LaCu_{6-x}Ag_{x}: A promising host of an elastic quantum critical point
L. Poudel, C. de la Cruz, M. R. Koehler, M. A. McGuire, V. Keppens, D. Mandrus, A. D. Christianson
LLaCu − x Ag x : A promising host of an elastic quantum critical point L. Poudel a,b,c,d, ∗ , C. de la Cruz b , M. R. Koehler e , M. A. McGuire f , V. Keppens e , D. Mandrus a,e,f , A. D.Christianson b,a a Department of Physics & Astronomy, University of Tennessee, Knoxville, TN-37996, USA b Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA c Department of Materials Science & Engineering, University of Maryland, College Park, MD 20742 d NIST Center of Neutron Research, Gaithersburg, MD-20899 e Department of Material Science & Engineering, University of Tennessee, Knoxville, TN-37996, USA f Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA
Abstract
Structural properties of LaCu − x Ag x have been investigated using neutron and x-ray diffraction, and reso-nant ultrasound spectroscopy (RUS) measurements. Diffraction measurements indicate a continuous struc-tural transition from orthorhombic ( P nma ) to monoclinic ( P /c ) structure. RUS measurements showsoftening of natural frequencies at the structural transition, consistent with the elastic nature of the struc-tural ground state. The structural transition temperatures in LaCu − x Ag x decrease with Ag compositionuntil the monoclinic phase is completely suppressed at x c = 0.225. All of the evidence is consistent with thepresence of an elastic quantum critical point in LaCu − x Ag x . Keywords:
Elastic quantum critical point, Quantum phase transition
1. Introduction
Quantum criticality continues to be a key pillarof research in condensed matter physics. Prototypeexamples include magnetic quantum critical points(QCP) in which quantum fluctuations of spins meltan ordered state of matter [1, 2, 3, 4]. However, inrecent years, interest is beginning to shift towardnew paradigms of quantum criticality [5, 6, 7, 8, 9].For example, SrTiO and KTaO have been iden-tified as being close to a ferroelectric QCP [6], andthe superconducting state in (Ca,Sr) Rh Sn is as-sociated with the nearby structural QCP [7]. Col-lectively these new examples of quantum criticalityprovide an opportunity for fresh perspectives andthe discovery of new unifying insights.Arising out of this emerging interest, an elasticQCP has been theoretically proposed [10]. Thistype of critical phenomenon is expected to occurwhen the quantum zero point motion of atoms gen-erates a residual strain in the lattice suppressingthe structural ground state. In this sense, the elas- ∗ Corresponding author
Email address: [email protected] (L. Poudel) tic QCP is fundamentally different from the mag-netic and other structural counterparts [10]. Fur-thermore, recent experimental work demonstratesthat LaCu − x Au x shows the promise of hosting anelastic QCP [11]. To provide a more comprehensiveunderstanding of the tunability of the structuralphase transition leading to an elastic QCP, here wepresent a study of the related series LaCu − x Ag x as a potential candidate of elastic QCP.In this paper, we present structural proper-ties of LaCu − x Ag x as a function of Ag compo-sition and temperature. The monoclinic phase ofLaCu − x Ag x is gradually suppressed with Ag sub-stitution. The structural transition is accompaniedby a gradual softening of some natural frequencies,as is expected for a continuous elastic phase transi-tion. Linear extrapolation of T S with x shows thata complete suppression of the monoclinic structureoccurs at the critical composition x QCP = 0.225.All of the measurements are consistent with thepresence of an elastic QCP in LaCu − x Ag x . a r X i v : . [ c ond - m a t . s t r- e l ] J un T (K) θ ( D e g ) (a) I n t e n s i t y ( a r b . un i t s ) T (K) c o s ( β ) × − (b)
50 100 150 200
T (K) F ( M H z ) (c) Figure 1: Characterization of the structural phase transition in LaCu − x Ag x . (a) Temperature dependence of laboratoryx-ray diffraction measurements of LaCu . Ag . showing splitting of Bragg peaks. Near T S (shown by vertical dottedline), due to the monoclinic distortion occurring in the orthorhombic ab plane, the Bragg peaks ( H K L ) with H (cid:54) = 0and K (cid:54) = 0 split into a pair of monoclinic peaks. The three peaks in the orthorhombic structure are (122), (220) and(221), which become (22 ± ±
2) and (21 ± ±
2) is the monoclinic (033), which becomes (303) inthe orthorhombic phase and overlaps with orthorhombic (221). (b) Temperature dependence of cos ( β ) in LaCu . Ag . . T S is obtained by linearly extrapolating cos ( β ) to zero. The red dotted line is a linear extrapolation of the data. (c) RUSmeasurement of LaCu . Ag . showing the square of a natural frequency ( F ) as a function of temperature. The blue andred dotted lines represent the slopes of F versus temperature in the orthorhombic and monoclinic phases, respectively. Thelines intersect at T S .
2. Experimental Details
Polycrystalline samples of LaCu − x Ag x ( x = 0,0.075, 0.1, 0.125, 0.135, 0.15, 0.155, 0.175, 0.2,0.225, 0.25, 0.3) were synthesized by arc meltingthe elements La, Cu and Ag in stoichiometric pro-portions. The phase purity of the sample wascharacterized by laboratory x-ray measurements atroom temperature. Samples with compositions x = 0.075, 0.1, and 0.125 were also measured on aPANalytical X’Pert Pro MPD powder x-ray diffrac-tometer using Cu K α, radiation ( λ = 1 . − x Ag x , x =0.135, 0.155, and 0.175 using a set-up as describedin the Ref. [12]. For the measurement, a rect-angular parallelepiped sample was held betweentwo transducers. Mechanical resonances within therange 10 - 1000 kHz were collected as a function oftemperature.Neutron diffraction measurements were per-formed on the samples of LaCu − x Ag x ( x = 0, 0.15,0.2, 0.225, 0.25) with the HB-2A powder diffrac- tometer at the High Flux Isotope Reactor (HFIR)of Oak Ridge National Laboratory (ORNL). Neu-trons of wavelength 1.54 ˚A were used for the mea-surement. Collimators containing parallel blades ofcadmium coated steel were positioned before themonochromator, sample, and detector with diver-gence of 12 (cid:48) − (cid:48) − (cid:48) respectively. The sample ofmass ≈ x = 0.15 and 0.2, diffraction pat-terns at several temperatures were obtained near T S . For x = 0.225 and 0.25, diffraction patternswere obtained at room temperature and at 4 K only.The structural parameters were obtained using Ri-etveld refinement with the FullProf Suite software[13].High resolution x-ray diffraction measurementsof LaCu . Ag . were performed using transmis-sion geometry with 11-BM at the Advanced Pho-ton Source at Argonne National Laboratory [14].A monochromatic x-ray of wavelength λ = 0 . ∆ QQ = 2 × − . The sample was finelyground inside a glove box filled with argon, whichwas then mixed with amorphous SiO in the molar2 igure 2: Phase diagram of LaCu − x Ag x . T S decreaseslinearly with Ag-substitution. No structural transition isobserved for x ≥ . ratio of 1:3 to minimize x-ray absorption. The mix-ture was packed inside a Kapton tube of 0.8 mmdiameter. The Kapton tube containing the samplewas spun at 60 Hz to achieve an efficient powderaveraging during the measurement.
3. Results and discussion
Analysis of the room temperature laboratory x-ray diffraction pattern shows that the samples areof high purity and are consistent with either the or-thorhombic (space group:
P nma ) ( x ≥ . P /c ) structure ( x = 0).The structural phase transition in LaCu − x Ag x wascharacterized by different methods: neutron and x-ray diffraction, and RUS measurements. Using x-ray diffraction, the temperature dependence of se-lected Bragg peaks was measured. At T S , someof the structural Bragg peaks pertaining to theorthorhombic structure split into two monoclinicpeaks, which is shown in Fig. 1(a). The split-ting occurs only for the Bijovet pairs ( H K L )with H (cid:54) = 0 and K (cid:54) = 0, which, due to the mon-oclinic distortion, acquire different d -spacing at T S .Neutron diffraction measurements were used to de-termine the temperature dependence of monoclinicangle β . Near T S , β gradually increases with low-ering temperature as expected for a second order Figure 3: (a) Synchrotron x-ray diffraction pattern (blackdots) from LaCu . Ag . with the fit (red line) of the or-thorhombic crystal structure. The fit was obtained from theRietveld refinement of the data using the FullProf software.(b) A perspective view of LaCu − x Ag x of the orthorhom-bic structure. The unit cell consists of four formula units ofLaCu − x Ag x . Cu atoms are distributed among five differ-ent sites. Ag atoms exclusively occupy the Cu2 site. Note:atoms closer to the observer are darker compared to the thosefurther away. phase transition, and consequently, there is a con-tinuous evolution of shear strain ( e ∝ cos( β )) inthe monoclinic phase. Therefore, cos ( β ) is linearlyextrapolated to zero for the estimation of T S , asshown in Fig. 1(b).RUS measures the resonances that occur whenthe frequency of an ultrasonic wave matches withthe natural frequency of the sample. The RUS mea-surements show that some of the natural frequen-cies gradually become soft as T S is approached.As the square of a natural frequency ( F ) is di-rectly proportional to the combination of elasticconstants, we have used F as an indirect probe ofelastic behavior in LaCu − x Ag x . T S is character-ized by the change in the slope of F versus temper-ature. An example is shown in 1(c), where the linesrepresenting the slope of F in the orthorhombicand monoclinic phases intersect at T S . The changein the slope of F versus temperature can be at-tributed to a complete softening of the C = C able 1: Structural parameters of LaCu . Ag . obtained from Rietveld refinement of high resolution synchrotron x-ray diffrac-tion measurement (Fig. 3(a)) at room temperature. The number in parentheses is the error in the last digit. a = 8.20655(5) ˚A, b = 5.13858(3) ˚A, c = 10.28100(4) ˚A R P = 16.3, R wp = 18, χ = 3.1Element Wyck. x/a y/b z/c B Occ.La1 4 c d c c c c c − X and evolution ofshear strain e [11].A phase diagram summarizing T S obtained fromdiffraction and RUS measurements is presented inFig. 2. T S in LaCu − x Ag x linearly decreases withAg composition. A linear extrapolation of T S withAg composition shows that the monoclinic phaseis completely suppressed near x c = 0.225. For thecompositions at and above x c , no structural phasetransition is observed above 4 K (marked as squaresin the phase diagram).For a detailed understanding of the orthorhom-bic structure in LaCu − x Ag x and in particular, toinvestigate the distribution of Ag atoms in the or-thorhombic unit cell, high resolution synchrotronx-ray measurements were performed for the com-position LaCu . Ag . . As expected from the phasediagram, the diffraction pattern is consistent withthe orthorhombic structure. In addition to thereflections from the main phase, small impuritypeaks consistent with elemental copper appear inthe diffraction pattern. The intensity of the im-purity peaks corresponds to only 0.39% of copperby weight. No additional impurities were detected.The measured pattern with the fit of orthorhombiccrystal structure is shown in Fig. 3(a). The or-thorhombic unit cell consists of four formula units,in which La and Cu/Ag are distributed in one gen-eral and five special sites. The crystal structureafter the Rietveld refinement is shown in Fig. 3(b).Details of the Rietveld refinements are presented inTable 1.The result presented here illustrates that thestructural properties of LaCu − x Ag x are micro-scopically similar to the related series LaCu − x Au x , which has recently been identified as a host ofan elastic QCP, and also to other members ofthe CeCu − x T x family [11, 17, 18]. In partic-ular, the Rietveld analysis of the x-ray diffrac-tion measurements shows that the crystal struc-ture of LaCu − x Ag x is virtually identical to thatof LaCu − x Au x [11, 17]. The substituent Agin LaCu − x Ag x exclusively occupies the specialcopper position Cu2, and T S decreases linearlywith chemical substitution as in the case ofLaCu − x Au x . Furthermore, RUS measurements in-dicate that the elastic properties of LaCu − x Ag x are similar to the related compound CeCu , indi-cating a similarity in structural properties. Thestructural resemblance of LaCu − x Ag x with theLaCu − x Au x and CeCu − x T x family indicatesthat the suppression of the monoclinic phase inLaCu − x Ag x results in an elastic QCP.
4. Conclusion
In conclusion, a structural phase diagram ofLaCu − x Ag x is constructed using neutron and x-ray diffraction and RUS measurements. High res-olution synchrotron x-ray diffraction measurementdemonstrates that LaCu − x Ag x bears a structuralresemblance to the related series LaCu − x Au x andCeCu − x T x . The phase transition in LaCu − x Ag x is driven by elastic instabilities, with the RUS mea-surement showing softening of natural frequenciesat T S . T S in LaCu − x Ag x can be suppressed withAg substitution, and the monoclinic phase is com-pletely terminated at the critical composition x QCP = 0.225. The evidence taken together suggests thatLaCu − x Ag x is a promising host of an elastic QCP.4 cknowledgement We acknowledge D. Singh for useful discussionsand M. Suchomel for assistance with the syn-chrotron x-ray measurements. The research atthe High Flux Isotope Reactor at Oak Ridge Na-tional Laboratory is supported by the ScientificUser Facilities Division, Office of Basic Energy Sci-ences, U.S. Department of Energy (DOE). MAMand DM acknowledge support from the U. S. DOE,Office of Science, Basic Energy Sciences, Mate-rials Sciences and Engineering Division. Use ofthe Advanced Photon Source at Argonne NationalLaboratory was supported by the U. S. Depart-ment of Energy, Office of Science, Office of BasicEnergy Sciences, under Contract No. DE-AC02-06CH11357. This manuscript has been authoredby UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of En-ergy. The United States Government retains andthe publisher, by accepting the article for publi-cation, acknowledges that the United States Gov-ernment retains a non-exclusive, paid-up, irrevoca-ble, world-wide license to publish or reproduce thepublished form of this manuscript, or allow oth-ers to do so, for United States Government pur-poses. The Department of Energy will providepublic access to these results of federally spon-sored research in accordance with the DOE Pub-lic Access Plan (http://energy.gov/downloads/doe-public-access-plan).
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