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Magnetic excitations in sinusoidally modulated spin structures
Journal of Magnetism and Magnetic Materials 22 (1980) 93-97 © North-Holland Publishing Company
MAGNETIC EXCITATIONS IN SINUSOIDALLY MODULATED SPIN STRUCTURES S.H. LIU Ames Laboratory-USDOE * and Department of Physics, Iowa State University, Ames, IA 50011, USA, and Institut far Theorie der Kondensierten Materie, Freie UniversitavtBerlin, Berlin, Germany
Received 21 July 1980
We show by using a simple model that in incommensurate sinusoidally modulated spin structures, such as those found in Er and Tm just below their magnetic ordering temperatures and in the ordered phase of Cr, the magnon modes are broadened due to the spatial fluctuations of the exchange and anisotropy energies.
[5 ] on Er has detected broad inelastic responses near the magnetic satellites, indicating overdamped magnetic excitations. Chromium also has sinusoidally modulated spin structures in the ordered phase [6]. The magnetic structure is longitudinal at low temperatures, but becomes transverse at higher temperatures [7,8]. The vector Q is near the zone boundary along a (100) direction. This implies that the magnetic moments in adjacent (100) planes tend to align antiparallel. Doping the crystal with a small amount of Mn shifts the Q vector to the zone boundary [9], thereby making the magnetic structure commensurate with the lattice. Until recently all measurements of the spin waves of Cr were carried out in the commensurate sample [10,11 ]. One observes that spin-wave branches with very high velocity emerge from the zone boundary, which is the magnetic satellite point. This result is consistent with the prediction of the itinerant model [10,12]. Fincher et al. [13] succeeded recently in performing inelastic neutron scattering experiment on the incommensurate Cr. They observed magnons near the magnetic satellite points, but in addition there was an absorption feature at 4 meV at the zone boundary points (0, 0, I) and (0, 1,0). The latter was unexpected, and it is not clear whether the itinerant model could explain this mode. Wolfram and Ellialtioglu [ 14] have demonstrated that a localized spin model with sinusoidally modulated moments could explain qualitatively the observed behaviour. The equations of the
1. Introduction The rare-earth metals Er and Tm have sinusoidally modulated spin structures for a range of temperature immediately below their magnetic ordering temperatures [1,2]. The spins in each hexagonal plane of the crystal are ferromagnetically aligned in the direction perpendicular to the plane, and the size of the ordered moment varies sinusoidally from plane to plane with a periodicity that is not an integral multiple of the crystal c-axis. This is a special case of a class of periodic spin structures which can be characterized by a wavevector Q. For Er and Tm the vector Q is not far from the centre of the Brillouin zone and points along the c-axis. The modulated spin structure is said to be longitudinal because the spins are in the direction of Q. Nishikubo and Nagamiya [3 ] showed that this kind of spin ordering arises due to an interplay of the longrange exchange interaction, ~hich favours a periodic spin arrangement, and a uniaxial anisotropy, which forces the.spins to lie in the c direction. Cooper [4] discussed the equations of motion of the spins in the sinusoidally modulated phase. He showed that the elementary excitations are not ordinary spin waves. The preliminary measurement of Wakabayashi and Nicklow * Operated for the US Department of Energy by Iowa State University under contract No. W-7405-Eng.-82. This research was supported by the Director for Energy Research, Office of Basic Energy Sciences, WPAS-AK-01-02. 93
S.H. Liu / Sinusoidally modulated spin structures
94
spin motion employed in this work are the same as those discussed by Cooper. On the other hand, whether the local moment model is applicable to the study of magnons in itinerant systems is not completely settled. Many authors have discussed this problem in recent years [15-17]. We show in this paper that for localized moment systems the spin equations of motion in the modulated spin phase can be solved so that the magnetic response function can be put in the form of an infinite continued fraction. For the low-lying modes, the infinite continued fraction can be evaluated to a good approximation. The results show that there are broadened magnon modes emerging from the zone centre as well as the magnetic reciprocal lattice points. The line broadening comes from the spatial fluctuations of the exchange and anisotropy energies. This conclusion is in qualitative agreement with the preliminary measurements [5]. For Cr the model predicts a low-lying magnon branch at the zone boundary in agreement with the observations. The magnons emerging from the magnetic satellite points have an intrinsic line width. It will require much higher resolution to verify this prediction.
In the last term the commutator is
=-2
J,jts
+ O [Sf S + + S+Sf ] .
Theory
i ~ Ga(t ) = 2 ( S t ) 6 i t 6 ( t ) + 2 ~. Jij[(Sf) Git(t) I
- all(t)] + 2D (Sf) an(t).
q
J'llSi " Sj - D ~
(S~ ) 2 .
i
(1)
We define the Fourier transform ofGa(t ) by
an(t) = 1 ~ a(k, k', 60) IV kk'
- i60t] ,
+ f(k - Q) G(k - Q, k', 60) +f(k + Q) a ( k + Q, k', co),
(6)
where (7)
and
( S/~> =
~(a) = .~]~j exp[~. (Ri - Rj)].
where Q = (0, O, Q) lies in the z direction, which is the c-axis of the crystal. We define the spin propagator by
air(t) = - i(TS+(t) Si- (0)),
60G(k, k', 60) = S[8~,~,+Q + StOe-Q]
f(k) =S[D + ~(Q) - ~ (k)],
The ordered moment at the site R i has the components ( s~ > = s co s((2. R ~),
(5)
then we can rewrite eq. (5) as
For the rare-earth systems we use a Hamiltonian which consists of a long-range exchange term and a uniaxial anisotropy term [2,18 ]: H = - ~
(4)
In the spin-wave theory we apply the mean.field approximation to decouple the spin products SrS 7 ~-, (S~) S~'. The mean-field approximation is good only for zero temperature where there are no thermally excited spin waves. The validity of this approximation to the sinusoidal phase of Er and Tm is doubtful because it is far from the ground state. Nevertheless, for lack of a better alternative, we shall proceed with this scheme and explore the consequences. One must bear in mind that the conclusions of this calculation are at best qualitative. Thus eq. (3) becomes
X exp [i(k • R i - k " R l ) 2.
s;-s';sf]
I
(2)
where S~ = S x + iS~. The equation of motion for the propagator
We define a symbol
p(k +-Q, k) = a ( k + Q, k + Q, 60)/G(k, k + Q, 60). (9) Then we find by setting k ' = k + Q in eq. (6) that
[60 - f ( k - Q) p ( k - Q, k ) - f ( k + Q ) p ( k + Q, k)] X G ( k , k + Q , 60) =S.
i -~Git(t ) = 2(S z) 8itS(t) + ( T[S~(t),H] ST(0)). (3)
(8)
1
(10)
The substitution o f k + Q for k and k' in eq. (6) leads
S.H. Liu / Sinusoidally modulated spin structures to
[60 - f ( k + 2Q) p(k + 2Q, k + Q)I p(k +-Q, k) = f(k).
(11) We combine eqs. (10) and (11) to obtain an expression of G(k, k + Q, 60) in terms of p(k + 2Q, k + Q). The procedure is repeated so that G will be expressed in terms of p(k + 3Q, k + 2Q) and so on. Since Q is not a multiple of the reciprocal lattice vector, we get an infinite chain of relations such that G ( k , k + Q, w ) = S / [ 6 0 - g ( k - Q ) - g ( k + Q)],
(12)
95
~(q) of the exchange interaction drops fairly steeply with increasing Iql. To stabilize the modulated structure it is necessary that ~(q) is maximum at q = Q. For k near the zone center, i.e. Ikl < < IQ[, we may approximate ~(k) -~ ~ ( 0 ) ,
~ ( k + Q) ~ ~ ( Q ) + a2k s ,
where for simplicity we assume isotropic k dependence near the maximum of if(q). Putting these into eq. (12) we obtain g(k +-Q) ~ f(O) (DS + ask2)~ [60 - (DS + a 2 k S ) x ] ,
(16)
where g(k -+ Q) are infinite continued fractions f(k) g(k + Q) = [60/f(k +-Q)]
1 [60/f(k +-2Q)1
(13)
1 D
[ 6 0 / f ( k _+ 3 0 ) ]
-
... "
where
In the limit of weak exchange interaction, i.e. f ( k + nQ) ~ D S for all n, we can evaluate the infinite
.x-
continued fractions in closed form:
[60/f(2Q)]
~ ( k +--Q) = D S x ,
1 - [60/f(3Q)] - . . .
(17)
We anticipate that 6o < < f ( n Q ) for n ~ 2..Then since all the quantities 60/f(nQ) are small, it is a good approximation that
where x = 1/(60/DS- x ) , or
x ~, 1/[60/f(2Q) - x]
x = 60/2DS + [(60/2DS) 2 - 1] in
which loads to x ~ i. Thus,
This gives
a(k,k
G(k, k + Q, 60) = S/( 602 - 4D2 S2) t n .
(14)
The neutron scattering cross-section is proportional to the imaginary part of G, i.e. - I m G(k, k + Q, 60) = S/(4D2S s - 602)1a, 1601< 2DS,
+ Q,60)
S/ (60 - 2f(O) (DS
+
a 2 kS)./[60 : i(DS
+
a s k2)] }.
(18) The mode frequency is solved from 60 [60 - i(DS + a2k2)] - 2f(0) (DS + a2k 2) = O,
with the result =0,
1601> 2DS. The finite linewidth comes from the sinusoidal variation of the anisotropy field. Otherwise, the magnetic excitation is dispersionless. In the opposite case where the anisotropy field is weak compared with the exchange field, an approximate reduction of the infinite continued fraction to a closed expression is possible if the Fourier transform
w ~ ~ {(DS + a 2 k 2) [2f(0) - DS - a 2 k 2] }t a + ½i(DS + a s k 2 ) .
(19)
The real part of the frequency can be shown to be small compared withf(2Q). The imaginary part,, which determines the linewidth, is partly due to the fluctuation in anisotropy, with the exchange field making an additional contribution which is more important for shorter wavelengths.
96
S.H. Liu / Sinusoidally modulated spin structures
In the intermediate case where the exchange and anisotropy energies are comparable, we may need to extend the continued fraction in eq. (13) to a higher order and approximate the rest by i. Numerical calculations may be necessary to assess the convergence and determine the necessary number of orders that must be kept. For k near 0 the approximations must be carried out differently because 6o is not small compared with f ( k - 20). We need to calculated g(k - 0 ) to one higher order a(k) h(g) - j---b(k)
+
(1/co)f(k)[f(k + Q) + f ( k - Q)] G(k, k', co)
+ (l[co)f(k + Q ) f ( k + 2 Q ) G ( k + 20, k', co).
This equation can be solved by the method used previously, with the solution
G(k, k + Q, co) = coS/[co - co (k) -
h(k + Q) -
h(k -
f(k) [co/f(k- 0)] -
I
(20)
[co/f(k - 20)] - i"
On the other hand we may take g(k + Q) = if(k).
(21)
The form of G is more complicated, but the mode frequency is again given by eq. (19). The same lineshape should be observed both near the zone centre and near the magnetic reflection point Q. In conclusion, we have shown that the magnon-like excitations in the sinusoidally modulated phase of Er and Tm are inherently broad because of the spatial fluctuation in the exchange and anisotropy energies. For Cr in the transverse spin-density-wave state the anisotropy appears to be quite complicated [ 13 ]. We will not attempt to analyse the anisotropy energy at this time, but will continue to approximate it by a uniaxial anisotropy as in eq. (1). This approach is also taken by Wolfram and Ellialtioglu [14]. The vector Q is now perpendicular to the spin axis, but it is not difficult to see that the equation of motion of the spins, eq. (6), is unaffected. However, since 0 is near the zone boundary,f(k) is nearly periodic in 20. This suggests that we must solve eq. (6) by successive iterations which advance the wavevector by 2 0 at a time. Iterating once, we find coG(k, k ' , co) = S[8~t¢+Q + 8~,k,_Q] + ( l / c o ) f ( k - Q ) f ( k - 2 Q ) G ( k - 2Q, k', co)
Q)],
(23)
where co~(k) = S 2 f ( k ) [ f ( k + Q) + f ( k - Q)],
-
(24)
a(k + 2Q) co2 _ b(k + 20)
S(k - 0 ) =
(22)
a(k + 40) ...... - co2 _ b(k + 4Q) - ... "
(25)
and a(k) = S4 f ( k - Q) f 2 ( k ) f ( k + Q ) ,
(26)
b(k) = S 2 f ( k + O ) [ f ( k ) + f ( k + 2 0 ) ] .
(27)
As in the rare-earth case, when the anisotropy energy dominates, the spin propagator is given by eq. (14). When the anisotropy energy is negligibly small compared with the exchange energy, we find that f ( Q ) = 0, and the magnon frequency given by the pole of the spin propagator in eq. (23) vanishes for k = 0 and +Q. This indicates that magnon branches emerge from the magnetic satellite points and the zone centre. Furthermore, for k near the zone boundary the magnon dispersion curve is approximately given by
co
V%oo (k),
(28)
where COo(k)is given in eq. (24). The intrinsic linewidth is very small. Because 0 is very near the zone boundary, the magnon frequency is low and has a maximum at the zone boundary, as depicted in fig. I. This result is in good agreement with the neutron data [13]. Far away from the magnetic satellite points the magnons are damped for the same reason as the rareearth case. It is tedious to write down the approximate expressions for the mode frequency. Also, the result is not of immediate interest to experimental workers because the present experimental resolution is not sufficiently high to detect the intrinsic linewidth of the magnons. The magnon dispersion curve in fig. 1 is rather dif-
S.H. Liu / Sinusoidally modulated spin structures
97
sions. The manuscript was prepared when the author was visiting the Free University Berlin. The hospitality o f Professor KJ-I. Bennemann and his group is especially acknowledged. A
t~
3
References
>,
tILl
o fz. o~ o
t
I
(o,o,Q) (o,o,~) (o,o,1 -o ) Magnon Wavevector k" Fig. 1. Schematic drawing to show the magnon dispersion curve near the zone boundary for Cr in the incommensurate phase. ferent from that predicted b y Wolfram and Ellialtioglu [14]. Whereas these authors attribute the 4 meV feature as the reciprocal lattice point to the anisotropy gap, we have shown the exchange interaction also comes into the picture. The difference between the two theories seems to lie in the details of the assumed long-range exchange interaction models.
Acknowledgements The author is indebted to Drs. S. Werner, T. Wolfram and N. Wakabayashi for informative discus-
[1] W.C. Koehler, in: Magnetic Properties of Rare Earth Metals, ed. RJ. Elliott (Plenum Press, London and New York, 1972) p. 187. [2] S.K. Sinha, in: Handbook on the Physics and Chemistry of Rare Earths, Vol. I, eds. K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1978) ch. 5. [3] T. Nishikubo and T. Nagamiya, J. Phys. Soc. Japan 20 (1965) 808. [4] B.R. Cooper, in: Solid State Physics, VoL 21, eds. F. Seitz, D. Turnball and H. Eltrenreich (Academic Press, New York, 1968) p. 393. [5 ] N. Wakabayashi and R.M. Nicklow, private communication (1979). [6] G. Shirane and W.J. Takei, J. Phys. Soc. Japan 17 Suppl. B-111 (1963) 35. [7] V.N. Bykov, V.S. Galovkin, N.V. Ageef, V.A. Levdik and S.I. Vonogradov, DokL Akad. Nauk SSSR 128 (1959) 1153. [8] G.E. Bacon, Acta Cryst. 14 (1961) 823. [9] W.C. Koehler, R.M. Moon, A.L. Trego and A.R. Mackintosh, Phys. Rev. 151 (1966) 405. [10] S.K. Sinha, SJ-I. Liu, L.D. Muldestein and N. Wakayashi, Phys. Rev. Lett. 23 (1969) 311. [11] J. Als-Nielsen, J.D. Axe and G. Shirane, J. Appl. Phys. 42 (1971) 1666. [12] S.H, Liu, Phys. Rev. B2 (1970) 2664. [13] C.R. Fincher, Jr., G. Shirane and S.A. Werner, Phys. Rev. Lett. 43 (1979) 1441. [14] T. Wolfram and S. Ellialtioglu, Phys. Rev. Lett. 44 (1980) 1295. [15] S.H. Liu, Phys. Rev. B13 (1976) 3962. [16] H. Capellmann, Z. Phys. B34 (1979) 29. [17] J. Hubbard, Phys. Rev. B19 (1979) 2826, B20 (1979) 4584.
MAGNETIC EXCITATIONS IN SINUSOIDALLY MODULATED SPIN STRUCTURES S.H. LIU Ames Laboratory-USDOE * and Department of Physics, Iowa State University, Ames, IA 50011, USA, and Institut far Theorie der Kondensierten Materie, Freie UniversitavtBerlin, Berlin, Germany
Received 21 July 1980
We show by using a simple model that in incommensurate sinusoidally modulated spin structures, such as those found in Er and Tm just below their magnetic ordering temperatures and in the ordered phase of Cr, the magnon modes are broadened due to the spatial fluctuations of the exchange and anisotropy energies.
[5 ] on Er has detected broad inelastic responses near the magnetic satellites, indicating overdamped magnetic excitations. Chromium also has sinusoidally modulated spin structures in the ordered phase [6]. The magnetic structure is longitudinal at low temperatures, but becomes transverse at higher temperatures [7,8]. The vector Q is near the zone boundary along a (100) direction. This implies that the magnetic moments in adjacent (100) planes tend to align antiparallel. Doping the crystal with a small amount of Mn shifts the Q vector to the zone boundary [9], thereby making the magnetic structure commensurate with the lattice. Until recently all measurements of the spin waves of Cr were carried out in the commensurate sample [10,11 ]. One observes that spin-wave branches with very high velocity emerge from the zone boundary, which is the magnetic satellite point. This result is consistent with the prediction of the itinerant model [10,12]. Fincher et al. [13] succeeded recently in performing inelastic neutron scattering experiment on the incommensurate Cr. They observed magnons near the magnetic satellite points, but in addition there was an absorption feature at 4 meV at the zone boundary points (0, 0, I) and (0, 1,0). The latter was unexpected, and it is not clear whether the itinerant model could explain this mode. Wolfram and Ellialtioglu [ 14] have demonstrated that a localized spin model with sinusoidally modulated moments could explain qualitatively the observed behaviour. The equations of the
1. Introduction The rare-earth metals Er and Tm have sinusoidally modulated spin structures for a range of temperature immediately below their magnetic ordering temperatures [1,2]. The spins in each hexagonal plane of the crystal are ferromagnetically aligned in the direction perpendicular to the plane, and the size of the ordered moment varies sinusoidally from plane to plane with a periodicity that is not an integral multiple of the crystal c-axis. This is a special case of a class of periodic spin structures which can be characterized by a wavevector Q. For Er and Tm the vector Q is not far from the centre of the Brillouin zone and points along the c-axis. The modulated spin structure is said to be longitudinal because the spins are in the direction of Q. Nishikubo and Nagamiya [3 ] showed that this kind of spin ordering arises due to an interplay of the longrange exchange interaction, ~hich favours a periodic spin arrangement, and a uniaxial anisotropy, which forces the.spins to lie in the c direction. Cooper [4] discussed the equations of motion of the spins in the sinusoidally modulated phase. He showed that the elementary excitations are not ordinary spin waves. The preliminary measurement of Wakabayashi and Nicklow * Operated for the US Department of Energy by Iowa State University under contract No. W-7405-Eng.-82. This research was supported by the Director for Energy Research, Office of Basic Energy Sciences, WPAS-AK-01-02. 93
S.H. Liu / Sinusoidally modulated spin structures
94
spin motion employed in this work are the same as those discussed by Cooper. On the other hand, whether the local moment model is applicable to the study of magnons in itinerant systems is not completely settled. Many authors have discussed this problem in recent years [15-17]. We show in this paper that for localized moment systems the spin equations of motion in the modulated spin phase can be solved so that the magnetic response function can be put in the form of an infinite continued fraction. For the low-lying modes, the infinite continued fraction can be evaluated to a good approximation. The results show that there are broadened magnon modes emerging from the zone centre as well as the magnetic reciprocal lattice points. The line broadening comes from the spatial fluctuations of the exchange and anisotropy energies. This conclusion is in qualitative agreement with the preliminary measurements [5]. For Cr the model predicts a low-lying magnon branch at the zone boundary in agreement with the observations. The magnons emerging from the magnetic satellite points have an intrinsic line width. It will require much higher resolution to verify this prediction.
In the last term the commutator is
=-2
J,jts
+ O [Sf S + + S+Sf ] .
Theory
i ~ Ga(t ) = 2 ( S t ) 6 i t 6 ( t ) + 2 ~. Jij[(Sf) Git(t) I
-
q
J'llSi " Sj - D ~
(S~ ) 2 .
i
(1)
We define the Fourier transform ofGa(t ) by
an(t) = 1 ~ a(k, k', 60) IV kk'
- i60t] ,
+ f(k - Q) G(k - Q, k', 60) +f(k + Q) a ( k + Q, k', co),
(6)
where (7)
and
( S/~> =
~(a) = .~]~j exp[~. (Ri - Rj)].
where Q = (0, O, Q) lies in the z direction, which is the c-axis of the crystal. We define the spin propagator by
air(t) = - i(TS+(t) Si- (0)),
60G(k, k', 60) = S[8~,~,+Q + StOe-Q]
f(k) =S[D + ~(Q) - ~ (k)],
The ordered moment at the site R i has the components ( s~ > = s co s((2. R ~),
(5)
then we can rewrite eq. (5) as
For the rare-earth systems we use a Hamiltonian which consists of a long-range exchange term and a uniaxial anisotropy term [2,18 ]: H = - ~
(4)
In the spin-wave theory we apply the mean.field approximation to decouple the spin products SrS 7 ~-, (S~) S~'. The mean-field approximation is good only for zero temperature where there are no thermally excited spin waves. The validity of this approximation to the sinusoidal phase of Er and Tm is doubtful because it is far from the ground state. Nevertheless, for lack of a better alternative, we shall proceed with this scheme and explore the consequences. One must bear in mind that the conclusions of this calculation are at best qualitative. Thus eq. (3) becomes
X exp [i(k • R i - k " R l ) 2.
s;-s';sf]
I
(2)
where S~ = S x + iS~. The equation of motion for the propagator
We define a symbol
p(k +-Q, k) = a ( k + Q, k + Q, 60)/G(k, k + Q, 60). (9) Then we find by setting k ' = k + Q in eq. (6) that
[60 - f ( k - Q) p ( k - Q, k ) - f ( k + Q ) p ( k + Q, k)] X G ( k , k + Q , 60) =S.
i -~Git(t ) = 2(S z) 8itS(t) + ( T[S~(t),H] ST(0)). (3)
(8)
1
(10)
The substitution o f k + Q for k and k' in eq. (6) leads
S.H. Liu / Sinusoidally modulated spin structures to
[60 - f ( k + 2Q) p(k + 2Q, k + Q)I p(k +-Q, k) = f(k).
(11) We combine eqs. (10) and (11) to obtain an expression of G(k, k + Q, 60) in terms of p(k + 2Q, k + Q). The procedure is repeated so that G will be expressed in terms of p(k + 3Q, k + 2Q) and so on. Since Q is not a multiple of the reciprocal lattice vector, we get an infinite chain of relations such that G ( k , k + Q, w ) = S / [ 6 0 - g ( k - Q ) - g ( k + Q)],
(12)
95
~(q) of the exchange interaction drops fairly steeply with increasing Iql. To stabilize the modulated structure it is necessary that ~(q) is maximum at q = Q. For k near the zone center, i.e. Ikl < < IQ[, we may approximate ~(k) -~ ~ ( 0 ) ,
~ ( k + Q) ~ ~ ( Q ) + a2k s ,
where for simplicity we assume isotropic k dependence near the maximum of if(q). Putting these into eq. (12) we obtain g(k +-Q) ~ f(O) (DS + ask2)~ [60 - (DS + a 2 k S ) x ] ,
(16)
where g(k -+ Q) are infinite continued fractions f(k) g(k + Q) = [60/f(k +-Q)]
1 [60/f(k +-2Q)1
(13)
1 D
[ 6 0 / f ( k _+ 3 0 ) ]
-
... "
where
In the limit of weak exchange interaction, i.e. f ( k + nQ) ~ D S for all n, we can evaluate the infinite
.x-
continued fractions in closed form:
[60/f(2Q)]
~ ( k +--Q) = D S x ,
1 - [60/f(3Q)] - . . .
(17)
We anticipate that 6o < < f ( n Q ) for n ~ 2..Then since all the quantities 60/f(nQ) are small, it is a good approximation that
where x = 1/(60/DS- x ) , or
x ~, 1/[60/f(2Q) - x]
x = 60/2DS + [(60/2DS) 2 - 1] in
which loads to x ~ i. Thus,
This gives
a(k,k
G(k, k + Q, 60) = S/( 602 - 4D2 S2) t n .
(14)
The neutron scattering cross-section is proportional to the imaginary part of G, i.e. - I m G(k, k + Q, 60) = S/(4D2S s - 602)1a, 1601< 2DS,
+ Q,60)
S/ (60 - 2f(O) (DS
+
a 2 kS)./[60 : i(DS
+
a s k2)] }.
(18) The mode frequency is solved from 60 [60 - i(DS + a2k2)] - 2f(0) (DS + a2k 2) = O,
with the result =0,
1601> 2DS. The finite linewidth comes from the sinusoidal variation of the anisotropy field. Otherwise, the magnetic excitation is dispersionless. In the opposite case where the anisotropy field is weak compared with the exchange field, an approximate reduction of the infinite continued fraction to a closed expression is possible if the Fourier transform
w ~ ~ {(DS + a 2 k 2) [2f(0) - DS - a 2 k 2] }t a + ½i(DS + a s k 2 ) .
(19)
The real part of the frequency can be shown to be small compared withf(2Q). The imaginary part,, which determines the linewidth, is partly due to the fluctuation in anisotropy, with the exchange field making an additional contribution which is more important for shorter wavelengths.
96
S.H. Liu / Sinusoidally modulated spin structures
In the intermediate case where the exchange and anisotropy energies are comparable, we may need to extend the continued fraction in eq. (13) to a higher order and approximate the rest by i. Numerical calculations may be necessary to assess the convergence and determine the necessary number of orders that must be kept. For k near 0 the approximations must be carried out differently because 6o is not small compared with f ( k - 20). We need to calculated g(k - 0 ) to one higher order a(k) h(g) - j---b(k)
+
(1/co)f(k)[f(k + Q) + f ( k - Q)] G(k, k', co)
+ (l[co)f(k + Q ) f ( k + 2 Q ) G ( k + 20, k', co).
This equation can be solved by the method used previously, with the solution
G(k, k + Q, co) = coS/[co - co (k) -
h(k + Q) -
h(k -
f(k) [co/f(k- 0)] -
I
(20)
[co/f(k - 20)] - i"
On the other hand we may take g(k + Q) = if(k).
(21)
The form of G is more complicated, but the mode frequency is again given by eq. (19). The same lineshape should be observed both near the zone centre and near the magnetic reflection point Q. In conclusion, we have shown that the magnon-like excitations in the sinusoidally modulated phase of Er and Tm are inherently broad because of the spatial fluctuation in the exchange and anisotropy energies. For Cr in the transverse spin-density-wave state the anisotropy appears to be quite complicated [ 13 ]. We will not attempt to analyse the anisotropy energy at this time, but will continue to approximate it by a uniaxial anisotropy as in eq. (1). This approach is also taken by Wolfram and Ellialtioglu [14]. The vector Q is now perpendicular to the spin axis, but it is not difficult to see that the equation of motion of the spins, eq. (6), is unaffected. However, since 0 is near the zone boundary,f(k) is nearly periodic in 20. This suggests that we must solve eq. (6) by successive iterations which advance the wavevector by 2 0 at a time. Iterating once, we find coG(k, k ' , co) = S[8~t¢+Q + 8~,k,_Q] + ( l / c o ) f ( k - Q ) f ( k - 2 Q ) G ( k - 2Q, k', co)
Q)],
(23)
where co~(k) = S 2 f ( k ) [ f ( k + Q) + f ( k - Q)],
-
(24)
a(k + 2Q) co2 _ b(k + 20)
S(k - 0 ) =
(22)
a(k + 40) ...... - co2 _ b(k + 4Q) - ... "
(25)
and a(k) = S4 f ( k - Q) f 2 ( k ) f ( k + Q ) ,
(26)
b(k) = S 2 f ( k + O ) [ f ( k ) + f ( k + 2 0 ) ] .
(27)
As in the rare-earth case, when the anisotropy energy dominates, the spin propagator is given by eq. (14). When the anisotropy energy is negligibly small compared with the exchange energy, we find that f ( Q ) = 0, and the magnon frequency given by the pole of the spin propagator in eq. (23) vanishes for k = 0 and +Q. This indicates that magnon branches emerge from the magnetic satellite points and the zone centre. Furthermore, for k near the zone boundary the magnon dispersion curve is approximately given by
co
V%oo (k),
(28)
where COo(k)is given in eq. (24). The intrinsic linewidth is very small. Because 0 is very near the zone boundary, the magnon frequency is low and has a maximum at the zone boundary, as depicted in fig. I. This result is in good agreement with the neutron data [13]. Far away from the magnetic satellite points the magnons are damped for the same reason as the rareearth case. It is tedious to write down the approximate expressions for the mode frequency. Also, the result is not of immediate interest to experimental workers because the present experimental resolution is not sufficiently high to detect the intrinsic linewidth of the magnons. The magnon dispersion curve in fig. 1 is rather dif-
S.H. Liu / Sinusoidally modulated spin structures
97
sions. The manuscript was prepared when the author was visiting the Free University Berlin. The hospitality o f Professor KJ-I. Bennemann and his group is especially acknowledged. A
t~
3
References
>,
tILl
o fz. o~ o
t
I
(o,o,Q) (o,o,~) (o,o,1 -o ) Magnon Wavevector k" Fig. 1. Schematic drawing to show the magnon dispersion curve near the zone boundary for Cr in the incommensurate phase. ferent from that predicted b y Wolfram and Ellialtioglu [14]. Whereas these authors attribute the 4 meV feature as the reciprocal lattice point to the anisotropy gap, we have shown the exchange interaction also comes into the picture. The difference between the two theories seems to lie in the details of the assumed long-range exchange interaction models.
Acknowledgements The author is indebted to Drs. S. Werner, T. Wolfram and N. Wakabayashi for informative discus-
[1] W.C. Koehler, in: Magnetic Properties of Rare Earth Metals, ed. RJ. Elliott (Plenum Press, London and New York, 1972) p. 187. [2] S.K. Sinha, in: Handbook on the Physics and Chemistry of Rare Earths, Vol. I, eds. K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1978) ch. 5. [3] T. Nishikubo and T. Nagamiya, J. Phys. Soc. Japan 20 (1965) 808. [4] B.R. Cooper, in: Solid State Physics, VoL 21, eds. F. Seitz, D. Turnball and H. Eltrenreich (Academic Press, New York, 1968) p. 393. [5 ] N. Wakabayashi and R.M. Nicklow, private communication (1979). [6] G. Shirane and W.J. Takei, J. Phys. Soc. Japan 17 Suppl. B-111 (1963) 35. [7] V.N. Bykov, V.S. Galovkin, N.V. Ageef, V.A. Levdik and S.I. Vonogradov, DokL Akad. Nauk SSSR 128 (1959) 1153. [8] G.E. Bacon, Acta Cryst. 14 (1961) 823. [9] W.C. Koehler, R.M. Moon, A.L. Trego and A.R. Mackintosh, Phys. Rev. 151 (1966) 405. [10] S.K. Sinha, SJ-I. Liu, L.D. Muldestein and N. Wakayashi, Phys. Rev. Lett. 23 (1969) 311. [11] J. Als-Nielsen, J.D. Axe and G. Shirane, J. Appl. Phys. 42 (1971) 1666. [12] S.H, Liu, Phys. Rev. B2 (1970) 2664. [13] C.R. Fincher, Jr., G. Shirane and S.A. Werner, Phys. Rev. Lett. 43 (1979) 1441. [14] T. Wolfram and S. Ellialtioglu, Phys. Rev. Lett. 44 (1980) 1295. [15] S.H. Liu, Phys. Rev. B13 (1976) 3962. [16] H. Capellmann, Z. Phys. B34 (1979) 29. [17] J. Hubbard, Phys. Rev. B19 (1979) 2826, B20 (1979) 4584.