arXiv:2007.02129v2 [cond-mat.str-el] 18 Nov 2020

Putative quantum critical point in the itinerant magnet ZrFe4Si2with a frustrated quasi-one-dimensional structure

M. O. Ajeesh,* K. Weber, C. Geibel, and M. NicklasMax Planck Institute for Chemical Physics of Solids, Nothnitzer Str. 40, 01187 Dresden, Germany

(Dated: November 19, 2020)

The Fe sublattice in the compound ZrFe4Si2 features geometrical frustration and quasi-one-dimensionality. We therefore investigated the magnetic behavior in ZrFe4Si2 and its evolutionupon substituting Ge for Si and under the application of hydrostatic pressure using structural, mag-netic, thermodynamic, and electrical-transport probes. Magnetic measurements reveal that ZrFe4Si2holds paramagnetic Fe moments with an effective moment 𝜇eff = 2.18 𝜇𝐵 . At low temperaturesthe compound shows a weak short-range magnetic order below 6 K. Our studies demonstrate thatsubstituting Ge for Si increases the unit-cell volume and stabilizes the short-range order into a long-range spin-density wave type magnetic order. On the other hand, hydrostatic pressure studies usingelectrical-resistivity measurements on ZrFe4(Si0.88Ge0.12)2 indicate a continuous suppression of themagnetic ordering upon increasing pressure. Therefore, our combined chemical substitution andhydrostatic pressure studies suggest the existence of a lattice-volume-controlled quantum criticalpoint in ZrFe4Si2.

I. INTRODUCTION

Itinerant magnetic systems with low dimensionalityand magnetic frustration exhibit enhanced quantum fluc-tuations leading to the emergence of novel and exoticphases displaying unconventional behaviors. Studyingsuch systems is of great importance as it is becomingincreasingly evident that quantum fluctuations play acrucial role in emergent phenomena, including uncon-ventional superconductivity and unconventional metal-lic phases1–9. In order to improve our understanding ofthese phenomena and the importance of dimensionalityand frustration on the ground-state properties investiga-tions of new candidate materials are highly desired.

In this regard, ternary intermetallic 𝐴Fe4𝑋2 (𝐴 = rareearth, 𝑋 = Si, Ge) compounds are interesting candi-dates due to their peculiar crystal structure. These com-pounds crystallize in the ZrFe4Si2-type structure withthe 𝑃42/𝑚𝑛𝑚 space group at room temperature10. Thecrystal structure consists of slightly distorted Fe tetrahe-dra, which are edge shared to form chains along the crys-tallographic 𝑐 axis, as illustrated in Fig. 1. The Fe tetra-hedra are prone to magnetic frustration, and the chainlikearrangement provides the quasi-one-dimensional charac-ter of the magnetic system, rendering the 𝐴Fe4𝑋2 com-pounds excellent candidate materials to study quantumfluctuations and their effect on the physical properties inlow-dimensional frustrated systems.

Previous studies on the 𝐴Fe4𝑋2 family mostly focusedon compounds with magnetic rare-earth ions 𝐴. Low-temperature neutron and x-ray diffraction studies on (Er,Dy, Ho, Tm)Fe4Ge2 revealed that the compounds un-dergo antiferromagnetic ordering at low temperatures re-sulting in complex spin arrangements due to competinginteractions between the magnetic rare-earth and Fe sub-lattices11–21. In all of the compounds, the magnetic or-dering is accompanied by a structural transition fromtetragonal to orthorhombic symmetry. 𝐴Fe4𝑋2 com-pounds with nonmagnetic rare-earth elements are even

FIG. 1: (a) Crystal structure of ZrFe4Si2 viewed along thecrystallographic 𝑐 axis. (b) The chainlike arrangement ofedge-shared Fe tetrahedra viewed along the 𝑏 axis.

less investigated as only powder neutron diffraction stud-ies on YFe4Ge2, LuFe4Ge2, and YFe4Si2 have been re-ported22,23. These compounds also order antiferromag-netically at low temperatures with a simultaneous struc-tural transition from tetragonal 𝑃42/𝑚𝑛𝑚 to orthorhom-bic 𝑃𝑛𝑛𝑚 symmetry. While the existing studies addressthe magnetic structure and the magneto-elastic transi-tions in these compounds, there are no reports on tuningthe magnetic to a non-magnetic ground state by an ex-ternal control parameter.

In this paper, we present an investigation on a mem-ber of the 142 family: ZrFe4Si2. Replacing the rare-earth ions with Zr not only reduces the lattice volumebut also changes the valency of the 𝐴 ion from 3+ to4+. This may lead to a significant change in the elec-tronic structure and, therefore, of the ground state com-pared to rare-earth-containing 142 compounds. Here,we studied the ground-state properties of ZrFe4Si2 usingmagnetic, thermodynamic, and electrical-transport mea-surements on polycrystalline samples. Our results revealshort-range magnetic ordering below 6 K which is stabi-lized into a spin-density wave (SDW) long-range order by

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substituting Ge on Si sites. In addition, we applied exter-nal hydrostatic pressure on ZrFe4(Si0.88Ge0.12)2 to tunethe antiferromagnetically ordered ground state toward anonmagnetic state. Finally, we discuss the temperature–lattice-volume phase diagram in which the magnetic or-dering is suppressed by decreasing lattice volume towarda putative quantum critical point (QCP).

II. METHODS

Polycrystalline samples of ZrFe4Si2 were synthesizedby a standard arc-melting technique on a copper hearth.At first stoichiometric amounts of the constituent ele-ments (at least 99.9% purity) were melted in an arc fur-nace under argon atmosphere, followed by several flippingand remelting of the resulting ingot to ensure homogene-ity. Then the as-cast samples were annealed at 1150∘Cunder a static argon atmosphere for a week. The phasepurity of the annealed samples was checked by powderx-ray diffraction (PXRD) using Cu K𝛼 radiation and ascanning electron micrograph (SEM). Energy dispersivex-ray (EDX) analysis was used to check the stoichiometryof the samples. SEM studies reveal only a small amount(up to 2%) of eutectic phase Fe3Si in our samples, en-abling us to study the intrinsic properties of ZrFe4Si2.PXRD patterns confirm the tetragonal 𝑃42/𝑚𝑛𝑚 struc-ture type with lattice parameters 𝑎 = 6.9916(5) A and𝑐 = 3.7551(5) A, in good agreement with those reportedin the literature10. Polycrystalline samples of the substi-tution series ZrFe4(Si1−𝑥Ge𝑥)2 were also synthesized fol-lowing the same procedure. EDX analysis provides theGe concentrations in the obtained samples as 𝑥 = 0.12(0.1), 0.23 (0.2), 0.34 (0.3), and 0.46 (0.4), where the cor-responding nominal Ge concentrations used in the syn-thesis are given in the parantheses. SEM studies revealedthat these samples contain up to 5% of impurity phasesmainly consisting of Fe3(Si1−𝑥Ge𝑥).

DC magnetization measurements were carried out inthe temperature range between 1.8 and 300 K and inmagnetic fields up to 7 T using a superconducting quan-tum interference device magnetometer (magnetic prop-erty measurement sysytem , Quantum Design). The spe-cific heat was recorded by a thermal-relaxation methodusing a physical property measurement system (PPMS;Quantum Design). The electrical transport experimentswere carried out in the temperature range between 2 and300 K and magnetic field up to 7 T also using a PPMS.The electrical resistivity was measured using a standardfour-terminal method, where electrical contacts to thesample were made using 25 − 𝜇m gold wires and silverpaint.

Electrical-resistivity measurements onZrFe4(Si0.88Ge0.12)2 under hydrostatic pressure wereperformed using a double-layered piston-cylinder-typepressure cell with silicon oil as the pressure transmittingmedium. The pressure inside the sample space wasdetermined at low temperatures by the shift of the

superconducting transition temperature of a pieceof Pb. Electrical resistivity was measured using anLR700 resistance bridge (Linear Research) working at ameasuring frequency of 16 Hz.

III. EXPERIMENTAL RESULTS

A. Physical properties of ZrFe4Si2

In order to understand the ground-state properties ofZrFe4Si2, we have carried out magnetic, thermodynamic,and electrical-resistivity measurements. The tempera-ture dependence of the DC magnetic susceptibility 𝜒(𝑇 )is shown in Fig. 2a. We note that our ZrFe4Si2 sam-ples contain up to 2% eutectic phase Fe3Si, which or-ders ferromagnetically above 800 K. Accordingly, thisimpurity phase induces in the magnetization a ferromag-netic (FM) contribution which, however, saturates at lowfields and is therefore field and temperature indepen-dent above 1 T and below 300 K, respectively24. Thus,the impurity contribution can easily be separated fromthe intrinsic contribution of ZrFe4Si2. Using magnetiza-tion measurements at different fields, this contribution(𝑀FM) is found to be rather small with a saturationmoment of the order of 1 × 10−4𝜇B/Fe. 𝑀FM is thensubtracted from the measured magnetization to obtainthe intrinsic susceptibility as 𝜒(𝑇 ) = [𝑀(𝑇 )−𝑀FM]/𝐻.At high temperatures, 𝜒(𝑇 ) follows a Curie-Weiss be-havior 𝜒(𝑇 ) = 𝐶/(𝑇 − 𝜃W), where 𝐶 and 𝜃W are theCurie constant and the Weiss temperature, respectively.A Curie-Weiss fit to the 𝜒−1(𝑇 ) data (Fig. 2a, right axis)for 100 K < 𝑇 < 300 K yields an effective moment𝜇eff = 2.18 𝜇𝐵 and a Weiss temperature 𝜃W = −85 K.The relatively large value of the effective moment is asignature of fluctuating Fe moments in the paramag-netic state. Moreover, the negative 𝜃W indicates thatthe dominant interactions between the moments are an-tiferromagnetic. At low temperatures, 𝜒(𝑇 ) presents aweak shoulder at around 50 K, followed by a broad peakcentered around 6 K. The specific heat data also showa rounded peak at around 6 K, corresponding to theanomaly in 𝜒(𝑇 ) (see Fig. 2b). However, there is noevident feature in 𝐶𝑝(𝑇 ) at 𝑇 ≈ 50 K, making the pres-ence of any phase transition in this temperature rangeunlikely. The rounded nature of the anomalies in suscep-tibility and specific heat at 𝑇 ≈ 6 K point to short-rangemagnetic order.

In the inset of Fig. 2b the specific-heat data are plottedas 𝐶𝑝(𝑇 )/𝑇 vs. 𝑇 2. The linear region observed between20 and 35 K was fitted with 𝐶𝑝(𝑇 ) = 𝛾𝑇 +𝛽𝑇 3 to obtainthe Sommerfeld coefficient 𝛾 = 150 mJ/molK2. This is avery large value for a transition metal compound, indicat-ing very strong electronic correlation effects. Using the 𝛾value and the low-temperature susceptibility, we obtain aSommerfeld-Wilson ratio 𝑅W = (𝜋2𝑘2𝐵𝜒1.8K)/(𝜇eff𝛾) of4.9, which is enhanced compared to 𝑅W = 1 for the free-electron gas. The enhanced Sommerfeld-Wilson ratio in-

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FIG. 2: (a) Temperature dependence of the DC magnetic sus-ceptibility 𝜒(𝑇 ) of ZrFe4Si2 (left axis). The intrinsic suscepti-bility is obtained by removing the FM impurity contribution,as explained in the main text. The inverse magnetic suscep-tibility 𝜒−1(𝑇 ) is shown on the right axis. The red curve isthe Curie-Weiss fit to the data in the temperature intervalbetween 100 and 300 K. (b) Temperature dependence of thespecific heat of ZrFe4Si2 plotted as 𝐶𝑝/𝑇 vs. 𝑇 . The insetshows 𝐶𝑝/𝑇 vs. 𝑇 2, where the red line is a linear fit to thedata between 20 and 35 K. (c) Temperature dependence ofthe electrical resistivity 𝜌(𝑇 ) of ZrFe4Si2. Inset: an enlargedview of the low temperature region of the 𝜌(𝑇 ) curve (leftaxis) along with the temperature derivative d𝜌(𝑇 )/d𝑇 (rightaxis).

dicates the presence of strong electron-electron magneticcorrelations in ZrFe4Si2.

The temperature dependence of the electrical resistiv-ity 𝜌(𝑇 ) of ZrFe4Si2 is shown in Fig. 2c. 𝜌(𝑇 ) decreasesmonotonically upon cooling with a strong negative curva-ture below 100 K, probably originating from strong mag-netic correlations. Preliminary Mossbauer and muon-spin relaxation (𝜇SR) experiments indicate the onset ofdynamic correlations below 100 K and the onset of weakstatic magnetic order below 8 K25. At low temperatures,

𝜌(𝑇 ) presents only an extremely weak feature around 6 K,as seen in the temperature derivative of the resistivityd𝜌(𝑇 )/d𝑇 plotted in the inset of Fig. 2c. Such a weakanomaly in resistivity is also consistent with short-rangemagnetic order.It is also important to note that, unlike other com-

pounds in the 𝐴Fe4𝑋2 family with trivalent 𝐴 whichshow a structural transition associated with the magneticordering, temperature-dependent PXRD data do not re-solve any structural transition in ZrFe4Si2 around 6 K26.Therefore, the low temperature properties of ZrFe4Si2 arenot related to a structural phase transition.

B. Tuning the ground state of ZrFe4Si2 by Gesubstitution

The weak, short-range ordered magnetic ground statein ZrFe4Si2 raises the question of whether the material issituated close to a QCP connected to the disappearanceof long-range magnetic order, especially since such long-range order has been observed in other members of the142 family22,23. To investigate this possibility, we havecarried out a Ge substitution study. As Ge is larger thanSi, varying the Ge content in ZrFe4(Si1−𝑥Ge𝑥)2 providesa tuning parameter for systematically increasing the unit-cell volume.To this end, polycrystalline samples of

ZrFe4(Si1−𝑥Ge𝑥)2 with Ge concentrations 𝑥 = 0.12,0.23, 0.34, and 0.46 were synthesized, and their magneticproperties were studied using various physical probes.We note that our attempts to synthesize samples with𝑥 = 0.5 and 0.6 resulted in phase separation, indicatingthat compounds with large Ge content are unstable.This is corroborated by the fact that, to our knowledge,pure ZrFe4Ge2 has not been reported in the literature.

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FIG. 3: Lattice parameters 𝑎 (left axis) and 𝑐 (right axis)of the investigated ZrFe4(Si1−𝑥Ge𝑥)2 samples plotted againsttheir Ge content 𝑥. The inset shows the change in the latticevolume 𝑉 with Ge substitution. The solid line in the inset isa linear fit to the data.

Figure 3 depicts the change in the lattice parametersof ZrFe4(Si1−𝑥Ge𝑥)2 with increasing Ge content 𝑥, ex-

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FIG. 4: Temperature dependence of (a) DC magnetic susceptibility 𝜒(𝑇 ), (b) specific heat 𝐶𝑝(𝑇 )/𝑇 , and (c) normalizedelectrical resistivity 𝜌(𝑇 )/𝜌300K of ZrFe4(Si1−𝑥Ge𝑥)2 for several Ge concentrations. The intrinsic susceptibility was obtainedby removing the FM impurity contribution, as explained in the main text. The position of the peak maxima in the 𝜒(𝑇 ) and𝐶𝑝(𝑇 )/𝑇 data were taken as the transition temperatures. Corresponding resistive transition temperatures, estimated from theminima in the temperature derivative d𝜌(𝑇 )/d𝑇 , are marked by the arrows.

tracted from PXRD measurements. Lattice parameters𝑎 (left axis) and 𝑐 (right axis) monotonically increase withincreasing Ge content. Ge substitution with 𝑥 = 0.46 re-sults in an increase of 𝑎 and 𝑐 by 0.6% and 1.2%, respec-tively. The unit-cell volume 𝑉 increases nearly linearly,reaching a 2.4% increase for the compound with 𝑥 = 0.46(see the inset of Fig. 3). These results confirm that, asexpected, the unit-cell volume of ZrFe4(Si1−𝑥Ge𝑥)2 con-tinuously increases with Ge substitution.

The physical properties of ZrFe4(Si1−𝑥Ge𝑥)2 with dif-ferent Ge concentrations were investigated using mag-netization, specific heat, and electrical-resistivity mea-surements. The temperature dependence of magneticsusceptibility is shown in Fig. 4a. The samples in thesubstitution series contain up to 5% impurity phases ofFe3(Si1−𝑥Ge𝑥), which order ferromagnetically between800 K and 600 K24,27. Their temperature indepen-dent, saturated magnetization contribution 𝑀FM wassubtracted to obtain the intrinsic susceptibility 𝜒(𝑇 ) =[𝑀(𝑇 ) − 𝑀FM]/𝐻. As discussed earlier, 𝜒(𝑇 ) of thestoichiometric ZrFe4Si2 sample has a broad shoulder atabout 50 K and a small anomaly around 6 K. Theshoulder-like feature at 50 K becomes weaker for the𝑥 = 0.12 and 0.23 samples and eventually vanishes for𝑥 = 0.34. The anomaly corresponding to the short-rangemagnetic order in ZrFe4Si2 shifts to higher temperaturesupon increasing Ge content. Moreover, the anomaly de-velops into a cusp-like feature indicating long-range an-tiferromagnetic ordering in compounds with larger Gecontents. These results are confirmed by the specific-heatdata presented in Fig. 4b. The peak in 𝐶𝑝(𝑇 )/𝑇 shiftsto higher temperatures and sharpens with increasing Gecontent, with the transition temperature 𝑇𝑁 reaching23 K at 𝑥 = 0.46. In the Ge-substituted samples, theanomaly in 𝐶𝑝(𝑇 )/𝑇 resembles a mean-field-type tran-sition into a long-range ordered phase. We further note

that the 𝐶𝑝(𝑇 )/𝑇 values at the lowest temperatures re-main large, in the range of 200 − 300 mJ/K2mol. Thusthe Sommerfeld coefficient stays large in the whole con-centration range, confirming the presence of strong elec-tronic correlations.The temperature dependent resistivity data

𝜌(𝑇 )/𝜌300K provide further details on the nature ofthe magnetic ordering (see Fig. 4c). Already at a lowGe substitution level of 𝑥 = 0.12, the 𝜌(𝑇 )/𝜌300K curveshows a noticeable upturn around 11 K, reminiscent of aSDW transition. The increase in resistivity is attributedto the formation of an energy gap at part of the Fermisurface due to the SDW formation. With increasing Gecontent, the upturn in resistivity becomes much morepronounced and shifts to higher temperatures. Theseresults reveal that Ge substitution stabilizes the weakshort-range magnetic order in ZrFe4Si2 into a long-rangeSDW-type magnetic order.

C. Tuning ZrFe4(Si0.88Ge0.12)2 by hydrostaticpressure

The previous results from the Ge substitution inZrFe4(Si1−𝑥Ge𝑥)2 study show that application of neg-ative chemical pressure stabilizes the magnetic order inZrFe4(Si1−𝑥Ge𝑥)2. This leads to the expectation that ex-ternal hydrostatic pressure suppresses the magnetic or-der and eventually drives the system toward an antifer-romagnetic QCP. In order to study this, we performedelectrical-resistivity measurements under external pres-sure. We decided to use the slightly Ge substituted com-pound ZrFe4(Si0.88Ge0.12)2 for the pressure study sincethe anomaly in the electrical resistivity of ZrFe4Si2 isonly weak and we do not have a well-developed long-range ordered state. In contrast to that, our data for

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ZrFe4(Si0.88Ge0.12)2 indicate long-range SDW order andshow a clear anomaly in 𝜌(𝑇 ) corresponding to the SDWtransition, making it an ideal sample for the pressure ex-periment. The relatively low 𝑇𝑁 = 11.4 K for 𝑥 = 0.12ensures also that moderate pressures will be sufficientto suppress the magnetic order in comparison with com-pounds with larger Ge concentrations.

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FIG. 5: (a) Normalized electrical resistivity 𝜌(𝑇 )/𝜌300K ofZrFe4(Si0.88Ge0.12)2 as a function of temperature for differentapplied pressures. (b) Enlarged view of the low temperatureregion of the 𝜌(𝑇 )/𝜌300K curves. (c) Temperature derivatived𝜌(𝑇 )/d𝑇 vs. 𝑇 . The arrows indicate the transition temper-atures 𝑇𝑁 determined by the minima in d𝜌(𝑇 )/d𝑇 .

The electrical resistivity of ZrFe4(Si0.88Ge0.12)2 hasbeen investigated under hydrostatic pressures up to𝑝 = 2.5 GPa and in the temperature range between2 and 300 K. The curves of the normalized resistivity𝜌(𝑇 )/𝜌300K for several pressures are plotted in Fig. 5a.One immediately recognizes that pressure has a sizableinfluence on the temperature dependence of the resistiv-ity. Comparing Fig. 5a with Fig. 4c, it is obvious thatapplying pressure has the opposite effect of substitutingGe for Si: The curvature in the temperature range be-tween 20 and 300 K increases with pressure, resulting ina larger slope d𝜌(𝑇 )/d𝑇 at 20 K (see also 5c). Since thisstrong curvature is very likely connected to the onset ofdynamical correlations observed in 𝜇SR25, applying pres-sure seemingly strengthens these dynamic correlations.Furthermore, with increasing pressure the anomaly cor-responding to the long range order at 11.4 K at ambi-

ent pressure shifts to lower temperatures. This can bebetter seen in Figs. 5b and 5c, where we plot the low-temperature parts of the resistivity and its temperaturederivative d𝜌(𝑇 )/d𝑇 . At 𝑝 = 0.02 GPa, a small humpin 𝜌(𝑇 ) associated with the SDW transition is observed.The transition temperature 𝑇𝑁 is determined from theminimum in the temperature derivative of 𝜌(𝑇 ). As pres-sure is increased, the anomaly in resistivity shifts to lowertemperatures, as marked by the arrows. Moreover, themagnitude of the upturn strongly reduces with increasein pressure. The anomaly shifts to 5.2 K at 1.67 GPa. At1.87 GPa, the anomaly becomes too small and not trace-able due to limited resolution of the data in the respec-tive temperature range. At further increased pressures,𝜌(𝑇 ) monotonously decreases upon decreasing temper-ature without any visible anomaly down to the lowestaccessible temperature in our experiments. These resultsconfirm that the SDW transition in ZrFe4(Si0.88Ge0.12)2is continuously suppressed to zero temperature by exter-nal pressure, which suggests the existence of a pressure-tuned QCP.The magnetoresistance MR(𝐻) = [𝜌(𝐻) − 𝜌(0)]/𝜌(0)

of ZrFe4(Si0.88Ge0.12)2 shows marked features connectedto the suppression of the magnetic order. Figure 6 de-picts MR(𝐻) measured at 𝑇 = 2 K for several pressures.At low pressures, MR(𝐻) continuously increases uponincreasing field exhibiting a quadratic field dependence,which is typical for a metallic system. For 𝑝 ≥ 1.59 GPa,MR(𝐻) decreases initially upon increasing field, displaysa broad minimum and increases again. This contrastingbehavior of the MR between the low- and high-pressureregions might be attributed to the enhanced magneticfluctuations associated with the suppression of the mag-netic order. The magnetic field quenches such fluctu-ations and reduces their scattering contribution, givingrise to the negative MR.

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FIG. 6: Magnetic field dependence of the magnetoresistanceMR(𝐻) = [𝜌(𝐻) − 𝜌(0)]/𝜌(0) of ZrFe4(Si0.88Ge0.12)2 mea-sured at 𝑇 = 2 K under several applied pressures. The grayline corresponds to MR = 0.

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IV. DISCUSSION

The results from the Ge substitution studies inZrFe4(Si1−𝑥Ge𝑥)2 are presented as a temperature–Ge-content phase diagram in Fig. 7a. The transition tem-peratures 𝑇𝑁 obtained from magnetic susceptibility, heatcapacity, and electrical resistivity data are plotted. In-creasing Ge concentration stabilizes the short-range mag-netic order present in ZrFe4Si2 into a SDW phase observ-able in Ge-substituted ZrFe4Si2. 𝑇𝑁 (𝑥) is continuouslyenhanced with increasing Ge content 𝑥, reaching 23 K at𝑥 = 0.46.As expected, external hydrostatic pressure produces

the opposite effect to that of the negative chemi-cal pressure from Ge substitution. The results ob-tained from the electrical-resistivity measurements onZrFe4(Si0.88Ge0.12)2 are summarized in the temperature–pressure phase diagram presented in Fig. 7b. At am-bient pressure, the compound orders antiferromagneti-cally at 𝑇𝑁 = 11.4 K. With increasing pressure, 𝑇𝑁 ismonotonously suppressed to lower temperatures, reach-ing 5.2 K at 𝑝 = 1.67 GPa. No traceable anomaly inresistivity can be resolved at higher pressures. However,an extrapolation of the experimental data proposes thatthe magnetic ordering is suppressed to zero temperatureat a critical pressure of 𝑝𝑐 ≈ 2.1 GPa, suggesting thepresence of a pressure-tuned antiferromagnetic QCP.

In order to compare the effect of Ge substitution andhydrostatic pressure on the magnetic ordering, it is use-ful to use the unit-cell volume as a common scale. High-pressure PXRD investigations on isostructural LuFe4Ge2revealed a nearly linear pressure dependence of the lat-tice volume, yielding a bulk modulus of about 160 GPa28.In ZrFe4Si2 a similar pressure dependence of the lat-tice volume and, therefore, a similar bulk modulus isexpected. Thus, we use a bulk modulus of 160 GPato estimate the lattice volume at different pressures inZrFe4(Si0.88Ge0.12)2. The transition temperatures deter-mined from electrical-resistivity data of the Ge substitu-tion series and that of ZrFe4(Si0.88Ge0.12)2 under hydro-static pressure are plotted against the unit-cell volume𝑉 in the combined temperature–lattice-volume phase di-agram shown in Fig. 7c. The change in 𝑇𝑁 with Gesubstitution and with hydrostatic pressure is rather con-sistent and evidences the lattice volume as the governingcontrol parameter. Furthermore, 𝑇𝑁 shows a continuoussuppression of the magnetic ordering with decreasing lat-tice volume toward a putative antiferromagnetic QCP at𝑉 ≈ 182 A3.

A highly debated question is whether the Fermi liquidbehavior expected for a metal at low temperatures breaksdown at a QCP, resulting in a non-Fermi liquid (NFL).In the electrical resistivity an NFL is characterized bya deviation from the quadratic dependence of 𝜌(𝑇 ) atlow temperatures (𝜌 = 𝜌0 + 𝐴𝑇𝑛, 𝑛 < 2), which is ex-pected for a Fermi liquid. Our data in the temperaturerange down to 2 K seem to indicate a temperature expo-nent smaller than 2 close to 𝑝𝑐 and a recovery of Fermi

180 182 184 186 1880

4

8

12

16

20

24

Ge sub. pressure

T (K

)

V (Å3)

AFM

PM

TN

(c)

0.4 0.3 0.2 0.1 00

5

10

15

20

25

ZrFe4(Si1-xGex)2

c Cp

r

T (K

)

Ge content x

AFM

PMTN

(a)

0 0.5 1 1.5 2 2.5 3

ZrFe4(Si0.88Ge0.12)2

p (GPa)

TN

AFM

PM

(b)

FIG. 7: (a) Temperature vs. Ge-content phase diagram. 𝑇𝑁

determined from magnetic-susceptibility, heat-capacity, andelectrical-resistivity data are included. (b) Temperature–pressure phase diagram of ZrFe4(Si0.88Ge0.12)2 determinedfrom the electrical-resistivity data. (c) Temperature–latticevolume phase diagram showing 𝑇𝑁 determined from electricalresistivity data from both Ge substitution and high-pressurestudies. The open symbol corresponds to the weak anomalyobserved in ZrFe4Si2. The solid lines are guide to the eyes.The dashed line is an extrapolation to the experimental data.

liquid behavior (𝑛 = 2) at higher pressures. However,these data give only a first hint. The temperature ex-ponent 𝑛 = d ln(𝜌− 𝜌0)/d(ln𝑇 ) shows a significant tem-perature dependence in the low-temperature region forall pressures. In addition, the increased noise in the low-temperature data makes the accurate determination ofthe exponent difficult. These first results indicate thatexperiments at lower temperatures are highly desirable.Magnetic QCPs in transition metal systems are, mean-

while, a well-established research topic. Early caseswere the ferromagnetic systems ZrZn2 and NbFe2

29,30.Presently, the most prominent examples are certainlythe Fe pnictides and chalcogenides because there thedisappearance of an antiferromagnetic (AFM) state re-sults in the onset of unconventional superconductiv-ity31,32. It is therefore interesting to compare ZrFe4Si2with well-studied transition metal systems close to a mag-netic QCP. The 𝑇 -dependence of the susceptibility, withCurie-Weiss behavior at high temperatures and a lev-eling out or maximum at lower temperatures is com-mon in transition metal systems close to a QCP. Alsothe 𝑇 -dependence of the resistivity, with a pronouncednegative curvature in the range 20 − 100 K is com-mon in such systems. However, there is one propertywhere ZrFe4Si2 stands out in comparison to most itin-

7

erant transition metal systems: it presents a huge Som-merfeld coefficient. In transition metal systems, QCPsdo not necessarily result in large 𝛾 values. In the proto-typical system BaFe2As2, e.g., 𝛾 reaches only a valueof 5 mJ/molK2 in the stoichiometric system and val-ues of about 25 mJ/molK2 at the substitution-inducedQCP33,34. In ZrFe4Si2, the value 𝛾 = 150 mJ/molK2 de-duced from the high-temperature (> 20 K) extrapolation(see Fig. 2b) is already one order of magnitude larger,and far above the values typically observed in transi-tion metal systems, even close to a QCP. Furthermorethis extrapolated 𝛾 value obviously misses a large part ofthe low energy excitations, since the 𝐶𝑝/𝑇 value at thelowest investigated temperature of 2 K is significantlylarger, about 290 mJ/molK2. The evolution of 𝐶𝑝/𝑇 at2 K as a function of Ge content evidences this value topresent a maximum at or near the putative QCP. To ourknowledge, within transition metal systems, the valueof 290 mJ/molK2 is only surpassed in the compoundLiV2O4, which presents a 𝐶𝑝/𝑇 value of 420 mJ/molK2

at low temperature35–37. Several mechanisms have beeninvoked to explain the huge 𝐶𝑝/𝑇 value in LiV2O4

38–42.All invoke strong geometrical frustration due to V atomsforming a pyrochlore sublattice. Notably, in many of thefurther itinerant systems presenting a very large 𝛾 value,there is compelling evidence for strong frustration too,as e.g., in YMn2 (𝛾 = 180 mJ/molK2)43, Mn3P (𝛾 =100 mJ/molK2)44, and 𝛽-Mn (𝛾 = 70 mJ/molK2)45.There is a second family of transition metal systemsshowing a large Sommerfeld coefficient, which includese.g. CsFe2As2 (𝛾 = 184 mJ/molK2)46, Ca2−𝑥Sr𝑥RuO4

(𝛾 = 250 mJ/molK2)47, and CaCu3Ir4O12 (𝛾 =175 mJ/molK2)48, but there the large 𝛾 coefficient issuggested to originate from the closeness to a Mott tran-sition. For ZrFe4Si2, the evolution of the resistivity as afunction of Ge substitution or pressure makes this sce-nario rather unlikely since it indicates the system be-comes more metallic when approaching the QCP. Thusthe huge electronic specific heat observed at low temper-ature in ZrFe4Si2 compared to values in transition metalsystems supports frustration being relevant in ZrFe4Si2.Already in the context of YMn2, Pinettes and Lacroixdemonstrated that frustration can strongly enhance the𝛾 coefficient close to a QCP in an itinerant system39.

V. SUMMARY

In conclusion, we have investigated ZrFe4Si2 usingmagnetization, thermodynamic, and electrical-transport

measurements and tuned its ground-state properties byGe substitution and by application of hydrostatic pres-sure. In the crystal structure of ZrFe4Si2 the Fe tetrahe-dra are prone to magnetic frustration, and their chain-like arrangement represent a quasi-one-dimensional mag-netic system, a combination which is expected to enhancequantum fluctuations. Despite having large paramag-netic Fe moments (𝜇eff = 2.18 𝜇𝐵) with dominantly an-tiferromagnetic interactions, ZrFe4Si2 shows short-rangemagnetic order below 6 K. Ge substitution on the Sisites acts as a negative chemical pressure and stabi-lizes the short-range magnetic order into a long-rangespin-density wave order. By applying hydrostatic pres-sure on ZrFe4(Si0.88Ge0.12)2 we continuously suppressedthe magnetic order to zero temperature, as shown bythe electrical-resistivity data. Therefore, our combinedchemical substitution and hydrostatic pressure studysuggests the presence of a lattice-volume controlled anti-ferromagnetic quantum critical point in ZrFe4Si2. In thehydrostatic pressure experiment on ZrFe4(Si0.88Ge0.12)2we can infer a critical pressure 𝑝𝑐 ≈ 2.1 GPa and, indeed,magnetoresistance data indicate enhanced magnetic fluc-tuations associated with the suppression of the magneticorder. Moreover, zero-field resistivity data point to abreakdown of the Fermi liquid description in the vicin-ity of 𝑝𝑐. In comparison to other transition metal sys-tems, ZrFe4Si2 presents a large specific heat at low tem-peratures, reaching a 𝐶𝑝/𝑇 value of 290 mJ/molK2 at2 K. This large electronic specific heat at low tempera-tures supports the relevance of magnetic frustration inZrFe4Si2. Therefore our results evidence ZrFe4Si2 as astrongly correlated electron system with a constellationof interesting properties and thus worth being investi-gated in depth.

Acknowledgment

This work was partly supported by DeutscheForschungsgemeinschaft (DFG) through the ResearchTraining Group GRK 1621.

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