The phase separation effects in a pseudospin-electron model
The two-sublattice pseudospin-electron model of high temperature superconductors is studied with respect to phase separations. Such a model can be used for the description of dielectric properties of YBaCuO-type crystals along c -axis (the pseudospins represent an anharmonic motions of apical ox...
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| Cite this: | The phase separation effects in a pseudospin-electron model / O.D. Danyliv, I.V. Stasyuk // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 523-529. — Бібліогр.: 17 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1206622025-06-03T16:30:09Z The phase separation effects in a pseudospin-electron model Ефекти фазового розшарування у псевдоспін-електронній моделі Danyliv, O.D. Stasyuk, I.V. The two-sublattice pseudospin-electron model of high temperature superconductors is studied with respect to phase separations. Such a model can be used for the description of dielectric properties of YBaCuO-type crystals along c -axis (the pseudospins represent an anharmonic motions of apical oxygen O4). The model is treated within different approximations. It is shown that the model has got phase transitions to the phase with pseudospin ordering as well as phase separation. The effect of separation on the appearance of the ordered state is discussed. Розглянуто двопідґраткову псевдоспін-електронну модель високотемпературних надпровідників з огляду на можливість появи фазових розшарувань. Така модель здатна описати діелектричні властивості кристалів типу YBaCuO вздовж осі c (псевдоспінами представлено ангармонічні локальні коливання вершинного кисню 04). Модель досліджено в різних наближеннях. Показано, що поряд з фазовим переходом у фазу із впорядкованою псевдоспіновою системою в моделі присутнє фазове розшарування. Обговорюється вплив розшарування на появу впорядкованої фази. This work was partially supported by the Fundamental Research Fund of the Ministry of Ukraine for Science and Education (Project No. 02.07/266). 2002 Article The phase separation effects in a pseudospin-electron model / O.D. Danyliv, I.V. Stasyuk // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 523-529. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 74.65.+n, 71.45.Gm DOI:10.5488/CMP.5.3.523 https://nasplib.isofts.kiev.ua/handle/123456789/120662 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The two-sublattice pseudospin-electron model of high temperature superconductors
is studied with respect to phase separations. Such a model can
be used for the description of dielectric properties of YBaCuO-type crystals
along c -axis (the pseudospins represent an anharmonic motions of
apical oxygen O4). The model is treated within different approximations. It
is shown that the model has got phase transitions to the phase with pseudospin
ordering as well as phase separation. The effect of separation on
the appearance of the ordered state is discussed. |
| format |
Article |
| author |
Danyliv, O.D. Stasyuk, I.V. |
| spellingShingle |
Danyliv, O.D. Stasyuk, I.V. The phase separation effects in a pseudospin-electron model Condensed Matter Physics |
| author_facet |
Danyliv, O.D. Stasyuk, I.V. |
| author_sort |
Danyliv, O.D. |
| title |
The phase separation effects in a pseudospin-electron model |
| title_short |
The phase separation effects in a pseudospin-electron model |
| title_full |
The phase separation effects in a pseudospin-electron model |
| title_fullStr |
The phase separation effects in a pseudospin-electron model |
| title_full_unstemmed |
The phase separation effects in a pseudospin-electron model |
| title_sort |
phase separation effects in a pseudospin-electron model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2002 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120662 |
| citation_txt |
The phase separation effects in a
pseudospin-electron model / O.D. Danyliv, I.V. Stasyuk // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 523-529. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT danylivod thephaseseparationeffectsinapseudospinelectronmodel AT stasyukiv thephaseseparationeffectsinapseudospinelectronmodel AT danylivod efektifazovogorozšaruvannâupsevdospínelektronníjmodelí AT stasyukiv efektifazovogorozšaruvannâupsevdospínelektronníjmodelí AT danylivod phaseseparationeffectsinapseudospinelectronmodel AT stasyukiv phaseseparationeffectsinapseudospinelectronmodel |
| first_indexed |
2025-11-26T20:20:25Z |
| last_indexed |
2025-11-26T20:20:25Z |
| _version_ |
1849885646339964928 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 3(31), pp. 523–529
The phase separation effects in a
pseudospin-electron model
O.D.Danyliv, I.V.Stasyuk
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received December 25, 2001
The two-sublattice pseudospin-electron model of high temperature super-
conductors is studied with respect to phase separations. Such a model can
be used for the description of dielectric properties of YBaCuO-type crys-
tals along c -axis (the pseudospins represent an anharmonic motions of
apical oxygen O4). The model is treated within different approximations. It
is shown that the model has got phase transitions to the phase with pseu-
dospin ordering as well as phase separation. The effect of separation on
the appearance of the ordered state is discussed.
Key words: phase separation, pseudospin, ferroelectric ordering
PACS: 74.65.+n, 71.45.Gm
1. Introduction
The role of apex oxygen O4 in high Tc superconductors is not well studied yet
as well as the mechanism of high values of superconducting phase transition tem-
peratures is not quite clear. The analysis of EXAFS and Raman spectra of the most
studied superconductor YBa2Cu3O7−δ [1–3] reveals the existence of a strongly anhar-
monic double well potential of apex oxygen. This conclusion agrees with the results
of X-rays diffraction experiments [4]. In some experiments YBaCuO was found to be
both pyroelectric and piezoelectric, implying the existence of macroscopic polariza-
tion directed along the c-axis [5]. This feature was connected with the existence of
anharmonic vibrations of O4. However, in [6] an alternative interpretation of EXAFS
data was given: it was suggested that the two-site O4 configurations may be related
to strong local distortions around single chain O1-vacancies.
To take into account the strong on-site Coulomb interaction in CuO planes and
the interaction of electrons with local vibrations of apex oxygen the pseudospin-
electron model (PEM) was used [7]. To explain the possible phase transitions con-
c© O.D.Danyliv, I.V.Stasyuk 523
O.D.Danyliv, I.V.Stasyuk
nected with the apex oxygen, it was proposed to consider the long range pseudospin-
pseudospin interaction [8]. To correctly describe the dielectric anomalies, PEM was
extended to a two-sublattice case [10].
The purpose of the current paper is to consider PEM from the point of view of the
phase separation and to study the influence of a separated state on the ferroelectric
type ordering which can be present it the system.
2. The Hamiltonian of model
In the paper we investigate the dielectric and thermodynamic properties of the
two-sublattice PEM, which takes into account a two-sublattice structure of an ele-
mentary cell of YBaCuO-type superconductors. Within the framework of this model,
the Hubbard model (applied to the description of electrons) is supplemented by a
pseudospin subsystem [9,10]:
H = He + Hs + He−s + Hs−s ,
He = −µ
∑
n,s
(ns
n1
+ ns
n2
) + U
∑
n
(n↑
n1
n↓
n1
+ n↑
n2
n↓
n2
),
Hs = −h
∑
n
(Sz
n1
− Sz
n2
),
He−s = g
∑
n,s
(ns
n1
Sz
n1
− ns
n2
Sz
n2
),
Hs−s = −J
∑
n
Sz
n1
Sz
n2
−
1
2
∑
n,n′
∑
α,β
Jαβ
nn′S
z
nαSz
n′β . (1)
Here, ns
nα is the operator of the number of electrons with a spin s while Sz
nα
stands for the operator of the pseudospin at the n cell in α plane (α = 1, 2 in the
two-sublattice case). He is the Hubbard Hamiltonian without a term describing the
transfer of electrons. Hs is the pseudospin part of the Hamiltonian and h describes
the asymmetry of the well potential. He−s is the term describing the interaction
between electrons and pseudospins. Hs−s gives the interaction between pseudospins;
the interaction −JSz
n1
Sz
n2
within one cell clusters is separated.
It is convenient to study the Hamiltonian based on the states |i, K〉 ≡ |i, 1, R〉⊗
|i, 2, S〉 which is the product of one-site states |i, α, R〉 ≡ |n̂↑
iα, n̂↓
iα, Ŝz
iα〉 (α = 1, 2).
Each one-site set of states includes eight components at a given site i and sublattice
α:
|1〉 = |0, 0, ↑〉, |1̃〉 = |0, 0, ↓〉,
|2〉 = |1, 1, ↑〉, |2̃〉 = |1, 1, ↓〉,
|3〉 = |0, 1, ↑〉, |3̃〉 = |0, 1, ↓〉,
|4〉 = |1, 0, ↑〉, |4̃〉 = |1, 0, ↓〉.
(2)
Hereafter we will consider the case of infinitely large Coulomb repulsion U . It
means that we are able not to consider the doubly occupied states |2〉 and |2̃〉.
524
The phase separation effects. . .
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Figure 1. h dependence of the temperature of ferroelectric phase transition Tc at
different values of parameter J11−J12 in the regime n=const: a) (J11−J12)/(J11+
J12) = −1, b) (J11 − J12)/(J11 + J12) = 0. Other parameters: J/(J11 + J12) = 1,
g/(J11 +J12) = 1, n = 0.4. Solid lines and dashed lines represent the second order
and the first order phase transitions, respectively. The widely spaced dashed line
corresponds to a one-loop approximation. Dots represent a separation area.
3. Phase separation induced by pseudospin-pseudospin
interaction
In [10] the phase transitions into ferroelectric phase (with the order parameter
η = 1/N
∑
n〈S
z
n1
− Sz
n2
〉 describing a macroscopic polarization) were investigated
in the mean field approximation and in the one-loop approximation. The regions of
parameter values of model (1) at which the ferrolelectric ordering takes place were
found. Nevertheless, the phase separation processes were not taken into account, as
far as they were found at zero temperature in the simple (one-sublattice) pseudospin-
electron model [12].
Figure 1 presents the phase diagrams for model (1) at different values of long
range interaction Jαβ =
∑
n′ J
αβ
nn′ and fixed concentration n = 0.4. In areas marked
by points, the system is separated into two regions with concentrations n1 and n2
(n1 < n < n2). Also, one can notice that phase separation takes place near the
border of stability region of two phases (the ordered with nonzero polarization and
the disordered). That is why the ordered phase becomes wider and is extended
up to the edge of the separated area. Figure 2 illustrates such a behaviour. The
dashed lines here represent the region of ferroelectric type instabilities. These lines
would separate the ferroelectric phase if there was no phase separation. Figure 2b
also shows that at a fixed value of anharmonicity parameter h with concentration
n < 0.75 (at low temperature), the ordered phase is possible only because the phase
separation takes place.
525
O.D.Danyliv, I.V.Stasyuk
Figure 2. Phase T − n diagram in a mean field approximation. The phase sep-
aration region is limited by solid lines. The dashed lines point to the region of
ferroelectric instabilities. Parameters: (J11−J12)/(J11 +J12) = 1, J/(J11 +J12) =
g/(J11 + J12) = 1; a) h/(J11 + J12) = 1, b) h/(J11 + J12) = 1.35 .
4. The role of electron transfer
Let us now consider the effect of electron transfer on the phase transitions. It
has been shown (in the simplest approximation Hubbard-I [11]) that at U → ∞ in a
one-sublattice case of model (1), one has in Hubbard-I approximation two subbands,
separated by the gap equal to g:
ε31
α (k) = ε41
α (k) = −(−1)α g
2
+ tk〈X
44
α + X11
α 〉0 ,
ε3̃1̃
α (k) = ε4̃1̃
α (k) = (−1)α g
2
+ tk〈X
4̃4̃
α + X 1̃1̃
α 〉0 . (3)
Here, 〈. . .〉0 represents an average over one-site part of Hamiltonian (1), tk =
1
N
∑
i−j
tije
ikrij – Fourier transformation of hopping integral and XRS
iα = |i, α, R〉〈i, α, S|
are Hubbard operators on the basis of states (2). We can improve this result following
the self-consistent generalized random phase approximation [13], which corresponds
to the summing of all one-loop contributions to Xpp in diagrammatic representation.
It gives us a shift of subbands by Q̃pq; the mean values of the Hubbard operators
then should be found self-consistently.
The changes in excitation spectrum (3) are demonstrated in figure 3b. There is
no more symmetry εpq
1
(k) = −εpq
2
(k). It is due to ferroelectric type ordering in the
system. The chemical potential behaviour near the edge of the band points to the
appearance of a phase separation. Figure 3 shows that in the case when the solution
of the system of equations indicates that the ferroelectric phase appears (figure 3a),
the separation takes place before that. This is also indicated by a certain concavity
in the free energy behaviour (tangent dashed lines in figure 3c link the points with
526
The phase separation effects. . .
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Figure 3. Dependence of the order parameter η (a), band spectrum (b), and
free energy (c) on concentration. The parameter values are: J11 = J12 = g/2,
T/g = 0.1, tij/g = 0.1, h/g = 0.5.
Y�Z Y Y�Z [ \�Z YY Z Y
Y Z [
\;Z Y
\;Z [
]�Z Y ^�_
`>a�b
ced
f�g;h cAi�c j g;klh:m cAn g
oqpTr s�t p�r s�t p�r uvt w�r sxtp�r p
p�r t
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w�r t
sTr p y{z
|>}T~
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Figure 4. Regions of phase separation at different values of interaction Jαβ, T/g =
0.1. Thick lines correspond to the case tij = 0; dashed lines correspond the the
case tij 6= 0. a) J11 = g, J12 = 0, tij/g = 0.2; b) J11 = J12 = g/2, tij/g = 0.1.
concentration values n1, n2 and n3, n4 on which the separation takes place). Hence,
there is a separation in the paraelectric and ferroelectric phases at concentration
values n1 < n < n2 and n3 < n < n4. The pure ferroelectric phase remains in the
concentration range n2 < n < n3.
Figure 4a and 4b show the boundaries of the separated phase in the presence
and in the absence of electron transfer. Their positions are temperature dependent.
At T = 0 the region of phase separation reaches the sides of the 0 6 h/g 6 1,
0 6 n∗ 6 2 [12] rectangle. The transfer effect is ambiguous. In the first case (4a) it
narrows the region of separation, whereas in the second case (4b) the transfer only
changes the shape of the region boundaries.
527
O.D.Danyliv, I.V.Stasyuk
5. Conclusions
It is shown that the phase transition to the ordered polar phase in the two-
sublattice pseudospin-electron model can be accompanied by a phase separation.
The system is separated into the regions with a different value of concentration and
polarization. The phase separation widens the area of the model parameter values
where polarization takes place: the borders of a separated phase become new borders
of the region with nonzero order parameter η.
It is shown that electron transfer has an effect on the phase separation caused by
pseudospin-pseudospin interaction. Electron hopping changes the shape of the region
boundaries. The reverse effect also takes place: the electron spectrum is changed by
a phase separation.
The obtained results correlate well with the experimental data on neutron scat-
tering for high-Tc superconductors which indicate the existence of stripe phases in
La2−xSrxCuO4 [14] as well as in YBa2Cu3O7−δ [15]. Also, there is a problem of a gen-
esis of structural inhomogeneities in a single crystal of the YBaCuO type observed
by the experiments using Raman spectroscopy [16] and mesoscopic structural date
[17]. Our results show the possibility of dielectric type mechanism of the appearance
of the mentioned instabilities.
6. Acknowledgement
This work was partially supported by the Fundamental Research Fund of the
Ministry of Ukraine for Science and Education (Project No. 02.07/266).
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