Longitudinal spin fluctuations in the antiferromagnet MnF 2 studied by polarized neutron scattering

Transcription

1 EUROPHYSICS LETTERS 1 November 22 Europhys. Lett., 6 (3), pp (22) Longitudinal spin fluctuations in the antiferromagnet MnF 2 studied by polarized neutron scattering W. Schweika 1,S.V.Maleyev 1,2,Th.Brückel 1,V.P.Plakhty 1,2 and L.-P. Regnault 3 1 Institut für Festkörperforschung des Forschungszentrums Jülich Jülich, Germany 2 Petersburg Nuclear Physics Institute - Gatchina, St. Petersburg 1883, Russia 3 CEA-Grenoble, DRFMC-SPSMS-MDN Grenoble Cedex 9, France (received 28 May 22; accepted 13 August 22) PACS Ds Spin waves. PACS Ee Antiferromagnetics. PACS Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.). Abstract. In neutron scattering experiments using polarization analysis we separated the spectra of transverse and longitudinal magnetic fluctuations in the anisotropic antiferromagnet MnF 2. While transverse modes are related to single-magnon scattering, the longitudinal part is essentially due to two-magnon scattering. They were measured at T = 3 K and 5 K well below the Néel temperature T N = 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 transfers ω by the density of states D(ω/2). Introduction. Well below the Néel 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 Harris et 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 () (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 below the critical region in the antiferromagnet MnF 2 (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 investigated in detail [3 8]. We obtain that there is a semi-quantitative agreement between the c EDP Sciences

2 W. Schweika et al.: Longitudinal spin fluctuations in MnF 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) where k f(i) and Q denote the final (initial) wave vector and the momentum transfer, respectively. S(Q,ω) is the van Hove scattering function related to the susceptibility χ(q,ω) by the simple expression π(1 e ω/t )S(Q,ω)=Imχ(Q,ω), where in our case χ is the longitudinal spin susceptibility (LSS) and Im χ is an odd function of ω. There are two processes that contribute 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 including T =, where absorption and scattering disappears. We consider both parts of S 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 simulations [1]. 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 q a 1, where q is the distance from the AF Bragg point and a is the lattice spacing, and for ω S JZ =SJ, where S is the spin value and Z is the number of nearest neighbors, we have S ex (ω) = V (SJ ) 2 2(2π) 3 S abs (ω) = V (SJ ) 2 2(2π) 3 dk 3 E 1 E 2 (N 1 + 1)(N 2 +1)δ(ω E 1 E 2 ), dk 3 E 1 E 2 N 1 N 2 δ(ω + E 1 + E 2 ), (1) where V is the unit cell volume; the indices 1 and 2 are short notations for the magnon wave vectors k + q/2, and k q/2, respectively, whereas E q = (cq) determines the spinwave energy. Here, c =SJ a/ 3 is the spin-wave velocity, and N(E) =(exp[e/t] 1) 1 is the Planck function of E. It is easy to show that the neutron can excite two spin-waves, if ω is larger than the threshold energy ω th (q) =2E q/2 = (cq) , (2) determined as the minimum of E(k 1 )+E(k 2 ), with q = k 1 + k 2. At the same time the two-magnon absorption increases the neutron energy on the amount larger than ω th (q). In other words the two-magnon continuum holds at ω< ω th (q) andω>ω th (q). At T much larger than the two-magnon excitation (annihilation) spectra should have a smooth maximum at ω E(q)+ (and ω E(q) ), which corresponds to the excitation (annihilation) of a magnon with k 1 q and magnons near the zone center k 2 having the highest thermal population. In the limit of q 1/a and ω>, S ex may be represented in the following simple form: S ex (ω) = 2(SJ ) 2 [ ( ) 2 ω ω 2 N +1] D 2 ( ω 2 ), (3) where D(ω) is the magnon density of states determined as D(ω) = V (2π) 3 d 3 kδ(ω E k ), (4)

3 448 EUROPHYSICS LETTERS and a similar expression is valid for the annihilation of two magnons for ω<. Indeed, eq. (3) holds if E q ω, i.e. the tail of the two-magnon continuum should be q-independent. We shall see below that this is the case in our experimental data. Further discussions of the general case of q 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 for T ω th (q) we can replace N by T/E and obtain readily a strong enhancement of the scattering intensity with increasing temperature. Neutron-magnon scattering, creation and annihilation. Again at q 1/a for the scattering case we have S sc (ω) = V (SJ ) 2 2(2π) 3 dk 3 E 1 E 2 N 1 (N 2 +1)δ(ω + E 1 E 2 ), (5) where the indices 1 and 2 stand for k + q/2 and2=k q/2, respectively. In this expression all k are allowed, and at k q we have E 1 E 2 = cq. This value determines the ω-range of the two-magnon scattering: cq < ω < cq. The quantity cq can be calculated from the single-magnon excitation energy E q, cq = E 2 q 2. (6) Again, we cannot evaluate S sc for the general case. We can only calculate S sc (q,ω)inthe limit of ω cq, ift is much larger than cq and δ, and obtain This simple expression will be used below. S sc (q,ω)= 3 3T 2 π 3 SJ (cq) 2. (7) Experimental. We choose a coordinate system with the x-axis along Q, and the z-axis perpendicular to the scattering plane, and the sublattice magnetizations of the ordered MnF 2 crystal parallel to the y-axis, see fig. 1. In this case, the longitudinal fluctuations are determined from the difference ( d 2 ) ( σ d 2 ) nsf ( σ d 2 ) nsf σ =, (8) dωdω dωdω P y dωdω P x without any contribution from the transverse fluctuations or nuclear background. Here, P x and P y denote a neutron polarization along the x and y axes, 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 MnF 2 with a volume of about 1.3 cm 3 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

4 W. Schweika et al.: Longitudinal spin fluctuations in MnF b z a x Q T z T x S y L c y Fig. 1 Scattering geometry for measuring longitudinal fluctuations along Q =(1+q,, )(in the xy scattering plane), which show up in the non spin-flip intensity if the neutron polarization is parallel to y ( Q), i.e. the axis of the preferred spin orientation. The nuclear background is determined by putting the neutron polarization parallel to x ( Q), while in both cases the transverse modes do not contribute. Here, the crystallographic axes a and c of MnF 2 are parallel to x and y, respectively. S y denotes the spin component of one sublattice pointing in the axis of the staggered magnetization, T z and T x are the transverse components of a spin-wave, and L represents a longitudinal fluctuation of S y, 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 (2) of the aluminum sample holder. Furthermore, on this nuclear scattering signal we checked the magnetic-field variation parallel and perpendicular to the scattering vector Q, which did not cause any significant changes in the flipping ratio within 3% accuracy. The energy resolution (FWHM) was 1.3 mev at ω =fork f =2.662 Å 1, increasing with ω. 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 neutron 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 () is a rather weak continuous 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 spinwave 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 = 5 K. Apart from this effect, the directional dependence of the spin-wave branches in MnF 2 leads to an intrinsic smearing of the gap at higher q. For h =(1.15,, ), where we measured also in the neutron energy gain mode (ω <), 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 is expected to be q-independent and should in principle reflect the magnon density of states, more precisely of D( ω 2 ). With regard to the Bose statistics, indeed, as expected for D ω2 a typical increase for low ω is found in all measured spectra, and significant intensities are observed up to approximately 1 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 3 K reweighting them using the Bose statistics for comparison with other spectra

5 45 EUROPHYSICS LETTERS h=(1.1,,) T=3K h=(1.15,,) T=5K h=(1.25,,) T=5K 1.2 counts / mon magnon (x.2) magnon (x.1) magnon (x.1) Fig. 2 Longitudinal spin fluctuation spectra, as measured for three different scattering vectors and two temperatures, T = 3 K and 5 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 5 K. This common representation of the data confirms instructively that there is a gap in the 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 longitudinal susceptibility is essentially in agreement with scattering processes involving two 1 counts/mon h=(1.25,,) h=(1.15,,) h=(1.5,,) [h=(1.1,,)] D(ω)~ω Fig. 3 The high-energy tail of the longitudinal frequency spectrum is approximately q-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) ω 2, indicates losses in the observed near the gap. (The data measured at 3 K, at h =(1.1,, ), have been reweighted according to the Bose statistic at 5 K for comparison.)

6 W. Schweika et al.: Longitudinal spin fluctuations in MnF two magnon excitation (absorption) ω th =(E ) 1/2 magnon scattering E cq=(e 2-2 ) 1/2 (creation and annihilation) q a π Fig. 4 Dispersion of the single-magnon excitation E(q)and regions of the longitudinal two-magnon spectra. The points denote the measured spin-waves at 5 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 low q, by the threshold energy ω th and cq, respectively. Therefore, a gap in the longitudinal (two-magnon)response, albeit fading out at higher q, 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 excitation (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 5 K, since the reduced temperature τ (T N T )/T N =.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 =andt = there is an infrared divergence (IRD) of the longitudinal susceptibility; in the 3D case, Im χ sgn(ω) and Re χ ln(j/ω), where J is the exchange energy. For nonzero T and ω T,wehave Im χ 1/ω and Re χ Im χ. For energies ω of the order of T, 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 the χ AF (q,ω) has to possess a weaker singularity than predicted by the twomagnon 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 totalspin 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 expressions ω should be replaced by the spin-wave gap. However, if the ratio J/ is large, the problem of the screening remains. In the present case of MnF 2 the IRD is strongly suppressed as the condition J is actually not fulfilled. Therefore, one can expect that the LSS is saturated

7 452 EUROPHYSICS LETTERS 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 in 2D antiferromagnets with very weak anisotropy in future experiments. The work was supported by RFBR (Grants No , , ) and Russian state programs: Collective and quantum effects in condensed matter and Quantum macroscopics. We wish to thank S. Freudenstein for support during the experiments, 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 Jülich for hospitality. REFERENCES [1] Harris A. B., Kumar D., Halperin B. I. and Hohenberg 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 annihilation operators. As a result, the interaction between spin-waves cannot change evenness or oddness of the intermediate states. So, this quality is determined by the structure of the spin operators only. The transverse operators S x and S y are odd and the longitudinal operator S z is even. [3] Low G. G., Okazaki A., Stevenson R. W. H. and Turberfield K. C., J. Appl. Phys., 35 (1964)998. [4] Okazaki A., Turberfield K. C. and Stevenson R. W. H., Phys. Lett., 8 (1964)9. [5] Turberfield K. C., Okazaki A. and Stevenson R. W. H., Proc. Phys. Soc. (London), 85 (1965)743. [6] Schulhof M. P., Heller P., Nathans R. and Linz A., Phys. Rev. B, 1 (197)234. [7] Schulhof M. P., Nathans R., Heller P. and Linz A., Phys. Rev. B, 4 (1971)2254. [8] Nicotin O., Lindgard P. A. and Dietrich O. W., J. Phys. C, 2 (1969)1168. [9] Braune S. and Maleyev S. V., Z. Phys. B, 81 (199)69. [1] Bunker A. and Landau D. P., Phys. Rev. Lett., 85 (2)261. [11] Kubo R., Phys. Rev., 87 (1952)568. [12] Toperverg B. P. and Yashenkin A. G., Phys. Rev. B, 48 (1993)1655. [13] Luzyanin I. D., Yashenkin A. G., Maleyev S. V., Zaitseva E. A. and Khavronin V. P., Phys. Rev. B, 6 (1999)R734.