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The van Hove singularity shift from the Fermi level as a function of composition
PHYSICA ELSEVIER
PhysicaC 257 (1996) 160-166
The van Hove singularity shift from the Fermi level as a function of composition J. Giraldo a,b,1, R. Baquero a,b,*,2 a Departamento de Fisica, Centro de Investigaci6n y de Estudios Aoanzados del LP.N., Apartado Postal 14740, 07000 M~xico D.F., Mexico b Departamento de F&ica, Universidad Nacional de Colombia, A.A. 14490, Santaf~ de Bogot?t, Colombia
Received 11 April 1995
Abstract The electronic band structure and the density of states of La2_~BaxCuO4 as a function of composition are evaluated in order to examine the dynamics of the van Hove singularity. We use the tight-binding method and the virtual crystal approximation. The shift of the van Hove singularity from the Fermi level has a minimum for the composition of highest T~ and follows an approximate linear behavior as a function of doping, which is consistent with an analysis of some experimental data.
1. Introduction Certain properties of the copper oxides in the normal (N) as well as in the superconducting (S) state have been found to be anomalous [1]. To explain these, alternative formulations to the traditional Fermi liquid (FL) description of the metallic phase have been proposed [2,3]. In this work we are concerned with a rather conventional approach where one assumes that a FL description is valid for La2_xBa~CuO4 at least at very low temperatures.
* Corresponding author. Fax: +52 5 747 7096; e-mail: [email protected]. i Permanent address: Departamentode Fisica, Universidad National de Colombia,A.A. 14490, Santaf6 de Bogota, Colombia. 2 Permanent address: Departamentode Fisica, Centro de Investigaci6n y de Estudios Avanzados del I.P.N., Apartado Postal 14740, 07000 M6xicoD.F., M6xico.
For the cuprates, the existence of a Fermi surface (FS) has been established experimentally by angleresolved photoemission spectroscopy (ARPES) [4], and other techniques. Theoretical calculations agree well with those results [5]. Broadly speaking, two main mechanisms consistent with a FL description are under examination. The electron-electron interaction mediated by antiferromagnetic spin fluctuations has been studied extensively [6]. This model has been associated with the d wave symmetry of the order parameter in the past. Very recently [7,8] there have been indications that d pairing might be consistent with other mechanisms as well [9,10]. The second approach is the phonon-mediated mechanism assisted by a high density of states (DOS) due to the quasi-two-dimensionality of the charge-carriers. Our work correlates with this direction of research. The situation regarding the role of phonons has evolved during the last years. Manifestations of the
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J. Giraldo, R. Baquero/ Physica C 257 (1996) 160-166
electron-phonon (e-p) interaction[ 11-14] have been found in different experiments as tunneling [15-17], inelastic neutron scattering [18], Raman [19-21] and infrared spectroscopy [20,22,23]. The coupling of the longitudinal phonon modes has been found evidence for experimentally by the group of Cardona [20]. Detailed calculations by, e.g., Pickett et al. [5,24], Freeman et .al. [25,26], Andersen et al. [27,28] and others [29] have predicted electronic band structures, Fermi surfaces and phonon modes for the cuprates that agree well with experiment [4,30,31]. Important new findings in tunneling experiments by Tsuda [32] strongly point to the possibility of a significant participation of the e - p interaction in the mechanism of high-T~ superconductors. The most striking feature of both theoretical and experimental data, is a flat band essentially pinned at the Fermi energy, E F. This fact is expected to have a large incidence on the physical properties of the cuprates, since it represents a very high electronic DOS. The approach based on the van Hove singularity (vHS) [33], a logarithmic singularity of the DOS in the 2D case [34-37], has been extensively studied. It has been invoked to explain some anomalies [38-40] and the isotope-effect variations [41]. ARPES studies on Y123, [42] Y124 [42,43] and Bi2212 [44] have found that the vHS is almost precisely located at the Fermi level at the composition of optimum T~. In LSCO and Y123 positron-annihilation experiments reveal the same feature [45,46]. The Russian school [47-51] and others [37,52] have emphasized the use of the Eliashberg formalism. The result is that the most striking anomalies in the N and in the S state of the cuprates are at least qualitatively understood. In particular, it was shown in Ref. [49] that an "extended Drude formula" is able to explain the linear T drop of the reflectivity, while in Ref. [50] a relaxation rate due to e - p interaction, in a region where the quasi-particle picture does not longer hold, leads to a different T dependence of the superconducting properties as compared to the BCS model. Also the anomalous electronic Raman scattering can be explained by using e - p coupling with an additional coupling to spin fluctuations in solving the Eliashberg equations [48]. The results are similar to those previously found in Ref. [22]. We underline on passing that the first generalization of the Eliashberg equations in-
161
cluding the effects of paramagnons has been made by Baquero et al. [53]. Many authors have proposed simplified models based on isolated CuO 2 planes to study the electronic properties of the copper oxides [2,54-58]. Comparison with early experimental results give some support to this simplification. With the progress achieved in the preparation of samples, experiments have revealed that coupling perpendicular to the CuO 2 plane is important, giving rise to a rather quasi-two-dimensional picture as opposed to a 2D one. In this work, we will use a tight-binding (TB) fit to a 3D ab initio calculation to obtain the electronic band structure of La2_xBaxCuO4 and use the virtual crystal approximation to introduce disorder. We evaluate as well the DOS and the vHS shift from the Fermi level as a function of composition. In the next section we present our calculations and results. We find that the evolution of the shift between the Evil (the energy at which the singularity appears) and the Fermi level is linear, as it is predicted in 2D models. The DOS peak varies as well almost linearly. Therefore a 3D formulation seems not to affect the essential feature. Using this fact and our results, we discuss in Section 3 the effect of the shift on T~. The last section is devoted to conclusions.
2. The van Hove singularity: calculations and results
An attractive program in high-T~ is to follow the variation of certain relevant properties in the N or in the S state as a function of composition (doping) for varied substitutions. This has been done on experimental grounds for different materials. Practical reasons constrain the program theoretically. Ab initio calculations, though very reliable, cannot be straight-forwardly applied to doped materials. Different procedures can be adopted. One approach is to build up a supercell, as has been done by Crockford and Yeung [59], Krakauer et al. [24] and Rodrlguez et al. [60] A difficulty with this approach is that one is limited to evaluate properties for a certain fixed composition. A complementary approach to the supercell is to use methods like the effective-medium theory (EMT) [61], that inhabits many important
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features of the full electron structure methods but has the advantage that it does not require much computer time. An improved EMT has been used recently [29] to evaluate the phonon modes and lattice parameters for YBa2Cu3_xCOxO 7 at different concentrations in good agreement with experiments [62]. An auxiliary tool is the tight-binding matrix parameterization introduced by Slater and Koster [63]. The tight-binding approach has the advantage that it can be combined with rather well-known techniques to introduce correlations. It can also be fitted to first-principles calculations to obtain the electronic properties and the phononic structure at any concentration. In a previous work the variation of the lattice parameters and the evolution of the five active Raman modes in Y123 have been followed as a function of Co concentration [64]. One can relate, e.g., the most interesting apex oxygen mode frequency to the critical temperature, as is done experimentally [62]. A similar relationship is established in this work beween the vHS and Tc. A basic ingredient of the so called van Hove scenario [34-37] of cuprate superconductivity is that the shifts of the saddle points with respect to the Fermi energy level as a funciton of composition correlates very well with the known variation of the critical temperature, and that the maximal Tc arises when the E F lies at or close to the energy of a vHS, Err i. Recent calculations and measurements provide considerable support to the idea that the vHS is playing an important role in the physics of the cuprates [37,42-44]. Most of the vHS based models use the logarithmic singularity in the DOS that appears in a 2D periodic system to give account of the high critical temperature [33-36,40]. As it is illustrated in this work, using a realistic 3D band calculation, one obtains as well a behavior of the DOS that supports the basic assumption of the van Hove scenario. It must be pointed out that recent photoemission measurements on Y123 [43] and Bi2212 [44] suggest that the DOS in these materials has a much stronger power-law divergence. Radtke and Norman [37] have carded out detailed calculations where it is found that while the weak-coupling solutions suggest a strong influence of the strength of the vHS on T~ and or, the strong-coupling solution shows less sensitivity to the singularity strength. In any case [65] one
obtains reasonable values for Tc. This is not the situation in the antiferromagnetic-spin-fluctuationmediated pairing models to the cuprates, where one is not able to explain the large critical temperatures, at least within Eliashberg theory [66]. In the following we present the calculated variation of the distance in energy of the vHS from the Fermi level in the La2_xBaxCuO4 system as a function of x. We have used the TB method [63] fitted to first-principles calculations [67] for La2CuO 4. To introduce disorder, we use the virtual-crystal approximation. We calculate the electronic band structure and the DOS near the Fermi level of La2_xBa~CuO 4 in the compositional range x = 0.0-0.3. Our TB parameters for the pure materials are taken from Ref. [67]. We have used a 31 basis set: 10 for La (2X5 d), 9 for Cu (s,p and d) and 12 for O orbitals (4X3 p). The composition is taken into account by a linear variation of the TB parameters between those of pure La2CuO 4 and pure Ba2CuO 4, following the virtual crystal approximation. The TB parameters for Lal.75Ba0.25CuO 4 and for the pure materials are shown in Table 1. As has been shown by DeWeert et al. [67] and as we have verified in our calculations, there is not an essential difference in the electronic band structure below the Fermi level if one uses a smaller basis that contains the CuO 2 plane and the apical oxygen, as much as one performs a new fitting to the ab initio calculations. Tt/e band structure varies smoothly as a function of x, as is seen in Fig. 1. There we show the bands near and below the Fermi energy for x = 0.0 and x = 0.3. The first 16 bands are occupied. The 17th band is partially empty. One notices that on doping this band goes up while the Fermi level decreases. For intermediate doping this causes a sharpening in the peak (cf. Fig. 2) near the Fermi level that corresponds to the vHS. The DOS has been calculated by direct integration on the irreducible first Brillouin zone using the well-known tetrahedral method. Fig. 2 exhibits the DOS in a region around the vHS peak for different x values. One can follow the variation with concentration of the vHS peaks and of the Fermi energy E F. For each concentration, E F is indicated by a heavy dot. The Fermi level varies almost in a linear way. Starting from E F = 0.5595 Ry it shifts to lower energies on increasing x with a constant slope
J. Giraldo, R. Baquero / Physica C 257 (1996) 160-166 Table l S l a t e r - K o s t e r parameters (in Ry) for La 2_ , B a x C u O 4
'°I
a. On-site parameters (M = La or Ba) La 2CuO4 La ~.75Bao.25 CuO4 Ba 2CuO4 M t 2g 1.0869 eg 0.7679 0(1) p 0.3359 0(2) p 0.3954 Cu s 0.9930 p 1.4462 t 2g 0.3581 e~ 0.4592
1.1280 0.8245 0.3341 0.3991 1.0278 1.4781 0.3133 0.4440
1.4160 1.2203 0.3218 0.4250 1.2717 1.7015 0.2356 0.3376
163
!.
,I
,,5 --~60
o.oo
i!
£K
t~
:t 1 tl
0 (13 4 0
'~i/
b. First-neighbor parameters (M = La or Ba) La2CuO4La j .75Bao.25CuO4Ba2CuO4 M-M ddty - 0.0215 dd'rr 0.0143 dd (5 - 0.0137 M-Cu ddo, -0.0112 ddqx -0.0019 dd~5 0.0019 M-O(1) dpc - 0.0380 dp,n 0.0754 M-0(2) dpcr 0.1762 dp'rr 0.1907 Cu-Cu sscr 0.0072 sp
v
•
- 0.0301 - 0.0001 - 0.0156 -0.0302 -0.0016 +0.0002 - 0.0333 +0.0811 +0.1935 + 0.2032 + 0.0324 + 0.0360 + 0.1898 + 0.1149 - 0.0220 -0.0181 +0.0340 -0.0146 + 0.0055 +0.0018 +0.1875 -0.1841 - 0.0364 +0.1008 +0.0616 + 0.0537 +0.1182 + 0.0751
o.to 0.20 COMPOSITION X
0")
- 0.0899 - 0.1007 - 0.0286 -0.1635 0.004 -0.0149 - 0.0006 0.1210 0.3142 0.2910 0.2086 0.2489 0.3395 - 0.2689 - 0.0321 -0.0378 0.0537 -0.0205 0.0226 -0.0051 0.1463 -0.1917 - 0.0398 0.0807 0.0580 0.0747 0.1313 160.0769
I ,
0 CI
:t
:
X=O.O0 . . . . . X=O.05
- -
= O . 1lO 5 '',i : :F . . . . . . ......... .. XX=O. , ...... X=0.20 - .... X=0.25 , - X=O.30 [
•
t
20
01
o.52
ttttttt
o. 4
I
o. 6
,
o.6o
Energy (Ry) Fig. 2. Electronic density of states for different concentrations around the van Hove singularities. These are indicated by dashes on the horizontal axes. A heavy dot on each DOS curve indicates the position of the Fermi level.
dE/dx
= 0.150 R y / h o l e . The shift 8 = I Eva - EFI is also almost linear as a function of concentration. Looking closely to the shape of the curve near the van Hove peaks (these are indicated by vertical arrows on the horizontal axis), one can infer that these are sharper near the concentration that yields the m a x i m u m Tc (approximately from x = 0.1 to x = 0.2). The inset in Fig. 2 illustrates the variation of 8 with x.
0.6
3. Comparison with experiment
0.4
In this section we use an equation for T~ within the van Hove scenario obtained as a function of t~, to judge our calculated values. It must be pointed out f a s t that we have started our calculations at x = 0.0, where one knows that the system is an antiferromagnetic insulator. L D A calculations predict a metallic behavior at this doping level. This failure of the band-structure calculations is very well known and has been an argument against the one-electron de-
>(.9 0.2 z
w o.o
r
(loo)
x
F
ZF
(loo)
x
r
z
Fig. 1. Electronic band structure along some symmetry lines for x = 0.0 (a) and x = 0.3 (b). Only bands below the Fermi level are shown.
164
J. Giraldo, R. Baquero / Physica C 257 (1996) 160-166
scription of the cuprates. Some attempts have been done to remove this drawback of LDA calculations [68]. We do not attempt to use our results in the underdoped regime ( x < 0.1) where our calculations are not reliable. As far as we are aware, there is not any model that can correctly predict even the qualitative behavior of T~ as a function of concentration for La2_~BaxCuO 4 [69]. The experimental results for the isotope coefficient a are even more intriguing. The situation is simpler in other systems. It is shown, e.g., in Ref. [41] that ot and T~ in La2_xSrxCuO 4 as predicted in the van Hove scenario follow qualitatively the experimental trends. A variation of 8 with doping can be estimated from the results of Newns et al. for Bi2Sr2CaCu20 s. These results were obtained using an Anderson lattice hamiltonian and slave-boson mean-field theory (2D). We have estimated from their Fig. 5 [34] that 8 is approximately represented by a square law, 8 ot x 2. When we examine our calculated DOS as a function of concentration in more detail, we observe that the shift could deviate slightly from a linear relationship but then it does not necessarily follow a simple power law. The important features of many calculations are that 8 has a minimum at the highest Tc and increases monotonically around T~. These features are reproduced in our calculations. We proceed now to a closer comparison with experiment. As we already mentioned, we use an expression for T~ that has been obtained in a two-dimensional van Hove scenario [41]:
([2
T ~ - l . 3 6 T F exp -
82
~ +
In ~
-1
.
4. Conclusions In conclusion, we have calculated the 3D band structure of La2_xBaxCuO 4 and the electronic DOS as a function of x using the tight-binding method. We find an approximate linear behavior of the shift 8 = I Evil -- EFI as a function of the concentration x that reproduces the overall features of other calculations. When a T~(8) equation established within the van Hove scenario is used, our calculated 8(x) values reproduce well the T~(x) experimental known results. We conclude that the quasi-linear behavior of 8 is consistent with the experimental data
Acknowledgements
1 +
at x = 0.15 in an almost parabolic way near T~r~x to 20 K at x = 0.20 in good agreement with the experimental results for To(x) [69]. We do not go further than x = 0.2 since Eq. (1) is not valid in the overdoped region ( x > 0.25), where ~ > 2kBT~. An important conclusion from the last two sections is that the TB fit, even within the crude estimate of the virtual-crystal approximation, is able to reproduce the main features of other calculations and is consistent with the experimental behavior of Tc as a function of concentration. Furthermore, from our analysis it is also straightforward to see that the rigid band model will not reproduce these features and that this approximation is inconsistent with our resuits.
(1)
Here N o is the DOS normalized to a flat band with a bandwidth of 2 E F and it is assumed that 8 < 2 k s T¢. Following the approach of Ref. [41], we estimate the parameters in the above expression in order to yield at 8 = 0 a T¢max = 30 K which is close to the experimental value [69]. We have used: NoV= 0.074, T~o = 700 K and Tv = 5500 K. Now one can use 5 ( x ) from Fig. 2 to calculate T~(x) from Eq. (1). One notices on interpolating that T~ decreases from 30 K
We acknowledge support from Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnologla, Colciencias, Colombia.
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PhysicaC 257 (1996) 160-166
The van Hove singularity shift from the Fermi level as a function of composition J. Giraldo a,b,1, R. Baquero a,b,*,2 a Departamento de Fisica, Centro de Investigaci6n y de Estudios Aoanzados del LP.N., Apartado Postal 14740, 07000 M~xico D.F., Mexico b Departamento de F&ica, Universidad Nacional de Colombia, A.A. 14490, Santaf~ de Bogot?t, Colombia
Received 11 April 1995
Abstract The electronic band structure and the density of states of La2_~BaxCuO4 as a function of composition are evaluated in order to examine the dynamics of the van Hove singularity. We use the tight-binding method and the virtual crystal approximation. The shift of the van Hove singularity from the Fermi level has a minimum for the composition of highest T~ and follows an approximate linear behavior as a function of doping, which is consistent with an analysis of some experimental data.
1. Introduction Certain properties of the copper oxides in the normal (N) as well as in the superconducting (S) state have been found to be anomalous [1]. To explain these, alternative formulations to the traditional Fermi liquid (FL) description of the metallic phase have been proposed [2,3]. In this work we are concerned with a rather conventional approach where one assumes that a FL description is valid for La2_xBa~CuO4 at least at very low temperatures.
* Corresponding author. Fax: +52 5 747 7096; e-mail: [email protected]. i Permanent address: Departamentode Fisica, Universidad National de Colombia,A.A. 14490, Santaf6 de Bogota, Colombia. 2 Permanent address: Departamentode Fisica, Centro de Investigaci6n y de Estudios Avanzados del I.P.N., Apartado Postal 14740, 07000 M6xicoD.F., M6xico.
For the cuprates, the existence of a Fermi surface (FS) has been established experimentally by angleresolved photoemission spectroscopy (ARPES) [4], and other techniques. Theoretical calculations agree well with those results [5]. Broadly speaking, two main mechanisms consistent with a FL description are under examination. The electron-electron interaction mediated by antiferromagnetic spin fluctuations has been studied extensively [6]. This model has been associated with the d wave symmetry of the order parameter in the past. Very recently [7,8] there have been indications that d pairing might be consistent with other mechanisms as well [9,10]. The second approach is the phonon-mediated mechanism assisted by a high density of states (DOS) due to the quasi-two-dimensionality of the charge-carriers. Our work correlates with this direction of research. The situation regarding the role of phonons has evolved during the last years. Manifestations of the
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J. Giraldo, R. Baquero/ Physica C 257 (1996) 160-166
electron-phonon (e-p) interaction[ 11-14] have been found in different experiments as tunneling [15-17], inelastic neutron scattering [18], Raman [19-21] and infrared spectroscopy [20,22,23]. The coupling of the longitudinal phonon modes has been found evidence for experimentally by the group of Cardona [20]. Detailed calculations by, e.g., Pickett et al. [5,24], Freeman et .al. [25,26], Andersen et al. [27,28] and others [29] have predicted electronic band structures, Fermi surfaces and phonon modes for the cuprates that agree well with experiment [4,30,31]. Important new findings in tunneling experiments by Tsuda [32] strongly point to the possibility of a significant participation of the e - p interaction in the mechanism of high-T~ superconductors. The most striking feature of both theoretical and experimental data, is a flat band essentially pinned at the Fermi energy, E F. This fact is expected to have a large incidence on the physical properties of the cuprates, since it represents a very high electronic DOS. The approach based on the van Hove singularity (vHS) [33], a logarithmic singularity of the DOS in the 2D case [34-37], has been extensively studied. It has been invoked to explain some anomalies [38-40] and the isotope-effect variations [41]. ARPES studies on Y123, [42] Y124 [42,43] and Bi2212 [44] have found that the vHS is almost precisely located at the Fermi level at the composition of optimum T~. In LSCO and Y123 positron-annihilation experiments reveal the same feature [45,46]. The Russian school [47-51] and others [37,52] have emphasized the use of the Eliashberg formalism. The result is that the most striking anomalies in the N and in the S state of the cuprates are at least qualitatively understood. In particular, it was shown in Ref. [49] that an "extended Drude formula" is able to explain the linear T drop of the reflectivity, while in Ref. [50] a relaxation rate due to e - p interaction, in a region where the quasi-particle picture does not longer hold, leads to a different T dependence of the superconducting properties as compared to the BCS model. Also the anomalous electronic Raman scattering can be explained by using e - p coupling with an additional coupling to spin fluctuations in solving the Eliashberg equations [48]. The results are similar to those previously found in Ref. [22]. We underline on passing that the first generalization of the Eliashberg equations in-
161
cluding the effects of paramagnons has been made by Baquero et al. [53]. Many authors have proposed simplified models based on isolated CuO 2 planes to study the electronic properties of the copper oxides [2,54-58]. Comparison with early experimental results give some support to this simplification. With the progress achieved in the preparation of samples, experiments have revealed that coupling perpendicular to the CuO 2 plane is important, giving rise to a rather quasi-two-dimensional picture as opposed to a 2D one. In this work, we will use a tight-binding (TB) fit to a 3D ab initio calculation to obtain the electronic band structure of La2_xBaxCuO4 and use the virtual crystal approximation to introduce disorder. We evaluate as well the DOS and the vHS shift from the Fermi level as a function of composition. In the next section we present our calculations and results. We find that the evolution of the shift between the Evil (the energy at which the singularity appears) and the Fermi level is linear, as it is predicted in 2D models. The DOS peak varies as well almost linearly. Therefore a 3D formulation seems not to affect the essential feature. Using this fact and our results, we discuss in Section 3 the effect of the shift on T~. The last section is devoted to conclusions.
2. The van Hove singularity: calculations and results
An attractive program in high-T~ is to follow the variation of certain relevant properties in the N or in the S state as a function of composition (doping) for varied substitutions. This has been done on experimental grounds for different materials. Practical reasons constrain the program theoretically. Ab initio calculations, though very reliable, cannot be straight-forwardly applied to doped materials. Different procedures can be adopted. One approach is to build up a supercell, as has been done by Crockford and Yeung [59], Krakauer et al. [24] and Rodrlguez et al. [60] A difficulty with this approach is that one is limited to evaluate properties for a certain fixed composition. A complementary approach to the supercell is to use methods like the effective-medium theory (EMT) [61], that inhabits many important
162
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features of the full electron structure methods but has the advantage that it does not require much computer time. An improved EMT has been used recently [29] to evaluate the phonon modes and lattice parameters for YBa2Cu3_xCOxO 7 at different concentrations in good agreement with experiments [62]. An auxiliary tool is the tight-binding matrix parameterization introduced by Slater and Koster [63]. The tight-binding approach has the advantage that it can be combined with rather well-known techniques to introduce correlations. It can also be fitted to first-principles calculations to obtain the electronic properties and the phononic structure at any concentration. In a previous work the variation of the lattice parameters and the evolution of the five active Raman modes in Y123 have been followed as a function of Co concentration [64]. One can relate, e.g., the most interesting apex oxygen mode frequency to the critical temperature, as is done experimentally [62]. A similar relationship is established in this work beween the vHS and Tc. A basic ingredient of the so called van Hove scenario [34-37] of cuprate superconductivity is that the shifts of the saddle points with respect to the Fermi energy level as a funciton of composition correlates very well with the known variation of the critical temperature, and that the maximal Tc arises when the E F lies at or close to the energy of a vHS, Err i. Recent calculations and measurements provide considerable support to the idea that the vHS is playing an important role in the physics of the cuprates [37,42-44]. Most of the vHS based models use the logarithmic singularity in the DOS that appears in a 2D periodic system to give account of the high critical temperature [33-36,40]. As it is illustrated in this work, using a realistic 3D band calculation, one obtains as well a behavior of the DOS that supports the basic assumption of the van Hove scenario. It must be pointed out that recent photoemission measurements on Y123 [43] and Bi2212 [44] suggest that the DOS in these materials has a much stronger power-law divergence. Radtke and Norman [37] have carded out detailed calculations where it is found that while the weak-coupling solutions suggest a strong influence of the strength of the vHS on T~ and or, the strong-coupling solution shows less sensitivity to the singularity strength. In any case [65] one
obtains reasonable values for Tc. This is not the situation in the antiferromagnetic-spin-fluctuationmediated pairing models to the cuprates, where one is not able to explain the large critical temperatures, at least within Eliashberg theory [66]. In the following we present the calculated variation of the distance in energy of the vHS from the Fermi level in the La2_xBaxCuO4 system as a function of x. We have used the TB method [63] fitted to first-principles calculations [67] for La2CuO 4. To introduce disorder, we use the virtual-crystal approximation. We calculate the electronic band structure and the DOS near the Fermi level of La2_xBa~CuO 4 in the compositional range x = 0.0-0.3. Our TB parameters for the pure materials are taken from Ref. [67]. We have used a 31 basis set: 10 for La (2X5 d), 9 for Cu (s,p and d) and 12 for O orbitals (4X3 p). The composition is taken into account by a linear variation of the TB parameters between those of pure La2CuO 4 and pure Ba2CuO 4, following the virtual crystal approximation. The TB parameters for Lal.75Ba0.25CuO 4 and for the pure materials are shown in Table 1. As has been shown by DeWeert et al. [67] and as we have verified in our calculations, there is not an essential difference in the electronic band structure below the Fermi level if one uses a smaller basis that contains the CuO 2 plane and the apical oxygen, as much as one performs a new fitting to the ab initio calculations. Tt/e band structure varies smoothly as a function of x, as is seen in Fig. 1. There we show the bands near and below the Fermi energy for x = 0.0 and x = 0.3. The first 16 bands are occupied. The 17th band is partially empty. One notices that on doping this band goes up while the Fermi level decreases. For intermediate doping this causes a sharpening in the peak (cf. Fig. 2) near the Fermi level that corresponds to the vHS. The DOS has been calculated by direct integration on the irreducible first Brillouin zone using the well-known tetrahedral method. Fig. 2 exhibits the DOS in a region around the vHS peak for different x values. One can follow the variation with concentration of the vHS peaks and of the Fermi energy E F. For each concentration, E F is indicated by a heavy dot. The Fermi level varies almost in a linear way. Starting from E F = 0.5595 Ry it shifts to lower energies on increasing x with a constant slope
J. Giraldo, R. Baquero / Physica C 257 (1996) 160-166 Table l S l a t e r - K o s t e r parameters (in Ry) for La 2_ , B a x C u O 4
'°I
a. On-site parameters (M = La or Ba) La 2CuO4 La ~.75Bao.25 CuO4 Ba 2CuO4 M t 2g 1.0869 eg 0.7679 0(1) p 0.3359 0(2) p 0.3954 Cu s 0.9930 p 1.4462 t 2g 0.3581 e~ 0.4592
1.1280 0.8245 0.3341 0.3991 1.0278 1.4781 0.3133 0.4440
1.4160 1.2203 0.3218 0.4250 1.2717 1.7015 0.2356 0.3376
163
!.
,I
,,5 --~60
o.oo
i!
£K
t~
:t 1 tl
0 (13 4 0
'~i/
b. First-neighbor parameters (M = La or Ba) La2CuO4La j .75Bao.25CuO4Ba2CuO4 M-M ddty - 0.0215 dd'rr 0.0143 dd (5 - 0.0137 M-Cu ddo, -0.0112 ddqx -0.0019 dd~5 0.0019 M-O(1) dpc - 0.0380 dp,n 0.0754 M-0(2) dpcr 0.1762 dp'rr 0.1907 Cu-Cu sscr 0.0072 sp
v
•
- 0.0301 - 0.0001 - 0.0156 -0.0302 -0.0016 +0.0002 - 0.0333 +0.0811 +0.1935 + 0.2032 + 0.0324 + 0.0360 + 0.1898 + 0.1149 - 0.0220 -0.0181 +0.0340 -0.0146 + 0.0055 +0.0018 +0.1875 -0.1841 - 0.0364 +0.1008 +0.0616 + 0.0537 +0.1182 + 0.0751
o.to 0.20 COMPOSITION X
0")
- 0.0899 - 0.1007 - 0.0286 -0.1635 0.004 -0.0149 - 0.0006 0.1210 0.3142 0.2910 0.2086 0.2489 0.3395 - 0.2689 - 0.0321 -0.0378 0.0537 -0.0205 0.0226 -0.0051 0.1463 -0.1917 - 0.0398 0.0807 0.0580 0.0747 0.1313 160.0769
I ,
0 CI
:t
:
X=O.O0 . . . . . X=O.05
- -
= O . 1lO 5 '',i : :F . . . . . . ......... .. XX=O. , ...... X=0.20 - .... X=0.25 , - X=O.30 [
•
t
20
01
o.52
ttttttt
o. 4
I
o. 6
,
o.6o
Energy (Ry) Fig. 2. Electronic density of states for different concentrations around the van Hove singularities. These are indicated by dashes on the horizontal axes. A heavy dot on each DOS curve indicates the position of the Fermi level.
dE/dx
= 0.150 R y / h o l e . The shift 8 = I Eva - EFI is also almost linear as a function of concentration. Looking closely to the shape of the curve near the van Hove peaks (these are indicated by vertical arrows on the horizontal axis), one can infer that these are sharper near the concentration that yields the m a x i m u m Tc (approximately from x = 0.1 to x = 0.2). The inset in Fig. 2 illustrates the variation of 8 with x.
0.6
3. Comparison with experiment
0.4
In this section we use an equation for T~ within the van Hove scenario obtained as a function of t~, to judge our calculated values. It must be pointed out f a s t that we have started our calculations at x = 0.0, where one knows that the system is an antiferromagnetic insulator. L D A calculations predict a metallic behavior at this doping level. This failure of the band-structure calculations is very well known and has been an argument against the one-electron de-
>(.9 0.2 z
w o.o
r
(loo)
x
F
ZF
(loo)
x
r
z
Fig. 1. Electronic band structure along some symmetry lines for x = 0.0 (a) and x = 0.3 (b). Only bands below the Fermi level are shown.
164
J. Giraldo, R. Baquero / Physica C 257 (1996) 160-166
scription of the cuprates. Some attempts have been done to remove this drawback of LDA calculations [68]. We do not attempt to use our results in the underdoped regime ( x < 0.1) where our calculations are not reliable. As far as we are aware, there is not any model that can correctly predict even the qualitative behavior of T~ as a function of concentration for La2_~BaxCuO 4 [69]. The experimental results for the isotope coefficient a are even more intriguing. The situation is simpler in other systems. It is shown, e.g., in Ref. [41] that ot and T~ in La2_xSrxCuO 4 as predicted in the van Hove scenario follow qualitatively the experimental trends. A variation of 8 with doping can be estimated from the results of Newns et al. for Bi2Sr2CaCu20 s. These results were obtained using an Anderson lattice hamiltonian and slave-boson mean-field theory (2D). We have estimated from their Fig. 5 [34] that 8 is approximately represented by a square law, 8 ot x 2. When we examine our calculated DOS as a function of concentration in more detail, we observe that the shift could deviate slightly from a linear relationship but then it does not necessarily follow a simple power law. The important features of many calculations are that 8 has a minimum at the highest Tc and increases monotonically around T~. These features are reproduced in our calculations. We proceed now to a closer comparison with experiment. As we already mentioned, we use an expression for T~ that has been obtained in a two-dimensional van Hove scenario [41]:
([2
T ~ - l . 3 6 T F exp -
82
~ +
In ~
-1
.
4. Conclusions In conclusion, we have calculated the 3D band structure of La2_xBaxCuO 4 and the electronic DOS as a function of x using the tight-binding method. We find an approximate linear behavior of the shift 8 = I Evil -- EFI as a function of the concentration x that reproduces the overall features of other calculations. When a T~(8) equation established within the van Hove scenario is used, our calculated 8(x) values reproduce well the T~(x) experimental known results. We conclude that the quasi-linear behavior of 8 is consistent with the experimental data
Acknowledgements
1 +
at x = 0.15 in an almost parabolic way near T~r~x to 20 K at x = 0.20 in good agreement with the experimental results for To(x) [69]. We do not go further than x = 0.2 since Eq. (1) is not valid in the overdoped region ( x > 0.25), where ~ > 2kBT~. An important conclusion from the last two sections is that the TB fit, even within the crude estimate of the virtual-crystal approximation, is able to reproduce the main features of other calculations and is consistent with the experimental behavior of Tc as a function of concentration. Furthermore, from our analysis it is also straightforward to see that the rigid band model will not reproduce these features and that this approximation is inconsistent with our resuits.
(1)
Here N o is the DOS normalized to a flat band with a bandwidth of 2 E F and it is assumed that 8 < 2 k s T¢. Following the approach of Ref. [41], we estimate the parameters in the above expression in order to yield at 8 = 0 a T¢max = 30 K which is close to the experimental value [69]. We have used: NoV= 0.074, T~o = 700 K and Tv = 5500 K. Now one can use 5 ( x ) from Fig. 2 to calculate T~(x) from Eq. (1). One notices on interpolating that T~ decreases from 30 K
We acknowledge support from Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnologla, Colciencias, Colombia.
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