- Email: [email protected]
Anti-renormalization of paramagnetic fluctuations in CsMnBr3
ARTICLE IN PRESS
Physica B 350 (2004) 59–62
Anti-renormalization of paramagnetic fluctuations in CsMnBr3 . a,*, C. Reicha, B. Roesslib, E. Rastellic P. Boni b
a Physik-Department E21, Technische Universitat Garching D-85747, Germany . Munchen, . Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland . c Dipartimento di Fisica dell’Universita" and Istituto Nazionale per la Fisica della Materia, Parco area, delle Scienze 7/A, 43100 Parma, Italy
Abstract The critical behavior of the spin-wave dispersion in the triangular antiferromagnet CsMnBr3 was investigated in the paramagnetic phase using inelastic neutron scattering. Unexpectedly, we find that the energy gap of the excitations increases with increasing temperature. This effect cannot be explained by perturbed spin wave theory taking only intrachain exchange interactions into account. We speculate that a description of the spin dynamics in terms of an extension of an approach for an S ¼ 12 model for a Heisenberg chain to S ¼ 52 may be more appropriate to explain our results. r 2004 Elsevier B.V. All rights reserved. PACS: 75.50.Ee; 75.40.Gb; 75.10.Jm; 75.30.Ds Keywords: Low dimension; Spin waves; Phase transition; Critical behavior
1. Introduction Low-dimensional systems have attracted a lot of attention during the last few years. The dimensionality of a system proves to be very important for its critical behavior since the 1D- and 2D-cases can exhibit very peculiar properties not found in 3D-systems. 3D-ordered systems forming spin chains with antiferromagnetic coupling usually exhibit an energy gap in their dispersion relation. An energy gap was also observed in CsMnBr3 in the ordered state [1,2] and was inter*Corresponding author. Fax: +49-89-289-14711. . E-mail address: [email protected] (P. Boni).
preted in terms of an effective Hamiltonian with XY-like anisotropy X X ~iþ1 þ 2Jc ~iþ1 ~i S ~i S S S H ¼ 2Jab i
þD
X
i
ðSiz Þ2
ð1Þ
i
with an in-plane exchange Jab ¼ 1:770:1 meV and an out-of-plane exchange Jc ¼ 0:8970:01 meV: The effective single ion anisotropy parameter D ¼ 1271 meV leads to a confinement of the spins to the a–b plane. It is expected that the energy gap decreases when approaching TN ¼ 8:3 K from below. An alternative explanation for the gap was recently provided by Hummel and Schwabl
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.253
ARTICLE IN PRESS
[3], who demonstrated that the gap may be caused by the dipolar interactions between the moments of Mn2þ : However, the observed 120 configuration is only reached asymptotically for vanishing dipolar interactions, a condition that may not be fulfilled for CsMnBr3 [3]. Several years ago, Mason et al. [4] indeed observed by means of two-axis neutron measurements the critical scattering at TN indicating a softening of at least one mode in CsMnBr3 : However, direct information on the T-dependence of the gap was not provided. In order to study the renormalization of the spin wave branches near TN ; we have performed inelastic neutron scattering experiments in the paramagnetic phase of CsMnBr3 : Unexpectedly, we found indications for an increase of the energy gap of the optic in-plane and the out-of-plane modes.
2. Experiment The experiments were performed on a single crystal of CsMnBr3 on the triple-axis spectrometer TASP at SINQ that is located at the end of a guide for cold neutrons. The energy of the incident neutrons was varied while the energy of the scattered neutrons was fixed at Ef ¼ 5:0 meV: Higher-order neutrons were removed by a cooled Be-filter before the analyzer. Furthermore, an 800 collimator was inserted before the sample to reduce the background. The monochromator and the analyzer were vertically and horizontally focusing, respectively. The single crystal of CsMnBr3 ð7 7 3 mm3 Þ was mounted on the cold finger of an ‘orange’ refrigerator providing a temperature stability of better than 0:1 K: The crystal structure of CsMnBr3 is hexagonal P63 =mmc with lattice ( and c ¼ 6:52 A: ( Each constants a ¼ b ¼ 7:609 A unit cell contains two molecules of CsMnBr3 : The Mn2þ -ions ðS ¼ 52Þ are positioned along the c-axis at the edges of the hexagonal unit cell. The wavevector for magnetic ordering was determined by Eibschutz . et al. to be ð13 13 1Þ; i.e. the moments have a 120 -ordering within the plane and antiferromagnetic order along the chain axis.
Neutron counts [monitor 50000]
P. Boni . et al. / Physica B 350 (2004) 59–62 400 #1283-01
350
25 K
300 250 200 150 100 50 0
Neutron counts [monitor 50000]
60
0
1
2
3 ω [meV]
4
5
6
400 #1301-01
350
35 K
300 250 200 150 100 50 0
0
1
2
3 4 ω [meV]
5
6
Fig. 1. Neutron scattering data measured at (0 0 1) showing spin wave peaks that move to higher E with increasing T: The curves are fits to the experimental data using a DHO-function.
A few representative measurements at (0 0 1) are shown in Fig. 1. The central peaks consist of quasielastic scattering and incoherent scattering. The spin wave peaks at finite energy were fitted by means of a damped harmonic oscillator function that was convoluted with the resolution function of the spectrometer. It is clearly seen that the energy gap increases with increasing T: This is a very peculiar behavior because usually the gap decreases with increasing T: In order to substantiate this observation we show in Fig. 2 the dispersion curve for several temperatures. Indeed, the spin-wave energy increases for 1pzo1:16 with increasing T: Between 20pTp40 K the gap at (0 0 1) can be parametrized by Eg ¼ 0:729 meV þ 0:058 T;
ð2Þ
where Eg is measured in meV and T in K. In Fig. 3 we have plotted the fitted linewidths Gq of the excitations. Interestingly there is no
ARTICLE IN PRESS P. Boni . et al. / Physica B 350 (2004) 59–62
LinewidthΓ [meV]
5 4.5 4
3
20 K
2.5
25 K
2
5
10 15 20 25 30 35 40 Temperature [K]
2.5
1
35 K
0.5
40 K
0
10 15 20 25 30 35 40 Temperature [K]
3
30 K
1.5
5
3.5
ω [meV]
ω [meV]
3.5
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
61
2
1.5 1
1
1.05
1.1
1.15
(0 0 ζ) [r.l.u.]
1.2
0.5 0
0
Fig. 2. Dispersion curve of CsMnBr3 near (0 0 1) showing the hardening of the excitations with increasing T: The gray line indicates the dispersion of the out-of-plane mode (not observed in this experiment) and the black line shows the in-plane mode as calculated in the ordered phase at T ¼ 6 K: On the right the T-dependence of Gq and Eg are shown.
Eg has also been observed at other positions in reciprocal space, in particular also at the magnetic zone centre ð13 13 1Þ [5].
1.6 40 K
Linewidth Γ [meV]
1.4 35 K 1.2
30 K
3. Conclusions
25 K 1 20 K 0.8
0.6
1
1.05
1.1
1.15
(0 0 ζ) [r.l.u.] Fig. 3. The figure shows that the line widths increase with increasing T: However, for a fixed temperature they are independent of q: The different symbols denote data taken at different temperatures, according to the notation of Fig. 2. The lines represent the average linewidths at different temperatures. Note that the zero of the vertical scale is suppressed.
q-dependence observed. Gq increases also with increasing T as somehow expected and the T-dependence can be parametrized by Gq ¼ 0:328 meV þ 0:027 T:
Our present results clearly show that only the acoustic in-plane mode renormalizes at TN while the optic in-plane mode and the out-of-plane mode renormalize upwards. An increase of an energy gap in an excitation spectrum was also observed experimentally in Sr2 Cu3 O4 Cl2 [6], however, only below the ordering temperature. This example of ‘‘order-by-disorder’’ [7] was ascribed to quantum fluctuations in systems with frustrated exchange between inequivalent sublattices. In CsMnBr3 the idea of a similar effect is not applicable since the interactions are not restricted to be 2-D [8]. Therefore, we have to rely on future spin-wave calculations in frustrated, hexagonal lattices that include effects of anisotropy as well as dipolar interactions based on an extension of a Heisenberg chain model from S ¼ 12 to 52:
ð3Þ
Clearly, for the T-range investigated the excitations are significantly under-damped Gq =Eg t2; i.e. the hardening of the energy-gap is a real effect. For completeness we point out that an increase of
Acknowledgements This work was performed at the spallation neutron source SINQ at PSI, Villigen, Switzerland.
ARTICLE IN PRESS 62
P. Boni . et al. / Physica B 350 (2004) 59–62
References [1] B.D. Gaulin, et al., J. Appl. Phys. 61 (8) (1987) 3409. [2] U. Falk, et al., Phys. Rev. B 35 (10) (1987) 4888. [3] M. Hummel, et al., Phys. Rev. B 63 (2001) 094425.
[4] [5] [6] [7] [8]
T.E. Mason, et al., Phys. Rev. B 39 (1) (1989) 586. C. Reich, et al., Phys. Rev. Lett. 91 (15) (2003) 157203. Y.J. Kim, et al., Phys. Rev. Lett. 83 (4) (1999) 852. J. Villain, et al., J. Phys. (France) 41 (1980) 1263. A. Carbognani, et al., Phys. Rev. B 62 (2) (2000) 1015.
Physica B 350 (2004) 59–62
Anti-renormalization of paramagnetic fluctuations in CsMnBr3 . a,*, C. Reicha, B. Roesslib, E. Rastellic P. Boni b
a Physik-Department E21, Technische Universitat Garching D-85747, Germany . Munchen, . Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland . c Dipartimento di Fisica dell’Universita" and Istituto Nazionale per la Fisica della Materia, Parco area, delle Scienze 7/A, 43100 Parma, Italy
Abstract The critical behavior of the spin-wave dispersion in the triangular antiferromagnet CsMnBr3 was investigated in the paramagnetic phase using inelastic neutron scattering. Unexpectedly, we find that the energy gap of the excitations increases with increasing temperature. This effect cannot be explained by perturbed spin wave theory taking only intrachain exchange interactions into account. We speculate that a description of the spin dynamics in terms of an extension of an approach for an S ¼ 12 model for a Heisenberg chain to S ¼ 52 may be more appropriate to explain our results. r 2004 Elsevier B.V. All rights reserved. PACS: 75.50.Ee; 75.40.Gb; 75.10.Jm; 75.30.Ds Keywords: Low dimension; Spin waves; Phase transition; Critical behavior
1. Introduction Low-dimensional systems have attracted a lot of attention during the last few years. The dimensionality of a system proves to be very important for its critical behavior since the 1D- and 2D-cases can exhibit very peculiar properties not found in 3D-systems. 3D-ordered systems forming spin chains with antiferromagnetic coupling usually exhibit an energy gap in their dispersion relation. An energy gap was also observed in CsMnBr3 in the ordered state [1,2] and was inter*Corresponding author. Fax: +49-89-289-14711. . E-mail address: [email protected] (P. Boni).
preted in terms of an effective Hamiltonian with XY-like anisotropy X X ~iþ1 þ 2Jc ~iþ1 ~i S ~i S S S H ¼ 2Jab i
þD
X
i
ðSiz Þ2
ð1Þ
i
with an in-plane exchange Jab ¼ 1:770:1 meV and an out-of-plane exchange Jc ¼ 0:8970:01 meV: The effective single ion anisotropy parameter D ¼ 1271 meV leads to a confinement of the spins to the a–b plane. It is expected that the energy gap decreases when approaching TN ¼ 8:3 K from below. An alternative explanation for the gap was recently provided by Hummel and Schwabl
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.253
ARTICLE IN PRESS
[3], who demonstrated that the gap may be caused by the dipolar interactions between the moments of Mn2þ : However, the observed 120 configuration is only reached asymptotically for vanishing dipolar interactions, a condition that may not be fulfilled for CsMnBr3 [3]. Several years ago, Mason et al. [4] indeed observed by means of two-axis neutron measurements the critical scattering at TN indicating a softening of at least one mode in CsMnBr3 : However, direct information on the T-dependence of the gap was not provided. In order to study the renormalization of the spin wave branches near TN ; we have performed inelastic neutron scattering experiments in the paramagnetic phase of CsMnBr3 : Unexpectedly, we found indications for an increase of the energy gap of the optic in-plane and the out-of-plane modes.
2. Experiment The experiments were performed on a single crystal of CsMnBr3 on the triple-axis spectrometer TASP at SINQ that is located at the end of a guide for cold neutrons. The energy of the incident neutrons was varied while the energy of the scattered neutrons was fixed at Ef ¼ 5:0 meV: Higher-order neutrons were removed by a cooled Be-filter before the analyzer. Furthermore, an 800 collimator was inserted before the sample to reduce the background. The monochromator and the analyzer were vertically and horizontally focusing, respectively. The single crystal of CsMnBr3 ð7 7 3 mm3 Þ was mounted on the cold finger of an ‘orange’ refrigerator providing a temperature stability of better than 0:1 K: The crystal structure of CsMnBr3 is hexagonal P63 =mmc with lattice ( and c ¼ 6:52 A: ( Each constants a ¼ b ¼ 7:609 A unit cell contains two molecules of CsMnBr3 : The Mn2þ -ions ðS ¼ 52Þ are positioned along the c-axis at the edges of the hexagonal unit cell. The wavevector for magnetic ordering was determined by Eibschutz . et al. to be ð13 13 1Þ; i.e. the moments have a 120 -ordering within the plane and antiferromagnetic order along the chain axis.
Neutron counts [monitor 50000]
P. Boni . et al. / Physica B 350 (2004) 59–62 400 #1283-01
350
25 K
300 250 200 150 100 50 0
Neutron counts [monitor 50000]
60
0
1
2
3 ω [meV]
4
5
6
400 #1301-01
350
35 K
300 250 200 150 100 50 0
0
1
2
3 4 ω [meV]
5
6
Fig. 1. Neutron scattering data measured at (0 0 1) showing spin wave peaks that move to higher E with increasing T: The curves are fits to the experimental data using a DHO-function.
A few representative measurements at (0 0 1) are shown in Fig. 1. The central peaks consist of quasielastic scattering and incoherent scattering. The spin wave peaks at finite energy were fitted by means of a damped harmonic oscillator function that was convoluted with the resolution function of the spectrometer. It is clearly seen that the energy gap increases with increasing T: This is a very peculiar behavior because usually the gap decreases with increasing T: In order to substantiate this observation we show in Fig. 2 the dispersion curve for several temperatures. Indeed, the spin-wave energy increases for 1pzo1:16 with increasing T: Between 20pTp40 K the gap at (0 0 1) can be parametrized by Eg ¼ 0:729 meV þ 0:058 T;
ð2Þ
where Eg is measured in meV and T in K. In Fig. 3 we have plotted the fitted linewidths Gq of the excitations. Interestingly there is no
ARTICLE IN PRESS P. Boni . et al. / Physica B 350 (2004) 59–62
LinewidthΓ [meV]
5 4.5 4
3
20 K
2.5
25 K
2
5
10 15 20 25 30 35 40 Temperature [K]
2.5
1
35 K
0.5
40 K
0
10 15 20 25 30 35 40 Temperature [K]
3
30 K
1.5
5
3.5
ω [meV]
ω [meV]
3.5
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
61
2
1.5 1
1
1.05
1.1
1.15
(0 0 ζ) [r.l.u.]
1.2
0.5 0
0
Fig. 2. Dispersion curve of CsMnBr3 near (0 0 1) showing the hardening of the excitations with increasing T: The gray line indicates the dispersion of the out-of-plane mode (not observed in this experiment) and the black line shows the in-plane mode as calculated in the ordered phase at T ¼ 6 K: On the right the T-dependence of Gq and Eg are shown.
Eg has also been observed at other positions in reciprocal space, in particular also at the magnetic zone centre ð13 13 1Þ [5].
1.6 40 K
Linewidth Γ [meV]
1.4 35 K 1.2
30 K
3. Conclusions
25 K 1 20 K 0.8
0.6
1
1.05
1.1
1.15
(0 0 ζ) [r.l.u.] Fig. 3. The figure shows that the line widths increase with increasing T: However, for a fixed temperature they are independent of q: The different symbols denote data taken at different temperatures, according to the notation of Fig. 2. The lines represent the average linewidths at different temperatures. Note that the zero of the vertical scale is suppressed.
q-dependence observed. Gq increases also with increasing T as somehow expected and the T-dependence can be parametrized by Gq ¼ 0:328 meV þ 0:027 T:
Our present results clearly show that only the acoustic in-plane mode renormalizes at TN while the optic in-plane mode and the out-of-plane mode renormalize upwards. An increase of an energy gap in an excitation spectrum was also observed experimentally in Sr2 Cu3 O4 Cl2 [6], however, only below the ordering temperature. This example of ‘‘order-by-disorder’’ [7] was ascribed to quantum fluctuations in systems with frustrated exchange between inequivalent sublattices. In CsMnBr3 the idea of a similar effect is not applicable since the interactions are not restricted to be 2-D [8]. Therefore, we have to rely on future spin-wave calculations in frustrated, hexagonal lattices that include effects of anisotropy as well as dipolar interactions based on an extension of a Heisenberg chain model from S ¼ 12 to 52:
ð3Þ
Clearly, for the T-range investigated the excitations are significantly under-damped Gq =Eg t2; i.e. the hardening of the energy-gap is a real effect. For completeness we point out that an increase of
Acknowledgements This work was performed at the spallation neutron source SINQ at PSI, Villigen, Switzerland.
ARTICLE IN PRESS 62
P. Boni . et al. / Physica B 350 (2004) 59–62
References [1] B.D. Gaulin, et al., J. Appl. Phys. 61 (8) (1987) 3409. [2] U. Falk, et al., Phys. Rev. B 35 (10) (1987) 4888. [3] M. Hummel, et al., Phys. Rev. B 63 (2001) 094425.
[4] [5] [6] [7] [8]
T.E. Mason, et al., Phys. Rev. B 39 (1) (1989) 586. C. Reich, et al., Phys. Rev. Lett. 91 (15) (2003) 157203. Y.J. Kim, et al., Phys. Rev. Lett. 83 (4) (1999) 852. J. Villain, et al., J. Phys. (France) 41 (1980) 1263. A. Carbognani, et al., Phys. Rev. B 62 (2) (2000) 1015.