Dipolar surface spin waves in antiferromagnets

Journal

264

DIPOLAR SURFACE B. LOTHI

of Magnetism

and Magnetic Materials 38 (1983) 264-268 North-Holland Publishing Company

SPIN WAVES IN ANTIFERROMAGNETS

and R. HOCK

Physikalisches Instirut der Universitiit, Robert - Mayer - Str. 2 - 4, D - 6000 Frankfurt a. M., Fed. Rep. Germany

We present calculations for dipolar surface spin waves in antiferromagnets at low temperatures. We calculate bulk and surface spin wave spectra for long wavelengths in the magnetostatic mode region. The different phases in an applied field along the easy axis are discussed with special emphasis on the spin flop- and paramagnetic phase. Nonreciprocal features of the surface spin waves are discussed. Numerical applications for GdAlO, are given.

1. Introduction Dipolar, both ferromagnetic and antiferromagnetic, surface spin waves exist for the long wavelength region, where a continuum approach can be used to describe these excitations. Exchange dominated surface spin waves, on the other hand, have to be described in a microscopic way for the short wavelength region. So far only dipolar ferromagnetic surface spin waves have been observed in a number of ferromagnetic materials [l], whereas no observation of surface spin waves has been reported for antiferromagnets. Recently, we presented a theory for dipolar antiferromagnetic surface spin waves (ASSW), applied it to real systems and discussed possible experiments [2]. We discussed mainly the antiferromagnetic region and mentioned only briefly the spin flop region. In this paper we extend the calculations for ASSW into the spin flop phase and the paramagnetic phase. The calculation is based on a continuum approach and applicable strictly only for T = 0. We shall give numerical applications for real systems such as MnF,, FeF, and GdAlO,.

sider uniaxial antiferromagnets with magnetic field applied along the easy direction. Other geometries are less interesting for ASSW [2] 2.1. Antiferromagnetic

phase

We deal with this phase very briefly because it was discussed in detail before [2]. The stability limit for this phase is H, = Q/]y], where (Q/y)’ = 2H,,H, + H,‘. Here H, is the externally applied field along the easy direction (z-axis), H, is the anisotropy field, H,, the exchange field and y the gyromagnetic ratio (negative number). The dispersion equation for bulk and surface spin waves read ]21: bulk: (:h)2=($)2+H,i+4nMH,sin28 2

k2

H,z E i i) Y

+ 4nMH, l/2

+ 4,rr2M2Hi

sin48

;

ASSW: 2. Theory

2 H, cos $I

=First we give a brief outline of the calculation for the antiferromagnetic phase (AF). Afterwards we present the calculation for the spin flop phase (SF) and the paramagnetic phase (PM). We con0304-8853/83/0000-0000/$03.00

0 1983 North-Holland

Hi sin29

1 + co&$ Hl sin4$ +_ (1 + cos2+)2

(la>

265

B. Liithi. R. Hock / Dipolar surface spin waoes in antiferromagnets

SF 14 kG

0

1

0

20

I

I

-I

40

60

Ho

180

90

P

(kG) Fig. 1. Bulk and surface spin waves as a function of applied field for GdAlO,. The shaded regions are the bulk spin wave bands. Full lines are stable surface spin waves in - q,- direction. Dotted line unstable region. H,, h,, and H3 are defined in the text.

Fig. 2. Angular dependence of surface spin waves in the AF-phase (lower part, H, = 10 kG) and in the SF-phase (upper part, H, = 14 kG). Some bulk spin wave modes are also given to indicate limits of stability.

2.2. Spin flop phase

(lb) Here 8 is the angle between the easy direction (z-axis) and the propagation direction q of the bulk spin wave. The angle r#~is measured from the x-axis to the propagation direction q,, of the ASSW. The coordinate system is chosen with x and z in the surface plane and the y-axis into the medium. Eq. (1) and the stability criteria have been discussed before [2]. The non-reciprocal feature for the ASSW is clearly seen from eq. (lb) in the first term. C#J = 0 and C#J = n give different ASSW excitation energies o,/]y]. Typical numerical examples are shown below in figs. 1 and 2. Note that due to the magnetostatic approximation wt, and o, are dependent only on the direction of q and not on its value.

Denoting the two sublattices by A, B we have for the Zeeman energy E, = -MH,,(cos 8, + cos 13,) for the anisotropy energy E, = +MH, x(sin2t9, + sin20,) and for the exchange energy E,, = MH,,cos( 0, + 8,). The stability region for the SF-phase is found (with (I, = 8, = 0) as H, =

2Hex - Ha 0 -
(2)

The field H2, which is smaller than the spin flop field Q/]y], is often called the supercooling field. For the equation of motion in the SF-phase it is convenient to introduce a coordinate system for each sublattice and to transform the external, anisotropy and exchange fields accordingly. The equation of motion gives two eigenfrequencies of the system. One, w-/v = 0, corresponds to the rotation around, let us say, the z-axis if the spins are in the (x - z)-plane, which does not cost any

266

B. L&hi, R. Hock / Dipolar surface spin waues in antiferromagnets

energy. The other frequency is w+/y = 2H,,(H,cos 8 + H,cos 28). Introducing dipolar fields gives dynamical susceptibilities

we obtain @,+-q’+aY

for @ = @Oe’-q’-a.v outside:

inside

and

@=

M, = CX,.,h,, - 4MH,, xxx = (o/Y)2

cos28

- (U./Y)’

(dY)2

2iM( w/y) cos 8 x,v

=

(w,y)2

_

(w+,y)2

=

$

-x:.x

=(H,,

+ H&OS

and leads to a modified

Walker

equation:

+

(4) where with curl h = 0, div b = div(h + 4&f) = 0 we introduce the scalar potential @ with h = -grad@. Inserting (3) into (4) gives the bulk spin waves in the SF region: 2 =

2 H,, (H,cos

8 + H,cos 20) 2

+

16pMH,,cos20

1+47rx,,,’

+ 8aM( H,,cos t3

In fig. 1 we show ~~/lyl as a function of applied field H,. The different propagation direction q give rise to a band of possible excitation frequencies. Because of the magnetostatic approximation w b is only dependent on the direction q and not on 141, as in the AF-phase. In this calculation only transverse fluctuations are considered. For a complete temperature dependent calculation longitudinal relaxation had to be introduced also [3]. Since we are only interested in surface spin waves at low temperatures we neglect this complication. Next we consider surface excitations. We take as the surface again the x-z plane and assume the canted spins to lie also in the x-z plane. With the usual boundary conditions h, and b, continuous,

4df)COS

0 +

sin2G

+

Eliminating cx and inserting sion equation

’

- (~+/Y)’

cos2G

1 + 47X,”

41:

- 2M( H,cos 8 + H,cos 20)

x .yy=

(1+ %Y,,)

ff2 -=

’

6

H&OS

COS

x,, gives the disper-

+

28

2 cos 8 cos +

(6)

Here B is the canting angle for the spins measured from the easy z-axis and $ is again the angle between the q,,-direction and the x-axis. If the equilibrium spin configuration is not parallel to the (xz)-plane, eq. (6) is modified. We shall give numerical results in the figures. Both bulk spin waves o b and surface spin waves w, exhibit ferromagnetic-like character in the SFphase. This is clearly shown in fig. 1, where one has two branches in the AF-phase but only one branch in the SF-phase. Also the nonreciprocal behaviour for the ASSW in the SF-phase is similar as for a ferromagnetic surface spin wave: w,/lyl is only positive for + G 180” and becomes unstable for a critical angle & where a/(-q,,) becomes zero, where from above a -= - 411

1 - 4aix,,cos

(p

1 + 47rx,,

(7)

ASSW are stable if CX/- q,, < 0. 2.3. Paramagnetic phase The magnetic phase (PM) is stable for H, > 2 H,, - H,. For this region we have for the canting angle B = 0, (a+/~)~ = (H, + H,)’ and for the spin waves bulk:

B. Liithi, R. Hock / Dipolar surface spin waves in antiferromagnets

ASSW: HII + H, 2

cos$J+

H0 + % 2 cos l$ *

(8b)

Here sin2# = (42 + q:)/q2, i.e. Ic, is the angle between the z-axis and the propagation direction q for bulk waves and + is again the angle between q,, and the x-axis for the surface waves. As in the SF-phase both bulk and ASSW exhibit ferromagnetic like character for the PM-phase. This is illustrated again in fig. 1. In addition as for the SF-phase the nonreciprocal behaviour for ASSW is also present in the PM-phase and similar as in ferromagnets. The dynamical susceptibilities xXx are equal in the PM-phase. Therefore, and XY.” there is no longer any anisotropy with respect to the spin canting plane.

3. Applications In a previous paper we have applied the results of subsection 2.1 (antiferromagnetic phase) to three representative antiferromagnetic materials: MnF,, FeF, and GdAlO,. It was found that GdAlO, was the most likely candidate to observe ASSW experimentally, because they were sufficiently removed from the bulk spin waves for certain angles (almost 1 kG) and because the spin flop field lies in an easily accessible region (12 kG). On the other hand GdAlO, is, due to its orthorhombic symmetry, not strictly a uniaxial antiferromagnet. However, an additional orthorhombic anisotropy could easily be incorporated into the formulas for bulk and surface spin waves. Therefore, we discuss here as numerical example solely the case of GdAlO, with the following parameters [2,4]: He, = 18.8 kG, Ha = 3.65 kG, M = 624 G, which gives for the spin flop field L’/]y] = 12.26 kG, for the supercooling field H2 = 10.099 kG and for H3 = 33.95 kG (demagnetising field always subtracted). In fig. 1 we show bulk and surface spin waves for the three phases as a function of the applied field Ho. The bulk waves are the shaded portions. Because of the angular dependence of w,,, the spin wave energies form bands. One notices clearly the two branches in the AF-phase and the single branch in the SF- and PM-phase. Also shown are

261

the surface spin wave o,/]y] for two fixed directions C#J = 0’ and 180” in + x-direction (see insert of fig. 1). In the AF-phase both directions give stable ASSW which were discussed previously [2]. In the SF- and PM-phase surface spin waves exist only in the C$= 180” direction. They are slightly above the bulk continuum as for the case of ferromagnetic surface spin waves. In the SF-phase the ASSW is unstable for smaller fields. This fact will be discussed below with the aid of figs. 2 and 3. In figs. 2 and 3 we show the angular dependence of bulk and surface spin waves in the three phases for various fields. The calculations have been performed again with the values of GdAlO, above. Starting with the lower part of fig. 2 we show the spin wave spectra for H = 10 kG, i.e. in the AF-phase. One notices the split bulk spin wave bands and the ASSW in between with the stable region near 180” and 0” and the unstable region (dotted line) in between. The nonreciprocal features for the ASSW are clearly seen from this figure. The splitting between bulk and surface spin waves at C#I = 180” becomes larger, the closer the field comes to the spin flop field HsF, where it amounts to more than 500 G. Since we discussed the AF-phase before [2] we shall move to the SF-phase. In the upper part of fig. 2 we show the angular dependence of bulk and surface spin waves for H,, = 14 kG, which is above HsF, but for which the ASSW is unstable for + = 180” (see also fig. 1). We have calculated the bulk waves for q = qv and for q = q,, and the surface spin wave for the canted spin structure in the x-z plane (denoted by ]I SSW) and in the y-z plane (denoted by I SSW). One notices that the stable region for SSW is the angular + region between 160” (where the bulk spin wave ob( q,,) touches the SSW) and 128” (where w,,(qy) is tangential to the SSW curve). The I SSW mode is unstable for all angles +. In the lower part of fig. 3 we show analogous spin wave spectra for H,, = 25 kG, which is still in the SF-phase, but where the stability region for the SSW extends to + = 180”. The reason for this is clear if one compares this figure with the upper part of fig. 2. In fig. 3 the bulk ob(q,,) does not touch the ]I SSW, therefore the stability region

B. L&hi, R. Hock / Dipolar surface spin waves in antiferromagnets

268

r

i%

4. Discussion

(kG 5;

ssw, ___\\j__

,’

5

jljll

-8’

PM 40 kG

r o,, botk

5c)

‘;

31

SF 25 kG

31

3c ;60

18 170 160 150 140

1 130

Ip

Fig. 3. Angular dependence of surface spin waves SF-phase (lower part, H,, = 25 kG) and in the PM-phase part, H, = 40 kc).

in the (upper

extends to C#I = 144’ where the bulk oi,( q,,) touches the ]I SSW. Likewise the stability region for I SSW extends from (p = 180” to 150’ where w, = wt,( q,,, 180’). As pointed out before the nonreciprocity features are similar as in ferromagnets. The region of stable SSW as a function of C#Iin the SF-phase becomes narrower the closer one moves towards H2 and c#+tends towards 90”. Finally the upper part of fig. 3 gives spin wave spectra for Ho = 40 kG in the PM-phase. wb(q,,) and w, are shown with the unstable region given again as a dotted line. Again, as in ferromagnets, stable surface spin waves are found only in a limited region near C$= 180’. The stability angle $I~ is at the minimum of the SSW where w,(q$) = wb( q,,, C#J= 1800). In our numerical example $J== 149”. The four different cases illustrated in figs. 2 and 3 exhibit all the typical features of surface spin waves in the antiferromagnetic-, spin flop- and paramagneticphases of uniaxial antiferromagnets.

The recent antiferromagnetic resonance work [5] has clearly shown that dipolar fields give also rise to magnetostatic modes in antiferromagnets. Therefore, in analogy to the ferromagnetic case the investigation of long wavelength dipolar antiferromagnetic surface spin waves is appropriate. In the previous section and in a previous paper [2] we have made a calculation of dipolar surface spin waves for some typical well characterised antiferromagnets. The results presented in figs. 1, 2 and 3 show that it should be possible to observe ASSW. Three different methods are possible to detect ASSW. One possible experiment is the resonant interaction of a surface acoustic wave with ASSW. If one matches frequency and wave-vector, one should observe attenuation and dispersion. One had to perform such an experiment in the vicinity of the spin flop transition. Because of effects of the spin flop transition on the surface acoustic wave [6] it is important to minimize its effect by choosing $I = 180’ and the field H,, < HSF. Another possible experiment is Brillouin scattering where one could choose the appropriate frequency in a wide range for one of the three phases. A third possibility is a microwave experiment using a grating to excite the surface modes.

Acknowledgements This research was supported in part Sonderforschungsbereich 65. We would thank H. Rohrer for useful discussions.

by the like to

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[2] (3)

[4] [5] (61

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