Phase diagram of a three-dimensional antiferromagnet with random magnetic anisotropy
Felio A. Perez, Pavel Borisov, Trent A. Johnson, Tudor D. Stanescu, David Lederman, M. R. Fitzsimmons, Adam A. Aczel, Tao Hong
PPhase diagram of a three-dimensional antiferromagnet withrandom magnetic anisotropy
Felio A. Perez, ∗ Pavel Borisov, Trent A. Johnson, Tudor D. Stanescu, and David Lederman † Department of Physics and Astronomy,West Virginia University, Morgantown, WV 26506-6315, USA
M. R. Fitzsimmons
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Adam A. Aczel and Tao Hong
Quantum Condensed Matter Division, Neutron Sciences Directorate,Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA (Dated: November 6, 2018)
Abstract
Three-dimensional (3D) antiferromagnets with random magnetic anisotropy (RMA) experimen-tally studied to date do not have random single-ion anisotropies, but rather have competing two-dimensional and three-dimensional exchange interactions which can obscure the authentic effectsof RMA. The magnetic phase diagram Fe x Ni − x F epitaxial thin films with true random single-ionanisotropy was deduced from magnetometry and neutron scattering measurements and analyzedusing mean field theory. Regions with uniaxial, oblique and easy plane anisotropies were identified.A RMA-induced glass region was discovered where a Griffiths-like breakdown of long-range spinorder occurs. PACS numbers: 75.30.Kz, 75.50.Ee, 75.10.Hk, 75.70.-i, 71.23.-k a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec he behavior of insulating antiferromagnets (AFs) can range from spin glass phases torandom exchange, random anisotropy, and random field Ising models [1]. As a consequence,these compounds have received much attention due to their use as possible experimen-tal realizations [2, 3] of theoretical models [4–6] for random magnets. Although AFs withpseudo random magnetic anisotropy (RMA) have been studied previously in Fe x Co − x Cl ,Fe x Co − x Br , Fe x Co − x TiO , and K Co x Fe − x F alloys, these systems only have approxi-mate three-dimensional (3D) order because the effective RMA actually consists of differentintra- and inter-layer magnetic exchange coupling constants [2, 3, 7–12]. To see why this isimportant, consider the spin Hamiltonian H = Σ i D ( S zi ) + Σ ij ∆ J ij S zi S zj + Σ ij J ij S i · S j , (1)where D is a single-ion anisotropy constant, ∆ J ij is the difference between intra- and inter-layer exchange coupling constants, and J ij is the intra- layer exchange coupling constant. Inthe mean field approximation, and taking into account only strongest neighbor interactions J , the Hamiltonian for a spin on the λ sublattice of an antiferromagnet becomes H λ = D ( S zλ ) + z ∆ J S zλ D S zλ E + zJ S λ · h S λ i , (2)where z is the number of neighbors located on the sublattice λ that interact with a spin S λ .The second term on the right hand side of Eq. 2, associated with an effective single-ion mag-netic anisotropy resulting from the anisotropic exchange interaction, is strongly temperaturedependent near the Néel temperature T N because h S zi i → T → T N . On the other hand,the first term, which represents a true single-ion anisotropy, is not temperature dependentand therefore dominates the physics in the vicinity of the magnetic phase transition. Con-sequently, the physics that governs a system with true random single-ion anisotropy nearthe phase transition will be in general different from the physics generated by an effectiveRMA produced by anisotropic exchange interactions. In this Letter, we report on the phasediagram of a solid solution of two tetragonal 3D AFs that have orthogonal anisotropies orig-inating solely from the single-ion anisotropies of each component, and thus represents a true3D RMA antiferromagnet.FeF and NiF share the rutile crystal structure with similar lattice parameters ( a = b = 4 . c = 3 . and a = b = 4 . c = 3 . atroom temperature ) [13, 14]. Both materials are 3D AFs with similar exchange interaction2 IG. 1. TRM for Fe x Ni − x F samples measured in H = 0 after field cooling in H F C = 100 Oe.Data for x = 0 and 0.10 were measured with H F C perpendicular to the c-axis; all others measuredwith H F C parallel to the c-axis. Inset: magnetic and crystalline structures of the parent compoundsNiF and FeF . Yellow, blue, and red dots are F + , Ni , and red Fe ions, respectively. strengths zJ S ( S +1) / T N s, 73.2K and 78.4K, for NiF and FeF ,respectively [15, 16]. Their magnetic anisotropies are, however, very different. FeF has astrong uniaxial anisotropy which results in its magnetic moments being aligned along thetetragonal c-axis, and is therefore considered an ideal realization of the 3D Ising model [16].In NiF , moments order antiferromagnetically in the a-b plane (Fig. 1) and are canted by ≈ . ◦ with respect to the a- or b-axis [15]. Weak ferromagnetism in NiF is due to thepresence of two non-equivalent magnetic sites in the NiF crystal lattice [17]. The similarityof crystal structures and magnetic exchange interactions in NiF and FeF suggests thatFe x Ni − x F is an ideal system to study RMA, which should vary from transition metal siteto site depending on whether it is occupied by Ni (favoring a-b plane ordering) or Fe (favoring c-axis ordering) [18, 19].In order to study the 3D RMA anisotropy problem, epitaxial (110) Fe x Ni − x F films weregrown with nominal thicknesses of 37 and 100 nm on (110) MgF substrates at 300 o C viamolecular beam epitaxy, as described previously [18, 20], and capped with a 10 nm BaF orPd layer to prevent oxidation. The Fe concentration x was determined using a quartz-crystalmonitor with an accuracy of ± .
05 [18, 20]. Thermal remanent magnetization (TRM) mea-surements were carried out which consisted measuring the magnetization M while increasing3 from T = 5 K after field-cooling (FC) from T = 300 K in a field H F C = 100 Oe (Fig. 1)along the in-plane [001] (c-axis) and [¯110] directions. The transition temperatures weredetermined by fitting the data near the phase transition with a rounded power-law I = I σ c √ π Z ∞ (1 − T /T c ) β e − ( T c − T c ) / σ c dT c , (3)where T c is a transition temperature, β is a critical exponent, σ c is the width of the transition,and I is an overall scaling factor [21, 22]. Magnetic hysteresis loops were measured as afunction of T and found to have large coercivities at low T that decreased with increasing T for 0 . < x <
1, in agreement with previous measurements of Fe x Ni − x F /Co bilayers(see Supplementary Materials) [18, 20]. FC and zero-field cooled (ZFC) measurements of M vs. T of all alloy samples behaved in a way that can be explained by the appearanceof a ferromagnetic multi-domain state during the ZFC process and its realignment afterfield-cooling (see Fig. 2 inset).TRM data in Fig. 1 show the general effect of alloying on M . Relatively small deviationsof x from the pure phases result in significant increases of M at low T , but these values aremuch smaller than would be expected for ferrimagnetic order [23], and are therefore due tomagnetic disorder.Examples of TRM phase transitions with H F C || c-axis, and H F C ⊥ c-axis for the x = 0 . T than the actual onset of the remanent magnetization, while the pure FeF andNiF samples only had one transition. The fits to the data using Eq. 3 with two transitionsfor the alloys and one transition for the pure samples, indicated that β ≈ . ± .
05 forall samples. The transition temperatures and transition widths obtained from the fits forall samples are shown in Fig. 4 and discussed further below. The presence of two phasetransitions was more clearly seen in the form of two minima, at T = T and T = T , inthe ∂M/∂T vs. T data, as shown in Fig. 2(b). When the TRM was measured in small H applied along the c-axis, the transition at T broadened substantially, while the transitionat T and the low temperature TRM remained unaffected. This phenomenon occurred forall samples with 0 . < x < .
0. For x = 0 .
1, a similar transition was observed with H ⊥ c ,indicating the existence of the easy-plane ordering similar to that of pure NiF . TRM datafor H || c had unusual behavior due to the existence of an oblique phase, as discussed below(see Supplementary Materials). 4 IG. 2. (a) TRM data ( H = 0) near the phase transition for four representative samples. Symbolsare data and red curves are fits to Eq. 3 with two transitions for x = 0 . , .
47 and one transitionfor x = 1 . x = 0 .
0. Inset: M measured while warming with H = 80 Oe applied along thec-axis after ZFC from T = 300 K to 5 K and during FC from 300 K to 5 K for the x = 0 . ∂M/∂T of the x = 0 .
47 TRM data measured under differentapplied fields. Vertical blue lines indicate transition temperatures T and T . The magnetic phase in the range T < T < T , where the magnetic structure is stronglycoupled to H , can be explained in two ways: (1) there is a first order spin-reorientationtransition from an Ising-like, single-axis anisotropy structure, similar to FeF , to a weaklyferromagnetic structure, similar to NiF , at T = T with increasing T , or (2) the transitionat T = T is from the FeF magnetic structure to a magnetically disordered structure. Inorder to determine which of these explanations is correct, neutron scattering was measuredin x = 0 . x = 0 . H F C = 60 Oe || c-axis. Once cooled to T = 4 K, H was removed and the integrated intensities of the magnetic (100) and (001) reflectionswith their background subtracted, I (100) and I (001) (corresponding nuclear reflections areforbidden), were measured as a function of increasing T . From neutron scattering selectionrules, I (100) ∝ L + L , where L b,c is the component of the staggered magnetization vector L
5f the AF along the c- or b-axis, while I (001) ∝ L , where L = L + L is the componentof L in the a − b plane. The staggered magnetization vector is defined by L = ( M − M ),where M , are the two sublattice magnetization vectors with M = M . Explanation (1)would result in I (001) = 0 only in the T < T < T temperature range. On the other hand,explanation (2) requires that I (100) , I (001) > T < T because lack of long-rangeorder in the T < T < T range would preclude the observation of magnetic scattering.Figure 3(a) shows I (100) ( T ) and I (001) ( T ). For both samples, the data indicate the presenceof a single phase transition. For the x = 0 . I (001) = 0 for 0 < T <
85 K, andtherefore the spins did not order antiferromagnetically in the a-b plane. For the x = 0 . I (100) and I (001) were non-zero at low T , and both → T → T . This indicatesthat L pointed in an oblique direction between the c-axis and the a-b plane. Fitting thedata to a rounded power law phase transition similar to Eq. 3, but with β → β to takeinto account the fact that I ∝ L , yielded the results shown in Fig. 3(b). The value of T N coincided with T measured for x = 0 . x = 0 . T > T N ≈ T for either sample, we conclude thatexplanation (2) is correct: there is a transition with increasing T from an AF with long-rangeorder to a disordered magnetic phase in the T < T < T range.The values of β from neutron scattering agreed with those from the TRM measurements.They are in better agreement with critical exponents corresponding the the 3D Ising, Heisen-berg, and random exchange models ( β ≈ .
35) [16] than with the 3D random field model( β ∼ .
1) [21, 24, 25]. To determine β more accurately, and thus identify the transition’suniversality class, measurements must be made of the lineshape as a function of scatter-ing wavevector, T and H to take into account possible incoherent scattering backgroundscommon in random magnetic systems [21, 24]. Significantly thicker samples than the onesused here, possibly bulk single crystals, would be required. Accurately determining theuniversality class is therefore beyond the scope of this paper.Whereas L a − b = 0 for the x = 0 . T range, this is not thecase for the x = 0 . x = 0 . T range, where L points at an angle θ away from the c-axis. The valueof θ can be determined using L = L c + L ab and assuming that L b = L a , i.e., oblique domainsare equally likely to tilt towards the a- or b-axis, which yields tan θ = (cid:16) I (001) /I (100) − / (cid:17) / .The dependence of θ on T calculated from this equation is shown in Fig. 3(c).6 IG. 3. (a) Neutron scattering intensity as a function of temperature for the x = 0 . x = 0 . T c . Black curves are fits to Eq. 3(with β → β ) and the resulting fitting parameters are shown in the graph. (c) Angle of thestaggered magnetization vector L as a function of T with respect to the c-axis for the x = 0 . x indicated in the graph. The phase diagram in Fig. 4, constructed from the TRM and neutron scattering data,can be understood using mean field theory (MFT). While MFT is inaccurate when predict-ing T N , it is relatively successful at predicting quantities which depend on changes in theeffective field rather than on its absolute value [26] and can describe, at least qualitatively,the concentration dependence of T N in mixed AF systems [27]. The spin Hamiltonian in-cluded single-ion anisotropy terms and Heisenberg-type exchange contributions, similar tothe model used by Moriya [17] to study weak ferromagnetism in NiF . Using mean field de-coupling for the exchange interactions while treating the single-site anisotropy terms exactlyyields an average of the η spin component for α -type ions (either Fe or Ni) on the sublattice λ h S ηα i λ = 1 Z αλ Tr [ S ηα exp ( − H αλ /k B T )] , (4)where the effective single-site Hamiltonian has the form H αλ = Σ η ≈ h ηαλ S ηα + D α (cid:16) S Zα (cid:17) (5)7ith the molecular field given by ≈ h ηαλ = z Σ β =Ni,Fe J αβ p β h S ηβ i λ . (6)In Eq. 4 the partition function is Z αλ = Tr [exp ( − H αλ /k B T )] and the spins S ηα are repre-sented by 3 × × T , h S ηα i λ was determined numerically using an iterative scheme. Convergence was checked in thelimit D α = 0 by comparing with analytic expressions obtained within the full decouplingscheme (see Supplementary Materials for more details). The exchange coupling constants J α j β k were non-zero for next-nearest neighbor sites j and k (between ions at the center ofthe tetragonal unit cell with those at the corners) and could take the values J NiNi , J FeFe , and J NiFe , corresponding to the different possible pairs of spins. Weaker exchange contributionswith other neighbors were neglected. The number of interacting neighbors was z = 8 and D α was positive for Ni ions and negative for Fe ions. Self-consistent calculations were carriedout until the upper end of the phase diagram in Fig. 4 was reproduced.Results of MFT calculations are shown in Fig. 4. The paramagnet (PM)-AF phase tran-sition boundary was reproduced by adjusting the exchange constants to J FeFe = 0 .
475 meV, J NiNi = 1 .
63 meV, and J NiFe = 0 .
94 meV, and using the known single-ion anisotropy constants D F e = − .
80 meV and D Ni = 0 .
54 meV [15, 28]. T N values for pure NiF and FeF sampleswere larger than expected from the bulk parameters, but this has been previously attributedto strain (piezomagnetism) [29, 30]. The exchange constants were therefore different thanthe bulk values ( J FeFe = 0 .
451 meV and J NiNi = 1 .
72 meV) [15, 28]. This non-monotonicdependence of T on x is due to an enhancement of the exchange between unlike ions, J FeNi = 0 .
88 meV > √ J FeFe J NiNi , similar to what has been observed in Fe x Mn − x F [26].Increasing J NiFe much further shifts the minimum to x = 0.MFT also predicts a region where oblique ordering occurs, similar to prior MFT resultsfor AF systems with anisotropic exchange couplings [4, 6]. The canting angle θ ( T ) wascalculated using the same model and is depicted by the black curves in Fig. 3(c) (see SM).The behavior was found to be extremely sensitive to x and remarkably good agreementwas found for x ∼ . x = 0 . ± .
05, but with J NiFe = 1 .
02 meV. This indicates that other exchange interactionsneglected by the model may play a role in determining θ ( T ).Regions of different types of order predicted by the calculations are indicated in Fig. 4.8 IG. 4. Magnetic phase diagram for Fe x Ni − x F . Regions are indicated by AF a-b (ordering in thea-b plane), AF c (ordering along the c-axis), AF AG (anisotropy glass phase), AF O (oblique phase),and PM (paramagnetic phase). The PM-AF phase boundary calculated using MFT is denoted bythe solid green curve. T and T were determined from fits to Eq. 3 of TRM data. Horizontalerror bars represent uncertainty in x from quartz crystal monitor measurements. Vertical errorbars correspond to the transition widths σ T C . Measurements taken for samples that had a responsewith H ⊥ to the c-axis ([¯110] direction, ) are also indicated. Magenta lines enclose the MFTAF O region. Transitions observed via neutron scattering are also indicated. The dark blue curveis a AF c - AF AG phase boundary drawn as a guide to the eye. Whereas the calculated PM/AF boundary agrees well with T , neutron scattering data in-dicate that long-range order disappears for T > T . This leads to the conclusion that aGriffiths-like [31–33] short-range order phase exists in the T < T < T region as a re-sult of the random single-ion anisotropy. Griffiths phases in other AFs usually result fromfrustration of their exchange interactions. For example, magnetic field-induced antiferromag-netic correlations have been reported in metamagnetic FeCl [34], in intraplanar frustratedFeBr [35], and in the dilute AFs Fe − x Zn x F [36] and Rb Co − x Mg x F [37]. Here we proposea mechanism where a breakdown of magnetic long-range order occurs at T , with the randomorthogonal single-ion magnetic anisotropy playing the role of an effective local random fieldthat leads to frustration. The emerging RMA-induced anisotropy glass region exists in theinterval T < T < T , where T is the upper phase transition determined by the average9xchange interaction strength of the alloy. The MFT used here is unable to reproduce thisregion because it does not take into account local fluctuations of the effective field.In conclusion, the magnetic structure of Fe x Ni − x F , an authentic 3D AF with randomsingle-ion magnetic anisotropy, transforms from easy a-b plane to the easy c-axis with in-creasing x via an oblique phase region at x = 0 . − .
14. Two phase transition temperatures, T and T , were identified for 0 . < x < .
9. Long-range order disappears for
T > T , butshort-range order persists up to T = T . The short-range order region is a result of theRMA which induces a magnetic glass phase for T < T < T . This phase is similar tomagnetic glassy states formed as a result of combining structural disorder with frustratedexchange interactions, but with randomly distributed single-ion anisotropies replacing ex-change frustration as the driving mechanism. These effects have not been observed beforein AFs because most AF systems studied to date do not have authentic single-ion RMA, butrather have an effective RMA induced by asymmetric exchange interactions which decreasesrapidly as T N is approached.The supports of the National Science Foundation (grant No. DMR-0903861) and theWV Higher Education Policy Commission (Research Challenge Grant HEPC.dsr.12.29) atWVU are gratefully acknowledged. Some of the work was performed using the WVU SharedResearch Facilities. Research conducted at ORNL and LANL was sponsored by the ScientificUser Facilities Division, Office of Basic Energy Sciences, US Department of Energy. ∗ Current address: Integrated Microscopy Center, The University of Memphis, Memphis, TN38152. † [email protected][1] D. S. Fisher, G. M. Grinstein, and A. Khurana, Physics Today , 56 (1988).[2] P. Z. Wong, P. M. Horn, R. J. Birgeneau, and G. Shirane, Phys. Rev. B , 428 (1983).[3] W.A.H.M. Vlak, E. Frikkee, A.F.M. Arts, and H.W. de Wijn, Phys. Rev. B , 6470 (1986).[4] F. Matsubara and S. Inawashiro, J. Phys. Soc. of Japan , 1529 (1977).[5] S. Fishman and A. Aharony, Phys. Rev. B , 3507 (1978).[6] F. Matsubara and S. Inawashiro, J. Phys. Soc. of Japan , 1740 (1979).[7] M. K. Wilkinson, J. W. Cable, E. O. Wollan, and W. C. Koehler, Phys. Rev. , 497 (1959).
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