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Neutron scattering study of anisotropic spin correlations in quasi two-dimensional antiferromagnet MnTiO3
Solid State Communications,
NEUTRON SCATTERING
Pergamon Press.
Vol. 15, pp: 1123- 1127, 1974.
STUDY OF ANISOTROPIC SPIN CORRELATIONS ANTIFERROMAGNET MnTiOs
Printed in Great Britain
IN QUASI TWO-DIMENSIONAL
Jun Akimitsu Institute
for Solid State Physics, University
of Tokyo Roppongi,
Minato-ku,
Tokyo 106, Japan
and Yoshikazu Ishikawa Physics Department,
Faculty of Science, Tohoku University
Sendai 980, Japan
(Received 5 July 1974 by T. Nagamiya) An anisotropic three-dimensional neutron critical scattering has been observed from a “quasi” two-dimensional antiferromagnet MnTiOs in the vicinity of the Neel temperature. When temperature increases, the scattering profile transforms gradually to a ridge-like form, indicating that “crossover” from three-dimensional to two-dimensional character appears at about 16 degrees above TN. RECENTLY, both theoretical and experimental attentions have been paid to low-dimensional magnetic systems, especially to their critical phenomena. There have been extensive experimental studies on the static and dynamical critical behavior of the nearly ideal two-dimensional ([2] ) antiferromagnet Kz NiFI .l In such [2] magnetic systems the phase transition is driven by an asymmetry of the exchange interaction or by the lattice dimensionality, since a [2] Heisenberg system cannot exhibit long range order at finite temperatures. According to a current theory,2 the critical exponent should depend on the symmetry of the order parameter or the lattice dimensionality but not on its details. Recent neutron scattering shows3*4 that the critical exponents of nearly ideal [2] Ising magnets K2CoF4 and RbzCoF4 really coincide with those from Onsager’s exact solution for the [2] Ising model. From the ratio of h” and Zf; (the anisotropy field and the interplanar exchange field, respectively), it is likely that the phase transition in MnTiOs is brought about by the three-dimensionality. The theory has also predicted that the critical behavior in such a system will be [3] in the neighbourhood of the critical temperature in spite of the directional nature of the interaction. To confirm this anticipation, neutron scattering experiment was performed on a single crystal of MnTiOs ’
We have reported in a previous letter 5 that the critical exponent of the sublattice magnetization, p, of MnTiOs is 0.32 + 0.01. This value almost coincides with the value of S/16 for the [3] Ising model. In this letter we report that [3] anisotropic critical scattering has been observed from MnTiOs and that it changes continuously to become of [2] character as temperature increases. The magnetic critical scattering from MnTiOs appears around both the (003) and (110) reciprocal lattice points (Fig. 1). Since the spins are directed along the [OOl] axis, the (003) scattering detects only transverse spin fluctuations, while the (110) scattering contains both longitudinal and transverse spin fluctuations. The quasi-elastic scattering cross-section around the (003) and (1 IO) points are given by a 2x1 do i G
1 101
a(1
+cos2q5)x1+sin2~xl~=
1.09x1+0.91xl, (1)
where $I is the angle given in Fig. 1 and xl and ~11are the transverse and longitudinal staggered usceptibilities, respectively. In the case of a [3] anisotropic 1123
1124
STUDY OF ANISOTROPIC
REAL
SPACE
Vol. 15, No. 6
SPIN CORRELATIONS
RECIPROCAL
REACTOR
SPACE
C’ A Q>
ONOCHROMATOR
SAMPLE
TABLE
OLLIMATOR(H) 3O’COLLIMATOR(V) (000:
FIG. 1. Diagram of a two-axis neutron
spectrometer. Black ellipses show resolution principle axis of the ellipses.
antiferromagnet, the staggered susceptibility x”“(Qe + 4) for wave vector Qe + 4 close to the antiferromagnetic wave vector Qe is given in the vicinity of the Neel temperature by &’
c
x‘YQo + 4) = (k2 + 4: + a& i 42) 1-q’2
(2)
with 2b2J’ o=3azJ,
contours and Q, is the
neutrons with monochromatized energies of 54 meV (I.S.S.P.) and 30 meV (TUNS.) were obtained by a distorted germanium crystal and a pyrolytic graphite, respectively. The size of the single crystal used in this experiment was approximately 1.5 cm X 0.5 cm*. The critical temperature TN was determined to be 64.04 + O.Ol”K from the disappearance of magnetic Bragg peaks. This value is in good agreement with the value obtained by specific heat measurement.8 The observed neutron
where J and J’ are the nearest-neighbour intraplanar and interplanar exchange constants, respectively, and Q, b are the corresponding.distances. The x, y, z directions are taken parallel to the crystal axes, y being along the c axis. Then the (003) and (101) scattering corss-sections take forms given by
I(401
=
JJJo(4
intensity -
40)
at 4. is given by
Re(4)dq
(4)
where o(4) is the scattering cross-section. The instrumental resolution function Re(q), which is approximated by a Gaussian form, was determined from (003) and (101) nuclear Bragg reflections. The resolution contour is shown in Fig. I schematically.
do 0E
003
(3)
do 0
dsz rer =
df(QoN2W
ck2 +‘;&2+ *
1.09t (k’ + qi)l +I”
with 42 = 4: + WC +42. t(= CL/C,,) is assumed to be unity in our analysis. A neutron scattering experiment was made with the 2 axis spectrometers of I.S.S.P. and of Tohoku University (TUNS.) at J.A.E.R.I. Japan. The incident
The magnetic critical scattering was measured at both the (003) and (101) reciprocal lattice points. The observed (003) and (101) cross-sections in the 4x and qy directions at (T - TN)/TN = 0.0156 are shown in Figs. 2 and 3. Solid lines in the figures are obtained by folding the cross-sections given by equation (3) with the instrumental resolution function. The apparent inverse correlation lengths K, and K~ were evaluated by the 4% and q,, scans, and the parameter (Ywas determined using the relation 01 = K:/K~. The value of (Ythus obtained is OL= 0.27 f 0.02. J’/J was evaluated to be 0.22 + 0.04 using equation (3). The transverse staggered susceptibility a(Qo) as well as the inverse correlation length K~ were analyzed using power laws
STUDY OF ANISOTROPIC
Vol. 15, No. 6
SPIN CORRELATIONS
1125
(003) ’7 x10* 12-
q,-scan(nqcc0.07)
=(x0156
x10* ,*
qy-scan(aqd=0.025)
2-
FIG. 2. Quasi-elastic scattering in MnTiOs around the (003) reciprocal lattice point at T = 6504°K measured with 30 meV incoming neutrons. Solid lines show least-squares fits by the first member of equation (3). Black circles are intensities corrected for the nuclear Bragg reflection. (101) qx-xanbqcr-aoz)
XI0
122. 23.
qy-scan b%=O.O)
18. 16.
FIG. 3. Quasi-elastic scattering in MnTi03 around the (101) reciprocal lattice point at T = 65.04OK measured with 30 meV incoming neutrons. Solid lines show the least-squares fit by the second member of equation (3). Black circles are intensities corrected for the nuclear Bragg reflection. The deviation from equation (3) in the q,-direction is described in the text.
xltQo>a CT- TL)-~ kl a (T - TL)” The critical temperature c depends on the anisotropy of the substance.’ Since the ratio of the anisotropy field and the exchange field is reported to be
1.2 X 10-j for MnTi03 ,’ the difference TN - Tl is estimated to be 0.077”K in the molecular field approximation. Therefore we assumed TL- TN = 0 in our analysis. The best tit to the data are attained with parameters
1126
STUDY OF ANISOTROPIC
SPIN CORRELATIONS
Vol. 15, No, 6
x (q=O)
I04
,0-i’
I$ 1
f 10-3
FIG. 4. Longitudinal
10 -z
10-l
(T-T+'TN
inverse correlation length K(I and staggered susceptibility xs(QO) measured with 30 meV incoming neutrons. Solid lines are least-squares fits by power laws.
FIG. 5. Scans in the qr direction
vl = 0.634 kO.08
l
l-67
"K
0
T-70
"K
o
T-75
"K
A
T-80'K
at various temperatures measured with 54 meV incoming guides to the eyes.
yl = 1.23 + 0.16
r7 = 0.2+-0.15 The value of q is slightly larger than that expected for the [3] Ising model. However the accuracy of the data is not high enough to conclude this with certainty. The values of 7 and v for the longitudinal component are obtained from equation (3) as 711 = 1.22kO.10
neutrons.
Solid lines are
“II = 0.63 + 0.04, which are displayed in the log-log plots of K11and x,,(Q,) vs temperature in Fig. 4. The values given above *for the two components are in good agreement with those for the [3] Ising model. As was mentioned above, the phase transition in MnTiOs is always accompanied by .a [3] critical scattering extending in the qy direction, which we will call
STUDY OF ANISOTROPIC
Vol. 15, No. 6
SPIN CORRELATIONS
anisotropic spin correlation. As temperature increases the line shapes of the q,-scans change gradually to become flat. The Fisher-Burford type cross-section given in equation (2) becomes inadequate for the (101) scattering in the qy direction at high temperatures. This tendency appears even in the vicinity of TN as shown in Fig. 3. The line shapes in the (ly direction at various temperatures are shown in Fig. 5. At approximately 80”K, the scattering intensity of the qy scans manifests flat proile, which exhibits short range correlations in
only two dimensions. This shows “cross-over” behavior lo from [3] to [2] region. The cross-over temperature is estimated to be 80’K. It should be noted that the values of y and v were obtained in the [3] critical region. Acknowledgements - The authors would like to express their sincere thanks to Prof. K. Hirakawa, Prof. Y. Ito and Dr. H. Ikeda for encouragement and useful discussions on this work. Thanks are also due to Dr. Tajima for assistance in a part of this experiment.
REFERENCES 1.
BIRGENEAU
2.
GRIFFITHS
3.
SAMUELSEN E.J.,Phys.
4.
IKEDA H. and HIRAKAWA
5.
AKIMITSU J., ISHIKAWA Y. and ENDOH Y., Solid State Cbmmun. 8,87 (1970).
6.
FISHER M.E. and BURFORD
7.
SCHULHOF M.P. et al., Phys. Rev. Bl, 2304 (1970).
8.
AKIMITSU J., Ph. D. thesis unpublished.
9.
DE JONGH L.J. and MIEDEMA A.R., to be published
10.
R.J., SKALY0.J.
and SHIRANE G.,Phys.
&v.B3,1736
(1971).
R.B., Phys. Rev. Lett. 24, 1479 (1970). Rev. Lett. 31, 1973 (1973). K., Solid State Cbmmun. 14,529 R.J., Phys. Rev. 156,583
LIU L.L. and STANLEY H.E., Phys. Rev. B&2279
1127
(1974).
(1967).
in Adv. Phys
(1973).
On a observe la diffusion critique anisotrope B trois dimensions des neutrons dans un antiferromagnet & “quasi” deux dimensions, MnTi03, au voisinage de la temperature de Neel. Toutefois, avec l’augementation de la temperature, le profile de la diffusion se transforme graduellement a une forme de “L’arrete”, ce qiri nous suggere que la phenomene de “cross-over” de trois dimensions a deux dimensions se montre vers 16” audessus de TN.
NEUTRON SCATTERING
Pergamon Press.
Vol. 15, pp: 1123- 1127, 1974.
STUDY OF ANISOTROPIC SPIN CORRELATIONS ANTIFERROMAGNET MnTiOs
Printed in Great Britain
IN QUASI TWO-DIMENSIONAL
Jun Akimitsu Institute
for Solid State Physics, University
of Tokyo Roppongi,
Minato-ku,
Tokyo 106, Japan
and Yoshikazu Ishikawa Physics Department,
Faculty of Science, Tohoku University
Sendai 980, Japan
(Received 5 July 1974 by T. Nagamiya) An anisotropic three-dimensional neutron critical scattering has been observed from a “quasi” two-dimensional antiferromagnet MnTiOs in the vicinity of the Neel temperature. When temperature increases, the scattering profile transforms gradually to a ridge-like form, indicating that “crossover” from three-dimensional to two-dimensional character appears at about 16 degrees above TN. RECENTLY, both theoretical and experimental attentions have been paid to low-dimensional magnetic systems, especially to their critical phenomena. There have been extensive experimental studies on the static and dynamical critical behavior of the nearly ideal two-dimensional ([2] ) antiferromagnet Kz NiFI .l In such [2] magnetic systems the phase transition is driven by an asymmetry of the exchange interaction or by the lattice dimensionality, since a [2] Heisenberg system cannot exhibit long range order at finite temperatures. According to a current theory,2 the critical exponent should depend on the symmetry of the order parameter or the lattice dimensionality but not on its details. Recent neutron scattering shows3*4 that the critical exponents of nearly ideal [2] Ising magnets K2CoF4 and RbzCoF4 really coincide with those from Onsager’s exact solution for the [2] Ising model. From the ratio of h” and Zf; (the anisotropy field and the interplanar exchange field, respectively), it is likely that the phase transition in MnTiOs is brought about by the three-dimensionality. The theory has also predicted that the critical behavior in such a system will be [3] in the neighbourhood of the critical temperature in spite of the directional nature of the interaction. To confirm this anticipation, neutron scattering experiment was performed on a single crystal of MnTiOs ’
We have reported in a previous letter 5 that the critical exponent of the sublattice magnetization, p, of MnTiOs is 0.32 + 0.01. This value almost coincides with the value of S/16 for the [3] Ising model. In this letter we report that [3] anisotropic critical scattering has been observed from MnTiOs and that it changes continuously to become of [2] character as temperature increases. The magnetic critical scattering from MnTiOs appears around both the (003) and (110) reciprocal lattice points (Fig. 1). Since the spins are directed along the [OOl] axis, the (003) scattering detects only transverse spin fluctuations, while the (110) scattering contains both longitudinal and transverse spin fluctuations. The quasi-elastic scattering cross-section around the (003) and (1 IO) points are given by a 2x1 do i G
1 101
a(1
+cos2q5)x1+sin2~xl~=
1.09x1+0.91xl, (1)
where $I is the angle given in Fig. 1 and xl and ~11are the transverse and longitudinal staggered usceptibilities, respectively. In the case of a [3] anisotropic 1123
1124
STUDY OF ANISOTROPIC
REAL
SPACE
Vol. 15, No. 6
SPIN CORRELATIONS
RECIPROCAL
REACTOR
SPACE
C’ A Q>
ONOCHROMATOR
SAMPLE
TABLE
OLLIMATOR(H) 3O’COLLIMATOR(V) (000:
FIG. 1. Diagram of a two-axis neutron
spectrometer. Black ellipses show resolution principle axis of the ellipses.
antiferromagnet, the staggered susceptibility x”“(Qe + 4) for wave vector Qe + 4 close to the antiferromagnetic wave vector Qe is given in the vicinity of the Neel temperature by &’
c
x‘YQo + 4) = (k2 + 4: + a& i 42) 1-q’2
(2)
with 2b2J’ o=3azJ,
contours and Q, is the
neutrons with monochromatized energies of 54 meV (I.S.S.P.) and 30 meV (TUNS.) were obtained by a distorted germanium crystal and a pyrolytic graphite, respectively. The size of the single crystal used in this experiment was approximately 1.5 cm X 0.5 cm*. The critical temperature TN was determined to be 64.04 + O.Ol”K from the disappearance of magnetic Bragg peaks. This value is in good agreement with the value obtained by specific heat measurement.8 The observed neutron
where J and J’ are the nearest-neighbour intraplanar and interplanar exchange constants, respectively, and Q, b are the corresponding.distances. The x, y, z directions are taken parallel to the crystal axes, y being along the c axis. Then the (003) and (101) scattering corss-sections take forms given by
I(401
=
JJJo(4
intensity -
40)
at 4. is given by
Re(4)dq
(4)
where o(4) is the scattering cross-section. The instrumental resolution function Re(q), which is approximated by a Gaussian form, was determined from (003) and (101) nuclear Bragg reflections. The resolution contour is shown in Fig. I schematically.
do 0E
003
(3)
do 0
dsz rer =
df(QoN2W
ck2 +‘;&2+ *
1.09t (k’ + qi)l +I”
with 42 = 4: + WC +42. t(= CL/C,,) is assumed to be unity in our analysis. A neutron scattering experiment was made with the 2 axis spectrometers of I.S.S.P. and of Tohoku University (TUNS.) at J.A.E.R.I. Japan. The incident
The magnetic critical scattering was measured at both the (003) and (101) reciprocal lattice points. The observed (003) and (101) cross-sections in the 4x and qy directions at (T - TN)/TN = 0.0156 are shown in Figs. 2 and 3. Solid lines in the figures are obtained by folding the cross-sections given by equation (3) with the instrumental resolution function. The apparent inverse correlation lengths K, and K~ were evaluated by the 4% and q,, scans, and the parameter (Ywas determined using the relation 01 = K:/K~. The value of (Ythus obtained is OL= 0.27 f 0.02. J’/J was evaluated to be 0.22 + 0.04 using equation (3). The transverse staggered susceptibility a(Qo) as well as the inverse correlation length K~ were analyzed using power laws
STUDY OF ANISOTROPIC
Vol. 15, No. 6
SPIN CORRELATIONS
1125
(003) ’7 x10* 12-
q,-scan(nqcc0.07)
=(x0156
x10* ,*
qy-scan(aqd=0.025)
2-
FIG. 2. Quasi-elastic scattering in MnTiOs around the (003) reciprocal lattice point at T = 6504°K measured with 30 meV incoming neutrons. Solid lines show least-squares fits by the first member of equation (3). Black circles are intensities corrected for the nuclear Bragg reflection. (101) qx-xanbqcr-aoz)
XI0
122. 23.
qy-scan b%=O.O)
18. 16.
FIG. 3. Quasi-elastic scattering in MnTi03 around the (101) reciprocal lattice point at T = 65.04OK measured with 30 meV incoming neutrons. Solid lines show the least-squares fit by the second member of equation (3). Black circles are intensities corrected for the nuclear Bragg reflection. The deviation from equation (3) in the q,-direction is described in the text.
xltQo>a CT- TL)-~ kl a (T - TL)” The critical temperature c depends on the anisotropy of the substance.’ Since the ratio of the anisotropy field and the exchange field is reported to be
1.2 X 10-j for MnTi03 ,’ the difference TN - Tl is estimated to be 0.077”K in the molecular field approximation. Therefore we assumed TL- TN = 0 in our analysis. The best tit to the data are attained with parameters
1126
STUDY OF ANISOTROPIC
SPIN CORRELATIONS
Vol. 15, No, 6
x (q=O)
I04
,0-i’
I$ 1
f 10-3
FIG. 4. Longitudinal
10 -z
10-l
(T-T+'TN
inverse correlation length K(I and staggered susceptibility xs(QO) measured with 30 meV incoming neutrons. Solid lines are least-squares fits by power laws.
FIG. 5. Scans in the qr direction
vl = 0.634 kO.08
l
l-67
"K
0
T-70
"K
o
T-75
"K
A
T-80'K
at various temperatures measured with 54 meV incoming guides to the eyes.
yl = 1.23 + 0.16
r7 = 0.2+-0.15 The value of q is slightly larger than that expected for the [3] Ising model. However the accuracy of the data is not high enough to conclude this with certainty. The values of 7 and v for the longitudinal component are obtained from equation (3) as 711 = 1.22kO.10
neutrons.
Solid lines are
“II = 0.63 + 0.04, which are displayed in the log-log plots of K11and x,,(Q,) vs temperature in Fig. 4. The values given above *for the two components are in good agreement with those for the [3] Ising model. As was mentioned above, the phase transition in MnTiOs is always accompanied by .a [3] critical scattering extending in the qy direction, which we will call
STUDY OF ANISOTROPIC
Vol. 15, No. 6
SPIN CORRELATIONS
anisotropic spin correlation. As temperature increases the line shapes of the q,-scans change gradually to become flat. The Fisher-Burford type cross-section given in equation (2) becomes inadequate for the (101) scattering in the qy direction at high temperatures. This tendency appears even in the vicinity of TN as shown in Fig. 3. The line shapes in the (ly direction at various temperatures are shown in Fig. 5. At approximately 80”K, the scattering intensity of the qy scans manifests flat proile, which exhibits short range correlations in
only two dimensions. This shows “cross-over” behavior lo from [3] to [2] region. The cross-over temperature is estimated to be 80’K. It should be noted that the values of y and v were obtained in the [3] critical region. Acknowledgements - The authors would like to express their sincere thanks to Prof. K. Hirakawa, Prof. Y. Ito and Dr. H. Ikeda for encouragement and useful discussions on this work. Thanks are also due to Dr. Tajima for assistance in a part of this experiment.
REFERENCES 1.
BIRGENEAU
2.
GRIFFITHS
3.
SAMUELSEN E.J.,Phys.
4.
IKEDA H. and HIRAKAWA
5.
AKIMITSU J., ISHIKAWA Y. and ENDOH Y., Solid State Cbmmun. 8,87 (1970).
6.
FISHER M.E. and BURFORD
7.
SCHULHOF M.P. et al., Phys. Rev. Bl, 2304 (1970).
8.
AKIMITSU J., Ph. D. thesis unpublished.
9.
DE JONGH L.J. and MIEDEMA A.R., to be published
10.
R.J., SKALY0.J.
and SHIRANE G.,Phys.
&v.B3,1736
(1971).
R.B., Phys. Rev. Lett. 24, 1479 (1970). Rev. Lett. 31, 1973 (1973). K., Solid State Cbmmun. 14,529 R.J., Phys. Rev. 156,583
LIU L.L. and STANLEY H.E., Phys. Rev. B&2279
1127
(1974).
(1967).
in Adv. Phys
(1973).
On a observe la diffusion critique anisotrope B trois dimensions des neutrons dans un antiferromagnet & “quasi” deux dimensions, MnTi03, au voisinage de la temperature de Neel. Toutefois, avec l’augementation de la temperature, le profile de la diffusion se transforme graduellement a une forme de “L’arrete”, ce qiri nous suggere que la phenomene de “cross-over” de trois dimensions a deux dimensions se montre vers 16” audessus de TN.