Transition from one-dimensional antiferromagnetism to three-dimensional antiferromagnetic order in single-crystalline CuSb{}_{2}O{}_{6}

Transition from one-dimensional antiferromagnetism to three-dimensional antiferromagnetic order in single-crystalline CuSbO

A. Rebello Physics Department, P. O. Box 173840, Montana State University, Bozeman, MT 59717-3840, USA    M. G. Smith Physics Department, P. O. Box 173840, Montana State University, Bozeman, MT 59717-3840, USA    J. J. Neumeier Physics Department, P. O. Box 173840, Montana State University, Bozeman, MT 59717-3840, USA    B. D. White Physics Department, P. O. Box 173840, Montana State University, Bozeman, MT 59717-3840, USA    Yi-Kuo Yu National Center for Biotechnology Information, 8600 Rockville Pike, Bethesda, MD 20894, USA
July 1, 2019
Abstract

Measurements of magnetic susceptibility, heat capacity and thermal expansion are reported for single crystalline CuSbO in the temperature range K. The magnetic susceptibility exhibits a broad peak centered near 60 K that is typical of one-dimensional antiferromagnetic compounds. Long-range antiferromagnetic order at = 8.7 K is accompanied by an energy gap ( = 17.48(6) K). This transition represents a crossover from one- to three-dimensional antiferromagnetic behavior. Both heat capacity and the thermal expansion coefficients exhibit distinct jumps at , which are similar to those observed at the normal-superconducting phase transition in a superconductor. This behavior is quite unusual, and is presumably associated with a Spin-Peierls transition occurring as a result of three-dimensional phonons coupling with Jordan-Wigner-transformed Fermions.

pacs:
75.40.-s, 75.30.Gw

I Introduction

Although a clear physical picture for three dimensional antiferromagnets has emerged, the understanding of one dimensional (1D) antiferromagnets (AFMs) is less clear.affleck () In these Heisenberg chain systems, quantum effects play an important role, but long-range order is not expected.affleck (); bonn () One of the interesting effects that occurs in 1D AFMs is a dimerization of the spins, known as the spin-Peierls transition,hase (); pouget () which leads to a decrease in the magnetic energy without long-range magnetic order as reported in CuGeO. However, the spin-Peierls transition can also be caused by three-dimensional phonons; in this case 3D antiferromagnetic order can occur.pytte (); jacobs ()

CuSbO is a green-coloredbystrom () insulator which orders antiferromagnetically at Néel temperature = 8.7 K.nakua (); nakua2 (); gibs (); kato (); pro () It possesses a nearly octahedral local environment of the Cu ions in contrast to other cuprates, which have a quasiplanar fourfold Cu-O coordination. muller () It undergoes a phase transition from an ideal trirutile tetragonal structure (-CuSbO) above 380 K to monoclinic (-CuSbO) below while largely maintaining the quasioctahedral environment.giere () The distortion of CuSbO to lower symmetry is in sharp contrast to the related trirutiles, SbO, which remain tetragonal ( = Co and Ni) and order antiferromagnetically at 12.64 K and 2.5 K, respectively. donaldson (); reimers (); ramos1 (); ramos2 (); ehrenberg (); nakua ()

The magnetic sublattice of CuSbO has been studied previously.nakua (); nakua2 (); gibs (); kato () Two investigationsnakua (); gibs () noted weak magnetic superlattice peaks below . The first was unable to refine the structure using Rietveld analysis. Two possible magnetic structures were proposed and the authors establishednakua2 () that the magnetic moment on the copper site was 0.5. The second investigation also proposed two magnetic structural models, and ultimately found only one to be consistent with magnetic susceptibility data.gibs () The single-crystal investigation by Kato et al. presents the highest-quality data, and their magnetic structure model appears to deserve the most confidence,kato () although some magnetic reflections could not be indexed. The suggested model involves Cu moments ordered ferromagnetically along with a magnetic wave vector of (/a,0,/c).

The 1D model for the magnetic susceptibility of linear chain spin- systems due to Bonner and Fisher agrees well with the magnetic susceptibilitynakua2 (); gibs () of CuSbO. The interchain-to-intrachain coupling ratio was estimatednakua () at 210. This is about 10 times smaller than reportedhase2 () for CuGeO. Electronic structure calculations for CuSbO reveal an unusual quasi-1D magnetic ground state driven by orbital ordering, which is attributed to the presence of competing in- and out-of-plaquette orbitals and strong electronic correlations. The 1D superexchange interaction (Cu-O-O-Cu) is along the [110] direction, via the apical oxygen ions, rather than the in-plaquette oxygens. This unique orbital exchange is believedkasi () to be at the core of the unusual 1D magnetism in CuSbO.

In this report, measurements of the magnetic susceptibility, heat capacity, and thermal expansion of CuSbO are reported. A broad peak at 60 K in the magnetic susceptibility indicates the presence of one-dimensional antiferromagnetism and three-dimensional antiferromagnetism appears at = 8.7 K. Subtraction of the heat capacity for the non-magnetic analog ZnSbO allows determination of the magnetic entropy, which clearly shows that the one-dimensional antiferromagnetism begins to appear at 115 K. Both the heat capacity and thermal expansion coefficients reveal distinct jumps at , similar to what is typically observed at a normal-superconductor phase transition. The heat capacity data reveals an energy gap below of magnitude = 17.48(6) K. The unusual character of this phase transition suggests that it is a crossover between one-dimensional antiferromagnetism and three-dimensional antiferromagnetic order that is associated with a Spin-Peierls transition occurring as a result of three-dimensional phonons coupling with Jordan-Wigner-transformed Fermions. The thermal expansion reveals extremely unusual and anisotropic behavior attributed to anharmonic lattice vibrations resulting from the presence of short-range antiferromagnetic order. The thermal expansion results are compared with publishedhei () x-ray data for CuSbO, demonstrating excellent agreement. Spin-flop transitions for magnetic fields applied along both the and axes are observed.

Ii experimental

Single crystals used in this work were grown by chemical vapor transport, as described elsewhere. pro () Two CuSbO crystals were characterized by magnetic and heat capacity measurements using a Quantum Design Physical Property Measurement System and they yielded identical results. The orientation of the single crystals was determined using Laue x-ray diffraction. We observed twinning in the CuSbO crystals. In most cases, the Laue diffraction on a sample selected for orientation confirmed the presence of two crystals (primary and secondary) with the same axis. The [100] and [001] directions of the primary crystal coincided with the [013] and [101] directions of the secondary crystal. The sample was then polished to eliminate the secondary crystal and therafter could be oriented along the principle crystallographic axes for measurements. The final Laue images of these polished samples revealed no twinning.

A polycrystalline sample of ZnSbO was preparedkatsui () for heat capacity measurements by reacting well-mixed, stoichiometric amounts, of SbO and ZnO in air at for 12 h. The powder was reground, pelletized, and sintered at in air for 12 h. X-ray diffraction revealed single-phase nature with lattice parameters = 4.66 Å and = 9.24 Å, in agreement with a prior report.mats () ZnSbO single crystals were grown using chemical vapor transport. Polycrystalline ZnSbO powder was sealed in an evacuated quartz tube and the hot and cold ends of the tube were kept at 800 and 750 C, respectively for 100 h. Well-faceted crystals were obtained, which exhibited good Laue imaged with no twinning.

Thermal expansion measurements were performed on a CuSbO sample with dimensions of about , , and (, , and axes, respectively). Measurements of a second sample confirmed the results. Thermal expansion was also measured on an oriented ZnSbO single crystal with dimensions 1.85, 1.22 and 0.38 mm (, , and axes, respectively). The quartz dilatometerJohn () used for the thermal expansion measurements has a sensitivity to changes in length of 0.1 Å. Measurements along each axis consist of 1700 data points and were repeated 2-3 times and averaged. The data are corrected for the empty-cell effect and for the relative expansion between the cell and the sample. Note that the relative resolution of our dilatometer cell is about 4 orders of magnitude higher than that possible with x-ray or neutron diffraction.

Iii results

Figure 1: (color online) (a) Temperature (T) dependence of magnetization () when the magnetic field is applied along the , and axes. The lower inset shows expanded near . The upper inset shows the dependence of derivative of magnetization near .

The main panel of Figure 1 shows the temperature (T) dependence of the magnetization () with the magnetic field () applied along each of the principal crystallographic axes. While cooling from 300 K, increases with decreasing and exhibits a broad maxima near 60 K. This behavior is typical of systems consisting of 1D chains of magnetic ions. Accordingly, we have used the Bonner-Fisher model bonn () for the spin 1/2 AFM Heisenberg chain to compare to the observed behavior with good agreement. The fits yield an exchange coupling strength = 47.2(2) K, g= 2.20(1), g= 2.20(1) and g= 2.24(2), which indicates a strong spin-orbit interaction and, perhaps, some minor g-factor anisotropy. Anisotropy in the g-factor was observed through electron-spin resonance measurements.hei () At = 8.7 K long-range antiferromagnetic (AFM) order is observed, as reported previously. kasi (); gibs () Thus, the transition at represents a crossover from one- to three-dimensional antiferromagnetic behavior

Prominent anisotropy is observed below in , as shown in the lower inset of Figure 1. This anisotropy is clearly evident in versus for the three field directions; for and exhibits an abrupt increase at at the AFM transition while cooling whereas it exhibits a small decrease for , as seen in the upper inset of Fig. 1; the subscript in the symbol denotes the direction along which was applied. This behavior is typical for AFM systems, and is largely consistent with the proposed ordering of the magnetic moments.gibs (); kato ()

Figure 2: (color online) Temperature dependence of magnetization at different magnetic fields applied along the (a) (b) and (c) axes. The numbers indicate the magnetic field strength() in T. The insets show the field dependence of for applied along the respective axes.

To further examine the AFM ordered state, the temperature dependence of at different magnetic fields applied along each crystallographic axis was measured. Fig. 2 shows for magnetic field along each axis. For field strengths 0.5 and 1.2 T along , exhibits a downward drop below = 8.7 K [Fig. 2(a)]. However, for 1.3 T, this downward drop is followed by an increase below 5 K while cooling, which could be attributed to a spin-flop transition where the spins turn from a parallel orientation to a perpendicular orientation, thereby lowering their energy. Furthermore, the spin-flop transition shifts towards higher temperatures with increasing field strengths. Note that the spin-flop transition is also manifested as abrupt jumps in , for temperatures below , as shown in the inset of Fig. 2(a); in these data, it is evident that the transition occurs at lower field for lower temperatures. This is due to the lowering of total energy of the spin-flop configuration compared to the antiferromagnetic state with increasing anisotropy and coupling strength as temperature decreases. Interestingly, and when is applied along [Fig. 2(b)] exhibit similar behavior, indicating that the spin-flop transition occurs in CuSbO for along both the and axes. Since Heinrich et al. [hei, ] observed a spin-flop transition for the -axis and not for directions perpendicular to , it was suggested that the easy antiferromagnetic axis lies along the direction. However, our results confirm that CuSbO orders below 8.7 K with easy antiferromagnetic axis along either the or directions, in agreement with a prior report,gibs () which may reflect the presence of significant magnetic disorder, or weak differences in the magnetocrystalline energy for these two directions. As expected, we do not observe any spin flop transition in and for along [Fig. 2(c)].

Figure 3: (a) Temperature (T) dependence of the linear thermal expansion () along the , , and axes. Each curve contains more than 1700 data points. Inset shows the expanded region around . (b) Comparison of temperature dependence of lattice parameters using dilatometer (solid lines, this work) and x-ray diffraction (open symbols, Ref. hei, ) techniques.

In Figure 3(a), the linear thermal expansion normalized to the length at 300 K, , is plotted for the , and axes. Here, is the length of the sample at 300 K. Upon warming from 5 K, along and increases. The magnitude of along b shows a variation of between 5 and 300 K. On the other hand, along decreases while warming from 5-150 K and thereafter increases with further increasing temperature. The inset of Fig. 3(a) displays along and on the left and right scale, respectively, for near . Note that a clear change of slope is visible at along (similar behavior occurs along ) and , as expected for a continuous phase transition.

The thermal expansion data are compared to data obtained using x-ray diffraction hei () (open symbols) in Fig. 3(b). Our data were multiplied by the respective lattice parameter at room temperature, and the result was then added to that lattice parameter in order to obtain the temperature dependence which appears as solid lines in the figure, which are actually about 1700 discrete data points. The lattice parameters and tend to merge near 350 K due to the well-known monoclinic to tetragonal structural transitiongiere () at 380 K. This comparison reveals the excellent agreement with the data obtained from diffraction as well as the outstanding resolution of our data.

Figure 4: (a) Temperature (T) dependence of linear thermal expansion coefficient (/) along the three crystallographic directions for CuSbO. Inset shows the expanded region around . Dotted lines are for the non-magnetic analog compound ZnSbO.

Fig. 4 shows the linear thermal expansion coefficient / for the , and axes. While along and is positive over the entire temperature range, along it is negative at temperatures below 175 K and positive at higher temperatures. Moreover, displays a broad peak along and and a broad minimum along near 50 K. The expanded view of near the AFM transition is given in the inset of Fig. 4. Abrupt jumps occur in at the AFM transition. While along and jumps downward at the paramagnetic-AFM transition on cooling, along shows an upward jump. Features in are expected at for a continuous phase transition, but the shape is unusual (see below). The main panel also shows for ZnSbO along the and axes of the tetragonal crystal as dotted lines. Note that this non-magnetic analog compound exhibits monotonic behavior of and fails to exhibit 0 along . The unusual temperature dependence of the data for CuSbO is likely associated with its 1D antiferromagnetism and lower crystallographic symmetry, this point will be addressed below.

The heat capacity () as a function of temperature is presented for CuSbO in Fig. 5. At the highest measurement temperature, = 395 K (Data above 100 K not shown.), reaches 202.4 J/molK which is close to 224.4 J/molK calculated from Dulong-Petit. A prominent anomaly occurs at , which will be discussed further below. Fitting to the equation over the range 9.2 K T 15.6 K yields 58.7(7) mJ/molK, 0.171(5) mJ/molK and a Debye temperature = 467(5) K.

The main panel of Fig. 5 also shows of ZnSbO, the non-magnetic analog of CuSbO. Fitting these data over the range 0.45 K T 15.6 K to the equation yields 0.02(4) mJ/molK 0.11(2) mJ/molK and a Debye temperature = 539(30) K. Subtraction of the ZnSbO data from the CuSbO data yields , which is shown in the insets of Fig. 5 as either or in expanded views. The broad peak in above corresponds to the contribution of short-range-magnetic ordering to the entropy. Integrating the versus curve (lower inset) over the region below 115 K (the temperature below which the CuSbO and ZnSbO data sets deviate from one another) provides the entropy change , which is plotted in the inset of Fig. 6. An entropy change of 3.85 J/molK (66.8% of ) is observed over this entire temperature range. If one considers only the region from just above to 115 K, 57.8% is obtained, so most of the entropy change occurs between 115 K and . Note that for the region 50 K to 115 K, the contribution to is only 12% Rln2. Some measurements of at 8 tesla, with the magnetic field parallel to the axis, were also conducted. The zero field ZnSbO data were subtracted from the data to obtain , which is shown in the insets of Fig. 5. These data extend only slightly above , where a small increase in is evident. The region will be discussed below.

Figure 5: Main panel displays the molar heat capacity versus for CuSbO and its non-magnetic analog ZnSbO (shown as a dotted line). The upper inset shows the temperature dependence of the magnetic contribution to the heat capacity () plotted as ln() versus 1/ for 0 and 8 tesla; the data uncertainties are similar to the data point size. The solid line represents the fit to the 0 tesla data to determine the energy gap in the equation . The lower inset shows at 0 and 8 tesla; the area under these curves is the entropy.
Figure 6: Molar heat capacity at zero magnetic field after subtracting the entropy contributions, (closed symbols) and versus ; 5510 J/molK. Inset shows , which was obtained by integrating the versus data over the region below 115 K.

Considering the thermodynamics of a continuous phase transition, the heat capacity in the immediate vicinity of can be written assou ()

(1)

Here, , , and are the molar entropy, pressure, molar volume and volume thermal expansion coefficient, respectively; the subscript signifies that Eq. (1) is valid near the Néel temperature. For this analysis, it is assumed that =(++)sin, near 8.7 K, since the crystallographic angle tends to saturate to 91.3 below 100 K (see Fig. 2 of Ref. [hei, ]). When the entropy contribution is eliminated by subtracting the term that is linear in from , the result, , is proportional to the product of and . As expected from Eq. (1) and shown in Fig. 6, close to , scales with following the relation . The excellent overlap of the two data sets is obtained with 5510 J/molK. Furthermore, the observation that the data sets scale suggests that the phase transition is continuous. The parameter can now be used to determine since . The molar density at 300 K provides as 6.8 10 m/mol, which then yields the pressure derivative = -0.11(1) K/GPa. Note that there are no direct measurements of available thus far.

Iv discussion

The data reveal a sizable term of magnitude 57.4(6) mJ/molK, that completely vanishes below . Since CuSbO is an electrical insulator, this term cannot be associated with the presence of conduction electrons and must have a purely magnetic origin. For a Heisenberg chain, theoryjohnston () reveals = 120 mJ/molK using the value of obtained from fitting our magnetic susceptibility data. Thus, the obtained is approximately half of the theoretical value. The decay of below is exponential, suggesting the presence of an energy gap. This gap is extracted from a linear fit to the data plotted as ln() versus 1/ (see upper inset of Fig. 5), suggesting the function . The gap = 17.48(6) K (1.51(1) meV) is obtained from the fit. This gap must be associated with the crossover from partially-ordered moments on non-interacting chains to three-dimensional, long-range order. Measurements of at an applied magnetic field of 8 T have a small influence on the obtained , yielding = 18.42(6) K (1.59(1) meV), which suggests that the low-temperature ordered phase is more robust in non-zero magnetic field.

The character of the phase transition at warrants some discussion. Normally the heat capacity and thermal expansion coefficient of a magnetic system displays -like character at the magnetic ordering temperature, which is associated with a divergence of measurable quantities, such as , , and at the critical point.sou (); white () The antiferromagnetic analog system CoSbO, for example, clearly exhibits -like character in at its AFM transition.aaron () This is not the case for CuSbO, where and display discontinuities at followed by an exponential decay in below . The character of these transitions is reminiscent of that observed at the normal-superconducting phase transition, the only phase transition known to exhibit distinct jumps at the critical temperature.sou () Note that otherjohnston (); win () 1D magnetic systems, such as CuGeO and NaVO exhibit -like character in and . These systems, however, possess strong lattice distortionspouget () at the spin-Peierls transition (in the case of CuGeO) and charge-ordering transition (in NaVO). This magnetic-lattice coupling may play a role in the large -like features in at their respective transitions. In the case of CuSbO, it exhibits a transition from 1D to 3D order at , without an obvious presence of strong magnetic-lattice coupling.

A possible scenario that agrees with the observed discontinuity of specific heat, the energy gap below , and a weak magnetic-lattice coupling is described as follows. The Spin-Peierls transition originates from the coupling between the 1-D Jordan-Wigner-transformed Fermions and 3-D phonons. Similar to the usual Peierls transition, it opens up an energy gap that is very much BCS-like.pytte (); jacobs () In our system, there is a collection of quasi-1D spin chains that run parallel to one another. The same group of phonons that couple spins on one chain may also couple spins on the other chains. This introduces an effective inter-chain coupling. If this coupling remains marginally weak, upon the dimerization (Spin-Peierls) transition, the energy gap opened should behave essentially like the case of a single chain while the dimerization ordering of all chains will then appear as a 3-D ordering. If this is the case, the specific-heat discontinuity occurs in the same manner as the BCS case and the spin-energy gap appears as an additional contribution of specific heat the same way as the heat capacity contribution of superconducting electron pairs. This ordered state is likely an alignment of the 1D chains, rather than a true 3D ordered state; this may be the source of difficulty in identificationnakua2 (); gibs (); kato () of the magnetic lattice below and the disorder suggested from the spin-flop transition’s occurrence along both the and axes.

The nearly tetragonal crystal structure of CuSbO results in almost identical positive thermal expansion behavior observed along the and axes below 200 K. Above this temperature differs for these two axes due to the well-known monoclinic to tetragonal structural transition at 380 K.giere () In contrast, the thermal expansion along is negative at temperatures below about 170 K. Negative thermal expansion can originate from unusual phonon modes that become active at low temperaturesbarerra () and are often observed in low-D systems.san (); alwyn () However, consideration of for the non-magnetic analog compound ZnSbO, along with the fact that 1D antiferromagnetism sets in below 115 K (see versus in the inset of Fig. 6) suggests that the unusual temperature dependence of in CuSbO is associated with the 1D antiferromagnetism.

The thermal expansion in solids is due to anharmonic vibrations, and in turn, anharmonic contributions to the elastic potential. This is a many-body potential, but pair potentials between neighboring atoms can play a significant role.barerra () In CuSbO the onset of local 1D magnetic order occurs on cooling below 115 K, as determined from the heat capacity . Local order would strongly affect the pair potentials in those regions of the sample where the order occurs. This could lead to stretching or tilting of bonds as well as changes in bond strength in these regions. Even if the affected pair potentials are harmonic, the result can be a potential for the sample that is anharmonic.barerra () Thus, the increase in magnitude of the thermal expansion coefficients below 115 K is most likely associated with changes in anharmonic contributions to the elastic potential of the sample as a result of the formation of short-range 1D antiferromagnetic order. Below 50 K the thermal expansion coefficients change behavior because a large proportion of the spins are ordered, and the anharmonic contributions reduce upon further cooling leading to a decrease in the magnitude of . The negative thermal expansion along is in strong contrast to the behavior along and , and is certainly connected to the 1D behavior between 8.7 K and 115 K and anharmonic lattice vibrations that specifically affect along this direction.

Other types of transitions can also lead to the development of anharmonic contributions to the thermal expansion, such as in LiMoO where at 28 K a crossover in dimension has been associated with two-particle hopping and the formation of bosons.san () CuGeO also exhibitswin () a prominent correlation between thermal expansion and the formation of 1D magnetic correlations.

V conclusions

The thermodynamic and magnetic properties of CuSbO were investigated. One-dimensional antiferromagnetic order among the spin Cu atoms is observed to occur below 115 K. This order has a strong impact on anharmonic lattice vibrations, and leads to a sizable change in the thermal expansion coefficients. At = 8.7 K, a crossover from one- to three-dimensional antiferromagnetic behavior occurs. This long-range antiferromagnetic order leads to distinct jumps in the heat capacity and thermal expansion coefficients, which are reminiscent of those observed at the normal-superconducting phase transition. The heat capacity data reveals the presence of a gap below of magnitude = 17.48(6) K. The occurrence of three-dimensional antiferromagnetic order is attributed to a Spin-Peierls transition occurring as a result of three-dimensional phonons coupling with Jordan-Wigner-transformed Fermions. Spin-flop transitions for magnetic field applied in and axes below indicate that the easy AFM axis lies in both the and directions, which may reflect the presence of significant magnetic disorder, or weak differences in the magnetocrystalline energy for these two directions.

We thank Anton Vorontsov for valuable discussions. This material is based on the work supported by the National Science Foundation under Contract No. DMR-0907036.

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