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Low energy phonons in the NTE compounds Zn(CN)2 and ZnPt(CN)6
ARTICLE IN PRESS
Physica B 385–386 (2006) 60–62 www.elsevier.com/locate/physb
Low energy phonons in the NTE compounds ZnðCNÞ2 and ZnPtðCNÞ6 Karena W. Chapmana, Mark Hagenb,,1, Cameron J. Keperta, Pascal Manuelc a
School of Chemistry, University of Sydney, New South Wales 2006, Australia b Bragg Institute, ANSTO, Menai, New South Wales 2234, Australia c ISIS Facility, Rutherford Appleton Lab., Oxon. OX11 0QX, UK
Abstract The compounds ZnðCNÞ2 and ZnPtðCNÞ6 both display negative thermal expansion (NTE) properties, that is to say they undergo a volume contraction with increasing temperature. In the case of ZnðCNÞ2 this volume contraction occurs over a temperature range from 25 to 375 K with a coefficient of thermal expansion a ¼ 16:9ð2Þ 106 K1 [A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030]. This phenomenon is believed to be related to the presence of low energy rigid unit modes (RUMS) in the phonon dispersion relations of ZnðCNÞ2 [A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030]. We have examined the low energy part of the phonon density of states in ZnðCNÞ2 and ZnPtðCNÞ6 using time of flight inelastic neutron scattering from powder samples. In ZnðCNÞ2 there is a strong peak in the density of states at 2 meV whose temperature dependence can be correlated with that of a. There is a similar peak in the density of states of ZnPtðCNÞ6 at 7.5 meV, which correlates with the smaller NTE effect in this compound. r 2006 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 63.20.Dj; 63.20.Ry Keywords: Negative thermal expansion; Rigid unit modes; Inelastic neutron scattering
1. Introduction While there are a number of materials that exhibit anisotropic negative thermal expansion (NTE) over a limited temperature range there are only a few that exhibit isotropic (volume) contraction over a wide range of temperature. The best known, and most widely studied, of these materials is ZrW2 O8 [1] which has a co-efficient of thermal expansion a ¼ 9 106 K1 [2]. ZnðCNÞ2 however displays an even larger NTE effect, with a value of a ¼ 16:9 106 K1 [3]. The origin of NTE can be explained in terms of the thermal excitation of phonons that have a negative Gruneisen parameter. It is well known [4] that a purely harmonic theory of lattice vibrations in a solid cannot explain the presence of thermal expansion (or contraction) and that anharmonic terms are required. The effect of these Corresponding author.
E-mail address: [email protected] (M. Hagen). Present address: Spallation Neutron Source, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge TN37831, USA. 1
0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.05.102
anharmonic terms are modelled in a quasi-harmonic way [4] in terms of the mode Gruneisen parameter gq;s given by gq;s ¼
qloge oq;s V qoq;s ¼ , oq;s qV qloge V
(1)
where V is the volume and oq;s is the frequency of the phonon (lattice vibration) mode with polarization s at wavevector q. The coefficient of thermal expansion is then given by 2 _oq;s kB X _oq;s a¼ nq;s exp (2) g , 3B q;s kB T kB T q;s where nq;s ¼ ð1 expð_oq;s =kB TÞÞ1 and B is the bulk modulus. Both ZrW2 O8 and ZnðCNÞ2 have cubic crystal structures containing relatively rigid molecular polyhedra. It is the transverse motion of these rigid polyhedra (rigid unit modes) that are believed to be the origin of the negative Gruneisen parameters in ZrW2 O8 [1,5] and ZnðCNÞ2 [3]. Both inelastic neutron scattering [2,6] and Raman scattering [7,8] have been used to examine phonon spectra in
ARTICLE IN PRESS K.W. Chapman et al. / Physica B 385–386 (2006) 60–62
ZrW2 O8 and generally support this hypothesis. In this paper we report the first results of inelastic neutron scattering measurements on powder samples of ZnðCNÞ2 and the closely related compound ZnPtðCNÞ6 . 2. Experimental details The experimental measurements reported in this paper were carried out using the PRISMA spectrometer at the ISIS Spallation Neutron Source, Rutherford Appleton Laboratory, UK. PRISMA can be operated as an indirect geometry time of flight spectrometer [9] and in these measurements it was set up to work with an analyzing energy E f ¼ 8 meV, which led to a resolution of DE ¼ 0:6 meV at the elastic line. Powder samples of ZnðCNÞ2 and ZnPtðCNÞ6 were loaded into cylindrical vanadium walled cans, 10 mm in diameter by 40 mm tall, and attached to the copper block of a closed cycle refrigerator that was mounted in the beam on PRISMA. The data were taken over a range in scattering angle from 129 to 109 and at temperatures of 20, 75, 150, 200 and 250 K for the ZnðCNÞ2 sample, and 8 and 50 K for the ZnPtðCNÞ6 sample. 3. Experimental results
giving rise to the peak form a relatively dispersionless branch over a large portion of the Brillouin zone. Such a description would closely fit the behaviour of an optic mode, a rigid unit mode. However given that the peak is at such low energy, it is perhaps more likely that it is a transverse acoustic mode, which has hybridized with a transverse optic mode. Such a speculation can only be verified by measurements on a single crystal specimen. The temperature dependence of the 2 meV low energy peak is shown in Figs. 2 and 3. As can be clearly seen in Fig. 2 the position of the low energy peak moves to higher energy with increasing temperature. This temperature dependence of the peak position is plotted in Fig. 3. Although the peak corresponds to a band of modes we can understand this temperature dependence by interpreting it as though it were a single mode. Since our measurements are performed at constant pressure rather than constant volume we need to express the temperature dependence of the phonon energy as [8] qoq;s qoq;s qV qoq;s ¼ þ , (3) qT P qT V qT P qV T where the two terms on the right-hand side physically represent the effects of the intrinsic anharmonicity of the mode and the expansion (contraction in this case) of the
20K 75K 150K 200K 250K
40
30 Counts (arb. units)
In Fig. 1 the measured spectrum from ZnðCNÞ2 at 20 K is shown for energy transfers up to 20 meV. There is a continuous spectrum of excitations in this energy range, with maxima at 8 meV and 11 meV. However the striking feature in the spectrum is the very strong low energy peak at 2 meV. It should be noted that since this is a powder sample and, because the spectrum corresponds to an integration over the scattering angle, this peak is not a single mode but instead represents a ‘‘band’’ of modes. However given the relative sharpness, and strength, of the peak it is reasonable to conclude that the band of modes
61
20
10
0 0
1
2
3
4
5
Energy transfer (meV) Fig. 1. The low energy phonon spectrum of ZnðCNÞ2 at 20 K.
Fig. 2. Temperature dependence of the 2 meV peak in ZnðCNÞ2 .
6
ARTICLE IN PRESS K.W. Chapman et al. / Physica B 385–386 (2006) 60–62 2.6
14
2.5
12
2.4
10 Counts (arb. units)
Energy (meV)
62
2.3 2.2 2.1
8
6
4
ZnPt(CN)6
2.0 0
50
100
150
200
250
300
Temperature (K)
Fig. 3. Peak position vs. temperature for the ZnðCNÞ2 low energy peak.
T=8K 2
0 0
lattice, respectively. If the definitions of the mode Gruneisen parameter gq;s and the coefficient of thermal expansion a are used then Eq. (3) can be re-written as [8] 1 qoq;s 1 qoq;s ¼ gq;s a. (4) oq;s qT P oq;s qT V For ZnðCNÞ2 we can estimate from Fig. 3 that the normalized total anharmonicity (the left-hand side of Eq. (4)) is 106 106 K1 for temperatures below 150 K. Since both a and gq;s are negative in Eq. (4) this means that the intrinsic (constant volume) normalized anharmonicity must be even larger than this value. We can set a scale for this by comparing to the results for a ‘‘regular’’ material such as aluminium which, using the results of Nakai et al. [10], has an intrinsic normalized (constant volume) anharmonicity of 4 106 K1 and ZrW2 O8 where Ravindran et al. [8] have determined a value of 27 106 K1 . The low energy spectrum of ZnPtðCNÞ6 is shown in Fig. 4 at a temperature of 8 K. The co-efficient of the thermal expansion for ZnPtðCNÞ6 is a 4 106 K1 [11] and as can be seen the strong low energy peak found in ZnðCNÞ2 is missing. The first peak in the spectrum that may correspond to a dispersionless band of rigid unit modes is at 7:5 meV. 4. Conclusions Inelastic neutron scattering measurements, reported in this paper, show the presence of a band of dispersionless rigid unit modes in ZnðCNÞ2 at low energy. The presence of these modes will make a significant contribution to the
5 10 Energy transfer (meV)
15
20
Fig. 4. The low energy phonon spectrum of ZnPtðCNÞ6 at 8 K.
large NTE that has been observed in ZnðCNÞ2 . From a study of the temperature dependence of this band it can be estimated that the normalized intrinsic anharmonicity of these modes is extremely large. The inelastic spectrum of the related ZnPtðCNÞ6 compound has also been measured. It shows a much smaller negative thermal expansion, and does not possess the large low energy peak observed in ZnðCNÞ2 . References [1] T.A. Mary, J.S.O. Evans, T. Vogt, A.W. Sleight, Science 272 (1996) 90. [2] G. Ernst, C. Broholm, G.R. Kowach, A.P. Ramirez, Nature 396 (1998) 147. [3] A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030. [4] N.W. Ashcroft, N.D. Mermin, Solid State Physics, CBS Publishing Asia, Hong Kong, 1987. [5] A.K.A. Pryde, K.D. Hammonds, M.T. Dove, V. Heine, J.D. Gale, M.C. Warren, J. Phys.: Condens. Matter 8 (1996) 10973. [6] R. Mittal, S.L. Chaplot, H. Schober, T.A. Mary, Phys. Rev. Lett. 86 (2001) 4692. [7] T.R. Ravindran, A.K. Arora, T.A. Mary, Phys. Rev. Lett. 84 (2000) 3879. [8] T.R. Ravindran, A.K. Arora, T.A. Mary, Phys. Rev. B 67 (2003) 064301. [9] M.J. Bull, M.J. Harris, U. Steigenberger, M. Hagen, C. Petrillo, F. Sacchetti, Physica B 234 (1997) 1061. [10] Y. Nakai, N. Kunitomi, M. Hagen, R.M. Nicklow, A. Onodera, Solid State Commun. 64 (1987) 783. [11] K.W. Chapman, P.J. Chupas, C.J. Kepert, J. Am. Chem. Soc. 128 (2006) 7009.
Physica B 385–386 (2006) 60–62 www.elsevier.com/locate/physb
Low energy phonons in the NTE compounds ZnðCNÞ2 and ZnPtðCNÞ6 Karena W. Chapmana, Mark Hagenb,,1, Cameron J. Keperta, Pascal Manuelc a
School of Chemistry, University of Sydney, New South Wales 2006, Australia b Bragg Institute, ANSTO, Menai, New South Wales 2234, Australia c ISIS Facility, Rutherford Appleton Lab., Oxon. OX11 0QX, UK
Abstract The compounds ZnðCNÞ2 and ZnPtðCNÞ6 both display negative thermal expansion (NTE) properties, that is to say they undergo a volume contraction with increasing temperature. In the case of ZnðCNÞ2 this volume contraction occurs over a temperature range from 25 to 375 K with a coefficient of thermal expansion a ¼ 16:9ð2Þ 106 K1 [A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030]. This phenomenon is believed to be related to the presence of low energy rigid unit modes (RUMS) in the phonon dispersion relations of ZnðCNÞ2 [A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030]. We have examined the low energy part of the phonon density of states in ZnðCNÞ2 and ZnPtðCNÞ6 using time of flight inelastic neutron scattering from powder samples. In ZnðCNÞ2 there is a strong peak in the density of states at 2 meV whose temperature dependence can be correlated with that of a. There is a similar peak in the density of states of ZnPtðCNÞ6 at 7.5 meV, which correlates with the smaller NTE effect in this compound. r 2006 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 63.20.Dj; 63.20.Ry Keywords: Negative thermal expansion; Rigid unit modes; Inelastic neutron scattering
1. Introduction While there are a number of materials that exhibit anisotropic negative thermal expansion (NTE) over a limited temperature range there are only a few that exhibit isotropic (volume) contraction over a wide range of temperature. The best known, and most widely studied, of these materials is ZrW2 O8 [1] which has a co-efficient of thermal expansion a ¼ 9 106 K1 [2]. ZnðCNÞ2 however displays an even larger NTE effect, with a value of a ¼ 16:9 106 K1 [3]. The origin of NTE can be explained in terms of the thermal excitation of phonons that have a negative Gruneisen parameter. It is well known [4] that a purely harmonic theory of lattice vibrations in a solid cannot explain the presence of thermal expansion (or contraction) and that anharmonic terms are required. The effect of these Corresponding author.
E-mail address: [email protected] (M. Hagen). Present address: Spallation Neutron Source, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge TN37831, USA. 1
0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.05.102
anharmonic terms are modelled in a quasi-harmonic way [4] in terms of the mode Gruneisen parameter gq;s given by gq;s ¼
qloge oq;s V qoq;s ¼ , oq;s qV qloge V
(1)
where V is the volume and oq;s is the frequency of the phonon (lattice vibration) mode with polarization s at wavevector q. The coefficient of thermal expansion is then given by 2 _oq;s kB X _oq;s a¼ nq;s exp (2) g , 3B q;s kB T kB T q;s where nq;s ¼ ð1 expð_oq;s =kB TÞÞ1 and B is the bulk modulus. Both ZrW2 O8 and ZnðCNÞ2 have cubic crystal structures containing relatively rigid molecular polyhedra. It is the transverse motion of these rigid polyhedra (rigid unit modes) that are believed to be the origin of the negative Gruneisen parameters in ZrW2 O8 [1,5] and ZnðCNÞ2 [3]. Both inelastic neutron scattering [2,6] and Raman scattering [7,8] have been used to examine phonon spectra in
ARTICLE IN PRESS K.W. Chapman et al. / Physica B 385–386 (2006) 60–62
ZrW2 O8 and generally support this hypothesis. In this paper we report the first results of inelastic neutron scattering measurements on powder samples of ZnðCNÞ2 and the closely related compound ZnPtðCNÞ6 . 2. Experimental details The experimental measurements reported in this paper were carried out using the PRISMA spectrometer at the ISIS Spallation Neutron Source, Rutherford Appleton Laboratory, UK. PRISMA can be operated as an indirect geometry time of flight spectrometer [9] and in these measurements it was set up to work with an analyzing energy E f ¼ 8 meV, which led to a resolution of DE ¼ 0:6 meV at the elastic line. Powder samples of ZnðCNÞ2 and ZnPtðCNÞ6 were loaded into cylindrical vanadium walled cans, 10 mm in diameter by 40 mm tall, and attached to the copper block of a closed cycle refrigerator that was mounted in the beam on PRISMA. The data were taken over a range in scattering angle from 129 to 109 and at temperatures of 20, 75, 150, 200 and 250 K for the ZnðCNÞ2 sample, and 8 and 50 K for the ZnPtðCNÞ6 sample. 3. Experimental results
giving rise to the peak form a relatively dispersionless branch over a large portion of the Brillouin zone. Such a description would closely fit the behaviour of an optic mode, a rigid unit mode. However given that the peak is at such low energy, it is perhaps more likely that it is a transverse acoustic mode, which has hybridized with a transverse optic mode. Such a speculation can only be verified by measurements on a single crystal specimen. The temperature dependence of the 2 meV low energy peak is shown in Figs. 2 and 3. As can be clearly seen in Fig. 2 the position of the low energy peak moves to higher energy with increasing temperature. This temperature dependence of the peak position is plotted in Fig. 3. Although the peak corresponds to a band of modes we can understand this temperature dependence by interpreting it as though it were a single mode. Since our measurements are performed at constant pressure rather than constant volume we need to express the temperature dependence of the phonon energy as [8] qoq;s qoq;s qV qoq;s ¼ þ , (3) qT P qT V qT P qV T where the two terms on the right-hand side physically represent the effects of the intrinsic anharmonicity of the mode and the expansion (contraction in this case) of the
20K 75K 150K 200K 250K
40
30 Counts (arb. units)
In Fig. 1 the measured spectrum from ZnðCNÞ2 at 20 K is shown for energy transfers up to 20 meV. There is a continuous spectrum of excitations in this energy range, with maxima at 8 meV and 11 meV. However the striking feature in the spectrum is the very strong low energy peak at 2 meV. It should be noted that since this is a powder sample and, because the spectrum corresponds to an integration over the scattering angle, this peak is not a single mode but instead represents a ‘‘band’’ of modes. However given the relative sharpness, and strength, of the peak it is reasonable to conclude that the band of modes
61
20
10
0 0
1
2
3
4
5
Energy transfer (meV) Fig. 1. The low energy phonon spectrum of ZnðCNÞ2 at 20 K.
Fig. 2. Temperature dependence of the 2 meV peak in ZnðCNÞ2 .
6
ARTICLE IN PRESS K.W. Chapman et al. / Physica B 385–386 (2006) 60–62 2.6
14
2.5
12
2.4
10 Counts (arb. units)
Energy (meV)
62
2.3 2.2 2.1
8
6
4
ZnPt(CN)6
2.0 0
50
100
150
200
250
300
Temperature (K)
Fig. 3. Peak position vs. temperature for the ZnðCNÞ2 low energy peak.
T=8K 2
0 0
lattice, respectively. If the definitions of the mode Gruneisen parameter gq;s and the coefficient of thermal expansion a are used then Eq. (3) can be re-written as [8] 1 qoq;s 1 qoq;s ¼ gq;s a. (4) oq;s qT P oq;s qT V For ZnðCNÞ2 we can estimate from Fig. 3 that the normalized total anharmonicity (the left-hand side of Eq. (4)) is 106 106 K1 for temperatures below 150 K. Since both a and gq;s are negative in Eq. (4) this means that the intrinsic (constant volume) normalized anharmonicity must be even larger than this value. We can set a scale for this by comparing to the results for a ‘‘regular’’ material such as aluminium which, using the results of Nakai et al. [10], has an intrinsic normalized (constant volume) anharmonicity of 4 106 K1 and ZrW2 O8 where Ravindran et al. [8] have determined a value of 27 106 K1 . The low energy spectrum of ZnPtðCNÞ6 is shown in Fig. 4 at a temperature of 8 K. The co-efficient of the thermal expansion for ZnPtðCNÞ6 is a 4 106 K1 [11] and as can be seen the strong low energy peak found in ZnðCNÞ2 is missing. The first peak in the spectrum that may correspond to a dispersionless band of rigid unit modes is at 7:5 meV. 4. Conclusions Inelastic neutron scattering measurements, reported in this paper, show the presence of a band of dispersionless rigid unit modes in ZnðCNÞ2 at low energy. The presence of these modes will make a significant contribution to the
5 10 Energy transfer (meV)
15
20
Fig. 4. The low energy phonon spectrum of ZnPtðCNÞ6 at 8 K.
large NTE that has been observed in ZnðCNÞ2 . From a study of the temperature dependence of this band it can be estimated that the normalized intrinsic anharmonicity of these modes is extremely large. The inelastic spectrum of the related ZnPtðCNÞ6 compound has also been measured. It shows a much smaller negative thermal expansion, and does not possess the large low energy peak observed in ZnðCNÞ2 . References [1] T.A. Mary, J.S.O. Evans, T. Vogt, A.W. Sleight, Science 272 (1996) 90. [2] G. Ernst, C. Broholm, G.R. Kowach, A.P. Ramirez, Nature 396 (1998) 147. [3] A.L. Goodwin, C.J. Kepert, Phys. Rev. B 71 (2005) 14030. [4] N.W. Ashcroft, N.D. Mermin, Solid State Physics, CBS Publishing Asia, Hong Kong, 1987. [5] A.K.A. Pryde, K.D. Hammonds, M.T. Dove, V. Heine, J.D. Gale, M.C. Warren, J. Phys.: Condens. Matter 8 (1996) 10973. [6] R. Mittal, S.L. Chaplot, H. Schober, T.A. Mary, Phys. Rev. Lett. 86 (2001) 4692. [7] T.R. Ravindran, A.K. Arora, T.A. Mary, Phys. Rev. Lett. 84 (2000) 3879. [8] T.R. Ravindran, A.K. Arora, T.A. Mary, Phys. Rev. B 67 (2003) 064301. [9] M.J. Bull, M.J. Harris, U. Steigenberger, M. Hagen, C. Petrillo, F. Sacchetti, Physica B 234 (1997) 1061. [10] Y. Nakai, N. Kunitomi, M. Hagen, R.M. Nicklow, A. Onodera, Solid State Commun. 64 (1987) 783. [11] K.W. Chapman, P.J. Chupas, C.J. Kepert, J. Am. Chem. Soc. 128 (2006) 7009.