Calculations of surface magnon dispersion by means of the matching procedure

ELSEVIER

J~H Journalof magnetism and magnetic ~ H materials Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

Calculations of surface magnon dispersion by means of the matching procedure M. Tamine

*

Laboratoire de Physique de L'Etat Condense, URA CNRS 807, Facultd des Sciences, Universir~ du Maine, Le Mans Cedex, France Received 4 July 1995; revised 24 July 1995

Abstract

We present a formalism for calculating the surface magnon dispersion which uses the matching procedure for the first time. In the present paper this method is detailed and applied in order to calculate the (001) surface magnon dispersion in the [010] and [110] directions in the case of a Heisenberg cubic antiferromagnet. First, a secular equation is derived which is shown to contain particularly all the information on the dispersion curves. Dispersion curves of the surface magnons are obtained within a single framework by matching the evanescent and travelling solutions, respectively, of the secular equation, satisfying surface boundary conditions. A significant result is that the surface magnon is found to be the sum of evanescent waves.

1. Introduction A large number of studies of magnetic excitations of ordered surfaces have been reported in the literature during the last decade. Many theoretical investigations have been developed by Wolfram and De Wames [1] to describe the magnetic excitations in the case of semi-infinite ferro- and antiferromagnetic cubic systems. From a theoretical point of view, the principal mathematical formulation used to describe the spin motion equation is based on the relevant surface Green function method, which can give information on the magnetic properties at a crystal surface, the spectral densities, the influence of surface anisotropy on the magnetization of the Heisenberg ferromagnet, and the thermodynamic properties of different systems [2-5]. The new theoretical approach for calculating the surface magnon dispersion curves proposed in this paper consists in matching the properties of bulk magnons with evanescent spin waves along the direction perpendicular to the surface. The feature that motivated this work to present the methodology based on the matching procedure, derives from the theoretical study of vibrational dynamics of an ordered surface exhibiting low

* Corresponding author. Fax: + 33-4383-3518. 0304-8853/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 0 5 2 3 - 4

M. Tamine/ Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

367

dimensionality. This method introduced by Feuchtwang [6] has not been favoured because it has been thought not well suited to resonance calculations. This point was studied again by Szeftel and Khater [7], and their results allow us to give a general definition of the resonance concept as well as a transparent analysis of the surface displacement spectral density, presented within a scattering bulk phonon-surface interaction. Up to now, this method has been extensively used to calculate surface phonon dispersion curves and surface resonances of semi-infinite crystals [8,9]. From these considerations, we have adapted this method to calculate the magnetic excitations of antiferromagnetic surface structure, leading to the dispersion curves of the bulk and surface magnons. The purpose of the present paper is to describe a new approach based on the matching technique allowing calculations of the surface magnon dispersion curves along the high symmetry directions at a surface. We present a method which provides a straightforward derivation of the equations of motion for any overlayer commensurate with two-dimensional (2D) periodicity. Consequently, to apply the matching procedure to the calculation of surface magnons in a semi-infinite antiferromagnetic structure with two sublattices, knowledge is required of the complete set of evanescent modes in the bulk region of the system limited on one side by a vacuum, and exhibiting (2D) periodicity along the surface. The application of the matching technique requires that the magnetic structure is divided into three parts, all exhibiting the same 2D periodicity parallel to the surface: (i) a surface region (n < s) consisting of a number of adsorbate, reconstructed or relaxed layers; (ii) an intermediate region of bulk matter (s < n < l) used to match bulk magnons with the boundary conditions imposed by the surface: its thickness increases with increasing range of interlayer interactions; and (iii) a bulk region (n > l) having three-dimensional (3D) periodicity where bulk magnon dispersion curves are first worked out.The integers s and l, which characterize the thickness in atomic layers n of a surface and matching regions, depend especially on the range of interlayer magnetic exchange interaction near the surface. Taking into account only the nearest and the next nearest neighbour exchange interactions, the surface region will he composed of at least the three first atomic layers (s = 3). The method is illustrated here on an antiferromagnetic (001) surface along the [010] and [110] directions when we consider (001) cubic plane as a mirror plane; such a consideration allows an exact treatment using Bloch's theorem [10]. According to the quantum theory of spin waves in insulators, the spin vectors of each magnetic ion precess along the direction of quantization.

2. Secular equations and bulk spin wave properties

Knowledge of a complete set of evanescent modes in the bulk region, along the direction normal to (001) surface plane, requires a description of the precessional field in the matching region. Owing to the periodic character of the bulk region, this precessional field can be reduced to spin-wave amplitude per layer, denoted by U,(A). Within the Hamiltonian exchange in the harmonic approximation when the magnon-magnon interaction is neglected, this amplitude reads Un<~) = E exp(ikll" rll)V(E,klI,n,A ) (R) E

• Q(E,klI),

(2.1)

where the superscript (A) indicates the corresponding magnetic sublattice containing the spin position R localized on the plane defined by n. Q is a normal coordinate associated with the magnon of the 2D wavevector kll and energy E. The summation over kll is confined within the 2D Brillouin zone associated with the surface. The components V(E, klI,n,A) determine a vector v(E,k u) which is an eigenvector of the dynamical matrix D(k).

M. Tamine/ Journal of Magnetism and MagneticMaterials 153 (1996)366-378

368

Denoting by p the phase factor per interlayer distance a z in the direction normal to the surface, the travelling and evanescent modes are then characterized in this representation by I pl -- 1, and I pl < 1, respectively, which entails p+ ' ( E , k , ) = exp( + i k z . a . ) ,

(2.2)

where the + signs correspond to waves travelling to and away from the surface, respectively. Eq. (2.2) allows us to write 8(E,kll ) = p(E,kll ) + p-'(E,kll ) = 2cos( k z . a z ) ,

(2.3)

which is the dispersion relation of bulk magnons along the b z direction in reciprocal space, when k z • a z varies in the range [0,~r ]. Consequently, there exists n t distinct bulk magnon branches since the secular equation (2.3) has ( E . n t) roots, each of which is a particular function of & Qualitative features and applications of the matching procedure are most easily illustrated in the case of a simple cubic antiferromagnetic structure with two sublattices, A and B. The structure is assumed infinite along the x and y directions with an extension from z = 0 to z = oo. The semi-infinite antiferromagnetic structure is described by n > 1 and the surface region extends from n = 0 to n = l = 3 (see Fig. 1). The coordinate system for a (001) surface is illustrated in Fig. 1. We consider here first neighbour and second nearest neighbour exchange interactions. Beyond the second layer, all exchange interactions have bulk value J~B(t,J) i~ • • for first neighbours, and J2[(i,i'), J2~(j,f)for second neighbours. The first and second nearest neighbours exchange interactions between spins on surface were defined by t~AB ~ , and t2~s ~AA , i2~s ~BB respectively, whereas the interactions between first and second layers by jl~ .L and JA2~.L, ~BBI 2v L . So, the Heisenberg Hamiitonian for this system can be expressed as

H = ~_, J ~ ( i , j ) " S}A)" s~B)+ Y', J2~(i,i')" St(.A)" St(.,A) "[(i,j)

(i,j')

+ Eg'txb'(Ho+H~)'s}g)+ i

E

j2~(j,f),

s}a). s~B)

(j,f)

E g ' t x b ' ( H o + H ~ ( ~ ] ) ' S } m + E D~,.)'(S}A'×S}m), j

(2.4)

(i,j)

with H(A) u(A) 1), a(n= 1) :/: ha(n>

]D~j(.= 1)[ :~: IDij(n> l)[,

(2.5a)

H(B) a(n= 1) 4="(a) "'a(n > 1),

[Di~. > 2)1~

(2.5b)

ID[,

where the subscript n describes the value of the anisotropic field Hta) a(n), and Dij(n) is the value of the Dzyalozhinsky-Moriya [11,12] term in the nth layer. The travelling and evanescent modes are given by solving following equation deduced from a system of equations, which is established for the bulk region in the Cartesian basis (x,y,z).

ho.)dl- J/~AaA Zii,')/ii,( k ) -- J~BSB Zij-- j22 SA Zii, ' ~ - g / . z b ( a 0 2t- n(aA ) )

(jtA~SBZijTij+2IDISAZijTij) •

ko

'~ hoj+gBaSBzj/yj/(k)_J2~SAZij

= o,

( JABSA Zij'Yij

2l olsA z,y3'ij)

--JBBSBZaf +gIZb(Ho+H(a B)) 2~ (2.6)

where the operators %° + and ilk+ are given by

Ol;(k,o.)) = E eik'ri'S+, k

[~;(k,o.))= E e ik-r~-s/, k

(2.7)

M. Tamine /Journal of Magnetism and Magnetic Materials 153 (1996)366-378

FACE [0 01)

369

j;la

n=l

n:2

i

~"

n:3.

i

l I

,J'~'P

•

/

I J:" ' J~" [

,' "

•

,

,-t

,

~

Ji

~,~'" I

II

/

n=4 .S S

J,u,

,~,,/

J

tl

n=5

I ~,,,t

•

JA.

Surface View

J~a

Y 2/

,e s.

J;,

J;,

Jr Fig. 1. (001) surface of a simple cubic antiferromagnet: the z-axis is normal to the surface, whereas the x- and y-axes lie in the surface (O spin up, O spin down). Beyond the second layer, all exchange interactions have the bulk values JA~(i,j) for nearest neighbours and 2v . .t / 2 ~ ( . #~ J~(I,I )' JBB J,J ] for next nearest neighbours. The nearest neighbour and next nearest neighbour exchange interactions between spins on the surface are defined by jA~s and JAA,2~sj e ~ s respectively, and the interactions between first and second layers by JA~X and JAA2v±, 2v± JBs

370

M. Tamine/ Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

with 1 j=6

Y] ei"(r'-ri ) = ½[cos( kxa ) + cos(kya) + cos( k z a ) ] , Zij j~ 1

(2.8a)

yij( k ) -~ - -

1 i'~ 12

~,,,(k) =--Zii' i'=E 1 e ik'(r:r:)

= 1 [COS( k x "1- ky)a "k"cos(k, + kz)a + cos( k x + k z ) a

+COS(ky- k z ) a

+ cos( k x - k z ) a + cos(k x - k , ) a ] ,

(2.8b)

1 /=12

e ik'(rj-r/) = 6 [cos(k x + ky)a + cos(ky + kz)a + cos( k~ + k z ) a

z j/ j,=

+cos(ky - k z ) a + cos( k x - k z ) a + cos( k~ - ky)a].

(2.8c)

A bulk magnon is characterized by p ± l(E,k u) -- exp( + ik z • az), with k z the component of the 3D wave vector which is not contained in the 2D Brillouin zone. The bulk magnon dispersion curve at a given E follows from inserting Eq. (2.2) into (2.7) and (2.8). We obtain also a p-secular equation expressed as E2-E

JAB(SA+SB)+JAASAZii, 1 - - - ( (

+cos(~x-~,)

Zi:

/)+J~S~z~: 1-

p + p-')(COSq~x + c O S ¢ y ) +cos(~ox + q~y)

(o+o-')(cOS~x+COS~,)+cos(~x+~,)

+ cos( q~x- q~y)))+gI-%(2Ho+(H~aA)+H~aB)))]

"~-JAAJBBSASBZii'Zjf

(O"~-p-I)(cOS~x'~COS~Oy)"~COS(~xdr~y)-~COS(~Ox--~Dy)} 2

--{(PdWP--I)(coS(Px"~-COS(pY)"~COS(~OXdW~DY)-~COS(~OX--~DY)

w + _ _ Zjf Zii,

+I

l ') + 2(cos~x +cos~,)f ] +J2.s~s.z~j [ l - z~{(p+p+JABZij 1--

.

( p + p - l ) ( C O S ~ O x +COS~Oy)

+cos(~x + ~,) + cos(~-

~,)(J~S~z,, + J.:~z.))]

+gI~bZii,((Ho + H(A))JBBSB-- (Ho + H(B')JAASA)

1-- ~ ( (

p + p-1)(COS CPx+COS Cpy)

M. Tamine / Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

+ cos( ~ + ¢,) + cos( ~x -

~,))

371

/

+g~bZ,jJ,,B(("o + "(a"')SA + ("o + "(~B')Ss) + ~2~("o + "(aA')("o + "(~') + 41DI2SASa{( p + p - l ) + 2(cos ex + cos q~y)}2 = 0.

(2.9)

When the determinant of the system (2.6) vanishes, there exists a simple solution which results from an algebraic secular equation of fourth degree in p. Both phase factors p and p-1 are solutions because this system is hermitian. The set of points (E, kll) is defined such that It)l-- 1, to describe the projected bulk magnons modes on the surface. Consequently, Eq. (2.9) represents the determinant of the dynamical matrix D(E,k), which is established for the bulk region (n > l) in the Cartesian basis (x,y,z) with ~0x = kxa and COy= k y a .

Unless its determinant vanishes, we obtain a trivial solution which yields a p-secular equation, that, for each point in space (E, kll), four pairs ( p,p- l) of roots, with n t roots # such that I pl = 1 describing bulk modes, and n¢ roots p such that [ pl < 1 describing evanescent modes. Space (E, kll) is hence divided into bulk magnons bands ( n t > 0), and zones ( n t < 0), for which the density of states of bulk magnons vanishes. The roots p so that I pl > 1, give divergent solutions at infinity; they are not accounted for in the present analysis. The projected bulk magnon density of states, denoted dp(E, kll), is defined by 0 - - ~ "

=z

(2.10)

I

The present procedure of numerical calculations of dp(E,kll) consists in using a mesh of kll vectors taken over the bidimensional first Brillouin magnetic cell. This integration leads to a histogram representing the number of bulk magnons whose energy is in the E < E < E + d E range. An illustration is given in Fig. 2. 1~

~

50

irIIJIJllllllllilllgl

LtlllJrr

,,,i1111 0

20

~

~

~

1~

1~

1~

1~

1~

200

Energy E OK)

Fig. 2. Illustration of ~ c pr~ccted bulk

magnondensity of states dp of magnons on ~e

(001)

surfaceobtained by ~ e

matching me~od.

M. Tamine / Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

372

3. Model surface spin waves

In general, from group theory including the translation operator properties [10] the precessional amplitude field of quantum spins in the bulk region can be described by a general linear development on a complete set of the n t + n e modes: n e + r/t

2~ R , C ( A , p , ) p " - t ( E , k , i ) ,

U(n,A) =

(3.1)

i=1

where the weighting coefficients R i characterize the contributions of the different modes in the bulk precessional amplitude field, and C(A,pi) are the corresponding polarisation vectors from the bulk dynamical matrix satisfying in this case tie+ rt t

Y'~ [C( A,pi)] 2= 1.

(3.2)

i=1

Let us consider the (001) surface of a semi-infinite antiferromagnetic simple cubic lattice with ISAI = ISBI = 5//2, which is described by a two-dimensional magnetic cell (a,b). Thus each atomic spin site on each sublattice (A) will be determined by three integers (r/,a,n), where (r/,a) are the coordinates in the basis of vectors (a,b), and n represents the increasing index of the monolayers of the semi-infinite crystal that are parallel to the surface, assuming n = 1 for the first layer. We now introduce some parameters, defined as h ¢ o - g" lZb

E=

. H o

JS

6 =

J

,

ei~=

j2vs

J

IDll

a=

J

e

=

2v ±

BB

=

J

,

6j~=

J

ID.~21

~ = ~ ,

J

,

~i

J

2v

JBB

,

J

2v

JAA

6u ,=

j

JaB

,

cjy

j

,

J2~=J.

(3.3)

The recurrent equations of motion for bulk layers (n > 3) read as follows: for n > 3,

[E-(

6+4gii''Y}l'- 126u' + g'JS/XbH(A)I]u(Al-47)~u' }B)]]a n

__ 4Ei{]li~

lrr(A) ~._,

__

-- ~ij'l"H(B)'/n-I - - "Yijlu(B-I'{

4,~ii,~i~, ur r.(+, A ! + 8t3rJ~v?) + 2 ~ r , ~ u n(~) - 1 + 2t3r,)

[E+(6+4~j/T):.-12gjf

g]-XtbH(B)]I(B)+w'--a H I' (A)+T~iU ' ("A]J-" ~Sj

__.,,_,

u.+,"(~)= o,

(3.4a)

_ +~,lrr(a)tijv,+,

+ 4Ejf yjJ; U(B] + 4~j/ yj~ U}B+] -- 8 flT}~U(. A ) - 2 flyi~. U(A_I -- 2 flYi~" "n+'(A)l =

O.

(3.4b)

The equations of motion for the surface layers (n = 1 and 2) read as follows: for n = 1,

e-

4~I~ + ~,~ + 4~ ii lii," - 4 4 , -

4~,~, +

JS

a(l))]

I

-- 4°°}~T)~U~", - 4°°,# ru'L '-'2"(A)_ 8i jx ,yij±U2(B)+ 8 a,y}~U~S, + 2/33ti~- U~B) = 0,

(3.5a)

M. Tamine / Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

.,.. ~a

$

373

•

I

E~"

= 0.15

lJ

Wave Vector

Wave V~tor

"L-....• .....

Wave Vector

.

Wave Vector

.g

Wave Vector

Wave Vector

ploUed in the [010] surface direction. The variation of acoustic and optical Fig. 3. Energies of bulk and surface spin waves for JAB ~~ /J~^(BB) 2 modes at the surface relative to the bulk magnon domain, as a function of the ratio ~!~= J~S/J,,B. The other parameters are assumed to be constant in this evolution.

M. Tamine/Journal of Magnetism and Magnetic Materials 153 (1996)366-378

374

[ E + (4e,~ + e./~ + 4~.11 ,v II .L j / 1 j / - 4¢J~, - 46j/

g JS "_.._~bH(B)/] ) + 4~,}~'y}~u[A ) -I- 4gj-) Tj~ U(B) a(l)]] u(B 1

"~- ~i~" ~/ij7 U(2A ) - 8°l~l]~U( A ) - ~1 171"~'l I[(A' -wrij ~2 = 0 ;

(3.5b)

for n = 2 ,

[ E - ( 5+'~i')" +4gii'~}l'-z'~ii'-4gi~

+ g'l&bH(A))] a

--~i~: "Yij-u(B)-- "YiJ"U3(B)- 48i~ Ti~' U( a ) - 4,~ii"Yi~' U(3A) + 8fl]//~-U2(B) + 2flTiJ- U(B) + 2 fl'Yi~ U3(B)

= 0,

(3.6a)

E + 5 + ~,~. + 4cj/~)~, - 8 ~ j / - 4~j)~

JS

+'si2 YiJ- U(A) + Yi~-U3(A) + 4,~j) "/jj: U(B) --I-4ej/yjj~ U3(B)- 8 fly]~U2(A)- 2fl~/i-). U( A ) - 2fl'yi j- U(A) = 0.

(3.68)

ii = 1.50

Z5

1

2~

2

2~

3

Wave Vector

Wsve Vector

II w

~

•

~ ' = 1.80

" = 2.50

• s

1

1.5

z

z~

a

Wsve Vector

Wxw V~'tor

Fig. 4. AS Fig. 3, for other values of 81~.

,-.,

375

M. Tamine/Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

In these equations, the factors ypq(k,p) are expressed by 1

E cik'(rl-rJ)'

(3.7a)

7 i J ( k ) ~- Zij (i,j)

(3.7b)

r,~(k): L [2(cos(k...) ÷ ~os(~,. o)) ÷ ( o ÷ o-')1, Zq

1 ~li'(i;)( k ) -~ ~

E

(3.7c)

ei/~'(r'(j~-r¢(/'),

Zii'(Jf) (i,i'(j,f)>

2

")'.'(s/)( /t) =

Zif(jf)

[( p + p-*)(cos(

k x • a) + c o s ( k , - a ) ) + cos( k, + k r). a + cos(k x - k y ) - a ] .

(3.7d) In order to deduce the surface magnon dispersion branches, we use the matching method, which requires the semi-infinite magnetic structure to be divided into three domains: (i) surface (n < 2), (ii) matching (1 < n < 3)

r t

!!!!!!!!!!'. ,

~

a..,

~,

,.~ Wave Vector

~

2-~

~4

l.z

IIIllllL!l !!! ..... ~

~-"

~

da

"~;. . . . .

.

A

1,4

.

.

.

.

.

.

.

.

.

.

:,'.,

W s W Vetlor

Fig. 5 . Energies of bulk and surface spin waves for J~B/J^A(aB) t~ ~ _ -- 10 plotted in the [l lO] surface direction. The variation of acoustic and -t v s /JAB. The other parameters are optical modes at the surface relative to the bulk magnon domain, as a function of the ratio ~)yI -- - J~a assumed to be constant in t h i s evolution.

376

M. Tamine /Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

....

.

.....

..........

.

~0 .............

."

~..:::..

Wive

,. :..

.....

..

.

•

.....

~0

,

Wsve

Vector

•

im .........

. ..........

G2

O.4

O.6

1

~8 Wave

12

":" .:'"~

::::'

"""

: :: "'

Vector

":!::::,-.':"i'" " ' "

"

"'**

1

t

i 00

•

14

Wave V c t ¢ o r

Vcclor

Fig. 6. As Fig. 5, for other values of e]~.

and (iii) bulk (n > 3) regions, all of which have the same 2D periodicity parallel to the surface, as mentioned in Section 1. The analysis of surface magnons branches requires the knowledge of a complete set of evanescent modes in the bulk region. These can be characterized by a complex phase factor which describes the decrease of the precessional amplitude with increasing penetration depth into the crystal [13,14]. Let us note by this factor in the normal direction of the surface: the bulk and evanescent modes of magnons can be characterized by I pl -- 1 and I pl < 1, respectively. Matching Eq. (3.1) applied to an antiferromagnetic structure with two sublattices A and B from the intermediate region of bulk matter (0 < l < 3) which is labelled by n = 3 and 4, gives: U ( n , A ( B ) ) = RIC(A(B ) ,p, )p~"-3) + R2C(A(B),p2)p~,-3).

(3.8)

So, the spin-wave amplitude for layer n = 3 is

u(3A)=RIC(A,pl) + R2C(A,p2),

U(3a)=RtC(B,p,) + R2C(A,p2 ),

(3.9)

and for layer n --- 4,

U,(A) = RIC(A,p,)P, + R2C(A,p2)P2,

U~B) = R,C(B,pl)P, + R2C(B,p2)P2"

(3.10)

M. Tamine/ Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

377

One can use a matching matrix M m (8 × 6), expressed as 'U~A)

U~ a) U~A) U~ B)

U~A)

U~ B)

U1AI UJ B)

=

0 0 0

0 0 0

0 0 0

1

0

0

0

0

C(A,p,)

C(A,p2)

0

0

0

C(B,p,)

C(B,P2)

0

0

0

0

0

0

0 0

C(A,p,)p, C(A,p,)R,

C(A,P2)P2 C(A,R2)R 2

1

0

0

0 0 0

1 0 0

0 1 0

0

0

0

U~A) U~B) U~ A )

U~ B)

(3.11)

R1 R2

The matrix which describes the localized states of magnons near a surface is also obtained by Ms(tg,K ) = M b ( 8 × 6 ) " M m ( 6 × 8 ),

O = 1 . . . . . 8, K= 1. . . . . 8,

(3.12)

with 0, K =f(E, kli,Eu~,Teq(kll)),where M b results from Eqs. (3.3)-(3.7) and Mm(E, J,~B, l~ 2~ E'uv, e ik~'") represents the matching matrix.The set of points (E, kll) which defines the surface magnon is given by det[ Ms(E,kll)] = 0 .

(3.13)

The shaded area which represents the bulk magnon band frequency is given by Eq. (2.9), and the localized surface magnon states are described by the numerical calculations of (3.13) for different values of the ratio ~}~ (for the sake of simplifying the calculations, the other parameters euo defined by (3.3) are assumed to be constant). These illustrations are shown for the [010] surface directions in Figs. 3 and 4, and for the [110] llv/12v ~- - 10. surface directions in Figs. 5 and 6, for one value of the ratio .,AB/OAA(BB)

4. Conclusions

The formalism of the matching procedure has been presented with the aim of developing a practical tool for calculating the dispersion curves of surface magnons. We have shown that this method, which was previously applied to calculate the surface phonon, can be successfully extended to describe the magnetic excitations at a surface. This feature was recently studied for an antiferromagnetic frustrated magnet system in Refs. [15,16]. This work is now being extended to frustrated ferrimagnetic systems which exhibit a competing interactions between first and second nearest neighbour exchange. Within this framework, surface magnons are analyzed in terms of an incoming component and scattered components: so that this feature may permit a description of the spectral density behaviour in the vicinity of a Van Hove singularity.

Acknowledgements

The author acknowledges Professor A. Khater and Dr J.M. Greneche for useful discussions.

References [1] T. Wolfram and R.E. De Wames, in: Progress in Surface Science, Vol. 2, ed. S.G. Davison (1972). [2] N.N. Chen and M.G. Cottam, Phys. Rev. B 44 (1991) 7466.

378 [3] [4] [5] [6] [7] [8] [9] [10] [l l] [12] [13] [14] [15] [16]

M. Tamine / Journal of Magnetism and Magnetic Materials 153 (1996) 366-378

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