# Jahn–Teller mechanism of stripe formation in doped layered La2 − xSrxNiO4 nickelates

**ABSTRACT** We introduce an effective model for eg electrons to describe quasi-two-dimensional layered La2 − xSrxNiO4 nickelates and study it using correlated wavefunctions on 8 × 8 and 6 × 6 clusters. The effective Hamiltonian includes the kinetic energy, on-site Coulomb interactions for eg electrons (intraorbital U and Hund's exchange JH) and the coupling between eg electrons and Jahn–Teller distortions (static modes). The experimental ground state phases with inhomogeneous charge, spin and orbital order at the dopings x = 1/3 and 1/2 are reproduced very well by the model. Although the Jahn–Teller distortions are weak, we show that they play a crucial role and stabilize the observed cooperative charge, magnetic and orbital order in the form of a diagonal stripe phase at x = 1/3 doping and a chequerboard phase at x = 1/2 doping.

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**ABSTRACT:**We review briefly several approaches used to investigate the stability of stripe phases in high temperature superconductors, where charge inhomogeneities arise from competing kinetic and magnetic energies. The mechanism of stripe formation, their consequences for the normal state and enhancement of pairing interaction triggered by charge inhomogeneities are briefly summarized. Finally, we demonstrate that orbital degeneracy ($i$) leads to a more subtle mechanism of stripe formation, and ($ii$) plays an important role and determines the symmetry of the superconducting state in pnictides.05/2012;

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arXiv:1112.3445v1 [cond-mat.str-el] 15 Dec 2011

Jahn-Teller mechanism of stripe formation

in doped layered La2−xSrxNiO4nickelates

Krzysztof Ro´ sciszewski1and Andrzej M. Ole´ s1,2

1Marian Smoluchowski Institute of Physics, Jagellonian University,

Reymonta 4, PL-30059 Krak´ ow, Poland

2Max-Planck-Institut f¨ ur Festk¨ orperforschung,

Heisenbergstrasse 1, D-70569 Stuttgart, Germany

E-mail: krzysztof.rosciszewski@uj.edu.pl; a.m.oles@fkf.mpg.de

Abstract.

dimensional layered La2−xSrxNiO4 nickelates and study it using correlated wave

functions on 8 × 8 and 6 × 6 clusters. The effective Hamiltonian includes the kinetic

energy, on-site Coulomb interactions for eg electrons (intraorbital U and Hund’s

exchange JH) and the coupling between egelectrons and Jahn-Teller distortions (static

modes). The experimental ground state phases with inhomogeneous charge, spin and

orbital order at the dopings x = 1/3 and x = 1/2 are reproduced very well by the

model.Although the Jahn-Teller distortions are weak, we show that they play a

crucial role and stabilize the observed cooperative charge, magnetic and orbital order

in form of a diagonal stripe phase at x = 1/3 doping and a checkerboard phase at

x = 1/2 doping.

Published in J. Phys.: Condens. Matter 23, 265601 (2011).

We introduce an effective model for eg electrons to describe quasi-two-

PACS numbers: 75.25.Dk, 75.47.Lx, 75.10.Lp, 63.20.Pw

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 2

1. Introduction

The nature and origin of stripe phases in transition metal oxides continuesly attracts

considerable attention in the theory of strongly correlated electrons. The phenomenon

of stripes occurs in doped materials and, as one of few developments in the physics

of superconducting cuprates, was discovered first in theory [1] before their existence

was confirmed by experimental observations [2]. In a system with dominant electron-

electron interactions novel phases with charge order in form of a Wigner crystal, or stripe

phases with nonuniform charge distribution are expected. In the well known example of

the cuprates charge order coexists with the modulation of antiferromagnetic (AF) order

between alternating domains [3,4]. The stripe phases in the realistic models for cuprates

were obtained, inter alia, by calculations using Hartree-Fock (HF) approximation [1],

correlated wave functions [5], dynamical mean-field theory [6] and slave bosons [7].

These studies have shown that correlation effects play an important role in the physical

properties of stripes.

The physical origin of stripes becomes more complex in systems with active orbital

degrees of freedom, as in the doped manganites with egorbital degrees of freedom [8–10],

and it was soon recognized that the mechanism of stripes has to involve charge and

magnetic order coexisting with certain type of orbital order [11,12]. The mechanism of

stripe formation in such systems is subtle and involves directional hopping between the

orbital wave functions, in particular for t2gorbital states [13], that could play a role in

the properties of doped iron pnictides [14].

In contrast to the stripes in cuprates [3], the stripes in nickelates are along diagonal

direction in the square lattice [15]. This by itself is puzzling and suggests that other

degrees of freedom may contribute. The properties of doped perovskite nickelates are

still not completely understood in spite of considerable effort put forward both in theory

[12,16–21] and in experiment [22,23]. The main difficulty in the theoretical description

of this class of compounds is related to simultaneous importance of numerous degrees

of freedom. Unfortunately, all of them contribute and one can not argue that only some

types of the interactions are essential and could be considered in a simplified approach,

while the others are of secondary importance and would be responsible for quantitative

corrections only. We argue that attempts to consider only some interactions, for instance

either including only strong on-site Coulomb and Hund’s exchange interactions and

neglecting the coupling to the lattice via the Jahn-Teller (JT) distortions, or studying

the coupling to the lattice distortions in absence of strong local Coulomb interactions,

are both not sufficient to account for the experimental situation. The phase situation

in this class of compounds is a result of a subtle balance between numerous factors.

On the theoretical side, the multiband models and/or approaches based on the ab

initio local density approximation (LDA) computations extended by static corrections

due to local Coulomb interaction U (within the LDA+U scheme) [19,21], seem to be the

most realistic and complete approaches to theoretical description of the nickel oxides

with a perovskite structure. The drawback of such approaches, however, is considerable

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 3

technical difficulty to work with doped systems especially for arbitrary doping levels x

because in such a case very big unit cells (or clusters) need to to considered.

Therefore it is quite helpful (and complementary to LDA+U approaches) to identify

first all physical mechanisms which are essential in doped nickelates and then develop a

simplified but still well motivated effective model to describe them. In the present paper

we use such an effective model (featuring only Ni sites renormalized by the presence of

surrounding oxygens) for the description of itinerant eg electrons which includes all

essential interactions present in layered nickelates. We do not develop new concepts

here but use the model well tested before in the manganite perovskites, including

also monolayer and bilayer systems [24]. Further justification of this model follows

from similar microscopic approaches developed earlier to describe doped nickelates

[11, 12, 16, 18]. With this microscopic model we investigate the nature and type of

coexisting magnetic, orbital and charge order in the ground state when local electron

correlations and the coupling to JT distortions are both included. In this paper we focus

on quasi two-dimensional (2D) monolayer nickelates La2−xSrxNiO4.

The paper is organized as follows.First, we introduce in section 2 a realistic

model for egelectrons in nickelate which includes the electron interactions and the local

potentials due to static JT distortions. In this section we also introduce the method

to treat electron correlation effects beyond the HF approximation, and discuss realistic

values of model parameters. The results obtained for various doping level are presented

and analysed in section 3. Next we concentrate on the role played by the JT distortions

(in section 4) and show that they are crucial for the stability of the stripe phase at the

x = 1/3 doping. Finally we present a short summary in section 5 as well as some general

conclusions.

2. The microscopic model and methods

2.1. The effective model Hamiltonian

We

La2−xSr1+2xNiO4(with doping 0 < x < 0.5), using an effective model describing only

Ni sites, where the state of surrounding oxygens is included by an effective potential at

each site. (A realistic effective Hamiltonian which acts in the subspace of low energy eg

states and the precise values of the Hamiltonian parameters can be derived by a pro-

cedure of mapping the results of HF, or LDA+U, or all-electron ab initio calculations

obtained within a more complete approach). A local basis at each nickel site is given

by two Wannier orbitals of egsymmetry, i.e., x2− y2and 3z2− r2orbitals.

In the present study we use a Hubbard-type Hamiltonian H for two egorbital states

defined on a finite cluster:

investigate stronglycorrelatedelectronsindopedmonolayernickelates

H = Hkin+ Hcr+ Hint+ Hspin+ HJT,

which consists of kinetic (Hkin), crystal field (Hcr), on-site Coulomb (Hint), spin (Hspin),

and JT (HJT) terms. The variants of this microscopic model were used to describe doped

(1)

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 4

monolayer, bilayer and perovskite manganites [24], where the same egorbital degrees of

freedom are present.

The kinetic part Hkinis expressed using two egorbitals:

|z? ≡ |(3z2− r2)/√6?,

per site, with anisotropic phase dependent hopping which depends on the orbital phases

on neighbouring sites,

Hkin= −1

{ij}||ab,σ

√3(d†

|x? ≡ |(x2− y2)/√2?.(2)

4t0

?

?

(3d†

ixσdjxσ+ d†

izσdjzσ)

±

ixσdjzσ+ d†

izσdjxσ)

?

.(3)

Here d†

site i. The {i,j} runs over pairs of nearest neighbours (bonds); ± is interpreted as plus

sign for ?ij? being parallel to the crystal axis a and minus for ?ij? being parallel with

to the axis b.

The kinetic energy is supplemented by the crystal field term which describes orbital

splitting

Hcr=1

2Ez

iσ

For the present convention and for negative values of crystal field parameter (Ez< 0)

the z orbital is being favored over the x orbital.

The Hintand Hspinstand for an approximate form of local Coulomb and exchange

interactions for electrons within degenerate egorbitals used before for monolayer, bilayer

and cubic perovskite manganites [24],

niµ↑niµ↓+ (U0−5

iµσare creation operators for an electron in orbital µ = x,z with spin σ =↑,↓ at

?

(nizσ− nixσ).(4)

Hint = U

?

iµ

2JH)

?

i

nixniz,(5)

Hspin= −1

2JH

?

i

(nix↑− nix↓)(niz↑− niz↓).(6)

On-site Coulomb interaction is denoted as U, the Hund’s exchange interaction constant

is JH. Here the spin symmetry is explicitly broken, the quantization axis is fixed in spin

space and the full Hund’s exchange interactions with SU(2) symmetry are replaced by

the Ising term. However, the above form suffices as it gives the same Hamiltonian in

the HF approximation as an exact expression for two egorbitals [25].

The simplified JT part HJTis:

?

HJT= gJT

?

i

Q1i(2 − nix− niz) + Q2iτx

i+ Q3iτz

i

?

+1

2K

?

i

?

2Q2

1i+ Q2

2i+ Q2

3i)

?

,(7)

where the pseudo-spin operators, i.e., {τx

τx

?

σ

i,τz

i} operators, are defined as follows

τz

?

σ

i=(d†

ixσdizσ+ d†

izσdixσ),

i=(d†

ixσdixσ− d†

izσdizσ). (8)

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 5

The Hamiltonian eq. (7) includes three different static JT modes, where {Q1i, Q2i,Q3i}

denote the JT static deformation modes of the i−th octahedron.

the harmonic constant of isotropic JT (breathing) mode Q1is assumed to be double

with respect to those corresponding to Q2 and Q3 unsymmetric modes as discussed

in refs. [8, 12]). Note that HJT taking the above form is a simplification: (i) first,

some anharmonic terms were omitted (compare for example the corresponding terms

in Ref. [26]), and (ii) second, all JT distortions are treated here as independent from

each other. On the contrary, in reality two neighbouring Ni atoms share one oxygen in

between them, and thus neighbouring JT distortions are not really independent. The

present approximation is the simplest one to describe the coupling to the local JT modes

in a model where oxygen ions are not explicitly included (for a more detailed discussion

see ref. [12]).

(For simplicity

2.2. Calculations within the Hartree-Fock approximation

We performed extensive calculations for clusters with periodic boundary conditions. A

systematic study of increasing doping from x = 0, through x = 1/8, 2/8, 3/8, up to

x = 4/8 was performed for 8×8 clusters, and was supplemented by doping x = 1/3 for

6 × 6 cluster. We investigate the ground state at zero temperature (T = 0).

First, the calculations within the single-determinant HF approximation were

performed to determine the ground state wave function. Also the gap between the

highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital

(LUMO), the HOMO-LUMO gap

∆ ≡ ELUMO− EHOMO, (9)

was extracted at this step. In the next step the HF wave function was modified to

include the electron correlations by employing local ansatz [27], see below.

Coming to details: in the first step HF computations were run starting from

one of several different initial conditions (on average a few thousands for each

set of Hamiltonian parameters are necessary as the identification of a true energy

minimum is difficult), i.e., from predefined (some symmetry fixed, but mostly random)

charge, spin and orbital configurations and several predefined sets of classical variables

{Q1i,Q2i,Q3i}. For each fixed set of starting parameters and starting initial conditions

we obtain on convergence a new HF wave function |ΨHF? which is a candidate for a

ground state wave function. This self-consistent procedure is assumed to provide energy

minimum also with respect to the classical {Q1i,Q2i,Q3i} variables [24].

2.3. Electron correlations

After completing the HF computations for a given wave function |Ψ?, we performed

correlation computations (second step) which provide the total energy. (For details see

refs. [5,24]). Namely the HF wave function |Φ0? was modified to include the electron

Page 6

Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 6

correlation effects. We used exponential local ansatz for the correlated ground state [27],

|Ψ? = exp

?

−

?

m

ηmOm

?

|Φ0?,(10)

where {Om} are local correlation operators. The variational parameters ηm(four singlet

and two triplet ones) are found by minimizing the total energy,

Etot=?Ψ|H|Ψ?

?Ψ|Ψ?

.(11)

Here for the correlation operators we use

Om=

?

i

δniµσδniνσ′, (12)

The sum in the above equation ensures smaller number of the variational parameters;

on the other hand the nonhomogeneous correlations are treated in an averaged way (this

procedure is somewhat similar to calculations performed within mean-field approaches).

The symbol δ in δniµσindicates that only that part of niµσoperator is included which

annihilates one electron in an occupied single particle state which belongs to the HF

ground state |Φ0?, and creates an electron in one of the virtual states. The above

local operators Om correspond to the subselection of most important two electron

excitations within the ab initio configuration-interaction method. We note that three,

four, etcetera, electron excitations (which are omitted here) are also important in the

strongly correlation regime. However, as yet there is no an easy way to implement them

for a larger systems, thus one is able to implement within theory only the leading part

of the correlation energy (which follows from two-particle excitations).

After obtaining the total energy for a given configuration, we repeat all the

procedure from the beginning, i.e., we take the second set of HF initial conditions

and repeat all computations to obtain the second candidate for a ground state wave

function. Other configurations for the third, fourth, etcetera, set of initial conditions

are investigated in a similar way. Finally, the resulting set of total energies was inspected

and the lowest one was identified as the best candidate for the true ground state. In

general, the correlations are found to be strong, in fact much stronger than those found

in the perovskite manganites [24].

2.4. Parameters of the model

The values of the Hamiltonian parameters used below follow closely the sets commonly

employed in the literature to study doped nickelates. For the effective d − d hopping

(ddσ) we assume t0= 0.6 eV, following ref. [18]. Local Coulomb interactions are strong,

and we assume 8 < U/t0< 12 (the lowering of the very large atomic value on nickel

is due to an appropriate screening of the egelectrons). Hund’s exchange coupling was

JH = 0.9 eV [18, 19]. The crystal field values we considered were either Ez = 0 or

Ez= −0.3 eV, with the latter value inducing a higher electron density in z orbitals, as

expected for a single NiO2plane (compare with refs. [12,18]).

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 7

Table 1. Parameters of the microscopic model eq. (1) used in most of the performed

calculations (all electronic parameters {t0,U,JH,Ez} in eV).

t0

UJH

Ez

gJT(eV˚ A−1)

0.67.0 0.9−0.3

K (eV˚ A−2)

13.0 3.0

Very little is known about JT interactions in nickelates, therefore the JT constant

K was fixed as K = 13 eV˚ A−2(like in manganite perovskites [24]), and the coupling

constant with the lattice distortions gJTwas assumed to be in the range 2 eV˚ A−1< gJT<

3.8 eV˚ A−1, following ref. [8].

The best set of the Hamiltonian parameters which allows one to reproduce the

experimental phase situation is given in table 1. In some cases, other parameter values

were used as discussed below. However, the parameters one should use are constrained

by the experimental observations. It is fortunate that the obtained types of spin, orbital

and charge order turn out to be rather sensitive on the parameter values. In fact, many

sets of the Hamiltonian parameters (different than those in table 1) were also considered

in test computations but they do not generate reasonable results (i.e., the ground states

with the ordering close to that reported in experiments). Although full phase diagram

would be very appropriate here, it is unfortunately too expensive to compute by the

present method.

3. Results

The computations were repeated for many sets of the Hamiltonian parameters and for

many (up to several thousands) charge, spin, orbital and JT distortions (i.e., for local

configurations required to start HF iterations). Extensive calculations were required to

establish the parameter values which are realistic for doped nickelates and allow one for

reproduction of basic experimental observations. Below we present the results obtained

for the realistic parameter set as given in table 1.

3.1. Low doping regime

The results obtained in low doping regime are presented in figures 1-2.

considered x = 1/8 doping. In La2−xSr1+2xCuO4 cuprates one finds at this doping

level stable stripe phases [3,5–7]. In the present case we observed isolated Ni ions with

holes doped in x orbitals, as shown in figure 1. At these ions the magnetic moments are

due to S = 1/2 spins and are therefore lower than magnetic moments at S = 1 sites,

thus appearing as spin defects in the G-AF phase. While both x and z orbitals are

occupied at undoped sites, one finds that only z orbitals are occupied at doped sites.

Of course, certain delocalization of electrons between the two sublattices takes place as

well in a doped antiferromagnet investigated here, but the above ionic picture applies to

First we

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Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 8

a good approximation. In fact, the obtained phase is insulating with a HOMO-LUMO

gap of 1.86 eV.

Local charge defects consist of z electrons occupying doped Ni3+sites which couple

to lattice distortions {Qi3}, as shown in the lower panel of figure 1. These distortions

act as self-trapping on the doped charges and suppress the kinetic energy of doped

holes. Therefore, this regime of doping is manifestly different from the observations

in cuprates, where JT distortions were inactive and self-organization of doped holes in

form of horizontal stripes takes place [1–3].

At a higher doping x = 1/4 the magnetic order changes to the C-AF phase shown

in figure 2, with relatively large magnetic cells. This phase is weakly insulating, with the

HOMO-LUMO gap ∆ = 0.28 eV. Horizontal ferromagnetic (FM) lines are characterized

by charge modulations between almost undoped Ni2+ions and doped Ni2+ions. The

latter doped sites are distributed far from one another both within FM chains and

between consecutive AF horizontal lines. This may be seen as a physical reason which

prevents the occurrence of stripes at this doping level. Furthermore, the doped sites

couple actively to local distortions, and can be identified by looking at the pattern of

JT distortions, see the lower panel in figure 2. It helps to identify imperfect long-range

order of doped holes which avoid each other, similar to the x = 1/8 doping considered

above.

3.2. Diagonal stripes at doping x = 1/3

The case of x = 1/3 doping was investigated with 6 × 6 cluster, see figure 3. Here the

number of doped sites is the same as the doping, i.e., x = 1/3, meaning that some of

doped sites have to be located at next-nearest neighbour positions. The resulting stripe

phase with diagonal lines of reduced charges (doped sites) may be therefore seen as a

modification of the C-AF phase obtained at the x = 1/4 doping by adding more doped

holes (Ni3+ions). Indeed, one finds that every third site in an AF row (FM column)

contains the reduced charge close to one added hole, and these sites form a (11) stripe

phase. Note that the horizontal and vertical lines are here interchanged as compared

with figure 2, but this configuration is of course equivalent to the one with FM horizontal

lines, and AF order between them. The large JT distortions accompany the sites with

minority charges, so the stripe pattern can be also recognized by analyzing them. It

reproduces the experimental results in refs. [22,23], with the long-range order coexisting

charge, spin and orbital order in form of diagonal stripes, present at the doping x = 1/3.

Also this phase is insulating, with a HOMO-LUMO gap ∆ = 0.57 eV.

We emphasize that the JT distortions play a crucial role in the observed stripe

phase at x = 1/3 doping. We have verified that when one switches off the JT coupling,

i.e., when one puts gJT= 0 with other parameters unchanged, the stripes vanish and the

ground state becomes conducting (HOMO-LUMO gap ∆ = 0.03 eV), see below. In this

case one finds the same magnetic phase with horizontally arranged FM lines coupled in

the vertical direction in the C-AF order, with slightly nonuniform charge distribution

Page 9

Jahn-Teller mechanism of stripe formation in doped layered La2−xSrxNiO4nickelates 9

Figure 1. Insulating G-AF (S=1) spin arrangement, with mostly uniform charge

distribution and equal electron densities within x and z egorbitals (no orbital order)

at undoped sites as obtained for x = 1/8 doping in 8 × 8 cluster. Note 8 doped

holes (charge minority sites with smaller spins) placed randomly and with z orbitals

occupied by electrons. Upper panel: At each site the circle diameter corresponds

to one-half of egon-site total charge; the arrow length to one-half of the egspin; the

horizontal bar length to charge density difference between x and z orbitals (longest

bar to the right corresponds to pure x, to the left to pure z, zero length to half by

half combination). All these values are expressed in approximate proportionality (as

generated by latex graphic package) to nearest neighbour site-site distance which is

assumed to be unity. Lower panel: JT distortions Q2i (in˚ A) for the same plane

shown as bars drown slightly to the left of each site and Q3i, shown by bars to the

right of each site. Isotropic (breathing) mode Q1iin between. Q2iand Q3ibars are

artificially enlarged (by a factor of 2) for a better visibility. The static JT distortions

(finite Q3mode, the other modes vanish) are only present at charge minority sites.

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Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates10

Figure 2. Weakly insulating C-AF phase (ferromagnetic horizontal lines coupled

antiferomagnetically in vertical direction) as obtained for the doing x = 1/4 within the

8 × 8 cluster.Charge majority (undoped) sites have uniform charge distribution, with

both x and z orbitals occupied and S = 1 spins; the charge minority sites are occupied

by z electrons with spins S = 1/2 and show small JT distortions, Q2and Q3. Meaning

of symbols and the values of parameters as in figure 1.

and slightly nonuniform orbital order (z orbital occupancy is then larger than x one on

most sites in the cluster).

3.3. Large doping regime 1/3 < x ≤ 1/2

Coming to large doping regime x > 1/3, we have found further modifications of the

C-AF spin order.First of all, the next doping level considered by us x = 3/8 is

incompatible with the stripes shown in figure 3. As shown in figure 4, in this case the

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Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates11

Figure 3.

coupled antiferomagnetically), with uniform charges on charge majority sites (spins

S = 1 snd both x and z orbitals occupied), as obtained for the x = 1/3 doping in

6 × 6 cluster. The charge minority sites form diagonal stripe boundaries. They are

occupied by z electrons with spins S = 1/2 and they show small uniform Q2and Q3

JT distortions. Meaning of symbols and the values of parameters as in figure 1.

Insulating C-AF phase spin arrangement (ferromagnetic vertical lines

ionic picture does not apply anymore and the electron charge is distributed uniformly.

Therefore the values of the magnetic moments are close to 1.6 µB per site, and the

egorbitals are occupied in a similar way at all cluster sites. Hole doping reduces the

electron density in x orbitals, therefore at the present doping level of x = 3/8 the ratio

of electron density in x and z orbitals is close to 1:2. Unlike for lower doping level

considered above, this electron distribution does not favor JT distortions at particular

cluster sites with minority charge, and one finds uniform and rather small JT distortions

at all the sites. Nevertheless, the electronic structure predicts that this ground state is

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Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates12

Figure 4.

uniform charges, with about 1:2 x to z orbital order for eg electrons and very small

JT distortions, as obtained for x = 3/8 doping in 8 × 8 cluster. The pattern of JT

distortions is almost uniform. Meaning of symbols and the values of parameters as in

figure 1.

Almost perfect C-AF spin arrangement (like in figures 2-3) but with

insulating, having a HOMO-LUMO gap ∆ = 0.72 eV.

The next physically interesting and experimentally much studied doping is half

filling, i.e., x = 1/2. One finds that the C-AF spin order remains unchanged, but charge

modulation develops. This doping level is compatible with a two sublattice structure in

the charge ordered state shown in figure 5. The alternating electron charges are close

to n = 2 and n = 1 on the majority and minority charge sites. Therefore, the ionic

picture applies again to this doping level, and the system may be viewed as alternating

charge order, with Ni2+and Ni3+ions on the two sublattices. This result reproduces

Page 13

Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates13

the experimental results [22,23], with long-range checkerboard-like charge order found

at half doping. Similar to all other cases, also this configuration is insulating, with a

HOMO-LUMO gap ∆ = 0.91 eV.

The charge order found for x = 1/2 doping is closely followed by JT distortions,

shown in figure 5. Large JT distortions at charge minority sites stabilize the

checkerboard charge order in this case. Large {Qi2} distortions stabilize the charge

order, and nonuniform electron distribution over egorbitals is stabilized by large {Qi3}

distortions. As a result, x orbitals are almost empty at charge minority sites, and the

occupancy of z orbitals is close to one electron at each site.

3.4. Maximal doping x = 1 for LaSrNiO4

Experimental studies have established that for the examined class of La2−xSrxNiO4

compounds with large doping x ≃ 1, the ground state is paramagnetic and conducting

[28], with a semiconductor-like conductivity (at low temperature close to zero Kelvin)

[29]. We investigated this case using the present model by numerical computations,

either using parameter set of table 1 or somewhat modified parameters. Extensive

calculations have shown that reasonable values of parameters do not give results in

agreement with experimental observations. In all considered cases one obtains perfect

AF order, with homogeneous charge distribution (one electron per site) and with purely

z-orbital occupation. In addition, the HOMO-LUMO gap is quite large ∆ = 4.5 eV (for

the parameters from table 1).

While this theoretical result is very reasonable and may be expected taking the

broken symmetry between two egorbitals in a monolayer LaSrNiO4system, it stimulates

a question whether this result could invalidate all the previous results reported above.

In our opinion the found disagreement of theory with experiment for such large doping

level is only an indication that the effective model presented in this paper (without extra

modifications) is too limited to perform satisfactorily. Our reasoning, based entirely

upon experimental data, is twofold.

First of all, the conductivity at x ≃ 1 could be likely due to a completely other

mechanism [29], namely to sample quality and near impossibility to grow perfect

LaSrNiO4crystals. In reality what one grows are samples with oxygen vacancies (at

vertices of NiO6octahedra). The level of oxygen vacancies is estimated to be at best

0.01 − 0.04 which seems enough to form a narrow donor band and therefore p-type

conduction could set in. How this p-band destroys the AF order is another and difficult

question. Still it is clear that the two-band model is not appropriate for the description

of LaSrNiO4.

Second, the model computations are based on a formal scheme: the doping level

is equal to fraction x in the actual chemical formula La2−xSrxNiO4. This is a common

knowledge that possibly this is not true, similar as in YBa2Cu3O6+xsuperconductors

[30], and most probably this simple-minded assumption works here reasonably well only

for small values of x. (Note that the same fault can be attributed to modelling done

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Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates14

Figure 5. Perfect checkerboard crystal-like order with two sublattices, as obtained

for x = 1/2 doping in 8×8 cluster. The C-AF spin order with alternating spin values

between S ≃ 1 and S ≃ 1/2 spins for charge majority/minority sites, and holes doped

into x orbitals at charge minority sites. Uniform electron distribution between x and

z orbitals results in no JT distortions at charge majority sites, while considerable JT

distortions at charge minority sites stabilize the orbital order with (almost) empty x

orbitals. Meaning of symbols and the values of parameters as in figure 1.

in doped cuprates and manganites). Clearly, for larger doping one can expect sizable

deviations between electron doping level and x fraction in chemical formula. These

deviations depend on the type of the crystal lattice, the atom-atom distances, on the

constituent atoms and no simple formula exists to compute them.

We suggest that the question whether the first or second explanation is more

appropriate cannot be answered at present. It could be that both are partly valid

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Jahn-Teller mechanism of stripe formationin doped layered La2−xSrxNiO4nickelates15

at the same time.

There are however some indications which can signal when and where the above

proportionality between doping level and the chemical concentration x could not be

obeyed. In our particular case such indications can be read out from ref. [31] where

the true quantum-chemistry ab-initio computations (HF, BLYP and PBE) for LaNiO3

were performed (using CRYSTAL06 computer code) and where Mullikan population

analysis was done. This is a different (three-dimensional) conducting substance, not

the one studied in this paper, but still the problem is more or less the same. The

Mullikan charges found on Ni atoms are ≃ +1.7 instead of formal +3 and on oxygens

the Mullikan charges are ≃ −1.2 instead of formal -2. It is true that Mullikan charges

are only a rather crude estimate but still from the above ab-initio data it follows that

the effective model with formal occupied with one d-electron-type Wannier function (per

site) is not correct for LaNiO3.

One can envisage the modification of our model which possibly could work properly

for the description of LaSrNiO4. It seems necessary to supplement the model with at

least one extra Wannier orbital per site. For the first trial, in the spirit of the Anderson

lattice model, this could be a simple s-type Wannier orbital of the oxygen type (assuming

spherically symmetric distribution of the oxygen p electrons around the central nickel).

The new extra Hamiltonian terms would be simple kinetic s-type nearest-neighbour

hopping supplemented by s−d hybridization. According to the Koster-Slater rules, the

s − d hybridization on single lattice site vanishes and one should consider it between

nearest-neighbour sites only. This goes however beyond the scope of the present paper

and is planned as a separate future investigation.

4. Crucial role played by Jahn-Teller distortions

4.1. Finite Jahn-Teller coupling gJT

We already remarked several times on the importance of JT coupling. Here we try to

draw some more general conclusions and provide extra information.

Test computations for the undoped La2NiO4 substance (x = 0) invariably

(independently of the initial conditions) give insulating G-AF ground state (for S = 1

spins) with uniform charge distribution and equal occupation of x and z orbitals, in

perfect agreement with experiment [32,33]. In this state no JT distortions can arise.

This result was obtained for almost any set of the Hamiltonian parameters with one

notable exception. For too large JT coupling, in particular for gJT> 3.5 eV˚ A−1(while

the other Hamiltonian parameters are those from the table 1) nonmagnetic phase starts

to develop. The emerging nonmagnetic sites feature equal occupation of 0.5 electron

per up and down spin both for x and z orbitals. This is accompanied by huge and

clearly unphysical JT distortions of the Q2type. For gJT= 3.8 eV˚ A−1the common

magnetic sites with spin S = 1 (and zero JT distortions) are already quite scarce and the

nonmagnetic sites do abound. We suggest that for so large values of gJTthe simplified

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