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Theory of surface spin-waves in a semi-infinite ferrimagnet
Volume 53A, number 2
PHYSICS LETTERS
2 June 1975
THEORY OF SURFACE SPIN-WAVES IN A SEMI-INFINITE FERRIMAGNET D.T. HUNG, I. HARADA and 0. NAGAI Department of Physics, Kobe University, Rokkodai, Kobe 657, Japan Received 1 March 1975 The surface spin-wave spectra of both the body-centered and simple cubic ferrimagnets are studied theoretically.
The theoretical studies of surface spin-waves in ferro- and antiferromagnets have been reported recently by several workers [1—4]. It is known that the surface
a lattice point in the B-sublattice is denoted by m. The quantity HAJ (or HAm) is the anisotropy field at the lattice site R• (or Rm) and H0 denotes the external
spin-wave mode may have an important role on the physical properties of magnetic systems [1,2,5]. In this letter we report on a study of the surface spinwave spectrum of a Heisenberg ferrimagnet by using a simple model for the crystal. Although actual fernmagnets have complicated structure as seen in ferrites and garnets, the general features of the spin-wave spectra described in this study will be found in the case of actual crystal. We first consider a two-sublattice body-centered cubic (bcc) ferrimagnet. A simple cubic (sc) ferrimagnet case will be discussed later in this letter. We divide the magnetic bcc lattice into two sc sublattices, A and B. Each lattice point of A-sublattice is occupied by an up spin of magnitude SA and each lattice point of other sublattice is occupied by a down spin of magnitudeSB.WeputSA=(1+P)SandSB(l—P)S.A parameter P stands for a numerical value which is proportional to the difference between SA and SB. consider the crystal lattice which is assumed to lie in the regionx >0 with a (100) surface of A spins. The spin quantization axis is taken to be the z-axis. The Hamiltonian which we consider here is written as
magnetic field applied parallel to the z-direction, p being the Bohr magneton. In what follows, we take the exchange integral between a surface spin and its nn to be .J~,and all the other exchange integrals to be J. It is known that the spin-wave spectrum in the case of antiferromagnet is not strongly affected by the surface anisotropy and the zero point spin contraction [6]. Hence, we adopt the following two assumptions in the present case: (i) HAJ = HA and HAm = HB. (ii) The average value of S1~, is equal to (1 +P)S and (Smz) = (1 If we use the above-mentioned assumptions and if we use the conventional methods [3] we obtain the following set of equations for the spin deviations:
5m
2p
~T~(H
+HAJ)SJz 0
i
,
—
°
(E+E’)U2 =-~E1(1—P)(U2 1+U2 +~), forn~1 ~ (E — E ) 0
n
n—
u2
=
n—
(1 +P)E1((J2
fl
),
2 + U2
n—
for n
~2
n
where E = (w pH0)/JS, w being the eigen-frequency of the spin deviations. The quantities U,~are the spinwave amplitudes for the nth layer. The surface is the layer n = 1. We have used the following abbreviations: 4cos (k~a/2) cos (k~a/2), 8(1 the F) lattice + EA and=E~ = 8(1 + F) + EB, where E a~ denotes constant and EA and EB are given by PHA/JS and PHB/JS respectively. In the above equations, we as—
—
H = (j,m) ~ ‘Tjm ~i ~ —
—
—
I~ (H0
2j~
HAm)Smz,
m
where is the exchange integral between a spin at the lattice site R1 and a nearest neighbor (nn) at Rm. A lattice point in the A-sublattice is denoted by! and
sumed the free surface(J1 J) for the sake of brevity. From these difference equations, E is calculated as a function of and k~. The solutions for the surface spin-wave frequency with k~= k~= 0 are given by 157
Volume 53A, number 2
\
PHYSICS LE~FERS
(a) j_~=~~_~ ~ ~E
Pr—I/3j
_______
~—1oH -i.oH
12
______
2510
~25
nrl
05/
_____________________________
I
0 2
0 4
0.6
0.8
0.2
—A
0.4
0.6
H
2
2 June 1975
Ho.5h~:
-~O5H
3
Fig. 2. The bccE~
—1/3 and
0.8
—A
mode may be rather interesting because the similar mode does not appear in the ferro- and antiferromagfigure net cases. represent Invalues fig. bulk 1,ofwe illustrate eigen-vector energyareas spectrum forinthe for several Tstates. = J1/J.The Thethe shaded the -12
-12
Fig. 1. The 2 (kyal2) energy COS2 of the (kza/2) surface forspin-wave various values versusofAT = J = 1 — c05 1/J and EA = EB = 0.01. (a) P = —1/3 and (b) P 1/3. The shaded areas correspond to bulk states for values of k~between 0 and lr/a for a given value of A.
2 + (EA +EB)] 1/2~ ET~= —4F±4[P
EB ~ Fl and if we expand E~.in obtainE~<= 81P1 andE~< = ---4EA/IPI forP< 0. On the other hand, the energy at the optical band bottom is equal to l6lFl and the energy at the acoustic band bottom is equal to —4EA/IPI. We have (U 20+1/U2~ i) = (1 +F)/(l —F) and(U2n/U2n1)—l forE=E~
If we assume EA
=
powers of EA, we
—
+
—(1 —P)/(1 +F)forE=E~<. Thus, onlyE5<. mode is physically meaningful in the case P < 0. Similarly, only E~>-mode,of which the energy is equal to 4EA/IPI, is meaningful in the case F> 0. In other words, if the surface contains spins of the sublattice of smaller spin magnitude, the surface spin-wave energy for the free surface is rather high and it depends not on the anisotropy energy but on the total magnetization. While, if the surface contains spins of the sublattice of greater spin magnitude, the surface spinwave mode of k~= k~= 0 is degenerate with the acoustic band bottom and its energy strongly depends on the anisotropy energy. The above-mentioned E~<
E~<-modeofk~= = 0 is shown in fig. 2 for the first four 1ayers~The E~>-mode for F> 0 is very similar to the well-known surface antiferromagnetic resonance mode [3]. Finally, we give a brief discussion on the surface spin-wave mode in a sc ferrimagnet with a (100) surface. There are many differences between the study of the (100) surface of the bcc crystal and that of the (100) surface of the sc crystal. This situation is the same plicated as that and lengthy in the antiferromagnet calculation, we case. obtain After two akinds cornmodes, besides the surface modes located above the acoustic and optical bands. One of them, which has the lower energy, is degenerate in energy at k~= = 0 with the acoustic band bottom arid the other, which has the higher energy, is degenerate of surface spin-wave
at k~=
=
0 with the optical band bottom.
One of the authors (I.H.) is indebted to Sakkokai Foundation for financial support.
=
-
158
References 11 D.L. Mills and A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. [2] D.L. Mills and W.M. Saslow, Phys. Rev. 171 (1968) 488. [3] R.E. De Wames and T. Wolfram, Phys. Rev. 185 (1969) 752 [4] T. Wolfram and R.E. Dc Wames, Phys. Rev. 185 (1969) 762. [51 F. Keffer and II. Chow, Phys. Rev. Lett. 31(1973)1061. [6] D.T. Hung, I. Harada and 0. Nagai, private communication.
PHYSICS LETTERS
2 June 1975
THEORY OF SURFACE SPIN-WAVES IN A SEMI-INFINITE FERRIMAGNET D.T. HUNG, I. HARADA and 0. NAGAI Department of Physics, Kobe University, Rokkodai, Kobe 657, Japan Received 1 March 1975 The surface spin-wave spectra of both the body-centered and simple cubic ferrimagnets are studied theoretically.
The theoretical studies of surface spin-waves in ferro- and antiferromagnets have been reported recently by several workers [1—4]. It is known that the surface
a lattice point in the B-sublattice is denoted by m. The quantity HAJ (or HAm) is the anisotropy field at the lattice site R• (or Rm) and H0 denotes the external
spin-wave mode may have an important role on the physical properties of magnetic systems [1,2,5]. In this letter we report on a study of the surface spinwave spectrum of a Heisenberg ferrimagnet by using a simple model for the crystal. Although actual fernmagnets have complicated structure as seen in ferrites and garnets, the general features of the spin-wave spectra described in this study will be found in the case of actual crystal. We first consider a two-sublattice body-centered cubic (bcc) ferrimagnet. A simple cubic (sc) ferrimagnet case will be discussed later in this letter. We divide the magnetic bcc lattice into two sc sublattices, A and B. Each lattice point of A-sublattice is occupied by an up spin of magnitude SA and each lattice point of other sublattice is occupied by a down spin of magnitudeSB.WeputSA=(1+P)SandSB(l—P)S.A parameter P stands for a numerical value which is proportional to the difference between SA and SB. consider the crystal lattice which is assumed to lie in the regionx >0 with a (100) surface of A spins. The spin quantization axis is taken to be the z-axis. The Hamiltonian which we consider here is written as
magnetic field applied parallel to the z-direction, p being the Bohr magneton. In what follows, we take the exchange integral between a surface spin and its nn to be .J~,and all the other exchange integrals to be J. It is known that the spin-wave spectrum in the case of antiferromagnet is not strongly affected by the surface anisotropy and the zero point spin contraction [6]. Hence, we adopt the following two assumptions in the present case: (i) HAJ = HA and HAm = HB. (ii) The average value of S1~, is equal to (1 +P)S and (Smz) = (1 If we use the above-mentioned assumptions and if we use the conventional methods [3] we obtain the following set of equations for the spin deviations:
5m
2p
~T~(H
+HAJ)SJz 0
i
,
—
°
(E+E’)U2 =-~E1(1—P)(U2 1+U2 +~), forn~1 ~ (E — E ) 0
n
n—
u2
=
n—
(1 +P)E1((J2
fl
),
2 + U2
n—
for n
~2
n
where E = (w pH0)/JS, w being the eigen-frequency of the spin deviations. The quantities U,~are the spinwave amplitudes for the nth layer. The surface is the layer n = 1. We have used the following abbreviations: 4cos (k~a/2) cos (k~a/2), 8(1 the F) lattice + EA and=E~ = 8(1 + F) + EB, where E a~ denotes constant and EA and EB are given by PHA/JS and PHB/JS respectively. In the above equations, we as—
—
H = (j,m) ~ ‘Tjm ~i ~ —
—
—
I~ (H0
2j~
HAm)Smz,
m
where is the exchange integral between a spin at the lattice site R1 and a nearest neighbor (nn) at Rm. A lattice point in the A-sublattice is denoted by! and
sumed the free surface(J1 J) for the sake of brevity. From these difference equations, E is calculated as a function of and k~. The solutions for the surface spin-wave frequency with k~= k~= 0 are given by 157
Volume 53A, number 2
\
PHYSICS LE~FERS
(a) j_~=~~_~ ~ ~E
Pr—I/3j
_______
~—1oH -i.oH
12
______
2510
~25
nrl
05/
_____________________________
I
0 2
0 4
0.6
0.8
0.2
—A
0.4
0.6
H
2
2 June 1975
Ho.5h~:
-~O5H
3
Fig. 2. The bccE~
—1/3 and
0.8
—A
mode may be rather interesting because the similar mode does not appear in the ferro- and antiferromagfigure net cases. represent Invalues fig. bulk 1,ofwe illustrate eigen-vector energyareas spectrum forinthe for several Tstates. = J1/J.The Thethe shaded the -12
-12
Fig. 1. The 2 (kyal2) energy COS2 of the (kza/2) surface forspin-wave various values versusofAT = J = 1 — c05 1/J and EA = EB = 0.01. (a) P = —1/3 and (b) P 1/3. The shaded areas correspond to bulk states for values of k~between 0 and lr/a for a given value of A.
2 + (EA +EB)] 1/2~ ET~= —4F±4[P
EB ~ Fl and if we expand E~.in obtainE~<= 81P1 andE~< = ---4EA/IPI forP< 0. On the other hand, the energy at the optical band bottom is equal to l6lFl and the energy at the acoustic band bottom is equal to —4EA/IPI. We have (U 20+1/U2~ i) = (1 +F)/(l —F) and(U2n/U2n1)—l forE=E~
If we assume EA
=
powers of EA, we
—
+
—(1 —P)/(1 +F)forE=E~<. Thus, onlyE5<. mode is physically meaningful in the case P < 0. Similarly, only E~>-mode,of which the energy is equal to 4EA/IPI, is meaningful in the case F> 0. In other words, if the surface contains spins of the sublattice of smaller spin magnitude, the surface spin-wave energy for the free surface is rather high and it depends not on the anisotropy energy but on the total magnetization. While, if the surface contains spins of the sublattice of greater spin magnitude, the surface spinwave mode of k~= k~= 0 is degenerate with the acoustic band bottom and its energy strongly depends on the anisotropy energy. The above-mentioned E~<
E~<-modeofk~= = 0 is shown in fig. 2 for the first four 1ayers~The E~>-mode for F> 0 is very similar to the well-known surface antiferromagnetic resonance mode [3]. Finally, we give a brief discussion on the surface spin-wave mode in a sc ferrimagnet with a (100) surface. There are many differences between the study of the (100) surface of the bcc crystal and that of the (100) surface of the sc crystal. This situation is the same plicated as that and lengthy in the antiferromagnet calculation, we case. obtain After two akinds cornmodes, besides the surface modes located above the acoustic and optical bands. One of them, which has the lower energy, is degenerate in energy at k~= = 0 with the acoustic band bottom arid the other, which has the higher energy, is degenerate of surface spin-wave
at k~=
=
0 with the optical band bottom.
One of the authors (I.H.) is indebted to Sakkokai Foundation for financial support.
=
-
158
References 11 D.L. Mills and A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. [2] D.L. Mills and W.M. Saslow, Phys. Rev. 171 (1968) 488. [3] R.E. De Wames and T. Wolfram, Phys. Rev. 185 (1969) 752 [4] T. Wolfram and R.E. Dc Wames, Phys. Rev. 185 (1969) 762. [51 F. Keffer and II. Chow, Phys. Rev. Lett. 31(1973)1061. [6] D.T. Hung, I. Harada and 0. Nagai, private communication.