mass dependence of the SOC [47, 48].The band structure shows the shift of the Fermi levelin CeRh As to lower energies w.r.t. SrPt As [45, 46]and LaPt Si [46]. In both latter compounds, the bandstructures around the Fermi level are associated with theatoms located in layers, i.e., Pt and As in SrPt As [45,46], as well as Pt and Si in LaPt Si [46]. Similarly, theband structure around the Fermi energy results mainlyfrom the Rh As layers in CeRh As . This is well visiblewhen we compare the band structure of real CeRh As FIG. 5. (a) The Fermi surface of CeRh As and (b)–(e) itsseparate pockets. Results in the presence of the spin-orbitcoupling. with an artificial system without Ce atoms (i.e. con-taining only two Rh As layers), cf. Fig. S3 in theSM [28]. The main features of the artificial Rh As sys-tem are conserved and well visible in the band structureof CeRh As . The differences arise only from 5 d electronlevels of Ce atoms occupied by one electron. Fermi surface. — The Fermi surface of CeRh As [Fig. 5(a)] is composed of four pockets [see Figs. 5(b)–5(e)]. Two of them exhibit three dimensional (3D) char-acter [cf. Fig. 5(b) and (e)]. The layered structure ofCeRh As leads also to emergence of two pockets withquasi-two dimensional features [cf. Fig. 5(c) and (d)].These features are observed also in other heavy fermionsystems, like CeCoIn [49–55]. However, modifications ofthe quasi-3D pockets caused by the magnetic field can bemore meaningful (because they are sensitive to the mag-netic field), and, as a result, these pockets can play moreimportant role in the physical properties of CeRh As . Role of f electrons. — Comparison of the bandstructures, when Ce 4 f electrons are treated as core orvalence electrons in the DFT calculations, are shown inFig. 4 (solid blue and gray lines, respectively). The bandsconnected with Ce 4 f electrons are located around 2 eV.This is also clearly seen in the electronic DOS shown inFig. 6, where contributions of Ce 4 f electrons are markedby the orange area. Thus, one concludes that, in prac-tice, a type of the treatment does not change electronicband structure or DOS around and below the Fermi level.As a result of this, the Ce 4 f electrons should play a lessimportant role in low energy description of the physicalproperties of CeRh As than in other heavy fermions sys-tems [56] (such as, e.g. URu Si [57] or CeRh Si [58] –both with I4/mmm symmetry), where bands constructedof 4 f electrons are located around the Fermi level [59].Regardless of this, the lattice constants found in thenumerical calculations depend on the description of 4 f electrons of Ce atoms (cf. Tables I and II in the SM [28]).Better agreement of numerical results with experimentaldata is obtained when the 4 f electrons are treated asvalence electrons. This behavior is well known and asso-ciated with a formation of stronger bonding in systems FIG. 6. Electronic DOS in the presence of the spin-orbitcoupling. Solid blue and dashed black lines correspond todifferent treatments of the 4 f electrons originating from Ceatoms (as valence and core electrons, respectively). For calcu-lation with Ce 4 f electrons as valence ones, their contributioninto DOS is marked by the orange color area. including f electrons, what leads to a decreased volume. Magnetic instability. — The GS found from the abinitio calculations is nonmagnetic. However, Ce atomshave non-zero magnetic moments of magnitude below0 . µ B . This can be a symptom of magnetic instabilityof the system, which could lead to magnetic ordering atlower temperatures. Regardless of the GS, the numeri-cal calculations with initially enforced different magneticorders uncover structures with lattice constants bettermatching experimental values (cf. Tab. I in the SM [28]).These enforced initial magnetic orders can discriminatebetween two Ce atomic sites and support the realizationof previously mentioned hidden orbital order. However,this case is out of the scope of this work. Lifshitz transition. — We emphasize that severalbands are located close to the Fermi level. Here, oneshould particularly point out the dispersionless bandalong the Γ–M direction or the band crossing the Fermilevel at the R point (solid and dashed red rectangles inFig. 4, respectively). External magnetic field, due to theZeeman effect and band splitting, can lead to emergenceof a new, fully polarized Fermi pocket or disappearanceof the existing one. This phenomenon, called magneticLifshitz transition (MLT), was described in the context ofhigh-magnetic field phase observed in FeSe [60]. Duringthe Lifshitz transition the modification of the Fermi sur-face topology is observed, which could be realized hereby the external magnetic field [60, 61].To mimic a tendency of CeRh As to the realizationof MLT, we performed calculations with non-equal num-ber of spin-up and spin-down electrons, i.e., in a spin-polarized state (cf. Fig. S4 in the SM [28]). In this case,we assume the spin imbalance equal to one electron perthe conventional unit cell. The new split bands (for dif-ferent spins) are shifted by approximately 0 .
13 eV eachfrom the initial band position. The artificially introducedpolarization in the system leads to a shift of some splitbands to lower energies and to emergence of a new Fermipocket formed by the band around R point. Finally, note that the mentioned band at the M point islocated nearly 0 .
05 eV above the Fermi level, which cor-responds to relatively large external magnetic field. How-ever, one expects that the arising magnetic order may inturn induce sufficient internal “effective” magnetic field.This resembles the antifferomagnetic-like order in otherheavy fermion compounds (in the presence of the externalmagnetic field). As an example one can mention here the Q -phase in CeCoIn [62–64]. We underline that the mag-netic field at which the MLT occurs has approximatelyconstant value [60, 61, 65]. Experimentally, the transi-tion observed in CeRh As occurs also at approximatelyconstant magnetic field [1]. Conclusions. — Summarizing, by using the ab ini-tio techniques we showed relatively weak impact of the4 f electrons of Ce atoms on the physical properties ofthe CeRh As compound. The specific crystal structureas well as the previous investigations on heavy fermioncompounds with the same P4/nmm symmetry suggest aprominent role of the Rh As layers in the properties ofthis compound. Their impact is visible in the phononspectra (where Ce modes are located only at lower fre-quencies due to the cage-like structure) and in the elec-tronic spectra, with Ce 4 f states ∼ As layers, theysupport emergence of hidden orbital ordering. Thus, twodistinguishable Ce sublattices with different effective “or-bitals” can be found. Orbital order should occur inde-pendently of the external magnetic field.From the observed small band splitting we concludethat the spin-orbit coupling is relatively weak. However,electronic band structure of CeRh As reveals specifictopological properties of this system. First, we can ob-serve band crossing (the Dirac points), approximately1 eV below the Fermi level. Second, the bands near theFermi level support the hypothesis of the magnetic Lif-shitz transition. At non-zero magnetic field, the new fullypolarized Fermi pocket can emerge, and the spin-tripletpairing (topological superconducting phase) could arise.We kindly thank Dominik Legut for insightful discus-sions. Some figures in this work was rendered using Vesta [66] and
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Electronic and dynamical properties of CeRh As :Role of Rh As layers and expected hidden orbital order Andrzej Ptok, Konrad J. Kapcia, Pawe(cid:32)l T. Jochym, Jan (cid:32)La˙zewski, Andrzej M. Ole´s, , and Przemys(cid:32)law Piekarz Institute of Nuclear Physics, Polish Academy of Sciences,W. E. Radzikowskiego 152, PL-31342 Krak´ow, Poland Faculty of Physics, Adam Mickiewicz University in Pozna´n,Uniwersytetu Pozna´nskiego 2, PL-61614 Pozna´n, Poland Institute of Theoretical Physics, Jagiellonian University,Profesora Stanis(cid:32)lawa (cid:32)Lojasiewicza 11, PL-30348 Krak´ow, Poland Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Dated: February 5, 2021)In this Supplemental Material we present additional results, in particular concerning: • The details of numerical calculations in the Section below. • Lattice constants obtained from presented ab initio calculations concerning Ce 4 f electrons as valence and coreelectrons, respectively (in Tables I and II). • Effects of the orbital order (i.e., lowering of the system symmetry) on the phonon band structure in Fig. S1. • Comparison of the band structures in the absence and in the presence of the spin-orbit coupling for CeRh As compound in Fig. S2. • Comparison between the band structures of nonmagnetic CeRh As compound and an artificial system withoutCe atoms (i.e, not existing in the nature Rh As system) in Fig. S3. • Comparison between the band structures of the non-polarized ground state and system with fixed polarization(for CeRh As compound) in Fig. S4. • Emergence of the Fermi surface along A–M direction during magnetic Lifshitz transition in Fig. S5.Phonon spectra in Fig. S1 are obtained with treatment of 4 f electrons of Ce atoms as valence electrons, while resultspresented in Figs. S2–S5 are obtained with 4 f electrons in Ce atoms treated as core electrons, which does not changequalitatively electronic structure of CeRh As system (cf. discussion in the main text). DETAILS OF NUMERICAL CALCULATION
The first-principles (DFT) calculations are preformed using the projector augmented-wave (PAW) potentials [24]implemented in the Vienna Ab initio Simulation Package (
Vasp ) code [17–19]. The calculations are made withingeneralized gradient approximation (GGA) in the Perdew, Burke, and Ernzerhof (PBE) parametrization [25]. Thesummation in reciprocal space was performed over 18 × × k -point grid generated with the Monkhorst–Packscheme [26]. The energy cutoff for the plane-wave expansion is set to 350 eV. The crystal structure as well as atompositions were optimized in the conventional unit cell with Ce 4 f electrons treated as valence electrons. As a breakcondition of the optimization loop, we take energy difference of 10 − eV and 10 − eV for ionic and electronic degreesof freedom, respectively.The optimized lattice constants read a = 4 . c = 9 . a = 4 . c = 9 . / , / , . / , / ,
0) and (1 / , / , . / , / , /
2) and (1 / , / , . c , a , c , b , and c , respectively.The electronic band structures are also evaluated within the Quantum Espresso [20, 21]. In this case, duringcalculations we use pseudopotentials developed in a frame of
PSlibrary [27]. Additionally, we use the cutoff forcharge density and wave function with the nominal value increased by 100 Ry.Phonon calculations were performed in the supercell with 40 atoms containing 2 × × Alamode software [22].Calculations are performed for the thermal distribution of multi-displacements of atoms at T = 50 K, generatedwithin hecss procedure [23]. Energy of the one hundred different configurations of the supercells and the Hellmann-Feynman forces acting on all atoms are determined using Vasp . In calculated dynamical properties, we includecontributions from harmonic and cubic interatomic force constants to phonon frequencies.
TABLE I. Experimental (taken from Ref. [1]) and theoretical ( ab initio ) lattice constants obtained in the absence of magnetism(NM), assuming antiferromagnetic (AFM) order (in two directions), and assuming ferromagnetic (FM) order (in two directions)as initial magnetic orders. During calculations, the final states conserved assumed magnetic orders (AFM and FM with finalmagnetic moments 0 . µ B and 0 . µ B at Ce atoms, respectively). In contrary to this, calculations starting from the NMstate (all magnetic moments equal zero) lead finally to small magnetic moments with magnitude smaller than 0 . µ B . Forthe theoretical results we show also δE , denoting difference of energy between ground state (GS) and given magnetic order.Results obtained with treatment of 4 f electrons of Ce atoms as valence electrons and in the presence of the spin-orbit coupling.exp. [1] NM AFM FM— M || ab M || c M || ab M || cδE (meV) — (GS) 0.000 3.135 4.082 4.447 4.606a (˚A) 4.2802 4.2801 4.2804 4.2802 4.2800 4.2798c (˚A) 9.9752 9.8616 9.9788 9.9773 9.9761 9.9766TABLE II. The same as in Table I, but from ab initio calculations with treatment of 4 f electrons of Ce atoms as core electrons.Independently of the initial magnetic moments configurations, the final states do not exhibit any non-zero magnetic moments.exp. [1] NM AFM FM— M || ab M || c M || ab M || cδE (meV) — 0.108 0.112 0.116 (GS) 0.000 0.047a (˚A) 4.2802 4.3243 4.3243 4.3243 4.3243 4.3243c (˚A) 9.9752 10.0442 10.0442 10.0442 10.0442 10.0442 FIG. S1. Effects of the hidden orbital order on phonon spectra — shown by width of line corresponding to the projection ofthe polarization vector on the 1st and 2nd Ce atoms. In the case of two non-distinguishable Ce atoms the
P4/nmm symmetryexists. Introduction of the orbital order leads to realization of two sub-lattices, distinguishable by realized pseudo-orbitals atCe atoms. A main role on realization of this orbital order is played by two non-equivalent Rh As planes surrounding Ceatoms from top and bottom. Assuming P4/nmm symmetry (a), the phonon projection on 1st and 2nd Ce atoms (red and bluecolor, respectively), are non-distinguishable (system is described by 64 symmetry operations). Introduction of orbital orderleads to lowering of the symmetry of the system to
P4mm symmetry (b). In this case system undergo only a half of symmetryoperations, i.e., 32 operations. This is well visible by projection of polarization vectors on 1st and 2nd Ce atoms in a form ofsplitting red and blue lines. From this, degeneracy of the band structure in high symmetry points are lift [e.g., cf. green circlesat (a) and (b)]. This effect is similar to observed in electronic band structure, e.g., as emergence of the Dirac semi-metals afterintroducing magnetic order.
FIG. S2. Comparison between the electronic band structures of the non-magnetic ground state of CeRh As in the absenceand in the presence of spin-orbit coupling (solid orange and blue lines, respectively). 4 f electrons of Ce atoms are treated ascore electrons. FIG. S3. Comparison between the electronic band structures of the non-magnetic ground state of CeRh As (solid gray lines)and an artificial structure without Ce atoms (blue solid lines). Zero-energy denotes the Fermi level of CeRh As . Green arrow(on the right) shows location of the Fermi level for the artificial system. Results obtained in the presence of the spin-orbitcoupling. 4 f electrons of Ce atoms are treated as core electrons. FIG. S4. Comparison between the electronic band structures between the non-magnetic ground state (i.e., in the absenceof magnetic polarization; solid gray lines) and nonphysically polarized state (difference between electrons with opposite spinsequal one electron per the conventional unit cell; dispersion curves for spin-up and spin-down electrons are shown by solid redand blue lines respectively). Results obtained in the absence of the spin-orbit coupling for CeRh As . 4 ff