- Email: [email protected]
Phonon spectra of the Nd1.85Ce0.15CuO4 superconductor
Physica C 172 (1990) 260-264 Norlh-Holland
Phonon spectra of the Ndl.85Ceo. 15CUO4superconductor V.N. P o p o v and V.L. Valchinov University of Sofia. CentreJor Space Research and Technology, 5A Ivanov Blvd. 1126 Sofia. Bulgarta Received 16 August 1989 Revised manuscript received 17 October 1990
Calculation of the phonon dispersion curves for the high-T¢ superconductor NdLssCeo ~sCuO4 using a shell model has been performed. The frequencies at the zone centre were classified according to their vibrational character. The parameters of the model were adjusted to reach reasonable agreement with the available optical data on Ndt ssCeo tsCuO4.
The high-T~ superconductor (HTS) Nd2_,CexCuO4 synthesized by Tokura et al. [ 1 ] has two prominent features. First of all, the superconductivity in this material is due to electrons [2] but not to holes as is the case in all previously discovered high-T~ superconductors. This was achieved by Cedoping of the initial pure substance Nd2CuO4. Secondly, the Cu- and O-ions form CuO4 squares instead of tetrahedral CuO6 or pyramidal CuO5 structures. Many physical properties of superconducting Nd_~_xCe~CuO4 have already been studied, but the reasons for quasiparticle pairing in it remain unclear. Whether the mechanism of superconductivity is phononic or not. the investigation of the optical spectra of this superconductor is of special interest in itself. Neutron scattering measurements of phonons in Nd2_,CexCuO4 have not yet been done because of the impossibility to obtain large enough single crystals. The first Raman spectra measurements wcrc carried out on ceramic samples of pure N d 2 C u O 4 by Sugai et al. [ 3 ] and yielded three of the four Raman-activc modes for this compound. Polarized micro-Raman observations by Hadjiev et al. [4] led to slightly different values for these modes, which were later confirmed in polarized Raman spectra measurements on single cry'stals by Heyen et al. [5]. Recently infrared-active modes have been detected in single crystals and have been assigned by Crawford et al. [ 6 ] and by Heyen et al. [ 7 ]. The rigorous analysis of the observed infrared spectra, carried out by the two groups, provides an unambigu-
ous picture of the position and the symmetry of the infrared-active modes. Here wc present the results of the calculation of the phonon frequencies of Nd2_xCe, CuO4 at the F-point using a shell model. The parameters of this model taken from other compounds with perovskite-like structures were slightly adjusted to reach agreement with the available optical data for this HTS. The phonon spectra in three symmetry, directions were also calculated. The crystal structure of Nd,_,Ce.~CuO4 was determined by Tokura et al. [ 1 ]. This compound belongs to the tetragonal system, space group 14/mmm ( D a17 n ) and crystallizes in the so called T' phase. The body-centered tetragonal unit cell of this material is shown in fig. 1. The atoms in the unit cell have coordinates: two Cu-atoms - (0,0,0), (½,~,~); four ( O , , O , . ) - a t o m s - (½,0,0), (0, ½,0), (0, ½, ½), ( ½,0.½ )' four Nd-atoms (O,O,+zNd), (O,O,--ZNj), (~,~,~+ZNd). (~,½,½--ZNO); four O.-atoms I I 3 (~,0,~ i (3,I 0 ,~), (0,2,~), (0,~,~), ). (The number of atoms is doubled because the unit cell is body-centered tetragonal.) The parameters of the unit cell are: a = 3 . 9 4 ~,, c=12.08 ,~, and the Nd-position is Z N , a = 0 . 3 5 2 . The site symmetry of the different atoms in the standard Wyckoff notation is shown in table I. Only the Cu-atom has the highest symmetry. The other atoms occupy positions with lower symmetry: the Ox- and Oy-atoms occupy position 4c, the Nd-atom position 4e, and the O~-atoms position 4d. Using the factor group theoretical analysis we determined the irreducible representations of the point
0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
V. N Popov. [:L. Valchinov /Phonon spectra of Ndt asCeo 15CuO,
Z
-•0
0 .0
~ Y -X 0
O~
0
O~ • q,x)---~ OI
•
Oq~o
Cu
Nd(Cel 0 0 •
0 0
vj Q
Fig. 1. The tetragonal unit cell of Nd2_~Ce~CuO4. Table I Classification of the normal modes.
Atomic position 2Cu 4 (Ox, Oy) 4Nd 40~
Irreducible representations 2a 4c 4e 4d
A2u + Eu A2u 4- B2u 4- Eu Ais+Azu+Es+Eu A2~+ B ~s+ Es + E~
Ftot=Ats+ Bt~ + B2~+ 4A2u + 2Es+ 5Eu F,c = A2u + Eu l-'opt= A~s+ Bts+ 3A2u + 2E8 + 4E u Raman modes: A~s, Bts , 2E s Infrared modes: 3A2~, 4E~ Silent modes: B2~
group D4h under which the normal modes are transformed. The results are displayed in table I. Of all 21 normal modes three are acoustic modes - A2u and E,, and the rest are optical ones. The infrared active modes are of the type A2u and E., the BEu is a silent mode and the A~g, the B~s and the Eg modes are Raman active ones. The lattice dynamics of Nd2_xCexCuO4 was described by a shell model, which has been often applied to crystals with a perovskite structure. In this model, in order to allow for ion polarization, each ion consists of a charged core and an undeformable charged shell around it, the sum of the core and the shell charges, X and Y, yielding the ionic charge Z. The interaction between the various cores and shells
261
consists of two parts. The long-range Coulomb interaction contains as parameters the ionic and shell charges only and can be calculated using a standard Ewald procedure. The short-range part of the interactions between the ions is well described by BornMayer potentials v ( r ) = a e -~', while the short-range interaction between the core and the shell of an ion is characterized by a force constant k. In the case of Nd 2 _ x C e x C u O 4 several simplifying assumptions were made, which reduced the number of model parameters. Namely, the ionic charges were fixed at their nominal values Z ( C u ) = + 2 . 0 0 and Z ( N d ) = +3.08 (accounting for the 8% Ce 4÷ ions present). The ionic charge of the O-ion was determined using the charge neutrality condition: Z ( O ) =2.04. The short-range interactions between the ions in the form of Born-Mayer potentials as well as the core-shell interactions were considered isotropic. In our calculations we have taken account of the short-range Cu-O, Nd-O and O - O interactions between the nearest (NN) and next-to-nearest ( N N N ) neighbours only. As established by Lehner et ai. [ 8 ], the short-range potentials for a great number of alkali halides and fluoridic perovskites are similar and can be transferred from one structure to another. This idea regarding the perovskite-like high7~ compounds was spread by Kulkarni et al. [9 ], who showed that for a number of HTS the shell model parameters are indeed closely situated in the parameter space. Following these propositions, we considered as an initial set of parameters of the shell model the corresponding calculated values for several perovskite-like structures [ 9,10 ]. The available measured Raman and infrared frequencies [3-7] were used to determine the shell model parameters in a least-squares fitting procedure. The Raman frequencies were weighted by a larger factor than the infrared ones to account for the greater experimental accuracy of the former. The equilibrium conditions were used to eliminate the aparameters for the three short-range interactions. Throughout the fitting procedure stability of the phonon spectra at the F, Z, X, and G~ points was also required. The results for the shell model parameters are given in table II. They can be compared with the calculated parameters of L a 2 C u O 4 [ 10 ] and a number of other HTS [9]. The former were derived using the avail-
E N. Popov, V.L. Valchinov / Phonon spectra of Ndl ssCeo tjCuO~
262
Table I I Parameters of the model: a, b - Born-Mayer constants; k - core-shell force constant; Y - shell charge; Z - ionic charge (v, - volume of the unit cell ). Interaction
a
Nd-O Cu-O O-O
b (A -~)
Ion
(eV)
Z (lel)
Y (lel)
k (e2/~)
3018 5236 167
3.13 4.38 2.95
Nd Cu O
3.08 2.00 - 2.04
1.63 3.02 - 2.44
314 474 247
able neutron scattering and optical data on La2CuO4, while the latter form an universal set of parameters obtained on the basis of optical data only. There is an agreement within 10% for the values of the b-parameters given in refs. [ 9,10 ] and the present work, except for the short-range C u - O interaction in which case in ref. [9] values reduced by about 25% were obtained and the corresponding a-parameters are several times smaller. The resulting Cu-O short-range forces are the largest ones and are comparable with these given in refs. [9,10]. The O - O forces are an order smaller than the rest. The shell charges of the Cu and O ions are very close to those in refs. [9,10], but the shell charge of the Nd ion is two times smaller than that of the La ion [ 10]. Using the shell charge Y and the core-shell force constant k, the free ion polarizability can be expressed as a = y2/k. This for-
mula yields 0.79, 1.80, and 2.24/k 3 for the Nd, Cu and O ions, which nearly coincide with the corresponding results in ref. [9]. In table III are given the calculated phonon frequencies at the F-point in comparison with those observed by several groups. Most of the frequencies obtained by us do not deviate by more than 5% from the experimental ones, but there are some that exceed these limits. These are modes including mainly Cu and O vibrations. A possible explanation of these results may be found in the simplicity of the accepted model, namely, in the assumed isotropy of the short-range Cu-O interactions as well as the isotropy of the core-shell interaction for the O-ions. The eigenvectors of all optical modes for Nd2_~,Ce.~CuO4 are illustrated in fig. 2. The entire phonon dispersion curves in three symmetry direc-
Table III Measured and calculated phonon frequencies at the F-point (the transverse and longitudinal frequencies for each infrared mode are separated by a slash; all frequencies are in c m - ~). Optical mode
Ais B~s E I
A2u
Calculated frequencies
Measured frequencies Ref. [ 3 ]
Ref. [ 4 ]
Ref. [ 6 ]
Ref. [ 7 ]
Ref. [ 7 ]
present work
230 344
229 330
228 333
228 328
488 134/146 269/440 505/555
480 134/144 282/433 516/559
129/140 300/342 350/437 509/595
132/139 304/341 353/432 512/593
230 328 161 479 138/148 320/462 492/506 289 136/157 349/360 399/446 538/560
228 331 147 479 142/145 260/488 527/534 306 131/162 333/347 348/439 484/554
-
_
494
488
-
B2u
Eu
V.N. Popov, V.L. Valchinov/Phonon spectra of Nd~ 8jCeo~sC'uO,
263
Romon ocfive
Eg I/.7
A Ig 228
81g 331
Eg 479 Sitenf
IR ecfive
I
A 2u 14211~5
A2u 2601488
A 2u 527153~
Eu 1311162
Eu 3331347
Eu 3481439
g2u 306
Eut,8415~
Fig. 2. Calculated eigenfrequenciesand eigenvectorsof all optical normal modes at the centre of the Brillouin zone (the transverse and longitudinal frequencies for each infrared mode are separated by a slash; all frequenciesare in cm - ~).
tions in the Brillouin zone are presented in fig. 3. In order to check the transferability of the model parameters we used them to calculate the p h o n o n frequencies of La2CuO4 at F. The results are in good agreement with the observations. For example, the R a m a n modes were obtained at 201 cm - t and 470 c m - ' (A, s modes); 76 cm - I and 299 c m - ' (E s modes). The disagreement for the R a m a n modes is not more than 15% and it shows that a more com-
plicated model is needed to describe the dynamical properties of the two similar solids on a c o m m o n basis. This could be carried out when complete optical and neutron scattering data on these HTS is available. In summary, we have obtained the shell model parameters for NdE_xCe~CuO4 which provide reasonable agreement with the measured R a m a n and infrared frequencies. For the unobserved R a m a n active mode Es ( N d ) a value of 147 c m - t is predicted. The
I.tN. Popov. V.L. Valchinov / Phonon spectra *)fNdl sjCeo if'uO,
264
600 t,, •
.
•
•
.
•
•
•
•
.
.
•
•
•
.
.
500
. •
.
•
B
ttt •
.
.
:
,
:
m|
. .
,.::!'i~!!!!
"t"
• °
-
•
• • • •
400 ~:::
E w
•
•
•
.
.
;:!':::::::
.... : . - '
300
g
. . . . .
.
.
.
.
.
.
.
.
.... G$ t-
200
. : t : t ~ . . . . •
• • • | t ~ •
. . . . . . . . . . . .
.
,
.
.
:,...
. . . . t
."
•
.
•I t •
• : 1. . ; . ; . |. % . . t. l. l t t • °~• •
it ill
I" •t
. . . . . . . . .
;:
~ t t : : : : ;
tl
oo°°
.° ° ° s ~ .
100 ° t ~ ° • • • • • ' • • :
oo r
: "
" "
°
o's [qO,O]
t,'
G,
oo F
I i
D
"
o's [q,q,O]
["
1o o o X F
. •
o's [O,O,q] Z
Fig. 3. Calculated phonon dispersion curves in the [ q, 0, 0 ], [ q, q, 0 ] and [ 0, 0, q ] directions of the Brillouin zone ( the phonon frequencies are in c m - t; the reduced wave vector components q are in units rt/a, a being the smallest lattice constant ).
phonon dispersion curves in three symmetry directions have also been calculated.
Acknowledgement The authors would like to acknowledge the useful discussions with I.Z. Kostadinov.
References [ 1 ] Y. Tokura, H. Takagi and S. Uchida, Nature 337 (1989) 345. [21H. Takagi, S. Uchida and Y. Tokura, Phys. Rev. Lett. 62 (1989) 1197. [3] S. Sugai, T. Kobayashi and J. Akimitsu, Phys. Rev. B40 (1989) 2686. [4] V.G. Hadjiev, I.Z. Kostadinov, L. Bozukov, E. Dinolova and D.M. Mateev, Solid State Commun. 71 (1989) 1093.
[5]E.T. Heyen, R. Liu, B. Gegenheimer, C. Thomsen, M. Cardona, S. Pifiol and D. Mckpaul, in: Phonons 89, eds. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, 1990) p. 346. [6]M.K. Crawford, G. Burns, G.V. Chandrashekhar, F.H. Dacol, W.E. Farneth, E.M. McCarron Ill and R.J. Smalley, Phys. Rev. B41 (1990) 8933. [7] E.T. Heyen, G. Kliche, W. Kress, W. K~nig, M. Cardona, E. Rampf, J. Prade, U. SchrOder, A.D. Kulkarni, F.W. de Wette, S. Pifiol, D. McK. Paul, E. Moran and M.A. AlarioFranco, Solid State Commun. 74 (1990) 1299. [8] N. Lehner, H. Rauh, K. Strobel, R. Geick, G. Heger, J. Bouillot, B. Renker, M. Rousseau and W.G. Stirling, J. Phys. Cl5 (1982)6545. [9] W. Kress, U. Schr6der, J. Prade, A.D. Kulkarni and F.W. de Wette, Phys. Rev. B38 ( 1988 ) 2906: J. Prade, A.D. Kulkarni, F.W. de Wette, U. SchriSder and W. Kress, Phys. Rev. B39 (1989) 2771A.D. Kulkarni, J. Prade, F.W. de Wette, W. Kress and U. Schr~Rier, Phys. Rev. B40 (1989) 2642. [10] M. Mostoller, J. Zhang, A.M. Rao and P.C. Eklung, Phys. Rev. B41 (1990) 6488.
Phonon spectra of the Ndl.85Ceo. 15CUO4superconductor V.N. P o p o v and V.L. Valchinov University of Sofia. CentreJor Space Research and Technology, 5A Ivanov Blvd. 1126 Sofia. Bulgarta Received 16 August 1989 Revised manuscript received 17 October 1990
Calculation of the phonon dispersion curves for the high-T¢ superconductor NdLssCeo ~sCuO4 using a shell model has been performed. The frequencies at the zone centre were classified according to their vibrational character. The parameters of the model were adjusted to reach reasonable agreement with the available optical data on Ndt ssCeo tsCuO4.
The high-T~ superconductor (HTS) Nd2_,CexCuO4 synthesized by Tokura et al. [ 1 ] has two prominent features. First of all, the superconductivity in this material is due to electrons [2] but not to holes as is the case in all previously discovered high-T~ superconductors. This was achieved by Cedoping of the initial pure substance Nd2CuO4. Secondly, the Cu- and O-ions form CuO4 squares instead of tetrahedral CuO6 or pyramidal CuO5 structures. Many physical properties of superconducting Nd_~_xCe~CuO4 have already been studied, but the reasons for quasiparticle pairing in it remain unclear. Whether the mechanism of superconductivity is phononic or not. the investigation of the optical spectra of this superconductor is of special interest in itself. Neutron scattering measurements of phonons in Nd2_,CexCuO4 have not yet been done because of the impossibility to obtain large enough single crystals. The first Raman spectra measurements wcrc carried out on ceramic samples of pure N d 2 C u O 4 by Sugai et al. [ 3 ] and yielded three of the four Raman-activc modes for this compound. Polarized micro-Raman observations by Hadjiev et al. [4] led to slightly different values for these modes, which were later confirmed in polarized Raman spectra measurements on single cry'stals by Heyen et al. [5]. Recently infrared-active modes have been detected in single crystals and have been assigned by Crawford et al. [ 6 ] and by Heyen et al. [ 7 ]. The rigorous analysis of the observed infrared spectra, carried out by the two groups, provides an unambigu-
ous picture of the position and the symmetry of the infrared-active modes. Here wc present the results of the calculation of the phonon frequencies of Nd2_xCe, CuO4 at the F-point using a shell model. The parameters of this model taken from other compounds with perovskite-like structures were slightly adjusted to reach agreement with the available optical data for this HTS. The phonon spectra in three symmetry, directions were also calculated. The crystal structure of Nd,_,Ce.~CuO4 was determined by Tokura et al. [ 1 ]. This compound belongs to the tetragonal system, space group 14/mmm ( D a17 n ) and crystallizes in the so called T' phase. The body-centered tetragonal unit cell of this material is shown in fig. 1. The atoms in the unit cell have coordinates: two Cu-atoms - (0,0,0), (½,~,~); four ( O , , O , . ) - a t o m s - (½,0,0), (0, ½,0), (0, ½, ½), ( ½,0.½ )' four Nd-atoms (O,O,+zNd), (O,O,--ZNj), (~,~,~+ZNd). (~,½,½--ZNO); four O.-atoms I I 3 (~,0,~ i (3,I 0 ,~), (0,2,~), (0,~,~), ). (The number of atoms is doubled because the unit cell is body-centered tetragonal.) The parameters of the unit cell are: a = 3 . 9 4 ~,, c=12.08 ,~, and the Nd-position is Z N , a = 0 . 3 5 2 . The site symmetry of the different atoms in the standard Wyckoff notation is shown in table I. Only the Cu-atom has the highest symmetry. The other atoms occupy positions with lower symmetry: the Ox- and Oy-atoms occupy position 4c, the Nd-atom position 4e, and the O~-atoms position 4d. Using the factor group theoretical analysis we determined the irreducible representations of the point
0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
V. N Popov. [:L. Valchinov /Phonon spectra of Ndt asCeo 15CuO,
Z
-•0
0 .0
~ Y -X 0
O~
0
O~ • q,x)---~ OI
•
Oq~o
Cu
Nd(Cel 0 0 •
0 0
vj Q
Fig. 1. The tetragonal unit cell of Nd2_~Ce~CuO4. Table I Classification of the normal modes.
Atomic position 2Cu 4 (Ox, Oy) 4Nd 40~
Irreducible representations 2a 4c 4e 4d
A2u + Eu A2u 4- B2u 4- Eu Ais+Azu+Es+Eu A2~+ B ~s+ Es + E~
Ftot=Ats+ Bt~ + B2~+ 4A2u + 2Es+ 5Eu F,c = A2u + Eu l-'opt= A~s+ Bts+ 3A2u + 2E8 + 4E u Raman modes: A~s, Bts , 2E s Infrared modes: 3A2~, 4E~ Silent modes: B2~
group D4h under which the normal modes are transformed. The results are displayed in table I. Of all 21 normal modes three are acoustic modes - A2u and E,, and the rest are optical ones. The infrared active modes are of the type A2u and E., the BEu is a silent mode and the A~g, the B~s and the Eg modes are Raman active ones. The lattice dynamics of Nd2_xCexCuO4 was described by a shell model, which has been often applied to crystals with a perovskite structure. In this model, in order to allow for ion polarization, each ion consists of a charged core and an undeformable charged shell around it, the sum of the core and the shell charges, X and Y, yielding the ionic charge Z. The interaction between the various cores and shells
261
consists of two parts. The long-range Coulomb interaction contains as parameters the ionic and shell charges only and can be calculated using a standard Ewald procedure. The short-range part of the interactions between the ions is well described by BornMayer potentials v ( r ) = a e -~', while the short-range interaction between the core and the shell of an ion is characterized by a force constant k. In the case of Nd 2 _ x C e x C u O 4 several simplifying assumptions were made, which reduced the number of model parameters. Namely, the ionic charges were fixed at their nominal values Z ( C u ) = + 2 . 0 0 and Z ( N d ) = +3.08 (accounting for the 8% Ce 4÷ ions present). The ionic charge of the O-ion was determined using the charge neutrality condition: Z ( O ) =2.04. The short-range interactions between the ions in the form of Born-Mayer potentials as well as the core-shell interactions were considered isotropic. In our calculations we have taken account of the short-range Cu-O, Nd-O and O - O interactions between the nearest (NN) and next-to-nearest ( N N N ) neighbours only. As established by Lehner et ai. [ 8 ], the short-range potentials for a great number of alkali halides and fluoridic perovskites are similar and can be transferred from one structure to another. This idea regarding the perovskite-like high7~ compounds was spread by Kulkarni et al. [9 ], who showed that for a number of HTS the shell model parameters are indeed closely situated in the parameter space. Following these propositions, we considered as an initial set of parameters of the shell model the corresponding calculated values for several perovskite-like structures [ 9,10 ]. The available measured Raman and infrared frequencies [3-7] were used to determine the shell model parameters in a least-squares fitting procedure. The Raman frequencies were weighted by a larger factor than the infrared ones to account for the greater experimental accuracy of the former. The equilibrium conditions were used to eliminate the aparameters for the three short-range interactions. Throughout the fitting procedure stability of the phonon spectra at the F, Z, X, and G~ points was also required. The results for the shell model parameters are given in table II. They can be compared with the calculated parameters of L a 2 C u O 4 [ 10 ] and a number of other HTS [9]. The former were derived using the avail-
E N. Popov, V.L. Valchinov / Phonon spectra of Ndl ssCeo tjCuO~
262
Table I I Parameters of the model: a, b - Born-Mayer constants; k - core-shell force constant; Y - shell charge; Z - ionic charge (v, - volume of the unit cell ). Interaction
a
Nd-O Cu-O O-O
b (A -~)
Ion
(eV)
Z (lel)
Y (lel)
k (e2/~)
3018 5236 167
3.13 4.38 2.95
Nd Cu O
3.08 2.00 - 2.04
1.63 3.02 - 2.44
314 474 247
able neutron scattering and optical data on La2CuO4, while the latter form an universal set of parameters obtained on the basis of optical data only. There is an agreement within 10% for the values of the b-parameters given in refs. [ 9,10 ] and the present work, except for the short-range C u - O interaction in which case in ref. [9] values reduced by about 25% were obtained and the corresponding a-parameters are several times smaller. The resulting Cu-O short-range forces are the largest ones and are comparable with these given in refs. [9,10]. The O - O forces are an order smaller than the rest. The shell charges of the Cu and O ions are very close to those in refs. [9,10], but the shell charge of the Nd ion is two times smaller than that of the La ion [ 10]. Using the shell charge Y and the core-shell force constant k, the free ion polarizability can be expressed as a = y2/k. This for-
mula yields 0.79, 1.80, and 2.24/k 3 for the Nd, Cu and O ions, which nearly coincide with the corresponding results in ref. [9]. In table III are given the calculated phonon frequencies at the F-point in comparison with those observed by several groups. Most of the frequencies obtained by us do not deviate by more than 5% from the experimental ones, but there are some that exceed these limits. These are modes including mainly Cu and O vibrations. A possible explanation of these results may be found in the simplicity of the accepted model, namely, in the assumed isotropy of the short-range Cu-O interactions as well as the isotropy of the core-shell interaction for the O-ions. The eigenvectors of all optical modes for Nd2_~,Ce.~CuO4 are illustrated in fig. 2. The entire phonon dispersion curves in three symmetry direc-
Table III Measured and calculated phonon frequencies at the F-point (the transverse and longitudinal frequencies for each infrared mode are separated by a slash; all frequencies are in c m - ~). Optical mode
Ais B~s E I
A2u
Calculated frequencies
Measured frequencies Ref. [ 3 ]
Ref. [ 4 ]
Ref. [ 6 ]
Ref. [ 7 ]
Ref. [ 7 ]
present work
230 344
229 330
228 333
228 328
488 134/146 269/440 505/555
480 134/144 282/433 516/559
129/140 300/342 350/437 509/595
132/139 304/341 353/432 512/593
230 328 161 479 138/148 320/462 492/506 289 136/157 349/360 399/446 538/560
228 331 147 479 142/145 260/488 527/534 306 131/162 333/347 348/439 484/554
-
_
494
488
-
B2u
Eu
V.N. Popov, V.L. Valchinov/Phonon spectra of Nd~ 8jCeo~sC'uO,
263
Romon ocfive
Eg I/.7
A Ig 228
81g 331
Eg 479 Sitenf
IR ecfive
I
A 2u 14211~5
A2u 2601488
A 2u 527153~
Eu 1311162
Eu 3331347
Eu 3481439
g2u 306
Eut,8415~
Fig. 2. Calculated eigenfrequenciesand eigenvectorsof all optical normal modes at the centre of the Brillouin zone (the transverse and longitudinal frequencies for each infrared mode are separated by a slash; all frequenciesare in cm - ~).
tions in the Brillouin zone are presented in fig. 3. In order to check the transferability of the model parameters we used them to calculate the p h o n o n frequencies of La2CuO4 at F. The results are in good agreement with the observations. For example, the R a m a n modes were obtained at 201 cm - t and 470 c m - ' (A, s modes); 76 cm - I and 299 c m - ' (E s modes). The disagreement for the R a m a n modes is not more than 15% and it shows that a more com-
plicated model is needed to describe the dynamical properties of the two similar solids on a c o m m o n basis. This could be carried out when complete optical and neutron scattering data on these HTS is available. In summary, we have obtained the shell model parameters for NdE_xCe~CuO4 which provide reasonable agreement with the measured R a m a n and infrared frequencies. For the unobserved R a m a n active mode Es ( N d ) a value of 147 c m - t is predicted. The
I.tN. Popov. V.L. Valchinov / Phonon spectra *)fNdl sjCeo if'uO,
264
600 t,, •
.
•
•
.
•
•
•
•
.
.
•
•
•
.
.
500
. •
.
•
B
ttt •
.
.
:
,
:
m|
. .
,.::!'i~!!!!
"t"
• °
-
•
• • • •
400 ~:::
E w
•
•
•
.
.
;:!':::::::
.... : . - '
300
g
. . . . .
.
.
.
.
.
.
.
.
.... G$ t-
200
. : t : t ~ . . . . •
• • • | t ~ •
. . . . . . . . . . . .
.
,
.
.
:,...
. . . . t
."
•
.
•I t •
• : 1. . ; . ; . |. % . . t. l. l t t • °~• •
it ill
I" •t
. . . . . . . . .
;:
~ t t : : : : ;
tl
oo°°
.° ° ° s ~ .
100 ° t ~ ° • • • • • ' • • :
oo r
: "
" "
°
o's [qO,O]
t,'
G,
oo F
I i
D
"
o's [q,q,O]
["
1o o o X F
. •
o's [O,O,q] Z
Fig. 3. Calculated phonon dispersion curves in the [ q, 0, 0 ], [ q, q, 0 ] and [ 0, 0, q ] directions of the Brillouin zone ( the phonon frequencies are in c m - t; the reduced wave vector components q are in units rt/a, a being the smallest lattice constant ).
phonon dispersion curves in three symmetry directions have also been calculated.
Acknowledgement The authors would like to acknowledge the useful discussions with I.Z. Kostadinov.
References [ 1 ] Y. Tokura, H. Takagi and S. Uchida, Nature 337 (1989) 345. [21H. Takagi, S. Uchida and Y. Tokura, Phys. Rev. Lett. 62 (1989) 1197. [3] S. Sugai, T. Kobayashi and J. Akimitsu, Phys. Rev. B40 (1989) 2686. [4] V.G. Hadjiev, I.Z. Kostadinov, L. Bozukov, E. Dinolova and D.M. Mateev, Solid State Commun. 71 (1989) 1093.
[5]E.T. Heyen, R. Liu, B. Gegenheimer, C. Thomsen, M. Cardona, S. Pifiol and D. Mckpaul, in: Phonons 89, eds. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, 1990) p. 346. [6]M.K. Crawford, G. Burns, G.V. Chandrashekhar, F.H. Dacol, W.E. Farneth, E.M. McCarron Ill and R.J. Smalley, Phys. Rev. B41 (1990) 8933. [7] E.T. Heyen, G. Kliche, W. Kress, W. K~nig, M. Cardona, E. Rampf, J. Prade, U. SchrOder, A.D. Kulkarni, F.W. de Wette, S. Pifiol, D. McK. Paul, E. Moran and M.A. AlarioFranco, Solid State Commun. 74 (1990) 1299. [8] N. Lehner, H. Rauh, K. Strobel, R. Geick, G. Heger, J. Bouillot, B. Renker, M. Rousseau and W.G. Stirling, J. Phys. Cl5 (1982)6545. [9] W. Kress, U. Schr6der, J. Prade, A.D. Kulkarni and F.W. de Wette, Phys. Rev. B38 ( 1988 ) 2906: J. Prade, A.D. Kulkarni, F.W. de Wette, U. SchriSder and W. Kress, Phys. Rev. B39 (1989) 2771A.D. Kulkarni, J. Prade, F.W. de Wette, W. Kress and U. Schr~Rier, Phys. Rev. B40 (1989) 2642. [10] M. Mostoller, J. Zhang, A.M. Rao and P.C. Eklung, Phys. Rev. B41 (1990) 6488.