Solid State Communications, Vol. 72, No. 8, pp. 763-766, 1989. 0038-1098/89 $3.00 + .00 Printed in Great Britain. Pergamon Press plc

ONE PARTICLE SPECTRAL DENSITIES FOR TH E CuO2 PLANES OF H I G H T,. SUPERCONDUCTORS

C.A. Balseiro* and M. Avignon,

LEPES-CNRS, BP 166-38042 Grenoble Cedex, associated with University Joseph Fourier Grenoble, France

and

E.R. Gagliano

Centro Atomico Bariloche, 8400 Bariloche, R.N. Argentina

(Received 21 April 1989, in revised form 25 July 1989 by P. Burlet)

We calculate numerically the one particle spectral densities for a small cluster of CuO2 including intra and interatomic correlations. We show that the stoichiometric system with one hole per unit cell is a charge transfer insulator. The XPS spectra is calculated for this stoichiometric situation and for system doped with electrons or holes. Our results indicate that electron doping produces in the XPS spectrum a large peak at the Fermi energy.

T H E C O M M O N feature of all high 7],. cuprate super- conductors are the CuO2 planes which play a crucial role in determining the electronic properties of these materials. In fact, there is now a general consensus that the relevant carriers are located in these planes. In all the systems studied so far these charge carriers are holes, due to the substitution of divalent ions (as Sr 2+ or Ba 2÷) for trivalent La in La2CuO4 or by adding oxygen in Y-Ba-CuO. Different spectroscopic results indicate that these holes have a predominantly oxygen - 2 p character [1-4] and that the on site coulomb interaction within the Cu-3d holes is large [5-7] as in most of the transition metal compounds, thus making Cu3+(d 8) very unfavourable energetically. Whether correlations are important on oxygen also [7] remains a matter of debate. However, recent estimates [8] seem to indicate fairly large values of coulomb interactions not only on Cu but also on oxygen, although smaller than on Cu. Concerning the nature of the p-orbitals involved the situation is still uncertain. Some authors [9-10] rule out O-2p= state for YBa2Cu307 and Bi2 Sr2CaCu208 but cannot distinguish between tr and n symmetry from Px.y orbitals in the CuO2 plane. Bianconi et al. [I 1] proposed a different scenario for Y-Ba-CuO in which holes reside on the p.. orbital of the O out of the CuO2 plane.

* On leave from: Centro Atomico Bariloche, 8400 Bariloche, Argentina.

Recently a new family ofcuprate superconductors have been discovered by Tokuda et al. [12]. The important difference between these systems and the previously known is the fact that, presumably in the new materials, the CuO2 planes are doped with elec- trons rather than with holes due to the substitution of trivalent rare earths in Ln2CuO4 (Ln = Pr, Nd, Sm) by Ce which presumably enters as tetravent or as intermediate valency ~3.5. It is then important to understand the difference in the electronic properties of CuO2 planes doped with electrons and holes.

Due to the large electron-electron correlation in the Cu ions [5-8], the conventional band structure calculations can not be used as a good starting point for the understanding of the electronic properties. In fact, this type of approach fails in predicting the true nature of the ground state and excitations of these materials. However, total energy calculations cor- rectly describe the two following features: the low energy electronic excitation involves the Cu 2 2 3d~-y and the O 2p.~ and 2py orbitals and an enhanced covalency in the Cu-O planes as compared to transition metal oxides.

These features support the idea that the simplest model Hamiltonian which correctly describes the elec- tronic structure of the CuO2 planes should include - at least as a starting point - the Cu and O orbitals, a strong hybridization and large on-site correlations as the basic ingredients.

In the usual notation, the model Hamiltonian

763

764 ONE PARTICLE SPECTRAL DENSITIES FOR C u O 2

8 8 8

r 1 8

18 8 ++,+ + =,,+ ++

8 8 8 Fig. 1. The CuO2 cluster with d(x2-y 2) and p, (x , y) orbitals on Cu and O sites.

reads:

H = ~ eini~ + ~+ tijC + Cja + E Uin, Tni~ ia ( i j> i

a

+ V ~+ ni+~nj,,, (1)

f fa"

where i runs over the sites of the CuO 2 planar lattice + (see Fig. 1). The operator ce, creates a hole with spin

a at the Cu or O sites. The parameters e, = ed, /3p and U~ = Ud, Up are the d and p orbitals energies and intra-atomic Coulomb repulsion respectively. In Eq. (1), tij is the hopping matrix element and V stands for the interatomic repulsion between holes at nearest neighbour sites. We have retained only the dx2_y:

, i i ,

~ 0 ~ 0 0.0 5.0

orbital on Cu, and the Px.y with cr symmetry among the p orbitals.

The stoichiometric reference systems like La2CuO4, Ln2CuO4 (Ln = Pr, Nd, Sm) or in ideal YBaCuO are insulating or semiconductors. Moreover, La2CuO4 and YBaCuO are antiferromagnetic [13] while the magnetic properties of Ln2fuO 4 have not been reported yet.

We first give a "zero order" description of these doped systems in terms of the parameters of Hamilton- ian (1). This is done by taking zero hopping (t U = 0). In the undoped materials there is one hole per unit cell in the subspace defined by equation (1). This hole is loc- alized in the Cu 3d shell (~d < ep) giving rise to Cu 2+ and 02- . When holes are added it is energetically favourable for these holes to go to the O 2p states. This is a consequence of the large intra-atomic Coulomb repulsion on the Cu3d shell (Ud > ep - ed). If this is the case, each added hole produces an O - . With electron doping, some holes are removed from the planes, and then each electron produces a Cu ÷ . This ionic picture makes evident the difference in the nature of the states, which are occupied by electrons or holes. When the hopping term is included, the description becomes much more complicated.

In what follows, we present exact results obtained by the numerical diagonalisation of a small CuO2 clus- ter. The cluster studied is shown in Fig. 1. In order to

Vol. 72, No. 8

n

101)

minimise finite size effects, we have worked using dif- ferent boundary conditions: periodic and antiperiodic along x and y directions. The results of Fig. 2 corre- spond to a superposition of spectral densities obtained with the four boundary conditions.

The ground state and wave functions were calcu- lated using a modified Lancz6s method and the one particle spectral densities were obtained using the method described in Ref. [14].

The one particle spectral densities are defined as the Fourier transform of (~bolfi(t)Ci+(O)l¢o) or (~bolf~+(t)Ci~(O)l¢o> where I~k0) is the ground state wave function and c~ (t) are the creation operators in the Heisenberg representation. These spectral den- sities are given by:

a,+(w) - ~ I(~lC,~l¢o>l=~(w - E, + Eo) a #

a~-(w) - ~, I<vlf,ol¢o>126(w - E, + go) try

and in our finite clusters we replace the f-functions by Lorenzians of width ?. The function Gi + (w) gives the

Fig. 2. One particle spectral densities for the stoi- chiometric case for different values ofe = 0.25 (a), 0.5 (b), 1 (c) and 2(d). The other parameters are Ua = 2Up = 8t and V = t. The full (dashed) line is the partial density for O(Cu).

Vol. 72, No. 8

1

©

ONE PARTICLE SPECTRAL DENSITIES FOR C u O 2

0.5 o ~ E/t

Fig. 3. The Cu-hole-occupation as a function of 8It. Other parameters are the same as in Fig. 2.

XPS spectrum while Gi-(w) is associated with the inverse photoemission spectra.

The total one-particle spectral densities are obtained by combining G + and G - . We define

Oi(W) = G i-(W) + Gi +( -W)

which for a non-interacting system gives the local density of states for electrons. The Fermi energy is obtained as an energy E r for which:

Gi (w) = 0 f o r w < EF

and

G i + ( - w ) = 0 f o r - w > EF.

The results obtained for stoichiometric systems are shown in Fig. 2 for Ua = 2Up = 8t, V = t and different values of ~ = (ep - Ca)~2. The values chosen for Ud, Up and V are consistent with recent estimates [8].

For large e the system is clearly a semiconductor with a gap which is of the order of 2e i.e. a charge transfer insulator. At e decreases, the gap decreases and eventually disappears although with this small cluster calculation, it is not possible to give correctly the detailed behaviour of the gap.

Moreover, at zero temperature, in the thermo- dynamic limit it is possible that even for small e the system has a non zero gap, due to long range magnetic order. This gap, however, should disappear at the Nrel temperature in contrast with the results obtained for large or intermediate values of e where the gap is almost independent of the spin configuration.

The total occupation of the Cu-3d orbital as a function of e is shown in Fig. 3. Clearly as e increases, the system becomes less covalent and Cu ions become more magnetic. We can summarize the results pre- sented for the stoichiometric systems in the following way:

765

(i) for intermediate or large values of e, the strongly correlated CuO2 planes are charge transfer insulators

(ii) In this parameter range the gap is of the order of A = 2e and is not very sensitive to the spin con- figuration. We have checked this by calculating the spectral densities for a cluster with total spin one.

(iii) as the charge transfer energy e decreases, the system becomes more covalent and the gap tends to disappear. This behaviour supports the idea of a metal-insulator transition as e decreases [15, 16].

The next question concerns the problem of doping. In the small clusters we studied we can not change continuously the number of carriers. Adding or sub- stracting a hole in the cluster corresponding to a large dopand concentration (x = 0.25) in L2_xAxCuO4. For the prototypes of hole and electron doping, L and A and La are Sr or Nd and Ce respectively.

We calculate the XPS spectra assuming that the energy of the X-rays is such, that electrons are photo- emitted from the Cu 3d shell and neglecting the energy dependence of matrix elements. In this simple approxi- mation, the XPS spectra is simply given by Gi + ( - w) for i being a Cu site. Experimentally, as noted by Fujimori et al. [5] the photoemission spectra at most energies should be dominated by the Cu 3d states, since the O 2p photoelectron cross section is much smaller than the Cu 3d cross section. Similar behav- iour is also obtained in UPS which resembles XPS more closely at higher photon energy [17].

In Figure 4, we present the results of stoichio- metric, hole and electron doped systems. In the figure we have shifted the frequency to measure the energies from the Fermi level (Er = 0).

Clearly, the effect of hole doping is just a shift of the spectrum towards the Fermi energy as observed experimentally. For the parameters used in the calcu- lation, all the features obtained for the stoichiometric sample are also observed in the doped compound. The reason is simply that when adding holes, these holes are formed in the oxygen derived band, thus lowering the Fermi energy. As a first approximation, there are only little change on the Cu-states which are almost Cu 2+ -d 9. In contrast, decreasing the number of holes (electron doping) produces important changes in the spectrum. The most pronounced feature being a large peak at EF. This density of states at the Cu sites is a consequence of the increase in the probability of find- ing Cu + . Now when electrons are added they go into the states with predominant Cu character (Cu ÷-d a°) which are situated at an energy ,,, A = (ep - 8d)/2 + V, above the oxygen derived states previously occupied in the stoichiometric case, A being the charge transfer

766 ONE PARTICLE SPECTRAL DENSITIES FOR CuO2 Vol. 72, No. 8

b

¢..

c

-150 -10.0 -5,0 0,0 50

Fig. 4. One particle Cu-spectral density (XPS part - see text) for stoichiometric (b), hole (a) and electron (c) doped systems; ~ = t and the other parameters are the same as in Fig. 2. The Fermi energy has been set equal to zero in the three cases.

energy i.e. the energy to transfer an electron from O to Cu, producing Cu+-O - from Cu2+-O =. Although the details of the spectrum may depend on the values of the parameters, the general behaviour should not be very sensitive to them, if we remain in the regime (~p - ed) < Ud. Consequently, if in the new materials the CuO2 planes are really doped with electrons, as the X-ray absorption measurements seems to indicate [18], the XPS spectrum should clearly reflect this fact. In this experiment, the signature of the electrons added to the CuO2 planes is a large peak at the Fermi energy.

Acknowledgements - One of us (C.A. Balseiro) acknowledge financial support of Institut Laue Langevin, Grenoble, France, where part of the work has been done

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