## Phase diagrams of intercalation materials with polaron-type carrier localization

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*Physics of the Solid State, Vol. 42, No. 3, 2000, pp. 434–436. Translated from Fizika Tverdogo Tela, Vol. 42, No. 3, 2000, pp. 425–427.*

Original Russian Text Copyright 2000 by Titov, Dolgoshein.
**SEMICONDUCTORS **
**AND DIELECTRICS**
**Phase Diagrams of Intercalation Materials **
**with Polaron-Type Carrier Localization**
**A. N. Titov and A. V. Dolgoshein**
*Ural State University, pr. Lenina 51, Yekaterinburg, 620083 Russia*
*e-mail: [email protected], [email protected]*
**Abstract**—A study is reported on the effect of the position of the polaron band relative to the Fermi level on

the phase diagram. The validity of the phase stability criterion obtained is tested in the specific example of sil-

ver-intercalated TiSe2 and TiTe2.

* 2000 MAIK “Nauka/Interperiodica”.*
Phase diagrams of intercalation compounds based
Because the main distinctive feature of the polaron-
on layered titanium dichalcogenides with a common
type carrier localization consists in the density-of-
states being temperature dependent, it appears reason-

*x*Ti

*X*2 (

*M* = Ag, Ti, Cr, Fe, Co, Ni;

*X* = Se, Te)
can be divided in two groups. The compounds with

*M* =
able to study the effect of the Fermi level position on its
Ti, Cr, Fe, Co, and Ni belong to the first group. For low
change induced by the polaron band collapse (the
intercalant concentrations, these materials exhibit the
presence of disordered solid-solution regions [1–3].

To do this, we performed simulation of the concen-
The second group comprises materials with

*M* = Ag,
tration dependences of the chemical potential, which
where insertion of arbitrarily small amounts of an inter-
were derived from the charge neutrality condition sim-
calant brings about decomposition into a phase
enriched in silver and the starting material [4, 5]. It was
shown [6, 7] that intercalation of titanium dichalco-genides with the above metals results in carrier local-
ization in the form of small-radius polarons, which can
provide a dominant contribution to thermodynamicfunctions. Viewed from this standpoint, the materials of
where

*p * is the hole concentration and

*N* is the intercal-
the first and second groups differ in the position of the
ant concentration. The density of states in the conduc-
polaron band relative to the Fermi level of the starting
tion band was written as a sum of the original and
material. In materials of the first group, the polaron
states lie below the Fermi level and can be probed with
X-ray photoelectron spectroscopy [6]. In materials
belonging to the second group, the kinetic properties
[8, 9] and the behavior of thermodynamic functions
[10] can be interpreted only by assuming that the
ρ =

*g*(

*x*)

*N*∆/{(ε –

*E *)2 ∆2
polaron band is located above the Fermi level of thestarting material. The difference between the phase dia-
grams of these two groups of materials should be obvi-
states for a quadratic dispersion law (i.e., the density of
ously associated with different concentration depen-
states in the starting material),

*g*(

*x*) is a function relating
dences of the thermodynamic functions, which is
the concentration of the localization centers to that of
related to differing positions of the polaron band rela-
the intercalant, depending on the extent of ordering of
tive to the Fermi level of the material to be intercalated.

the latter, ∆ is the width of the impurity band, and

*Ep* is
Electronic structure parameters derived by fitting the edge of the homogeneity region for Ag

*x*TiSe2 and Ag

*x*TiTe2 together with
1063-7834/00/4203-0434$20.00 2000 MAIK “Nauka/Interperiodica”

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
PHASE DIAGRAMS OF INTERCALATION MATERIALS
its position. Because structural studies of Ag–TiSe
Ag–TiTe2 do not reveal intercalant ordering along the

*c*axis, the polaron center concentration can be written inthe gas approximation (as in [9]), and the

*g*(

*x*) function
can be presented in the form

*g*(

*x*) = 2

*x*(1 –

*x*), where

*x*
The concentration dependences of the chemical
potential were calculated in this way for the cases of awide impurity band (2 eV) and a narrow one (0.001 eV)for the temperatures below (100 K) and above (550 K)its collapse, respectively (Fig. 1). All the other parame-ters were fixed.

The numerical values of the effective electron and
hole masses and the character of disorder on the inter-calant sublattice were chosen close to the realistic ones

**Fig. 1.** Schematic illustration of the polaron band collapse

The concentration dependence of the polaron shift
obtained by subtraction of these two dependences (see
Fig. 2) reveals the presence of a maximum and acquiresnegative values for low carrier concentrations. By defi-nition, ∆µ/∆

*T* is the entropy of the electron subsystem,
which, taking into account the constancy of the vibra-
tional contribution [14] and assuming a constant con-figurational contribution (i.e., the absence of an order-disorder transition), can be equated to the total entropyof the material within the temperature interval includ-
ing the polaron band collapse. Obviously enough, only
the concentration region for which ∆µ/∆

*T* > 0 has aphysical significance. Therefore, the condition ∆µ/∆

*T* = 0(i.e., ∆µ = 0) at the boundary of the single-phase region
(

*x* = 0.22 in Ag–TiSe2 [4] and

*x* = 0.55 in Ag–TiTe2 [5])permits one to determine the parameters of the carrier

**Fig. 2.** Variation of the chemical potential (polaron shift) for

energy spectrum in these systems. The values of the
Ag

*x*TiSe2 (curve

*1*) and Ag

*x*TiSe2 (curve

*2*) with concentra-
tion. Vertical lines identify the boundaries of single-phase
parameters obtained by this method are listed in the
table together with literature data.

As is seen from the table, for Ag

*x*TiTe2, one observes
a satisfactory agreement between the obtained value of

*Ep* and the data quoted in [10]. The discrepanciesbetween the carrier effective masses for Ag
obtained in this work and in [10] are probably associ-ated with the fact that the authors of [10] neglected the
concentration dependence of the vibrational contribu-tion, which is apparently wrong, because a change inthe impurity concentration entails a change in the num-
ber of phonon modes. By contrast, the comparison withexperiment in this work was made at one point (at the
boundary of the single-phase region), where the vibra-
tional contribution at a fixed concentration is constant[14]. The discrepancy between the parameters obtained

**Fig. 3.** Schematic illustration of the dependence of the

and the literature data for TiSe2 can be caused by the
polaron shift on the position of the polaron band (

*Ep*) rela-
fact that the effective carrier masses in [12] were deter-
tive to the chemical potential (µ).

mined for the low-temperature charge-density-wavestate (

*T* ~ 200 K), whereas the present study was carriedout above the decomposition temperature of this state
Fig. 3, determines the stability criterion of phases with
the polaron type of carrier localization. For compounds
Thus, the dependence of the polaron shift on the
intercalated by transition metals µ >

*Ep*, i.e., the com-
mutual position of µ and

*Ep*, presented graphically in
pound is stable for any intercalant concentration. If,
PHYSICS OF THE SOLID STATE Vol. 42 No. 3 2000
6. A. Titov, S. Titova, M. Neumann,

*et al.*, Mol. Cryst. Liq.

*p* (as is the case with Ag

*x*TiSe2), the com-
pound is stable starting from the concentrations corre-
Cryst.

**311**, 161 (1998).

7. V. G. Pleshchev, A. N. Titov, and A. V. Kuranov, Fiz.

Tverd. Tela

**39**, 1618 (1997) [Phys. Solid State

**39**, 1442

(1997)].

8. V. M. Antropov, A. N. Titov, and L. S. Krasavin, Fiz.

Support of the Russian Foundation for Basic
Tverd. Tela

**38**, 1288 (1996) [Phys. Solid State

**38**, 713

(1996)].

Research (grant no. 97-03-33615a) and of the Ministryof Education (grant no. 97-0-7.1-169) is gratefully
9. A. N. Titov and S. G. Titova, J. Alloys Comp.

**256**, 13

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**40**,

1187 (1998) [Phys. Solid State

**40**, 1081 (1998)].

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**34**, 3593

(1992) [Sov. Phys. Solid State

**34**, 1924 (1992)].

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*et al.*, J. Solid
State Chem.

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2. O. Yu. Pankratova, L. I. Grigor’eva, R. A. Zvinchuk,

*et* *al.*, Zh. Neorg. Khim.

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13. D. K. G. de Boer, C. F. van Bruggen, G. W. Bus,

*et al.*,
Phys. Rev. B

**29**, 6797 (1984).

3. Y. Tazuke and T. Takeyama, J. Phys. Soc. Jpn.

**66**, 827

14. A. S. Alexandrov and N. F. Mott,

*Polarons and Bipo-*
*larons* (World Scientific, Singapore, 1995).

4. A. N. Titov and S. G. Titova, Fiz. Tverd. Tela

**37**, 567

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5. A. N. Titov, Neorg. Mater.

**33**, 534 (1997).

PHYSICS OF THE SOLID STATE Vol. 42 No. 3 2000

Source: http://www.interkal.narod.ru/Articles/PD-polaron_en.pdf

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