PhysicaC 216 (1993) 12-16
North-Holland
Electronic structure of mercury-based high- TCcompounds:
H@aKan-ICu,%+2 (n= L&3)
C.O. Rodriguez
IFLYSIB,
Grupo Fisica de1 S&do, C.C. 565, La Plata (1900). Argentina
N.E. Christensen
and E.L. Peltzer y Blanch
Institute of Physics and Astronomy, Aarhus University, DIM000 Aarhus, C, Denmark
Received 14 July 1993
A fist principles determination of the electronic structure of the tetragonal mercury-based compounds containing one Hg plane,
HgBarCuO, (Hgl201), HgBa2CaCua06 (Hgl212) and HgBaaCarCusOs (Hgl223), has been made within the local-density approximation using the full potential linear muffin tin orbital method. The band structures exhibit overall features similar to those
of other cuprate superconductors with a marked peculiarity: an antibonding band coming mostly from the hybridization of Hg - p
andd and apicaloxygen (02) -p states whichlies slightly above the Fermilevel for Hgl201 and below it for Hgl212 andHg1223,
but whose exact position depends on the relative z-position of apical 02 atoms within the unit cell. This detail is important to
establish the degree of self-doping in these compounds.
1. Introduction
The report by Putilin et al. [ 1] of superconductivity at 94 K in HgBazCu04+, stimulated considerable interest in Hg-based systems. The crystal
structure of Hgl20 1 is multi-layered, each unit containing one copper oxide layer. Experience with
analogous copper oxide compounds based on thallium or bismuth showed that the transition temperature increased as the number of copper oxide layers
n in the unit cell increased from n= 1 to 3. A similar
effect was expected for the mercury family with more
than one copper oxide layer. &hilling et al. [2] reported magnetic and resistivity measurements confirming a maximum critical temperature of 133 K in
a material containing HgBazCazCu908+,, HgBazCaCuzOs+,., and an ordered superstructure composed of a sequence of the unit cells of these two
phases. The highest-T, superconducting phase has not
yet been identified.
In this paper we present a fist principles determination of the electronic structure for the stoichiometric structures for the three members of the family Hg1201, Hg1212 and Hg1223 within the local
0921-4534/93/$06.00
density approximation (LDA) using the full potential linear muffin tin orbital (FP-LMTO) method
[ 3 1. Structural properties and a detailed study of the
electronic structure of Hgl201 have already been reported [ 4 1, and this serves as a basis for the analysis
in the present study. One can compare the behavior
of the mercury family HgBaZCan_-ICunOZn+Z
to that
of TlzBazCa,_ 1C~n02n+4,n= 1, 2, 3 (notice that, as
pointed out in ref. [51,the formal valence of two
T13+02- layers is the same as that of a single Hg2+
layer). The major difference lies in the oxygen stoichiometry of the HgO, and TlO,_, layers, respectively (x=0 for the studied stoichiometric compounds). From theoretical predictions in such Tlbased materials [ 6-8 ] a Tl-0 derived band dips below the Fermi energy (&) for the entire series making the behavior of all these materials self-doped. The
minimum of the “dopand” band is 0.14, 0.20 and
0.25 eV below EF,for the one-, two-, and three-layer
materials, respectively. A similar behavior cannot be
ruled out for the Hg family, since a hybridization of
Hg-states with the apical oxygens and Ba may develop, so as to make this Hg-0 derived band cross
or stay close to Ep The structural and internal struc-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
CO. Rodriguez et al. /Electronic structureo/HgBa2Ca,_ ICun02n+l
tural parameters of Hgl201 are well established
[ 1,9]. In this case, the Hg-0 derived band lies
slightly above EF (0.13 and 0.09 eV at X and R, respectively ) . The precise character of this band, where
it is close to EF, is Hg( p,)-02 (p,), whereas it is
mainly Hg( dZ2)-02(p,) in the rest of the Brillouin
zone. A structural model was set up for Hg12 12 and
Hg1223, based on the relative positions of atoms in
the infinite-layered compound CaCuO, [ lo] and
those of Hg 120 1 [ 1,9], scaled according to the experimentally known lattice parameters. This allowed
a systematic study of the band structure to be made
and in particular made it possible to determine the
position of the Hg-0 derived band for the three
members of the Hg family, and its dependence on
the z-position of apical 02 atoms.
Using our model structures for Hg1212 and
Hg1223, the band minimum lies below EF at the X
point by an amount of 0.09 and 0.10 eV, respectively. The values at the R point are very similar. We
have also used for Hg1223 the known structure of
the analogue, T1Ba2CazCujOg compound [ 111 substituting the TlO plane by Hg, resealing for the experimental c and a lattice parameters and found the
Hg-0 band crossing below EF at 0.146 eV. We have,
however, studied the variations of that particular
band for small changes of the 02 z-position, and
found minor modifications of the dipping relative to
EF. This means that variations from our model
structure of the 02-Hg distance will not modify the
qualitative picture of the band structure presented in
this paper for Hg1212 and Hg1223. The question as
to the degree of the self-doping for the two- and threelayered compounds can only be answered when the
actual crystal structures are known.
13
eluded explicitly for 15 6, inside the MT spheres. The
charge density in the interstitial region is obtained
by matching a linear combination of atom-centered
Hankel functions with II 6 and for two different rc2
values (formal envelope kinetic energy [ 12 ] ) , to the
values and slopes of the density on the sphere. The
“tails” with I> 6 of the Hankel functions extend into
the spheres. The final density is then continuous and
smooth and includes angular momentum terms with
1 up to infinity. Poisson’s equation can then be solved
analytically for the interstitial region and by numerical integration inside the spheres.
For the contribution from the interstitial potential
to the matrix elements, the product of two Hankel
functions is fitted in the interstitial region by a linear
combination of Hankel functions as for the charge
density [ 13 1. The basis functions are defined using
a triple+ set. The choice of kinetic energies IC’,associated with the envelope Hankel functions is not
critical and we have taken them to be -0.25, - 1.0
and -3.0 Ry. The core electrons are allowed to relax; the core charge density is recalculated at each
iteration in the self-consistency loop. All calculations
were done within the LDA using the Barth-Hedin
parametrization of the exchange-correlation energy
and potential [ 141. The basis set included 19 1, 256
and 276 LMTO’s per cell for the one-, two- and threelayer compounds, respectively. To obtain the desired
precision it was necessary to use 84 points plus linear
tetrahedral interpolation in the irreducible Brillouin
zone of the tetragonal structure. The calculations were
carried out in two panels including the following orbitals as semicore: Cu (3s and 3p), Ba (Ss), Hg (5~).
3. Results
2. Method of calculation
We’ have used the FP-LMTO technique which
makes no shape approximation for the charge density or for the potential [ 31. Space is divided into
non-overlapping atomic spheres and an interstitial
region (muffin-tin (MT) geometry). The basis consists of MT orbitals, which are Hankel functions outside the MT spheres, augmented inside by numerical
solution of Schrijdinger’s equation [ 12 1. Spherical
harmonic components of the charge density are in-
The tetragonal structures of Hg 120 1, Hg 12 12 and
Hg1223 contain (parallel to the xy plane) n=l, 2
and 3 CuOZ planes, respectively, and n - 1 Ca atoms
are located above the centres of Cu02 squares. These
sandwiches are bounded by a pair of BaO planes.
Oxygen atoms in the Ba planes will be called 02. Hg
ions then separate these basic layered sheets. 02 is
directly on top of Cu and the position of Hg is directly above 02. The lattice parameters are:
az3.8797,
3.93, 3.93 A and ~~9.5087, 12.7, 16.1 8,
for Hg1201, Hg1212 and Hg1223, respectively. The
C.O. Rodriguez et al. /Electronic structure of HgBa2Ca,,_ IC~nOZn+2
14
position of the atoms for Hg12 12 and Hg 1223, based
on our model structures are given in ref. [ 15 1. The
band structure of Hg1201 for the experimental lattice constant is shown in fig. 1. Figures 2 and 3 correspond to the band structures of Hg1212 and
Hg1223.
The dispersion in the z direction can be seen by
comparing left and right parts of figs. l-3 which are
joined by the T-Z line. The left part shows the bands
at k,=O in different directions between the high
symmetry point r, M and X. The right part depicts
the corresponding symmetry points with kpn/c:
Z,
Fig. 3. Band structure for Hg1223.
-6
r
M
X
r
z
A
R
2
Fig. 1. Band structure for Hg1201. The symmetry points are
r=(O,O,O),M=(l,
1,O)r/aandX=(l,O,O)n/a,Z=(0,0,
1)x/c, A= (1, 1, a/c)rr/a and R= (1, 0, a/c)x/a, (a, care the
tetragonal lattice constants).
r
z
“,x
I
0
I
7
0
x
E!?
3
-2
-4
-6
4
: /
--ny
~
~
-IOl-
M
X
rZ
I
~
A
Fig. 2. Band structure for Hg1212.
R
R
Fig. 4. Intersection of the LDA Fermi surface of Hg12 12 with the
upper half of the Brillouin zone boundaries. The dashed line shows
the Fermi surface in the nearest-neighbor two-dimensional tightbinding model for half-tilling.
4
5
I
Z
Fig. 5. Same as tig. 4 but for Hg1223.
CO. Rodriguez et al. /Electronic structure of HgBa,Ca._ IC%&+J
A, R. Figures 4 and 5 show the Fermi surface (FS)
cross-sections of Hg12 12 and Hg1223 in the symmetry planes (the “box” laid out). Again, the dispersion in the z direction can be observed by the difference of crossings in the two squares. The band
structure is highly two-dimensional. Our band structures agree for Hg1201 and Hg1223 with recent calculations by Singh [ 5,17 ] using the full potential linear augmented plane wave (FP-LAPW). The FS of
Hg 120 1 has only one sheet denoted by (I as shown in
fig. 2 of ref. [ 41, but for the case of the two and three
layer compound three and four sheets appear, denoted by ul, u2, b and a,, a,, u3 and b, respectively.
The a, (n= 1, 2 or 3) FS-sheets come from linear
combinations of the n Cu02-derived bands characteristic of the cuprate superconductors. Each CuO,
band can be understood as the antibonding pdo band
from a simple tight-binding model, in the nearestneighbor approximation, which includes 0 1 (p,),
Cu(dx2_,,2 ), 01 (p,,). The corresponding two-dimensional FS for the case of half filling is a square,
with saddle points (van Hove singularities) at the
points k=X and Y exactly at the Fermi level and
would have “perfect nesting”. Both of these features
disappear when next-nearest neighbor hopping between 01 (p,) and (p,) are included together with
a weak hybridization of these orbitals with Cu (s) and
01 (s). The shape of the FS is more like a square
turned 45” with respect to that of the simplest tightbinding model. This is very similar to what happens
in YBa2Cu30, (YBCO) (see ref. [ 16 ] ). In the case
of Hg1201 the saddle points in the u sheet lie far at
0.52 and 0.37 eV below EF, at X and R respectively,
but get closer to EF in the other two compounds due
to the interaction between the planes, which also
causes the interaction with the Hg-0 derived band
to increase. For the case of Hg1201 a band sets very
close to E,:O. 13 and 0.09 eV at points X and R, respectively. It is also highly two-dimensional. This
band is mainly derived from the Hg-apical 02 interaction, and this hybridization is such as to make
this band lie close to the Fermi level. The position
of this band changes and even crosses EF in both
Hg12 12 and Hg1223 for our proposed structural
models. This brings out the b FS-sheet in figs. 4 and
5. A finer analysis of the character of this band for
Hg1223 shows that from M to X and from X to XT/
2 is Hg(p,)-02(p,)-Ol(p,)
and Ba(d,*). The
15
character changes in the region I to M, XI-/2 to F,
where the states are predominantly Hg (d,z)-02 (p,).
Similar behaviors have the corresponding states with
k,= n/c due to the highly two-dimensional character
of these states.
The corresponding band for Hgl201 and Hg 12 12
shows very similar behavior. Since the precise atomic
coordinates are not yet known for the two- and threelayered compound, we have studied the position of
the Hg-0 derived band as 02 atoms were moved
closer to and away from the Hg plane. As already
mentioned the dipping was slightly modified. The
deformation potential is such that changes in the zposition of 02 from our model atomic positions cannot make the minimum of this band lie on top of EF.
4. Conclusions
We have presented a detailed study of the electronic structure of Hg1201, Hg1212 and Hg1223
within the LDA using the FP-LMTO method. A description of the character of the states close to and
at the FS was given. We have seen that due to the
small dependence of the position of the Hg-0 derived band with the z-position of the 02 atoms, only
the question to the degree of self-doping in Hg1212
and Hg1223 will remain open until the precise structure is known.
Acknowledgements
We have benefited from the interaction with W.E.
Pickett and D.J. Sir@. This work was supported by
Consejo National de Investigaciones Cientificas y
Ttcnicas de la Republica Argentina, by the Commission of the European Communities, Contracts no.
S/CIl*-913141 and CIl*-CT92-0086 and by the
Danish Research Council under grant No. 11-96851SE.
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16
C. 0. Rodriguezet al. /Electronic structureof HgBaJCa._D.O2,,+2
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[ 151The atomic z-coordinates are z(Ca)=O, r(Cu)=0.1245,
z(O1)=0.1245,
z(Ba) =0.2763,
2(02)=0.3453,
z(Hg)=0.5
for Hg1212 and z(Cul)=O,
z(Oll)=O,
z(Cu)=O.1993,
z(O1)=0.1993,
z(O2)=0.3771,
z(Ba)=0.3271, z(Hg)=0.5 for Hg1223, in units of the
corresponding c lattice parameter.
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O.K. Andersen, J. Comp. Phys., to be published.
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