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Exotic magnetic phenomena in transition metal materialsLummen, Tom Theodorus Antonius

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Chapter 5

Cascade of Magnetic Phasesin Frustrated MagnetCuFeO2

The magnetic phase diagram of CuFeO2 as a function of applied magnetic fieldand temperature is explored through pulsed magnetic field experiments. The typ-ical metamagnetic staircase of CuFeO2 is reproduced and extended toward highermagnetic field strengths at various temperatures, both for magnetic fields appliedparallel and perpendicular to the material’s c-axis. Magnetic susceptibility mea-surements as a function of temperature at various constant magnetic fields serveto complement the resulting experimental phase diagrams. An additional, highfield metamagnetic transition is revealed for both magnetic field configurations,above which CuFeO2 retrieves virtually complete isotropic behavior, signalingthe recovery of the undistorted triangular lattice at high magnetic fields. Also,at temperatures just below TN2 (11 K), a newly observed intermediate state isfound between two collinear phases, at approximately 18 T. A phenomenologi-cal rationalization of the intricate B,T phase diagrams of CuFeO2 is providedon the basis of a consistent classical spin model, involving isotropic Heisenbergspins and both a field-dependent, distortion-induced easy-axis anisotropy and aspin-phonon coupling. The latter two are found to be prerequisites for adequatesimultaneous modeling of the magnetization process in both field configurations.

This chapter is in part based on the results published in: T.T.A. Lummen, C. Strohm,H. Rakoto, A.A. Nugroho and P.H.M. van Loosdrecht, ”High-field recovery of the undistortedtriangular lattice in the frustrated metamagnet CuFe2”, Physical Review B 80, 012406 (2009)(4 pages).

5

118 Cascade of Magnetic Phases in Frustrated Magnet CuFeO2

5.1 Introduction

One of the richest and most fascinating phenomena in magnetic systems, geo-metrical frustration, was introduced in the preceding Chapter (Chapter 4). Inshort, the phenomenon occurs when the geometry of a specific lattice preventsthe simultaneous minimization of all magnetic exchange interactions within thesystem. With the primary interactions of the magnetic system unable to selecta unique magnetic ground state, the magnetic behavior of frustrated systems isdominated by secondary, often weaker interactions, which can vary strongly evenacross closely related materials. Consequently, the field of frustrated magnetismis characterized by its vast richness and diversity, exotic magnetic states and lowtemperature physics [1, 2, 3, 4]. In Chapter 4, the occurrence of this phenomenonand its consequences for the rare earth titanates were investigated. The com-peting interactions on the pyrochlore lattice of corner-sharing tetrahedra of thesetitanates, a classic lattice geometry in which frustration occurs, are found to dom-inate the (low-temperature) magnetism of the materials.

In this Chapter, the focus is on another classic geometry in which geometricalfrustration readily manifests itself; that of the triangular lattice (Fig. 4.1a). Likein the pyrochlore system, antiferromagnetic interactions on this geometry of edge-sharing triangles lead to a macroscopic ground state degeneracy. In absence ofsignificant secondary interactions, the spins of the triangular lattice antiferromag-net (TLA) compromise in their ’desire’ to align antiparallel and typically adopta noncollinear 120◦ spin configuration at low temperatures, an underconstrainedground state which can be realized in an enormous amount of ways [5, 6]. Thesituation can be quite different, however, in systems where secondary interactionsare important, such as in the stacked delafossite material CuFeO2, which con-sists of hexagonal Fe3+, O2− and Cu+ layers (space group R3m, a = b = 3.03A,c = 17.17A). As the Fe3+ (3d5, S = 5/2) ions are the system’s only magneticconstituents (Cu+ and O2− have filled electronic shells), and their spins interactantiferromagnetically, the system corresponds to an archetypical TLA (Fig. 5.1a).Strikingly though, in contrast to other delafossite TLAs like LiCrO2, AgCrO2 andCuCrO2 [5, 6], CuFeO2 adopts a collinear ground state at low temperatures.

As was the case for the Gd3+ ion in Gd2Ti2O7 (Chapter 4), the antiferro-magnetic exchange interactions within the CuFeO2 system are expected to beisotropic, based on the spin-only electronic ground state of the Fe3+ ion (6S5/2,L = 0). One would thus expect CuFeO2 to behave as a pure Heisenberg TLA.The presence of a substantial spin-lattice coupling however (the secondary inter-action), induces a structural distortion in CuFeO2 through the ’spin Jahn-Teller’effect [7, 8, 9, 10], thereby reducing the spin state degeneracy in the system. Thisspin-lattice coupling has its origin in the distance dependence of the magneticexchange coupling between two sites i and j, J(rij)Si · Sj , and is also knownas magnetoelastic exchange. The symmetry lowering effect of this coupling isanalogous to the spin-Peierls effect (dimerization) in one-dimensional spin chainsand is strongly reminiscent of the collective Jahn-Teller effect in crystalline solids.

5

5.1 Introduction 119

Analogous to the situation in the traditional Jahn-Teller case, where the electronicground state is degenerate, the high symmetry of the system induces a degener-acy in the spin states here. In accordance with the original Jahn-Teller theorem,the system can thus lower its ground state energy by undergoing a structural dis-tortion. Because in this case it are the spin variables that cause the degeneracyand drive the distortion, it is termed the spin Jahn-Teller effect. Structurally, thesymmetry of the CuFeO2 lattice is first lowered from the hexagonal R3m groupto the monoclinic C2/m group at TN1 ≈ 14 K and subsequently further reducedto a lower monoclinic symmetry at TN2 ≈ 11 K [11, 12, 13, 14]. The strongpressure dependence of these transition temperatures discloses their spin-latticeorigin [15, 16]. Magnetically, CuFeO2 undergoes a transition from its paramag-netic (PM) phase to a partially disordered, incommensurate (PDIC) magneticphase at TN1 where a sinusoidally amplitude-modulated magnetic structure witha temperature dependent propagation wave vector ((q q 0), where q varies from≈ 0.19 to ≈ 0.22) is adopted [17, 18]. Another phase transition at TN2 brings thesystem into its collinear four-sublattice (4SL) ground state, in which the spinsalign (anti-)parallel to the c-axis, adopting a two-up two-down order with thecommensurate propagation wave vector ( 1

414

32 ), as illustrated in Figure 5.1b

[19, 20]. To avoid confusion, we will refer to crystallographic directions using thehexagonal description depicted in Figure 5.1 throughout the Chapter.

The stabilization of the collinear 4SL state in CuFeO2 proved to be one ofits most puzzling issues. Initially, the system was described as a two-dimensional(2D) Ising TLA with exchange interactions up to the third nearest-neighbors.The first (J1), second (J2) and third (J3) in-plane nearest-neighbor interactionswere estimated to compare as J2/J1 ≈ 0.5 and J3/J1 ≈ 0.75 in this model[18, 20, 21, 22, 23], while J1 was estimated to be of the order of 1.2 meV [21, 24, 25].The fact that the second and third in-plane nearest-neighbor interaction strengthsare comparable to that of the first nearest-neighbor interaction seems quite pe-culiar, since the corresponding interatomic distances are 3.03, 5.25 and 6.04 A,respectively. Moreover, there is a priori no physical justification for the assumedIsing nature of the magnetic moments. This assumption is also inconsistent withmagnetic susceptibility measurements, which show highly isotropic behavior aboveTN1 in CuFeO2 [21, 25, 26, 27]. Nonetheless, the magnetic properties below TN1

are unmistakably strongly anisotropic. The recent discovery of the small lowtemperature structural distortion (<0.4 %) offers an alternate picture, as it re-sults in a lattice of scalene triangles in the basal plane, which splits the firstnearest-neighbor interaction within every triangle into three unequal exchangeinteractions [11, 12]. Moreover, the lattice is stretched along [110], which en-larges the nearest neighbor distance, thereby lowering the ferromagnetic directexchange. Simultaneously, the Fe-O-Fe bond angle along that direction is slightlyincreased (from 96.89◦ to 97.52◦), which changes the superexchange through oxy-gen. As the Goodenough-Kanamori rules state the latter interaction can only beantiferromagnetic between two high-spin d5 cations (all d-orbitals are half filled),the overall effect is an increase of the strength of the antiferromagnetic coupling

5


a

b

c Cu+

Fe3+

O2-

O2-

Fe3+

Cu+

O2-

O2-

Fe3+

Cu+

O2-

O2-

Fe3+(a) (b)

a

bc

[110]

J’’J

’J

scalene

(c) [110]

a

bc

J’’J

’J

scalene

# ferromagnetic n.n pairs along [110]: 0%

(d) [110]

a

bc

JJ

’J

isosceles

# ferromagnetic n.n pairs along [110]: 20%

(e) [110]

a

bc

JJ

’J

isosceles

# ferromagnetic n.n pairs along [110]: 33.3%

Figure 5.1: a) Schematic crystal structure of CuFeO2, space group R3m, a = b = 3.03A,c = 17.17A. Since only the Fe3+ ions are magnetic, the magnetic structure (in red) iscomposed of stacked triangular layers, separated by nonmagnetic O-Cu-O tri-layers. b-e) Spin structures and lattice symmetries in various phases of CuFeO2, when subjectedto an applied magnetic field B ‖ c. The number of ferromagnetic nearest neighbor (n.n)

pairs along the [110] direction is indicated for the collinear phases. b) (B‖ < B‖c1)

Magnetic ordering in the four sublattice (4SL) phase, where the magnetic momentsalign parallel to the c axis, adopting a two-up two-down order, with nearest neighborsin the [110] direction coupled antiferromagnetically. c) (B

‖c1 < B‖ < B

‖c2) Proper-

helical magnetic order in the ferroelectric incommensurate (FEIC) phase of (doped)CFO. Spins are in the plane perpendicular to the helical axis ([110] direction), orderingwith in-plane modulation vector (q,q,0), where q varies between ≈ 0.19 and ≈ 0.22

[17, 18]. For clarity one Fe3+-chain along [110] has been isolated. d) (B‖c2 < B‖ <

B‖c3) Experimentally determined collinear three-up two-down magnetic order in the five

sublattice (5SL) phase. e) (B‖c3 < B‖ < B

‖c4) Analogous collinear two-up one-down

order proposed as the magnetic structure in the 3SL phase.

5

5.1 Introduction 121

along [110] with respect to that along [100] and [010]. This in turn lowers theenergy of the 4SL state [11, 12, 13, 14]. The collinear 4SL order is further sta-bilized by a small easy axis anisotropy, which has been argued to be induced bythe scalene triangle distortion as well, through an anisotropy in the Fe-3d,O-2phybridization and the charge transfer between Fe and O [28]. Experimentally, asmall single-ion anisotropy interaction was estimated by fitting a 3D HeisenbergHamiltonian with single anisotropy term to the spin-wave dispersion along c belowTN1, which supports the picture of the distortion-induced anisotropy [24, 29]. Aswill be confirmed below, the combination of this weak magnetic anisotropy andthe relatively strong spin-phonon coupling in CuFeO2 can explain its observedIsing-like behavior [9, 10].

Arguably the most fascinating physical properties arise when CuFeO2 is sub-jected to an external magnetic field below TN2. Upon increasing applied mag-netic field along the c axis (B ‖ c), the material has been shown to undergoa series of metamagnetic transitions at B

‖c1 ' 7 T, B

‖c2 ' 13 T, B

‖c3 ' 20 T

and B‖c4 ' 34 T, before ultimately reaching saturation around B

‖sat ' 70 T

[21, 25, 27, 28, 30, 31, 32]. Corresponding magnetic structures between the succes-sive transitions were shown to be a noncollinear phase with an incommensuratewave vector, which also carries a ferroelectric moment (B‖

c1 < B‖ < B‖c2, vide

infra), and a collinear five-sublattice (5SL) phase where the spins again align par-allel to the c axis, adopting a three-up two-down order with a ( 1

515 0) propagation

wave vector (B‖c2 < B‖ < B

‖c3), as shown in Figure 5.1d [30, 33]. Spin structures

at higher fields have not yet been experimentally determined due to the demand-ing experimental requirements. Nevertheless, the magnetic phases at high fieldshave been suggested to have a three-sublattice (3SL, B

‖c3 < B‖ < B

‖c4, two-up

one-down, Figure 5.1e) and a canted three-sublattice magnetic order (c3SL, B‖

> B‖c4), based on the corresponding relative magnetization values. Yet another

indication of the strong spin-lattice coupling in CuFeO2 is the strong correlationbetween the crystal structure and the various magnetic phases observed in ap-plied magnetic field [28, 32]. Particularly worth noting is the increase in crystalsymmetry observed at B

‖c2, where the scalene triangle distortion is (partially) re-

lieved, resulting in a lattice of isosceles triangles [13].Illustrating the low temperature anisotropy in the material, the magnetism in

CuFeO2 evolves quite differently when it is subjected to a magnetic field perpen-dicular to the c axis (B ⊥ c) below TN2, showing only two transitions up to 40T, at B⊥

c1 ' 24 T and B⊥c2 ' 30 T [21, 25, 28, 30]. Apart from the zero-field 4SL

structure, the corresponding magnetic structures have not yet been experimen-tally determined. Based on magnetization measurements, the magnetic structurehas been proposed to undergo a spin rearrangement from a canted 4SL phase(with spins tilted away from the c-direction) to a 3SL phase (with spins in thebasal plane) at B⊥

c1, and subsequently to a canted 3SL phase at B⊥c2. Additionally,

synchrotron x-ray diffraction measurements in applied magnetic field have shown,

5


also for this configuration, that the crystal structure is strongly correlated withthe system’s magnetization [28].

A particular surge of scientific interest has gone to the noncollinear, incom-mensurate phase (B‖

c1 <B‖ <B‖c2), after Kimura et al. showed the multiferroic

nature of this phase [27]. The stability range of this ferroelectric, incommensurate(FEIC) phase has proven specifically dependent on nonmagnetic Al3+ substitu-tion [34, 35, 36, 37, 38]. Further evidence of the dependence of the propertiesof CuFeO2 on its exact stoichiometry had been reported earlier on [39]. For aparticular range of substitution (CuFe1−xAlxO2, 0.014 < x < 0.030) the FEICphase is the ground state even in zero applied field [40, 41]. Recently, nonmag-netic Ga3+ substitution has also been shown to induce the FEIC phase in zerofield [42]. Due to the occurrence even at zero field, the magnetic structure couldbe determined as a proper helical magnetic order with in-plane modulation vector(q,q,0), in which the spins are perpendicular to the helical axis, as is illustrated inFigure 5.1c [43, 44, 45]. The microscopic origin of the multiferroicity is expectedto be different from the so-called ’inverse’ Dzyaloshinsky-Moriya (DM) interaction[46, 47], which has been used to explain the ferroelectricity in transition metaloxides with cycloidal-type magnetic order (where the propagation vector and ro-tation axis of the spiral are orthogonal), like Ni3V2O8, RMnO3 (where R=Tb,Tb1−xDyx), CoCr2O4 and MnWO4 [48, 49, 50, 51, 52, 53, 54]. This inverse DMmechanism predicts an induced ferroelectric polarization proportional to eij ×(Si × Sj), where eij is the unit vector connecting two neighboring magneticspins Si and Sj . In the proper screw type magnetic order of the FEIC phasein (substituted) CuFeO2, eij is on average parallel to (Si × Sj), ergo there isno net induced polarization through this mechanism. Instead, the term thoughtresponsible (written as (Si · eij)Si - (Sj · eij)Sj) originates from the variationof the 3d-2p hybridization with spin-orbit coupling, which induces an imbalancein the charge transfer between neighboring Fe-O pairs, resulting in a net uniformpolarization along the [110] direction for the (q,q,0) proper screw magnetic orderin the FEIC phase of CuFeO2 [43, 55, 56, 57]. The same variation in the hy-bridization may also induce a single-ion anisotropy, as mentioned above. Also,the mechanism predicts a relation between the electric polarization and the spinhelicity of a proper helical magnet, which was recently experimentally confirmedin the FEIC phase of CuFe1−xAlxO2 [45].

As is clear from above disquisition, the magnetic behavior of CuFeO2 as afunction of temperature and applied magnetic field has proven very rich and hasyielded unanticipated, fascinating new insights. This work aims to expand thescope to even higher magnetic fields, in a search for possible new metamagnetictransitions and exotic magnetic phases. Simultaneously, the B,T phase diagramsare thoroughly mapped out and extended up to 58 T, for both for the B ‖ cand the B ⊥ c configuration. Thus, we have found an additional, high magneticfield metamagnetic transition for both magnetic field orientations, at B

‖c5 ' 53.3

T and B⊥c3 ' 51.6 T, respectively. Above both these transitions, which exhibit

5

5.2 Experiment 123

the characteristic first order, step-like hysteretic behavior, CuFeO2 retrieves vir-tually complete isotropic behavior. The intricate phase diagram of CuFeO2 isdiscussed in terms of a consistent phenomenological model, which rationalizes the(high) field dependence for both configurations. Simulations based on this clas-sical model, which includes spin-phonon and magnetic anisotropy terms, suggestthe retrieval of the undistorted triangular structure at high fields, in line with theexperimentally observed high field isotropy.

5.2 Experiment

5.2.1 Sample preparation

The single crystals studied in this chapter were prepared at the Zernike Insti-tute for Advanced Materials, in close collaboration with Agung Nugroho. A highquality, single crystalline rod of CuFeO2 was synthesized using the floating zonetechnique, following the procedure described by Zhao et al. [26] A 57Fe-enrichedstarting material (57Fe2O3, 57Fe > 95.5%) was used in the synthesis, to facilitatethe nuclear forward scattering experiments described in chapter 6 of this thesis.An extended description of the single crystal growth can be found there. X-rayLaue diffraction, using a Philips PW 1710 diffractometer equipped with a PolaroidXR-7 system, was employed to orient the single CuFeO2 crystal, while simultane-ously confirming the sample’s single crystallinity and R3m space group (at roomtemperature). Next, small cuboid samples (5 x 1 x 1 mm3), with long sidesoriented parallel (35.9 mg) and perpendicular (42.1 mg) to the crystallographicc-axis, respectively, were prepared from the single crystal. Further characteriza-tion, including 57Fe Mossbauer spectroscopy, Raman spectroscopy and SQUIDmagnetometry, also yielded experimental data in excellent agreement with liter-ature on CuFeO2, confirming the high sample quality. The same samples wereused in all measurements reported here and in a previous work [10].

5.2.2 Instrumentation

Magnetization measurements

High (pulsed) magnetic field magnetization measurements, up to a maximum fieldof 58.27 T were performed at the ’Laboratoire National des Champs MagnetiquesPulses’ in Toulouse, France, under direction of Harison Rakoto†. The obtainedmagnetization data were accurately scaled through a least squares fit to low fieldmeasurements (up to 10 T), performed by C. Strohm and D. Maillard on a wellcalibrated static (dc) magnetic field setup (using the extraction technique) ofthe ’Institut Neel’ in Grenoble, France. The accuracy in the scaling procedurewas such that it introduces an uncertainty of ± 0.3% in all magnetization valuesdetermined from the pulsed field experiments.

5


SQUID magnetometry

The temperature dependence of dc magnetic susceptibilities in various constantmagnetic fields (up to 7 T) was measured at the Zernike Institute for AdvancedMaterials using a Quantum Design MPMS magnetometer equipped with a su-perconducting quantum interference device (SQUID). Samples were prepared byplacing an oriented single crystalline cuboid in a gelcap and fixing it in the ap-propriate orientation (B ‖ c or B ⊥ c, respectively) using nonmagnetic cottonwool. After zero-field cooling to 4 K, the sample’s magnetic susceptibility wasmeasured in a particular constant field while slowly heating the sample from 4 to25 K. Accordingly, magnetic susceptibility curves were obtained in various appliedfields for both sample orientations.

5.3 Results and Discussion

5.3.1 Magnetization in pulsed magnetic fields

Parallel field configuration (B ‖ c)

Figure 5.2 depicts the magnetization curves up to 58.3 T for various tempera-tures below TN1, where the applied magnetic field B is parallel to the c-axis (B‖).As B‖ increases, several successive metamagnetic steps are observed, in excellentagreement with previously reported results [21, 25, 27, 28, 30, 31, 32]. At 1.5 K,the system shows metamagnetic phase transitions at B

‖c1 ' 7.2 T (4SL to FEIC

phase transition), B‖c2 ' 13.0 T (FEIC → 5SL), B

‖c3 ' 19.7 T, B

‖c4 ' 32.4 T

and B‖c5 ' 53.3 T. The three transitions at lowest critical fields B

‖c1, B

‖c2 and B

‖c3

are all accompanied by large magnetization steps and exhibit significant hysteresis(B‖

c1↑ = 7.27 T, B‖c1↓ = 7.15 T, B

‖c2↑ = 13.44 T, B

‖c2↓ = 12.51, B

‖c3↑ = 20.32 T and

B‖c3↓ = 19.08 T at 1.5 K), indicating their first order nature. In contrast, at the

fourth magnetic transition (B‖c4 ' 32.4 T), the M(B‖)-curve shows only a change

in slope, suggesting this transition is of second order, which is consistent withsynchrotron x-ray diffraction results [28, 32]. In the 4SL phase, the magnetiza-tion is close to zero, as expected for the two-up two-down structure. In the FEICphase, M increases linearly with B‖ as observed before [25, 27, 28, 36], signalinga continuous reorientation of the spin system. In the 5SL phase, M is almostconstant, at a value approximately equal to one-fifth of the 5 µB/Fe3+ saturationvalue, in good agreement with the three-up two-down structure. Between B

‖c3 and

B‖c4, M is again almost independent of B‖, having a value close to 1/3rd of the

saturation-value, while between B‖c4 and B

‖c5 the magnetization again increases

linearly with B‖, indicating another continuous reorientation of the spin system.Based on these observations, these two phases have been proposed to correspondto a collinear three-sublattice (3SL, two-up one down, Figure 5.1e) and a canted

5

5.3 Results and Discussion 125

0 5 10 15 20 25 30 35 40 45 50 55 600.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

B//

c5

B//

c4

B//

c3

B//

c2B//

c1

5SL

FEIC

12.5 K10 K

9 K

7 K4 K

M (

B/F

e3+)

B// (T)

1.5 K

B // c

4SL0

25

50

75

100

125

150

175

200

(A m

2 kg-1 = em

u g-1)

Figure 5.2: Magnetization measurements in pulsed magnetic fields at various tempera-tures. The magnetic field is applied in the direction parallel to the c-axis. The variouscurves are offset by consecutive multiples of 0.5 µB/Fe3+ with increasing temperaturefor clarity. Thick and thin lines represent sample magnetization in increasing and subse-quently decreasing magnetic field, respectively. Dashed arrows indicate the temperaturedependence of the various metamagnetic transitions (See also Fig. 5.3).

2 4 6 8 100.85

0.90

0.95

1.00

1.05

1.10

c3

c2

c1

//c5

//c3

//c4

//c2

2nd order

Bc(

T)/B

c(1.

5 K

)

T (K)

B B

B B

B B

B B

1st order //c1

Figure 5.3: Temperature dependence of the critical fields corresponding to the variousmetamagnetic transitions, for both configurations (B ‖ c and B ⊥ c). Critical fieldvalues (midpoints of hysteresis loop for first order transitions) are normalized by theircorresponding 1.5 K values.

5


three-sublattice phase (c3SL), respectively [28, 32]. Next to the aforementioned,previously observed transitions, an additional metamagnetic transition is revealedby the magnetization measurements, at B

‖c5 ' 53.3 T (sample temperature 1.5

K). This feature has been overlooked in previous magnetization data recorded byAjiro et al., who measured the magnetization of a powder sample of CuFeO2 at 8K in a single turn coil measurement up to 100 T [21]. In fact, though it is obscuredin their M ,B-curve due to the polycrystalline nature of the sample, a clear featurecan be seen around ∼ 52 T in the corresponding (dM/dB) vs. B graph. At B

‖c5,

the system undergoes a first order transition where the magnetization jumps to' 3.3 µB/Fe3+, which is quite close to 2/3rds of saturation. The most straight-forward corresponding collinear phase would have a six-sublattice unit cell, wherethe ratio of spins aligned parallel to B‖ to those aligned antiparallel is five to one.However, as M increases continuously below and above the transition (there is noplateau behavior), such a collinear state appears incompatible and a noncollinearspin structure seems the more likely candidate. Above B

‖c5, the magnetization

shows a steady linear increase up to the highest field measured, 58.27 T. At thispoint M has taken a value of 3.54 µB/Fe3+ (at 1.5 K), close to the ' 3.7 µB/Fe3+

value for the powder sample measured at 8 K by Ajiro et al. [21]As the temperature increases, the general features of the M ,B-curve survive,

though metamagnetic steps spread out over an increasingly wide field range, hys-teresis widths are reduced and plateau phases acquire increasing slopes. As thetemperature approaches TN2, the characteristic staircase features of the magne-tization smooth out and M increases (quasi-)linearly with B, and deviates fromthis behavior only at high magnetic fields. The fact that this appears to occuralready just below TN2 is ascribed to a slight offset of the temperature sensor atthese temperatures, as transition temperatures measured in susceptibility exper-iments on the same sample (see below) are in accordance with literature values.A striking feature is the temperature dependence of the various magnetic tran-sitions (indicated by the dashed arrows in Figure 5.2). Figure 5.3 shows therelative variation of the corresponding critical magnetic fields with temperature.With the exception of the lowest field-induced transition, all (first order) transi-tions exhibiting hysteresis show identical behavior; a continuous decrease of thecorresponding critical field (B‖

c2, B‖c3 and B

‖c5, respectively) with increasing tem-

perature. In contrast, the critical field of the second order transition (B‖c4) proves

rather temperature independent, once more indicating its different nature.

Perpendicular field configuration (B ⊥ c)

Figure 5.4 shows the magnetization process up to 58.3 T for various tempera-tures below TN1, for the perpendicular configuration (B ⊥ c). As for the parallelconfiguration, the magnetization curves are in excellent agreement with earlier ob-servations [21, 25, 28, 30]. With increasing B⊥, the magnetization shows a steadylinear increase up to B⊥

c1 (' 24.8 T at 1.5 K), suggesting a continuous canting of

5


the 4SL spins from the c direction, toward the basal (a,b) plane. Indeed, neutrondiffraction data have confirmed the stability of the 4SL magnetic structure up toat least 14.5 T [30]. At B⊥

c1, the system undergoes a first order metamagnetictransition to a plateau state, which exhibits significant hysteresis (at 1.5 K, B⊥

c1↑= 25.40 T and B⊥

c1↓ = 24.27 T). The magnetization in this plateau state is ratherindependent on B⊥ at an average value of ' 1.53 µB/Fe3+, close to 1/3rd of thesaturation value, implying a three-sublattice state with spins in the basal plane,directed along B⊥. After undergoing a second order phase transition at B⊥

c2 '30.0 T (at 1.5 K), M once again increases (quasi-)linearly with B⊥, which inturn implies a continuous reorientation of the moments away from collinearity.The slope in this phase differs from that in the same field interval for the parallelconfiguration (see below). At B⊥

c3 ' 51.6 T (1.5 K), an additional metamagnetictransition is revealed, similar to that at B

‖c5 in the parallel configuration. As in

that case, the additional transition observed here consists of a first order meta-magnetic step, which exhibits hysteresis (at 1.5 K, B⊥

c3↑ = 52.02 T and B⊥c3↓ =

51.18 T). At 1.5 K, M jumps to ' 3.1 µB/Fe3+ at B⊥c3↑, after which it resumes a

steady increase. Combined with the absence of plateau behavior, this suggests anoncollinear spin arrangement in the high field phase also in this field configura-tion. The fact that the additional transition occurs at slightly lower critical fieldin the perpendicular configuration (B⊥

c3 ' 51.6 T vs. B‖c5 ' 53.3 T) explains the

broadness of the feature at ∼ 52 T in the aforementioned (dM/dB),B-curve ofthe powder sample of Ajiro et al. [21]

With increasing temperature, the general features of the M ,B-curve remainintact, although the plateau phase acquires an increasing slope. Furthermore, asfor the parallel case, the transition features are smoothed out upon approachingTN2, where M increases linearly with B, and deviation from this behavior onlyoccurs at very large B. Again, the small apparent temperature mismatch withrespect to susceptibility measurements (below) is attributed to a slight offset ofthe temperature sensor at temperatures close to TN2. Also here, the temperaturedependence of the various critical fields correlates to the nature of the correspond-ing transitions (See Fig. 5.5); first order transitions (at B⊥

c1 and B⊥c3) exhibit the

same identical relative decrease with temperature as B‖c2, B

‖c3 and B

‖c5, while the

second order transition (at B⊥c2) shows a much weaker temperature dependence.

Retrieval of magnetic isotropy

As mentioned in the introduction, the strong coupling between spin and latticedegrees of freedom is a key ingredient in the description of CuFeO2. Recently,Terada et al. [28, 32] showed the strong correlation between the lattice param-eters and the magnetization in applied field in both configurations. For B ‖ c,coinciding with the metamagnetic steps at B

‖c1, B

‖c2, and B

‖c3, the lattice un-

dergoes corresponding discontinuous contractions along the [110] direction, whilechanges in the [110] direction are much smaller. The lattice parameter along [110]

5


0 5 10 15 20 25 30 35 40 45 50 55 600.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Bc3

Bc2

Bc1

B c

(A

m2 kg

-1 = emu g

-1)

M (

B/F

e3+)

B (T)

9K8K

7K6K4.2K

1.5K

0

25

50

75

100

125

150

175

200

Figure 5.4: Magnetization measurements in pulsed magnetic fields at various tempera-tures. Here, the applied magnetic field is perpendicular to the c direction of the crystal.The various curves are offset by consecutive multiples of 0.5 µB/Fe3+ with increasingtemperature for clarity. Thick and thin lines represent sample magnetization in increas-ing and subsequently decreasing magnetic field, respectively. Dashed arrows indicatethe temperature dependence of the various metamagnetic transitions.

0 5 10 15 20 25 30 35 40 45 50 55 600.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

30 35 40 45 50 55 60

1.82.02.22.42.62.83.03.23.43.6

0.360.400.440.480.520.560.600.640.680.72

B//c5

Bc3B//

c3

Bc2

B//c5

B//c4B//

c3B//c2

B//c1

B c

B // c

M (

B/F

e3+)

B (T)

Bc11.5 K

M/M

sat.

Figure 5.5: Magnetization process in pulsed magnetic fields at 1.5 K, for both theparallel (red lines) and the perpendicular configuration (blue lines). Thick and thinlines represent sample magnetization in increasing and subsequently decreasing mag-netic field, respectively. Vertical dotted (dashed) lines indicate critical fields for themetamagnetic transitions in the parallel (perpendicular) configuration. Inset: high fieldzoom-in, showing the additional high field metamagnetic transitions.

5


mirrors the behavior of the magnetization in applied field; within the (collinear)4SL and 5SL phases it remains constant and in the (noncollinear) FEIC phasethe lattice continually contracts with increasing field (and magnetization). Thisstriking observation can be rationalized as follows: in zero field, the scalene trian-gle distortion (corresponding to a stretching of the triangular lattice in the [110]direction) is induced through the spin-lattice coupling, increasing the antiferro-magnetic coupling along [110] and thus optimizing the magnetic exchange energyof the 4SL phase, which has only antiferromagnetically aligned nearest neighbor(n.n) spin pairs along [110]. In addition, the same distortion induces a magneticeasy axis along the c direction, which further stabilizes the 4SL state. As B‖ in-creases however, the Zeeman energy term in the system’s Hamiltonian successivelystabilizes different magnetic structures with increasing M and a correspondinglygrowing tendency for parallel spin alignment in the field direction. The latter isclearly reflected in the increasing amount of ferromagnetic n.n. spin pairs alongthe [110] direction (See Figure 5.1). Thus, the gain in magnetic exchange energy,which arises from the stronger antiferromagnetic coupling along [110], is succes-sively reduced with B‖. Consequently, along with every spin rearrangement, thesystem rebalances the magnetic energy gain and elastic energy loss, resulting in alattice contraction which follows the changes in magnetization. In addition to thecontraction along [110], the lattice has also been shown to increase its symmetryat B

‖c2, where the scalene triangle distortion is partially relieved, resulting in a

lattice of isosceles triangles (Figure 5.1d), with two equal exchange couplings Jalong the a and b directions and a different exchange coupling J ′ along the [110]direction [13]. In other words, as the degree of magnetic frustration is reducedwith increasing magnetic field, the extent of the spontaneous lattice distortion -which occurred to partially relieve this spin frustration - is successively reduced.

Since the induced magnetic anisotropy in the material is also directly coupledto the lattice distortion, one may expect the strength of the induced single-ionanisotropy to diminish accordingly with M , undergoing steps across first ordertransitions and continuously decreasing in (quasi-)linear phases. Indeed, as shownin Figure 5.5, which shows the M ,B-curves for both the parallel and perpendicu-lar configuration at 1.5 K, the system’s response to an applied field becomes moreand more isotropic as B increases. Moreover, above both high field transitions atB‖c5 ' 53.3 T and B⊥

c3 ' 51.6 T, the system even behaves completely isotropic(within the experimental accuracy) up to the highest field measured (58.3 T). Inline with the concept of progressive symmetry increase and anisotropy reductionwith B, the absence of magnetic anisotropy at these fields suggests a completesymmetry recovery, corresponding to the retrieval of the undistorted lattice ofequilateral triangles.

5


5.3.2 Magnetic susceptibility in constant fields

In order to supplement the magnetic phase diagrams of CuFeO2 and to furtherelucidate its magnetic behavior, the temperature dependence of dc magnetic sus-ceptibilities in various constant magnetic fields was measured in both field con-figurations. Panels a)-h) of Figure 5.6 compare the low temperature magneticsusceptibility curves for the two field orientations in applied fields of 0.01, 1, 2,3, 4, 5, 6 and 7 T, respectively. Consistent with previous measurements, bothχ‖M (B ‖ c) and χ⊥M (B ⊥ c) show a diffuse maximum at TN1 ' 13.5 K and a

subsequent abrupt drop at TN2 ' 11.2 K (at low field) upon decreasing temper-ature [21, 25, 26, 37]. Above TN2 the susceptibility is isotropic, for all appliedfields measured. As expected for an ordered antiferromagnet, χ

‖M approaches

zero with decreasing temperature below TN2, while χ⊥M remains almost constantafter the initial drop at TN2. The field dependence of the magnetic susceptibility,visualized in panels i) and j ) for the parallel and perpendicular configuration,respectively, shows the invariance of TN1 with applied field for both configura-tions. Though relatively field independent for the perpendicular configuration,TN2 shifts to lower temperatures as the applied magnetic field approaches B

‖c1

(' 7.2 T) in the parallel case. This difference can be regarded as a consequenceof the lower susceptibility in the ordered phase for B ‖ c, which is unfavorabletoward the Zeeman interaction, which becomes increasingly strong with B. Thus,the magnetic ordering transition at T

‖N2 is shifted to lower temperature. For the

perpendicular case, the susceptibility drop across T⊥N2 is only marginal, ergo thistemperature down-shift is far less pronounced.

As is clear from panels h) and i), the transition at T‖N2 and 7 T acquires a

double feature, indicating the process becomes two-stepped. This points towardthe presence of an intermediate phase between the two steps. Based on the con-structed phase diagram presented below (Figure 5.7), this intermediate phase isidentified as the helical FEIC phase, as the phase boundaries of both the 4SL andFEIC phases bend toward the T

‖N2(B) line at these temperatures.

5.3.3 Phase diagrams

With the phase transition data obtained above in hand, the experimental phasediagrams of CuFeO2 as a function of applied field and temperature can be as-sembled for both configurations. Magnetic transition temperatures are definedthrough the position of corresponding anomalies in the derivatives of the pulsedfield magnetization (∂M/∂B) and susceptibility (∂χM/∂T ) curves, respectively.

Parallel configuration (B ‖ c)

Figure 5.7 shows the B,T phase diagram for CuFeO2 that can be constructedbased on aforementioned experiments and other, currently available data [25, 27,28, 30, 31, 33, 40], for the parallel configuration (B ‖ c). The diagram features all

5


0.0

0.1

0.2

0.3

0.4

4 6 8 10 12 14 16 18 20 22 24

4 6 8 10 12 14 16 18 20 22 240.0

0.1

0.2

0.3

0.4

B // c B c

TN1

B = 4 T

B = 3 T

B = 2 T

B = 1 T

T (K)

B // c B cM

(cm

3 /m

ol)

B = 0.1 T

TN2

a)4 6 8 10 12 14 16 18 20 22 24

0.0

0.1

0.2

0.3

0.4

B // c B c

B = 7 T

B // c B c

f)

TN1

B c

B // c

B = 6 T

B = 5 T

M (cm3/m

ol)

T (K)

B // c B c

0.0

0.1

0.2

0.3

0.4 b)

TN1

TN2

TN2

M (c

m3 /

mol

)

0.0

0.1

0.2

0.3

0.4g)

TN1

M (cm3/m

ol)

TN10.0

0.1

0.2

0.3

0.4

d)

c)

TN1

B // c B c

M (c

m3 /

mol

)

0.1

0.2

0.3

0.4

0.5

0.1 T 4 T 1 T 5 T 2 T 6 T 3 T 7 T

h)

M (cm3/m

ol)

Two-stepped

0.0

0.1

0.2

0.3

0.4

TN1TN2

B // c B cM

(cm

3 /m

ol)

8 9 10 11 12 13 14 15 160.34

0.36

0.38

0.40

0.42

0.1 T 4 T 1 T 5 T 2 T 6 T 3 T 7 T

j)

i)

M (cm3/m

ol)

T (K)4 6 8 10 12 14 16 18 20 22 24

0.0

0.1

0.2

0.3

0.4

TN2

e)

TN1T //

N2

B // c B cM

(cm

3 /m

ol)

T (K)

TN2T //N2

TN2T //N2

TN2T //N2

M (cm3/m

ol)

Figure 5.6: Magnetic susceptibility versus temperature in various constant magneticfields for both the parallel and the perpendicular configuration. Panels a)-h) comparethe magnetic susceptibility for B ‖ c and B ⊥ c as a function of temperature at differentfields up to 7 T. Vertical dotted/dashed lines indicate magnetic transition temperatures,as deducted from corresponding dχM/dT data. The zoom-ins on the data in panelsi) and j) depict the field dependence of the magnetic susceptibility in parallel andperpendicular configurations, respectively.

5


Figure 5.7: Top panel: Differential magnetization curves for B ‖ c. Solid lines de-pict (normalized) dM/dB-curves as measured in increasing and subsequently decreasingmagnetic field, respectively. dM/dB-curves measured in decreasing magnetic field areinverted for clarity. Additionally, each curve has an offset equal to its correspondingtemperature. Lower panel: B,T phase diagram of CuFeO2 for the case where B isparallel to the c-axis. Large, black squares depict magnetic transitions as observed inthis work and smaller gray symbols indicate previously reported transitions. Open andclosed symbols represent transitions observed in increasing and decreasing (B or T )ramps, respectively. Solid lines correspond to proposed phase boundaries. The grayshaded region I corresponds to an intermediate state (see text).

5


the previously confirmed phases; the paramagnetic phase (PM) above TN1 ' 14K, the partially disordered, incommensurate phase (PDIC) for TN2 ≤ T ≤ TN1

and the collinear four-sublattice phase (4SL) below TN1 in zero field, and withincreasing applied magnetic field below TN2 the ferroelectric, incommensuratephase (FEIC) between B

‖c1 and B

‖c2 and the collinear five-sublattice phase (5SL,

B‖c2 ≤ B‖ ≤ B

‖c3). Between B

‖c3 ∼ 20 T and B

‖c4 ∼ 32.5 T, a plateau phase at

approximately one third of saturation exists, which has previously been ascribedto an analogous collinear three-sublattice (3SL) spin configuration (Fig. 5.1e) [28,32]. The same interpretation is adopted here, as it is consistent with pulsed fieldnuclear forward scattering experiments (Chapter 6, [58]) and a nonzero magneticanisotropy (plateau behavior). At B

‖c4, M starts increasing steadily with B after

a second order phase transition, which suggests a continuous canting of the 3SLspins away from the c axis. Thus, this phase is labeled the canted three-sublatticephase (c3SL). In similar manner, we label the phase above the high field transitionat B

‖c5 a canted High Field (cHF) phase, based on its continuous linear growth of

M . Also worth noting is the fact that the transition from the 5SL to the 3SL phase(at B

‖c3) appears to split up into a two-step transition with temperature, implying

an intermediate spin state (indicated by the gray colored region in the phasediagram). In this newly observed intermediate phase I, which occurs betweenthe collinear 3SL and 5SL phases and only at temperatures approaching TN2, themagnetization of the system deviates continuously from the 5SL plateau value(see the 7 K line in Figure 5.2), before the abrupt rearrangement to the 5SLspin configuration. This behavior reflects the aforementioned reduction of themagnetic anisotropy with applied magnetic field at these temperatures.

Perpendicular configuration (B ⊥ c)

Figure 5.8 shows the analogous B,T phase diagram resulting from above exper-iments and earlier reported data [27, 28, 30, 40] for the case where B ⊥ c. Thediagram includes the zero field phases; the paramagnetic phase (PM) above TN1,the partially disordered, incommensurate phase (PDIC) between TN2 and TN1

and the four-sublattice phase (4SL) below TN1. As B increases below TN1, thelinearly increasing magnetization implies a canted four-sublattice (c4SL) phasethat undergoes a first order transition at B⊥

c1 ' 24 T to a plateau state withM ∼ 1/3rd of the saturation value. Based on the same arguments as mentionedin the parallel configuration case, latter state will be labeled as a collinear threesublattice phase (3SL), though spins are presumably oriented in the basal planehere. Analogously, the continuous evolution of M below and above B⊥

c3 ∼ 51T encourages a canted three sublattice (c3SL) and a canted High Field (cHF)interpretation for the phases between B⊥

c2 and B⊥c3 and above B⊥

c3, respectively.

5


0 5 10 15 20 25 30 35 40 45 50 55 600123456789

1011121314151601234567891011

0 5 10 15 20 25 30 35 40 45 50 55 60

cHFc3SLc4SL 3SL

This work Terada et al.27 Petrenko et al.30 Kimura et al.26 Seki et al.40

T (K

)

B (T)

PDIC

PM

4SL

1.5 K

B (T)

9 K8 K7 K6 K

4 K

dM/dB

(a.u.)

T (K

)

Figure 5.8: Top panel: Differential magnetization curves for B ⊥ c. Solid lines de-pict (normalized) dM/dB-curves as measured in increasing and subsequently decreasingmagnetic field, respectively. dM/dB-curves measured in decreasing magnetic field areinverted for clarity. Additionally, each curve has an offset equal to its correspondingtemperature for clarity. Bottom panel: B,T phase diagram of CuFeO2 for the B ⊥c case. Large, black squares depict magnetic transitions as observed in this work andsmaller gray symbols indicate previously reported transitions. Open and closed symbolsrepresent transitions observed in increasing and decreasing (B or T ) ramps, respectively.Solid lines correspond to proposed phase boundaries.

5


(a)

rij

ujui

J(r)

3

a

b

c

[110]

J†

J‡

1

2

3

3

3

3

3

3

3

2

2

2

2

2

2

2

1

1

1

1

1

1

1

J

(b)

(c)

30 40 50 60 70 80 900.00

0.01

0.02

B ^ c

Bsat.

B//

c4

D(m

eV

)

B (T)

B^

c2

B // c

Figure 5.9: Features of the classical spin model. a) Illustration depicting the dependenceof the magnetic exchange coupling J on the atom displacements (spin-lattice interaction).b) Exchange interactions in the 3SL spin structure on the lattice of isosceles triangles.Numbers indicate the three magnetic sublattices, while dotted gray lines indicate themagnetic unit cell. First- (J), second- (J†) and third-neighbor (J‡) interactions areindicated. c) Schematic illustration of the assumed field dependence of the magneticanisotropy D. For both field configurations, D is approximated to be anti-proportionalto M(B); as M starts increasing (at B

‖c4 and B⊥

c2, respectively), D starts decreasing.Ultimately, as M approaches saturation, D vanishes accordingly.

5.3.4 Classical spin model

Additional experimental information on the high field spin structures is unavail-able at present, as conventional neutron and x-ray based techniques require static(dc) fields, which do not exceed 40 T at this time. Furthermore, more specializedtechniques, such as nuclear forward scattering in pulsed magnetic fields, make useof custom magnets and operation above 35 T is not currently feasible. Ergo, weresort to a classical spin model to investigate the magnetic behavior of CuFeO2

at fields which are experimentally inaccessible by above-mentioned techniques (inthe 3SL phases and above). To adequately describe the system, its primary in-teractions must be incorporated into the spin Hamiltonian. Thus, along with thebasic magnetic exchange and Zeeman interaction terms, the strong spin-phononcoupling and the magnetic isotropy in CuFeO2 should be included. Indeed, thecombination of the latter two seems key to capture the Ising-like behavior of thesystem, as was recently shown [9, 10].

5


Spin-lattice interactions are typically incorporated into the Hamiltonian throughthe distance dependence of the exchange coupling J(r) (See Fig. 5.9a) [9, 59, 60].Ergo, in general for a system with isotropic exchange interactions the effectiveHamiltonian becomes:

Heff. = J∑

〈ij〉Si · Sj(1− αuij) + Hdef.({ui}), (5.1)

where the ui are the displacement vectors, the uij (=(ui − uj)·rij/|rij |) are thecorresponding relative changes in bond length between sites i and j, α is thespin-lattice constant (to first approximation equal to J−1 ∂J/∂r) and Hdef. cor-responds to the deformation energy cost associated with the atom displacementsui, which is thus dependent on the phonon model still to be chosen. For the sakeof simplicity we will consider the bond-phonon (BP) model here, which treats thebond lengths uij as independent variables [59, 60]. The Hamiltonian then takesthe form:

Heff. = J∑

〈ij〉Si · Sj(1− αuij) +

k

2

∑

〈ij〉u2

ij , (5.2)

where k is an elastic constant. To simplify this effective Hamiltonian one maysimply minimize with respect to the bond lengths uij in this case, to find:

∑

〈ij〉uij =

αJ

k

∑

〈ij〉Si · Sj . (5.3)

As the bond lengths are independent variables and the spin and lattice degreesof freedom are directly coupled, we can set uij=αJ

k Si ·Sj for each bond and canthus construct the resulting general spin Hamiltonian [59, 60] (containing onlymagnetic contributions) based on eq. 5.1:

Hs = J∑

〈ij〉Si · Sj − bJ(Si · Sj)2, (5.4)

where b = α2J/k. In effect, the spin-phonon coupling in the BP model thus in-troduces a biquadratic spin interaction of strength bJ . Furthermore, since neigh-boring bond lengths uij are independent here, the biquadratic term is restrictedto nearest neighbor couplings only. Due to the quadratic nature of the term, ei-ther parallel or antiparallel spin configurations are favorable, which explains thetendency of spin-lattice coupling to stabilize collinear spin states.

With above expression for the spin-lattice coupling, the general spin Hamilto-nian for CuFeO2 can now be written as:

Hs = −gµB ·∑

i

Si +∑

i,j

JijSi · Sj

−∑

〈ij〉bJij(Si · Sj)2 −D(B)

∑

i

S2iz, (5.5)

5


where B is the applied magnetic field, Jij is the exchange interaction between sitesi and j, b is the biquadratic coupling constant and D is the magnetic anisotropyconstant. As discussed above, the latter is strongly coupled to the lattice dis-tortion in CuFeO2, and since this in turn strongly correlates with the system’smagnetization, D is also field dependent (Fig. 5.9c). The Zeeman and anisotropyterms sum over all sites i , the biquadratic term couples only nearest neighborspin pairs i and j, and the exchange term includes all spin pair interactions in thesystem.

Experimentally, the magnetization process of CuFeO2 at high fields is quali-tatively equivalent for both field configurations; after exhibiting a 1/3rd magne-tization plateau, implying a three sublattice (3SL) structure, M starts growingsteadily, which indicates a continuous reorientation of those 3SL spins. In orderto capture the magnetic behavior of CuFeO2 in both configurations, we thereforeconsider the spin Hamiltonian (eq. 5.5) of the 3SL structure on a single two di-mensional sheet of isosceles triangles [61]. Discarding further neighbor couplings,one finds three unique first- (J), second- (J†) and third-neighbor (J‡) interactionsfor each of the three spins in the magnetic unit cell of the 3SL structure (Fig.5.9b). Considering the spins as classical, justified by the large S = 5/2 value, wewrite Si = eiS (where e is a unit vector), thus obtaining:

Hs = −2µBSB ·∑

i

ei + CS2(p12 + p13 + p23) + 9J†S2

−GS4(p212 + p2

13 + p223)−D(B)S2

∑

i

e2i,z, (5.6)

where g is taken as 2 and spin-spin couplings are written as pij (= ei · ej). Theexchange (C) and spin-phonon (G) parameters are defined as C = (3J + 3J‡)and G = 3bJ , respectively. Note that including further neighbor exchange inter-actions only has scaling effects; the first and third neighbor interactions J and J‡

generate contributions of exactly the same form and can thus be combined, whilethe second neighbor interactions (J†) produce a constant term and thus merelyshift the total energy as a whole.

To determine the magnetic ground state of the 3SL system, the spin Hamilto-nian of the 3SL system per magnetic unit cell (eq. 5.6) is numerically minimizedas a function of the three independent spin directions e1, e2 and e3 at any givenfield B. Since the lattice distortion persists up to at least 40 T for both field con-figurations [28], a nonzero magnetic anisotropy D may be expected up to thesefields at least. Moreover, to incorporate its field dependence (a priori unknown),D is approximated to be proportional to Msat.−M(B). Thus, D mirrors the field-dependence of M , vanishing as the system approaches saturation, see Figure 5.9c.Upon examination of the simulated magnetization process of the correspondingmagnetic 3SL system, one finds that having both the anisotropy constant D andthe spin-lattice constant G nonzero is a prerequisite for good qualitative agree-ment with experiment. Without a finite D in the collinear 3SL state, the model,and thus the simulated magnetization process, becomes isotropic, contrasting the

5


(a) (b)

Figure 5.10: Comparison of experimental (left panel) and simulated (right panel) mag-netization curves in CuFeO2. In general, red lines correspond to the parallel field con-figuration (B ‖ c) and blue lines correspond to the perpendicular configuration (B ⊥c). Dotted vertical lines indicate transition fields observed in both experimental andsimulated processes, dashed lines indicate the experimental high field transitions notobserved in the simulations. a) Experimental magnetization curves at 1.5 K (for in-creasing magnetic field). b) Simulated magnetization curves based on a classical spinmodel for the 3SL structure (eq. 5.6, see text). The magnetization in the both the fielddirection (left axis) and the direction perpendicular to B (right axis) are plotted.

z

y

x S3

S ,1 S2B

z

y

xS3

S ,1 S2

B

Figure 5.11: Evolution of individual sublattice spins in the magnetic unit cell for theparallel (left panel) and perpendicular configuration (right panel), as given by the sim-ulations. For the perpendicular configuration the magnetic field was aligned with they-direction.

5


experimental results. Simultaneously, a nonzero G is required to stabilize a mag-netization plateau at 1/3rd of saturation for the perpendicular field configuration(B⊥c). Matching the simulated and experimental magnetization curves, by vary-ing the model parameters C, D (at 25 T) and G, yields numerical estimates forthe exchange, anisotropy and spin-phonon parameters of C = 1.32 meV, D(25T)= 0.021 meV (plateau) and G = 0.0074 meV, respectively. Considering onlyfirst-neighbor interactions (J† = J‡ = 0), JS2 can be estimated to be ∼ 2.76meV (32.0 K) and DS (at 25T) to be ∼ 0.052 meV (0.6 K), in line with previousestimates [9, 24, 25]. For GS4 we estimate ∼ 0.29 meV (3.4 K), which sets therelative strength of the dimensionless biquadratic spin interaction b at ∼ 0.0056.With these model parameters the simulated magnetization process is in strikingagreement with experiment, as depicted in Figure 5.10. The spin-phonon interac-tion (G), which favors collinear spin states but does not specify any spin directions(non-directional), stabilizes the 1/3rd magnetization plateau in both configura-tions. The dependence of the plateau-width on field direction is then a naturalconsequence of the interplay with the directional interactions in the system, theZeeman interaction and the magnetic anisotropy. In essence, the Zeeman inter-action combines with the spin-phonon term, thus providing the preferred axis ofcollinearity, while the anisotropy term either further stabilizes or destabilizes thecollinear structure, dependent on the field direction. For (B ‖ c), D promotes thecollinear structure, thus widening the plateau, whilst it opposes collinearity in theperpendicular configuration, thereby inducing a narrowing of the correspondingplateau. At fields above the plateau, the increasingly dominant Zeeman term hasstrengthened enough to induce a gradual spin canting, in order to increase M .There, the spin-lattice term induces a positive second derivative (∂2M/∂B2) inthe magnetization curve, as is also observed in experiment.

The evolution of the spins on the three sublattices in the magnetic unit cell,as determined by the numerical minimization, is depicted in Figure 5.11. Forboth configurations the plateau state (3SL) corresponds to two spins (S1 and S2)being aligned with the applied field and one being antiparallel (S3). From thesecond order phase transition (at B

‖c4 and B⊥

c2, respectively) on, the antiparallelspin starts gradually tilting toward the field direction, in order to increase thetotal magnetization. The two parallel spins respond by initially canting awayfrom B in the opposite direction, thereby optimizing the overall magnetic energy,before gradually returning after the antiparallel spin has passed the direction nor-mal to B. Naturally, the tilting-direction of the individual spins with respect toan external coordinate system is arbitrary within the symmetry of the system’sHamiltonian (eq. 5.6): C∞v for B ‖ c and C2v for B ⊥ c. The spin-phonon andanisotropy terms ensure the three spins stay aligned in a single plane, and keepthe two parallel spins collinear throughout the magnetization process. Moreover,the system thereby acquires a small magnetization component in the directionorthogonal to the field, which quickly grows and slowly decreases with B abovethe second order transition, as depicted in the right panel of Figure 5.10. This in-

5


30 40 50 60 70 80 90

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Bc5

Bc3

E (m

eV /

unit

cell)

B (T)

Figure 5.12: Estimated magnetic energy gain (∆Em) upon lattice distortion for both theparallel (black) and perpendicular field configuration (grey), as approximated using eq5.6 (see text). Dashed vertical lines indicate the experimental transition fields observedat 1.5 K.

plane magnetization component is not expected to be observed in a macroscopicsample, however, as this contribution is likely to average out as a result of the for-mation of many domains. Thus, based on the classical spin Hamiltonian (eq. 5.6),the spin structure of the canted 3SL phase (c3SL) in the parallel and perpendicu-lar configuration is as depicted in Figure 5.13b and e, respectively. Although themagnetization process is qualitatively equivalent for both configurations, quanti-tative details (plateau widths, transition fields, slopes) differ slightly due to theaforementioned interplay between directional and nondirectional interactions.

5.3.5 High field transition

Although accurately describing the 3SL and c3SL phases for both field configu-rations, the classical spin model does not directly account for the additional highfield transitions observed in experiment (Fig. 5.10). Instead, for both configura-tions, the model predicts a continuous evolution toward full alignment along Bat Bsat.. However, the model assumes a distortion-induced magnetic anisotropyD(B) that is finite up to saturation (Fig. 5.9c), while the deformation energy costassociated with that lattice distortion (Hdef.) is not included in the all-magneticHamiltonian (eq. 5.6). Experimentally, the lattice distortion has been confirmedto persist up to at least 40 T for both field configurations [28], justifying theassumption of a finite D up to those fields. At higher fields, however, the spin-frustration-driven lattice distortion (the spin Jahn-Teller effect) will only persistas long as its associated magnetic energy gain (∆Em) outweighs the correspond-

5


ing deformation energy cost (Hdef.). The amount of magnetic energy the systemgains upon having this distortion (and thus finite D) may be approximated bytaking the difference between the energy of an isotropic spin configuration (e.g. acanted 120◦ configuration, see Fig. 5.13c and e) and the minimum energy c3SLconfiguration of eq. 5.6 (Fig. 5.13b and e). As shown in Figure 5.12, ∆Em di-minishes as B increases, for both field configurations. Thus, the first order highfield transitions at B

‖c5 and B⊥

c3 can then be interpreted as the point where ∆Em

no longer prevails, upon which the system undergoes a ’reversed spin Jahn-Tellertransition’, reverting to a lattice of equilateral triangles. This notion is consistentwith the above-mentioned trend of progressive symmetry increase and anisotropyreduction with B, and is corroborated by the observed isotropy retrieval aboveB‖c5 (See Fig. 5.5). Intuitively, this is also what one would expect at high fields;

as the frustrated component of the spin interactions diminishes, at some pointthe frustration-induced lattice distortion is relieved. Interesting to note is thefact that at the fields corresponding to this transition (B‖

c5 and B⊥c3 respectively),

∆Em is approximately 0.42 meV per magnetic unit cell. As this unit cell con-tains three spins, this corresponds very closely to the temperature scale of theexperimental data; 3kT at 1.5 K is ∼ 0.39 meV. This suggests an involvement ofthe system’s temperature in the high field transition, which is consistent with theobserved temperature dependence of B

‖c5 and B⊥

c3 (See Fig. 5.3). Above thesetransitions, in the canted High Field (cHF) phase, a classical spin model (withD = 0) predicts a canted 120◦ spin configuration. In this umbrella-like spin struc-ture, all three spins are tilted away from the field-direction, while their projectionsin the orthogonal plane keep mutual 120◦ angles, as is shown in Figure 5.13c andf for B ‖ c and B ⊥ c, respectively.

Despite its satisfactory and intuitive results, the classical spin model doeshave its limitations. Although the low field collinear phases (4SL and 5SL) canbe described using eq. 5.5, being a phenomenological model meant to describethe high field phases, it does not capture the complex helical ferroelectric phase(FEIC). A fully accurate and quantitative description of CuFeO2 would requirethe inclusion of all additional interactions in the system that could play a role.Possibly the most important additional aspect is the interplane coupling withinthe system. Recent inelastic neutron scattering work shows significant spin-wavedispersion along the hexagonal axis, estimating the interplane interaction to besubstantial (Jz/J1 ≈ 0.3) [24, 25, 29, 38]. In addition, the observed finite dis-persion of calculated electronic bands also points to non-negligible interlayer cou-plings [62]. The incorporation of finite temperature, a more realistic phonon model(yielding longer range biquadratic interactions [9, 60]) and quantum spin effectsmay also improve the quantitative understanding of the system. Furthermore,more exotic interactions may play a role in stabilizing the incommensurate spiralstate [63].

5


(b) 3SLc [110]

a

bc

JJ

’J

isosceles

//

(c) HFc

a

bc

[110]

JJ

J

equilateral

//

[110](d) 3SL

a

bc

JJ

’J

isosceles

^

[110](e) 3SLc

a

bc

JJ

’J

isosceles

^

^(f) HFc

a

bc

[110]

JJ

J

equilateral

(a) 3SL [110]

a

bc

JJ

’J

isosceles

//

Figure 5.13: Magnetic 3SL spin structures in various phases of CuFeO2, as given bythe classical spin model. A single triangle is highlighted in yellow in each phase forclarity. Proposed lattice symmetries in the phases are depicted on the right of each spinstructure. a) (B

‖c3 < B‖ < B

‖c4) Collinear two-up one-down order in the three sublattice

phase (3SL) (B ‖ c). b) (B‖c4 < B‖ < B

‖c5) Spin structure in the canted three sublattice

(c3SL) phase (B ‖ c), where the magnetic moments undergo a gradual canting (see text,

Fig. 5.11). c) (B‖ > B‖c5) Magnetic ordering in the canted High Field (cHF) phase (B ‖

c), where the spins adopt a canted 120◦ structure. d) (B⊥c1 < B⊥ < B⊥

c2) Collinear threesublattice (3SL) phase for the perpendicular configuration, where two spins are parallelto the applied field and a third is antiparallel. e) (B⊥

c2 < B⊥ < B⊥c3) Spin configuration

in the canted three sublattice (c3SL) phase for B ⊥ c, where spins continuously reorientwith the applied field (see text, Fig. 5.11). f) (B⊥ > B⊥

c3) Spin structure in the cantedHigh Field (cHF) phase for the perpendicular configuration, where the spins adopt acanted 120◦ structure. The three inequivalent spins in red are drawn with mutual originfor clarity.

5


5.3.6 Metamagnetism crossover

It is instructive to regard the magnetic field dependence of CuFeO2 from the view-point of metamagnetism. The term metamagnetism is typically used in referenceto any system that, upon a small variation of the externally applied magneticfield, exhibits a (dramatic) change in magnetization. The corresponding magneticphase diagrams can in general be (qualitatively) rationalized according to the de-gree of magnetic anisotropy in the materials [64]. In highly anisotropic systems,spins are effectively restricted to align (anti-)parallel to the magnetic easy-axisand magnetic transitions typically involve discontinuous spin reversals, leadingto first-order type metamagnetic transitions. For isotropic (weakly anisotropic)systems this directional restriction is relieved (strongly reduced), thus transitionsin such materials often reflect the onset of a continuous, second order type reori-entation of the spins.

In terms of metamagnetism, the behavior of CuFeO2 in the parallel config-uration field corresponds to a crossover of regimes. At low magnetic fields, thespin-phonon coupling and magnetic anisotropy, both promoting collinearity (alongthe c direction), essentially combine to yield a highly anisotropic spin system. Asthe Zeeman interaction is too weak at these fields to stabilize canted spin states,the system successively adopts different collinear spin states with increasing mag-netization and field-induced magnetic transitions involve discontinuous spin rever-sals. Thus, at low fields the system behaves as a highly anisotropic metamagnet,exhibiting abrupt, first-order transitions with significant hysteresis and plateauphases. However, as the distortion-induced magnetic anisotropy diminishes withincreasing field, so the does the preferred directionality along the c axis. Ef-fectively, the system thus crosses over to the limit of weak anisotropy/isotropy,since the spin-phonon interaction is non-directional. Consequently, at high mag-netic fields, the now dominant Zeeman interaction overrules the tendency towardcollinearity and spins undergo gradual reorientations with B. The correspondingtilting angles then result from the continuously evolving balance of the Zeeman,exchange, spin-phonon and anisotropy terms. Because the high field first ordertransition does not correspond to an abrupt spin-flip, but is a consequence of thestructural relaxation of the system, it does not contradict the concept of metam-agnetism crossover.

For the perpendicular configuration, the situation is somewhat different. Hereat low fields, the spin-phonon coupling and magnetic anisotropy again combine tofavor collinear states with spins along the c-axis. Due to the fact that in this casethe Zeeman interaction promotes a net magnetization in the direction orthogonalto c, however, a spin canting is induced even at low fields. At intermediate fields,the magnetic anisotropy is overcome and there Zeeman interaction combines withthe spin-phonon coupling to induce a preferred axis of collinearity along B. Athigh magnetic fields, the dominant Zeeman interaction once again overrules thetendency toward collinearity and continuous spin reorientations occur.

5


5.4 Conclusions

In conclusion, we have expanded the experimental field of view in CuFeO2 byperforming magnetization experiments at various temperatures below TN2 up tohigh magnetic fields. The present pulsed field magnetization data at varioustemperatures extend the metamagnetic staircase characteristic of CuFeO2 up to58.3 T and reveal an additional metamagnetic phase transition for both magneticfield orientations. Above these transitions, the system exhibits virtually com-plete isotropic behavior, indicative of an undistorted triangular structure and inline with the trend of progressive anisotropy reduction and symmetry increaseand with magnetic field. Additionally, a newly observed intermediate phase wasfound at temperatures close to TN2 (11 K). Magnetization data suggest a tilting of(some) spins in this intermediate phase, occurring in between the collinear three-and five sublattice states. Correspondingly, we have thoroughly mapped out theexperimental B,T phase diagrams of CuFeO2 for both the parallel (B ‖ c) and per-pendicular (B ⊥ c) configurations and expanded them into previously unchartedterritory. Through numerical minimizations of a simple classical Heisenberg modelthat includes the system’s primary interactions, the experimental magnetizationprocess was satisfactorily simulated and a set of corresponding spin Hamiltonianparameters was obtained. A consistent phenomenological rationalization of thespin system’s field dependence is given and spin structures are proposed for allexperimentally observed phases. Combining the classical spin model with our ex-perimental results, the recovery of a lattice of equilateral triangles is anticipatedat high fields. The underlying intuitive concept of progressive symmetry increaseas the degree of frustration in spin Jahn-Teller distorted systems diminishes isfound to be more generally applicable, as it is based exclusively on energy ar-guments. Thus, corresponding distortion-relieving high field transitions (inversespin Jahn-Teller transitions) may be anticipated in similar distorted systems gov-erned by competing interactions of comparable strength, such as members of thechromium spinel family, for example. Indeed, a similar high field transition hasrecently been observed in the frustrated pyrochlore system HgCr2O4, whose ori-gin has thus far remained unclear [65]. Corresponding high field transitions maybe expected in other members of the chromium spinel family, such as CdCr2O4

and ZnCr2O4.

5

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