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Microscopic collective dynamics in liquid tellurium
Journal of Non-Crystalline Solids 353 (2007) 1005–1007 www.elsevier.com/locate/jnoncrysol
Microscopic collective dynamics in liquid tellurium M.D. Ruiz-Martı´n
a,* ,
M. Jime´nez-Ruiz a, F.J. Bermejo b, R. Ferna´ndez-Perea
c
a Institut Laue Langevin, 6 Rue Jules Horowitz, BP 156, F-38042 Grenoble cedex 9, France C.S.I.C., Department of Electricity and Electronics, UPV/EHU, Box 644, 4880 Bilbao, Spain Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı´ficas, Serrano 123, Madrid, Spain b
c
Available online 20 February 2007
Abstract The spectra of collective excitations of liquid tellurium have been studied by means of inelastic neutron scattering. Here we report on the dynamics of liquid Te as measured at two different temperatures, just above melting (Tm = 723 K) and at 1000 K. Estimates for the velocity of propagating excitations for both temperatures have been obtained from the experimental data and a contrasting behavior is found with respect to anomalies shown by the adiabatic sound velocity measured by ultrasound methods. 2007 Elsevier B.V. All rights reserved. PACS: 61.25.Mv; 62.60.+v; 61.12.–q Keywords: Phonons; Liquid alloys and liquid metals; Neutron diffraction/scattering; Transport properties – liquids
1. Introduction The physical properties of liquid tellurium (l-Te) have been studied extensively during the last decades. Crystalline tellurium, as well as selenium, is a twofold coordinated semiconductor that crystallizes into a hexagonal structure formed by chains along the crystal c-axis. The intrachain bonding is purely covalent whereas interactions between chains are known to be of Van der Waals type. Also, in contrast with Se, it becomes a poor conductor upon melting [1]. Models concerning the structure of l-Te have been a matter of debate for quite some time. First, a threefold coordinated random network was proposed [2], but the model currently in use, described by Bichara et al. [3] points out the existence of twofold coordinated chains at melting, which survive from the crystal. The length of these chains becomes significantly shorter than those characteristic of Se, and have about ten atoms [4]. The coordination number increases up to 2.8 at T 1000 K, due to a break-
*
Corresponding author. Tel.: +33 4 76 20 70 04; fax: +33 4 76 20 76 88. E-mail address: [email protected] (M.D. Ruiz-Martı´n).
0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.12.087
down of the interchain structure, which is accompanied by the formation of longer covalent bonds. The temperature dependence of the macroscopic sound velocity of l-Te is also anomalous if compared to other chalcogenide elements. As reported from an ultrasound study, the sound velocity increases with increasing temperature from some 890 m s1 up to 1100 m s1 at 1100 K [5]. The result cannot be correlated with any density anomaly upon melting, since such a quantity decreases with increasing temperature. The purpose of this work is to find out whether the macroscopic anomaly just referred to has a microscopic correlate in the frequency of the excitations supported by the liquid, and therefore a correlate in bonding patterns may explain such an anomaly. Here we report on the spectra of excitations of l-Te, measured near the melting point and 1000 K by means of inelastic neutron scattering. 2. Experimental The experiment was performed on the thermal three-axis spectrometer IN8, at the Institut Laue Langevin (ILL). The experiment was carried out in a constant final wave-vector
1006
M.D. Ruiz-Martı´n et al. / Journal of Non-Crystalline Solids 353 (2007) 1005–1007
˚ 1 achieving a resolution in energy mode with kf = 4.1 A transfers measured as the full width at half maximum (FWHM) of the elastic line using a vanadium rod of 0.9 meV. A high purity tellurium sample was heated up above the sample melting point Tm = 723 K. Two thermodynamic states were studied, one a few degrees above melting (773 K) and another at 1000 K, where the sound velocity of the melt reaches its maximum. Several inelastic scans at constant momentum transfer ˚ 1 up to 0.7 A ˚ 1. At the latter were performed from 0.3 A value, the excitations became too broad to provide us some information. The multiple scattering contribution, which was estimated by means of computer simulations using a modified version of the MSCAT code [6], was negligible. 3. Results The measured intensity can be described as IðQ; xÞ / SðQ; xÞ RðQ; xÞ þ N bcgr , where RðQ; xÞ is a Gaussian function which stands for the spectrometer resolution, Nbcgr is a background term and S(Q, x) represents the dynamic structure factor, a quantity that contains all the information about the liquid dynamics, and which can be
modeled in terms of the sum of a quasielastic and an inelastic contribution. The quasielastic part model the stochastic processes by means of a Lorentzian function. The inelastic part, which contains the information about the collective dynamics, can be modeled in terms of a damped harmonic oscillator S inel ðQ; xÞ ¼ H ðQÞx½nðxÞ þ 1
4xC 2
ðx2 X2 Þ þ 4x2 C2
;
ð1Þ
where X is a renormalised excitation frequency, X2 = x2 + C2, and x and C stand for the bare oscillator frequency and the damping, respectively. H(Q) is an amplitude coefficient and n(x + 1) is the Bose occupation factor. A set of corrected and fitted spectra is shown in Fig. 1. The quasielastic linewidths obtained from the fitting parameters are close to those observed in previous studies, and display the quadratic wave-vector dependence Dx = 2DsQ2, where Ds represents the self-diffusion coefficient. Our estimate for this coefficient was 2.3(6) · 105 cm2 s1, which can be compared to the value of 2.(6) · 105 cm2 s1found by Chiba et al. [7]. The wave-vector dependence of the fitting parameters, x and C, is displayed in Fig. 2. From there, estimates for the sound velocity and the damping coefficient were obtained.
Fig. 1. Several experimental spectra measured at 773 K and 1023 K, fitted with the model described in the text. The multiple scattering contribution is represented by a dashed line in the lower frames.
M.D. Ruiz-Martı´n et al. / Journal of Non-Crystalline Solids 353 (2007) 1005–1007
1007
neutron scattering. The measured frequencies follow a linear dispersion, the slope of which approximately follows the high-frequency sound velocity given by c1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3k B T =M þ 3x2E R20 =10 [8], where xE is the Einstein frequency and R0 the atomic diameter. Equating c1 to the values obtained in this work, xE takes the values of 6.12 meV and 5.81 meV for the measurements at 773 K and 1023 K, respectively. These values come close to the maximum of the spectral frequency distribution studied by means of neutron scattering in [9], and also are closer to those exhibited by the crystalline solid than those derived from ultrasound. The high sound velocities here obtained are understood on the basis of the elastic response of the liquid to a highfrequency probe. Within our frequency window, processes that may be far less important to lower frequency probes such as the tear and repair of the directional bonds with s ps or whole chain motions, all of them having frequencies of the order of a meV [10,11], become highly relevant. The temperature dependence of the sound velocity here found is however at odds with that reported from ultrasound. At present it is not clear what the cause of such discrepancy may be. However, the present results serve to rule out the presence of any microscopic anomaly as responsible for the anomalous behavior found at long wavelengths (ultrasound), and therefore the origin of such an anomaly has to be sought at far larger length and time scales. References Fig. 2. Dispersion curves for the liquid at the studied temperatures. Squares and circles represent the frequency and the width of the excitations, respectively. The dotted line represents the macroscopic measurements. The dashed-dotted line represents the dispersion curve for the solid. The values obtained for the sound velocity and the damping coefficient for both temperatures can be read in the frame.
4. Discussion and conclusions The values obtained for the sound velocity lay well above those measured by ultrasound methods. Moreover, the anomaly present in the macroscopic measurements has not been found in the microscopic scale explored by
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
J.C. Perron, Adv. Phys. 16 (1967) 657. B. Cabane, J. Friedel, J. Phys. 32 (1971) 73. C. Bichara, J.-Y. Raty, J.-P. Gaspard, Phys. Rev. B 53 (1996) 206. T. Aoyama, Y. Hiwatari, A. Ueda, J. Phys. Soc. Jpn. 68 (1999) 857. M.B. Gitis, I.G. Mikhailov, Sov. Phys. 12 (1966) 14. J.R.D. Copley, P. Verkerk, A. van Well, H. Fredrikz, Comput. Phys. Commun. 40 (1986) 337. A. Chiba, Y. Ohmasa, M. Yao, J. Chem. Phys. 119 (2003) 9047. D.L. Price, M.L. Saboungi, F.J. Bermejo, Rep. Prog. Phys. 66 (2003) 407. A. Chiba, Y. Ohmasa, Y. Kawakita, M. Yao, H. Endo, J. Non-Cryst. Solids 312–314 (2002) 384. R.M. Martı´n, G. Lucovski, K. Helliwell, Phys. Rev. B 13 (1976) 1383. B.C. Pan, Phys. Rev. B 65 (2002) 085407.
Microscopic collective dynamics in liquid tellurium M.D. Ruiz-Martı´n
a,* ,
M. Jime´nez-Ruiz a, F.J. Bermejo b, R. Ferna´ndez-Perea
c
a Institut Laue Langevin, 6 Rue Jules Horowitz, BP 156, F-38042 Grenoble cedex 9, France C.S.I.C., Department of Electricity and Electronics, UPV/EHU, Box 644, 4880 Bilbao, Spain Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı´ficas, Serrano 123, Madrid, Spain b
c
Available online 20 February 2007
Abstract The spectra of collective excitations of liquid tellurium have been studied by means of inelastic neutron scattering. Here we report on the dynamics of liquid Te as measured at two different temperatures, just above melting (Tm = 723 K) and at 1000 K. Estimates for the velocity of propagating excitations for both temperatures have been obtained from the experimental data and a contrasting behavior is found with respect to anomalies shown by the adiabatic sound velocity measured by ultrasound methods. 2007 Elsevier B.V. All rights reserved. PACS: 61.25.Mv; 62.60.+v; 61.12.–q Keywords: Phonons; Liquid alloys and liquid metals; Neutron diffraction/scattering; Transport properties – liquids
1. Introduction The physical properties of liquid tellurium (l-Te) have been studied extensively during the last decades. Crystalline tellurium, as well as selenium, is a twofold coordinated semiconductor that crystallizes into a hexagonal structure formed by chains along the crystal c-axis. The intrachain bonding is purely covalent whereas interactions between chains are known to be of Van der Waals type. Also, in contrast with Se, it becomes a poor conductor upon melting [1]. Models concerning the structure of l-Te have been a matter of debate for quite some time. First, a threefold coordinated random network was proposed [2], but the model currently in use, described by Bichara et al. [3] points out the existence of twofold coordinated chains at melting, which survive from the crystal. The length of these chains becomes significantly shorter than those characteristic of Se, and have about ten atoms [4]. The coordination number increases up to 2.8 at T 1000 K, due to a break-
*
Corresponding author. Tel.: +33 4 76 20 70 04; fax: +33 4 76 20 76 88. E-mail address: [email protected] (M.D. Ruiz-Martı´n).
0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.12.087
down of the interchain structure, which is accompanied by the formation of longer covalent bonds. The temperature dependence of the macroscopic sound velocity of l-Te is also anomalous if compared to other chalcogenide elements. As reported from an ultrasound study, the sound velocity increases with increasing temperature from some 890 m s1 up to 1100 m s1 at 1100 K [5]. The result cannot be correlated with any density anomaly upon melting, since such a quantity decreases with increasing temperature. The purpose of this work is to find out whether the macroscopic anomaly just referred to has a microscopic correlate in the frequency of the excitations supported by the liquid, and therefore a correlate in bonding patterns may explain such an anomaly. Here we report on the spectra of excitations of l-Te, measured near the melting point and 1000 K by means of inelastic neutron scattering. 2. Experimental The experiment was performed on the thermal three-axis spectrometer IN8, at the Institut Laue Langevin (ILL). The experiment was carried out in a constant final wave-vector
1006
M.D. Ruiz-Martı´n et al. / Journal of Non-Crystalline Solids 353 (2007) 1005–1007
˚ 1 achieving a resolution in energy mode with kf = 4.1 A transfers measured as the full width at half maximum (FWHM) of the elastic line using a vanadium rod of 0.9 meV. A high purity tellurium sample was heated up above the sample melting point Tm = 723 K. Two thermodynamic states were studied, one a few degrees above melting (773 K) and another at 1000 K, where the sound velocity of the melt reaches its maximum. Several inelastic scans at constant momentum transfer ˚ 1 up to 0.7 A ˚ 1. At the latter were performed from 0.3 A value, the excitations became too broad to provide us some information. The multiple scattering contribution, which was estimated by means of computer simulations using a modified version of the MSCAT code [6], was negligible. 3. Results The measured intensity can be described as IðQ; xÞ / SðQ; xÞ RðQ; xÞ þ N bcgr , where RðQ; xÞ is a Gaussian function which stands for the spectrometer resolution, Nbcgr is a background term and S(Q, x) represents the dynamic structure factor, a quantity that contains all the information about the liquid dynamics, and which can be
modeled in terms of the sum of a quasielastic and an inelastic contribution. The quasielastic part model the stochastic processes by means of a Lorentzian function. The inelastic part, which contains the information about the collective dynamics, can be modeled in terms of a damped harmonic oscillator S inel ðQ; xÞ ¼ H ðQÞx½nðxÞ þ 1
4xC 2
ðx2 X2 Þ þ 4x2 C2
;
ð1Þ
where X is a renormalised excitation frequency, X2 = x2 + C2, and x and C stand for the bare oscillator frequency and the damping, respectively. H(Q) is an amplitude coefficient and n(x + 1) is the Bose occupation factor. A set of corrected and fitted spectra is shown in Fig. 1. The quasielastic linewidths obtained from the fitting parameters are close to those observed in previous studies, and display the quadratic wave-vector dependence Dx = 2DsQ2, where Ds represents the self-diffusion coefficient. Our estimate for this coefficient was 2.3(6) · 105 cm2 s1, which can be compared to the value of 2.(6) · 105 cm2 s1found by Chiba et al. [7]. The wave-vector dependence of the fitting parameters, x and C, is displayed in Fig. 2. From there, estimates for the sound velocity and the damping coefficient were obtained.
Fig. 1. Several experimental spectra measured at 773 K and 1023 K, fitted with the model described in the text. The multiple scattering contribution is represented by a dashed line in the lower frames.
M.D. Ruiz-Martı´n et al. / Journal of Non-Crystalline Solids 353 (2007) 1005–1007
1007
neutron scattering. The measured frequencies follow a linear dispersion, the slope of which approximately follows the high-frequency sound velocity given by c1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3k B T =M þ 3x2E R20 =10 [8], where xE is the Einstein frequency and R0 the atomic diameter. Equating c1 to the values obtained in this work, xE takes the values of 6.12 meV and 5.81 meV for the measurements at 773 K and 1023 K, respectively. These values come close to the maximum of the spectral frequency distribution studied by means of neutron scattering in [9], and also are closer to those exhibited by the crystalline solid than those derived from ultrasound. The high sound velocities here obtained are understood on the basis of the elastic response of the liquid to a highfrequency probe. Within our frequency window, processes that may be far less important to lower frequency probes such as the tear and repair of the directional bonds with s ps or whole chain motions, all of them having frequencies of the order of a meV [10,11], become highly relevant. The temperature dependence of the sound velocity here found is however at odds with that reported from ultrasound. At present it is not clear what the cause of such discrepancy may be. However, the present results serve to rule out the presence of any microscopic anomaly as responsible for the anomalous behavior found at long wavelengths (ultrasound), and therefore the origin of such an anomaly has to be sought at far larger length and time scales. References Fig. 2. Dispersion curves for the liquid at the studied temperatures. Squares and circles represent the frequency and the width of the excitations, respectively. The dotted line represents the macroscopic measurements. The dashed-dotted line represents the dispersion curve for the solid. The values obtained for the sound velocity and the damping coefficient for both temperatures can be read in the frame.
4. Discussion and conclusions The values obtained for the sound velocity lay well above those measured by ultrasound methods. Moreover, the anomaly present in the macroscopic measurements has not been found in the microscopic scale explored by
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
J.C. Perron, Adv. Phys. 16 (1967) 657. B. Cabane, J. Friedel, J. Phys. 32 (1971) 73. C. Bichara, J.-Y. Raty, J.-P. Gaspard, Phys. Rev. B 53 (1996) 206. T. Aoyama, Y. Hiwatari, A. Ueda, J. Phys. Soc. Jpn. 68 (1999) 857. M.B. Gitis, I.G. Mikhailov, Sov. Phys. 12 (1966) 14. J.R.D. Copley, P. Verkerk, A. van Well, H. Fredrikz, Comput. Phys. Commun. 40 (1986) 337. A. Chiba, Y. Ohmasa, M. Yao, J. Chem. Phys. 119 (2003) 9047. D.L. Price, M.L. Saboungi, F.J. Bermejo, Rep. Prog. Phys. 66 (2003) 407. A. Chiba, Y. Ohmasa, Y. Kawakita, M. Yao, H. Endo, J. Non-Cryst. Solids 312–314 (2002) 384. R.M. Martı´n, G. Lucovski, K. Helliwell, Phys. Rev. B 13 (1976) 1383. B.C. Pan, Phys. Rev. B 65 (2002) 085407.