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Vibrational properties and atomic Debye temperatures for YBa2Cu3O7 from neutron powder diffraction
Physica C 175 ( 1991 ) 615-622 North-Holland
Vibrational properties and atomic Debye temperatures for YBa2Cu307 from neutron powder diffraction George H. Kwei, A.C. Lawson, W.L. Hults a n d J a m e s L. Smith Los Alamos National Laboratory, Los Alamos, N M 87545, USA
Received 29 October 1990 Revised manuscript received 19 February 1991
Neutron powder diffraction data taken at twelve sample temperatures have been used to determine both isotropic and anisotropic thermal parameters for atoms in YBa2Cu307.At fit of these parameters to a Debye-Wallermodel in turn yields the corresponding isotropic and anisotropicDebye-Wallertemperatures. The anisotropicthermal parameters for the Y, Ba and the planar Cu(2) and 0(2) atoms all exhibit a stronger temperature dependence, and therefore a lower Debye-Waller temperature, for motion alongthe c-axis;in addition, the chain O( 1) atoms showa similar strongtemperature dependencefor motion perpendicular to the chains. No discontinuities in the thermal parameters, suggestiveof possible structural changesor phonon anomalies near Tcare observed.
1. Introduction There continues to be a great deal of interest in the existence of structural instabilities or phonon anomalies near the superconducting transitions of high temperature superconductors, in particular YBa2CuaO6+6. Within the last few years many different kinds of experiments have provided somewhat conflicting evidence for instabilities or anomalies. These range from specific heat measurements of the Debye temperature [ 1,2], acoustic measurements of elastic constants [3,4], Raman and infrared spectroscopic studies of line intensities and frequency shifts [ 5-8 ], structural studies using X-ray and neutron diffraction [ 9-13 ], X-ray absorption spectroscopy [ 14,15 ], neutron recoil measurements [ 16,17 ], ion-channeling measurements [ 18 ], and inelastic neutron scattering measurements of the phonon densities of states [ 19 ] and phonon dispersion curves [20]. At present, there is no consensus from the different experiments as to which instabilities or anomalies actually exist, or are in fact important. We have been using pulsed-neutron powder diffraction data to measure the anisotropic thermal parameters of ions in crystals to study their anisotropic
vibrational motions. Of special interest has been both the static and dynamic displacements of the constituent ions in the high-To oxide superconductors and the possible relationship between these and their superconducting behavior [21,22]. Recently, we have been successful in fitting a simple Debye-Waller model to the measured temperature dependence of the thermal parameters [23] and carded out a detailed comparison of atomic Debye-Waller temperatures obtained for La2CuO4 with other experimental data and with the results of a lattice dynamical calculation [24]. The extensive work on YBa2Cu306÷6 provides an opportunity to extend our determinations of vibrational properties from diffraction data to a more complex system, to compare the information provided with the results from other techniques, and to gain insights into the behavior of superconducting materials.
2. Experimental details, data analysis and results The sample preparation and characterization has been described earlier [ 12]. Briefly, 15 gm of YBa2Cu307 was prepared using standard oxide synthesis techniques [25 ] and DC susceptibility mea-
0921-4534/91/$03.50 © 1991 - Elsevier SciencePublishers B.V. (North-Holland)
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G.H. Kwei et al. / Atomic Debye temperatures for YBa2CusO7
surements showed the Tc at onset for this sample to be 93 _+ l K, with a midpoint o f 85 _+ 1 K. The shielding volume fraction at 7 K and 100 Oe was greater than 90%, estimating a sample volume from the weight and the theoretical density. Time-of-flight neutron powder-diffraction data were collected on four detector banks ( _+ 153 ° and _ 90 ° ) o f the high intensity powder diffractometer ( H I P D ) at the Manuel Lujan Jr., Neutron Scattering Center ( L A N S C E ) at Los Alamos. In all, data were collected at twelve temperatures. Rietveld refinement of the structural model was c a r d e d out using the generalized structure analysis system (GSAS) developed by Larson and Von Dreele [26]; these gave values for the lattice constants, atomic positions, oxygen occupancies and either isotropic or anisotropic thermal parameter #l. The refined values for the oxygen occupancies gave the overall oxygen stoichiometry, YBa2Cu307. Because we were interested in comparing the refined thermal parameters, we were careful to refine data taken at the different temperatures in as consistent a way as possible. Therefore, parameters which are not temperature dependent (diffractometer constants, profile coefficients and absorption coefficients) were set to the average o f the values o f these parameters obtained in earlier refinements and not further refined. The resulting isotropic thermal parameters for YBa2Cu307 are shown in fig. 1 and the anisotropic thermal parameters are shown in fig. 2. The thermal parameters can be thought o f as meansquare atomic displacements which arise from both thermal motion and static displacements. Thus the temperature dependence o f the thermal parameters can be assumed to fit the Debye-Waller formula and can be fit with a two-term model [23,24],
( u 2) . . . . . ~ =
(u2)i~o.1 + (U2)off~,,
(1)
where (U2)ideal c a n be related to the Debye-Waller temperature and the temperature-independent (U2)off~t includes contributions from both static displacements and systematic errors. #1 We use the notation of O ( 1) for the chain oxygens, O (2) and 0(3) for the plane oxygens (with 0(2) lying above and below the O ( 1) ), and O ( 4 ) for the axial oxygens connecting the chains and planes. An ORTEP plot of the structure with this notation is shown as fig. 2 in ref. [ 12].
Table 1 Debye-Waller temperatures and offsets for the isotropic thermal displacements for the constituents of YBa2Cu3OT.Units for the Debye-Waller temperature are K and observed offsets are in 103 /k2. Atom
ODw
Observed offset
Y Ba Cu(1) Cu(2) O(1) 0(2) 0(3) 0(4)
345(6) 231(3) 487(12) 402(5) 497(7) 710(14) 719(14) 671(10)
2.1(1) 2.8(1) 0.1(1) -0.3(1) 3.1(2) 1.6(2) -1.6(2) -0.4(1)
The values o f the Debye-Waller temperature o f offsets for the constituents o f YBa2Cu307 fit to this model are presented in table 1 and shown graphically in fig. 1. When these Debye-Waller temperatures are properly mass weighted (e.g. m ~/20), most o f the ions, with the exception o f O ( 1 ), have comparable interatomic force constants. On the other hand, as had been found in the earlier neutron diffraction studies [ 9, l 0 ], O ( 1 ) has a considerably lower value o f the interatomic force constant than the other constituents. Since the motions of the constituents o f YBa2Cu307 are known to be anisotropic [21 ], especially the oxygen ions, we have also fit anisotropic Debye-Waller temperatures to the anisotropic thermal parameters. The Debye-Waller temperatures and offsets are given in table 2 and shown in fig. 2. These reveal much more about the vibrational properties o f the constituent atoms in YBa2Cu307 and provide greater opportunity for comparison with optical and inelastic neutron studies of the p h o n o n frequencies, densities o f states and dispersion curves, and with ion-channeling studies #2 In the refinements for YBa2Cu3OT, we discovered that the axial thermal parameters U3a for 0 ( 2 ) and O ( 3 ) are fairly strongly correlated, with a typical correlation coefficient o f approximately - 0 . 7 at all sample temperatures. Correlation coefficients o f this #2 The observed thermal motion of individual atoms in a crystal arise from the superposition of many normal vibrational modes of the crystal. Thus, this information is complementary to intensities of individual normal modes involving many atoms.
G.H. Kwei et al. /Atomic Debye temperatures for YBa2Cu~O7 lS
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Atom
ODw
Observed offset
Y
U. U22 U3a
369(22) 350(22) 234(6)
-1.5(3) 4.9(4) 5.1(3)
Ba
Un U22 U33
252(11) 233(9) 191(5)
2.1(3) 0.6(4) 5.6(3)
Cu(l)
Un U22 U33
334(12) 338(13) 400(19)
-1.5(4) -3.0(4) 6.1(3)
Cu(2)
Un /-/22 U33
663(70) 595(50) 242(3)
-1.6(3) 0.7(3) 1.6(2)
O(1)
Un U22 U33
404(11) 666(39) 388(10)
1.3(6) 3.9(6) 5.3(7)
0(2)
Ut~ U22 U33
1400(300) 880(60) 345(5)
3.3(5) --1.3(4) 3.4(5)
0(3)
U~, U22 U33
708(30) 734(33) 1600(500)
3.5(4) -0.2(4) -2.2(8)
0(4)
Ut~ U22 /-]33
530(16) 670(30) 728(30)
0.4(4) 1.2(4) -0.8(4)
magnitude are thought to result in less reliable, but nevertheless useful, refined parameters. Thus the relatively low resolution data available from HIPD may not be sufficient to distinguish reliably between axial thermal motion of the oxygens in the CuO2 planes and the errors in the resulting values are certainly larger than the reported standard deviations. However, there may be a more interesting reason for these correlations: the least squares procedure used in fitting the data assumes a harmonic model for independent constituent ions and both the correlation and the difficulty in fitting Ua3 for 0 ( 2 ) and 0 ( 3 ) may in fact result from substantial anharmonicity in the vibrational motion or a coupling of the axial vibrational motions of the two planar oxygens. In addition, the positional parameters z for O (2) and 0 ( 3 ) , the thermal parameters U 2 2 for 0 ( 2 ) and U~1 for O (3), and the thermal parameters U1] and U22 for Ba, were all found to be slightly correlated, but the refined values do not show the scatter shown by U33 for 0 ( 2 ) and 0 ( 3 ) . Figure 2 shows the anisotropic thermal parameters and the fits to the two-term Debye-Waller model. The data show the large anisotropic thermal parameters along the c-axis observed by us for Ba, Cu (2) and 0 ( 2 ) in our earlier joint X-ray/neutron study of YBa2Cu307 at room temperature and reinforce the existence of a soft optic mode involving these atoms [21 ]. In addition, the thermal displacements for Y also indicate a lower Debye-Waller temperature along the c-axis. Thus, all cations with the exception of Cu ( l ), show relatively soft-mode behavior along
618
G.H. Kwei et al. / Atomic Debye temperatures for YBa2Cu307
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Fig. 2. Mean-square anisotropic thermal displacements for all constituents in YBa2Cu3OT. Error bars representing one standard deviation are omitted for points where they are smaller than these size of the points. Curves represent fits to the two-term Debye-Waller model. Fits to U33 for O ( 2 ) and O (3) are included despite the correlation between these parameters in the Rietveld refinement; the erratic behavior of U33 for these atoms leads to poor fits to the model.
G.H. Kweiet al. /Atomic Debyetemperaturesfor YBa2Cu307
the c-axis. Two of the oxygens exhibit significant soft mode behavior: the planar oxygen O (2) is extremely soft along the c-axis and the chain oxygen O ( 1 ) is soft in both directions transverse to the Cu(1 ) O( l )-Cu( 1 ) bonds. Two "modes" involving oxygens in the CuO2 planes appear to be extremely stiff: O (2) has a Debye-Waller temperature of ~ 1400 K along the a-axis and O (3), whose thermal parameter along the c-axis is strongly correlated, has a DebyeWaller temperature of ~ 1600 K along this direction. The thermal displacements for the axial oxygen O (4) show it to be nearly isotropic and moderately stiff. Thus O(1 ) and 0 ( 2 ) appear to be relatively soft perpendicular to the Cu ( 1 )-O ( 1 ) -Cu ( 1 ) bonds and the c-axis, respectively, while 0 ( 3 ) and 0 ( 4 ) are both nearly isotropic and relatively stiff.
3. Discussion
As in all the cuprate superconductors, the C u O 2 sheets are thought to be responsible for carrying the supercurrents. In YBa2Cu307, the Cu ( 1 ) - O ( 1 ) chains provide charge reservoirs for the CuO2 sheets and are coupled to them through the axial oxygens O (4) [ 27 ]. The earlier joint X-ray/neutron diffraction studies at room temperature have shown that both CuO2 sheets and Cu( 1 ) - O ( 1 ) chains display relatively large and anisotropic thermal motion: along the c-axis for Cu(2) and transverse to the Cu(1 ) O ( 1 )-Cu ( 1 ) chains for O ( 1 ) [ 21 ]. Ikeda et al. [16] have reported surprisingly high "atomic" Debye temperatures of 1500 and 2000 K for the Cu ions in La2CuO4 and YBaECU307, respectively, and 1400 K for the O ions in both. It was suggested that these large values arise from high-frequency modes which may in turn play a role in the mechanism for superconductivity. These atomic Debye temperatures were determined using the Debye model from effective mean kinetic energies or temperatures (for example, Tefr= 750 K for Cu and 550 K for O in YBazCu307, compared with Teff= 319 K for Cu in copper); the latter in turn were determined from the recoil spectra of neutrons in the eV energy range. In typical phonon-mediated superconductors, the superconducting transition temperature is typically of the order of 10% of the Debye temperature. Thus Ikeda et al.'s finding that the Cu and O ions in
619
La2CuO4 and YBa2Cu307 have extremely high atomic Debye temperatures was quite intriguing. However, we find that the isotropic Debye-Waller temperatures obtained from the more direct diffraction measurements are much lower: averaging 487 K for Cu in YBa2Cu307. This value is closer to the value of 297_+ 28 K obtained by Mook et al. [ 17 ] from neutron resonance absorption experiments for 65Cu in Yla2Cu307. Similarly, the average value of the isotropic Debye-Waller temperatures for O is 677 K for YBa2CuaO7, which is again substantially lower than that given by Ikeda et al. The high atomic Debye temperatures obtained by Ikeda et al. reflect the high values of the effective temperatures obtained by them; their value of Tefffor Cu in YBa2Cu307 is much higher than that recently reported by Mook et al., despite their essential agreement on Tefffor Cu metal, and we suspect that Ikeda et al.'s results for La2CuO4 and YBa2Cu307must be incorrect. Our estimates for the Debye-Waller temperatures are lower than those given by Ikeda et al., and they conform more closely to other estimates of the atomic Debye temperatures for these materials [ 1,24,28 ]. For example, our earlier study of La2CuO4 gave a Debye-Wallter temperature of 430 K for Cu and an average value of 587 K for the oxygens, in reasonably good agreement with the results of lattice dynamical calculations [ 24 ]. The individual Debye-Waller temperatures can be averaged (with an m ~/2 weighting) to give an effective Debye temperature for YBa2Cu307, m 1/20~fr= ( 1 / n ) ~ n i m ~ / 2 0 ( i ) o w ,
(2)
where m is the average atomic mass, n the number of atoms per formula unit, n; the multiplicity for a given atom type, and the mi are the individual atomic masses. Using the Debye-Waller temperatures given in table l gives a value of 396 K, which is in excellent agreement with typical values of approximately 400 K obtained by fitting low-temperature specific-heat data [ l ]. Recently, substantial progress has been made in the measurement of generalized phonon densities of states [19] and phonon dispersion curves [20] for YBa2Cu306+~. These show little dependence on temperature, but significant changes when 0 is varied. The generalized phonon densities of states show that, as oxygen is depleted and 0 decreases from 1 to 0, there is ( 1 ) a stiffening of the phonon spectrum at
620
G.H. Kwei et al. / Atomic Debye temperatures for YBa2Cuj07
low energies (below 18 meV) associated with transverse vibrations of the O ( 1 ), (2) an appearance of a high-frequency peak near 80 meV, indicating a considerable stiffening of the planar CuO2 and Cu ( 1 ) - O (4) bond stretching vibrations, and ( 3 ) a stiffening in the 35-60 meV range corresponding to the various Cu-O bending modes [19]. These effects are also observed, although to a lesser degree, when small amounts of Zn are substituted for Cu or Pr is partially substituted for Y. Thus for YBa2Cu306+6, the increase of T~ with the increase of oxygen concentration or the decrease in cation dopant concentration is accompanied by a softening of the oxygen phonons. The most pronounced changes with increases in the oxygen stoichiometry are a softening of both the in-plane and out-of-plane Cu ( 2 ) O ( 2 ) , 0 ( 3 ) bending vibrations (35 to 60 meV) and of the high-frequency modes associated with the Cu ( 2 ) - O (2), O ( 3 ) and Cu ( 1 ) - O (4) stretching vibrations (ca. 80 meV) [20]. Our values of the anisotropic Debye-Waller temperatures can be correlated with the results from studies of the phonon densities of states and dispersion curves. The low values of the Debye-Waller temperatures associated with the transverse motion of the O ( 1 ) and the axial motion of the Y, Ba, Cu(2) and 0 ( 2 ) ions agrees well with the softness of the transverse modes of O( 1 ) at low energies and the bending modes for C u ( 2 ) - O ( 2 ) , 0 ( 3 ) observed for YBa2Cu307 in the 35-60 meV region, respectively [ 19,20]. Ion-channeling experiments on YBa2Cu306+~ and ErBa2Cu306+a have perhaps provided some of the most intriguing evidence for phonon anomalies below Tc [ 18 ]. These experiments have shown that ionchanneling widths along the [001 ] axis broaden discontinuously below To; this onset of broadening, which shifts with T¢ when ~ is decreased, is attributed to a sudden decrease of the transverse thermal (or static) displacements in the Cu-O (but not the Y-Ba or Er-Ba) rows along the channeling axes. This stiffening, which appears to be at odds with the results from both inelastic neutron scattering and the structural results described above, has been attributed to a strong coupling of the transverse Cu-O phonons in the superconducting state. Since it is not yet clear which atoms are associated with these transverse displacements, it is difficult to compare the values obtained from ion channeling with the
mean-square displacements obtained from our neutron diffraction data. Nevertheless, the observed changes at T¢ are not reflected in the neutron results, as all the isotropic- or the transverse mean-square displacements shown in figs. 1 and 2 exhibit a smooth temperature dependence without abrupt changes at
To. Other evidence for phonon anomalies near Tc have been found in heat capacity measurements [ 2 ] and in infrared and Raman spectroscopic studies [ 5-8 ]. A number of the low lying modes have been observed to soften slightly below Tc. One of the more intriguing observations has been the observation of unusual spectral intensities: where it is found that the lowest energy phonon observed at 155 cm-1, assigned to the Blu "barium" mode, involving the Cu( 1 ) - O ( 1 ) chains, the 0 ( 4 ) axis oxygens and the Ba ions on the basis of lattice-dynamical calculations [ 5,8 ], has an intensity approximately 15 times higher than expected. Unfortunately, this mode does not show any appreciable temperature dependence. There have also been many indications for the existence of s t r u c t u r a l instabilities in YBa2Cu307 at or near T¢. However, the exact nature of these instabilities has not been elucidated. Detailed X-ray diffraction studies of an untwinned single crystal as a function of temperature have revealed lattice parameter anomalies near Tc, which are most pronounced for the lattice parameter b and c [ 13 ]. Recent X-ray absorption studies [ 14,15 ] suggest a special role for the axial oxygens in structural instabilities near T~. However, the details of the two studies differ in that the mean-square displacements involving the axial oxygen ( O ( 4 ) ) bonds increase in the former [14], while they decrease in the latter [ 15 ]; furthermore, the former suggests that the Cu ( 2 ) - O (4) bond decreases with decreasing temperature and that the 0 ( 4 ) resides in a double well potential with nearly equal population near T¢, while the latter observes no change in the Cu ( 2 ) - O (4) bond length. Our more direct diffraction experiments show no such structural anomalies involving the axial oxygens [ 12 ]. Analysis of both Bragg and diffuse scattering of neutrons from the related superconductor T12Ba2CaCu2Oa [ 11 ] does not suggest any anomaly involving the axial oxygens but instead finds evidence for correlated axial displacements of both O
G.H. Kwei et al. /Atomic Debye temperatures for YBa2Cu307
and Cu in the C u O 2 sheets at the onset of superconductivity. In summary, mean-square displacements obtained from structural refinement of powder neutron diffraction data provide a detailed picture of the vibrational properties of YBaECU307. These suggest the participation of 0 ( 2 ) and Cu(2) in the soft axial vibrational modes for YBa2Cu307, and, to a lesser extent, the participation of Y and Ba as well. The motion of O ( 1 ), transverse to the Cu ( 1 )-O ( 1 ) Cu ( 1 ) bond, is also soft and has components along the c-axis. None of the thermal parameters display any anomalies near To, at least at the temperature resolution used in these experiments. These observations correlate well with measurements of the phonon densities of states and the phonon dispersion curves obtained from inelastic neutron scattering experiments [ 19,20 ]. However, no stiffening of the transverse displacements involving Cu-O near To, as suggested by the ion-channeling experiments [ 18 ], is observed; this discrepancy is not understood at present, perhaps the channeling experiments may be more sensitive to local fluctuations. Values of the Debye temperature derived from the thermal parameters agree well with results from specific heat measurements [ 1 ] and show that estimates of the Cu and O atomic Debye temperatures obtained from the analysis of the Doppler broadening of nuclear resonances in eV range neutron scattering are too large by factors of 2-3 [ 16]; however, more recent experiments studying the 65Cu r e s o n a n c e s in YBaECU307 yield effective temperatures that agree better with our results [ 17 ].
Acknowledgements We thank P.B. Allen for several interesting discussions while he was the Bernd T. Matthias Visiting Scholar at Los Alamos and L.E. Rehn of Argonne National Laboratory for a discussion of his ionchanneling experiments. We thank R.B. Von Dreele for his role as instrument responsible scientist for HIPD. This work was done under the auspices of the United States Department of Energy and funded in part by its Office of Basic Energy Sciences, Division of Material Sciences.
621
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G.H. Kwei et aL /Atomic Debye temperatures for YBazCu307
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Vibrational properties and atomic Debye temperatures for YBa2Cu307 from neutron powder diffraction George H. Kwei, A.C. Lawson, W.L. Hults a n d J a m e s L. Smith Los Alamos National Laboratory, Los Alamos, N M 87545, USA
Received 29 October 1990 Revised manuscript received 19 February 1991
Neutron powder diffraction data taken at twelve sample temperatures have been used to determine both isotropic and anisotropic thermal parameters for atoms in YBa2Cu307.At fit of these parameters to a Debye-Wallermodel in turn yields the corresponding isotropic and anisotropicDebye-Wallertemperatures. The anisotropicthermal parameters for the Y, Ba and the planar Cu(2) and 0(2) atoms all exhibit a stronger temperature dependence, and therefore a lower Debye-Waller temperature, for motion alongthe c-axis;in addition, the chain O( 1) atoms showa similar strongtemperature dependencefor motion perpendicular to the chains. No discontinuities in the thermal parameters, suggestiveof possible structural changesor phonon anomalies near Tcare observed.
1. Introduction There continues to be a great deal of interest in the existence of structural instabilities or phonon anomalies near the superconducting transitions of high temperature superconductors, in particular YBa2CuaO6+6. Within the last few years many different kinds of experiments have provided somewhat conflicting evidence for instabilities or anomalies. These range from specific heat measurements of the Debye temperature [ 1,2], acoustic measurements of elastic constants [3,4], Raman and infrared spectroscopic studies of line intensities and frequency shifts [ 5-8 ], structural studies using X-ray and neutron diffraction [ 9-13 ], X-ray absorption spectroscopy [ 14,15 ], neutron recoil measurements [ 16,17 ], ion-channeling measurements [ 18 ], and inelastic neutron scattering measurements of the phonon densities of states [ 19 ] and phonon dispersion curves [20]. At present, there is no consensus from the different experiments as to which instabilities or anomalies actually exist, or are in fact important. We have been using pulsed-neutron powder diffraction data to measure the anisotropic thermal parameters of ions in crystals to study their anisotropic
vibrational motions. Of special interest has been both the static and dynamic displacements of the constituent ions in the high-To oxide superconductors and the possible relationship between these and their superconducting behavior [21,22]. Recently, we have been successful in fitting a simple Debye-Waller model to the measured temperature dependence of the thermal parameters [23] and carded out a detailed comparison of atomic Debye-Waller temperatures obtained for La2CuO4 with other experimental data and with the results of a lattice dynamical calculation [24]. The extensive work on YBa2Cu306÷6 provides an opportunity to extend our determinations of vibrational properties from diffraction data to a more complex system, to compare the information provided with the results from other techniques, and to gain insights into the behavior of superconducting materials.
2. Experimental details, data analysis and results The sample preparation and characterization has been described earlier [ 12]. Briefly, 15 gm of YBa2Cu307 was prepared using standard oxide synthesis techniques [25 ] and DC susceptibility mea-
0921-4534/91/$03.50 © 1991 - Elsevier SciencePublishers B.V. (North-Holland)
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G.H. Kwei et al. / Atomic Debye temperatures for YBa2CusO7
surements showed the Tc at onset for this sample to be 93 _+ l K, with a midpoint o f 85 _+ 1 K. The shielding volume fraction at 7 K and 100 Oe was greater than 90%, estimating a sample volume from the weight and the theoretical density. Time-of-flight neutron powder-diffraction data were collected on four detector banks ( _+ 153 ° and _ 90 ° ) o f the high intensity powder diffractometer ( H I P D ) at the Manuel Lujan Jr., Neutron Scattering Center ( L A N S C E ) at Los Alamos. In all, data were collected at twelve temperatures. Rietveld refinement of the structural model was c a r d e d out using the generalized structure analysis system (GSAS) developed by Larson and Von Dreele [26]; these gave values for the lattice constants, atomic positions, oxygen occupancies and either isotropic or anisotropic thermal parameter #l. The refined values for the oxygen occupancies gave the overall oxygen stoichiometry, YBa2Cu307. Because we were interested in comparing the refined thermal parameters, we were careful to refine data taken at the different temperatures in as consistent a way as possible. Therefore, parameters which are not temperature dependent (diffractometer constants, profile coefficients and absorption coefficients) were set to the average o f the values o f these parameters obtained in earlier refinements and not further refined. The resulting isotropic thermal parameters for YBa2Cu307 are shown in fig. 1 and the anisotropic thermal parameters are shown in fig. 2. The thermal parameters can be thought o f as meansquare atomic displacements which arise from both thermal motion and static displacements. Thus the temperature dependence o f the thermal parameters can be assumed to fit the Debye-Waller formula and can be fit with a two-term model [23,24],
( u 2) . . . . . ~ =
(u2)i~o.1 + (U2)off~,,
(1)
where (U2)ideal c a n be related to the Debye-Waller temperature and the temperature-independent (U2)off~t includes contributions from both static displacements and systematic errors. #1 We use the notation of O ( 1) for the chain oxygens, O (2) and 0(3) for the plane oxygens (with 0(2) lying above and below the O ( 1) ), and O ( 4 ) for the axial oxygens connecting the chains and planes. An ORTEP plot of the structure with this notation is shown as fig. 2 in ref. [ 12].
Table 1 Debye-Waller temperatures and offsets for the isotropic thermal displacements for the constituents of YBa2Cu3OT.Units for the Debye-Waller temperature are K and observed offsets are in 103 /k2. Atom
ODw
Observed offset
Y Ba Cu(1) Cu(2) O(1) 0(2) 0(3) 0(4)
345(6) 231(3) 487(12) 402(5) 497(7) 710(14) 719(14) 671(10)
2.1(1) 2.8(1) 0.1(1) -0.3(1) 3.1(2) 1.6(2) -1.6(2) -0.4(1)
The values o f the Debye-Waller temperature o f offsets for the constituents o f YBa2Cu307 fit to this model are presented in table 1 and shown graphically in fig. 1. When these Debye-Waller temperatures are properly mass weighted (e.g. m ~/20), most o f the ions, with the exception o f O ( 1 ), have comparable interatomic force constants. On the other hand, as had been found in the earlier neutron diffraction studies [ 9, l 0 ], O ( 1 ) has a considerably lower value o f the interatomic force constant than the other constituents. Since the motions of the constituents o f YBa2Cu307 are known to be anisotropic [21 ], especially the oxygen ions, we have also fit anisotropic Debye-Waller temperatures to the anisotropic thermal parameters. The Debye-Waller temperatures and offsets are given in table 2 and shown in fig. 2. These reveal much more about the vibrational properties o f the constituent atoms in YBa2Cu307 and provide greater opportunity for comparison with optical and inelastic neutron studies of the p h o n o n frequencies, densities o f states and dispersion curves, and with ion-channeling studies #2 In the refinements for YBa2Cu3OT, we discovered that the axial thermal parameters U3a for 0 ( 2 ) and O ( 3 ) are fairly strongly correlated, with a typical correlation coefficient o f approximately - 0 . 7 at all sample temperatures. Correlation coefficients o f this #2 The observed thermal motion of individual atoms in a crystal arise from the superposition of many normal vibrational modes of the crystal. Thus, this information is complementary to intensities of individual normal modes involving many atoms.
G.H. Kwei et al. /Atomic Debye temperatures for YBa2Cu~O7 lS
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Atom
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Observed offset
Y
U. U22 U3a
369(22) 350(22) 234(6)
-1.5(3) 4.9(4) 5.1(3)
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252(11) 233(9) 191(5)
2.1(3) 0.6(4) 5.6(3)
Cu(l)
Un U22 U33
334(12) 338(13) 400(19)
-1.5(4) -3.0(4) 6.1(3)
Cu(2)
Un /-/22 U33
663(70) 595(50) 242(3)
-1.6(3) 0.7(3) 1.6(2)
O(1)
Un U22 U33
404(11) 666(39) 388(10)
1.3(6) 3.9(6) 5.3(7)
0(2)
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1400(300) 880(60) 345(5)
3.3(5) --1.3(4) 3.4(5)
0(3)
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708(30) 734(33) 1600(500)
3.5(4) -0.2(4) -2.2(8)
0(4)
Ut~ U22 /-]33
530(16) 670(30) 728(30)
0.4(4) 1.2(4) -0.8(4)
magnitude are thought to result in less reliable, but nevertheless useful, refined parameters. Thus the relatively low resolution data available from HIPD may not be sufficient to distinguish reliably between axial thermal motion of the oxygens in the CuO2 planes and the errors in the resulting values are certainly larger than the reported standard deviations. However, there may be a more interesting reason for these correlations: the least squares procedure used in fitting the data assumes a harmonic model for independent constituent ions and both the correlation and the difficulty in fitting Ua3 for 0 ( 2 ) and 0 ( 3 ) may in fact result from substantial anharmonicity in the vibrational motion or a coupling of the axial vibrational motions of the two planar oxygens. In addition, the positional parameters z for O (2) and 0 ( 3 ) , the thermal parameters U 2 2 for 0 ( 2 ) and U~1 for O (3), and the thermal parameters U1] and U22 for Ba, were all found to be slightly correlated, but the refined values do not show the scatter shown by U33 for 0 ( 2 ) and 0 ( 3 ) . Figure 2 shows the anisotropic thermal parameters and the fits to the two-term Debye-Waller model. The data show the large anisotropic thermal parameters along the c-axis observed by us for Ba, Cu (2) and 0 ( 2 ) in our earlier joint X-ray/neutron study of YBa2Cu307 at room temperature and reinforce the existence of a soft optic mode involving these atoms [21 ]. In addition, the thermal displacements for Y also indicate a lower Debye-Waller temperature along the c-axis. Thus, all cations with the exception of Cu ( l ), show relatively soft-mode behavior along
618
G.H. Kwei et al. / Atomic Debye temperatures for YBa2Cu307
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Fig. 2. Mean-square anisotropic thermal displacements for all constituents in YBa2Cu3OT. Error bars representing one standard deviation are omitted for points where they are smaller than these size of the points. Curves represent fits to the two-term Debye-Waller model. Fits to U33 for O ( 2 ) and O (3) are included despite the correlation between these parameters in the Rietveld refinement; the erratic behavior of U33 for these atoms leads to poor fits to the model.
G.H. Kweiet al. /Atomic Debyetemperaturesfor YBa2Cu307
the c-axis. Two of the oxygens exhibit significant soft mode behavior: the planar oxygen O (2) is extremely soft along the c-axis and the chain oxygen O ( 1 ) is soft in both directions transverse to the Cu(1 ) O( l )-Cu( 1 ) bonds. Two "modes" involving oxygens in the CuO2 planes appear to be extremely stiff: O (2) has a Debye-Waller temperature of ~ 1400 K along the a-axis and O (3), whose thermal parameter along the c-axis is strongly correlated, has a DebyeWaller temperature of ~ 1600 K along this direction. The thermal displacements for the axial oxygen O (4) show it to be nearly isotropic and moderately stiff. Thus O(1 ) and 0 ( 2 ) appear to be relatively soft perpendicular to the Cu ( 1 )-O ( 1 ) -Cu ( 1 ) bonds and the c-axis, respectively, while 0 ( 3 ) and 0 ( 4 ) are both nearly isotropic and relatively stiff.
3. Discussion
As in all the cuprate superconductors, the C u O 2 sheets are thought to be responsible for carrying the supercurrents. In YBa2Cu307, the Cu ( 1 ) - O ( 1 ) chains provide charge reservoirs for the CuO2 sheets and are coupled to them through the axial oxygens O (4) [ 27 ]. The earlier joint X-ray/neutron diffraction studies at room temperature have shown that both CuO2 sheets and Cu( 1 ) - O ( 1 ) chains display relatively large and anisotropic thermal motion: along the c-axis for Cu(2) and transverse to the Cu(1 ) O ( 1 )-Cu ( 1 ) chains for O ( 1 ) [ 21 ]. Ikeda et al. [16] have reported surprisingly high "atomic" Debye temperatures of 1500 and 2000 K for the Cu ions in La2CuO4 and YBaECU307, respectively, and 1400 K for the O ions in both. It was suggested that these large values arise from high-frequency modes which may in turn play a role in the mechanism for superconductivity. These atomic Debye temperatures were determined using the Debye model from effective mean kinetic energies or temperatures (for example, Tefr= 750 K for Cu and 550 K for O in YBazCu307, compared with Teff= 319 K for Cu in copper); the latter in turn were determined from the recoil spectra of neutrons in the eV energy range. In typical phonon-mediated superconductors, the superconducting transition temperature is typically of the order of 10% of the Debye temperature. Thus Ikeda et al.'s finding that the Cu and O ions in
619
La2CuO4 and YBa2Cu307 have extremely high atomic Debye temperatures was quite intriguing. However, we find that the isotropic Debye-Waller temperatures obtained from the more direct diffraction measurements are much lower: averaging 487 K for Cu in YBa2Cu307. This value is closer to the value of 297_+ 28 K obtained by Mook et al. [ 17 ] from neutron resonance absorption experiments for 65Cu in Yla2Cu307. Similarly, the average value of the isotropic Debye-Waller temperatures for O is 677 K for YBa2CuaO7, which is again substantially lower than that given by Ikeda et al. The high atomic Debye temperatures obtained by Ikeda et al. reflect the high values of the effective temperatures obtained by them; their value of Tefffor Cu in YBa2Cu307 is much higher than that recently reported by Mook et al., despite their essential agreement on Tefffor Cu metal, and we suspect that Ikeda et al.'s results for La2CuO4 and YBa2Cu307must be incorrect. Our estimates for the Debye-Waller temperatures are lower than those given by Ikeda et al., and they conform more closely to other estimates of the atomic Debye temperatures for these materials [ 1,24,28 ]. For example, our earlier study of La2CuO4 gave a Debye-Wallter temperature of 430 K for Cu and an average value of 587 K for the oxygens, in reasonably good agreement with the results of lattice dynamical calculations [ 24 ]. The individual Debye-Waller temperatures can be averaged (with an m ~/2 weighting) to give an effective Debye temperature for YBa2Cu307, m 1/20~fr= ( 1 / n ) ~ n i m ~ / 2 0 ( i ) o w ,
(2)
where m is the average atomic mass, n the number of atoms per formula unit, n; the multiplicity for a given atom type, and the mi are the individual atomic masses. Using the Debye-Waller temperatures given in table l gives a value of 396 K, which is in excellent agreement with typical values of approximately 400 K obtained by fitting low-temperature specific-heat data [ l ]. Recently, substantial progress has been made in the measurement of generalized phonon densities of states [19] and phonon dispersion curves [20] for YBa2Cu306+~. These show little dependence on temperature, but significant changes when 0 is varied. The generalized phonon densities of states show that, as oxygen is depleted and 0 decreases from 1 to 0, there is ( 1 ) a stiffening of the phonon spectrum at
620
G.H. Kwei et al. / Atomic Debye temperatures for YBa2Cuj07
low energies (below 18 meV) associated with transverse vibrations of the O ( 1 ), (2) an appearance of a high-frequency peak near 80 meV, indicating a considerable stiffening of the planar CuO2 and Cu ( 1 ) - O (4) bond stretching vibrations, and ( 3 ) a stiffening in the 35-60 meV range corresponding to the various Cu-O bending modes [19]. These effects are also observed, although to a lesser degree, when small amounts of Zn are substituted for Cu or Pr is partially substituted for Y. Thus for YBa2Cu306+6, the increase of T~ with the increase of oxygen concentration or the decrease in cation dopant concentration is accompanied by a softening of the oxygen phonons. The most pronounced changes with increases in the oxygen stoichiometry are a softening of both the in-plane and out-of-plane Cu ( 2 ) O ( 2 ) , 0 ( 3 ) bending vibrations (35 to 60 meV) and of the high-frequency modes associated with the Cu ( 2 ) - O (2), O ( 3 ) and Cu ( 1 ) - O (4) stretching vibrations (ca. 80 meV) [20]. Our values of the anisotropic Debye-Waller temperatures can be correlated with the results from studies of the phonon densities of states and dispersion curves. The low values of the Debye-Waller temperatures associated with the transverse motion of the O ( 1 ) and the axial motion of the Y, Ba, Cu(2) and 0 ( 2 ) ions agrees well with the softness of the transverse modes of O( 1 ) at low energies and the bending modes for C u ( 2 ) - O ( 2 ) , 0 ( 3 ) observed for YBa2Cu307 in the 35-60 meV region, respectively [ 19,20]. Ion-channeling experiments on YBa2Cu306+~ and ErBa2Cu306+a have perhaps provided some of the most intriguing evidence for phonon anomalies below Tc [ 18 ]. These experiments have shown that ionchanneling widths along the [001 ] axis broaden discontinuously below To; this onset of broadening, which shifts with T¢ when ~ is decreased, is attributed to a sudden decrease of the transverse thermal (or static) displacements in the Cu-O (but not the Y-Ba or Er-Ba) rows along the channeling axes. This stiffening, which appears to be at odds with the results from both inelastic neutron scattering and the structural results described above, has been attributed to a strong coupling of the transverse Cu-O phonons in the superconducting state. Since it is not yet clear which atoms are associated with these transverse displacements, it is difficult to compare the values obtained from ion channeling with the
mean-square displacements obtained from our neutron diffraction data. Nevertheless, the observed changes at T¢ are not reflected in the neutron results, as all the isotropic- or the transverse mean-square displacements shown in figs. 1 and 2 exhibit a smooth temperature dependence without abrupt changes at
To. Other evidence for phonon anomalies near Tc have been found in heat capacity measurements [ 2 ] and in infrared and Raman spectroscopic studies [ 5-8 ]. A number of the low lying modes have been observed to soften slightly below Tc. One of the more intriguing observations has been the observation of unusual spectral intensities: where it is found that the lowest energy phonon observed at 155 cm-1, assigned to the Blu "barium" mode, involving the Cu( 1 ) - O ( 1 ) chains, the 0 ( 4 ) axis oxygens and the Ba ions on the basis of lattice-dynamical calculations [ 5,8 ], has an intensity approximately 15 times higher than expected. Unfortunately, this mode does not show any appreciable temperature dependence. There have also been many indications for the existence of s t r u c t u r a l instabilities in YBa2Cu307 at or near T¢. However, the exact nature of these instabilities has not been elucidated. Detailed X-ray diffraction studies of an untwinned single crystal as a function of temperature have revealed lattice parameter anomalies near Tc, which are most pronounced for the lattice parameter b and c [ 13 ]. Recent X-ray absorption studies [ 14,15 ] suggest a special role for the axial oxygens in structural instabilities near T~. However, the details of the two studies differ in that the mean-square displacements involving the axial oxygen ( O ( 4 ) ) bonds increase in the former [14], while they decrease in the latter [ 15 ]; furthermore, the former suggests that the Cu ( 2 ) - O (4) bond decreases with decreasing temperature and that the 0 ( 4 ) resides in a double well potential with nearly equal population near T¢, while the latter observes no change in the Cu ( 2 ) - O (4) bond length. Our more direct diffraction experiments show no such structural anomalies involving the axial oxygens [ 12 ]. Analysis of both Bragg and diffuse scattering of neutrons from the related superconductor T12Ba2CaCu2Oa [ 11 ] does not suggest any anomaly involving the axial oxygens but instead finds evidence for correlated axial displacements of both O
G.H. Kwei et al. /Atomic Debye temperatures for YBa2Cu307
and Cu in the C u O 2 sheets at the onset of superconductivity. In summary, mean-square displacements obtained from structural refinement of powder neutron diffraction data provide a detailed picture of the vibrational properties of YBaECU307. These suggest the participation of 0 ( 2 ) and Cu(2) in the soft axial vibrational modes for YBa2Cu307, and, to a lesser extent, the participation of Y and Ba as well. The motion of O ( 1 ), transverse to the Cu ( 1 )-O ( 1 ) Cu ( 1 ) bond, is also soft and has components along the c-axis. None of the thermal parameters display any anomalies near To, at least at the temperature resolution used in these experiments. These observations correlate well with measurements of the phonon densities of states and the phonon dispersion curves obtained from inelastic neutron scattering experiments [ 19,20 ]. However, no stiffening of the transverse displacements involving Cu-O near To, as suggested by the ion-channeling experiments [ 18 ], is observed; this discrepancy is not understood at present, perhaps the channeling experiments may be more sensitive to local fluctuations. Values of the Debye temperature derived from the thermal parameters agree well with results from specific heat measurements [ 1 ] and show that estimates of the Cu and O atomic Debye temperatures obtained from the analysis of the Doppler broadening of nuclear resonances in eV range neutron scattering are too large by factors of 2-3 [ 16]; however, more recent experiments studying the 65Cu r e s o n a n c e s in YBaECU307 yield effective temperatures that agree better with our results [ 17 ].
Acknowledgements We thank P.B. Allen for several interesting discussions while he was the Bernd T. Matthias Visiting Scholar at Los Alamos and L.E. Rehn of Argonne National Laboratory for a discussion of his ionchanneling experiments. We thank R.B. Von Dreele for his role as instrument responsible scientist for HIPD. This work was done under the auspices of the United States Department of Energy and funded in part by its Office of Basic Energy Sciences, Division of Material Sciences.
621
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