Tuning the in-plane electron behavior in high-T
c
cuprate superconductors via apical atoms:
A first-principles Wannier-states analysis
Wei-Guo Yin
1,
* and Wei Ku
1,2,†
1
Department of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, New York 11973, USA
2
Department of Physics, State University of New York, Stony Brook, New York 11790, USA
Received 7 April 2009; published 12 June 2009
Using a recently developed first-principles Wannier-states approach that takes into account large on-site
Coulomb repulsion, we derive the low-energy effective one-band Hamiltonians for several prototypical cuprate
superconductors. The material dependence is found to originate primarily from the different energy of the
apical atom p
z
state. Specifically, the general properties of the low-energy hole state, namely, the Zhang-Rice
singlet, are significantly modified, via additional intrasublattice hoppings, nearest-neighbor “super repulsion,”
and other microscopic many-body processes. Implications on modulations of local pairing gaps, charge distri-
bution, hole-hopping range, electron-phonon interaction, and multilayer effects in cuprate superconductors are
discussed.
DOI: 10.1103/PhysRevB.79.214512 PACS numbers: 74.72.-h, 71.10.-w, 74.25.Jb, 74.40.+k
I. INTRODUCTION
The origin of high-T
c
superconductivity HTSC remains
under fierce debate notwithstanding monumental effort for
two decades. Although it is generally agreed that the most
relevant electron behavior is confined within the common
CuO
2
plane,
1–8
T
c
max
T
c
at optimal doping is strikingly af-
fected by modulation of the layering pattern along the less
essential third direction.
9–12
Hence, clarifying the material
dependence of the in-plane electron behavior, especially with
material-specific first-principles approaches, is an essential
step toward the resolution of the HTSC mechanism and the
quest for higher-T
c
superconductors.
A recent influential advancement along the line was made
by Pavarini et al.:
12
within the local-density approximation
LDA of density-functional theory DFT for a number of
hole-doped cuprates, they derived a one-band noninteracting
Hamiltonian in which |t' / t|t and t' are the in-plane first-
and second-nearest-neighbor hopping integrals, respectively
appeared to correlate with T
c
max
and was related to out-of-
plane apical atoms. Taken as the kinetic part of the effec-
tive one-band t-J or Hubbard model, the most studied model
for the CuO
2
plane,
1–7
this single band has been widely used
to compare with angular-resolved photoemission spectros-
copy ARPES.
8,13,14
A particular puzzle thus caused is that the LDA |t' / t|
=0.18 for La
2
CuO
4
and 0.12 for Ca
2
CuO
2
Cl
2
Ref. 12 con-
tradict with their diamondlike and squarelike Fermi surfaces,
respectively, as observed in ARPES.
8,13,15
In addition, the
suggested t' effect on T
c
Refs. 10–12 turned out to be con-
troversial in several numerical studies of the extended t-J or
Hubbard model.
16–18
More importantly, all LDA derived one-
band metallic picture suffers from a fundamental deficiency
of single-particle pictures, namely, the doped hole residing in
Cu d orbital that hybridizes with O p. This drastically con-
tradicts with the well-established experimental fact that
doped holes reside primarily in the oxygen atoms.
19–21
In
fact, it is now commonly accepted that the doped holes form
spin-singlet states with the intrinsic holes in Cu, the so-called
Zhang-Rice singlet ZRS, in the underdoped region—it is
based on ZRS that the one-band t-J or Hubbard model has
been justified.
1–7
Thus, it is important to examine the mate-
rial dependence of general electron behavior in the CuO
2
plane by deriving an effective interacting Hamiltonian with a
first-principles approach that takes into account, from the
beginning, large Coulomb repulsion on the Cu sites. Such
studies would not only produce a more appropriate character
of low-energy states but also reveal new material-dependent
physical effects derived from the charge-transfer nature and
the complexity of strong electronic correlation.
In this paper, we address this issue by extending the re-
cently developed first-principles Wannier-states WSs
approach
22,23
to low-energy scale 1 eV. The material de-
pendence is found to originate primarily from different
Pz
,
the energy of the apical atom p
z
state, in accord with the fact
that the apical coordination of Cu ions is the main structural
variation relevant to the CuO
2
plane.
24
In the present work,
the entire range of
Pz
approaching the Fermi level E
F
from
infinity is represented by four prototypical cuprates:
Nd
2
CuO
4
, Sr
2
CuO
2
F
2
, Ca
2
CuO
2
Cl
2
, and La
2
CuO
4
, where the
apical sites are empty, F, Cl, and O, respectively. The prox-
imity of
Pz
to E
F
is found to suppress |t' / t| of ZRS, resolv-
ing the aforementioned puzzle in existing first-principles re-
sults on Ca
2
CuO
2
Cl
2
. More interestingly, this proximity
generates several crucial many-body effects, including local
site-dependent potentials and a remarkable intersite “super-
repulsion,” which directly modulate local pairing gaps and
charge distribution, and imply an intriguing realization of
electron-phonon coupling. These findings would shed light
on the general material dependence and microscopic under-
standing of HTSC.
II. METHODS
A three-step approach is used to reduce systematically the
energy scale of the relevant Hilbert space:
A. At full-energy scale
The full-energy electronic structures of the undoped sys-
tem are obtained within the LDA+ U method with U
PHYSICAL REVIEW B 79, 214512 2009
1098-0121/2009/7921/2145129 ©2009 The American Physical Society 214512-1