Effect of pressure on magnetoacoustic resonance in uniaxial antiferromagnets I. E. Dikshtein, V. V. Tarasenko, and V. G. Shavrov Institute of Radio Engineering and Electronics, USSR Academy of Sciences (Submitted March 26, 1974) Zh. Eksp. Teor. Fiz. 67, 816-823 (August 1974) The effect of pressure and magnetic field on the magnetoacoustic oscillation spectrum is considered for uniaxial antiferromagnets with positive or negative anisotropy constants. The dependences of the propagation velocity of quasielastic waves on field strength and pressure are calculated for a wide range of variation of these quantities. It is shown that, with respect to both magnetic field strength and pressure, the strongest (about 100%) variation of the propagation velocity of transverse, long-wave, quasielastic oscillations occurs in the vicinity of the phase transitions. The ratio of the damping to the frequency of the magnetoelastic wave is maximal at the phase-transition points, but is always less than unity. 1. INTRODUCTION For simplicity we assume the antiferromagnet to be isotropic in its elastic and magnetoelastic properties. A strong dependence of the velocity of transverse A pressure applied along the x axis leads to the apsound on the value of the magnetic field applied in the pearance of an effective anisotropy in the basal plane. basal plane of the crystal has been discovered in exIt follows from the minimum of the ground-state energy perimental study of magnetoacoustic resonance in hemathat the vector L is established in the following way tite a-Fe20s and iron borate FeBOs [1- 4 ]. The possibility in the absence of a magnetic field (H = 0): of a similar strong dependence of the velocity of transLllz, if HA,=HrH 8(Il >0, (2a) verse sound in ferromagnetics has been pointed out in Lllx, if IlA,<oandllp>O, (2b) theoretical papers.[S-9] The case of a ferrite with colLily, if IlA,<oandIlp<O, (2c) linear magnetic mo-ments of the sub lattices has been studied by Bar'yakhtar and Yablonskii. [10] On the other where HA = ({3- {3')Mo, Hp = )'lPMo/ pst. Mo = IM11 = 1M2 I, hand, as follows from the work of Maksimenkov and P = -a is the external pressure (P > 0 corresponds to Ozhogin, [1] the resonance properties of hematite are very compression), and sensitive to the influence of directional pressure. { 1 at ~>O 8(;)= 0', at ~<O. In the present study we consider magnetoacoustic resonance in uniaxial antiferromagnets over wide ranges A. If the field is directed along the z axis and the of magnetic field and pressure. It is shown that the inequality (2a) is satisfied, then the collinear phase coupling between the elastic and spin waves depends L II H is stable in the range of fie Ids 0 < H < H2 and significantly on the value of the pressure. In phase the noncollinear phase is stable in the range H1 < H < HE. transitions of the first and second order in the field and Here H1zH2z,fHA1HE, HE=2oMo. the pressure, singularities arise in magnetic crystals In the analysis of the oscillations of the magnetic in the velocity of propagation of magnetoelastic waves. moments and of the lattice, we represent M1(r, t), M2(r, t) and u(r, t) in the form 2. MAGNETOELASTIC WAVES IN ANTIFERRO( 3) MAGNETS WITHOUT WEAK FERROMAGNETISM where Mjo and u~~ are the equilibrium values deterWe use the following Hamiltonian to calculate the mined from the energy minimum of (1). From the equadispersion equation of coupled acoustic and spin waves tions of motion of the magnetic moments of the sublatin antiferromagnets of the considered type: tices with the relaxation term in the form Rj =-os[Mj, de=dem+deme+dee, Mj l/Mo, where Os is the quantity which determines the relaxation of the spin waves, and from the equations of de m= du -ex - - + - + ex'----+oM,M, elasticity, we can easily obtain the dispersion equation 2 iix, iix, iix, iix, of the coupled magnetoelastic waves of the antiferro- ~ ~(M,.'+M"')-~'M"M,,-(H,M,+M2)}' (1) magnet. p) p J { 1 ((OM,), (OMz)'] iiM,iiM2 deme=~ fdv {y(M'-L')u,,+21I L ,L.u..+21,M,M.u,,}, 4 • de =+ J dv{p[';"+(s,'-s,')u,,'+2s.'u,:1-2au xx}. Here a, a' and 0 are exchange integrals, {3 and (3' are the anisotropy constants, )" )'1, Y2 the magnetostriction constants, L = M1- M2 and M = M1 + M2 are the antiferromagnetic and ferromagnetic vectors, respectively, Ml and M2 are the magnetizations of the sublattices, H the external magnetic field, p the density of the medium, sl and St the velocities of longitudinal and transverse sound in the antiferromagnet, u(r, t) the elastic displacement vector, and a the tensor of the external elastic stresses. 404 SOy. Phys.-JETP, Vol. 40, No.2 We first consider the case of the noncollinear phase (H1 <H <HE). For waves propagating parallel to the antiferromagnetic vector L (L II x if Hp > 0, and Lily if Hp < 0), we have the following dispersion equation: (w'-w ..')' (w'-w",') (w'-w".') (w'-w".) -(01'(0", '(w'-(O,,') '( W'-(OI,.') (4) -(0,'(0,,' «0'_(0,,2) «02-W,,') (w'-w".) -(0,'(0,,' «0'_0)",') X(w'-w,,') (w'-(O".) ""0. The quantities wtk and Wlk are the frequencies of transverse and longitudinal sound, Wlsk = W~sk - iW~sk = Y- elC:~17 W2sk = ffi~Sk - iW;~k = V £38 4 are the frequencies of the first and second branches of the spin oscillations, Copyright © 1975 American Institute of Physics 404 8t~WE-i6.(i), St E.=g[ (a+a' cos 2ft)k'M,+HEcos· ft+JJ m.-l/.u l-i6,m, E,=g[ (a-a') k'M.+ll m. sin' ft+ I Hp Il-i6.m, E,=g[ (a-a' cos 2tt) k'M.+HEsin' ft l-i6.m, g is the gyromagnetic ratio,k the quasimomentum, wE = gHE, Hme = 2i'~~/J.lSL J = Jl = 8z, cos 8 = H/HE, 81 and 8z are the polar angles of the magnetic moments of the sublattices, m.'=gHm.E, <tl3 (~1 o H, HE H FIG. I. Dependence of St on H for HAl> O. I) kilL, polarization of the sound ell z; 2) kilL. e 1 z; a) P = 0, b) P'* 0; 3) kilL. e II x for Hp > 0 or ell y for Hp < 0; 4) k II L.e II g for Hp > 0 or ell x for Hp < 0; k 1 L, el L. -1)' sin' 2ft, ,=g H meet (sin . , Au- - ~ Y, cos '-,,)' u In the derivation of the dispersion equation, we started out from the approximation ( 5) Analysis of the dispersion equation (4) shows that the magnetostriction interaction removes the degeneracy of the transverse sound. The transverse sound, polarized along the z axis, interacts most effectively with the first branch of the spin oscillations in the vicinity of the phase transition point of the first order from the noncollinear phase to the collinear. In the vicinity of the phase transition point (H = HI), at small values of the wave vector k(wtk« wiso), the following expressions are obtained for the corresponding frequencies of the coupled magnetoelastic waves: FIG. 2. Dependence of Q-I on H for HAl ~O, kilL. e 1 z. k. I) IHpl + (~') k2MO < 1/2Hme , Q~ax = IlsWtk {3y3g IIHpl + (a-a') k2M o 1r'; 2) IHpl + (a-a') k 2 Mo > 1/2Hme , QH = H = IlsWtkHme {2g [IHpl + + (a-a')k 2 Mo lYz [IHpl + (a-a')k 2Mo +'Hme l3/2r ' . der in the pressure (vanishing of the anisotropy in the basal plane).ll (6) In the case considered, Q-;.1 for g'l (a-a')M.Hm • sin ti lHpl ¢: (a-a') k'M,¢:Hm• sin' tt, At the phase transition point H = HI the disperSion law of the quasielastic oscillations changes-it becomes quadratic, The velocity of these oscillations sf decreases to zero as k - 0 (Fig. 1, curve 1). The ratio = 2WIIk"/ WIIk' is at maximum at H = HI and is expressed as Q~' Figure 2 shows the dependence of at various values of the pressure, Qil -1 Qu 6,s, . g[ (a-a')M,ll ... l'" (7) If Hme -1 Oe, st-10 5 -cm/sec, gM o-10 1O sec-I, a-a' _W- 1Z cm 2 , 6s -10- 5 , then glH.1 'hOll.1 +Hrn. sin' tt)'" (a-a')k'M,¢:Hm• sin' tt-lH.I. Qit - lO- z• Transverse sound polarized in the basal plane turns out to be strongly coupled with the second branch of the spin waves throughout the entire range of fields of the noncollinear phase, with the exception of the vicinity of the point HE. The corresponding frequencies of the coupled magnetoelastic waves in the approximation wtk «Wzso have the following form: Qii on (9) for H for this case At the point of collapse of the magnetic moments of the sublattices H '" HE. the magnetoelastic gapwzso vanishes. As a result the strong interaetion of the acoustic and spin waves also disappears. The interaction of the spin system with the longitudinal sound is maximal for k directed at an angle 1f/4 to L and lying in the basal plane (for HI < H < HE) or in the plane (LUH) (for H"'H 1), If P=O,then sl'=(si-sn I/Z• The results are. of course, invariant under permutation of the directions of propagation and polarization of the elastic wave. If the magnetic moments of the sublattices are collinear (0 <H <H2), the dispersion equation for coupled magnetoelastic waves propagating along the z axis (k=kz ) has the form (m'-m,.') ·(Ctl'-m,.') ({J)'-m:•• ) (m'-{J)~ •• ) In the range of fields HI <H < HE, the velocity of the longwave quasielastic transverse oscillations s£ is very sensitive to the value of the pressure applied in the basal plane of the crystal. With increase in pressure, increases from zero to St: St -gHm.{J),.'(m'-m,.') (Ctl'-m,,') (B,+B,) "'0, (10) where m,\,,""/,{Q,'+Q,'+2g'H'±[(Q,'-Q,')'+8g'H'(Q,'+Q,') l"·}. Q.'= {g[ (a-a')k'M,+Hm.+HA'+ IHpll-imll.}mE' Q.'={g[ (a-a') k'M.+ll m.+H A,l-iCtl6.}mK' (8') B,.,=mx(m'-Q:.,) . This is shown in Fig. 1, curves 2a and 2b. The point Hp = 0 is the point of a phase transition of the first or- Under the condition wtk« Wjso, the frequencies of the coupled magnetoelastic waves are determined by the following expressions: 406 J. E. Dikshtein et a I. Sov. Phys.-JETP, Vol. 40, No.2 406 (11) Transverse sound polarized along the easy (for L) direction in the basal plane (along the x axis if Hp >0, or along the y axis if Hp < 0) interacts mo st strongly with the spin system. The corresponding variation of the velocity sf as a function of H is shown schematically by Fig. 1, curve 3. B. We consider further the case in which H II x and the condition (2b) is satisfied. In an antiferromagnet of the easy-plane type (HA < 0), the collinear phase is stable in the range of fields 0 < H < H~ and the noncollinear phase, in which Lily, is stable in the field range H{ < H < HE; H{ '" H~ ",y HpHE' The frequencies of magnetoelastic resonance are determined from Eqs. (4)-(11) (for the case Hp >0), in which we make the substitutions x - y, y -z, Z -x, 8-CfJ=CfJ1=-(/J2, HA1-Hp, IHpl-IHAI, H1 -H{, H2 -H~, (CfJ1 and CfJ2 are the azimuthal angles of the magnetic moments of the sublattices). In an antiferromagnet of the easy-axis type (condition (2b) is satisfied in this case for O<HA <Hp), the collinear phase L II H is stable in the range of fields o< H < H::' and the noncollinear phase L II z is stable in the range of fields Hr < H < HE; H{' '" H~' '" Y IHA1IHE. The frequencies of the coupled magnetoelastic waves can be obtained from Eqs. (4)-(11) (for Hp>O) with the help of the substitutions x-z, z-x,8 -8'=(81-8 2)/2, HAl -IHA11, IHpl -IHAI, H1 -Hr, H2 -H~'. = In the vicinity of the easy-plane-antiferromagnet easy-axis-antiferromagnet phase transition, i.e., upon change of the sign of the magnetic anisotropy HA, and in magnetic fields that exceed the field of reversal of the magnetic moments of the sublattices, H{, the low-frequency branch of spin waves interacts effectively with transverse sound which propagates along the z(y) axis and is polarized along the y(z) axis. (H II x). As before, in examining the oscillations of the system, we represent M1, M2 and Uik in the form (3). For equilibrium values of the polar angles 8 1 and 8 2 and the azimuthal angles CfJ1 and CfJ2 of the magnetic moments of the sublattices, we obtain lIHD 2) for IIp>O andf[~H" or for cos <p=1, Hp<O /}.fJ)= (H+HD)/He. (14) Here CfJ=lh(CfJ1+CfJ2), <I>=lh(CfJ1-CfJ2), Hp = Y66PMO/C66 , and Hn =dMo. In the derivation of these formulas, we have assumed Hn, IHAI, Hp, H all «HE. It can be shown that the low-frequency branch of spin waves W1sk interacts most strongly with sound propagating along the x axis. In the approximation Slk« W1S0, we introduce the corresponding frequencies of the coupled magnetoelastic waves: LInthecase Hp>O and H::SH c : , 2 w"""w,,,=g { (a-a) k MoHe+lfm,He -Ik Wn, Ill' "" - - . Y2 [(Il,+HD)2-H2] (H.'-H') '/1 } H,(H.+H D) _H2 {S,2+Ss' (l-s) ±[ (s.'+(i-s)ss')2-4s,2Ss' -I-4ss,' (ss' sin' 2<p+S,2 cos' 2<p) l"'}''\ ( 15) (i)IVk-;::::;;S~k, where Analysis of expressions (15) shows that in the vicinity of the phase transition of second order H - Hc, CfJ - 0 (the magnetic moment lies along the direction of the magnetic field H), the transverse sound polarized along the y axis and the low-frequency branch of the magnon spectrum interact effectively. At the phase-transition point, 3. MAGNETOELASTIC WAVES IN ANTIFERROMAGNETS WITH WEAK FERROMAGNETISM ( 16) We now consider the features of magnetoacoustic resonance in an easy-plane antiferromagnet with weak ferromagnetism. The Hamiltonian of such an antiferromagnet can be written in the form (1), where J'6'm= aM,)2 aM, aM, a . + (8M2)2] - - -I- a' -- -I- 6M,M Sdv {[( {Ix, {Ix, {Ix, ax, 2 -I- ~ ~ (M,.'+M2,2) +~' M"ilJ,,-I-d[M,XM 2L- (H, M,+M 2 ) } , 2 J'6'me= -+ S dv{L'[ 112 (uxx-l-u yy ) -I-1"U,,]-L.'[ (112-1") (ux,-I-u yy ) ( 12) -1- (1"-1") u,,] +2166 (Lx2uxx+L,'uyy-l-2LxLyuxy) +41"L, (Lxu,,+Lyu,,)} S Thus, the dispersion of the quasielastic wave changes character at the phase-transition point H = Hc and the velocity of this wave vanishes. The damping is maximal at the phase-transition point, and the ratio Qi{=2WIIk"/wllk' is determined by Eq. (7). As in the case of an antiferromagnet without weak ferromagnetism, estimates give Qi{ _10- 2. The coupling of longitudinal sound with the low-frequency branch of the spin waves is maximal at H=(Hr/2+HpHE)1/2-Hn/v'2 (CfJ=1T/4). Here the frequencies of the quasielastic waves are equal to . (17) 1 dv[pu 2+c lI (uxx'-I-u yy') +c"u"'-I-2c,, (u xx +u,,) u" J'6'e ="'2 +2CI2U=Ully+4c66U:e1l2+4co\;~ (u:t:t2+UyZ 2) -2ou:xx]. Here d is the nzyaloshinskil constant, y the magnetostriction constant, and c the elastic modulus, and the remaining designations correspond to those introduced earlier. We shall not take account of the anisotropy in the basal plane of the crystal, and shall assume that L2 »M . L, M2. We consider the case that was realized experimentally in ell in measurement of the antiferromagnetic resonance, when the magnetic field is applied in the basal plane of the crystal parallel to the pressure 406 SOy. Phys.-JETP, Vol. 40, No.2 II. For Hp>O and H2:Hc (or for Hp<O and H2:0), the transverse sound polarized along the y axis interacts strongly with the low-frequency branch of the spin waves. The frequencies of the coupled magnetoelastic waves (wtk« W1S0) have the following form (18) It follows from Eqs. (18) that the maximum magnetoI. E. Dikshtein et al. 406 general principle according to which the character of the dispersion of at least one of the branches of the oscillations of the system should change in the phase transitions. Inasmuch as the frequency of neither of the branches of magnon oscillations vanishes at the phase-transition points, because of the presence of the spontaneous magnetoelastic gap,[lll the dispersion law of one of the quasielastic waves changes at these points (from linear to quadratic). s'·I0-~ em/sec D He 4 H,kOe FIG, 3, Dependence of st' and sl' on H in hematite with L 1 z. I) H II x, Hp > 0, P"'" I kbar, k IJ x: a) for st' (e IJ y), b) for sl'; 2) H II z, kilL, e II y for Hp > or e II x for Hp < 0, IHpl = (l/3)Hme for st'. The authors express their sincere gratitude to A. S. Borovik- Romanov, K. B. Vlasov, V. 1. Ozhogin and E. A. Turov for a discussion of the work. ° ° elastic coupling for Hp > takes place in the vicinity of the phase transition H = Hc (or H = 0 for Hp < 0) on the approach to which (from the high-field side) b -[l-(CI! - CI! ')k2Mo/Hme] and consequently the velocity of the coupled magnetoelastic wave 0 as k - O. St - The longitudinal sound is weakly coupled with the magnetic subsystem in this case. The dependence of the velocity of the longwave transverse and longitudinal vibrations on H in hematite is shown in Fig. 3. We turn our attention to another interesting case: an easy-plane antiferromagnet (with or without weak ferromagnetism) in a field H II z. Here, over the entire range of fields from zero to HE, a situation will exist that is similar to what occurs in an easy-axis antiferromagnet in the noncollinear phase, i.e., in the range of high fields HI < H < HE (curves 2 on Fig. 1). In this case, strong interaction of transverse sound with kilL, polarized in the basal plane of the crystal, with the lowfrequency oscillations of the magnons should be observed near the phase transition point in the pressure Hp = 0 over a wide range of magnetic fields 0 < H < HE (see (8'), footnote 1) and curve 2 in Fig. 3). The presence of strong singularities in the quasielastic-oscillation spectrum at first and second order phase-transition points is in correspondence with the 407 SOY. Phys ..JETP, Vol. 40, No.2 l)To account for the crystallographic anisotropy in the basal plane, HA'in Eqs. (8) and (8') should be replaced by IHpl-+ IHp + HA'I. Here the phase transition will take place at Hp = -HA'· lp. P. Maksimenkov and V. 1. Ozhogin, Zh. Eksp. Teor. Fiz. 65, 657 (1973) [Sov. Phys.-JETP 38, 324 (1975)]. M. H. Seavey, SoUd State Commun. 10, 219 (1972). 3 V . I. Ozhogin and P. P. Maximenkov, Digests of INTERMAG Conf. 49-4, Kyoto, 1972; IEEE Trans. Magn., Mag. 8, 645 (1972). 4y. 1. Shcheglov, Fiz. Tverd. Tela 14, 2180 (1972) [Sov. Phys.-Solid State 14, 1889 (1973)]. 51. Ya. Korenblit, ibid. 8, 2579 (1966) [8, 2063 (1967)]. 61. Iensen, Int. J. Magn. 1,271 (1971). 7 D . T. Vigren and S. H. Liu, Phys. Rev. B5, 2719 (1972). 8 B . W. Southern and D. A. Goodings, ibid. B7, 534 (1973) . 9 H. Chow and F. Keffer, ibid. B7, 2028 (1973). lOY. G. Bar'yakhtar and D. A. Yablonski'l, Ukr. Fiz. Zh. 18,1491 (1973). Ill. E. Dikshte'in, V. V. Tarasenko, and V. G. Shavrov, Fiz. Tverd. Tela 16, 2192 (1974). 2 Translated by R. T. Beyer 94 I. E. DikshteYn et al. 407