Dispersive magnetic excitations in the S=1 antiferromagnet Ba{}_{3}Mn{}_{2}O{}_{8}

Dispersive magnetic excitations in the antiferromagnet BaMnO

Abstract

We present powder inelastic neutron scattering measurements of the dimerized antiferromagnet BaMnO. The  K magnetic spectrum exhibits a spin-gap of  meV and a dispersive spectrum with a bandwidth of approximately  meV. Comparison to coupled dimer models describe the dispersion and scattering intensity accurately and determine the exchange constants in BaMnO. The wave vector dependent scattering intensity confirms the proposed dimer bond. Temperature dependent measurements of the magnetic excitations indicate the presence of both singlet-triplet and thermally activated triplet-quintet excitations.

pacs:
75.10.Jm, 75.40.Gb, 75.30.Et

I Introduction

Low-dimensional and gapped quantum magnets based upon strongly coupled spin pairs or dimers with weaker interdimer interactions have become especially topical systems. This is primarily due to the relevance of experimentally accessible quantum critical points (1); (2); (3); (4). For antiferromagnetic intradimer exchange, the ground state of such systems is a product of singlets, but strong magnetic fields can close the spin-gap to excited triplet states via Zeeman splitting of the triplet (5); (6); (7). Such systems thus provide an elegant realization of a lattice gas of hardcore bosons in which the external magnetic field plays the role of the chemical potential and the interdimer coupling determines both the kinetic and potential energy of the delocalized triplets (8); (9). Depending on the balance of these energy scales the triplets will either crystallize or condense at low temperatures(10) or, under the right set of conditions, form a supersolid (11). BaMnO is a particularly promising candidate material for the detailed study of magnetic field dependent quantum critical points. BaMnO has been identified as a dimerized antiferromagnet and the low-temperature phase diagram has been examined using thermodynamic measurements (12); (13); (14). However, a measure of the dominant exchange constants has been notably absent.

Figure 1: (Color online) Crystal structure of BaMnO. Oxygen atoms are not shown for clarity. Grey dashed lines indicate chemical unit cell. (a) Polyhedral representation of structure. Light green and dark green polyhedra around Ba sites represent the two types of Ba coordination that alternate along the -axis, 12 and 10-fold coordination respectively. Red tetrahedra illustrate the Mn coordination. (b) Simplified structure figure illustrating dominant exchange connectivity in BaMnO within and between the dimers. and exchange correspond to distances of 4.569 and 5.711 Å respectively.

Low-field magnetic susceptibility and pulsed field magnetization measurements of BaMnO agree with weakly coupled dimer models with antiferromagnetic dimer exchange between 1.50 and 1.65 meV and a zero-field spin-gap of  meV (12); (15). Unfortunately, comparisons to isolated dimer models only yield results in terms of weighted sums of interdimer exchange. Specific heat measurements on powder samples have been performed in magnetic fields up to  T to describe the low-temperature magnetic phase diagram of BaMnO(13). At  K, a critical field of  T is required to induce magnetic ordering. Following the thermodynamic phase transition to lower temperatures, the experiments extrapolate to a zero-temperature  T quantum critical point. These measurements also indicate two phase transitions near the lower critical field implying the existence of two magnetic long-range-ordered phases in close proximity to the quantum critical point.

More recent measurements using single crystals confirm the presence of two distinct ordered phases and provide a detailed map of the phase diagram (14). EPR measurements indicate a zero-field splitting of the triplet states, attributed in part to the effects of single ion anisotropy (16), and implying more complex magnetic structures than previously suggested(15).

The applicable Hamiltonian for this system is not yet fully characterized, and the magnitude and extent of dimer and interdimer exchange interactions are important for further understanding of BaMnO in both zero and applied magnetic fields. Here we describe temperature dependent inelastic neutron scattering (INS) measurements on polycrystalline samples to determine the extent of dispersive magnetic excitations and exchange constants in BaMnO.

BaMnO is hexagonal (space group ) with room temperature lattice constants and  Å(18). The Mn ions reside in a distorted tetrahedral environment, as shown in Fig. 1(a), resulting in an effective moment. Dimer and interdimer magnetic interactions are considered to be antiferromagnetic () and Heisenberg with a small single-ion anisotropy, . Including a Zeeman term for applied magnetic fields along the -axis, the Hamiltonian is

(1)

where and designate coordinates of individual interacting spins, . The nearest neighbor sites along the -axis,  Å, have been proposed as strongly coupled antiferromagnetic dimers. The dimers form an edge-sharing triangular lattice bilayer in the -plane. Neighboring planes along the -axis are separated by alternating layers of oxygen coordinated Ba sites, ten and twelve site coordination respectively, as illustrated in Fig. 1(a). Weaker interdimer exchange within the -plane, , and between the bilayers, , have also been proposed based upon the crystal structure. These are illustrated in Fig. 1(b) along with the dimer exchange .

Ii Experimental Methods

Powder samples of BaMnO were synthesized using stoichiometric amounts of BaCO and MnO in a solid state reaction. The reactants were calcined under flowing oxygen at  C for 30 h. The resulting green powder was then reground and sintered between and  C under flowing oxygen for approximately 7 days. This growth procedure is similar to that described in Ref. (12). Single crystals of appropriate mass are unfortunately not yet available for studies of the dispersion using INS.

SQUID magnetization measurements as a function of temperature for  Oe and as a function of magnetic field for  K  did not reveal any measurable impurities due to MnO or MnO which are ferrimagnetic below  K and antiferromagnetic below  K respectively(19). The temperature dependent magnetic susceptibility compares well with previously published data with a rounded peak at  K and an activated low-temperature susceptibility characteristic of antiferromagnetic spin-gap systems (12); (15). We also characterize the  K magnetic susceptibility via a Curie-Weiss law with  K.

Inelastic neutron scattering measurements were performed on 76 grams of BaMnO in an 18 mm diameter and 100 mm tall cylindrical aluminum sample can. The sample was produced from five separately prepared batches of BaMnO. Each batch was checked for impurity phases using SQUID magnetization and powder X-ray diffraction. Both measurements found no measurable impurity phases. Inelastic neutron scattering measurements were performed using the direct geometry chopper spectrometer, DCS, at the NIST Center for Neutron Scattering. Spectra were measured for temperatures between and  K for one hour in each configuration. Temperature control was provided by a liquid He flow cryostat. Two incident wavelengths, , were used.  Å measurements probed energy transfers up to  meV and wave vectors up to  Å at the elastic position.  Å measurements provide improved energy and wave vector resolution up to  meV and  Å. The calculated full width at half maximum (FWHM) energy resolution at the elastic position is and  eV for the and  Å incident wavelengths respectively. Background measurements were made for each incident wavelength using an empty sample can at  K. A  K vanadium standard was measured for calibration of detector sensitivity. A scattering angle dependent absorption correction for the cylindrical sample geometry was also applied to the inelastic scattering intensity. Unless otherwise noted, these backgrounds and normalization are applied to our presented results.

Neglecting Bose occupation and Debye-Waller factors, the magnetic neutron-scattering cross section is proportional to the scattering function, ,

(2)

where and are the magnitude of the final and initial neutron wave vectors and is the magnetic ion form factor. We plot our measured scattering intensity in units of , but we do not normalize the data by the magnetic form factor.

Iii Results and Discussion

Figure 2 shows the scattering intensity as a function of wave vector transfer in the vicinity of the elastic position,  meV, at several temperatures. Nuclear Bragg peak positions and intensity compare well with the previously determined room-temperature structure (18). We note that for  K there are two additional weak Bragg peaks at and  Å. These are likely associated with either MnO or MnO impurities as discussed earlier. We do not observe any contribution from spin-waves due to these impurities in the inelastic portion of the spectra.

Figure 2: (color online) Temperature dependent diffraction for BaMnO measured in the  Å configuration. Data are integrated between -0.1 and 0.1 meV energy transfer. Data have not been background subtracted or corrected for absorption. Higher temperature data are offset vertically from the  K data for presentation. Several characteristic Bragg peaks are indexed in the figure.
Figure 3: (color online) Temperature dependent lattice parameters of BaMnO. (a) -axis lattice constant versus temperature. (b) -axis lattice constant versus temperature. (c) as a function of temperature. Solid points from Ref. (18) for K are shown for comparison. Lattice constants were obtained from the weighted mean peak position of the (003), (006), (009) and (110) Bragg peaks for and  Å. Error bars are the error in the mean. Solid lines in (a) and (b) are linear fits described in the text. Solid line in (c) is calculated based upon the linear fits for the and -axis lattice constant.

Although wave vector resolution was not optimized for diffraction ( for  Å), we fit the , , and Bragg peak positions for each incident wavelength to determine lattice constants as a function of temperature, c.f. Fig. 3(a) and (b). These values are consistent with the previously determined room temperature structure. There are no apparent structural phase transitions from  to 1.4 K. Both the and lattice vectors contract at lower temperatures, but there is only a change in the -axis lattice constant and an even smaller change in the -axis lattice constant, , from  K to  K. Linear fits to these data along with the  K values are shown in Fig. 3(a) and (b) along with calculated values of and the respective calculated curve in panel (c). The fitted lines provide a good description of the data with coefficients of linear expansion: and  K. Including quadratic terms does not substantially improve the fits. The weak lattice parameter temperature dependence indicates that any changes in the magnetic excitation spectra with temperature are not likely associated with changes in exchange due to changes in distance between individual spins.

Figure 4: (Color online) Measured and calculated INS intensity for BaMnO at  K. Panels (a) and (b) are data measured for incident wavelengths of and  Å respectively. Panels (c) and (d) are the calculated scattering intensities corresponding to the coupled dimer model described in the text. Panels (e) and (f) are the energy dependent integrated scattering intensity of the data and model calculations shown in panels (a)-(d) for wave vectors  Å and  Å for and  Å respectively. Model lineshapes in (e) and (f) are fit to the data including an elastic Gaussian lineshape and a constant background as discussed in the text.

The  K excitation spectra for the and  Å configurations are shown in Fig. 4(a) and (b). There is a single dispersive excitation with a gap of  meV and a bandwidth of approximately 1.5 meV. The decreasing scattering intensity with increasing wave vector transfer immediately suggests the spectrum is magnetic. The increase in scattering intensity near meV in Fig. 4(a) for small wave vectors is not intrinsic to the sample and has been seen in background measurements. Figure 4(e) and (f) show the low-temperature wave vector integrated scattering intensity. There is a peak in the magnetic density of states in the vicinity of the spin-gap energy, but there is no peak at the top of the dispersive band of excitations. The spectrum of one-dimensional Heisenberg gapped quantum antiferromagnets has characteristic Van-Hove singularities at the top and bottom of the band. The absence of any peak at higher energy transfers implies the magnetic excitations in BaMnO are at least two-dimensional in their connectivity.

For comparison to our results, we calculate the INS cross section of antiferromagnetic dimers with weak interdimer interactions. At finite temperature the scattering function is written as (20)

(3)

where , is the partition function and the sum is over the initial and final states and with energies and . For a dimer, Eq. 3 becomes

(4)

where and are the respective crystallographic coordinates of the spins in the dimer and are the spin operators.

An isolated antiferromagnetic dimer with intradimer exchange will have a non-magnetic (total spin ) singlet ground state at an energy of with triplet and quintet excited states at energies and . This results in triplet and quintet spin-gaps of and . However, magnetic INS will only probe the singlet-triplet and triplet-quintet cross-sections, i.e. . Including structure factors from Eq. 4, the scattering function is a sum of two terms,

(5)

where is the bond vector between the spins of the dimer. Although the calculated matrix elements of the triplet-quintet transitions are larger than the singlet-triplet transitions, the triplet-quintet transitions are thermally activated with less spectral weight and will only be populated at higher temperatures.

Equation 5 does not account for interdimer correlations, i.e. dispersive excitations. The random phase approximation (RPA) has been successful in describing the dispersion of weakly coupled dimers in spin-gap systems. This has been illustrated for varying numbers of interactions and spin-quanta in several experimental systems including KCuCl(21); (22), TlCuCl(23), CsCrBr (24), CsCrI(25), PHCC(4) and BaCuSiO(26). The RPA dispersion for Heisenberg exchange coupled dimers is

(6)

where is the transition matrix element (), is the Fourier sum over interactions beyond dimer exchange, and is the thermal population difference between the ground and excited states. For antiferromagnetic dimers,

(7)

considering only singlet-triplet excitations. BaMnO has a single dimer per unit cell and the interdimer interactions propagate the triplet excitation leading to the Fourier sum

(8)

Recent EPR measurements have revealed a zero-field splitting of  meV, although both modes will have an identical dispersion(16). Examination of thermodynamic measurements have included an exchange constant, , which couples neighboring dimers in a bilayer from spin-1 of a dimer to spin-2 of a second dimer(15). This exchange term represents a change in phase of the triplet excitation between dimers in the plane, and would change the prefactor of the term in Eq. III to be . We have chosen to omit the exchange from the current analysis. Because of its large spin-spin distance ( Å) and out of plane coupling, it is presumably much weaker than and . In addition, recent calculations examining the relative strength of exchange constants in BaMnO have shown that the exchange constant is effectively zero (17).

The scattering function must also be modified to account for the dispersive excitations. The single mode approximation (SMA) has been successfully applied to several dispersive disordered gapped antiferromagnets with interdimer exchange included in the Hamiltonian (28); (4); (29); (26); (27). This results in an additional multiplicative term in such that considering only singlet-triplet scattering for BaMnO the scattering function becomes

(9)

where is given by Eqs. 6-III. The SMA is appropriate for the case of BaMnO for since only singlet-triplet excitations will be effectively probed via INS in this temperature regime.

For comparison of our high resolution ( Å) polycrystalline measurements we numerically spherically average Eq. 9,

(10)

This process was also recently employed successfully in examination of polycrstyalline measurements of a gapped antiferromagnet using the same instrumentation(27). The interpretation of our measurements is more straightforward given the absence of hydrogen or contamination from phonons in our spectrum. We calculate over spherical shells in space at fixed values of energy transfer with fixed as the proposed dimer bond vector. This was done for a series of , and values. This spectrum was combined with an identical spectrum shifted in energy transfer by the value  meV. The spectrum was then multiplied by   (30); (31) and convolved with a Gaussian representation of the mean instrumental energy and wave vector resolution over the energy and wave vector range of the magnetic excitation. A constant background and multiplicative prefactor were used as fitting parameters of the calculated spectrum in comparison to the measured data. This procedure yields best fit exchange constants   and  meV(32). The corresponding best fit INS scattering intensity for both instrument configurations is plotted in Fig. 4(c) and (d), and agrees very well with the dispersion and intensity modulation observed in the measurement. The determined value of and the corresponding spin-gap based upon the dispersion,  meV, are both in the vicinity of values from thermodynamic measurements.

Figures 4(e) and (f) show the fitted wave vector integrated lineshapes compared to the measured data. Fit parameters for the calculated lineshapes include an elastic Gaussian, an overall multiplicative prefactor and a constant background. The calculations based upon both incident wave lengths agree very well with the measurement.

We plot the low-temperature singlet-triplet dispersion relation in Fig. 5. Overall minima in the dispersion occur at the and wave vectors for and and wave vectors for where is an integer and . The nonzero value of results in a finite dispersion along the direction with a periodicity of three reciprocal lattice units (rlu). This is shown in Fig. 5(b) for the curves plotted using the top axis.

Figure 5: (Color online)  K singlet-triplet single crystal dispersion relation determined for BaMnO based upon powder average analysis. (a) Contour figure of dispersion in the hexagonal (HK0) plane. (b) Dispersion as a function of reciprocal lattice units (rlu) along other primary directions of the hexagonal structure. Dispersion along the and directions is plotted using the top axis.
Figure 6: (Color online) First moment of scattering intensity integrated between and meV for the K and 2.9 Å measurements shown in Fig. 4(a) and (b). Solid lines correspond to fits described in the text with the fitted value of  Å.

The powder average of the first frequency moment of the energy integrated scattering function,

(11)

provides direct information regarding the length of the dimer bond, . Figure 6 shows the first moment of the measured  K scattering intensity for both incident wavelengths. These data are fit to Eq. 11 including an overall constant and multiplicative prefactor. A simultaneous fit of both data yields a good representation of the measured results with  Å. This value agrees with the description of the dimer bond being the short vertical bond between Mn moments in the BaMnO crystal structure Å, c.f. Fig. 1.

Figure 7: (Color online) Energy dependent scattering intensity for  Å for a series of temperatures as measured in the  Å configuration. Data for different temperatures are vertically offset. Solid lines are fits to the Lorentzian functions described convolved with the energy dependent resolution of the instrument. Inset illustrates scattering intensity of the  K measurement plotted on a logarithmic intensity scale. Dotted red line corresponds to the spectral contribution of the lower energy mode and the dashed blue line corresponds to the higher energy mode.

The magnetic spectra of BaMnO as a function of temperature for  K was also measured. In order to consider a range of energy transfers beyond the singlet-triplet band, we examine the  Å  configuration data. Constant wave vector scans for  Å are shown in Fig. 7 for several temperatures for both neutron energy gain and loss. This wave vector was chosen because it is in the vicinity of both the peak in the density of states and the overall minimum in the measured dispersion. From the  K  spectrum in Fig. 4(b), this wave vector also provides minimal interference from higher energy portions of the singlet-triplet spectrum. The  K spectrum consists of a single peak at  meV with no significant scattering intensity on the neutron energy gain side of the spectrum. As temperature is increased, the single peak broadens, moves to slightly larger energy transfers and additional scattering intensity develops in the vicinity of 3 meV. There is also additional scattering intensity which develops on the neutron energy gain side of the spectrum at elevated temperatures.

In order to distinguish singlet-triplet and thermally activated triplet-quintet excitations, we model the constant wave vector scans using two excitations. We fit to two Lorentzian functions of the form (33)

(12)

where , , and are the half width at half maximum and energy of two respective excitations with the Lorentzian width reflecting temperature dependent broadening of the spectrum. The Bose factor enforces detailed balance of the scattering intensity. Equation 12 and a variable width and amplitude Gaussian peak at the elastic position were convolved with the energy transfer dependent instrumental energy resolution with fitting parameters , , and . A time independent background of the form , as typically used for time-of-flight direct geometry INS measurements and a constant background were determined from the  K data and held fixed. The two excitation fits are shown as solid lines in Fig. 7. The lineshapes agree well with the energy dependent distribution of scattering intensity for both neutron energy loss and gain. To further illustrate the presence of the triplet-quintet excitation, we plot the  K data and fitted lineshape including the individual contributions from each mode in the inset of Fig. 7.

Figure 8: (Color online) Temperature dependent integrated intensity and characteristic energy determined from fits to data shown in Fig. 7. (a) Total and neutron energy gain and loss portions of the integrated scattering intensity for both modes. The solid black line is a fit of the temperature dependent total scattering intensity of the isolated antiferromagnetic dimer including both singlet-triplet and thermally activated triplet-quintet excitations. The dashed line only includes singlet-triplet excitations. Solid blue and green curves are calculated from the singlet-triplet and triplet-quintet fitted scattering intensity. (b) Mode energy as a function of temperature. The K spin-gap value determined from fitted exchange parameters is shown as a single red diamond. Solid line is a calculation of the RPA model temperature dependent spin-gap using the determined values of exchange constants.

Figure 8(a) shows the total integrated scattering intensity for both modes as well as the neutron energy gain and loss portions as a function of temperature. We fit the total scattering intensity to the temperature dependent isolated dimer scattering function in Eq. 5 with an overall multiplicative prefactor and the exchange as fitting parameters. Considering only singlet-triplet excitations yields the dashed red line in Fig. 8(a) and a value of meV. However, including both singlet-triplet and triplet-quintet modes yields a much better fit (solid black line) and a more appropriate value of meV. The improved fit implies that there is a non-negligible contribution of thermally activated triplet-quintet excitations at higher energy transfers. The calculated temperature dependent scattering intensity for the neutron energy gain and loss portions of the spectrum based upon the singlet-triplet, triplet-quintet model also agrees well the data shown in Fig. 8(a).

The fitted values  and  are plotted in Fig. 8(b). The value is not the spin-gap energy, rather it is the characteristic energy of the lower energy mode found for the constant wave vector scan. For comparison, we also plot the determined spin-gap energy based upon the  K powder average analysis. The calculated temperature dependent spin-gap of the singlet-triplet excitation based upon Eqs. 6-III and the fitted exchange constants is plotted as a solid line in Fig. 8(b). The RPA coupled dimer description agrees well with the temperature dependence of the energy scale of the singlet-triplet excitation. We also note that above base temperature the RPA function agrees with the temperature dependence of the activated triplet-quintet excitation, , although shifted to higher energy transfers, implying that these excitations may share a similar temperature dependent dispersion renormalization.

Iv Conclusions

Through INS measurements we have shown that there exists a well-defined singlet-triplet spectrum in BaMnO. Although the measured bandwidth of the magnetic spectrum is larger than the spin-gap, the exchange constants indicate that BaMnO can be considered a triangular lattice of weakly coupled dimers. Comparison to an appropriate RPA coupled dimer description of the scattering function is able to determine the exchange constants. In addition, the wave vector dependent scattering intensity agrees with the dominant dimer bond being the predicted short vertical bond illustrated in Fig. 1(b).

The examination of temperature dependent scattering intensity indicates that both singlet-triplet and triplet-quintet excitations are observed in the INS spectrum (there is no INS cross-section for antiferromagnetic singlet-quintet excitations). Based upon the relative energy scales of the two observed excitations, we can estimate the mean energy of singlet-quintet excitations as  meV. Single-crystal INS measurements may be able to determine the dispersion associated with thermally activated triplet-quintet excitations or perhaps observe multi-particle excitations, quintet-triplet decay or interference of single- and multi-particle excitations (34). The existence of singlet-quintet and triplet-quintet excitations may also be able to explain the heat capacity above  K which can not be accounted for by single excitation models(13).

We also point out that the currently determined exchange constants may be able to further describe the magnetic field dependent phase diagram at low-temperatures or place limits on the nature of the proposed long-range-ordered phases. The difference between the observed phase diagrams for and must be related to the single ion anisotropy term in the Hamiltonian, Eq. 1. As we will show elsewhere(14), this term induces an effective exchange anisotropy in the low-energy Hamiltonian that results from projecting the original Hamiltonian into the subspace generated by the singlet and triplet of each dimer. These two states can be described with a pseudospin 1/2 variable. The combined effect of geometric frustration and anisotropy leads to the appearance of a new phase for that will be discussed in Ref. (14).

V Acknowledgments

MBS and MDL acknowledge valuable discussions with I. Zaliznyak and A. Zheludev. ORNL is managed for the US DOE by UT-Battelle Inc. under contract DE-AC05-00OR22725. This work utilized facilities supported in part by the National Science Foundation under Agreement No. DMR-0454672. work at Stanford was supported by the National Science Foundation, under grant DMR 0705087.

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