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166 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
with respect to that at the oxygens in the planes [147]. Equally interesting, X-
ray absorption spectroscopy on single crystals of infinite-layer compounds of Ca
doped YBCO compounds have shown that superconductivity might appear in
this compound only when holes are present at the apical oxygen sites [148].
Going also in this direction, it was argued recently that upon hole doping,
the static charge attraction between the apical oxygens and the d orbitals could
lead to a reduction of this distance [149]. Theoretically, it was shown by the
first-principles variational calculations of the spin-density-functional approach
that the optimized distance between apical O and Cu in La 2−x Sr x CuO 4 which
minimizes the total energy decreases upon Sr doping. As a result, the elongated
CuO 6 octahedrons by the Jahn-Teller interactions, shrink by doping holes. This
shrinking effect was called anti-Jahn-Teller effect.
We propose to address the issue of the presence of apical oxygen, in the context
of circulating orbital currents, and we use the same variational frame-work as
developed in the previous section. We consider the extended 6-band Hubbard
model including the two additional apical oxygens surrounding a copper atom
and also the additional d 3z2−r2 orbitals:
H = ∑ ɛ α n m,ασ + ɛ p + ∑ n k,aσ +
∑ (
t αp d
†
m,ασ p iσ + c.c. ) +
m,ασ
k,σ
〈m,i〉,ασ
∑ (
t za d
†
m,zσ a kσ + c.c ) ∑ (
)
+ t pp p † iσ p ∑ (
)
jσ + c.c. + t pa p † iσ a kσ + c.c. +
〈m,k〉,σ
〈i,j〉,σ
〈i,k〉,σ
∑
∑
∑
U d n m,α↑ n m,α↓ + U p n i,p↑ n i,p↓ + U a n k,a↑ n k,a↓ + ( U xz − 1J ) ∑
2 xz n mx n mz +
mα
i
k
m
∑ (
)
J xz d † m,x↑ d† m,x↓ d ∑
m,z↓d m,z↑ + c.c. − 2J xz s mx · s mz + ∑ U αp n mα n ip +
m
m
〈m,i〉
∑
U αa n mα n ka
〈m,k〉,α
(6.22)
Here the d † mxσ and d† mzσ creates a hole respectively in the d x2−y2 and d 3z2−r2
orbitals; p iσ refer to the orbitals lying inside the plane, and a iσ refer to the apical
oxygen orbitals. Furthermore, S mx and S mz are spin operators for the d x2−y2 and
d 3z2−r2 orbitals. The Hamiltonian contains inter-orbitals Hund’s coupling J xz
which reduces the on-site Coulomb repulsion U xz . The first three terms specify
the reference hole atomic energies. The hopping elements t αp (α = x, z), t za , t pp ,
t pa stand for the copper-oxygen-in-plane, copper-Apical-oxygen, oxygen-oxygenin-plane,
oxygen-in-plane-Apical-oxygen hoppings respectively. The on-site intraorbitals
Coulomb repulsion are U d , U p and U a respectively, while the inter-orbital
Coulomb and exchange energies at copper sites are U xz and J xz . Only the most
important nearest neighbor Coulomb repulsions U dp and U da were considered.
Moreover, we consider a set of realistic parameters [145, 147, 150, 151]: