Magneto-structural coupling in ilmenite-type NiTiO_3: a combined diffraction and dilatometry study
Kaustav Dey, Sven Sauerland, Bashir Ouladdiaf, Ketty Beauvois, Hubert Wadepohl, Rüdiger Klingeler
MMagneto-structural coupling in ilmenite-type NiTiO : a combined diffraction and dilatometry study K. Dey, ∗ S. Sauerland, B. Ouladdiaf, K. Beauvois, H. Wadepohl, and R. Klingeler
1, 4 Kirchhoff Institute of Physics, Heidelberg University, INF 227, 69120, Heidelberg, Germany Institut Laue-Langevin, CS20156, 38042 GRENOBLE Cedex 9 - France Institute of Inorganic chemistry, Heidelberg University, 69120, Heidelberg, Germany Center for Advance Materials (CAM), Heidelberg University, INF 227, D-69120, Heidelberg, Germany (Dated: February 19, 2021)We report the ground state magnetic structure and in-field magnetostrictive effects of NiTiO studied bymeans of zero field and in-field single crystal neutron diffraction, magnetization and high-resolution dilatometryexperiments. Zero-field neutron diffraction on NiTiO single crystals reveal an easy-plane antiferromagnetwith a multidomain ground state. Upon application of external magnetic fields, neutron diffraction shows theevolution of domains with spins perpendicular to the applied field. The rotation of spins in the multidomainstate exhibits pronounced lattice changes in the magnetostriction measurements. We see magnetization andmagnetostriction measurements sale with each other in the multidomain state revealing the strong coupling ofspins to the lattice. PACS numbers:
I. INTRODUCTION
Layered honeycomb magnets have been a great av-enue for exciting and rich physics since time immemorial.The recent theoretical and experimental studies into Kitaevquantum spin-liquid in Co-based honeycomb materials ,Dirac magnons and topological spin excitations in hon-eycomb ferromagnets, non-reciprocal magnons in honey-comb antiferromagnets , zig-zag and incommensurate spinground states or 2D magnetism in Van-der-Waals materials have resulted in enormous interest in these class of mate-rials. Moreover, the spin-lattice coupling in several hon-eycomb magnets such as Fe Nb O , Na Ni SbO , andCo Nb O have resulted in significant magnetoelectric cou-pling and hence motivating possible technological applica-tions.Ilmenite titantes with chemical formula M TiO ( M = Mn,Fe, Co, Ni) form an isostructural series of antiferromagnetic(AFM) compounds where magnetic M ions in the basal ab -plane exhibit a buckled honeycomb-like structure. The M ions are interconnected via oxygen ions (O − ) leading to M -O- M as the dominant superexchange pathway . Along c axisthe crystal structure exhibits alternating layers of corner shar-ing TiO and M O octahedra resulting in relatively weaker M -O-Ti-O- M superexchange pathways. Depending on thesingle ion-anisotropies of the respective metal ions, variousmagnetic ground states are realised in ilmenites, for exampleuniaxial AFM ground state with spins pointing along c -axis inMnTiO whereas an easy-plane type AFM with spins lyingin the ab -plane for NiTiO and CoTiO respectively.Although, these compounds have been rigorously inves-tigated since 1950s , recent studies evidencing lin-ear magnetoelectric coupling in MnTiO , large spontaneousmagnetostriction in FeTiO , magnetodielectric and magne-toelastic coupling in NiTiO and CoTiO , respectively,as well as the observance of Dirac magnons in CoTiO have peaked enormous interest in these class of materials.The least investigated compound among the ilmenitesfamily, i.e., NiTiO , develops long-range AFM order at T N = 22.5 K . Recent studies of the dielectric per-mittivity and the thermal expansion show a pronounced mag-netodielectric effect as well as distinct significant magne-toelastic coupling . Notably, at T N , there is single energyscale dominantly driving the observed structural, magneticand dielectric anomalies . In this report, we study in detailthe magneto-structural coupling of NiTiO by means of sin-gle crystal X-ray and neutron diffraction and high-resolutiondilatometry. We observe by means of single-crystal neutrondiffraction that the macroscopic structural symmetry( R -3) isretained down to the lowest measured temperature of 2 Kwithin the experimental resolution. In addition, the mag-netic ground state of NiTiO is solved. At T N , in addi-tion to long-range AFM order, a significant lattice distor-tion evolves revealing large spontaneous magnetostriction inNiTiO . In applied magnetic fields, the multi-domain groundstate evolves to a spin-reoriented single domain state charac-terized by spins aligned perpendicular to the applied magneticfield. Magnetostriction measurements in the low-field re-gion show pronounced effects due to magnetoelastic domainsand remarkably scales with magnetization measurements con-firming both significant magneto-structural coupling and themagneto-structural domain model in NiTiO . II. EXPERIMENTAL METHODS
Macroscopic single crystals of NiTiO have been grownby means of the optical floating-zone technique in a fourmirror optical floating-zone furnace (CSC, Japan) equippedwith 4 ×
150 W halogen lamps. Details of the growth pro-cess and characterization the single crystals have been pub-lished previously . Single crystal X-ray intensity data wereobtained at 100 K with an Agilent Technologies Supernova-ECCD 4-circle diffractometer (Mo-K α radiation λ =0.71073 ˚A,micro-focus X-ray tube, multilayer mirror optics). Static mag-netisation χ = M/B was studied in magnetic fields up to5 T in a Quantum Design MPMS-XL5 SQUID magnetome-ter. The relative length changes dL i /L i were studied on a a r X i v : . [ c ond - m a t . s t r- e l ] F e b cuboid-shaped single crystal of dimensions × . × mm by means of a three-terminal high-resolution capacitancedilatometer . Magnetostriction, i.e., field-induced lengthchanges dL i ( B ) /L i , was measured at several fixed tempera-tures in magnetic fields up to 15 T and the longitudinal mag-netostriction coefficient λ i = 1 /L i · dL i ( B ) /dB was derived.The magnetic field was applied along the direction of the mea-sured length changes.Single crystal neutron diffraction experiments were per-formed up to 6 T magnetic fields on the D10 beamline of theInstitut Laue-Langevin (ILL) at Grenoble, France. To deter-mine the magnetic ground state at B = 0 T, the four-circleconfiguration was used with a 96 ×
96 mm two-dimensionalmicrostrip detector. An incident wavelength of 2.36 ˚A usinga vertically focusing pyrolytic graphite (PG)(002) monochro-mator was employed. A pyrolytic graphite filter was used inorder to suppress higher-order contamination to 10 − timesthat of the primary beam intensity. Measurements were madein the temperature range 2-50 K. The magnetic field-drivenevolution of the magnetic structure at T = 2 K was studied bymounting the sample in a 6 T vertical cryomagnet and alignedto within 1 ◦ of magnetic field. The magnetic field was appliedalong the b -axis limiting the scattering to the ( H, , L ) plane. III. EXPERIMENTAL RESULTSA. Single-crystal X-ray Diffraction
To the best of our knowledge, the earlier studies of theilmenite-type NiTiO crystal structure have been limited topowder diffraction experiments only . We have re-investigated the crystal structure by means of single-crystalhigh resolution XRD at 100 K, using Mo K α radiation ( λ =0.71073 ˚A). A single crystal splinter of size 0.16 × × was broken of from larger specimen and used fordata collection. A full shell of intensity data was collectedup to 0.4 ˚A resolution (24180 reflections, 1028 indepen-dent [ R int = 0 . ] of which 1024 were observed [ I > σ ( I ) ]). Detector frames (typically ω , occasionally φ -scans,scan width 0.5 ◦ ) were integrated by profile fitting . Datawere corrected for air and detector absorption, Lorentz andpolarization effects and scaled essentially by application ofappropriate spherical harmonic functions . Absorptionby the crystal was treated numerically (Gaussian grid) .An illumination correction was performed as part of the nu-merical absorption correction . Space group R − wasassigned based on systematic absences and intensity statis-tics (refined obverse centered unit cell on hexagonal axes,Hall group − R , a = 5 . , c = 13 . ˚ A , V = 301 . ˚ A , Z = 6 ). This choice was confirmedby analysis of the symmetry of the phases obtained ab ini-tio in P1. The structure was solved by intrinsic phasing and refined by full-matrix least-squares methods based on F against all unique reflections . Three somewhat differentmodels were employed for the atomic structure factors f at within the ISA approximation: conventional f at calculatedwith neutral atoms for Ni, Ti and O (model A) and two “ionic” models ( f at for Ni , Ti taken from ref.41 andO − from ref.42 (model B) or ref.43, respectively (model C)).An empirical secondary extinction correction was appliedin each case but proved insignificant. The different modelsrefined to essentially the same structure, with only insignifi-cant differences in key parameters like atom coordinates, R factors, U eq for all atoms and residual electron density. Ni-Oand Ti-O bond lengths agreed within one standard deviation.There was no evidence of cation mixing and fully occupiedsites were employed for all atoms. The results confirm theassignment of the space group and improve on the accuracyof the crystallographic parameters previously obtained frompowder XRD and neutron data . Fractional atomic co-ordinates, Wyckoff positions, site occupation and equivalentisotropic displacement parameters for model A are listed intable I . TABLE I: Fractional atomic coordinates, Wyckoff positions, site oc-cupation and equivalent isotropic displacement parameters ( ˚A ) forNiTiO at 100 K as obtained from refinement of model A. (Note: (1)These co-ordinates are correct but do not form uniquely bonded set;(2) a U eq is defined as one third of the trace of the orthogonalized U ij tensor. The anisotropic displacement factor exponent takes theform: − π [ h a ∗ U + ... + 2 hka ∗ b ∗ U ] .)Atom Site x y z sof U eqa Ni 6c 0 0 0.35051(2) 1 0.00308(2)Ti 6c 0 0 0.14422(2) 1 0.00297(3)O 18f 0.35198(8) 0.03455(8) 0.08662(2) 1 0.00421(4)
B. Single-crystal neutron diffraction
The crystal structure at lower temperatures and the mag-netic ground state of NiTiO were determined by means ofsingle-crystal neutron diffraction. At 50 K, 110 nuclear Braggreflections were collected. Appropriate correction for extinc-tion, absorption, and Lorentz factor was applied to all the nu-clear Bragg peaks. All the nuclear peaks at 50 K were success-fully indexed in the R -3 space group with lattice parameters a = 5.03 ˚A and c = 13.789 ˚A.In order to clarify the magnetic structure, preliminaryreciprocal-space scans (not shown here) were performed at2 K along the (0 , , L ) , ( H, , , and ( H, K, directions.The scans reveal a peak of significant intensity emerging at(0,0,1.5), indicative of the propagation vector k = (0,0,1.5).In order to determine the detailed magnetic structure, inte-grated intensities of 187 nuclear reflections allowed within thespace group R -3 and 292 satellite magnetic reflections werecollected at 2 K. The nuclear structure was firstly refined us-ing FULLPROF program within the R -3 space group. Theresults of refinement are listed in table II and the observedand calculated intensities from the Rietveld fits are shown inFig. 2(a). No peak splitting or significant broadening was ob-served within the experimental resolution in respective 2 Knuclear reflections as compared to 50 K, indicating that themacroscopic R -3 symmetry is maintained until the lowestmeasured temperatures. The nuclear Bragg peaks show notemperature dependence between 2 K and 50 K excluding k =(0,0,0). TABLE II: Parameters for the nuclear structure of NiTiO measuredat 2 K obtained from refinements of single-crystal neutron diffractiondata. The isotropic temperature factors ( B ) of all atoms were refined.[Space group: R -3 (148); Lattice parameters: a = b = 5.0229(1) ˚A, c = 13.7720(1) ˚A, α = β = 90 ◦ , γ = 120 ◦ .Atom Site x y z B iso ( ˚A )Ni1 6c 0 0 0.3537(2) 0.00748Ti1 6c 0 0 0.1338(5) 0.06643O1 18f 0.3344(6) 0.0052(1) 0.2466(2) 0.09830 ( 0 , 0 , 1 . 5 ) f i t Int. Intensity (a.u.)
T ( K )
I ~ ( T N - T ) (cid:1) T N = 2 2 ( 1 ) K (cid:1) = 0 . 3 5 ( 1 ) counts (per 105) W ( d e g . )
2 K 3 0 K( 0 , 0 , 1 . 5 )
FIG. 1: Temperature dependence of the integrated intensity of the(0,0,1.5) magnetic Bragg peak. The dashed black curve is a fit to thedata with the power law I ∼ ( T N − T ) β . The inset shows the Ω scanthrough the magnetic (0,0,1.5) peak at 2 K and 30 K respectively. Thesolid blue line is the Gaussian fit to the peak at 2 K. See the text formore details. All the finite intensity magnetic peaks are observed at thegeneral position ( H, K, L ) + (0,0,1.5) with H, K, L satisfy-ing the reflection conditions of the R -3 space group and henceconfirming k = (0,0,1.5). A few of the observed high-intensitymagnetic peaks are listed in Table III. The largest diffractionintensity occurs for the magnetic Bragg peak (0,0,1.5) indi-cating that the Ni -moments lie in the ab plane which hadbeen suggested by previous magnetization measurements .The temperature dependence of the integrated intensity of thecommensurate reflection (0,0,1.5) in Fig. 1 shows finite inten-sity below the magnetic ordering temperature. A power law fitin the critical region using I ∝ M ∝ τ β where M is the or-der parameter and τ = 1 − T /T N results in T N = 22(1) K and β = 0 . . The obtained value T N from the power law fitagrees to the one from previous macroscopic studies .The obtained critical parameter indicates that Ni -spins inNiTiO are of 3D Heisenberg nature. TABLE III: Observed intensities( I obs ) of several high-intensity mag-netic peaks as measured in D10 at 2 K and their corresponding cal-culated intensities( I cal ) as discussed in the text.Q I obs I cal (0,2,2.5) 975(17) 917(0,2,5.5) 1410(27) 1307(0,-1,5.5) 2481(36) 2685(0,0,4.5) 2809(22) 3348(1,-2,-1.5) 1923(20) 2045(1,-2,4.5) 1755(22) 1521(0,-1,2.5) 1787(17) 1965(-1,2,4.5) 1812(48) 1521(0,-1,8.5) 1729(109) 1679(0,0,1.5) 4366(21) 3942 I cal (*103) I o b s ( * 1 0 ) ( a ) N u c l e a r 2 K R F = R F (cid:1)(cid:1) = (cid:1) I cal (*102) I o b s ( * 1 0 ) ( b ) M a g n e t i c 2 K R F = R F (cid:1) = (cid:1)(cid:1)(cid:1) ( c ) N iT i
FIG. 2: Comparison between the observed and calculated integratedintensities of the non-equivalent nuclear (a) and magnetic (b) reflec-tions, respectively, at 2 K, and (c) easy-plane type magnetic structureof NiTiO as determined from the refinements at 2 K. The knowledge of the propagation vector k = (0,0,1.5) withthe Ni -moments lying in the hexagonal ab -plane points to-wards two possible magnetic models for NiTiO : (a) FMlayers stacked antiferromagnetically along the c -axis or (b)AFM layers with the spins aligned ferromagnetically alongthe c -axis. Previous static magnetic susceptibility χ = M/H vs. T measurements reveal the decrease of χ ab below T N whereas χ c stays nearly constant . Moreover, the mag-netic model (b) implies a zero magnetic structure factor at theposition Q = (0 , , . contrary to our observation. Hencemodel (a) is most suitable to describe the magnetic structureof NiTiO . Hence, the obtained magnetic peaks at 2 K wererefined against model (a) and a very good fit was obtainedas shown in Fig. 4(b). The obtained magnetic structure ofNiTiO re-confirms the structure proposed by Shirane et al. based on powder neutron data as early as 1959 . The cal-culated intensities of several high-intensity peaks are listed inTable III and the complete magnetic structure of NiTiO isschematically represented in Fig. 2(c). At T = 2 K, the or-dered moment amounts to 1.46(1) µ B .The crystal symmetry of the basal hexagonal planes ismarked by the presence of two sets of three two-fold axes.Hence, the rotated by 120 ◦ in-plane spin-configurations areexactly equivalent leading to the presence of spin domains(i.e., three domains). Since the refinements are usually per-formed using the average of the integrated intensities of theequivalent reflections, the directions of the spins cannot beuniquely determined using single-crystal neutron diffractionalone, similar to the problem existing in the powder diffrac-tion experiments . However, excellent agreement of the inte-grated intensities between the equivalent reflections ( R int =1 . indicates that there are likely three spin-domains ofequal population with spins rotated by 120 ◦ in between theneighbouring domains. C. Magneto-structural-dielectric coupling
FIG. 3: (a) Relative length changes dL ∗ i = ( L i − L Ki ) /L Ki measured along the principle crystallographic a - and c -axis, respec-tively, by means of high-resolution dilatometry. (b) Normalised dis-tortion parameter δ/δ , with δ = ( dL ∗ a − dL ∗ c ) / ( dL ∗ a + dL ∗ c ) . (c)Scaling of non-phononic linear thermal volume expansion volume( dV (cid:48) /V ) with the normalized dielectric permittivity digitzed fromref . The vertical dashed lines indicate T N . The magneto-structural coupling in NiTiO has been stud-ied by means of high-resolution capacitance dilatometry. Theuniaxial relative length changes dL ∗ i = ( L i − L Ki ) /L Ki ( i = a, c ) (Fig. 3(a)) versus temperature show abrupt changesat T N , i.e., shrinking of the c -axis and expansion along the a -axis, which demonstrates significant magnetoelastic couplingin NiTiO . At higher temperatures T ≥ K, isotropic ther-mal expansion coefficients results in similar rate of increaseof dL ∗ i along the a - and the c -axis, respectively. To furtherelucidate lattice changes at T N , the normalized distortion pa-rameter δ/δ K , with δ = ( dL ∗ a − dL ∗ c ) / ( dL ∗ a + dL ∗ c ) , is shownin Fig. 3(b).As evidenced by the distortion parameter, different be-haviour of the a - and c -axis starts to evolve gradually below50 K while δ sharply jumps at T N (Fig. 3(a)). Evidently, on-set of long-range AFM order is associated with a large spon- taneous magnetostricton effect and it implies strong magneto-structural coupling. Large spontaneous magnetostriction hasalso been observed in other ilmenites such as FeTiO , Thelatter, however, shows a reversed magnetostrictive effect, i.e.,there is an expansion of the c -axis and shrinking of the a -axis . We attribute this difference to the differing magneticground states in FeTiO and NiTiO and corresponding vari-ation in magneto-crystalline anisotropy. Finite distortion δ upto 50 K evidences a precursor phase with short-range orderwell above T N . Due to the observed strong magnetoelasticcoupling we conclude the presence of short-ranged spin corre-lations persisting up to twice the transition temperature. Thisis corroborated by previous specific heat measurements onNiTiO which reveal that nearly 20% of magnetic entropyis consumed between T N and 50 K. In addition, it has beenshown that q -dependent spin-spin correlations couple to thedielectric response via the coupling of magnetic fluctuationsto optical phonons, thereby causing a significant magnetoca-pacitive effect . Accordingly, we conclude the significantmagnetocapacitance of 0.01% and finite magnetostriction re-cently observed in NiTiO well above T N is due to persistingspin-spin correlations . D. Spin-reorientation
The effect of magnetic fields applied within the ab -planeon the crystal and magnetic structure of NiTiO is studied bymeans of in-field neutron diffraction at 2 K. Specifically, themagnetic field is applied vertically along b -axis and the scat-tering vector lies in the ( H, , L ) plane. Several nuclear andmagnetic reflections were measured with rocking curve scansin magnetic-fields up to 6 T. As will be discussed below, thereis a considerable decrease in intensity upon application of themagnetic field for all magnetic peaks while in contrast thereis no magnetic field effect on the nuclear peak intensities. Arepresentative scan through the magnetic peak Q = (-1,0,-2.5)is shown in the inset to Fig. 4.The magnetization curve displays a non-linear dependenceon magnetic fields applied along the ab -plane as evidencedby the magnetic susceptibility χ = ∂M/∂B in Fig. 4(b).The maximum in χ at B = 1 . T is indicative of a spin-reorientation transition. Correspondingly, the integrated mag-netic intensity (Fig. 4(a)) shows a continuous decrease inmagnetic fields up to 2 T above which it stays nearly constantat a finite value. Since the magnetic neutron diffraction inten-sity is proportional to the component of the magnetic momentsperpendicular to the scattering vector, this observation indi-cates that in magnetic field the spins are rotated smoothly fromthree magnetic domains to a single domain state with spins arealigned perpendicular to fields above 2 T. Between 2 and 6 T,negligible field dependence indicates a very small canting ofspins towards magnetic field. The FWHM calculated usingGaussian fits to nuclear peaks show negligible broadening upto 6 T indicating that the magnetostriction effects on latticeparameters corresponding to the spin-reorientation is belowthe experimental resolution.
FIG. 4: (a) Integrated intensity of the magnetic (-1,0,-2.5) peak asa function of magnetic field (up and down) and (b) the derivative ofstatic magnetization with respect magnetic field ∂M/∂B as a func-tion of magnetic field (from ref. 22) at 2 K. The inset to (a) showsthe Ω -scans through the magnetic (-1,0,-2.5) peak at 0 T and 5.9 T.The solid lines in blue and red are Gaussian fits to the peaks at 0 Tand 5.9 T respectively. E. Magnetostriction
Applying magnetic fields along the ab -plane yields a pro-nounced increase of the associated lattice parameter in thelow-field region ( B < B ∗ = 2 T) while there is only smallmagnetostriction at higher fields (see Fig. 5). Magnetostric-tion is also reportedly small for fields applied along c -axis .We conclude that this behaviour is associated with the field-driven collective rotation of spins as discussed above and ev-idenced by Fig. 4. However, as will be discussed below,the magnetisation changes do not scale with magnetostric-tion and the maxima in ∂M/∂B and ∂L a /∂B do not matcheach other (see Fig. 8(a)). The magnetostriction data hencedo not correspond to what is expected for a thermodynamicspin-reorientation transition. Instead, the presence of domainshas to be involved and in the following we will present clearevidence that the data represents the change from a low-fieldmulti-domain state to a high-field uniform mono-domain one.In order to further investigate the effect of in-plane mag-netic fields, the intensity evolution of two equivalent magneticBragg peaks (3,0,1.5) and (-3,0,-1.5) belonging to two differ- FIG. 5: Relative length changes dL a /L a , at different temperatures,versus the square of the magnetic field applied along the crystal-lographic a -axis for (a) magnetic fields up to 14 T, i.e., includ-ing the high-field single-domain (homogeneous) phase, and (b) for B ≤ . T which is the low-field multi-domain phase (see thetext). The solid black lines are corresponding linear fits. The in-set to (b) shows the relative length changes versus applied magneticfield. (c) Integrated intensity of the equivalent magnetic Bragg peaks(3,0,1.5) and (-3,0,-1.5) vs. magnetic field applied along the b -axisand ( dL a /L a ) for fields along a -axis, at T = 2 K. The verticaldashed line separates the multi-domain and the mono-domain (ho-mogeneous) regions. See text for more details. ent magnetic domains is displayed in Fig. 5(c). In the mul-tidomain state, the antiferromagnetic vector is uniform withina single domain and has different directions in different do-mains. The observed isotropic decrease in intensity of bothmagnetic peaks upon application of the magnetic field impliesthat the spins of both domains rotate perpendicularly to the ex-ternal field direction. The spin-rotation process is completedat B ∗ which hence signals the formation of a spin-reorientedmonodomain state. Accordingly, no significant changes in thepeak intensities are observed above B ∗ up to 6 T. IV. DISCUSSION
FIG. 6: Temperature dependence of the square of AFM order param-eter, i.e., the normalised integrated intensity of the (0,0,1.5) magneticBragg peak, the negative non-phononic volume changes dV (cid:48) /V , andthe normalized dielectric permittivity digitized from Ref 21. Comparison of the magnetic order parameter and the rela-tive volume changes with the reported data of the dielectricfunction by Harada et al. elucidates the coupling mecha-nism between the lattice and the dielectric degrees of free-dom in NiTiO . As displayed in Fig. 6(a), the non-phononicrelative volume changes dV (cid:48) /V = 2( dL a /L a ) + ( dL c /L c ) which are obtained by subtracting the phononic contributionfrom dV /V (cf. Ref. 22) show a very similar temperature de-pendence, below T N , as the normalized dielectric permitivitty.Note, that the polycrystalline sample studied in Ref. 21 dis-plays a slightly lower T N than the single crystals studied athand. Note, that in general length changes can directly af-fect the experimentally measured permitivitty via the relation (cid:15) = Cd/(cid:15) A , where C, (cid:15) , d and A are sample capacitance,vacuum permittivity, sample thickness and area, respectively.However, the changes in sample dimensions at T N are on theorder of 10 − , while the relative change in permitivitty an or-der higher, implying that spontaneous magnetostriction is notthe driving mechanism for the observed dielectric changes at T N . Interestingly, the normalized dielectric permitivitty variesas square of the antiferromagnetic order parameter ’ L ’ repre-sented by the normalized integrated intensity of the magnetic(0,0,1.5) Bragg peak in Fig. 6(a). In order to discuss this, werecall the Landau expansion of the free energy F , in termsof polarization P , and the sub-lattice magnetization L at zeromagnetic-field : F = F + αP + α (cid:48) L + βP L + γP L − P E (1)The dielectric function is obtained as ∂ F/∂P = (cid:15) ∝ γL . Hence Fig. 6(a) qualitatively evidences the presence ofmagnetodielectric coupling in NiTiO . On top of the spin anddielectric changes, the structural changes exhibits similar be-haviour below T N . Previously reported magnetic Gr¨uneisenanalysis evidences that the entropic changes at T N to be ofpurely magnetic nature. In our opinion the spin-phonon cou-pling is responsible for observed dielectric changes at T N . Inthe presence of spin-phonon coupling the phonon frequency ω can be affected by spin-correlation as ω = ω + λ < S i .S j > resulting in modification of permitivitty via the Lyddane-Sachs-Teller equation (cid:15) = ω L /ω T (cid:15) ∞ , where (cid:15) and (cid:15) ∞ are the permitivitty at zero frequency and optical frequency,respectively and ω L and ω T are the long-wavelength longi-tudinal and transverse optical phonon modes respectively.It is noteworthy that apart from spontaneous magnetostric-tion, an exchange-striction (ES) mechanism may in principlealso lead to spontaneous lattice deformation at T N and be apotential source for dielectric anomaly at T N . Magnetodi-electricity fueled by ES mechanism have been observed inseveral systems for example Y Cu O and TeCuO . InFeTiO a combination of ES and magnetostriction mecha-nisms have been suggested for the spontaneous lattice defor-mation at T N . In particular for NiTiO an ES mechanismwould imply a change in Ni-O-Ni bond angle in the ab planecloser to 90 ◦ favouring ferromagnetic super-exchange. How-ever, diffraction experiments reveal that the bond angle in-creases from 90.34 ◦ at 100 K to 90.36 ◦ at 2 K (supplementaryFig. 2), contrary to predictions of ES. Hence, ES mechanismis excluded as the origin of lattice distortion at T N in NiTiO .The crystallographic symmetry of the easy hexagonal planein NiTiO suggests the presence of three domains with spinsrotated by 120 ◦ in different domains. Such a spin struc-ture with three domains is often observed in easy-plane-type hexagonal antiferromagnets such as CoCl , NiCl , andBaNi V O . In NiTiO , the magnetostriction data implythat the field-driven changes of the domain structure is asso-ciated with structural changes. Indeed, orientational AFM do-mains are magnetoelastic in nature and have previouslybeen observed in various systems, for example in cubic an-tiferromagnets RbMnF , KNiF and KCoF , NiO ,iron-group dihalides CoCl and NiCl , the quasi-two-dimensional AFM BaNi V O , YBa Cu O . etc. In par-ticular, Kalita and co-workers have developed phenomenolog-ical theories describing the effect of domain re-distribution onthe magnetostriction for CoCl and NiCl . Note, thatboth NiCl and CoCl are easy-plane-type antiferromagnetswith similar crystalline symmetry, i.e., trigonally distortedoctahedral environment surrounding metal ions, similar toNiTiO and CoTiO . In the following, we will describethe field-dependency of the lengths changes in NiTiO basedon the phenomenological theories developed by Kalita and co-workers.Both at low magnetic fields B || a ≤ T and at high fieldsthe field-induced striction dL a /L a varies as the square of theapplied magnetic field as shown in Fig. 5(a,b). In the latter,i.e., the mono-domain state, this is predicted by calculating theequilibrium elastic strain by energy minimization of the mag-netoelastic and the elastic contributions to the free energy .The magnetostriction in the mono-domain state is describedby ( dL a /L a )( T, B ) = α ( T )( B ) + ( dL a /L a ) s ( T, B = 0 T) (2)where α ( T ) is the temperature dependent slope and ( dL a /L a ) s ( T, H = 0) is the spontaneous magnetostrictionof the mono-domain state obtained by extrapolating the linearfit to B = 0 T. Eq. 2 fits well with dL a /L a at different tem-peratures as shown by the solid black lines in Fig. 5(a). Theobtained fit parameters are listed in Table IV. ( dL a /L a ) s cor-responds to a hypothetical spontaneous striction that would beobserved if the magnetoelastic domains did not appear at lowfields, i.e., if the total spontaneous magnetostriction was notcompensated on the whole by summation of strains in differ-ent directions in each of the domains.The magnetostrictive response upon application of mag-netic fields in the multi-domain state is governed by domain-wall motion. Specifically, magnetostriction is large due to theassociated facilitated rotation of spins. The motion of mag-netoelastic domain-walls is predominantly reversible in na-ture and the associated lengths changes again exhibit asquare-dependence on the magnetic field which is expressedby ( dL a /L a )( T, H ) = ( dL a /L a ) s ( T, H = 0)(
H/H d ) . (3)Here, H d is an empirical parameter obtained from the fits(see Table IV). As shown in Fig. 5(b), the experimental dataare well described by Eq. 3 which is in-line with the predic-tions of phenomenological models . Although the mag-netoelastic domains are predominately reversible in nature, asmall irreversibility may arise due to pinning of domain wallsby crystal defects and imperfections in the crystals. A smallremanent striction amounting to ∼ . × − , at T = 2 K,is indeed observed in our data (see the Supplement Fig. 1)which indicates the presence of predominately mobile domainwalls in NiTiO . TABLE IV: Parameters obtained from fits to the magnetostrictiondata (Fig. 5(a,b)) using Eqs. 2 and 3. ( dL a /L a ) s is the spontaneousmagnetostriction (see the text). T ( dL a /L a ) s (10 − ) H d (T) α ( − ) (T )2 K 4.79 1.41 3.810 K 3.55 1.38 7.618.3 K 1.73 1.55 12.8 Unlike uniaxial antiferromagnets which show an abruptmagnetization jump at the spin-flop transition as, e.g., ob-served in MnTiO , the magnetization in NiTiO follows asickle-shaped field dependence in the non-flopped phase andthe reorientation transition is associated with smooth rightbending in M vs. B (see Fig. 7). Such characteristic smoothnon-linear variation of magnetization in low-fields is a man-ifestation of the multi-domain state where spin-reorientation c e B S i m u l a t e d
M ( m B/f.u.)
B ( T )
T = 2 KB | | a
FIG. 7: Magnetization M , at T = 2 K, versus applied magneticfield B || a -axis. The solid blue line represents a linear fit to M in thehigh-field region and the dashed black line shows a simulation to M at low fields (see the text for more details). takes place gradually by displacement of domain walls . Thisis described by M = (1 / χ e B [1 + ( B/B d ) ] (4)where χ e is the high-field magnetic susceptibility. A lin-ear fit to the M vs. B curve at B > T yields χ e =0 . µ B /f.u.T which is represented by the solid blue line inFig. 7. Using H d from the analysis of the magnetostrictiondata (see Table IV) enables to deduce the field dependence of M . The simulation using Eq. 4 is shown by the dashed linein Fig. 7. It yields a good description of the field-driven evo-lution of the magnetization in the multi-domain state, therebyfurther confirming the applied phenomenological model. Theblue line in Fig. 7 represents the expected magnetization in asingle-domain easy-plane AFM with no in-plane anisotropy. ¶ M/ ¶ (B) ( m B / f.u. T)
B ( T )
B | | a ( a )
T = 2 K l ||a (10-5 / T) B | | aT = 2 K ( b )
H * ¶ (M/B)/ ¶ ( B) (10-2 m B/f.u.*T2)
B ( T )
FIG. 8: (a) Scaling of ∂M / ∂B , (b) ∂ ( M/B ) / ∂B and λ (cid:107) a versus B at T = 2 K. The field-driven disappearance of the multi-domain stateyields different behaviour of the magnetic susceptibility ∂M/∂B and the magnetostriction ∂L/∂B . This is demon-strated in Fig. 8(a) where the derivative of the magnetiza-tion and the longitudinal magnetostriction coefficient λ || a =(1 /L a ) ∂L a /∂ ( µ H ) are shown at T = 2 K as a function of B . The data are scaled to match the corresponding peak val-ues. According to the Ehrenfest relation ∂B ∗ /∂p i = V m ∆ λ i / ∆[ ∂M/∂ ( µ H )] (5)Using molar volume V m = /mol and B ∗ = (1 /B ∗ ) ∂B ∗ /∂p = 0 . kbar − . Positive magnetostric-tion in the mono-domain phase reveals that (see also Fig. 5(a)) for each domain the in-plane distortion in magnetic fieldis such that the lattice expands perpendicular to the spin-direction. Hence, applying a uniaxial pressure p will inducean anisotropy in plane favouring domains with spins nearlyparallel to p in the multi-domain phase.The scaling of ∂ ( M/H ) /∂ ( µ H ) and λ || a at 2 K in Fig. 8(b)shows that the quantities vary proportional to each other in themulti-domain state peaking at B ∗ . The proportional variation d ( m/H ) /dH ∼ λ || a is consistent with equation 3 and is man-ifestation of magnetoelastic nature of the domains. The be-haviour is expected from phenomenological theories of mag-netoelastic domains which describe the variation of magneti-zation and length changes by means of a single domain co-alignment parameter and it’s variation with magnetic field .Apart for large magneto-crystalline anisotropy which dic-tates the easy-plane spin structure in NiTiO , an additionalsmall in-plane anisotropy may arise due to frozen strains in thedomain walls . Small in-plane anisotropy has been previ-ously observed in other easy-plane type antiferromagnets likethe dihalides NiCl2 ( ∼ ∼ and in CoTiO ( ∼ respectively. Althoughbond anisotropic exchange interaction pinning the order pa- rameters to the crystal axes was suggested as the responsi-ble mechanism for small in-plane gap in CoTiO we believethat a small in-plane anisotropy to be present in NiTiO cor-responding to magnetoelastic domain walls. V. SUMMARY
In summary, we have studied in detail the magneto-structural coupling in magnetodielectric NiTiO by means ofsingle crystal neutron-diffraction and high-resolution dilatom-etry. Zero-field neutron diffraction reveals multidomain A-type spin antiferromagnetic ordering with preservation ofcrystallographic R -3 symmetry down to 2 K. Zero-field ther-mal expansion measurements reveals spontaneous lattice de-formation at T N . The dielectric permitivitty (cid:15) scales with thesquare of magnetic order parameter L in line with predictionsof Landau theory and hence indicating finite magnetodielec-tric coupling in NiTiO . Our analysis suggests the presenceof spin-phonon coupling as a responsible mechanism for di-electric anomaly at T N in NiTiO . In-field neutron diffrac-tion shows the evolution of magnetic domains with spins per-pendicular to the applied field. The effect of magnetic do-mains on magnetostriction have been discussed in light ofphenomenological multi-domain theories. We see magneti-zation and magnetostriction scale with each other in the mul-tidomain state revealing strong coupling of spins to the lattice. Acknowledgments
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