Gradual pressure-induced enhancement of magnon excitations in CeCoSi

CeCoSi is an intermetallic antiferromagnet with a very unusual temperature-pressure phase diagram: at ambient pressure it orders below $T_{\mathrm{N}} = 8.8$ K, while application of hydrostatic pressure induces a new magnetically ordered phase with exceptionally high transition temperature of $\sim40$ K at 1.5 GPa. We studied the magnetic properties and the pressure-induced magnetic phase of CeCoSi by means of elastic and inelastic neutron scattering (INS) and heat capacity measurements. At ambient pressure CeCoSi orders into a simple commensurate AFM structure with a reduced ordered moment of only $m_{\mathrm{Ce}} = 0.37(6)$ $\mu_{\mathrm{B}}$. Specific heat and low-energy INS indicate a significant gap in the low-energy magnon excitation spectrum in the antiferromagnetic phase, with the CEF excitations located above 10 meV. Hydrostatic pressure gradually shifts the energy of the magnon band towards higher energies, and the temperature dependence of the magnons measured at 1.5 GPa is consistent with the phase diagram. Moreover, the CEF excitations are also drastically modified under pressure.


I. INTRODUCTION
Ce-based intermetallic compounds represent a rich playground for exploration of quantum critical phenomena [1][2][3]. The ground state of these materials originates quite often from a competition between RKKY interaction and Kondo screening, which tend to create long-range magnetically ordered and nonmagnetic heavy-fermion states, respectively. The delicate balance between RKKY and Kondo effects can be quite easily tuned by an external tuning parameters, e.g. composition, uniaxial or hydrostatic pressure, magnetic field etc. Usually, application of hydrostatic pressure enhances the coupling between the conduction electrons and the localized Ce moments J cf , and therefore, drastically increases the strength of the Kondo effect (T K ∝ exp(− 1 2J cf )) leading to a reduced magnetic ordering temperature and shifting the ground state of the material closer towards a nonmagnetic heavy-fermion state [4,5].
However, in several recent works it was shown, that CeCoSi represents an intriguing counterexample to this paradigm [6,7]. This material crystallizes in the tetragonal CeFeSi structure (space group P4/nmm) and the cerium moments order antiferromagnetically below the T N = 8.8 K [8,9]. Results of powder neutron diffraction measurements revealed a commensurate antiferromagnetic structure in isostructural CeCoGe with a simple antiferromagnetic stacking of FM Ce planes along the c-axis [10], but the information about the magnetic structure of CeCoSi is absent to the best of our knowledge. Resistivity measurements under hydrostatic pressure [6] have shown that the application of rather moderate pressure of only ∼ 0.6 GPa induces a new magnetically ordered phase with exceptionally high transition temperature T c ≈ 40 K (see the phase diagram in Fig. 9). The pressure-induced phase has a dome shape and the T c changes only slightly up to ∼ 1.7 GPa, whereas upon further pressure increase T c gets rapidly suppressed and a * nikitin@cpfs.mpg.de quantum critical point, characterized by a divergence of resistivity parameters A and ρ 0 , was found at ∼ 2.2 GPa [6]. A nonmagnetic Fermi-liquid state was observed at higher pressures.
In a recent study on single crystals, a very weak anomaly was observed in the specific heat and in the susceptibility at about 12 K and was proposed to be quadrupolar order [11]. Subsequent NMR and NQR results at high pressure indicate that the high-T transition under pressure is a weak structural transition [12]. Its primary order parameter was also proposed to be an antiferroquadrupolar one. However Ce 3+ is a Kramers ion, and in solids its J = 5/2 multiplet is split into 3 Kramers doublets, which do not bear a quadrupolar degree of freedom. A quadrupolar order is then only possible by mixing excited CEF doublets, which requires the excited CEF states to be at low energy, of the order of the quadrupolar ordering temperature. However preliminary results indicated the CEF splitting to be much larger, larger than 100 K [6,11], at least at ambient pressure. That would make a standard quadrupolar ordering not only at 12 K, but also at 35 K very unlikely. In order to clarify this question, reliable information on the CEF excitation energies is crucial.
It is worth noting that such a jump-like drastic increase of the transition temperature under the application of very moderate pressure is highly unusual for Ce-based metals and has no simple explanation in terms of RKKY/Kondo competition, and some authors [6] proposed a meta-orbital transition scenario to describe the appearance of a pressure-induced ordered phase. The concept of the meta-orbital transition was proposed by Kazumasa Hattori [13]. He investigated a two-orbital Anderson lattice model (orbital energy splitting is induced by the CEF effect) with Ising orbital intersite exchange interactions using a dynamical mean-field theory. It was shown, that if the hybridization between the groundstate f -electron orbital and conduction electrons is smaller than the one between the excited f -electron orbital and conduction electrons at low pressures, the occupancy of the two orbitals changes steeply upon application of pressure.
In other words, the excited CEF excitations, which typically had been ignored, because in most cases the lowest excited CEF state is well separated to the ground state, may start to contribute to the ground state properties and induce the transition. Such a meta-orbital transition has been theoretically predicted to happen in CeCu 2 Si 2 [14], but no experimental verification exists so far in any compound. Therefore, knowledge of the CEF splitting scheme, the magnon excitations and their pressure evolution can provide crucial information about the unusual physics of CeCoSi, which might be the first realization of a material exhibiting a meta-orbital transition.
To address these questions we synthesized polycrystalline samples of CeCoSi and its nonmagnetic counterpart LaCoSi. Then, we characterized the samples using neutron diffraction and specific heat measurements. The magnetic excitation spectra were investigated by means of elastic and inelastic neutron scattering under hydrostatic pressures up to 1.5 GPa.

II. EXPERIMENTAL DETAILS
The polycrystalline samples of CeCoSi and its nonmagnetic counterpart LaCoSi were synthesized from elemental Ce (La), Co and Si materials mixed in stoichiometric ratios using arc-melting technique, and then annealed for ∼2 weeks at a temperature close to 1200 • C (the details are given in [6]). The resulting materials were examined using x-ray powder diffraction and energy dispersive x-ray spectroscopy analysis (EDX). The EDX measurements have shown that after the annealing, the majority of the sample consists of CeCoSi phase, with a small inclusion of an elemental Ce and CeCo 2 Si 2 phase, but according to powder diffraction, the concentration of impurity phases is below 2 %.
Neutron powder diffraction measurements were performed at the diffractometer E6 (HZB facility). The powder diffraction patterns were collected at T = 1.7 and 20 K with λ = 2.41 Å. Inelastic neutron scattering (INS) measurements at ambient pressure were carried out at the timeof-flight (TOF) spectrometers IN4 and IN6 of the Institut Laue-Langevin in the temperature range 1.7-150 K. The incident neutron energies were fixed to E i = 31.95 meV and E i = 3.86 meV at IN4 and IN6 experiments, respectively. In these experiments we measured ∼ 10 g of powder samples.
To study the effect of hydrostatic pressure on the spin excitations in CeCoSi we performed two INS experiments using the cold TOF spectrometers LET [15] at ISIS neutron source and CNCS [16,17] at SNS, ORNL. In order to apply  At CNCS experiment we measured magnetic excitations with two neutron incident energies E i = 6.15 meV and 25.23 meV to study magnon and CEF excitations, respectively. The measurements were performed at the base temperature of the orange cryostat, T = 1.7 K, and at three pressures of P = 0.2, 0.6 and 1 GPa. The pressure cell used in the CNCS experiment had an optical window, which allowed us to monitor the pressure by means of a ruby fluorescence method [18,19].
For our experiment on CeCoSi the LET time-of-flight spectrometer had the special advantage of the so-called multirepetition mode [15], which allows one to perform the measurements with several incident neutron energies at the same pulse. Thereby, we could optimize the incident neutron energies in a way to simultaneously measure magnon and CEF excitations, and therefore decrease the counting time needed for a scan at a given temperature and pressure by a factor of two. In our experiment we collected data with three E i = 3.43, 6.8 and 19 meV in the high flux mode [20]. To further decrease the background scattering we used a small radial collimator with Gd 2 O 3 painted blades and acceptance diameter of ∼ 4 mm, which was installed directly on the pressure cell inside the cryostat. The data were collected in the temperature range 1.7-100 K. The pressure was calculated from the applied press force taking into account the data from CNCS experiment.
The recorded data were reduced and analyzed using JANA2006 [21], DAVE [22], MANTID [23] and LAMP [24] software packages. Specific-heat measurements were carried out using a commercial PPMS from Quantum Design at temperature range 1.8-300 K.

A. Magnetic structure at ambient pressure
To characterize the magnetic structure of CeCoSi we measured neutron powder diffraction using the E6 diffractometer at HZB. The powder diffraction patterns were collected at T = 1.7 and 20 K, i.e. below and above the T N and the experimental results are shown in Fig. 2(a). One can see that with decreasing temperature a new weak magnetic satellite appears at 2θ ≈ 34 • (see inset in Fig. 2(b)). The peak can be indexed as k = (100) (note that the (100) nuclear reflection is forbidden for the P4/nmm space group).
We performed magnetic group representation analysis using JANA2006 software and found that the magnetic symmetry group P mm n provide the best fit of our dataset. The low-temperature diffraction pattern along with the calculated curve are shown in Fig. 2 (b), and one can see a good agreement (R nuc = 2.45 % and R mag = 4.84 %). The lattice parameters of the CeCoSi at T = 1.7 K were determined to be a = 3.9967(8) Å and c = 6.937(1) Å with the space group P4/nmm (values in brackets denote the 1σ error of the least-squares fitting throughout the paper).
The magnetic structure (schematically shown in Fig. 1) turned out to be a collinear antiferromagnetic stacking of ferromagnetic Ce layers along the c-axis, with the moments pointing along the [100] direction. The ordered Ce moment is as small as m Ce = 0.37(6) µ B . It is worth noting that even though our results are consistent with data obtained for the isostructural CeCoGe [10], both analysis are based on a single (100) magnetic reflection, and therefore should be considered with care. Further single-crystal neutron diffraction experiments are highly desirable to confirm the proposed magnetic structure.

Magnon excitations
To explore the low-energy excitations of CeCoSi we performed powder INS measurements at the spectrometer IN6 at ILL at T = 1.7 − 100 K. Figure 3 shows the energy spectra collected with E i = 3.86 meV and integrated within Q = [1-1.5] Å −1 . The low temperature spectrum consists of a strong gapped magnon band at E ≈ 2.3 meV. With increasing temperature above T N the gap closes and the spectral weight transfers to the quasielastic channel as expected for a conventional antiferromagnet.
Note that the characteristic energy of the magnetic excitations in CeCoSi is ∼ 2.3 meV, which is approximately three time higher than the energy associated with the Neel ordering of Ce moments (T N = 8.8 K ≈ 0.75 meV). This may indicate the presence of magnetic frustration or lowdimensional magnetic behavior of the system. Unfortunately, the powder spectrum appears to be almost featureless, which does not allow us to extract specific details of the underlying magnetic interactions. Therefore, the determination of the low-energy spin Hamiltonian, which should contain at least 3 exchange interaction plus 3 parameters describing the anisotropy of the exchanges, requires further detailed single-crystal INS measurements.

CEF excitations
Ce 3+ in CeCoSi has a J = 5/2 ground state multiplet, which splits into three doublets under the action of a tetragonal CEF. Thereby, one can expect to observe two CEF transition in an INS spectrum. To characterize the CEF Hamiltonian in CeCoSi we performed INS measurements of CeCoSi and LaCoSi at the TOF instrument IN4 of the Institut Laue-Langevin. The spectra of both samples collected at T = 1.7 K with E i = 31.95 meV are displayed in Figs. 4(a, b) [25]. The spectrum of LaCoSi shows strong optical phonon bands, with their intensities increasing with Q because of the phonon form factor. The spectrum of CeCoSi shows similar phonon bands at large Q, but in addition exhibits broad magnetic excitations at energies E ≈ 10-20 meV.
To obtain the magnetic signal of CeCoSi -S M (Q, ħ hω) we directly subtracted the scaled phonon contribution estimated using the LaCoSi data [26]. To find the scaling coefficient α, we took an energy cut at high momentum, which is dominated by the phonon contribution in both La and Ce sam- ples, because of the phonon and magnetic form-factors. To compensate the difference of the sample masses and scattering lengths we scaled the LaCoSi dataset to get the best agreement between the spectra. Then, we used the obtained coefficient α to scale the LaCoSi spectrum in the whole Q-range and subtract it from the CeCoSi spectrum S M (Q, ħ hω) = S Ce (Q, ħ hω) − αS La (Q, ħ hω). The magnetic spectrum after subtraction is displayed in Fig. 4 (c).
To qualitatively extract the positions of CEF peaks we integrated the magnetic spectrum at Q = [1-3] Å −1 . Two representative curves taken at T = 1.7 and 15 K are shown in Fig. 4(d). One can see, that the peak shape is rather asymmetric and can not be fitted with a single peak function and therefore, to qualitatively extract the peak positions we fitted the curves with two Gaussian peaks. We found that the peaks are located at E 1 = 10.49(6) meV and E 2 = 14.1(2) meV at T = 15 K, i.e. above T N , and their positions slightly shift in the antiferromagnetic phase at T = 1.7 K (E 1 = 11.78(6) meV and E 2 = 14.8(3) meV) due to the splitting of the ground state doublet by an exchange field. It is worth noting that the CEF excitations are broader then the instrumental resolution, which may be due to the interaction with phonons [27], hybridization with the conduction band electrons or magnetic dispersion.

C. Specific heat
To check whether the broad asymmetric peak observed in the INS spectra indeed consists of two CEF excitations we carefully measured the heat capacity of the CeCoSi and LaCoSi samples over a wide temperature range T = 1.8-300 K using a PPMS. Specific heat of LaCoSi sample was used as a blank to estimate the phonon contribution and calculate the magnetic contribution in CeCoSi.
The raw data and the magnetic heat capacity C mag after subtraction of the phonon contribution are shown in Fig. 5. C mag (T ) exhibits two anomalies: a sharp peak at T N and a broad Schottky-like anomaly with a maximum at T * = 51.5 K. However, the absolute value of the specific heat at T * is larger compared to the 3.65 J/mol·K, expected for a simple Schottky anomaly, indicating that the anomaly is not caused by a single CEF excitation. To qualitatively calculate the magnetic specific heat of CeCoSi we used the standard equation for the specific heat of a discrete n-level system: where E i are energies of states and is a partition function. Using Eq. (1) and transition energies E 1 = 10.49(6) meV and E 2 = 14.09(21) meV determined by INS above T N we calculated the magnetic specific heat of CeCoSi, and the results are plotted in Fig. 5 by the red line. The good agreement between calculated and measured specific heat curves provides another evidence that the CEF transition energies determined by INS are valid. The low-temperature part of the specific heat contains information about the magnon density-of-state due to the magnetic ordering. For instance, the specific heat of the 3D Heisenberg AFM follows a simple power law C ∝ T 3 due to the 3D gapless dispersion with ħ hω ∝ k. On the other hand, if the system has a magnon gap one would expect an activation behavior C ∝ e −∆/k B T . We fitted the low-T part of our specific heat curve (T < 2 3 T N ) to different models and found that the best agreement can be obtained using a combination of exponential and power law functions plus electronic contribution: The fitted curve is shown in Fig. 5(b) by orange line, and one can see the perfect agreement between the experimental and calculated curves. The fitted parameters were found to be α = 2.1(2), γ = 10.8(9) mJ/mol·K 2 and ∆/k B = 9(1) K. It is interesting to note that the gap determined from the specific heat measurements is very close to the ordering temperature of CeCoSi.

D. Magnetic excitations under hydrostatic pressure
We start our presentation of the pressure-induced evolution of the spin dynamics in CeCoSi with the spectra collected at the LET spectrometer. Note that the pressure cell produces a massive background signal. In order to determine the nonmagnetic scattering we used the LaCoSi spectrum measured under similar conditions and the procedure described in Sec. III B 2 assuming that S M (Q, ħ hω) = S Ce (Q, ħ hω) − α · S La (Q, ħ hω). However, even without the subtraction a strong broad excitation band at E ≈ 4 meV is clearly seen in the spectrum (the raw spectra obtained on the LET spectrometer are presented in Appendix A, Fig. 11(ad)). As was discussed above, in this experiment we did not have a pressure sensor in the cell, and the pressure of 1.5 GPa was calculated from the applied press force taking into account ∼ 10 % loss, while cooling down to 1.7 K, which results in the relatively large estimated uncertainty of the pressure determination of ∼ 0.25 GPa. For this reason we decided to study the T dependence of the observed mode at fixed P. We subtracted the background and Bose-corrected all obtained spectra measured with E i = 6.8 meV. The resulting χ (ħ hω) curves integrated within Q = [0.5-2.5] Å −1 are shown in Fig. 6 (a). Increasing temperature induces a decrease of the mode intensity, and slightly shifts down the peak position. Fits of these parameters are presented in Fig. 6(b, c) and one can see that the magnon mode intensity disappears below the detection limit at T = 30 K. It is interesting to note that the spectral intensity does not decrease monotonically, but rather has two regimes, which may be related to the existence of the second magnetically ordered phase at lower temperature T c ≈ 12 K (see phase diagram in Fig. 9). In order to check the consistency of our results with the zero pressure data we also measured the spectra at almost ambient condition (P ≤ 0.1 GPa) at 1.7 K. The resulting spectrum along with the 1.5 GPa data and results of the IN4 experiment are shown in Fig. 7. The position of the CEF excitations obtained in the LET experiment perfectly coincides with the IN4 results indicating that we can reliably extract information about both CEF and magnon excitations from the LET data. It is interesting to note that the pressure of P = 1.5 GPa significantly shifts or suppresses the intensity of the CEF excitations as clearly seen in Fig. 7.
We now focus on the pressure dependence in more detail and present data obtained on the CNCS spectrometer. Figure 8 shows the summary of the background subtracted signal at 0.2, 0.6 and 1.0 GPa collected with two incident neutron energies. The E i = 6.15 meV spectra presented in the left panel display a rather strong magnon peak, which position gradually shifts upon increasing pressure. It is worth noting that already at the lowest pressure of 0.2 GPa the position of the peak is slightly higher than the one obtained in our IN6 experiment at ambient pressure.
The high-energy data have much stronger background due to the phonon scattering from the pressure cell. The data after subtracting the background contribution are shown in panel (b) of Fig. 8. At P = 0.2 GPa we found a weak peak at an energy of ∼ 13 meV. Its position is close to E 1 = 11.78(6) meV and E 2 = 14.81(26) meV observed in the IN4 experiment at ambient pressure. The position of the peak also shits to higher energies with pressure. However, the signal-to-noise ratio is much worse in the 25.23 meV dataset compared to the 6.15 meV one, as can be seen from the ratio between the neutron count rate and errorbars in the two panels of Fig. 8, and the 13 meV peak has an intensity, which exceeds the background level by 2-4 standard deviations only. Taking into account that the positions and intensities of the peaks would depend on the details of the subtraction procedure, we would like to point out that the obtained result should be considered with a reasonable caution, because we can not unambiguously proof a magnetic origin of the observed peak, which can be just an artefact of the background subtraction procedure [28].
On the other hand, the fitting of the low-energy peak ( Fig. 8 (a)) is rather robust and self-consistent, independent on subtraction details. Accordingly, we can conclude that the energy of the magnon mode indeed increases with the pressure, whereas the observation of the CEF excitations and their P dependence is much more questionable.

IV. DISCUSSION
Our experimental work on CeCoSi has a dual aim: (i) to characterize the magnetic ground state and excitations of CeCoSi at ambient pressure using a combination of different techniques and (ii) to study how the magnetic excitations evolve with pressure. Analyzing the results of neutron powder diffraction in the AFM phase and at ambient pressure we detected only one weak magnetic satellite peak, which appears below T N and can be indexed as the (100) reflection. This result is consistent with a simple commensurate antiferromagnetic structure, previously proposed for isostructural CeCoGe [10]. The Ce moments are aligned along the [100] direction and carry an ordered moment of only m Ce = 0.37(6) µ B , which is significantly reduced compared to the moment of free Ce 3+ with m Ce = 2.14 µ B . However, in Ce systems the CEF is comparatively strong and therefor the J = 5/2 multiplet is split in such a way that the energy of the first excited CEF level is in general much larger than T N . Then, the size of the ordered moment is limited to that of the CEF ground state doublet, which for the easy CEF direction is in the range 1 -2.5 µ B . But even compared to the lower bound, the observed value is small. This is striking, because the 4 f entropy collected just above T N is close to Rln2, indicating the absence of correlations above T N . Thus, the standard scenarios invoked to account for a reduced size of ordered moments, the presence of Kondo interaction or frustration, does not apply. Another alternative, which is presently discussed for a number of ferromagnetic systems, the ordering along the hard CEF direction, where the available CEF moment can be very small [29,30], seems to be unlikely because susceptibility data do not indicate a strong anisotropy [11]. Thus, the origin of the strong reduction of the ordered moment is a further mystery in this system.
Our INS and specific heat results revealed two CEF levels at E 1 = 10.49(6) meV and E 2 = 14.1(3) meV in the paramagnetic phase which evolve upwards to higher energies below T N . In fact, these results along with m Ce determined from the neutron diffraction can be used to calculate three B m l parameters of the CEF Hamiltonian, but more accurate data on anisotropy and temperature dependence of the magnetic susceptibility are desirable to shrink the parameter space.
In the low-energy spectrum we observed magnon excitations with a characteristic energy of E * ≈ 2.5 meV (29 K) at ambient pressure. It is worth noting that the excitation energy scale of the magnons exceeds by more than three times the ordering temperature of T N = 8.8 K. One possible explanation is a quasi-2D magnetic structure of the material (see Fig. 1) with much stronger exchange interactions within the ab-planes and only weak coupling along the c direction (J c < J ab ). In that case, short-range fluctuations within the a b plane will survive at temperatures above the T N . Indeed, we were able to resolve a broad paramagnon inelastic peak at 9 and 10 K, inline with such a scenario, whereas at higher temperatures all spectral weight is transferred to the T e m p e r a t u r e ( K ) Q C P Fig. 9. Temperature-Pressure phase diagram of CeCoSi from Ref. [6] and the P-dependence of the magnon mode (solid circles) at T = 1.7 K. Shaded areas represent different phases -AFM and two pressure-induced ordered phases (PIOP). Note that the right and left axis are shown on the same scale E = k B T . quasielastic channel.
Summarizing the results of the low-energy INS experiments under pressure, we found that the energy of the magnon mode gradually evolves from 2.5 meV at ambient pressure to ∼ 4 meV at 1.5 GPa (see Fig. 9). At this pressure the energy scale of the magnetic excitations is comparable to the transition temperature of the pressure-induced phase (4 meV ≈ 46 K). Note that these results are not in favor of the metaorbital transition scenario, because the last implies a sharp, abrupt change of the ordered moment and the magnon excitation energy as a consequence, which is in contrast to the gradual pressure-induced evolution observed in our measurements.
The pressure dependence of the CEF excitations is less clear: in the low-pressure CNCS experiment at 0.2 GPa we found a weak peak, close to the positions of the CEF excitations observed at zero-pressure measurements. The peak position changes only slightly with pressures up to 1 GPa. On the other hand, the results of the LET experiment unambiguously showed that at 1.5 GPa the CEF levels move out of their original location. One possible explanation is much worse signal-to-noise ratio in the high-energy CNCS measurements, which cast some doubts on the CNCS results. However, if one looks at the signature in the resistivity, the high-T ordering appears to be something new which appears quite abruptly at P ≥ 1.4 GPa, while the observed effects at lower pressure are different and order of magnitude smaller. Therefore, P 1 = 1.4 GPa was explicitly introduced to highlight this strong change in [6]. Thus the appearance of the strong anomaly in ρ(T ) at P ≥ P 1 may be related to the dramatic change of CEF excitations between 1 and 1.5 GPa, indicating that there is a real strong difference between the orderings above and below the P 1 , as was suggested in [6]. A single crystal neutron diffraction under pressure should be performed to resolve this question and clarify the order parameter of the PIOP.

V. ACKNOWLEDGMENTS
We acknowledge A. S. Sukhanov for stimulating discussions and C. Goodway for help with the high-pressure experiments at ISIS Neutron and Muon source. This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by Oak Ridge National Laboratory. S.E.N. acknowledges support from the International Max Planck Research School for Chemistry and Physics of Quantum Materials (IMPRS-CPQM). Figures 10 and 11 show the raw S(Q, ω) of CeCoSi and LaCoSi samples and the magnetic spectra S M (Q, ω) after phonon/background subtraction measured on CNCS and LET spectrometers, respectively.