Longitudinal spin fluctuations in the antiferromagnet MnF

Longitudinal spin fluctuations in the antiferromagnet MnF

studied by polarized neutron scattering

W. Schweika1, S. V. Maleyev1,2, Th. Br¨uckel1, V. P. Plakhty1,2

and L.-P. Regnault3

1 Institut f¨ur Festk¨orperforschung des Forschungszentrums J¨ulich 52425 J¨ulich, Germany

2 Petersburg Nuclear Physics Institute - Gatchina, St. Petersburg 188300, Russia

3 CEA-Grenoble, DRFMC-SPSMS-MDN - 38054 Grenoble Cedex 9, France

(received 28 May 2002; accepted 13 August 2002) PACS.75.30.Ds – Spin waves.

PACS.75.50.Ee – Antiferromagnetics.

PACS.75.40.Gb – Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dy- namic scaling, etc.).

Abstract. – In neutron scattering experiments using polarization analysis we separated the spectra of transverse and longitudinal magnetic fluctuations in the anisotropic antiferromagnet MnF₂. While transverse modes are related to single-magnon scattering, the longitudinal part is essentially due to two-magnon scattering. They were measured atT = 30 K and 50 K well below the N´eel temperatureTN= 67 K. The dynamic magnetic response due to two-magnon creation or annihilation is separated by a gap centered near the spin-wave frequency from the central peak corresponding to neutron-magnon scattering (creation of one magnon and annihilation of another). The longitudinal energy spectrum extends to about twice the frequency of the zone boundary modes. This tail at high energies is fairly independent of momentum transfer.

The observed longitudinal spectra are in qualitative agreement with the theory for two-magnon processes and are determined for large energy transfersωby the density of statesD(ω/2).

Introduction. – Well below the N´eel temperature T_N the transverse spin fluctuations, namely spin-waves, are well studied both theoretically and experimentally in a large amount of antiferromagnets (AF). Particularly, it was shown in the seminal paper by Harriset al. [1]

that the interaction of the spin-waves is very weak and slightly renormalizes the spin-wave velocity. Hence, the linear spin-wave theory is a very good approximation for the description of the experimental data. However, there have to be longitudinal spin fluctuations (LSF) (those along the sublattice magnetization), which consist of excitation and absorption of an even number of spin-waves [2].

In this study we investigated experimentally the LSF below the critical region in the an- tiferromagnet MnF₂ (T_N = 67 K). We have chosen this material as it is very well studied experimentally. Particularly, its spin-wave spectrum and the critical fluctuations were inves- tigated in detail [3–8]. We obtain that there is a semi-quantitative agreement between the

c EDP Sciences

two-magnon theory and the experiment. Unfortunately, the accuracy of our data does not allow us to make more precise comparisons.

As is well known, the magnetic scattering cross-section is proportional to (k_f/k_i)S(Q, ω) wherek_f(i) and Qdenote the final (initial) wave vector and the momentum transfer, respec- tively. S(Q, ω) is the van Hove scattering function related to the susceptibilityχ(Q, ω) by the simple expressionπ(1−e^−ω/T)S(Q, ω) = Imχ(Q, ω), where in our caseχ is the longitudinal spin susceptibility (LSS) and Imχis an odd function ofω. There are two processes that con- tribute to the scattering function: i) two-magnon excitation (absorption) and ii) absorption of one magnon and excitation of another one, which can be considered as neutron-magnon scattering. Excitations take place at all temperatures includingT = 0, where absorption and scattering disappears. We consider both parts ofS separately and show that they take place in different parts of the (Q, ω)-space. Details of such calculations for the two-dimensional case can be found in [9]. Below we discuss the final results only. It should be noted also that a gap in the longitudinal frequency spectrum was predicted from Monte Carlo simula- tions [10]. Quantitative comparisons, however, will not be particularly meaningful, because in these simulations severe finite-size effects determine the observable magnon spectrum.

Two-magnon excitations and absorption. For qa⁻¹, where q is the distance from the AF Bragg point anda is the lattice spacing, and forω SJ Z = SJ0, where S is the spin value andZ is the number of nearest neighbors, we have

Sex(ω) = V₀(SJ₀)² 2(2π)³

dk³

E1E2(N1+ 1)(N2+ 1)δ(ω−E1−E2), Sabs(ω) = V0(SJ0)²

2(2π)³

dk³

E1E2N1N2δ(ω+E1+E2), (1) whereV0 is the unit cell volume; the indices 1 and 2 are short notations for the magnon wave vectorsk+q/2, and k−q/2, respectively, whereasE_q =

(cq)²+ ∆² determines the spin- wave energy. Here,c= SJ₀a/√

3 is the spin-wave velocity, andN(E) = (exp[E/T]−1)⁻¹is the Planck function ofE. It is easy to show that the neutron can excite two spin-waves, ifω is larger than the threshold energy

ωth(q) = 2E_q/2=

(cq)²+ 4∆², (2)

determined as the minimum of E(k1) +E(k2), with q = k1+k2. At the same time the two-magnon absorption increases the neutron energy on the amount larger than ωth(q). In other words the two-magnon continuum holds atω <−ωth(q) andω > ωth(q).

AtT much larger than ∆ the two-magnon excitation (annihilation) spectra should have a smooth maximum atω≈E(q)+∆ (andω≈ −E(q)−∆), which corresponds to the excitation (annihilation) of a magnon withk1≈q and magnons near the zone centerk2≈0having the highest thermal population.

In the limit ofq1/aandω >0,S_exmay be represented in the following simple form:

S_ex(ω) =2(SJ0)² ω²

N

ω

2

+ 1 ₂

D

ω

2

, (3)

whereD(ω) is the magnon density of states determined as D(ω) = V₀

(2π)³

d³kδ(ω−E_k), (4)

and a similar expression is valid for the annihilation of two magnons forω <0. Indeed, eq. (3) holds ifE_q ω,i.e.the tail of the two-magnon continuum should beq-independent. We shall see below that this is the case in our experimental data. Further discussions of the general case of q = 0 based on analytical evaluations and convoluted with the resolution are hardly reasonable at present, due to the low accuracy of our data. However, at least forT ωth(q) we can replaceN byT/Eand obtain readily a strong enhancement of the scattering intensity with increasing temperature.

Neutron-magnon scattering, creation and annihilation. Again atq1/afor the scattering case we have

Ssc(ω) =V0(SJ0)² 2(2π)³

dk³

E1E2N1(N2+ 1)δ(ω+E1−E2), (5) where the indices 1 and 2 stand fork+q/2 and 2 =k−q/2, respectively. In this expression all k are allowed, and at k q we have E₁−E₂ =cq. This value determines theω-range of the two-magnon scattering: −cq < ω < cq. The quantity cq can be calculated from the single-magnon excitation energyE_q,

cq=

E_q²−∆². (6)

Again, we cannot evaluate Ssc for the general case. We can only calculate Ssc(q, ω) in the limit ofωcq, ifT is much larger thancq andδ, and obtain

Ssc(q, ω) = 3√ 3T²

π³SJ0(cq)². (7)

This simple expression will be used below.

Experimental. – We choose a coordinate system with thex-axis alongQ, and thez-axis perpendicular to the scattering plane, and the sublattice magnetizations of the ordered MnF₂ crystal parallel to they-axis, see fig. 1.

In this case, the longitudinal fluctuations are determined from the difference d²σ

dΩdω

LSF

= d²σ

dΩdω _nsf

Py

− d²σ

dΩdω _nsf

Px

, (8)

without any contribution from the transverse fluctuations or nuclear background.

Here,PxandPydenote a neutron polarization along thexandyaxes, respectively, while nsf stands for non–spin-flip processes. In an analogous manner, the transverse fluctuations are measured by the spin-flip scattering.

The neutron scattering experiments have been performed on the triple-axes spectrometer IN22 at the ILL, Grenoble. A sample of MnF₂ with a volume of about 1.3 cm³ was prepared by gluing 8 smaller single crystalline grains together on an Al support plate. The sample was mounted in a cryostat with the tetragonal a and c axes in the horizontal scattering plane. A Heusler crystal was used to polarize the incident beam. The final polarization was also determined by a Heusler crystal in combination with a π-spin flipper, which allows for measuring in fixed k_f geometry, and to suppress higher-order contaminations by a graphite filter. The desired neutron polarization at the sample position was achieved by appropriate nutation due to magnetic fields of vertical and horizontal coils around the sample position.

Comparing the spin-flip and non–spin-flip intensities at the single-magnon energies, a flipping

Q

ax bz

cy Sy

L Tx

Tz

✑✑✑✑✑✑✑✑✸

✻

❍❍❍❍

❍❍❍❍❥

✑✑✑✑✑✑✸

❍❍❍❍❍❥❍❍❥✑✑✻✸

Fig. 1 – Scattering geometry for measuring longitudinal fluctuations alongQ= (1 +q,0,0)(in thexy scattering plane), which show up in the non–spin-flip intensity if the neutron polarization is parallel toy (⊥Q),i.e.the axis of the preferred spin orientation. The nuclear background is determined by putting the neutron polarization parallel tox(Q), while in both cases the transverse modes do not contribute. Here, the crystallographic axesaandcof MnF₂are parallel toxandy, respectively. Sy

denotes the spin component of one sublattice pointing in the axis of the staggered magnetization,Tz

andTx are the transverse components of a spin-wave, andLrepresents a longitudinal fluctuation of Sy, which, however, does not exist in the linear spin-wave theory.

ratio of 19±1 was found. A similar result was observed for the Debye-Scherrer ring (200) of the aluminum sample holder. Furthermore, on this nuclear scattering signal we checked the magnetic-field variation parallel and perpendicular to the scattering vectorQ, which did not cause any significant changes in the flipping ratio within 3% accuracy. The energy resolution (FWHM) was 1.3 meV atω= 0 forkf = 2.662 ˚A⁻¹, increasing withω.

Results. – The measured longitudinal-fluctuation spectra are displayed in fig. 2. The data have been corrected for the wavelength-dependent monitor sensitivity for the incoming neu- tron flux. As compared to the transverse spin-waves, which are also shown (open symbols) on a reduced scale, the signal due to the longitudinal spin fluctuations (LSF) is a rather weak con- tinuous spectrum in energy. The characteristic features are the wave-vector– and temperature–

dependent central peak and the dip underneath the relatively strong peak due to single spin- wave excitation. Evidently, without polarization analysis this feature would not have been seen. The gap is affected by the experimental resolution, in particular, for the data taken at T = 50 K. Apart from this effect, the directional dependence of the spin-wave branches in MnF₂ leads to an intrinsic smearing of the gap at higher q. For h= (1.15,0,0), where we measured also in the neutron energy gain mode (ω < 0), we observed equivalent minima on both sides. Taking into account the Bose statistics and the (k_f/k_i) dependence of the dynamic scattering cross-section, we obtain a longitudinal susceptibility that is antisymmetric in energy.

According to eq. (3), which holds for small q(1/a) the high-energy tail of the LSF is expected to be q-independent and should in principle reflect the magnon density of states, more precisely ofD(^ω₂). With regard to the Bose statistics, indeed, as expected forD∝ω² a typical increase for low ω is found in all measured spectra, and significant intensities are observed up to approximately 10 meV, which agrees nicely with twice the energy of the highest magnon energies at the zone boundary. Note that in fig. 3, we have also included the data measured at 30 K reweighting them using the Bose statistics for comparison with other spectra

-5 0 5 10 ω [meV]

h=(1.15,0,0) T=50K

LSF magnon (x0.1)

0 5 10 ω [meV]

h=(1.25,0,0) T=50K

LSF magnon (x0.1)

0 5 10 ω [meV]

0 0.2 0.4 0.6 0.8 1 1.2

counts / mon

h=(1.1,0,0) T=30K

LSF magnon (x0.2)

Fig. 2 – Longitudinal spin fluctuation spectra, as measured for three different scattering vectors and two temperatures,T = 30 K and 50 K. The predicted gap (gray-shaded area)separating the regions of two-magnon excitation (absorption)from magnon scattering coincides with the observed minima.

For comparison, the measured transverse magnon peaks are also shown, however, on a reduced scale (open symbols).

measured at 50 K. This common representation of the data confirms instructively that there is a gap in the LSF near the single-magnon excitations. With respect to any further detailed comparisons with the measured low-temperature density of states (Nicotin [8]), one has to account for that, in addition to the limited statistical accuracy and resolution of our results, any structures due to van Hove singularities apparently may fade out at higher temperatures.

Discussion and conclusion. – From our results we can conclude that the observed lon- gitudinal susceptibility is essentially in agreement with scattering processes involving two

0 5 10

ω [meV]

-0.2 0 0.2 0.4 0.6 0.8 1

counts/mon

h=(1.25,0,0) h=(1.15,0,0) h=(1.05,0,0) [h=(1.1,0,0)]

D(ω)~ω²

Fig. 3 – The high-energy tail of the longitudinal frequency spectrum is approximatelyq-independent and is determined by the density of states D(ω/2). Its scaling behaviour with temperature is in agreement with eq. (3). The dashed line, corresponding to D(ω/2) ∝ ω², indicates losses in the LSF observed near the gap. (The data measured at 30 K, at h = (1.1,0,0), have been reweighted according to the Bose statistic at 50 K for comparison.)

0

q a 0

6

ω [meV]

two magnon excitation (absorption)

magnon scattering

E

π ω_th=(E²+3∆²)^1/2

cq=(E²-∆²)^1/2

(creation and annihilation)

∆

Fig. 4 – Dispersion of the single-magnon excitationE(q)and regions of the longitudinal two-magnon spectra. The points denote the measured spin-waves at 50 K. The regions of possible excitation of two magnons and scattering are limited by twice the energy of the highest single-magnon excitation (zone boundary mode), and as derived for lowq, by the threshold energyωth and cq, respectively.

Therefore, a gap in the longitudinal (two-magnon)response, albeit fading out at higherq, is to be found near to the much stronger scattering due to single-magnon excitations.

magnons. We have found evidence for the gap, which is separating the two-magnon excita- tion (absorption) from the scattering region as displayed in fig. 4. The observed longitudinal response near the elastic line displays a temperature– and wave-vector–dependence which is in qualitative agreement with the expected two-magnon scattering, see eq. (7). For further quantitative comparisons the numerical integration of the two-magnon processes based on spin-wave models is possible; however, with respect to the low accuracy of the data this is hardly sensible at present. Contributions due to critical fluctuations are expected to be still weak,T ≤50 K, since the reduced temperatureτ−≥(TN−T)/TN= 0.25 is large.

From the naive point of view the two spin-wave processes should saturate the longitudinal magnetic susceptibility and higher-order processes can be neglected. According to [9, 11], this is not necessarily the case for the longitudinal staggered susceptibility (LSS)χ_AF(q, ω). It was shown that in the isotropic exchange approximation at q = 0 and T = 0 there is an infrared divergence (IRD) of the longitudinal susceptibility; in the 3D case, Imχ ∝sgn(ω) and Reχ ∝ln(J/ω), where J is the exchange energy. For nonzero T and ω T, we have Imχ∝1/ω and ReχImχ. For energiesω of the order ofT, there is a crossover between the two regimes. It means that in the longitudinal scattering signal the interaction between spin-waves is expected to become strong, processes involving many spin-waves should screen this IRD, and theχAF(q, ω) has to possess a weaker singularity than predicted by the two- magnon theory. Unfortunately, there is not any existing theory describing this screening. The IRD does not appear in the uniform susceptibility due to cancellations connected to the total- spin conservation law (TSCL) which holds in the exchange approximation. A similar situation occurs for ferromagnets. Magnetic dipolar interactions violate the TSCL, and the IRD appears in the form as above [12]. Unfortunately, in both cases the problem of the screening of the IRD has not been solved theoretically, although it was demonstrated experimentally in the case of a ferromagnet [13]. Magnetic anisotropy suppresses the IRD and in the above expressionsω should be replaced by the spin-wave gap ∆. However, if the ratioJ/∆ is large, the problem of the screening remains. In the present case of MnF₂ the IRD is strongly suppressed as the condition ∆J is actually not fulfilled. Therefore, one can expect that the LSS is saturated

by two-magnon processes, as demonstrated for the present case. The IRD is expected to become stronger in lower dimensions [9, 11]. Hence, with respect to the screening of the IRD, it will be very interesting to measure the LSF in 2D antiferromagnets with very weak anisotropy in future experiments.

∗ ∗ ∗

The work was supported by RFBR (Grants No. 00-02-16873, 02-02-16981, 00-15-96814) and Russian state programs: “Collective and quantum effects in condensed matter” and

“Quantum macroscopics”. We wish to thank S. Freudenstein for support during the ex- periments, Dr. J. Baruchel for providing us with single crystals, and Dr. B. Toperverg for his interest and discussions. SVM and VPP are thankful to the Forschungszentrum J¨ulich for hospitality.

REFERENCES

[1] Harris A. B., Kumar D., Halperin B. I.andHohenberg P. C.,Phys. Rev. B,3(1971)961.

[2] In zero magnetic field the Hamiltonian is an even function of the spin-wave creation and annihi- lation operators. As a result, the interaction between spin-waves cannot change evenness or odd- ness of the intermediate states. So, this quality is determined by the structure of the spin opera- tors only. The transverse operatorsSxandSyare odd and the longitudinal operatorSz is even.

[3] Low G. G., Okazaki A., Stevenson R. W. H.andTurberfield K. C.,J. Appl. Phys.,35 (1964)998.

[4] Okazaki A., Turberfield K. C.andStevenson R. W. H.,Phys. Lett.,8(1964)9.

[5] Turberfield K. C., Okazaki A.andStevenson R. W. H.,Proc. Phys. Soc. (London),85 (1965)743.

[6] Schulhof M. P., Heller P., Nathans R.andLinz A.,Phys. Rev. B,1(1970)2304.

[7] Schulhof M. P., Nathans R., Heller P.andLinz A.,Phys. Rev. B,4(1971)2254.

[8] Nicotin O., Lindgard P. A.andDietrich O. W.,J. Phys. C,2(1969)1168.

[9] Braune S.andMaleyev S. V.,Z. Phys. B,81(1990)69.

[10] Bunker A.andLandau D. P.,Phys. Rev. Lett.,85(2000)2601.

[11] Kubo R.,Phys. Rev.,87(1952)568.

[12] Toperverg B. P.andYashenkin A. G.,Phys. Rev. B,48(1993)16505.

[13] Luzyanin I. D., Yashenkin A. G., Maleyev S. V., Zaitseva E. A.andKhavronin V. P., Phys. Rev. B,60(1999)R734.