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Surface spin waves in ferromagnetic thin films with nn and nnn exchange
Journal of Magnetism and Magnetic Materials i114- !117 (1992) 17117-17118 North-Holland
Surface spin waves in ferromagnetic thin films with nn and nnn exchange P. Politi ", A. Rettori b and M . G . Pini ~ a Dipartimento di Fisica. Unicersita" di Firenze, 1-50125 Firenze. Italy t, istituto di Fisica, Uni~'ersita' di Siena. 1-53100 Siena. Italy " lstituto di Eiettronica Quantistica, CNR, 1-50127 Firenze, Italy
The spectrum of the spin-wave excitations for a multilayer of spins inleracting via the Heisenberg ttamiltonian with nn and nnn exchange is investigated within a microscopic approach. Two types of surface magnons (monotonic and oscillating) are found and their relevance fol the thermodynamic properties is shown.
In the study of surface spin wavcs in thin films [1] many experimental results can be well accounted for within a continuum approximation, provided that the wavelength is sufficiently long (magnetostatic region). In thc exchange-dominated region (k > l0 s m I) a microscopic approach is more appropriate. We present a theoretical study of the linear excitations of a thin ferromagnetic film consisting of N layers of spins, parallel to thc (100) surface of an sc lattice, interacting via the Hciscnbcrg Hamiltonian: 37z~ = -JlY'.(i.j}Si " Sj - J2Eti.,,,~Si " S,,,. It is wcll known [2] that thc inclusion of both nn (J~ > 0) and nnn ( J , > 0) interactions leads to the excitation of surfacc spin waves ,ith encrgy lower than the bulk ones. While such a model has bccn thoroughly investigated i~l the scmi-infinitc limit [2], a dctailcd study of the surface modes for films of finite thickncss is, to our knowledge, still lacking. Moreover, the case of J2 4:0 appcars physically sound: e.g. for epitaxial thin films of bcc Fe/Au(100), the n n n interaction is the only onc rcsponsible for the long range order obscrvcd at room temperature in the monolayer limit [3]. The main rcsult of our analysis is the prcdiction, for sufficiently high wavc vectors, of oscillating surfacc m a g n o n s in addition to the usual monotonic surface ones. While both of them arc characterized by an exponential decay of thc amplitude of the spin fluctuations, the oscillating ones present a "rr phase variation when passing from a layer to the adjacent one. Cicarly such oscillating modes can only bc obtaincd within a microscopic approach, not limited to the Iow-k rcgion. We assume the magnetization to lie along the z direction, in the x z plane parallel to thc surfaces. Wc iinearize the H e i s e n b e r g equations of motion for the transversal components ( I S ' l = I S ' l ) of thc spin operators in the frec spin wavc approximation. Taking into account that the translational invariancc is preserved in the xz plane, we arc led to the N × N
eigenvalue problem (l, m = 1, 2 . . . . . N): ¢oSt(kll)= Y'.,,, A t.,,,(k II)S,,,( k II)' where k II = ( k .,, k. ) with - w < k ,, k . < rr and l, m denote the plane indexes. The frequencies of the excitations are given by the eigenvalues of the real symmetric tridiagonal matrix: At.,,,(kll ) = g t . t ~ H C n ( l ) ' 6 t , , , - SJ(kll; i - m ) , where for / = 1, N one has H~d(I) = H, = H + S ( 5 J t + 8J2)/gl.ti~ whilc H C f t ( l ) = H i = H + S ( 6 J I + 1 2 J , ) / g # B for ! = 2 . . . . . N - 1. The only nonzero Fourier transforms of the exchange arc J(kll; 0) = S ( 4 J l y ~ + 4J2y_,) and J(kll', 1) = S ( J I + 4J_,yl), with Yl = (cos k, + cos k : ) / 2 and y_, = cos k~ cos k:. The cigcnvcctors have definite parity, i.e. they arc symmetric ( + ) or antisymmctric ( - ) with respect to the center of the multilaycr, and lake the form: St(k,) =x~t i~+x~,X t~. D e p e n d i n g on the expression of x. they can bc classified into three different kinds of excitations: monotonic (M) surface lk~r x real positive (x = c - " ) , oscillating (O) surface for x real ncgativc (x = - c - " ) and bulk magnons for x complex with modulus I (x = ei"). From thc boundary conditions, the value of o¢ is d c t e r m i n c d as thc solution of the equation:
= - s,_(k~)/S,(kLt
) + (.; + ,
').
In fig. 1 we represent schematically the evolution, within the first Brillouin zonc, of thc number and type of surface excitations for different vaiucs ,ff N a~ld tff thc ratio z =J1/(AJ,)._ For a fixed value fo k, I. one can have up to two surface modes. With increasing the wave vector, their character changes from monotonic to oscillating. The energy of the surface modes is: oJ~ = g ~ B H i - J ( k l l ; 0 ) • 2J(kll; 1)cosh o~, (the signs and + refer to the M and O surface magnons, respectively), i.c. it is always below that of the bulk ones, which is given by: oot = g t x B H i - J ( k N, 0 ) - 2J(ktt; 1)
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
P. Politi et al. / Surface spin wal'es in .ferromagnetic thin films
17118
_&
cos c~. For kll << 1, analytical expressions can bc found for the energies of the lowest mode (a monotonic surfacc magnon) and the bulk ones:
0
(o~( kll ) = glxnH + 4,I~S( I - y~ ) + 4,I2S( 1 - Y2)
+
81~.s(1
-
I/N)(i
-
r,),
i .,,,.,
O Q t I O g a a ~
,or( kll ) = gtztin + 2S( J, + 4.I,)[1 - cos i f ( l ) ]
I
0
+ 4JiS(! - 7 t ) + 4J2S(l - 72)
×[(l-I/U)cosff(i)--1/N],
A(i) = S - < S : ( G = i ) > ~
d-'kSf'(kll; r,.=i)
×
-
kz
(21
',
kz 2M
a
I
20
(1)
with i f ( l ) = l w / N (1 = O, 1. . . . . N - 1). In the limit N >> I we recover the results already obtained by Mills and Maradudin [2]. For J, = 0 no surface modes are present and D6ring's [4] result is obtained. Notice that in correspondence of the curve (dashed line in fig. I) of equation: J(kll; I ) = 0, which divides the monotonic surface region from the oscillating one, one has ~ ~ ~, i.e. the two surface modes are completely localized (on the first and the last plane) and degenerate in energy: w s = g l ~ B H s - J ( k l l ; 01; the energy of the bulk modes w t = gl.LBH i -- J(kll; 0) presents an (N - 2) degeneracy. Using a boson approach, the spin-wave deviation of the magnetization of the /-plane from the saturation value is expressed in terms of the eigcnvcctors and cigcnvalucs of the matrix A as:
(,)/
I
Fig. 2. Effect of the surface modes on the magnetization profile of an N = 43 multilayer.
+ 8J, S ( l - 71)
= E /
I
10
IM
I
b
w h e r e S/(kll; ry = i) represents the amplitude of the
oscillation on the i-plane when the /-mode, with frequency ~t(klt), is excited. For real systems, such a property can be measured by means of conversion electron M6ssbauer spectroscopy [5]. In fig. 2 we show the normalized magnetization profile [ A ( i ) - A ( c ) ] / A(c) (where c = ( N + 11/21 versus the plane index i for a N = 43 film with JiS = 1, J2S = 0.5 at a fixed t e m p e r a t u r e T = 0.5JefrS, w h e r e Jcrr =Ji + 4J2[1 - 1 / (2N)]. A small magnetic field ( g t z n H = 10-4jiS) was introduced in the dispersion relation in order to obtain convergence. The effect of the surface modes, present only for J2 ~ 0 (full triangles), is apparent from the comparison with the magnetization profile obtained for an N = 43 film with J, = 0 and rcscaled parameter JIS =J~rrS = 2.91 (open circles). The localizcd modes act essentially on the two outermost layers, while for the inner layers the spin deviations arc almost the same as in the absence of surface modes. In conclusion, using a microscopic approach wc have shown that thin films with ferromagnetic nn and nnn exchange can present up to two surface magnons, whose character changes from monotonic to oscillating with increasing wave vector. The energies of :.he surface modes are found to iic below that of the bulk ones and their effect on the magnetization profile has been calculated. References
,,.20-.. t a 3
!_ ff
0
g
x
Fig. I. Schematic plot for the k-dependence of the number and type of surface modes for different values of N and r = J t / 4 , l , : (a) r > N : (b) i < r < N : (c) ! / N < r < l ; (d) r< I/N.
[1] M.G. Cottam and D.R. Tillcy, Introduction to Surface and Superlattice Excitations (Cambridge Univ. Press, Cambridge, !q8q) chaps 3 and 4. [2] D . L Mills and A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. 't3] W. Diirr, M. Taborclli, O. Paul, R. Gcrmar, W. Gudat, D. Pescia and M. Landolt, Plays. Rev. l,ett. 62 (1989) 206. [4] W. D6ring, Z. Naturf. 16a (1961) i146. [5] M. Przybylsky, J. Korecki and U. G:-~dmann, Appl. Phys. A 52 (19911 33.
Surface spin waves in ferromagnetic thin films with nn and nnn exchange P. Politi ", A. Rettori b and M . G . Pini ~ a Dipartimento di Fisica. Unicersita" di Firenze, 1-50125 Firenze. Italy t, istituto di Fisica, Uni~'ersita' di Siena. 1-53100 Siena. Italy " lstituto di Eiettronica Quantistica, CNR, 1-50127 Firenze, Italy
The spectrum of the spin-wave excitations for a multilayer of spins inleracting via the Heisenberg ttamiltonian with nn and nnn exchange is investigated within a microscopic approach. Two types of surface magnons (monotonic and oscillating) are found and their relevance fol the thermodynamic properties is shown.
In the study of surface spin wavcs in thin films [1] many experimental results can be well accounted for within a continuum approximation, provided that the wavelength is sufficiently long (magnetostatic region). In thc exchange-dominated region (k > l0 s m I) a microscopic approach is more appropriate. We present a theoretical study of the linear excitations of a thin ferromagnetic film consisting of N layers of spins, parallel to thc (100) surface of an sc lattice, interacting via the Hciscnbcrg Hamiltonian: 37z~ = -JlY'.(i.j}Si " Sj - J2Eti.,,,~Si " S,,,. It is wcll known [2] that thc inclusion of both nn (J~ > 0) and nnn ( J , > 0) interactions leads to the excitation of surfacc spin waves ,ith encrgy lower than the bulk ones. While such a model has bccn thoroughly investigated i~l the scmi-infinitc limit [2], a dctailcd study of the surface modes for films of finite thickncss is, to our knowledge, still lacking. Moreover, the case of J2 4:0 appcars physically sound: e.g. for epitaxial thin films of bcc Fe/Au(100), the n n n interaction is the only onc rcsponsible for the long range order obscrvcd at room temperature in the monolayer limit [3]. The main rcsult of our analysis is the prcdiction, for sufficiently high wavc vectors, of oscillating surfacc m a g n o n s in addition to the usual monotonic surface ones. While both of them arc characterized by an exponential decay of thc amplitude of the spin fluctuations, the oscillating ones present a "rr phase variation when passing from a layer to the adjacent one. Cicarly such oscillating modes can only bc obtaincd within a microscopic approach, not limited to the Iow-k rcgion. We assume the magnetization to lie along the z direction, in the x z plane parallel to thc surfaces. Wc iinearize the H e i s e n b e r g equations of motion for the transversal components ( I S ' l = I S ' l ) of thc spin operators in the frec spin wavc approximation. Taking into account that the translational invariancc is preserved in the xz plane, we arc led to the N × N
eigenvalue problem (l, m = 1, 2 . . . . . N): ¢oSt(kll)= Y'.,,, A t.,,,(k II)S,,,( k II)' where k II = ( k .,, k. ) with - w < k ,, k . < rr and l, m denote the plane indexes. The frequencies of the excitations are given by the eigenvalues of the real symmetric tridiagonal matrix: At.,,,(kll ) = g t . t ~ H C n ( l ) ' 6 t , , , - SJ(kll; i - m ) , where for / = 1, N one has H~d(I) = H, = H + S ( 5 J t + 8J2)/gl.ti~ whilc H C f t ( l ) = H i = H + S ( 6 J I + 1 2 J , ) / g # B for ! = 2 . . . . . N - 1. The only nonzero Fourier transforms of the exchange arc J(kll; 0) = S ( 4 J l y ~ + 4J2y_,) and J(kll', 1) = S ( J I + 4J_,yl), with Yl = (cos k, + cos k : ) / 2 and y_, = cos k~ cos k:. The cigcnvcctors have definite parity, i.e. they arc symmetric ( + ) or antisymmctric ( - ) with respect to the center of the multilaycr, and lake the form: St(k,) =x~t i~+x~,X t~. D e p e n d i n g on the expression of x. they can bc classified into three different kinds of excitations: monotonic (M) surface lk~r x real positive (x = c - " ) , oscillating (O) surface for x real ncgativc (x = - c - " ) and bulk magnons for x complex with modulus I (x = ei"). From thc boundary conditions, the value of o¢ is d c t e r m i n c d as thc solution of the equation:
= - s,_(k~)/S,(kLt
) + (.; + ,
').
In fig. 1 we represent schematically the evolution, within the first Brillouin zonc, of thc number and type of surface excitations for different vaiucs ,ff N a~ld tff thc ratio z =J1/(AJ,)._ For a fixed value fo k, I. one can have up to two surface modes. With increasing the wave vector, their character changes from monotonic to oscillating. The energy of the surface modes is: oJ~ = g ~ B H i - J ( k l l ; 0 ) • 2J(kll; 1)cosh o~, (the signs and + refer to the M and O surface magnons, respectively), i.c. it is always below that of the bulk ones, which is given by: oot = g t x B H i - J ( k N, 0 ) - 2J(ktt; 1)
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
P. Politi et al. / Surface spin wal'es in .ferromagnetic thin films
17118
_&
cos c~. For kll << 1, analytical expressions can bc found for the energies of the lowest mode (a monotonic surfacc magnon) and the bulk ones:
0
(o~( kll ) = glxnH + 4,I~S( I - y~ ) + 4,I2S( 1 - Y2)
+
81~.s(1
-
I/N)(i
-
r,),
i .,,,.,
O Q t I O g a a ~
,or( kll ) = gtztin + 2S( J, + 4.I,)[1 - cos i f ( l ) ]
I
0
+ 4JiS(! - 7 t ) + 4J2S(l - 72)
×[(l-I/U)cosff(i)--1/N],
A(i) = S - < S : ( G = i ) > ~
d-'kSf'(kll; r,.=i)
×
-
kz
(21
',
kz 2M
a
I
20
(1)
with i f ( l ) = l w / N (1 = O, 1. . . . . N - 1). In the limit N >> I we recover the results already obtained by Mills and Maradudin [2]. For J, = 0 no surface modes are present and D6ring's [4] result is obtained. Notice that in correspondence of the curve (dashed line in fig. I) of equation: J(kll; I ) = 0, which divides the monotonic surface region from the oscillating one, one has ~ ~ ~, i.e. the two surface modes are completely localized (on the first and the last plane) and degenerate in energy: w s = g l ~ B H s - J ( k l l ; 01; the energy of the bulk modes w t = gl.LBH i -- J(kll; 0) presents an (N - 2) degeneracy. Using a boson approach, the spin-wave deviation of the magnetization of the /-plane from the saturation value is expressed in terms of the eigcnvcctors and cigcnvalucs of the matrix A as:
(,)/
I
Fig. 2. Effect of the surface modes on the magnetization profile of an N = 43 multilayer.
+ 8J, S ( l - 71)
= E /
I
10
IM
I
b
w h e r e S/(kll; ry = i) represents the amplitude of the
oscillation on the i-plane when the /-mode, with frequency ~t(klt), is excited. For real systems, such a property can be measured by means of conversion electron M6ssbauer spectroscopy [5]. In fig. 2 we show the normalized magnetization profile [ A ( i ) - A ( c ) ] / A(c) (where c = ( N + 11/21 versus the plane index i for a N = 43 film with JiS = 1, J2S = 0.5 at a fixed t e m p e r a t u r e T = 0.5JefrS, w h e r e Jcrr =Ji + 4J2[1 - 1 / (2N)]. A small magnetic field ( g t z n H = 10-4jiS) was introduced in the dispersion relation in order to obtain convergence. The effect of the surface modes, present only for J2 ~ 0 (full triangles), is apparent from the comparison with the magnetization profile obtained for an N = 43 film with J, = 0 and rcscaled parameter JIS =J~rrS = 2.91 (open circles). The localizcd modes act essentially on the two outermost layers, while for the inner layers the spin deviations arc almost the same as in the absence of surface modes. In conclusion, using a microscopic approach wc have shown that thin films with ferromagnetic nn and nnn exchange can present up to two surface magnons, whose character changes from monotonic to oscillating with increasing wave vector. The energies of :.he surface modes are found to iic below that of the bulk ones and their effect on the magnetization profile has been calculated. References
,,.20-.. t a 3
!_ ff
0
g
x
Fig. I. Schematic plot for the k-dependence of the number and type of surface modes for different values of N and r = J t / 4 , l , : (a) r > N : (b) i < r < N : (c) ! / N < r < l ; (d) r< I/N.
[1] M.G. Cottam and D.R. Tillcy, Introduction to Surface and Superlattice Excitations (Cambridge Univ. Press, Cambridge, !q8q) chaps 3 and 4. [2] D . L Mills and A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. 't3] W. Diirr, M. Taborclli, O. Paul, R. Gcrmar, W. Gudat, D. Pescia and M. Landolt, Plays. Rev. l,ett. 62 (1989) 206. [4] W. D6ring, Z. Naturf. 16a (1961) i146. [5] M. Przybylsky, J. Korecki and U. G:-~dmann, Appl. Phys. A 52 (19911 33.