Field dependence of the Spin State and Spectroscopic Modes of Multiferroic BiFeO 3
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Field dependence of the Spin State and Spectroscopic Modes of Multiferroic BiFeO Randy S. Fishman
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Dated: September 24, 2018)The spectroscopic modes of multiferroic BiFeO provide detailed information about the verysmall anisotropy and Dzyaloshinskii-Moriya (DM) interactions responsible for the long-wavelength,distorted cycloid below T N = 640 K. A microscopic model that includes two DM interactions andeasy-axis anisotropy predicts both the zero-field spectroscopic modes as well as their splitting andevolution in a magnetic field applied along a cubic axis. While only six modes are optically activein zero field, all modes at the cycloidal wavevector are activated by a magnetic field. The threemagnetic domains of the cycloid are degenerate in zero field but one domain has lower energy thanthe other two in nonzero field. Measurements imply that the higher-energy domains are depopulatedabove about 6 T and have a maximum critical field of 16 T, below the critical field of 19 T for thelowest-energy domain. Despite the excellent agreement with the measured spectroscopic frequencies,some discrepancies with the measured spectroscopic intensities suggest that other weak interactionsmay be missing from the model. PACS numbers: 75.25.-j, 75.30.Ds, 75.50.Ee, 78.30.-j
I. INTRODUCTION
Due to the coupling between their electric and mag-netic properties, mutliferroic materials have intriguedboth basic and applied scientists for many years. Multi-ferroic materials would offer several advantages over mag-netoresistive materials in magnetic storage devices. Mostsignificantly, information could be written electricallyand read magnetically without Joule heating . Hence,a material that is multiferroic at room temperature hasthe potential to radically transform the magnetic storageindustry. As the only known room-temperature multifer-roic, BiFeO continues to attract intense scrutiny.Because BiFeO is a “proper” multiferroic, its ferro-electric transition temperature T c ≈ T N ≈
640 K. Below T N , a long-wavelength cycloid witha period of 62 nm enhances the electric polarization byabout 40 nC/cm . Although much smaller than the verylarge polarization P = 100 µ C/cm above T N but below T c , the induced polarization can be used to switch be-tween magnetic domains in an applied electric field .The availability of single crystals for both elastic and inelastic neutron-scattering measurements hasstimulated recent progress in unravelling the microscopicinteractions in BiFeO . Based on a comparison with thepredicted spin-wave (SW) spectrum, inelastic neutron-scattering measurements were used to obtain the an-tiferromagnetic (AF) nearest-neighbor and next-nearestneighbor exchanges J ≈ − . J = − . a ≈ .
96 ˚ A . Whenweaker interaction energies are suppressed by strain ,non-magnetic impurities , or magnetic fields above H c ≈
19 T, the exchange interactions produce a G-typeAF with ferromagnetic (FM) alignment of the S = 5 / spins within each hexagonal plane. In pseudo-cubicnotation, the AF wavevector is Q = ( π/a )(1 , , H c , the much weaker anisotropy andDzyaloshinskii-Moriya (DM) interactions producethe distorted cycloid of bulk BiFeO . For most ma-terials with complex spin states, neutron scatteringcan be used to determine the competing interac-tions. But for BiFeO , the cycloidal satellites at q = (2 π/a )(0 . ± δ, . , . ∓ δ ) with δ ≈ . Q . Because it lacks sufficient resolutionin q space, inelastic neutron-scattering measurements at Q reveal four broad peaks below 5 meV. Each of thosepeaks can be roughly assigned to one or more of the SWbranches averaged over the first Brillouin zone .By contrast, THz spectroscopy provides very pre-cise values for the optically-active SW frequencies atthe cycloidal wavevector Q . With polarization along z ′ = [1 , ,
1] (all unit vectors are assumed normal-ized to 1), the three magnetic domains have wavevec-tors Q = (2 π/a )(0 . δ, . − δ, .
5) (domain 1), Q = (2 π/a )(0 . δ, . , . − δ ) (domain 2), and Q =(2 π/a )(0 . , . δ, . − δ ) (domain 3). The local coor-dinate system { x ′ , y ′ , z ′ } for each domain is indicated inFig.1(c).In zero field, the four spectroscopic modes observed be-low 45 cm − were recently predicted by a model witheasy-axis anisotropy K along z ′ and two DM interactions.While the DM interaction D along y ′ is responsible forthe cycloidal period, the DM interaction D ′ along z ′ produces the small tilt τ in the plane of the cycloidalspins shown in Fig.1(b). The tilt alternates in sign fromone hexagonal plane to the next. In the AF phase above H c , D ′ produces a weak FM moment perpendicular to z ′ due to the canting of the moments within each hexag-onal plane.This microscopic model with parameters D , D ′ , and K also predicts the mode splitting and evolution of thespectroscopic modes with field. Due to mode mixing, allof the SWs are optically active in a magnetic field. Com-paring the predicted and observed field dependence al-lows us to unambiguously assign the spectroscopic modesof BiFeO . Despite the remarkable agreement betweenthe predicted and measured mode frequencies, however,discrepancies between the predicted and measured spec-troscopic intensities suggest that other weak interactionsmay be missing from the model.We have organized this paper into five sections. Sec-tion II discusses the spin state of BiFeO in a magneticfield, with results for the wavevector, domain energies,and magnetization. In Section III, the spectroscopic fre-quencies are evaluated as a function of field and comparethose results with measurements. The spectroscopic se-lection rules and intensities are discussed in Section IV.Section V contains a summary and conclusion. A shortdescription of the theory for the spectroscopic modes wasrecently presented by Nagel et al. . II. SPIN STATE
In a magnetic field H = H m , the spin state and SWexcitations of BiFeO are evaluated from the Hamiltonian H = − J X h i,j i S i · S j − J X h i,j i ′ S i · S j − K X i ( S i · z ′ ) − D X R j = R i + a ( x − z ) y ′ · ( S i × S j ) − D ′ X R j = R i + a x ,a y ,a z ( − R iz ′ /c z ′ · ( S i × S j ) − µ B H X i S i · m . (1)The nearest- and next-nearest neighbor exchange in-teractions J = − . J = − . between 5.5 meV and 72 meV. Onthe other hand, the small interactions D , D ′ , and K thatgenerate the cycloid can be obtained from spectroscopicmeasurements below 5.5 meV (44.3 cm − ).For a given set of interaction parameters, the spin stateof BiFeO is obtained by minimizing the energy E = hHi over a set of variational parameters. With the same spinstates in hexagonal layers n and n + 2, the spin states inlayers n = 1 and 2 are parameterized as S x ′ ( R ) = A ( n ) ( R ) sin µ cos τ ( n ) sin(2 πδR x ′ /a + γ ( n )1 )+ s p ( n ) x ′ , (2) S y ′ ( R ) = A ( n ) ( R ) sin µ sin τ ( n ) sin(2 πδR x ′ /a + γ ( n )2 )+ s p ( n ) y ′ , (3) S z ′ ( R ) = A ( n ) ( R ) cos µ F ( n ) ( R ) + s p ( n ) z ′ , (4)where F ( n ) ( R ) = X l =1 C l − cos (cid:0) l − πδR x ′ /a (cid:1) + X l =1 C l cos (cid:0) lπδR x ′ /a + Γ ( n ) (cid:1) (5) x ´ y ´ z ´ = [1,1,1] τ S x´ S y´ (b) (a) J J z ´ z y x domain 1: x ´ = [1,-1,0], y ´ = [1,1,-2] domain 2: x ´ = [1,0,-1], y ´ = [-1,2,-1] domain 3: x ´ = [0,1,-1], y ´ = [-2,1,1] E / N ( m e V ) H (T) -76.94 -76.91 -76.92 -76.935 -76.915 -76.925 -76.93 (e) M ( µ B ) (f) δ H (T) (d) H (T) h = [1,-1,0], e = [1,1,0] h = [1,1,0], e = [1,-1,0] (c) M * a FIG. 1: (Color online) (a) The pseudo-cubic cell with S = 5 / ions are at the corners. The exchange interactions J and J as well as the polarization direction z ′ cutting throughtwo hexagonal planes are indicated. (b) For any of the threemagnetic domains, a schematic of the spins in zero field show-ing their rotation about y ′ . Due to the DM interaction D ′ = D ′ z ′ , spins rotate by τ about z ′ in the x ′ y ′ plane. (c)The magnetic domains and THz field orientations. (d) Thewavevector parameter δ versus field with vertical lines show-ing their critical fields. (e) The energy per site E/N versusfield. (f) The magnetization M along the field direction. Thethin dashed line shows the nonzero intercept M ∗ . For (d), (e),and (f), the field is applied along [0,0,1], domain 1 is indicatedby solid curves and domains 2 or 3 by dashed curves. and we take C = 1. Notice that the unit vectors p ( n ) and tilt angles τ ( n ) can be different for layers 1 and 2.Four different phases γ ( n )1 and γ ( n )2 enter S x ′ ( R ) and S y ′ ( R ). In zero field, the higher odd harmonics C l +1 > in F ( n ) ( R x ′ ) are produced by either the anisotropy K orthe DM interaction D ′ . Even harmonics C l are producedby the magnetic field. Because C l falls off rapidly with l , we neglect harmonics above l = 4. For each layer, Γ ( n ) allows the even and odd harmonics to be out of phase.On layer n and site R , the amplitude A ( n ) ( R ) is fixedby the condition that | S ( R ) | = S , which is satisfied by aquadratic equation for A ( n ) ( R ). The lower root is usedfor layer 1; the upper root is used for layer 2.Fixing δ = 1 /q , where q ≫ E is mini-mized over the 17 variational parameters ( µ , τ ( n ) , γ ( n ) i ,Γ ( n ) , p ( n ) , C l ≤ , and s ) on a unit cell with q sites along x ′ and two hexagonal layers. An additional minimizationloop is then performed over q to determine the cycloidalwavevector as a function of field. In zero field, q = 222.We verify that the corresponding spin state provides atleast a metastable minimum of the energy E by checkingthat the classical forces on each spin vanish. Anothercheck is that the SW frequencies are all real.Bear in mind that the variational parameters are notfree but rather are functions of the interaction parameters D , D ′ , and K , and the magnetic field H . In zero field,the spin state reduces to the one used in Ref.[21]. Amuch simpler variational form for the spin state wouldhave been possible were the field oriented along the high-symmetry axis z ′ = [1 , ,
1] rather than along a cubicaxis.Although the number of variational parameters is farsmaller than the 4 q ≈
888 degrees of freedom for thespins in a unit cell, it may be possible to construct amore compact form for the spin state with fewer varia-tional parameters. Unlike a variational state with too fewparameters, however, a variational state with too manyparameters does not incur any penalty aside from theadditional numerical expense.With m = [0 , , | m · x ′ | and | m · y ′ | are the same fordomains 2 and 3. Therefore, the equilibrium and dynam-ical properties of domains 2 and 3 are identical. Fig.1(d)plots δ versus field for the three domains. The predictedcritical field H (2) c = 16 . H (1) c = 20 . H (2) c , thecycloid for domains 2 and 3 has a significantly longer pe-riod than the cycloid for domain 1. The variation of H (1) c with m was predicted for a purely harmonic cycloid andrecently reported for BiFeO .In zero field, all three domains have the same energy.But in a nonzero field, domain 1 has lower energy thandomains 2 and 3, as seen in Fig.1(e). At 5 T, the energydifference between domains is about 0.9 µ eV/site. Basedon a comparison between the measured and predictedspectroscopic frequencies discussed below, we conjecturethat domains 2 and 3 are depopulated above about 6 T.Assuming that the magnetic field is perpendicular to z ′ , the weak FM moment M of the AF phase canbe obtained by extrapolating the linear magnetization M ( H > H (2) c ) back to H = 0. In Ref.[21], the presumedmoment M = 0 . µ B of the AF phase was used to fix D ′ = M J /µ B S = 0 .
054 meV. For the tilted cycloid inzero field, the spin amplitude parallel to y ′ is then givenby S = M / µ B = 0 .
015 and the tilt angle τ is 0 . ◦ .But neither experimental group applied a magneticfield perpendicular to z ′ . As seen in Fig.1(f) for m =[0 , ,
1] and D ′ = 0 .
054 meV, the intercept M ∗ = 0 . µ B is then slightly smaller than the measured intercept M ∗ = 0 . µ B . Unlike M , M ∗ ( m ) depends on the orien-tation m of the magnetic field and reaches a maximum of M only when m · z ′ = 0 or when the field is in the (1 , , D ′ = 0 .
065 meV would produce the observed M ∗ ( m ) for m = [0 , , M ∗ are rather imprecise and because the predicted spec-troscopic frequencies evaluated using D ′ = 0 .
054 meVagree quite well with the measured frequencies. We shallreturn to this issue in the conclusion.As also indicated in Fig.1(f), the magnetization M ( H )of domains 2 and 3 is lower than that of domain 1.A hump in the magnetic susceptibility χ = dM/dH observed below 6 T may signal the depopulation of do-mains 2 and 3. III. SPECTROSCOPIC FREQUENCIES
Generally, the spin-spin correlation function S αβ ( q , ω )may be expanded in a series of delta functions at eachSW frequency ω m ( q ): S αβ ( q , ω ) = X m δ (cid:0) ω − ω m ( q ) (cid:1) S ( m ) αβ ( q ) , (6)which assumes that the SWs are not damped. The modefrequencies ω m ( q ) and the corresponding intensities S ( m ) αβ are solved by using the 1 /S formalism outlined in Ref.[27]and in Appendix A of Ref.[21]. With δ = 1 /q , the unitcell contains 2 q sublattices.Some of the SW modes are optically active with non-zero magnetic dipole (MD) matrix elements h δ | M | i ,where M = 2 µ B P i S i is the magnetization operator, | i is the ground state with no SWs, and | δ i is an excitedstate with a single SW mode at the cycloidal wavevec-tor Q . A subset of the MD modes have non-zero electricdipole (ED) matrix elements h δ | P ind | i , where the in-duced electric polarization P ind = λ X R i , R j = R i + e ij n x ′ × (cid:0) S i × S j (cid:1)o , (7)of BiFeO is produced by the inverse DMmechanism . Within each (1 , ,
1) plane, e ij = √ a x ′ connects spins at sites R i and R j . In the absence of tilt, h | S i × S j | i is parallel to y ′ and h | P ind | i is parallel to z ′ . Analytic expressions for h δ | M α | i and h δ | P ind α | i areprovided in Appendix B of Ref.[21]. There is no simplerelationship between the SW intensities S ( m ) αα ( δ ) at thecycloidal wavevector and the matrix elements h δ | M | i and h δ | P ind | i .For zero field with δ = 1 / the in-teraction parameters of BiFeO to fit the four spectro-scopic mode frequencies ν observed by Talbayev et al. .Fixing D ′ = 0 .
054 meV, we obtained the parameters D = 0 .
107 meV and K = 0 . .To label the spectroscopic modes at q = Q or η = δ , wehave modified the notation of de Sousa and Moore , whostudied the case where D ′ = K = 0 so that the cycloidis coplanar and purely harmonic. In an extended zone
0 0.2 0.4 0.6 0.8 1 η / δ Φ Ψ Φ Ψ Φ Ψ Ψ Φ Ψ Φ (b)
0 0.2 0.4 0.6 0.8 1 η / δ ω ( m e V ) (a) Φ Φ Ψ Φ Ψ Ψ / Φ Ψ Φ Ψ FIG. 2: (Color online) The mode frequencies versus q for (a)0 T and (b) 6.9 T in domain 1. Optically-active modes at η = δ are denoted by filled circles, inactive ones by whitecircles. Recall that 1 meV = 8.065 cm − . scheme, they labeled the SW modes at wavevector nQ as Ψ n and Φ n . Corresponding to excitations within thecycloidal plane, Φ n = Φ | n | is a linear function of n . Theout-of-plane modes satisfy the relation Ψ n = Φ √ n .Due to the higher harmonics of the cycloid producedby D ′ or K , Ψ n and Φ n ( n >
0) each split into two modesthat we label as Ψ (1 , n and Φ (1 , n .Any mode with a nonzero MD matrix element h δ | M α | i must also have a nonzero SW intensity S ( m ) ββ ( δ ) at the cy-cloidal wavevector. When D ′ = 0 and H = 0, the cycloidis coplanar and there is a sharp distinction between in-plane and out-of-plane modes. For a coplanar cycloid,the in-plane Φ n modes may have nonzero MD matrix el-ements with component α = y ′ and nonzero SW intensi-ties with components β = x ′ and z ′ . By contrast, out-of-plane Ψ n modes may have nonzero MD matrix elementswith components α = x ′ or z ′ and nonzero SW intensi-ties with component β = y ′ . When D ′ = 0, the cycloid istilted out of the x ′ z ′ plane but the distinction between thein-plane and out-of-plane modes is maintained, at leastfor the relatively small tilting angles considered here: theΦ n modes only have SW intensities S ( m ) ββ ( δ ) with β = x ′ and z ′ while the Ψ n modes only have SW intensity with β = y ′ . Of course, the distinction between in-plane andout-of-plane modes is lost in a magnetic field.In Fig.2, the SW frequencies are plotted versus q =(2 π/a )(0 . η, . − η, .
5) for domain 1 and H = 0 or 6.9T. The gaps between the Φ (1 , n> and Ψ (1 , n> modes at η = δ are enlarged in a magnetic field but the mode splittingsfall rapidly off with increasing n and cannot be seen forΦ (1 , and Ψ (1 , . Repulsion between SW branches alsooccurs away from η/δ = 0 or 1, such as at η/δ = 1 / with Ψ (1 , n > Φ (1 , n is restored.As shown in Fig.2(a) for zero field, only the six modesΦ , Ψ , Φ (1)1 , Ψ (2)1 , Ψ (1)1 , and Φ (1)2 are optically activeat η = δ . At a very small but nonzero frequency, Φ isoutside the range of THz measurements. For either D ′ =0 or K = 0, the anharmonicity of the cycloid splits Φ (2)1 ( ν = 10 . − ) from Φ (1)1 ( ν = 16 . − ) and Ψ (2)1 ( ν = 20 . − ) from Ψ (1)1 ( ν = 22 . − ). BesidesΦ , only Ψ (1)1 has a nonzero ED matrix element in zerofield. While Φ (1)2 ( ν = 27 . − ) is activated by the3 Q harmonic of the cycloid, which mixes Φ (1)2 with Φ ,Ψ and Φ (1)1 are activated by the tilt of the cycloid outof the x ′ z ′ plane, which mixes Ψ with Ψ (1)1 and Φ (1)1 with Φ . The nearly-degenerate Ψ and Φ (1)1 modes areresponsible for the observed spectroscopic peak at ν = 16 . − .In nonzero field, all of the SW modes at the cycloidalwavevector Q are optically active with nonzero MD ma-trix elements, as indicated in Fig.2(b) for 6.9 T. Noticethat the near degeneracy between Φ (1)1 and Ψ is brokenby the magnetic field.With m = [0 , , (1)1 ( H ) and Ψ ( H ) ( ν = 16 . − ) are splitlinearly by the field below about 4 T. For domain 1,Φ (1)1 ( H ) ≈ ν + 0 . µ B H and Ψ ( H ) ≈ ν − . µ B H ;for domains 2 and 3, the frequencies are slightly higherwith Φ (1)1 ( H ) ≈ ν + 1 . µ B H and Ψ ( H ) ≈ ν − . µ B H .Some magnon softening at Q occurs close to the criticalfields H ( i ) c for each domain.Spectroscopic frequencies measured by Nagel et al. are plotted in Fig.3(b). The THz magnetic field wasaligned along either h = [1 , − ,
0] or h = [1 , , e = [1 , ,
0] or e = [1 , − , h δ | h i · M | i and the ED ma-trix elements h δ | e i · P ind | i . The observed transition tothe AF phase occurs at about 18.9 T. Due to instrumen-tal limitations, no THz data is available for fields above12 T and frequencies below about 12 cm − . We believethat the energy difference between domains is responsi-ble for depopulating domains 2 and 3 above about 6 T,indicated by a dashed vertical line. To reflect this be-havior, we have cut off the predicted mode frequencies ofdomains 2 and 3 in Fig.3(b) above 6 T.The agreement between the measured and predictedmode frequencies in Fig.3(b) is astonishing. For smallfields, the slopes of Φ (1)1 ( H ) and Ψ ( H ) are quite close tothe predicted slopes for all three domains. The predictedsplitting of Φ (1 , ( H ) ( ν = 27 . − ) is clearly seen inFig.3(b). Also in agreement with predictions, Ψ (1)1 ( H )( ν = 22 . − ) is slightly lower in domains 2 and 3than in domain 1.However, our model cannot explain the field-independent excitation at about 16.5 cm − midway be-tween Φ (1)1 ( H ) and Ψ ( H ). Spectroscopic modes nevercross with field due to their coupling and mixing (al-though the coupling becomes very weak for some higher-frequency modes). Since it appears immune to mode no THz data H (T) ν ( c m - )
0 5 10 15 20 25 Φ Ψ Φ Ψ Ψ Ψ Φ Ψ Φ Φ Ψ Φ Ψ Φ Ψ Ψ Φ Ψ ν ( c m - ) (a) (b) α β α β FIG. 3: (Color online) (a) The predicted spectroscopic fre-quencies for domain 1 (solid) and domains 2 and 3 (dashed).The critical fields are indicated by dashed vertical lines. (b)The measured spectroscopic frequencies with THz field h (circles) or h (triangles). The predicted mode frequenciesfrom domain 1 (solid) and domains 2 and 3 (dashed) are alsoshown. We argue that contributions from domains 2 and 3stop at 6 T, indicated by a dashed vertical line. repulsion, the 16.5 cm − excitation may have some otherorigin, such as an optical phonon.In contrast to the domain depopulation indicated byTHz measurements, domains 2 and 3 appear to sur-vive up to about 16 T in electron spin resonance (ESR)measurements . As reported in Ref.[20], the predictedΦ (2)1 ( ν = 10 . − ) for domains 2 and 3 agrees quitewell with a mode detected by ESR measurements.We predict that the AF phase has two low-frequencymodes labeled α and β in Fig.3. As expected, α and β do not depend on the domain of the cycloid below thecritical field. Notice that β ( H ) is quite close to the Lar-mor frequency 2 µ B H for an isolated spin . For domains2 and 3, α ( H ) is predicted to vanish at the critical field H (2) c = 16 . indicate that α ( H ) ≈ . − at 16 T and that α ( H ) is projected to vanishbetween 10 and 12 T. This suggests that the true criticalfield H (2) c for domains 2 and 3 may be as low as 10 Tand that the spin state in those domains is metastablebetween 10 and 16 T. Even if the critical field for domains2 and 3 is 16 T, the depopulation of domains 2 and 3 at10 T would explain the optical anomalies observed atthat field. Above H (1) c , α ( H ) ∼ ( H − H (2) c ) / is sensitiveto the precise location of H (2) c , which may be shifted byquantum fluctuations or other interactions not includedin our model. IV. SPECTROSCOPIC SELECTION RULESAND INTENSITIES
In zero field, each optically-active mode is associatedwith a single MD component h δ | M α | i . Besides Φ , theoptically-active modes are:Ψ ( ν = 16 . − ) : |h δ | M x ′ | i| /µ B = 2 . (1)1 ( ν = 16 . − ) : |h δ | M y ′ | i| /µ B = 1 . (2)1 ( ν = 20 . − ) : |h δ | M z ′ | i| /µ B = 3 . (1)1 ( ν = 22 . − ) : |h δ | M x ′ | i| /µ B = 4 . (1)2 ( ν = 27 . − ) : |h δ | M y ′ | i| /µ B = 1 . (2)1 ( ν = 10 . − ) and Φ (2)2 ( ν = 27 . − ) are not optically active in zero field.The only mode with a nonzero ED matrix element in zerofield is Ψ (1)1 with |h δ | P ind y ′ | i| /λ = 12 . Q .For example, Φ (2)1 ( ν = 10 . − ) is not optically activeand has no SW intensity in zero field. But the SW inten-sities S αα ( δ ) plotted in Fig.4(a) for domain 1 grow like H . As shown in Fig.4(b), Φ (2)1 develops significant ma-trix elements |h δ | M x ′ | i| ∝ H and |h δ | M y ′ | i| ∝ H . De-spite the distortion of the cycloid in a magnetic field, Φ (2)1 remains primarily an in-plane cycloidal mode: S y ′ y ′ ( δ ) isquite small and h δ | M y ′ | i is the dominant MD matrixelement. But the significant matrix element h δ | M x ′ | i indicates that Φ (2)1 mixes with the nearby Ψ mode. Ex-perimentally, Φ (2)1 appears above about 3 T.Similar conclusions hold for Φ (1 , ( ν = 40 . − )and Ψ (1 , ( ν = 43 . − ), which are also activatedby the field and appear above about 5 T. The predictedsplitting of both modes can be observed above 10 T.Generally, the spectroscopic intensities of any mode inTHz fields h i and e i ( i = 1 or 2) are given byMD( h i ) = (cid:12)(cid:12)(cid:12) h δ | h i · M | i /µ B (cid:12)(cid:12)(cid:12) , (8) H (T)
0 1 2 3 4 5 6 | < δ | M α | > | / µ B α = z´ x´ y´ S αα ( δ ) Φ , ν = 10.8 cm -1 (a) (b) FIG. 4: (Color online) The field dependence of (a) the SWintensities S αα ( δ ) (with a different scale used for α = y ′ ) and(b) the matrix elements |h δ | M α | i| /µ B , where α = x ′ (solid), y ′ (medium dash), or z ′ (small dash) for Φ (2)1 in domain 1. ED( e i ) = (cid:12)(cid:12)(cid:12) h δ | e i · P ind | i /λ (cid:12)(cid:12)(cid:12) . (9)These expressions generalize those given in Ref.[21] forzero field, when each mode was associated with only asingle matrix element h δ | M α | i . The total spectroscopicintensity is a function of MD( h i ) and ED( e i ) that mayalso involve the non-reciprocal cross term containingthe product h δ | h i · M | i h | e i · P ind | δ i . We expect thatMD( h i ) dominates the spectroscopic intensity becausethe induced polarization for BiFeO is so small. But mea-surement of non-circular magnetic dichroism under anexternal magnetic field along z ′ can, at least in principle,be used to isolate ED( e i ) for any mode.To evaluate the spectroscopic weights, we must ex-press h i and e i in terms of the local coordinate system { x ′ , y ′ , z ′ } of the cycloid in each domain: h = x ′ , h = ( y ′ + √ z ′ ) / √ , (10)in domain 1 with x ′ = [1 , − ,
0] and y ′ = [1 , , − h = x ′ / − √ y ′ / , h = x ′ / √ y ′ / p / z ′ , (11) in domain 2 with x ′ = [1 , , −
1] and y ′ = [ − , , − h = − x ′ / − √ y ′ / , h = x ′ / − √ y ′ / p / z ′ , (12)in domain 3 with x ′ = [0 , , −
1] and y ′ = [ − , , e = h and e = h .The MD and ED weights of the first seven modes aboveΦ are plotted versus field in Fig.5. Because they haveno appreciable ED matrix elements, the ED weights ofΦ (2)1 and Ψ are not shown. In domain 1 with e = x ′ ,ED( e ) = 0 because P ind has no component parallel to x ′ . The sharp features in these figures can be attributedto the avoided crossings of the spectroscopic modes withfield. Experimentally, the contributions of domains 2 and3 can be suppressed by applying and then removing ahigh field above H (1) c .As shown in Fig.5(a) for Φ (2)1 ( ν = 10 . − ),MD( h i ) is much larger for domains 2 and 3 than for do-main 1. Within domain 1, MD( h i ) is stronger for THzfield h due to the dominant matrix element |h δ | M y ′ | i| plotted in Fig.4(b). Since |h δ | M y ′ | i| grows linearly withfield, MD( h ) ≈ |h δ | M y ′ | i| / and Ψ (1)1 . By 10 T, the ED intensity ofΨ (1)1 has fallen by about 66% while the ED intensitiesof several other modes have become significant. For do-main 1, we predict that the ED intensity of Φ (2)2 becomescomparable to that of Ψ (1)1 at about 10 T.However, a close comparison with measurements re-veals that the intensities of some activated modes areunderestimated by our model . For example, after aver-aging over domains, MD( h ) for Φ (1 , is predicted to beabout 25 times smaller than MD( h ) for Ψ (1)1 . But exper-imentally, Φ (1 , has twice the intensity of Ψ (1)1 . For THzfield orientation h , Fig.5(e) predicts that the MD inten-sity of Ψ (2)1 should vanish at H = 0. But experiments indicate that Ψ (2)1 survives for THz field orientation h inzero field, albeit with the h intensity reduced by about90% compared to the h intensity.Experimentally , the h intensities of Φ (1)2 and Φ (2)2 at H = 0 are larger for the field-treated sample than forthe non-field-treated sample. This implies that MD( h )is larger for domain 1 than for domains 2 and 3. But theonly nonzero MD matrix element for Φ (1)2 in zero field is h δ | M y ′ | i . So as shown in Figs.5(h) and (j) for Φ (1)2 andΦ (2)2 , MD( h ) = |h δ | M x ′ | i| → H → V. CONCLUSION
The remarkable agreement between the predicted andmeasured spectroscopic mode frequencies of the cycloidalphase leaves no doubt that a model with DM interactions M D ( h i ) M D ( h i ) M D ( h i ) M D ( h i ) M D ( h i ) M D ( h i ) M D ( h i ) E D ( e i ) E D ( e i ) E D ( e i ) E D ( e i ) E D ( e i ) H (T)
H (T) (a) (c) (e) (f) (h) (j) (b) (d) (f) (g) (i) (k) Φ , ν = 10.8 cm -1 Ψ , ν = 22.2 cm -1 Ψ , ν = 20.4 cm -1 Φ , ν = 16.5 cm -1 Ψ , ν = 16.4 cm -1 Φ , ν = 27.4cm -1 Φ , ν = 27.4cm -1 M D ( h i ) (c) Φ , ν = 16.5 cm -1 ( ) , (j) E D ( e i ) (i) ( i ) (b) Ψ , ν = 16.4 cm -1 ( ) , FIG. 5: (Color online) The spectroscopic intensities MD( h i )and ED( e i ) versus field for the lowest 7 modes. Domain 1(solid) and domains 2 and 3 (dashed) are indicated along withTHz fields polarizations i = 1 (thick curve) and 2 (thin curve).Side by side MD and ED plots refer to the same mode, indi-cated on the left. along y ′ and z ′ and easy-axis anisotropy along z ′ providesthe foundation for future studies of multiferroic BiFeO .However, the previous section exposed several discrepan-cies between the predicted and observed mode intensitieswhich must be addressed. Specifically, modes that are ac-tivated by the anharmonicity and tilt of the cycloid arestill too weak compared to measurements. Whereas ourmodel predicts that Φ (1 , should not appear in zero fieldfor domain 1 with THz field h , experiments indicatethat Φ (1 , are actually stronger in domain 1 than in do-mains 2 and 3.As mentioned above, we have used a smaller value of D ′ than warranted by the observed, weak FM moment M of the AF phase. For magnetic field along a cubicaxis, D ′ = 0 .
054 meV corresponds to the zero-field inter-cept M ∗ = 0 . µ B , smaller than the intercepts 0 . µ B and 0 . µ B obtained by Tokunaga et al. and Park etal. , respectively. Our value S = 0 .
015 for the cy-cloidal spin amplitude parallel to y ′ is roughly half whatRamazanoglu et al. estimated from elastic neutron-scattering measurements. Recall that the weak FM mo-ment of the AF phase is predicted to be M = 2 µ B S .A larger value for D ′ requires a commensurately largervalue for the anisotropy K to preserve the same zero-fieldsplittings of Φ (1 , and Ψ (1 , produced by the anhar-monicity of the cycloid. For example, when S = 0 . D ′ = 0 .
090 meV, the best fits to the zero-field fre-quencies are obtained with K = 0 . τ = 0 . ◦ when S = 0 . τ = 0 . ◦ when S = 0 . found that the matrix elements h δ | M x ′ | i and h δ | M y ′ | i of the tilt-activated modes Ψ and Φ (1)1 ( ν = 16 . − ) scale like S in zero field. Sothe intensities MD( h i ) of Ψ and Φ (1)1 are larger by a fac-tor of 25 / ≈ . S = 0 .
025 than for S = 0 . D ′ and K do not resolve the most serious discrep-ancies between the predicted and measured intensities inzero field. In particular, they do not generate nonzeromatrix elements h δ | M x ′ | i for the in-plane Φ (1 , modesor for the out-of-plane Ψ (2)1 mode at H = 0.Another set of weak interactions may possibly explainthe enhanced spectroscopic intensities. There are at leasttwo candidates for such interactions. The small rhombo-hedral distortion ( α = 89 . ◦ ) of BiFeO will change thenext-nearest neighbor exchange J within each hexag-onal plane compared to the interaction between differ-ent planes. Due to magnetoelastic coupling, easy-planeanisotropy perpendicular to y ′ may compete with the D ′ interaction, permitting much larger values for D ′ con-sistent with the observed moment M of the AF phase.Either set of additional interactions may modify the MDmatrix elements and change the spectroscopic intensitiesof the activated modes.To conclude, the spectroscopic frequencies and intensi-ties provide very sensitive probes of the weak microscopicinteractions that control the cycloid and induced polar-ization in BiFeO . We are confident that future workbased on the model presented in this paper will lay thegroundwork for the eventual technological applications ofthis important material.I gratefully acknowledge conversations with Nobuo Fu-rukawa, Masaaki Matsuda, Shin Miyahara, Jan Musfeldt, Urmas Nagel, Satoshi Okamoto, Toomas R˜o˜om, Rogeriode Sousa, and Diyar Talbayev. 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