Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system
Li₁₊xTi₂₋xO₄ The phonons of two end members of Li₁₊xTi₂₋xO₄ spinel system, i.e., the metallic LiTi₂O₄ and the insulating Li[Li₁/₃Ti₅/₃]O₄, are calculated in superspace symmetry approach using a short-range force constant model. The composition dependence of zone-centre optical phonons in Li₁₊...
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Інститут фізики конденсованих систем НАН України
2004
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irk-123456789-1187132017-06-01T03:03:42Z Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system Buletsa, E.P. Ivanyas, O.F. Kindrat, V.J. Nebola, I.I. Shkirta, I.M. Shtejfan, A.J. Li₁₊xTi₂₋xO₄ The phonons of two end members of Li₁₊xTi₂₋xO₄ spinel system, i.e., the metallic LiTi₂O₄ and the insulating Li[Li₁/₃Ti₅/₃]O₄, are calculated in superspace symmetry approach using a short-range force constant model. The composition dependence of zone-centre optical phonons in Li₁₊xTi₂₋xO₄ near phase transition and Li₁₋xMgxTi₂O₄ has been investigated using different models of substitution. One- and two-mode behavior is therefore predicted for F₁u and F₂u modes in case of tetrahedral and octahedral substitution, respectively. У надпросторовому підході, використовуючи модель короткодіючих силових постійних, розраховані фонони двох крайніх членів шпінелевої системи Li₁₊xTi₂₋xO₄: металічної LiTi₂O₄ та діелектричної Li[Li₁/₃Ti₅/₃]O₄. Застосовуючи різні моделі заміщення, досліджена композиційна залежність оптичних фононів в центрі зони Бриллюєна для Li₁₊xTi₂₋xO₄ в околі фазового переходу та Li₁₋xMgxTi₂O₄ . Одно- та двомодову поведінку передбачено для F₁u та F₂u мод у випадку тетраедричного та октаедричного заміщення відповідно 2004 Article Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system / E.P. Buletsa, O.F. Ivanyas, V.J. Kindrat, I.I. Nebola, I.M. Shkirta, A.J. Shtejfan // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 53-62. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 61.50.Ah, 63 DOI:10.5488/CMP.7.1.53 http://dspace.nbuv.gov.ua/handle/123456789/118713 en Condensed Matter Physics 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 |
Li₁₊xTi₂₋xO₄
The phonons of two end members of Li₁₊xTi₂₋xO₄ spinel system, i.e., the
metallic LiTi₂O₄ and the insulating Li[Li₁/₃Ti₅/₃]O₄, are calculated in superspace
symmetry approach using a short-range force constant model. The
composition dependence of zone-centre optical phonons in Li₁₊xTi₂₋xO₄
near phase transition and Li₁₋xMgxTi₂O₄ has been investigated using different
models of substitution. One- and two-mode behavior is therefore predicted
for F₁u and F₂u modes in case of tetrahedral and octahedral substitution,
respectively. |
| format |
Article |
| author |
Buletsa, E.P. Ivanyas, O.F. Kindrat, V.J. Nebola, I.I. Shkirta, I.M. Shtejfan, A.J. |
| spellingShingle |
Buletsa, E.P. Ivanyas, O.F. Kindrat, V.J. Nebola, I.I. Shkirta, I.M. Shtejfan, A.J. Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system Condensed Matter Physics |
| author_facet |
Buletsa, E.P. Ivanyas, O.F. Kindrat, V.J. Nebola, I.I. Shkirta, I.M. Shtejfan, A.J. |
| author_sort |
Buletsa, E.P. |
| title |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system |
| title_short |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system |
| title_full |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system |
| title_fullStr |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system |
| title_full_unstemmed |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system |
| title_sort |
phonon spectra near metal-insulator phase transition in li₁₊xti₂₋xo₄ system |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118713 |
| citation_txt |
Phonon spectra near metal-insulator phase transition in Li₁₊xTi₂₋xO₄ system / E.P. Buletsa, O.F. Ivanyas, V.J. Kindrat, I.I. Nebola, I.M. Shkirta,
A.J. Shtejfan // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 53-62. — Бібліогр.: 15 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T14:30:43Z |
| last_indexed |
2025-07-08T14:30:43Z |
| _version_ |
1837089482777559040 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 1(37), pp. 53–62
Phonon spectra near metal-insulator
phase transition in Li1+xTi2−xO4 system
E.P.Buletsa, O.F.Ivanyas, V.J.Kindrat, I.I.Nebola, I.M.Shkirta,
A.J.Shtejfan
Uzhgorod National University, 46 Pidhirna Str., Uzhgorod, 88000 Ukraine
Received November 22, 2002, in final form January 10, 2004
The phonons of two end members of Li1+xTi2−xO4 spinel system, i.e., the
metallic LiTi2O4 and the insulating Li[Li1/3Ti5/3]O4, are calculated in super-
space symmetry approach using a short-range force constant model. The
composition dependence of zone-centre optical phonons in Li1+xTi2−xO4
near phase transition and Li1−xMgxTi2O4 has been investigated using dif-
ferent models of substitution. One- and two-mode behavior is therefore pre-
dicted for F1u and F2u modes in case of tetrahedral and octahedral substi-
tution, respectively.
Key words: superspace symmetry, phonon spectra, spinel, Li1+xTi2−xO4
PACS: 61.50.Ah, 63.
1. Introduction
A large class of solid spinels XY2O4 are of technological and geological interest
having widely variable occupations of two (tetrahedral A and octahedral B) sites
by various cations (figure 1). Normally in most spinels there are occupations of the
octahedral and tetrahedral sites preferred by different atoms.
Among a large number of spinel compounds only a few show metallic behavior
and Li1+xTi2−xO4 system is the only one to be superconducting with a TC ≈ 13 ◦K.
The replacement of Ti ions by additional Li ions carries the system through a
metallic-insulator transition at x ≈ 0.1 − 0.15 [1–3]. Although many experimen-
tal studies have been done in connection with the transition [1–4], the origin of
the metal-insulator transition is still not fully understood. In order to study the
mechanism of superconductivity and metal-insulator transition in Li1+xTi2−xO4 the
investigations of its phonons are quite important. Therefore, the phonons of two end
members of Li1+xTi2−xO4 spinel system, i.e., the metallic LiTi2O4 and the insulating
Li[Li1/3Ti5/3]O4, are calculated using a simple short-range force constant model.
The simple force constant models are widely used in studying the phonons in
spinel type oxides, sulphides and selenides [5,6]. Lattice dynamic calculations of
c© E.P.Buletsa, O.F.Ivanyas, V.J.Kindrat, I.I.Nebola, I.M.Shkirta, A.J.Shtejfan 53
E.P.Buletsa et al.
Figure 1. Primitive cell of XY2O4 spinel structure.
inverse spinels had been performed in [7]. The phonons in superconducting oxide
spinel LiTi2O4 were also investigated using three short-range force constants α1,
α2, α3 by Gupta et al. [8]. The theoretical Raman frequencies and superconducting
transition temperature TC agree satisfactorily with the observed values in paper [8].
In accordance with the above, the simple model with three central force con-
stants for interatomic interaction between the first, second and third nearest neigh-
bors is used in the present work to investigate the phonons in Li[LixTi2−x]O4 and
Li1−xMgx[Ti2]O4 systems.
The analysis of the phonon spectra of the Li1+xTi2−xO4 spinel system, in par-
ticular, as well as of many other spinels, becomes complicated by disordering of the
cations between A and B sites, which may depend upon the temperature, conditions
of synthesis and chemical composition [9].
In this paper for the first time the Superspace Symmetry Approach (SSA) [10–
12] has been used in constructing the simplified dynamic matrix of spinel structure.
The dynamic matrix elements are calculated, using the experimental data of Raman
spectra of Li1+xTi2−xO4 solids near metal-insulator phase transition.
The SSA approach takes into account the composition freedom of the sites
occupied by different atoms. Thus, we have computed the compositional varia-
tions of zone-centre phonon frequencies with different kinds of substitutions in
Li[LixTi2−x]O4 and Li1−xMgx[Ti2]O4 lithium spinel systems.
2. Theory
XY2O4 type crystals have the cubic lattice (space group Fd3m (Oh)) the primi-
tive cell of which contains 14 atoms (figure 1).
One can consider a spinel structure as a suitable model object for the study of
phonon spectra in SSA approach due to the presence of equidistant sites, occupied
by various sorts of atoms and vacancies.
We can regard the real structure in terms of SSA as a compositional modulated
54
Phonons in Li1+xTi2−xO4 system
one from the basic “simpler” structure with a smaller period of translations [13]. So,
the complex crystal is the compositional natural superlattice in this approach.
Let us select a Volume-Centered Cubic (VCC) lattice as the basic structure.
(3+d)-dimensional superspace description [10–13] of given crystals is realized, using
the basis in the direct space:
a1
a2
a3
a4
a5
a6
=
−a a a b/4 −b/4 −b/4
a −a a −b/4 b/4 −b/4
a a −a −b/4 −b/4 b/4
0 0 0 0 b b
0 0 0 b 0 b
0 0 0 b b 0
, (1)
where a is a parameter of basic one atom lattice; b/4 is a lattice parameter in the
additional 3-dimensional phase space.
3-dimensional components of the last three vectors in reciprocal basis define the
elementary modulation vectors:
q1 = (− π
4a
,
π
4a
,
π
4a
), q2 = (
π
4a
,− π
4a
,
π
4a
), q3 = (
π
4a
,
π
4a
,− π
4a
). (2)
The linear combinations of these vectors in the limits of basic structure BZ will
derivate the set of 32 modulation vectors.
The primitive spinel cell has 18 vacancies concerning a VCC lattice of the basic
structure. Let us consider that the vacancy contains an atom with zero mass and is
surrounded by a zero force field whether atoms are included in first, second or third
coordination group or not. Such an atom does not carry changes into a dynamic
matrix.
In terms of SSA approach a phonon spectrum of crystal lattice is defined by the
solutions of the generalized eigenvalue problem [11]:
|DSSA(k) − ω2(k)M| = 0, (3)
where M is a matrix of the mass defect operator [10–12], responsible for mass mod-
ulation of basic one-atom lattice. The structure of DSSA(k) matrix is [12]:
DSSA(k) =
Dq1
(k − q1) Dq2
(k − q1) · · · Dq32
(k − q1)
Dqi
(k − q2) Dq1
(k − q2) · · · Dqm
(k − q2)
...
...
. . .
...
Dqj
(k − q32) Dqn
(k − q32) . . . Dq1
(k − q32)
. (4)
Here Dqj
(k − qi) – j-th fragment of a dynamic matrix of basic one-atom structure,
averaged at mass and force field, defined in k − qi points of BZ (i, j = 1, 2, . . . , 32).
We can write the set of matrices Dqj
(k − qi), using simple short-range force
constants model. Each force constant parameter describes the interaction in a cor-
responding coordination group. The mass modulation and modulation of force con-
stants in the averaged basic structure leads to the correct parameters of the complex
crystal.
55
E.P.Buletsa et al.
Hereinafter we shall view the interaction in the limits of the first three coordi-
nation groups of a VCC basic lattice, that corresponds to interactions X–O, Y–O,
and Y–Y (O–O).
Using the position and modulation vectors rj and qi we get the Fourier transfor-
mation F
Fij =
1√
32
exp(iqirj) (5)
and apply in the equation (3). We obtain:
|F−1DSSA(k)F − ω2(k)F−1MF | = |Dcl(k) − ω2(k)M | = 0. (6)
M is diagonal mass matrix. The structure for the force part of dynamic matrix
Dcl is:
Dcl =
Dcl
11 0 Dcl
12 0 · · · Dcl
1m
0 0 0 0 · · · 0
Dcl
21 0 Dcl
22 0 · · · Dcl
2m
0 0 0 0 · · · 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dcl
n1 0 Dcl
n2 0 · · · Dcl
nm
. (7)
The zero columns and strings correspond to the places, in which the vacancies
are disposed, and coincide with zero diagonal elements of mass matrix M . Therefore,
these columns and strings can be eliminated. Such a block matrix becomes equivalent
to a classical dynamic matrix. Apparently, its order equals 3 × (32 − 18) = 42.
The group theoretical treatment of the optical zone-centre (k=0) phonon modes
for the spinel structure yields [14]
Γ(0) = A1g(R) + Eg(R) + 3F2g(R) + 4F1u(IR) + F1g + 2A2u + 2Eu + 2F2u. (8)
Five modes A1g, Eg and 3F2g are Raman-active (marked as R) and four 4F1u are
infrared active (marked as IR).
As mentioned above, in the present investigation the dynamic matrix elements
are calculated using a simple short-range force constant model. For this purpose
we have obtained the evaluation of three parameters of interatomic interaction in
the way pointed in [8], based on the experimental Raman data [2] by analytical
expressions [6]:
{
mOω2
A1g
= α1 + α2 + 8α3 ,
mOω2
Eg
= α2 + 2α3 ,
(9)
where ωA1g
and ωEg
are the observed Raman A1g and Eg modes; mO is the mass
of oxygen atom; α1, α2, α3 represent the central force constants for interatomic
interaction between the first, second and third nearest neighbors. We can take the
values of the O–O and Ti–Ti force constants to be α3 = 20.0 N/m, based on earlier
studies [8] of oxides spinel and the semi-empirical treatment by Oda et al. [15].
56
Phonons in Li1+xTi2−xO4 system
Table 1. Experimental phonon modes A1g; Eg [2,8] and calculated short-range
force constants α1, α2, α3 for LiTi2O4 and Li[Li1/3Ti5/3]O4 spinel systems.
Spinel A1g , Eg , α1 , α2 , α3 ,
system cm−1 cm−1 N/m N/m N/m
LiTi2O4 [8] 628 429 80.0 135.0 20.0
Li[Li1/3Ti5/3]O4 675 [2] 427 [2] 139.5 133.1 20.0
The force constants for different spinel systems calculated by using the value
α3 = 20.0 N/m and Raman data [2,8] are given in table 1.
As an example of using SSA approach, in a short-range force constant model we
calculate the phonon frequencies in spinel systems in the following cases:
(i) The zone-centre phonon frequencies of lithium spinel LiTi2O4;
(ii) The dispersion curves of phonons for the end members of Li1+xTi2−xO4 spinel
system above and below the point of metal-insulator phase transition;
(iii) The composition dependence of zone-centre phonons in Li1+xTi2−xO4 system
(0 6 x 6 1/2) and Li1−xMgxTi2O4 system (0 6 x 6 1/2) with a different type
of substitutions.
3. Results and discussion
Using three force constants (see table 1), the phonon frequencies of LiTi2O4
at the zone centre are calculated in the framework of superspace symmetry and
classified according to irreducible representations. These are given in table 2 along
with the observed Raman measurements [2] and calculations of Gupta et al. [8] with
the same parameters α1, α2, α3.
A simple short-range force constant model, as was shown in papers [5–8], in
spinel cases gives the similar frequencies for zone centre vibrations and Raman data.
The consideration of a long-range Coulomb interatomic interaction may reduce the
existing difference. One can see that the present results also practically coincide with
calculations [8]. It testifies to the equivalence of dynamic matrices constructed in
different approaches.
The phonon dispersion curves calculated for LiTi2O4 and Li[Li1/3Ti5/3]O4 are
shown in figure 2. The mass of the introduced Li ion in Li[Li1/3Ti5/3]O4 system is av-
eraged on all occupied octahedral sites. We have made our calculations of LiTi2O4 for
comparison with the corresponding one, which used a short-range Buckingham po-
tential and a long-range Coulomb interaction [4]. The given model enables us to ob-
serve the qualitative difference in the phonon spectrum of insulator Li[Li1/3Ti5/3]O4
and metallic LiTi2O4 spinels.
The detailed analysis of our calculations shows (see figure 2) that the movement
from Γ point in Γ–R direction is associated with the splitting of each of triple
57
E.P.Buletsa et al.
Table 2. Observed [2] and calculated zone-centre frequencies of LiTi2O4 in cm−1.
Species Measurements [2] Calculations [8] Our calculations
A1g 628 628.0 628.4
A2u — 664.6 664.5
— 323.8 323.7
F1g — 429.0 429.3
F1u — 668.3 668.3
— 508.6 508.5
— 424.9 424.9
— 210.3 210.2
F2g — 652.4 652.5
494 542.4 542.3
339 344.2 344.2
F2u — 542.5 542.4
— 165.0 165.0
Eg 429 429.0 429.3
Eu — 603.3 603.1
— 236.8 236.7
generated modes 3F2g, 4F1u, F1g, 2F2u into doubly generated and single generated.
It leads to 28 dispersion curves in the direction Γ–R (14 doubly generated and 14
single generated branches).
The movement from point Γ in Γ–X direction is associated with similar splitting
of triple-generated modes and additional splitting of doubly generated modes 2Eu.
In the total we have 30 branches in Γ–X direction (12 doubly generated and 18 single
generated).
In Γ–K direction one can observe 42 modes (all modes are single generated).
The Li1+xTi2−xO4, spinel system (0 6 x 6 1/3) undergoes a metal-insulator
transition near x ≈ 0.15. In LiTi2O4 Li and Ti ions occupy tetrahedral and octahe-
dral sites, respectively. In Li[Li1/3Ti5/3]O4 an additional substitution of Ti ions by
Li leads to a loss of metallic properties [1–3].
In order to determine an effect of mass distribution in cationic sublattice on the
phonons in superconducting lithium spinel LiTi2O4, we have carried out a series of
model examinations of substitution in octahedral and tetrahedral sites (figure 3).
The masses of atoms in the sites of substitutions are calculated by assuming the
linear variation from one atom mass to the other with composition x :
mo(Li1+xTi2−xO4) = (1 − x)mo(LiTi2O4) + xmo(Li[Li1/3Ti5/3]O4), (0 6 x 6 1/2),
mt(Li1−xMgxTi2O4) = (1 − x)mt(LiTi2O4) + xmt(MgTi2O4), (0 6 x 6 1/2),
where mo and mt are the masses of atoms of corresponding systems in octahedral
58
Phonons in Li1+xTi2−xO4 system
(a) (b)
(c)
Figure 2. Phonon spectrum of crystals in symmetry directions, calculated
using a short-range force constant model. Wave vector defined in terms of
π/4a. a) Li[Ti2]O4, (α1 = 80.0 N/m, α2 = 135.0 N/m, α3 = 20.0 N/m);
b) Li[Li1/3Ti5/3]O4, (α1 = 139.5 N/m, α2 = 133.1 N/m, α3 = 20.0 N/m); c) 1/8
of Brillouin zone for FCC spinel lattice with lattice parameter 4a. The length of
an edge of an octant along the x-axes is π/4a.
and tetrahedral sites, locally involved in the given substitutions.
The force constants are taken corresponding to the end members of spinel sys-
tems Li[LixTi2−x]O4 and Li1+xMgx[Ti2]O4 with x = 0 (In both cases LiTi2O4 (see
table 1)). We do not change them during the whole composition range.
By modeling different types of substitution we can find the anomaly of the
phonon frequencies in the system Li[LixTi2−x]O4, which undergoes a metal-insulator
transition contrary to Li1−xMgx[Ti2]O4 with the saved metallic properties [2,4].
As we can see in figure 3a the energy of some modes (F1u) sharply increases in
the field of anomaly. It does not provide advantage to the system as a whole. The
light atoms Li (MLi = 6.9) are not inclined to occupy the octahedral sites relative to
oxygen framework (MO = 16.0) together with heavy atoms of Ti (MTi = 47.9). One
can assume the F1u modes will indicate the phase transition by taking into account
the adequate model of interatomic interaction.
The calculations shown in figure 3a should not be viewed as the ones simulating
the metal-insulator phase transition. The present model of lattice dynamics is very
simple in predicting or simulating any phase transition. The anomaly just shows
59
E.P.Buletsa et al.
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�����
�����
�����
���
�����
�����
� ���
���� ����� ��� � ��� � � � �����
ω
���
!
" #�$ #�# #�$ %�& #�$ &�##
' #�#
%�#�#
( #�#
) #�#
&�#�#
* #�#
+ #�#
, #�#
-�. / 0 1 24351 6 7�. 8 9�:�;
ω
<=>
?@
A
(a) (b)
Figure 3. Compositional variation of zone-centre phonon frequencies in mixed
systems: a) Li[LixTi2−x]O4; b) Li1−xMgx[Ti2]O4.
the dynamical instability of spinel lattice with given parameters when octahedral
substitutions occur by light atoms. It can be balanced by the renormalization of
force constants that one can interpret as a phase transition.
In Li1−xMgx[Ti2]O4 the substitution of cations occurs in tetrahedral sites. Fig-
ure 3b shows no anomaly of phonon branches. The given lattice model in this case
does not guarantee any transition in a spinel system.
Finally, as we can see in figure 3a that some phonon branches (F1u and F2u) of
the system Li[LixTi2−x]O4 exhibit two-mode behavior in which two sets of phonon
frequencies are observed for the whole composition range. On the other hand, on-
ly one set of phonon frequencies varies with the concentration of the components
in Li1−xMgx[Ti2]O4 system (see figure 3b). Thus, our calculations in mixed spinel
systems predict for F1u and F2u modes, one- and two-mode behaviors in case of
tetrahedral and octahedral substitution, respectively. However, precise infrared and
inelastic-neutron-scattering experiments are required to confirm the predicted be-
havior of phonons in Li[LixTi2−x]O4 and Li1−xMgx[Ti2]O4 systems.
60
Phonons in Li1+xTi2−xO4 system
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E.P.Buletsa et al.
Фононний спектр в околі фазового переходу
метал-діелектрик системи Li1+xTi2−xO4
Е.П.Булеца, О.Ф.Іваняс, В.Й.Кіндрат, І.І.Небола,
І.М.Шкирта, А.Й.Штейфан.
Ужгородський національний університет, 88000 Ужгород,
вул.Підгірна, 46, Україна
Отримано 22 листопада 2002 р., в остаточному вигляді –
10 січня 2004 р.
У надпросторовому підході, використовуючи модель короткодіючих
силових постійних, розраховані фонони двох крайніх членів шпіне-
левої системи Li1+xTi2−xO4 : металічної LiTi2O4 та діелектричної
Li[Li1/3Ti5/3]O4 . Застосовуючи різні моделі заміщення, досліджена
композиційна залежність оптичних фононів в центрі зони Бриллює-
на для Li1+x Ti2−xO4 в околі фазового переходу та Li1−xMgxTi2O4 .
Одно- та двомодову поведінку передбачено для F1u та F2u мод у
випадку тетраедричного та октаедричного заміщення відповідно.
Ключові слова: надпросторова симетрія, фононний спектр,
шпінель, Li1+xTi2−xO4
PACS: 61.50.Ah, 63.
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