広島大学学位請求論文

Monte Carlo Simulation of Atomic Distribution of Guest Atoms
Intercalated into Layered TiS2 Crystals(層状結晶TiS2にインターカレーションしたゲスト原子の原子分布のモンテカルロ・シミュレーション)

根岸 彩子

目

次

1.主論文
Monte Carlo Simulation of Atomic Distribution of Guest Atoms Intercalated into
Layered TiS2 Crystals(層状結晶TiS‑にインターカレーションしたゲスト原子の原子分布のモンテカルロ・シミュレーション)
根岸 彩子

2.公表論文(1) Simulation of Atomic Distribution of Guest Atoms in Layered lT‑TiS2 Crystal using
Monte Cairo Method and Its Effect on the Magnetic Properties and Local Structures
S. Negishi, H. Negishi, M. Sasaki, and M. Inoue
Journal of the Physical Society of Japan 69 (2000) 25 14‑2530.

(2) MonteCarlo Simulation of Intercalated Atomic Distributions in Layered Dichalcogenide
S. Negishi, H. Negishi, M. Sasaki, and M. Inoue
Molecular Crystals and Liquid Crystals 341 (2000) 69‑74.

3.参考論文( 1 ) Observation of the Fracture Plane of Potassium Iodide Formed at the Pressure‑Induced
Phase Transition

M. Nomura, S. Negishi, Y. Okano, and H. Fujiwara
Journal of the Physical Society of Japan 57 (1988) 1341‑1343.

(2)二源同時斜め蒸着Fe膜の磁気異方性と柱状構造

慶徳 進,根岸 彩子,土取 功,後藤 道太
日本応用磁気学会誌14 (1990) 217‑220.
(3) Large Negative Magnetoresistance of Intercalation Compound Mn TiS‑
M. Inoue, H. Negishi, H. Yamada, S. Negishi, and M. Sasaki
Molecular Crystals and Liquid Crystals 311 (1998) 137‑ 142.

(4) Possibility of Quantum Hall Effect in Quasi‑Two‑Dimensional 77‑Mo4On Crystal
M. Sasaki, N. Miyajima, H. Negishi, S. Negishi, M. Inoue, H. Kadomatsu, and
G. Machel
Solid State Communications 109 (1999) 357‑362.

(5) Impurity Scattering by Guest 3d Metals and Their Impurity Band Formation in M TiS9 (M=
3d Metals)
H. Negishi, S. Kakita, H. Yamada, S. Negishi, M. Sasaki and M. Inoue
Solid State Communications 112 ( 1999) 275‑279.

(6) A New Type of Bulk Quantum Hall Effect in Bi^Sn^Teg Crystals
M. Inoue, N. Miyajima, M. Sasaki, H. Negishi, S. Negishi, H. Kadomatsu, and
V. A. Kulbachinskii
Physica B: Condensed Matter 284‑288 (2000) 1718‑1719.

(7) EX∬S and XANES Studies on 3d Transition Metal Intercalation Compounds MJTiS.
H. Negishi, S. Negishi, M. Sasaki, M. Inoue, and S. Ohara
Molecular Crystals and Liquid Crystals 341 (2000) 63‑68.
(8)PossibilityofChargeDensityWaveTransitioninTelluriumMolybdenumOxideTeMo‑O'5‑16
H.Negishi,S.Negishi,M.Sasaki,M.Inoue,K.Takase,Y.Kubota,Y.Takano,and
K.Sekizawa
JapaneseJournalofAppliedPhysics39(2000)(inpress).

主論文

Monte Carlo Simulation of Atomic Distribution of Guest Atoms
Intercalated into Layered TiS2 Crystals

Saiko Negishi

Abstract

Monte Carlo simulations of atomic distribution of intercalated guest atoms in thelayered lT‑TiS2 have been performed by taking into account attractive or repulsive pair‑interactions between the neighboring guest atoms in the a‑axis plane (Val and Va2) and along
the c‑axis of the crystal lattice (Vcl and V2c); lattice size for computation: 18 × 18 × 6. X‑ray
diffraction patterns for FexTiS2 are calculated from the atomic distributions obtained using
these four pair‑interactions, in qualitative agreement with the experimental data of the 2a x 2a
x 2c short‑range ordered structure forズ= 0.15, 2寸3a x 2a x 2c superlattice for jc = 0.25, and寸3a x J3a x 2c superlattice for x ‑ 0.333 with fractional site occupancy. From the calculated
atomic distributions, we have evaluated the number of neighboring guest atoms, the formationof clusters, and percolation cluster, as well as its dimension, all of which are responsible forthe dynamical relaxation behaviors of the thermoremanent magnetization observed in thespin‑ and cluster‑glass phases of Fe;cTiS2. In particular, one‑ and two‑dimensional percolation
clusters are formed by the third nearest neighbors in the a‑axis plane for x = 0.15 and 0.25,
respectively, and a two‑dimensional one by the second nearest neighbors for x = 0.333, whichcorresponds to the magnetic phase diagram [spin‑glass (x = 0.15) and cluster‑glass (ズ= 0.25

and 0.333)]. Using the present results and EXAFS data, we have further discussed on thechange in the local structures near the host sulfur atoms by intercalation of the guest atoms in
FeJCTiS2 and CoxTiS2.
Similar calculations have been made for another type of intercalation compound ofstage‑2 AgJCTiS2 (x = 0.15) that shows a phase transition from the ordered J3a x J3a x 2c todisordered structure around 250‑300 K. The formation of stage‑2 structure can be reasonably
simulated by additional incorporation of repulsive pair‑interaction, Vc2, along the c‑axis
direction, and by taking into account the considerably weak pair‑interaction strengths,
compared to those of Fe;cTiS2, in satisfactory accord with the observed X‑ray patterns.
Furthermore, the order‑disorder phase transition can be understood by considering thetemperature dependence of atomic distributions, the thermal displacements of Ti, S, and Ag
atoms, and entropy change due to the phase transition. The essential difference between stage‑
1 Fe^TiSj and stage‑2 Ago 15TiS2 is discussed based on a local lattice deformation produced
by intercalation. The size of the guest atom plays a crucial role to the formation of stage‑1 or ‑0

2 compound; guest atoms with crystal radius larger than 1.0 A are found to produce
preferentially the stage‑2 structure in the host TiS2. Such a large deformation may be theorigin to exert repulsive pair‑interaction on the second nearest neighboring sites in the nextlayers.

Contents
Abstract
1 Introduction

1

2 Calculations

7

2. 1 Model Hamiltonian7

2. 2 Atomic distributions

8

2.3 X‑ray diffraction intensities
3 Calculated Results

10ll

3. 1 Effect of pair‑interaction Val on atomic distributionll

3.2 Effect of pair‑interactions Va2 and Vcl on atomic distribution

12

3.3 Atomic distributions in FeJTiS.

20

3.4 X‑ray diffraction patterns for Fe.cTiS221

3・5 Atomic distributions in stage‑2 Ago, 15TiS229

3.6 X‑ray diffraction patterns for Ago i5TiS2

en

4 Discussions38

4. 1 FeJTiS,

38

4. 1.1 Formation of clusters and percolationeft

4・ 1 ・2 Magnetic properties
4. 1.3 Local structures

ER

4.2 Phase transition in Ago i5TiS250

4.3 Comparison between FeJTiS2 and Ago 15TiS2

51

43

5 Conclusion

54

Acknowled gements

56

Reference s

57

Chapter 1
Introduction

Layered materials have been extensively studied from the scientific and technical interests
due to their low dimensional natures in crystal structures and electronic properties, and wide
possibility to form new and exotic materials by the insertion of the various guest atoms and/or

molecules into their van der Waals gaps between the constituent layers, so‑called "intercalation",
leading to a variety of applications such as a secondary battery, electrochromism, and catalyses.
As a host material there are known many layered materials such as graphite, transition metal
dichalcogenide TX2, molybdenum oxides, and clays [ 1‑5]. Their physical and chemical proper‑
ties are modified strongly upon intercalation of guest species. Schematic structures of the interca‑

lation compounds are shown in Fig. 1. 1, where the guest atoms are inserted into van der Waals
gaps per n host layers along the stacking direction. According to the periodicity of n, they arereferred to as stage‑n intercalation compounds; namely, stage‑1, stage‑2, and stage‑3 structures.
Such staging phenomena are often observed in various intercalation compounds, suggesting theexistence of rather long‑range interactions along the stacking direction.
Among others, lT‑TiS2 is a typical example of host material with a simple lT‑Cdl2 typelayered structure, where Ti atom is surrounded by six S atoms, forming constituent layers in thea‑axis plane stacked along the c‑axis by weak van der Waals force with the intralayer latticeO

spacing a ‑ 3.41 A and the interlayer spacing c = 5.70 A [Fig. 1.2(a)], and thus various guest
species G can be inserted into the van der Waals layers, forming intercalation compounds GxTiS2[Fig. 1.2(b)], where G is typically Ag, alkaline metal A (= Li, Na, K,...), 3d transition‑metal M,
and organic molecules [2‑10]. In particular, extensive studies ofMcTiS2 (= Ti, V, Cr, Mn, Fe, Co,
Ni, and Cu) have been performed, whose salient features are summarized as follows: (1) Upon
intercalation, the 3d metals M occupy the octahedral sites in the van der Waals layers to form
stage‑1 intercalation compound, and the intralayer lattice spacing a does not change substan‑tially, while the interlayer spacing c depends strongly on the guest M and its concentration x(large expansion for Mn guests and remarkable contraction for Co and Ni ones) [8,9]. Extended
X‑ray absorption fine structure (EXAFS) measurements of K‑edges ofM and Ti atoms [1 1, 12]have con丘rmed that the interatomic distance between the guest M and S atoms in M TiS2, R(M‑S), depends strongly on the kind of M but less depends on the concentration x, while that between
the host Ti and S atoms, /?(Ti‑S), is increased with x, meaning the small expansion of the hostlayers upon intercalation of the guest atoms. (2) Intercalation induces a charge transfer from theguest atom M to the host Ti 3d conduction band, leading to the change in the Fermi energy EF andthe density of states at JF, as found by specific heat measurements [13,14], photoemission spec‑
troscopic studies [15‑23], and transport measurements [24‑28]. (3) Various magnetic phases arefound in these compounds MxTiS2 (M ‑ V, Cr, Mn, Fe, Co, and Ni), such as paramagnetic (P),ferromagnetic (F), weak ferromagnetic (WF) , and antiferromagnetic (AF) phases [29‑37]. Amongthem, the Fe intercalate FtxTiS2 shows spin‑glass (SG, 0.01≦ x ≦ 0.20), cluster‑glass (CG, 0.20 <host layer
guest layer

c

.2
.l.J

°
<u
.≡

てI
el
こ
.i:
u
te
め

(a) stage‑1(b) stage‑2(c) stage‑3

Fig. 1. 1 Schematic structures of intercalation compounds with (a) stage‑ 1, (b) stage‑2, and (c) stage‑3.

l車‑ a 」‑1(b) GxTiS2

Fig. 1.2 Perspective and top views of the crystal structure of (a) TiS2 and (b) GxTiS2; the octahe‑dral sites for the guest atoms G are arranged to form 2‑dimensional triangular lattice in the a‑axis
plane.

2

Fig‑ 1 ‑3 Magnetic phase diagram of Fe;tTiS2; paramagnetic (P), spin glass (SG), clus‑
ter glass (CG), and ferromagnetism (F) [33].

x ≦ 0.40), and ferromagnetic phases (F, x > 0.40), as shown in Fig. 1.3 [33‑37]. In the SG and CG
phases the temperature dependence of ac magnetic susceptibility shows a cusp or peak, whichshifts to the lower temperature side when some static magnetic field is superposed, meaning thatthe observed cusp or peak are characteristic of the freezing effect due to the frustration between
spins or clusters, since the superposition of static field suppresses the fluctuation effects between
spins. (4) These electronic and magnetic properties of ATcTiS2 are understood primarily in termsof an ̀itinerant electron'picture rather血an a ̀localized'one, as predicted by band calculations
using self‑consistent APW methods [38‑42], and photoemission studies [15‑23], which have re‑vealed a strong 'hybridization'among the guest M 3d orbitals and the host Ti 3d and S 3p orbitals.
(5) In SG and CG phases of Fe;cTiS2, long‑time relaxation phenomena of the thermoremanent
magnetization Mr have been measured using an anomalous Hall effect to analyze the observed
time decay curves ofMf based on a domain theory [43‑46]. As a result, we have found that thesematerials have very broad equilibrium relaxation spectra, suggesting an important role of the
distribution of the guest Fe atoms, or the formation of their clusters.
Furthermore, the structural studies [47] by X‑ray diffraction for single crystals of FexTiS2
have shown the short‑range order of 2a x 2a x 2c for x = 0. 15, the superlattices of 2寸3a x 2a x 2c
for x ‑ 1!4, and J3a x J3a x 2c for x ‑ 1/3 (Fig. 1.4). According to neutron powder diffraction
patterns [48,49], the analysis by the Rietveld profile‑fitting method has revealed that the site
occupancy at each Fe site is not unity but some fractional, as shown by fan‑shaped black area in
Fig. 1.4. In the left side are shown schematic stacking layers of Ti, S, and Fe atoms along the c‑axis for (a) x = 0.15, (b)ズ= 1/4, and (c) x = 1/3, where the guest Fe atom layers at the position z

are 、indicated by bold solid and broken lines. The site occupancies for jc = 0. 15 [Fig. 1.4(a)】 aredetermined to be 0.0 and 0.2 at the sites la and 3c in the layer z = 1/4, and 0.3 and 0.1 at the sites

3

(a)x =0.15
Ti
S
Fe
S
Ti
S
Fe
S
Ti

z= 3/4z=1/4○(う○(う○(?(5<3O O(∋⑦9(53⑦<7>C>⑦⑦
Q‑‑⑦‥Q (3 0
(3‑‑0‑‑3 ⑦ 0
巧(5 (5 (*> (?)
わゥ(D⑦⑦
C1‑0‑0 0 0
0‑Cr‑O O O
1a

3c

z=1/4

lb

3d

z =3/4(b)x= 1/4
Fe
S
Ti
S
Fe
S
Ti
S
Fe

(c)x=1/3
T
S
Fe
S
Ti
S
Fe
S
Ti

z=3/4

<3・ゥ(5・⑦⑦●①⑦(50サO O●⑦①●⑦
CD3Aeゥサ

z=1/4

ゥ"cfi cD (5
2a

ォ?

z=1/4

z=3/4

0●○○●
○○●○○

00●○○
●○○●○

Fig. 1.4 (a) Short‑range ordered structure la x 2a x 2c forx ‑ 0.15, (b) 2寸3a x 2a x 2c superlattice forx = 1/4, and(c) V3a x寸3a x 2c superlattice forx = 1!3, determined from the neutron diffraction for Fe TiS2 [48]. In the left side
are shown schematic stacking layers of Ti, S, and Fe atoms along the c‑axis, where the guest Fe atom layers at theposition Z are indicated by bold solid and broken lines. In the right side are shown the intralayer structures ofFe
atoms at different positions. Broken lines mark the unit cell forx = 0.15, solid lines forx= 1/4 and 1/3, and thin solidlines for the psuedo‑hexagonal 2a x 2a x 2c superlattice with x = 1/4. Fan‑shaped black area indicates the fractionalsite occupancy at each Fe site determined experimentally. Solid and open circles indicate the full site occupancy andvacancy, respectively, in the ideal superlattices forx = 1!4 at z = 0 and 1/2, and for x = 1/3 at z = 1/4 and 3/4.

4

lb and 3d in z = 3/4 of the short‑range ordered lattice, respectively. Those for x = 1/4 [Fig. 1.4(b)]are determined to be 0.63, 0.19, and 0.18 at the sites 2a, 2b, and 4e in z = 0, and 0.20, 0.25, and0.25 at the sites 2c, 2d, and 4fin z ‑ 1/2 of the 2寸3a x 2a x 2c superlattice, respectively. Since theoccupancies of the 2b and 2c sites are nearly equal to those of the 4e and 4f sites, respectively, theFe lattice is regarded to be in pseudo‑hexagonal symmetry, forming approximately the la x 2a x
2c superlattice, as indicated by thin solid lines in Fig. 1.4(b). Forュ = 1/3 [Fig. 1.4(c)], the site
occupancies are 0.16, 0.75, and 0.05 at the sites 2a, 2c, and 2d in both z = 1/4 and z = 3/4 of the寸3a x J3a x 2c superlattice, respectively. These results suggest that in the real crystal of Fe^TiS^

the guest Fe atoms may have very complicated distributions in the van der Waals gaps.
Structural studies ofAg∫TiS2 have been extensively done 【50‑57】. Ag guest atoms areinserted into the octahedral sites of the van der Waals gaps in the host TiS2 to produce the stage‑
2 AgxTiS2 for the intermediate guest concentration range, 0.13 ≦ x ≦ 0.19 [54] or 0.17≦ I ≦ 0.22[56], below and above which the stage‑1‑ and stage‑ 1 Ag;cTiS2 are formed [54,56]. As shown in
Fig. 1.5(a), the interlayer spacing cln, which is given by lattice parameter c divided by stage
number n, increases with x up to x ‑ 0.12 for the stage‑1'and above x = 0.30 for the stage‑1, while
it is almost constant for the stage‑2 between 0.15 ≦ x ≦ 0.33; we can see a clearjump between
stage‑2 and stage‑1. The c‑axis structure in the stage‑2 Ag;cTiS2 with jc = 0. 18‑0. 19 determined by
X‑ray diffraction [5 1] is illustrated in Fig. 1.5(b). Ag atom intercalation expands van der Waals
Oロ

gap remarkably from 2.85 A to 3.56 A and shifts the Ti layers to off‑centered position between
the S layers. The X‑ray diffraction and heat capacity measurements revealed that an in‑planeorder‑disorder phase transition of the guest atoms in AgxTiS2 occurs around 250‑300 K [56,57] ,and another transition from stage‑2 to stage‑1 structure does around 1200 K [54]. Furthermore,
Kuroiwa et al [58] measured the precise X‑ray diffraction patterns of stage‑2 Ago 15TiS2, as well
as stage‑1 Ago 25TiS2. They found that Ago 15TiS2 shows the abrupt change of c‑axis parameteraround 250 K, rodlike diffuse scattering parallel to the c ‑axis at (1/3 1/3 0), (2/3 2/3 0) and theirequivalent positions, and the two‑dimensional disordered state above 250 K. Using the 2‑dimen‑
sional short‑range order parameters for Ago 15TiS2 obtained experimentally from the diffuse seaレ

tenngs, the in‑plane local arrangements ofAg atoms have been calculated by the Monte Carlo(MC) technique to show the variation with temperature from the ordered states of v3a x寸3a tothe disordered states [58]. However, there is no report, to our knowledge, that calculate the equi‑
librium atomic distributions of this material taking account of some guest‑guest interactions orinterlayer interactions, as done for GIC [59,60]. Different type MC calculations 【6 1 ] are made for
Li distribution in Li TiSe2 to compare with the experimental results of 77Se and 7Li NMR stud‑les, where a Boltzmann type thermal relaxation is introduced to allow a lithium atom to jump toan available empty site.
In the present work, computer simulations for the equilibrium atomic distribution of inter‑calated guest atoms in lT‑TiS2 using the MC method have been performed. Here we have taken
into account pair‑interactions between neighboring guest atoms in the triangular lattices stacked
with six layers (lattice size for computation: 18 × 18 × 6) with the periodic boundary condition

along the a‑ and c‑axis directions; the details of the calculations are presented in Chapter 2. 1
shall show the effect of various types of pair‑interactions (between nearest neighbors and/or
next‑nearest neighboring guest atoms) on their atomic distributions in GxTiS2. In particular, Fe

atomic distributions in Fe^T^ (x ‑ 0.15, 1/4, and 1/3) have been calculated successfully to5reproduce the observed short‑range order or superlattice with fractional Fe occupancy, and thosein Ago 15TiS2 to obtain the stage‑2 structure showing X‑ray diffuse rods. These calculated re‑suits are collected in Chapter 3. In Chapter 4, based on the calculated Fe atomic distributions, weshall discuss on the formation of guest atom clusters, percolation cluster, its dimensionality,
magnetic properties of FeJCTiS2, and the change in the local structures near the host sulfur atoms

by intercalation in MxTiS2. The order‑disorder phase transition in the a‑axis plane for Ago 15TiS2
will be discussed from the temperature variations in the calculated Ag atomic distributions. Fur‑thermore, the pair‑interaction strengths employed to obtain the atomic distributions for FexTiS2
and in Ago 15TIS2 are discussed. The main results obtained in the present study are summarizedin Chapter 5.

1 .4 0(b)

1 .4 5
1 .7 8

1 .7 8
1 .4 5
1 .4 0C = 5.695‑
2 .8 6TiS,

AgxTi S2

Fig. 1 ・5 (a) Lattice parameter c normalized by the stage number n, c/n, for Ag;cTiS2 plotted against the guest
concentration x [56] ; circle and square symbols correspond to the stage‑ 1 (ォ= 1) and stage‑2 (n = 2), respectively.

(b) The c‑axis structures in TiS2 and stage‑2 Ag^TiSj with x ‑ 0. 1 8‑0. 19; the interlayer distances are given in units
ofA【51].

6

Chapter 2
Calculations
昏2.1 Model HamiltonianIn the intercalation compound GxTiS2, the stable atomic distribution of the guest atoms
minimizes the total energy including many‑body interactions among all guest and host atoms
within the system. In the present calculations, we have simply taken into account the pair‑inter‑actions between the neighboring guest atoms G, which lie at the octahedral sites arranged in two‑
dimensional triangular lattices in the a‑axis planes stacked along the c‑axis, and performed theMC simulations for their atomic distributions.
We use the following Hamiltonian for the system,

H= ‑∑ ∑Vrkninj(1)<i,j> kwhere nt (‑ 1 or 0) is an occupancy of the /‑th site, njl an occupancy of they‑th site located at thek‑th neighboring position, and Vk a pair‑interaction strength between the guest atoms in the Jt‑thneighbor. Here k = al, a2, a3,... stand for the intralayer sites of the first nearest neighbors (1stNNs), the second (2nd NNs), the third (3rd NNs),〜, in the a‑axis layer [Fig. 2.1(a)], k = cl, c2,・.., for the interlayer sites of the lst NNs, 2nd NNs,... in the nearest layers separated by the lattice
constant c, and k = 2c for the lst NNs in the next nearest layers separated by 2c along the c‑axis[Fig. 2.1(b)]. In the above Hamiltonian we consider here at most five terms ofk = al, a2, cl, c2,
and2c,as,V,n(2),alJ*l+va2nf+Vclnfl+Vc2nf+Von
2cj2c

⑳◎

alゥ〇
◎○○

aL竺a@ 」fJ(a) Intralayer sites(b) Interlayer sites

Fig.2. 1 Definition of lattice sites used for the MC simulations, (a) Intralayer sites around a guest atom G
in the a‑axis layer for GxTiS2, where al, a2, and a3 stand for the lst, 2nd, and 3rd NN sites, respectively.
(b) Interlayer sites along the c‑axis, where cl and c2 stand for the lst and 2nd NN sites in the nearest
layers, respectively, and 2c the site in the next nearest layers.

7ァ2.2 Atomic distributions

For the triangular lattice stacked with six layers (lattice size: iV= 18 × 18 × 6) with theperiodic boundary condition along the a‑ and c‑axis directions of G TiS2, we have carried throughthe numerical calculations using Metropolis method, one of MC methods for canonical ensemble【62‑65].
In the initial state, Nx guest atoms are arranged following the relation,
サ,.=! (1≦i≦Nx),

fi:=0(Nx<i≦N).

(3)

The example of the initial atomic distribution in the six consecutive layers (m = 1‑6) of GxTiS2
with x = 0.25 are shown in Fig. 2.2. Next, we select a vacant 5‑th site to transit from an occupied
i‑th site using random numbers, and estimate an energy change Mi caused by this atomic transi‑tion; Aだ= H(ni = 0, ns = 1) ‑ //(ォ・= 1, ns ‑ 0). Then the transition probability W irom i‑th site to∫‑th site is given by,

W=exp(‑AH/T) for Atf>O
for AH≦0.

(4)

ForanegativevalueofAH,atomictransitionfromi‑thsitetos‑thsiteoccurs(W=1),whileforapositivevalueofAHitoccurswithprobabilityW=exp(‑A///T),whichisperformedtoallow
theatomictransitiononlywhentherandomnumberproducednewlyislessthanW.InoneMonte
Carlostep(MCS),suchanprocessismadesuccessivelyfortheNxguestatoms,yieldingone
atomicdistributionpatternforMCS=1.Thuswecanobtainanequilibriumatomicdistribution
forGyTiS2atgiventemperatureTwithpair‑interactionstrengths,V^,asparametersafterappro‑
pnateMCSs;V^munitsofKisusedthroughthiswork.
InFig.2.2areshowntypicalatomicdistributionsofGTiS2(x=0.25)calculatedwitha
repulsivepair‑interactionstrengthVal(=‑10,000K)atMCS=0,1,and100,wherethetempera‑
tureistakentobeT=870C=1,143K,thegrowthtemperatureofFe^T^crystalbyachemical
vaportechnique【8].Wi血increasingMCSnu血ber,山eguestatomsarescatteredtobeanearly
uniformatomicdistribution.InordertomakeMCSeffectmoreclear,thevaluesofEINxare
plottedagainstMCSnumberforthreecasescalculatedwithfixedpair‑interactionstrengthVj(=
1,000K,‑1,000K,and‑10,000K)inFig.2・3,fromwhichitcanbeseenthatthetotalenergyEof
thesystembecomesalmostconstantbeyond60MCSwithincalculationalerrorsforallcases.In
thepresentstudy,therefore,calculationsupto100MCSwereperformedtoobtainanequilibrium
atomicdistribution.
Furthermore,tocharacterizetheobtainedatomicdistributions,wehaveevaluatedtheav‑eragednumberofthek‑thneighboringatom,z^(k=al,a2,a3,cl,and2c),definedbyzk‑志亨I>;fijk.
j(5)ThevalueofzkforonepatternisobtainedastheaveragedvalueoverMCS=61‑100,andthenthe
mostprobablevalueandthedeviationofzkareobtainedfrom10patternsproducedfromdifferent
randomseeds.

8MCS=0*‑

MCS=1 ‑*‑MCS=100

Fig.2.2 Typical examples of the variation in guest atomic distributions with Monte Carlo step (MCS)
from the initial state (MCS ‑ 0) for GxTiSz (x ‑ 0‑25) with interaction strengths Va, = ‑10,000 K.

9

6000

4000

2000

■ー

お:
ヽ‑′

0

〉く

Z
i=さ

u

‑2 000

20

40

60

80

1 00

MCS

Fig.2.3 Typical examples of the variation in the total energy E with MCS‑ The values ofJ divided by
the number of the guest atoms Nx, E/Nx, are plotted against the MCS number for GxTiS2 (x ‑ 0.25)
with three different interaction strengths Val = ‑10,000 K (solid squares), ‑1,000 K (open squares),and +1 ,000 K (open triangles).

曇2.3 X‑ray diffraction intensities

Using the atomic distributions obtained by MC simulations, we have calculated the X‑ray
diffraction intensities. The X‑ray intensity at a reciprocal lattice point (<j; 77 」) in the unit cell of
TiS2 is expressed by

1 1+cosz2β

I(giO= ‑72
y2¥F(Z 7? Ol(6)where v is a volume of the lattice corresponding to a normalized factor, (1+ cos220)/2 is thepolarization factor, and the structure factor F(E, 7] 」) is given by the summation of the product of
the atomic scattering factorfAi16) and the geometrical factor FAB, 77 」) for the constituent ele‑ments A = Ti, S, and guest element G as, F(g ri Q = ZfA(2e) FA(帥」). For the atomic distribution
obtained with the site occupancy of the i‑th site ni and its position (x‑ yt z‑), FG(% 77 0 is written as,
FG(」nQ=Xn.exp[2冗i ^+ n?,‑+ 」*,‑)],(7)from which the intensity of any reciprocal lattice point (」 rj Q can be evaluated.
We should note here that the calculated X‑ray intensities丘om a single pattern of the atomic
distribution in the 18 × 18 × 6 lattice depend strongly on the ̀random seeds'used, but the aver‑aged number of the NN atoms, zk, is almost independent of the ・random seeds'. This means thatthe lattice size 18 x 18 x 6 we employed may be too small to reproduce the structural details in thereal system. Therefore, we have taken the averaged intensities of ten patterns simulated using
different random seeds to obtain reasonable intensity profiles, which are insensitive to the ran‑dom seeds used.

10Chapter 3

Calculated Results
蚤3.1 Effect of pair‑interaction V x on atomic distribution
At first, we have calculated the atomic distribution by considering only the pair‑interaction
Val between the lst NN guest atoms in GxTiS2 over the whole concentration x = 0‑1, where thetemperature is taken to be 7= 870 ‑C = 1,143 K, the growth temperature ofM TiS2 crystal by achemical vapor technique [8]. The interaction strength V j is taken to be from ‑10,000 K to 10,000
Kby a step of 1,000 K; the ratio of Val to T(= 1,143 K), Val/T, is nearly from‑9 to 9. As typicalresults, the calculated equilibrium distributions in the six consecutive layers (m = 1‑6 from bot‑
tom to top; see Fig・ 2.2) of GxTiS2with guest concentration x = 0.15, 0.25, 0.333, and 0.50 are
shown in Figs. 3. 1‑3.4, respectively, for丘ve cases of attractive interaction (Val = 10,000 K, 1,000K), no interaction (Val ‑ 0 K), and repulsive interaction (V , ‑ ‑1,000 K, ‑10,000 K). In the caseof ^al = ‑ K'the atoms are distributed randomly [Figs. 3. 1(c)‑3.4(c)]. For the strong attractive
interaction of l㌔ = 10,000 K, they remain in the same pattern as the initial condition 【Figs.

3. 1 (a)‑3.4(a)], because the transition probability is vanishingly small [exp(‑AJ7/7) = exp(‑60,000
K/1,143 K) ‑ 10‑22]. For the attractive force of V , = 1,000 K comparable to T, they stand togather to form some 'clusters'in each layer [Figs. 3. 1(b)‑3.4(b)]. While those for the repulsive
interactions of Val are scattered a little for Val = ‑1,000 K [Figs. 3.1(d)‑3.4(d)], and homoge‑
neously over the lattice sites for the strong repulsion of V j ‑ ‑10,000 K [Figs. 3.1(e)‑3.4(e)].
Thus we see that the calculated distribution patterns are strongly dependent on the paiFinterac‑tion strength Val. Moreover, we can see some ordering atomic arrangements in the a‑axis planesin the patterns calculated with the repulsive pair‑interaction‰, which are nearly to the super
structures 2a x 2a forュ‑ 0.25 [Fig. 3.2(e), Val = ‑10,000 K] and J3a x V3a forx = 0.333 [Fig.

3.3(e), Val = ‑10,000 K], and labyrinth‑like patterns forx = 0.50 [Fig. 3.4(d), Val = ‑1,000 K; Fig

3.4(e), ㌔ =‑10,000K].
The averaged number of the neighboring atoms zk [Eq. (5)】 (k = al, a2, a3, cl, and 2c ) isplotted against Val for different guest concentrations x = 0. 15, 0.25, 0.333, and 0.50 in Fig. 3.5.
The values of each zk are not changed for the attractive interaction V , larger than the systemtemperature T (= 1, 143 K), Val≧ 2,000 K, where the calculated atomic distributions are the sameas the initial patterns because of the vanishingly small transition probability (< 3 × 10‑5), as
described above for Val = 10,000 K [Figs. 3.1(a)‑3.4(a)]. For V&1≦ 1,000 K, on the other hand, zk
depends remarkably on Val along the a‑axis direction, while it is constant along the c‑axis direc‑

tion. With decreasing Val, zal decreases, za2 shows a minimum at Val = 0 K, and za3 shows aplateau between ‑2,000 K and 1,000 K. The value ofzk for k = cl and 2c is proportional to jc, as theresults of the random distributions along the c‑axis direction.
In Fig. 3.6 are depicted the guest concentration dependence of the averaged number of theneighboring atoms zk calculated with different pair‑interaction strengths Val. For Val = 0 K (solid
circles), zkis proportional tox {zk = 6xfor k = al, a2, and a3, andzk‑ 2xfor/:‑ cl and 2c). Thisllresult means that the atomic distributions are random both in the a‑axis plane and along the c‑axis. However, the concentration dependence of the averaged numbers zal , za2, and za3 in the a‑axis plane is quite different for attractive and repulsive pair‑interactions. For Val = 1,000 K and
10,000K,

thevalues

of

/a一,

za2,

and

za3

increaseprofoundly

with

x

compared

with

those

for

Va]

= 0 K. For a repulsive pair‑interaction, zal is decreased by varying from V , = ‑1,000 K to ‑10,000
K. In particular, for Val = ‑10,000 K, we get za] ‑ 0 up tox = 0.333, while the curves forza2 shows
two humps at x = 0.333 and 0.666, where that ofza3 has two dips. These two characteristic
concentrations are corresponding to the structural ordering寸3a x J2>a in the a‑axis plane ob‑served experimentally. On the other hand, the values ofzcl and z2c along the c‑axis are almost
independent of ¥㌔1・ These results indicate that the pair‑interaction between the lst NNs in血e d‑axis plane also affects the distribution of the 2nd and 3rd NNs in the planes. According to the
EXAFS spectra forMxTiS2(M=Mn, Fe, Co, and Ni, 0 ≦x ≦ 0.33) [11,12], zal ‑ 0 in the realsystem, which suggests that there exists a very strong repulsive interaction, in accordance withour calculations (㌔ = ‑ 10,000 K).
喜3.2EffectofVa2andVclonatomicdistributions
Furthermore,wehaveinvestigatedtheeffectofpair‑interactionsofthe2ndNNsinthea‑
axisplane,Va2,andofthelstNNsinthenearestlayersalongthec‑axis,Vj,ontheatomic
distributions.Forsimplicity,intheHamiltonianweconsideronlytwotermsofk=alanda2inthea‑axisplane,andk=alandclalongthec‑axis,respectively,as
H‑‑I^V/^+Vtfiij<i,j>誠),andtf‑‑Xit,.(valn/1<i.7>」‑ii";c).(8)Withthestrongrepulsivepair‑interaction(Val=‑10,000K)obtainedabove,wehavecalculated
theatomicdistributionandaveragednumberzkforGxTiS2(・*=0.15,0.25,0.333,and0.50),with㌔andl㌔Iasparameters.
Theaveragednumberszkfork=al,a2,a3,cl,and2careplottedagainstVa2andVclinthe
leftandrightsidesofFig.3.7,respectively.Ascanbeseen,withincreasingVa2,thevaluesofza2
showasharpincreasearoundVa2‑0,andthoseofza3asharpdecrease,butthevaluesofzal,ZCV
andz2carealmostunchanged.WithincreasingVcl,ontheotherhand,bothzclandz2cshowan
appreciablechangenearVcl‑0,butzal,za2,andza3remainunchanged.Foranysetofthesetwo
pair‑interactionstrengthsValand7a2orValandyc],however,theatomicdistributionhavingthe
2c‑periodicityalongthec‑axisforFeJtTiS2observedexperimentallycannotbereproduced,sug‑gestmganeedforfurtheradditionalpair‑interactionbetweentheguestatomsseparatedby2c,
'2c‑

12

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1^ォT重量買主L≒竜三(a)+10,000K (b)+1,000K(c)0 K(dト1,000K (e)‑10,000 K

Fig. 3.1 Atomic distributions of guest atoms G in the six consecutive o‑axis layers (m = 1‑6 from bottom to top) for
GxTiS2 (x = 0.15), calculated with (a) attractive interaction Val = +10,000 K, (b) Val = +1,000 K, (c) no interaction
Val = 0 K, (d) repulsive interaction Val = ‑1,000 K, and (e) Val = ‑10,000 K.

13

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11三II二Ill…k‑(a)+10,000K (b)+1,000K

(c)OK

≡::

i

(d)‑1,000K (e)‑10,000K

Fig. 3.2Atomic distributions for GxTiS2 (x ‑ 0.25), calculated with (a) Val = +10,000 K, (b) Val ‑ +1,000 K, (c) Va]=OK,(d) Val =‑1,000K,and(e) Val =‑10,000K.

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Fig. 3‑3 Atomic distributions for GxTiS2 (x = 0.333), calculated with (a) Val = +10,000 K, (b) Va, = +1,000 K, (c) Va,=OK, (d) Val =‑1,000K, and(e) Va, =‑10,000K.

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mft,¥gQ:is^t<mm>.ws&

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jpヨI:;‑ ‑;…‥し三^k"蝣 二(a)+10,000 K (b)+1,000K(c)OK(dト1,000 K (e)‑10,000K

Fig. 3.4Atomic distributions for GxTiS2 (x = 0.50), calculated with (a) Val = +10,0㈱ K, (b) Vai = +1,000 K, (c) Val=OK,(d)㌔ =‑1,㈱OK,and(e)㌔, =‑10,0∝IK・

16

‑10

‑5

10

Val (103K)

‑10

‑5

10

Val (103K)

Fig. 3.5 Averaged number of neighboring guest atoms in the atomic distributions for GrTiS, with x =0.15, 0.25, 0.333, and 0.50, zk (k = al, a2, a3, cl, and 2c), plotted against pair‑interaction strength V ,.

17

(d)

(e)

0

0.2

0.4

0.6

0.8 1.0

X

0

0.2

0.4

0.6

0.8 1.0

X

Fig. 3.6 Concentration dependence of zk for GxTiS2, calculated with the different fixed pair‑interac‑tion strength V j (= ‑10,000 K, ‑1,000 K, 0 K, +1,000 K, and +10,000 K).

IIS

‑10

‑5

10

Va2 (103K)

‑10

‑5

10

Vcl (103K)

Fig. 3.7 Dependence ofzk on V^ (left side) and on Vcl (right side) for G^TiS, (x ‑ 0.15, 0.25, 0.333,
and 0.50), calculated with the fixed pair‑interaction strength V , (= ‑10,000 K).

19

喜3.3AtomicdistributioninFeJTiS.
Inordertoreproducetheobservedshort‑range2c‑periodicityalongthec‑axisforx=0.15
andthelong‑rangeoneforx‑1/4and1/3[47],wehavetakenfourpair‑interactionsVal,Va2,Vc],
andK2c,as
H‑一享1Kin,‑al+y,'ary*2nja2+vcln/l+V2cォ.2c(9)SOQS
Hereafterwecallitmultiplepair‑interactionmodel.UsingasetofparametersVk(k=al,a2,cl,and2c)rangingfrom‑10,000Kto10,000Kbyastepof1,000K,wehavecalculatedtheatomic
distributionsandX‑raypatternsforFexTiS2withx=0.15,0.25,and0.333.Andfromthecom‑
parisonoftheseresultswiththeX‑raypatternsobservedexperimentally[47]andthosecalculated
withtheformfactorsusingthefractionaloccupanciesdeterminedfromtheneutronpowderdif‑
fractions[48],wehaveobtainedthebest‑fitvaluesofV^,aslistedinTableI(seelaterindetails).
InFigs.3.8(a)‑3.8(c)aredepictedtheatomicdistributionscalculatedwiththebest‑fitvaluesVk
forFexTiS2withx=0.15,0・25,and0.333,respectively,inthesixconsecutivelayers.
Reflectingthestrongrepulsiveinteractions(Va]=‑10,000K)inthea‑axisandalongthec‑axis(^cl=‑5'‑0‑K),theFeatomsarescatteredwithnonearestneighboringguestatomsineachlayerandalongthec‑axisforallsamples.Inthecaseofx=0.15[Fig.3.8(a)】,manyFeatomsarearrangedtohavethe3rdNNatomsseparatedbythedistance2aeachotherinthea‑axisplane,
showingtheshort‑rangeorderedstructureof2ax2a.Forx=0.25[Fig.3.8(b)],almostallofthe
Featomsarearrangedtohavethe3rdNNsinthea‑axisplane,andthearrangementsinthe2nd
NNlayers(k=2c)alongthec‑axisareingoodagreementwiththeobservedsuperlattice2寸3ax
2ax2corlax2ax2c.Furthermore,inthecaseofx=0.333[Fig.3.8(c)],theFeatomshavethe
2ndNNsinthea‑axisplaneand2c‑periodicityalongthec‑axis,formingtheJ3axJ3ax2C
superlattice.FromTableI,itisnotedthattheattractiveinteraction(V2c=+5,000K)isnecessary
toproducethenearlyregularpatternswiththelong‑ranged2c‑periodicityalongthec‑axis.In
addition,weseethatVa2isnegative(repulsiveinteraction)forx=0.25,whileitispositive(aトtractiveone)forx=0.333,andthusthesignofV^determineswhichsites(2ndNNor3rdNNsites)aremorestableandpreferableinthea‑axisplaneofFeTiS?.SuchvariationsofVoandV,
du.K2cwithguestconcentrationxmightbeascribedtothesimpleapproximationofthemany‑bodyinteractionsinFexTiS2bythemultipletwo‑bodyinteractionsbetweentheguestatoms.
BasedontheforegoingatomicdistributionsoftheguestFeatoms,wehaveevaluatedthe
averagednumberzkforthefc‑thneighboringatoms(k=al,a2,anda3inthea‑axisplaneandk=cland2calongthec‑axis).TheresultsforFeTiS2withJ=0.15,0.25,and0.333areshownin
theleft,center,andrightcolumnsofFig.3.9,respectively,wherethevaluesofzk(indicatedby
rectangles)areplottedagainstk(hereafterreferredtoasnumberspectrum)forfourdifferent
distributionpatterns,fromthetoptothefourthrows,calculatedwiththeattractiveinteractionVal(=+1,000K)[Figs.3.1(b)‑3.3(b)],nointeraction(Val=0K)[Figs.3.1(c)‑3.3(c)],therepulsive
oneva¥(=‑10,000K)[Figs.3.1(e)‑3.3(e)],andmultiplepair‑interaction[Figs.3.8(a)‑3.8(c),Table
I1.
InthecaseofVal=0K,thevalueofzkisnearlyequaltozk=6xfork=al,a2,anda3,and

20

Table I Best‑fit values of the multiple pair‑interaction strengths (in units ofK), Vaj, Va2, Vc], and V2c, used for calculation of atomic distribution of theguest Fe atoms in FexTiS2 (x = 0.15, 0.25, and 0.333).

x=0.15x=0.25x=0.333‑ 10,000‑10,000‑10,000

l 'iLl‑5,000‑5,000+5 ,000cl‑5 ,000‑5 ,000‑5,0000+5 ,000+5 ,000

r:> [

K2c

zlc = 2x for k = cl and 2c, as described earlier. For the attractive interaction, the number zk is
reduced in the order, zal > za2 > za3, which suggests the formation of some clusters of the lst
NNs, as described later. For the repulsive one, zal becomes nearly zero but za2 has the largest
value in the a‑axis plane, and zcl and z2c have the same values as those for Val = 0 and + 1,000 K.
Meanwhile, the case of the multiple pair‑interaction model shows a different behavior. For x =0. 15, zal and za2 vanish but only za3 has a large value, which indicates that the guest Fe atoms of
the lst and 2nd NNs are scattered, but the 3rd NNs form the short‑range order of the form la x 2ain the a‑axis plane. Forx = 0.25 and 0.333, za3 andZa2 have the values as large as 4 and 5,
respectively, while zcl is zero and z2c ‑ 2, indicating that their structures are similar to the la x 2aand V3<2 ×寸3a superstructures, respectively, in the a‑axis plane with the 2c‑periodicity along thec‑axis.
Similarly, we have evaluated zk for the short‑range ordered structure and the superlattices
with fractional occupancies for J = 0. 15, 0.25, and 0.333 (Fig. 1.4), and the ideal superlattices forx ‑ 0.25 and 0.333 having site occupancy of unity (solid circles) or zero (open circles) 【Figs.
1.4(b) and (c), respectively], as shown in the lower two rows ofFig. 3.9. For the lattices withfractional occupancies ofx = 0.15 and 0.25, zk = 1‑2 for k = al, a2, and a3, whose spectra re‑semble to those for the random distributions (Val = 0 K). This similarity may be due to the small
difference in the fractional occupancies at the guest atom sites. For x = 0.333, the value of the 2nd
NNs m the a‑axis plane is largest among others (z^ = 3.7サzal, za3), showing the large differ‑ence in the Fe occupancies. For the ideal superlattices, on the other hand, the guest atoms arearranged regularly to have za3 = 6 andz2c = 2 forズ= 0.25, andza2 = 6 andz2c = 2 forx= 0.333.

蚤3.4 X‑ray diffraction patterns for FexTiS2Experimental results of X‑ray diffraction of Fe^T^ crystals with x = 0. 15, 1/4, and 1/3[47] are illustrated in Fig. 3. 10, where the diffuse scattering and superlattice reflection points aremarked in the reciprocal lattice of TiS2 in the left side, and their profiles parallel to the c ‑axis

along (3/20 Q forx=0.15, (0 1/2 0 forx= 1/4, and (1/3 1!3 」) forx= 1/3 aredepictedintherightside. It is noted that the profile forュ = 0.15 is very broad centered on 」= 0.5, while those forズ=

1/4 and 1/3 have sharp peaks at Q = 0, 0.5, and 1.0.

21In order to compare with these experimental results of Fejt.TiS2, we have selected the recip‑
rocal latticepoints along (ァ1/2 1/2) and (3/2 0 Q forx= 0.15, (」 1/2 1/2) and (0 1!2 0 forズ=
0.25, and (」 1/3 1/2) and (1/3 1/3 」) for x = 0.333, and calculated the X‑ray diffraction profiles
from the obtained atomic distributions above using Eqs. (6) and (7). For the superlattices, wehave employed the values of the fractional Fe occupancy ni determined experimentally [48,49] tocalculate FFe(^ 77 Q in Eq. (7).
The results are shown in Figs. 3.1 1‑3.13 for x = 0.15, 0.25, and 0.333, respectively, where
the intensity profiles parallel to the a ‑axis and to the c ‑axis are arranged in the left and rightcolumns, respectively. Here are shown three or four X‑ray patterns, from the top row to thebottom, for the distributions calculated with V x = ‑10,000 K [Figs. 3. 1(e)‑3.3(e)], multiple pair‑interaction model (Fig. 3.8 and Table I), the superlattice with fractional occupancy, and ideal
superlattice (Fig. 1.4). For the strong repulsive interaction (Val = ‑ 10,000 K), the X‑ray intensities
for three samples are too weak, as shown in the top rows of Figs. 3.ll‑3.13, which means that
there is no obvious short‑ or long‑range ordered structure in the a‑axis plane and along the c‑axis.

The X‑ray intensities calculated by the multiple pair‑interaction model show characteristic pro‑files (second rows ofFigs. 3.ll‑3.13): we note that (i) forx = 0.15, three peaks around号= 0, 0.5,
and 1 along (」 1/2 1/2) and avery broad peak around 」= 0.5 along (3/2 0 」), 00 forx = 0.25, threepeaks around 」=0, 0.5, and 1 along (」 1/2 1/2) and threepeaks at 」=0, 0.5, and 1 along (0 1/2 a,and (in) forx= 0.333, apeak around号= 0.33 along (g 1/3 1/2) andthreepeaks at 」= 0, 0.5, and1 along (1/3 1/3 Q. These calculated results are in qualitative agreement with those observed
experimentally (Figs. 3. 10) and also those calculated for the short‑range ordered lattice (x = 0. 1 5)
and superlattices with the fractional Fe occupancy (x = 0.25 and 0.333) (third rows of Figs. 3. 1 1‑
3.13).
The very broad peak around 」= 0.5 along (3/2 0 」) for x = 0.15 is quite similar to theexperimental profile in Fig. 3. 10(a), revealing that the calculated atomic distribution has not along‑range order of 2c but a short‑range order along the c‑axis. Furthermore, it is noted that thecalculated peaks along the a ‑axis are broader than those for the superlattices with fractional
occupancy, which indicates a small deviation from the long‑range ordered structure, or a random‑
ness of the Fe atoms in the a‑axis plane. Such randomness may have been omitted in the calcula‑tions for the superlattices with fractional Fe occupancy obtained by the Rietveld profile fitting.
The X‑ray intensities for the ideal superlattice with the occupancy of unity or zero (bottom
rows of Figs. 3. 12 and 3.13) are much stronger than those for other distributions, by 10 times forx= 0.25 and double forx= 0.333. Especially, forx = 0.25 there is no peak at g= 0.5 and 」= 0 or
[Figs. 3.12 (d) and 3.12 (h)], the latter of which is quite different from the pattern observed
experimentally [Figs. 3. 10(b)].

Thus, from our MC simulations based on the multiple pair‑interaction model, we haveobtained reasonable X‑ray diffraction patterns for FexTiS2 (x = 0. 15, 0.25, and 0.333), in satisfac‑
tory agreement with the experiments.

22

(a)x=0.15(b) x = 0.25(c) x = 0.333

Fig. 3.8 Atomic distributions of the guest Fe atoms in FeA.TiS2 with (a) x = 0.15, (b) x = 0.25, and (c) x =0.333, calculated with the multiple pair‑interaction model (Table I).

23x=0.15x = 0.25(a)Va1=+1,000K(f) V al = +1,000 K(I) V a1 = +1,000 K

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司開 閉 ‑l.

開 閉 闘 ■ ー
2V3ax
2ax2c
(ideal)

(k)

al a2

a3
k

. ー

d

2c

ai a2

a3

T

d

■

I
2c

k

Fig‑ 3.9 Averaged number zk (k = al‑2c) for different atomic distributions in Fe^TiS, withx = 0.15 (leftcolumn), x = 0.25 (center column), and x= 0.333 (right column), calculated with Va] = +1,㈱O K (top row), Val= 0 K (second row), Val = ‑10,000 K (third row), the multiple pair‑interaction model (fourth row), the short‑
range ordered structure or superlattices with fractional site occupancy (fifth row), and the ideal superlattices(bottom row).

24

0.4

(a)x=0.15

倭(b)x=1!4∴

0.3(3/2 0 ^

. .

サー
0.1

"

.〜
%

〜
...

0
5
J一1‑ll.

3

ヽt一′

(0

4

1 /2

g

i

.
‑

」3
2 2

∫
、

ll i
0

A

⊥

5
! (1 /3 1 /3 0(c)x= 1/3

4
.
‑

3

、
、
〜

2
1

A

▲

0

0

0.2 0.4 0.60.8 1.0⊆

Fig. 3. 10 Diffuse scattering and superlattice reflections of Fe^TiS2 single crystals with (a) x ‑ 0. 15,(b) x = 1/4, and (c) x ‑ 1/3 in the reciprocal lattice ofTiS2[47]. Bragg reflections, diffuse scatteringand superlattice reflections are marked by solid circles, large and small open circles, respectively.
Right parts of each figure are profiles of diffuse scattering and superlattice reflections parallel to thec ‑axis along (a) (2/30 Qforx‑0‑15, (b) (0 1/2 」)for*‑ 1/4, and(c)(l!3 1/3 0forx= 1/3.

25

iM?

0.8

(盲1/2 1/2)

(a)

0.6

1.0

(d)

(3/2 0 C)

0.4

i

0.2

‑ ー
.〜
‑J l一一 一.

0
0.8

′ヽ
コ

′一1ヽ

且

且0.6

コ

」

0.4

∽
こ

の
」

a

aj 0.2

に

に

(e)
∴ III

Fig. 3.ll X‑ray diffraction intensities for FeJ‑rIiS2 (x = 0.15) along (盲1/2 1/2) parallel to the a ‑axis [left side, (a)‑(c)] and along (3/2 0 」) parallei to the c ‑axis 【right side, (d)‑(f)], calculated
from three different atomic distributions with Val ‑ ‑10,000 K (top row), the multiple pair‑interac‑tion model (second row), and the shorトrange ordered structure 2a x 2a x 2c with fractional site
occupancy (bottom row).

26

4
3

4

(a )

(g l′
2 1/2)
3

2

2

I

1

Jーl l ーl 一一(e)(0 1/2 0

0
4
3
2

′′■ヽ

′■■ヽ

冨1

∃

ヽt一′

1

ヽ‑メ

」・

0

.石

4

̲>
"55

亡

0
4

.ON

I3

i

3

2
1

Fig. 3.12 X‑ray diffraction intensities for Fe;tTiS2 (x = 0.25) along (￨ 1/2 1/2) Cleft side, (a)‑(d)】and along (0 1/2 Q [right side, (e)‑(h)], calculated from four different atomic distributions with"al = ‑10,000 K (top row), the multiple pair‑interaction model (second row), the superlattice 2寸3a
x 2a x 2c with fractional site occupancy (third row), and the ideal superlattice 2寸3a x 2a x 2C
(bottom row).

27

2 5
2 0

2 5

(a)

(盲
1/3 1/2)

1 5

15

1 0

10

5

5

0
5

0
5

15

15

‑. 10

■一1ヽ

(1/31/30(e)e l m .‑ I 題■

10

コ

CO

̲>
'萌

2 0

0
25

普

5

>ヽ

0
25

に

.t=
坊
こ

a) 20

B

20

c
^^^^^^1 r*j

00.20.40.60.8 1.0
g

Fig. 3.13 X‑ray diffraction intensities for FeJCT]S2 (∫‑ 0.333) along (号1/3 1!2) [left side, (a)‑(d)]and along ( 1/3 1/3 g [right side, (e)‑(h)], calculated from four different atomic distributions with"al = ‑10,000 K (top row), the multiple interaction model (second row), the superlattice V3<3 ×

寸3a x 2c with fractional site occupancy (third row), and the ideal superlattice V3a x J3a x 2C
(bottom row).

28

ァ3.5 Atomic distributions in stage‑2 Ago i5TiS2According to the X‑ray diffraction studies [58], the stage‑2 Ago 15TiS2 crystal undergoes aphase transition from ordered J3a x J3a x 2c to disordered structure around 250‑300 K. The
formation of¥3a x J3a structure itself is also realized in the a‑axis planes of FeJCTiS2 (x = 0.333),
as revealed by our MC calculations using four kinds of pair‑interactions (Val , Va2, Vcp and V2c)in the Hamoltonian [Eq. (9)】. With these parameters, however, only stage‑1 structures in GxTiS2
with any concentrations x are obtained, but not stage‑2, along the c‑axis direction.
In order to reproduce the stage‑2 Go i5TiS2 with V3a x J3a structure, we have taken into
account an additional term in the Hamiltonian, repulsive pair‑interaction, Vc2, between the 2nd
NN guest atoms in the nearest layers separated by la + c¥, which plays a crucial role to the atomic
distributions in stage‑2 Ago 15TiS2, as described below. In addition, the phase transition occur‑
ring around 250‑300 K suggests that these pair‑interaction strengths may be much weaker thanthose for Fe^T^ (x = 0.333). With these in mind and by considering that the transition probabiト

ity is given by W= exp(‑AH/7) [Eq. (4)】 we have employed a set of weaker pair‑interaction
strengths forAg0 15TiS2as, Val = ‑360 K, Va2 = +180K, Vcl = ‑180 K, and V2c = +180 K, where
the best‑fit values obtained for FeO 333TIS2 (Table I) are used but their magnitudes are reduced byabout l!30 with keeping the ratio Vk IVal unchanged.
Figure 3. 14 illustrates variation in the atomic distribution of GQ 15TiS2 at T = 200 K with
Vc2 (= 0 , ‑45, ‑90, and ‑180 K). In the case of Vc2 = 0 K, the guest atoms distribute in every layer,
forming the stage‑ 1 structure with ‑寸3a x J3a arrangements, as expected. While for the repulsive
interaction Vc2, the guest atoms are arranged having the in‑plane J3a x J3a arrangements, which
are stacked every two layers along the c‑axis to form the stage‑2 structure. Some domains ofstage‑2 structure are obtained for Vc2 = ‑45 K, and the domain size becomes larger with increas‑
ing IVc2l. These distributions are quite different from that for the stage‑1 Feo 15TiS2 with short‑
range order of2α × 2α × 2C 【Fig. 3.8(a)】.

With increasing temperature, the guest atoms vary in the a‑axis layers to destroy the in‑plane y3a x V3a ordered states, as can be seen from the temperature variations in atomic distribu‑
tions in the stage‑2 GQ 15TiS2 (Vc2 = ‑45 K; Figs. 3.15) and stage‑1 GoASTiS2 (Vc2 = 0 K; Figs.
3. 16). The guest atom arrangements are changed from the in‑plane J3a x寸3a structures below
200 K to the disordered states above 250 K. Temperature dependence of the averaged numbers ofneighboring atoms zk (k = al and a2 in the a‑axis planes, and k = cl, c2, and 2c along the c‑axis)
for four different pair‑interaction strengths Vc2 (= ‑180, ‑90, ‑45, and 0 K) are shown in Fig. 3.17.
At T = 100 K the numbers za2 and z2c, corresponding to attractive interactions Va2 and V2c, haverather large values, za2 = 4.2 and z2c = 1‑6, but those to repulsive interactions Val and Vc] aresmall, zal ‑ 0 and zcl ≦ 0.2. The value ofzc2 is strongly dependent on Vc2 from zc2 = 2.9 for Vc2= 0 K toz C2 = 0.0 for Vc2 = ‑180 K. These results give us the reason why the stage‑2 structurewith nearly J3a x J3a x 2c arrangement is formed by the interaction strengths used for our
calculation; the repulsive interaction V2c is needed to satisfy the necessary condition for the stage‑
2 structure, zc2 ‑ 0. With increasing temperature, the values of za2 and z2c decrease, while others
increase, and they tend to approach the values of random distributions, zal ‑ za2 ‑ 6x = 0.9, zcl ‑zc2 ‑ 2x = 0.3, and z2c ‑ 12x = 1.8. Here we should notice that the averaged number za2, which isstrongly related to the in‑plane寸3a x V3a structure, decreases gradually with temperature, which29means that the number z^ is useful as an prospect of the in‑plane ordered states but not enough to
be an order parameter specifying the order‑disorder transition of the y3a x J3a structure.
It should be emphasized that this is for the first time to construct such stage‑2 structure
only with pair‑interactions between neighboring atoms, to our knowledge; some long‑range
interlayer and/or intralayer interactions are employed for the explanation of the stage‑n structuresin GIC 【59, 66】.

喜3.6 X‑ray diffraction patters for Ago 15TiS2Now, we shall discuss more quantitatively the phase transition around 250‑300 K in
Ago 15TiS2 from the X‑ray scattering patterns calculated using the atomic distributions obtainedby our MC simulations. Figure 3. 18 shows the experimental results of (a) diffuse scattering pat‑terns (called diffuse rod) in Ago 15TiS2, parallel to the c ‑axis direction centered on (5/3 2/3 0)
and (b) those of in‑plane diffuse scattering along [1 1 0] direction at (5/3 2/3 1/2); the integrated
intensity and full width at half maximum of the latter are plotted against temperature in Figs.
3. 1 8(c) and (d), respectively. It is noted that the diffuse rod is very anisotropic elongated along thec ‑axis direction and it has small peaks at / of half integers. The integrated intensity is decreased
remarkably above 200 K with inflection point around 240 K, above which the width becomes
broader. We have calculated the X‑ray profiles along (5/3 2/3のusing Eqs. (3) and (4), and thecalculated atomic distributions at T= 200 K in Fig.3.14; the results for Vc2 = ‑180 K, ‑90 K, ‑45
K, and 0 K are shown in Fig. 3.19. With increasing ¥Vc2l, the intensities at /of integer increase,but those at / of half‑integer decrease oppositely. The calculated pattern with Vc2 = ‑45 K [Fig.
3. 19(c)] is best agreement with the experimental profile for Ago 15TiS2.
Temperature variations of diffuse scattering intensity profiles parallel to the c ‑axis direc‑
tion centered on (5/3 2/3 0) are shown in Fig. 3.20(a). For ♂of integers and half integers very
sharp peaks appear at low temperatures, whose intensities take maximum values around 200 K,
and decrease with increasing temperature, as shown in Fig. 3.20(b). The results along [1 1 0]
direction at (5/3 2/3 1/2) are illustrated in Fig. 3.21 ; the calculated profiles are very sharp com‑pared with those along the c ‑axis direction and its integrated intensity shows similar temperaturedependence to that shown in Fig. 3.20(b). Thus, overall behavior of the diffuse rod observed in
the stage‑2 Ago 15TiS2 are reproduced successfully using the atomic distributions in Fig. 3. 15
simulated with the repulsive Vc2 = ‑45 K and other four pair‑interactions.

30

(a)‑180 K(bト90 K(c) ‑45 K(d) 0 K

Fig. 3. 14 Variation in atomic distribution of guest atoms for Go j5TiS2 at T‑ 200 K with paiトinteraction strength

Vc2, when the values ofotherpair‑ interactions are fixed to be Va, = ‑360 K, V^ = +180 K, Vcl = ‑180 K, and V2c=+180K.

31

裏翫頭wb蝣nftJWWUWl'.wVKj.*k=:^t.‑:*>.::io:o:c‑‑:j.サーサ.サー・."ォ・.サ,*・.サ.‑.‑サ・.蝣‑SO'
IV*‑A'*サォサ*'*サ**4*サV買崇崇蜜等確8m;薬事璽盛蛍芸i5^*サ;?
^^S^^:蝣・蝣"・‑‑%‑サーーー・‑・‑‑・‑*・ササ‑蝣
IIL‑‑.・‑蝣‑‑‑'‑‑.‑サ・‑・‑蝣‑Vォ‑ォサォ‑;‑Ay.‑;V"y;vJ￨.サ.‑一・̲.w̲‑V.C*:
E*S:tt.vォv.‑‑.‑3fSSJ'蝣‑‑‑一蝣一蝣一・一・‑.^闘蝣AV.
'蝣蝣.'蝣・・・."m禁き.'.¥S享苛・・・'て::粟至:潜・、I簡塁::::.葦亭主尊皇i¥m豊二軍蓋.葦至=三三‑‑‑=ヨ固蕉

k= V.V=V^AV^V^＼筆茸*享:. *.:・Va

可ゴ濫yiv.;塁墓

歪賑堅

V* <亘葦至軍v.ォ "&:」措港:、IiE∃轟^tht^‑‑r^A.‑r‑^r‑^‑7^^^.^^ii^.‑5^:^"‑^ife"^^(a) 100 K(b)200 K(c)250 K(d) 300 K(e) 350 K

Fig. 3. 15 The temperature variation in the atomic distributions of the guest atoms in the stage‑2 GQ 15TiS2 with V 2= ‑45 K shown in Fig. 3. 14; these distributions can reproduce the observed X‑ray patterns for Ago i5TiS2.

32

蕪㌶X T ‑‑5*‑i‑
vuv¥A/UU‑¥fVV¥ォA‑ゝJ¥JWV7¥/‑17‑WW¥/W¥^yW¥A^A
等顎等芦蒜等;芋芋等軍事^j¥si」¥j¥j¥j ¥s¥s‑¥s¥r¥s¥s v¥Fsv¥pt
恐さ莞蓋語
顎還貰貰顎=題等還芦筑'宣/vM¥JWWV¥/¥A/WWVjV^
漂害悪Ⅹ芳Ec鑑〇〇〇〇缶¥A/VVVVV¥^fwni/¥/V¥/V¥AA¥Aj^¥ZVW‑WWWW¥ォ/V=tJ.I,※

翌

rcs‑sex:

至芸吉葉:こi.*匡討.蛸壷潜ぎ串・4こ:宝飾

ォ/V¥Au¥A/W¥WAA/¥A/UW¥A/¥A/¥A琵琶⊆琵琶竺竺逆さLでA/WWVUflflWWWV闇
IA;蒜el;.;;ま...蝣ぢ告3:i.ォs:ks:ォ.ォ:;:ご寺;^^^^^^^Vi l*.ォー¥'VWlさ′'A*.V<照AVAWiW.棚̲ヽJj断AWふむ等̲蕉主.宰享痛蕪耳憲責葦妄、.I.::;呈栄華畠、̲I、:囁f掌蒜
"i"vt^^^^^Kwi'i瓶'・,w‑i・ふ^^^^^・Vjj."重臣逆逆夢.^"ssi^COXOISSS^‑XXSksKSS!海亀、
毎主・¥導・軽≒主t:葦壷*笥'*・‑

TL
露療露褒轟壇軍萱K㍊yv/yy‑yi=>ss.:k:=5:::=5ViVi>‑?

K‑wg‑‑.‑‑‑i‑.‑‑‑i‑J‑.‑.‑jot.一・.‑.・.サ̲‑."。H室

重臣華土‑,:一・**.亨去蔓;一十.;; ‥諦
隼 .*Vア.'
薫=;o>‑::."o:o"5s 闇蓋ヨ:軽重ヨ獲sSs整 童墓室(a) 100 K(b)200 K(c)250 K(d) 300 K(e) 350 K

Fig. 3. 16 The temperature variation in the atomic distributions of the guest atoms in the stage‑1 Go i5TiS2 with Vc2=OKshown inFig・ 3.14.

33

flG

1.0

¥w

0.8

4= 1.0

0.6
▼tl
o
N0.4

0K

0.2
O

0

5

3

4

3

3

cM
o
N

2

1

1
0

0
1.5
c<1.0
0.5
0
1 00

200

300

400

T (K)

Fig. 3. 17 The variation in the temperature dependence ofzk {k = al, a2, cl, c2, and 2c) for Go 15TiS2
with four different Vc2.

34

1.2

(c)

ノー

完1.0

ヽtl′

>1
0.8
c
0)

c
.D

0.6

m 0.4
品

嘗0.2
0

(d)
0.10
■ー0.08も￡ 0.06
ォ≧ 0.04
0.02
0
1 00

200

300

400

T (K)

Fig. 3. 18 (a) X‑ray diffraction profiles of the diffuse rod observed for Ago 15TiS2 pararell to the c*‑axis
direction centered on (5/3 2/3 0) and (b) those of in‑plane diffuse scattering along [1 1 0] direction at (5/3 2/
3 1/2) at different temperatures [58]. Temperature dependence of (c) the integrated intensity and (d) the full
width at half maximum of the in‑plane diffuse scattering.

35

10
8
6
4
2
0
8
6

S 4
且 2
.蝣E

0

lカ

5
王
●l■

8
6
4
2
0
8
6
4

l(aト
1.80K I ■ IT=l Kl‑I ‑I… 症‑‑‑‑‑. ‑9I0Kl ‑山Jl) ‑ ーI ー1 血
■
l ー
(cト
4l5K
出血
IllLIL
J OK
F .
l ‑ I
血 ー‑ 1 ー‑ I.血‑2
2

Fig. 3.19 Variation in X‑ray diffraction profiles of the diffuse rod with Vc2 for Go 15TiS2pararell to the c ‑axis direction centered on (5!3 2/3 0) at T = 200 K. The result calculated with
Vc2 = ‑45 K resembles well to the observed profiles forAg0 15TiS2shown in Fig. 3.18.

36

8

(a)
T=100Kl…一一
日I ‑ I‑…Mil
200K
一
…‑
日一
250K
]Lllh II‑ I‑ .lLll,

4

0

4
′d.ヽ

完

0

'‑こ.

>ヽ

葛

4

1=
o

‑E 0

.

3 00 K4

蝣蝣I l l

蝣I l l3 50 K

‑6

‑4

‑2

0

2

4

6

c

Fig. 3.20 (a) Temperature variation in X‑ray diffraction profiles of the diffuse rod centered on (5/3 2/3 0) for
Ago 15TiS2 and (b) the values of the peak intensity at (5/3 2/3 0) and (5/3 2/3 1/2) plotted against tempera‑
ture.

メ

コ
dl

芸20
.I:
岬
に
.IO
こ
てI

B
d

I̲

訂10

11■

こ

‑0.04

‑0.02

0.02

8h

0.04

100

T

200

300

400

(K)

Fig. 3.21 (a) Temperature variation in X‑ray diffraction profiles of the in‑plane diffuse scattering along [ 1 10] direction at (5/3 2/3 1/2) for Ago ,5TiS2 and (b) the values of the integrated intensity plotted against
temperature.

37

Chapter 4

Discussion

ァ4.1 Fe,TiS2

4.1.1 Formation of clusters and percolation

We shall now discuss on the formation of clusters of the guest atoms in Fe;cTiS2 in the a‑axis plane and along the c‑axis. Fig. 4.1 illustrates typical atomic distributions in the first layer (m= 1) obtained with the multiple pair‑interaction model for (a) x = 0.15, (b) 0.25, and (c) 0.333,
where Fe‑occupied sites are marked by solid points. Solid lines (top row) mark the clusters thatare formed by connecting the lst NNs (k = al), which we refer to as the al‑cluster for short, Since
the number ofzal for x = 0.15 is zero 【Fig. 3.9(d)], there is no al‑cluster, while forx = 0.25 and0.333 very small al‑clusters exist that are isolated each other. Similarly, the a2‑clusters are formedby connecting the lst and 2nd NNs (k = al and a2) (middle row). The a2‑clusters for x = 0.15 and0.25 are small, while those for x = 0.333 are connected each other throughout the a‑axis plane likea web pattern, forming so‑called a percolation cluster [67]・ As shown in the bottom row of Fig.
4.1, the a3‑clusters formed by the lst, 2nd, and 3rd NNs for ∫ = 0.15 are large, and those for x =0.25 and 0.333 extend to form a large percolation cluster having a web‑like connection. Further‑more, it is noted that the dimension of the percolation cluster of the a3‑cluster appearing in somelayers for x = 0.15 is one‑dimensional (ID), while those of the a2‑ and a3‑clusters forx = 0.25 and0.333 are two‑dimensional (2D), as listed in Table II for three samples.
We here define a cluster size, st as the number of the connected atoms in the it‑cluster anda total number, N(sk), as the number of total guest atoms in its cluster of size sk within the sample
lattice considered (18 × 18 × 6). Fig. 4.2 plots the total number N(sk) versus cluster size sk (k ‑ al‑) for (a) x = 0.15, (b) 0.25, and (c) 0.333 ‑ so‑called cluster‑size distribution‑. Forx = 0.15,
we get a large value ofN(sk) with sal ‑ 1 and sa2 ≦ 2, which indicates that the al‑ and a2‑clusters

are almost isolated, while the cluster‑size distribution of the a3‑cluster is broad Oa3 = 1 ‑ 48). For
x = 0.25, the cluster size is increased (^al ≦ 3 and s.a2 ≦ 10), and the a2‑cluster has a particularly
broad distribution. In addition, for the a3‑cluster we get sa3 = 80, nearly equal to the total numberof the guest atoms per layer, ‑ Nx/6, which shows that the a3‑cluster is formed by almost allintercalated guest atoms. For further intercalation (x = 0.333), we have a little broad distribution
with larger cluster size (sal ≦ 7) for the al‑cluster and a single distribution with sa2 ‑ sa3 ‑108 (=Nx/6) for the a2‑ and a3‑clusters.
Moreover, the cl‑ (top row), 2c‑ (second row), and c2‑clusters (third row) for Fe^TiS2 withx = 0.15, 0.25, and 0.333 are formed along the c‑axis, as illustrated in Figs. 4.3(a), (b), and (c),respectively, where atomic distributions are depicted in the αc‑plane. In all samples, there is no

cl‑cluster and the Fe atoms are completely isolated (top row), while the 2c‑clusters form linear
chains along the c‑axis (second row), and thus these chains are ID percolation clusters. In con‑38

trast, the c2‑clusters have long‑range zigzag chains for x = 0. 15, linear chains with partiallyzigzag connection for x = 0.25, and wide web‑like network for jc = 0.333 (third row). In addition,
the bottom row of Fig. 4.3 shows these c2‑clusters projected on the a‑axis plane (m = 1), where
solid points indicate the Fe atoms in this layer and open circles those in the NN layers (m = 2 and6, because of the periodic condition of the lattice)I From these projected patterns, it is readily seenthat the dimension of the percolation cluster of c2‑cluster is 3D for all samples (Table II).
Figure 4.4 shows the cluster‑size distributions of the cl‑ (top row), 2c‑ (middle row), andc2‑clusters (bottom row) for (a) x = 0.15,仲) 0.25, and (c) 0.333. We see a large single peak at scl= 1, which indicates that the Fe atoms formed with the lst NNs are completely isolated. The
distributions for the 2c‑clusters (middle row) are broad (s2c = 1‑3). On the other hand, the c2‑
clusters have large peaks at sc2 ‑ 250, 460, and 650 for x = 0.15, 0.25, and 0.333, respectively,
which are nearly equal to the total guest atoms Nx. Thus the cluster formation and the dimensionof the percolation cluster are strongly dependent on the type and the number of neighboring guest
atoms in both directions.

(a)x=0.15(c) x = 0.333(b)x = 0.25

・

・

・

・

・

・

・・
・
・
・
・
・
・
・
・

・

・

・

tY・:・:・:
●
●
●
・

・

・

・
・

:・:㌔.′′:・:
・

a1 ‑cluster

・

・

・

・・・
・・・′●●●● ′・・ ・●■′ .・・・・

g:
* i
as
a2‑cl uster

匪'王,ォサ<サォi

サォサ‑‑‑<サォサ
L,‑ォサォサォ蠎‑ォ蠎一重If̲

Fig. 4‑1 Formation of the al‑ (top row), a2‑ (middle row), and a3‑clusters (bottom row) in the a‑axis plane (m ‑ 1) for
FexTiS2 with (a)x = 0.15, (b) x = 0.25, and (c) x ‑ 0.333. Solid points mark the Fe atoms and solid lines clusters
formed.

39

Table II Dimension of the percolation cluster of each ^‑cluster (k = al,a2, a3, cl, 2c, and c2) formed in FejfTiS2 (a; = 0.15, 0.25, and 0.333),
obtained with the multiple pair‑interaction model.

jc=0.15x=0.25x=0.333

al
a2

1D

a3
C1

ID
3D

2c
c2

(a)x=0.15
500
400
300
200
100

1 2

3

4

5

6

7

8

0

910

sal

500
400
300
200
100

1 2

3

4

5 6
2&

7

8

0

910

0 1 2

3

4

5

6

7

8910

sa2

500
400
300
200
100

1 0

20

30

40

50

0

SE

Fig. 4.2 Total number of the guest Fe atoms A^) in the ^‑cluster of size sk in the a‑axis layer involved in the wholelattice site (18 × 18 × 6) for Fe^TiSj with (a) x = 0.15, (b) x = 0.25, and (c) x = 0.333, plotted against the cluster size・S」 (&= al, a2, and a3).

40

(a)x=0.15(b)x =0.25

・

・

・

・

・

・
●

・

c1 ‑cluster

・

・

・

(c) x = 0.333

・・●(ac‑plane)

・

・

・

●

・
・

・

・

・
・

・

・

・

・

・

・

・・

・

・

・

・

・

・・・三三三2c‑cl uster(ac‑piane)c2‑cl uster(ac‑plane)c2‑cluster(a‑plane)Fig. 4.3 Formation of the cl‑ (top row), 2c‑ (second row), and c2‑clusters (third row) for FeJTiS2 with (a)* ‑ 0.15, (b)x ‑ 0.25, and (c) x = 0.333. Solid points mark the Fe atoms in the ac‑plane and solid lines the clusters formed. In thebottom row are shown the projections of the Fe atoms and clusters on the first a‑axis plane (m = 1), where solid andopen circles indicate the Fe atoms in the plane and in the next nearest planes, respectively.

41￨x=0.15(b) x = 0.25

Ix = 0.333

Fig. 4.4 Total number N(sk) of the guest Fe atoms within the ^‑cluster of size sk plotted against cluster size sk (k = cl ,2c, and c2) formed along the c‑axis for Fe^TiSj with (a)ェ= 0.15, (b)x = 0.25, and (c)x = 0.333.

42

4.1.2 Magnetic properties
As described in Chapter 1, Fe;tTiS2 shows SG (0.01 ≦x ≦ 0.20), CG (0.20くx ≦ 0.40), andF (0.40 < x ≦ 1) phases having a strong anisotropy with the easy axis along the c‑axis. The

positive Curie‑Weiss temperature observed in the paramagnetic phase above the phase transition
temperature [36] indicates the existence of strong fe汀omagnetic exchange interaction between
the Fe spins in the a‑axis plane and some weak but ferromagnetic exchange interaction between
the Fe spins along the c‑axis. In addition, we have found interesting behaviors in the relaxation

phenomena of thermoremanent magnetization 〟" of the SG and CG phases. Some of these mag‑netic properties can be understood from the present results of computer simulations as follows:(1) The ferromagnetic exchange interaction between the Fe atoms is considered as due tothe 2nd and 3rd NN atoms in the a‑axis layers, but not due to the lst NNs, since z , ‑ 0 for all
samples (x < 0.333). (2) The observed CG phase is primarily stemmed from the web‑like 2D
percolation cluster formed for x = 0・25 and 0.333 in the a‑axis plane (see Fig. 4.1). On the other
hand, the SG phase (x = 0.15) has no 2D percolation cluster but its percolation cluster is one‑dimensional. (3) In the relaxation behavior of Mr, we have found that the equilibrium relaxation
spectra for x = 0.333 is much broader than those for x = 0.25 [44‑46]. Such a difference is now
attributable to the difference in the type of the percolation cluster, where 2D percolation cluster isformed by the 3rd NNs forx = 0.25 and by the 2nd NNs forx= 0.333. In the relaxation process ofMT, an Fe spin reverses gradually its direction after a magnetic field applied along the c‑axis isswitched off. In the present case, this process corresponds to the removal of some Fe atoms froma given atomic distribution. This in turns means that the percolation cluster for x = 0.25 is de‑stroyed into some smaller clusters which no longer percolate over the α‑axis plane. But the 3D

percolation cluster for x = 0.333 may not be destroyed, because in this case it is formed with the
2nd NNs. (4) Since the c2‑cluster is a 3D percolation cluster for x = 0.15‑0.333, the exchangeinteraction between the c2‑cluster along the c‑axis may be much weaker that those between the
2nd and 3rd NNs in a‑axis plane.
For further quantitative discussion on the magnetic properties in SG and CG phases, espe‑
cially on their spin dynamics, simulations taking into account the exchange interactions between
the Ac‑th neighboring Fe atom (k = a2, a3, c2, and 2c) will be needed.

4.1.3 Local structures
EXAFS studies of MxTiS2 [ 1 1 , 1 2】 have revealed that the interatomic distance between theguest atoms M and host sulfur atoms S, R(M‑S), depends strongly on the kind ofM but not on its
concentration x, while the Ti‑S distance, /?(Ti‑S), increases with x (Fig. 4.5). Such changes in thelocal structures are in qualitative agreement with the concentration dependence of the lattice

parameters a and c ofAf TiS2 (Fig. 4.6) [8]. More detailed discussion has been done by consider‑ing that the S atoms change their position [A(n), H(n)] depending on the number of the surround‑mg guest atoms, n (integer), as shown in Fig. 4.7, using a random guest atom distribution [12]. In
the present work, we shall give further discussion on these local structures based on our calcu‑
lated atomic distributions in Af TiS2.

EE

′ー

<

2.40

ヽ一′′

1‑‑1ヽ‑∽
I≡

富 2.38

0.0

0.1

0.2

0.3

0.4

X

Fig. 4.5 Interatomic distances (a) R(M‑S) and (b) #(Ti‑S) obtained
from M and Ti K‑edges EXAFS experiments for MxTiS2 (M = Fe
and Co) plotted against the guest concentration jc [1 1,12].

Fig. 4.6 Lattice parameters a and c plotted against the guest concen‑

trationxforMxTiS2 (M= V, Cr, Mn, Fe, Co, and Ni) [8】.

44=主二二壬r=2
n=O

nこ

n=2

Fig. 4‑7 Schematic top view for the displacement of the host S atom (large circ一e) according to thenumber of the surrounded guest atoms (solid circles), n (= 0, 1, 2, and 3) for MxTiS2; open small
circle indicates unoccupied site. Corresponding cross‑section for the position of the S atom, [A(ォ),H(n)] from guest atom; the S atoms with no surrounded guest atom (n = 0) are assumed to be at H{ri)=c/4.

As a typical result, Figure 4.8 illustrates the atomic distribution of the guest Fe atoms
calculated with the multiple pair‑interaction model [solid circles; bottom row of Fig. 3.8(b)] andof the host S atoms in the first layer (m = 1) for Fe^T^ (x = 0.25). The S atoms with the numberof its nearest Fe atoms, n ‑ 0, 1, and 2 are marked by open circles, hatched circles, and open
squares, respectively. In this case, there is no S atom with n = 3. For numerical treatment of theseatomic distribution, we here introduce the probability that one host S atom has 〟 nearest guest

atoms, which we express as,p(n) = N(n)/Ns,(10)

where N(n) is the total number of the S atoms associated with the n nearest Fe atoms and Ns (= 18
× 18 × 6 × 2 = 3888) the total nu血ber of the S atoms in the whole lattice.
In Fig. 4.9 are shown the calculated values ofp(n) plotted against n for all samples (x =0.15, 0.25, and 0.333) obtained from various atomic distributions (Figs. 3.1‑3.3, and 3.8). We
should note that the p(n)‑n plot obtained by the multiple pair‑interaction model for each sample isnearly similar to those obtained with the repulsive interaction (Val = ‑10,000 K). This result is ingood agreement with the fact that the averaged number of the lst NN guest atoms zal is nearlyzero (zal ‑ 0), which arises from the strong repulsive pair‑interaction Val in the a‑axis plane, as
mentioned above (Fig. 3. 10). Using the probability p(ri) obtained with the multiple pair‑interac‑tion model and random distribution, and the observed lattice constants a and c [Fig. 4.6], we haveevaluated the Ti‑S distance Z?(Ti一声), or the change in the local structure near the S atoms. For

simplicity, we assume that the host Ti atoms array regularly with the periodicity of the lattice

spacmgs a and c, and express the S position as, [A(n), H(n)], from the guest atom M in the plane45

including the direction of〟‑S‑Ti bondings. Based on this model, we obtain the relations, A(3) =寸3a/3, H(0) ‑ c/4, andA2(n) + H2(n) ‑ R2(M‑S) for n ‑ 1, 2, and 3; therefore, i/(3) = [R2(M‑S) ‑az/3] lu. In addition, the displacement H(n) of the S atom from H(0) to H(3) is not always linear
to n, since the strength of the bonding between the guest M and S atoms is often much stronger
than that between the vacant site and S atoms. In facts, semiempirical analysis of the guest con‑centration

dependence

of

the

interlayer

spacing

c

based

on

the

conceptof…resonance"inchemi‑

cal bonding, have revealed that the force constant for the MS bond is larger than that for thevacancy‑S bond in the van der Waals gap in MxTiS2 [8]. Therefore, we assume a nonlinear depen‑
dence ofH(n) on n as,
H(n) = H(0) + [H(3) ‑H(0)】 (n/3) {(1+S)/[1+耳n/3)]}.

(1 1)

Here d is a parameter to specify a nonlinear dependence of H(n) on the number n; for 5 = 0 H(n)changes from H(O) to H(3) linearly, while for a positive S it does concavely with n when H(3) <H(0), and oppositely for a negative ∂. Then, /?(Ti‑S) can be written as,

R(Ti‑S) ‑ ∑ p(n) Rn(Ti‑S),

(12)

n

where p(n) is the probability obtained above and Rn(Ti‑S) is the interatomic distance for n, which
isgivenbyRn(Ti‑S) = (n/3){A2(n) + [(c/2) ‑ H(n)] 2] m

+ (1‑ nl3){B¥n) + [{ell) ‑H(n)] 2}m,(13)

using 5(0) ‑寸3a/3, B(l) ‑ {(a/2)2 + [(寸3a/2) ‑A(l)12} 1/2, and 」(2) ‑ (寸3a/2) ‑[A2(2) ‑ (a/2)2]l/
2. The first term in Eq. (13) is for the Ti atoms above or below the guest atoms and the second
term for those around the unoccupied sites. The best‑fit values of 5, A{n), and H(n), are listed inTable HI for two samples (x = 0.25 and 0.333). We see that the nonlinear parameter <5is very small(‑0.02 for x = 0.25 and ‑0.15 forx = 0.333), which means that the change in the S atom position

along the c‑axis is almost linear to the number of the nearest Fe atoms n Fig. 4. 10(a) and (b])]. In
addition, with increasing n, A(n) is almost unchanged but H(n) is reduced. However, no clear
difference in the values ofA(n) and H{n) are seen among these samples, regardless of the type ofthe distributions used.

Similar calculations for other intercalation compounds MxTiS2 have been made in the same
way as done above. The best‑fit values for CoJCTiS2 are summarized in Table IV, where the prob‑
ability of FexTiS2 is used, since much less is known about the structural studies of this compound

to our knowledge. We have found that among others Co^T^ has the largest value of ∂ 5 = 3.8
for x = 0.25 and 2.2 for x = 0.333 (multiple pair‑interaction model) and 9.8 for x = 0.25 and 6.3 for∫ = 0.333 (random distribution). Thus, we have clarified that atomic distribution of the guest
atoms around their host S atoms play a crucial role on the local structures in the intercalation
compounds. Such large values of d for Co guest atoms means that the host S atoms can be shiftedeasily along the c‑axis by even one Co atom [Fig. 4.10(c) and (d)], which may be due to a verystrong bonding between Co and S atoms. From these analyses, variations in the local structure forFe^T^ and Co^T^ are illustrated schematically in the (1 1 0) cross‑sections ofFig. 4. 1 1(a) and46

●Fe o S(n=0) ◎S(n=1) □S(n=2)

Fig. 4.8 Atomic distribution of the host S atoms with n nearest Fe guest atoms (solid circles) in the a‑axis layer (m = 1) for Fe;[TiS2 (x = 0.25), calculated with the multiple pair‑interaction model. Opencircles, hatched circles, and open squares mark the host S atoms with the number of its nearest Fe
atoms, n = 0, 1, and 2, respectively.

x = 0.333

UK
ME

(a )

(i)

V a 1 = + 1,000 K

O.6
ME
OP

閉

Pm

国

一

O

1.0
￨JE

(b

V a1 = O K

閉. 閉 閉

(j)

O.6
0.4

左

0.2
c

*M*t

0

l.0
0.8
0.5
0.4
0P
O
l.0
0.8
0.6
0.4
0.2
0

(c )

v a i = ‑10,000 K

(g)

図防

廟

(d)

(h)

M ultiple V L

国国
1

2

同.
3

0

1

2

3

n

n

Fig. 4.9 Probability p(n) defined by Eq. ( 10) plotted against the number of nearest Fe atoms around the host
S atoms for different atomic distributions in Fe^TiSj with x ‑ 0. 15 (left column), x = 0.25 (center column),
andx = 0.333 (right column), calculated with Val = +1,000 K (top row), Val = 0 K (second row), V j = ‑10,000 K (third row), and the multiple pair‑interaction model (bottom row).

47

Table III Best‑fit values of nonlinear parameter S, probability p(n), and positions A(n) andO

H(n) (in units of A) of the host S atom with n nearest guest Fe atoms in Fe TiS7 (x ‑ 0.25
and 0.333), obtained from the atomic distribution with the multiple pair‑interaction modeland random distribution.

n P(n)

Multiple pair‑interaction Random distribution
A (n)
H(ri)
p(n)
A (n)
H(n)

x= 0.2501

2
3∂= ‑0.022
0.295
0.660
0.045
0.000

x=0.333
0
1
2
3

1.943
1.959
1.974

8 = ‑0.028
1.428
1.407
1.385
1.363

0.422

∂ =‑0.15
0.050
0.900
0.050
0.000

‑
1.940
1.958
1.977

1.428

0.422
0.141
0.016

1.943
1.959
1.974

1.407
1.385
1.363

∂ =‑0.20
1.429
1.406
1.381
1.353

0.297
0.444
0.222
0.0370

1.429

1.940
1.957
1.977

1.407
1.382
1.353

(b), respectively. In Fe;cTiS2 the contraction due to S‑Fe‑S bonding and the expansion of the hostlayer, which caused mainly due to charge transfer from Fe atom to Ti atom, compensate eachother and the interlayer spacing c is less changed upon intercalation, while in Co^T^ the con‑traction due to S‑Co‑S bonding overcomes the expansion of the host layer, leading to the contrac‑
tion of the interlayer spacing c (Fig. 4.6).
Furthermore, the difference in S between FexTiS2 and Co;cTiS2 is attributable to the differ‑ence in the bond nature between the guest atom and the surrounding S atoms or Ti atoms. As
pointed outbyN. Suzuki etal. [40], inAf TiS2 (M=Mn, Fe, Co, andNi; x= 1/3 and 1) theM3dγ
states are hybridized strongly with S 3p states, and the bond order between them is large andplays a dominant role in the chemical bondings, leading to a large value of 8. In Fe TiSo, how‑ever, the Fe 3de states are also hybridized strongly with Ti 3de states, and the bond order of Fe
3de ‑Ti 3dJ bonding along the oaxis direction is fairly large comparable to that of Fe 3dγ‑S 3pbonding, which may be the main reason why the parameter 8 specifying the nonlinearlity in the
Fe‑S bonding along the c‑direction is very small in the Fe intercalates.

48

Table IV Best‑fit values of nonlinear parameter 5 , and positions A(n) and H(n) (in units of
0A) of the host S atoms with n nearest guest Co atoms in Co^T^ (x ‑ 0.25 and 0.333),
obtained丘om the atomic distribution with the multiple paiトinteraction model and random
distribution, where the probability p(n) for Fe;tTiS2 (Table V) is used.

Multiple pair‑interaction Random distributionn A (n)
H(n)
A(n)
H(n)
x=0.25∂ =3.80

5 =9.8

2

1.410
1.352
1.336

3

1.328

1

x=0.333

8 =2.2

0
1
2
3

1.410
1.341
1.332
1.328

8=6.3
1.410
1.358
1.336
1.325

1.410
1.343
1.330
1.325

Fig. 4.10 Variation in the displacement 〟(〟) of the S atoms along the c‑axis direction with ‑earest guest
atoms for (a) Fe0 25TiS2, (b) Fe0 33TiS2, (c) Coo 25TiS2, and (d) Coo 33TiS2. Solid and open circles shows
the values obtained from the atomic distribution with the multiple pair‑interaction model and those丘・om
random distribution, respectively.

49

.′ I‑ヽ
′
ヽ

、
、

■

、
、l.＼

T i

m獲

Fe
‑

& 」/

′ ̀ヽ
/
ヽ

S
、

,L

)

I

J
、
V ・
"‑'′
′
P,

(b) CoxTiS2(a) FexTiS2

Fig‑ 4.ll Schematic (1 1 0) cross‑section for the variation in the 一ocal structure in (a)Fe;cTiS2 and (b) Co^TiSj upon intercalation of the guest atoms.

蚤4.2 Phase transition in Ago 15TiS2
Now we shall discuss phase transition in stage‑2 Ago i5TiS2. As described above, the atomic
distribution in Ago.isTiS2 shows in‑plane ^3a x J3fl structure at low temperatures, and the or‑dered structure is destroyed with raising temperature [Fig. 3. 15]. This order‑disorder transition
occurs due to the small values of pair‑interactions V^ of the order of 100 K in the present Simula‑tion. Here, we should point out two experimental evidence, thermal displacement of血e host andguest atoms in‑plane and along the c‑axis determined by X‑ray analysis [5 1,53], and entropy ofphase transition evaluated from the specific heat measurements [56,57].
The c‑axis length for one silver layer in stage‑2 Ago 15TiS2 has a spacing elongation of

0.71A compared with the van der Waals gap of the host TiS2, as illustrated in Fig. 1.5 [51].
Furthermore, temperature factors of the constituent Ti and S atoms and guest Ag atom, Bjv Bs,
and BA‑, for TiS2 and Ago i5TiS2 are listed in Table V, as well as their mean square thermal

displacements, fJ^ A, eva^uated using the relation BA ‑ ^ifi^jc for atomA (= Ti, S, and Ag).
From these results, we can see that upon intercalation ofAg atoms, vibrational amplitude of Ti
atoms becomes larger both in‑plane and along the c‑axis, while that of S atoms is almost un‑changed, which means that the expansion of the host layers leads to the large vibration of Ti
atoms. Furthermore, it is noted that the value of2?Ag is very large, especially in the a‑axis planes(J5A = 3.0 A2 ‑ jBTj, Bs). This fact also supports that the pair‑interactions between the neighbor‑
ing Ag atoms are rather weak, and the in‑plane arrangements of the Ag atoms are easily destroyedby the thermal fluctuation of the order of 300 K into disordered states, as shown in Fig. 3. 15.
Specific heat measurements for AgxTiS2 (0.09 ≦ x ≦ 0.34) [56,57] confirmed that the tran‑
sition temperature is about 280 K, which is less dependent on Ag concentration x, and its entropy
per 1 mol of guest Ag atoms due to this transition is evaluated to be ∫ = 9.9 ± l.O JK‑lAg‑mol‑1
from the extra specific heat shown in Fig. 4.12. The obtained value is close to the expected value

Jf?ln3 (= 9.13 JK^mol‑1) when the Ag atom is accessible only to three sites a, fi, and /within a
J3a x寸3a unit cell [56].

50

0
‑=1
°
X
fc im
I

さ∠
「I

5

ヽ一■ll

∈
丘
o
<

0

300
T (K)

Fig. 4.12 Temperature dependence of the extra specific heat observed for
Ag;tTiS2 with x = 0.0902 (open square), 0.181(solid circle), and 0.361 (opencircle) 【57].

Table V Temperature parameters BA foxA atoms (= Ti, S, and Ag) in the a‑axis andalong the c‑axis for TiS2 and Ag/TiS2 with x = 0. 18 at room temperature [5 1,57]. The
mean square thermal displacement, A^h4'is evaluated using BA = Sn2〃tM
directionoV*」S(A2)(A2)(A2)篭Ti篭,s)^th,Ag

TiS2
AgxTi S 2

(A)

a

0.76

1.00

c

1.78

1.6

0.150

0.14

71

1.18

1.08

3.0

0.122

0.117

0.195

c

1.9

1.6

1.6

0.16

0.14

0.14

0.098

0.113

喜4.3 Comparison between FexTiS2 and Ago 15TiS2Finally, we shall compare the results for Fe^l^ and Ago 15TiS2. The intercalation com‑pound Fe^T^ shows stage‑1 structure over whole Fe concentrations x = 0‑1, but AgxTiS2 shows
stage‑2 structure in the restricted Ag concentration range 0. 1 ≦ x ≦ 0.3. The interaction strengths

Vk (k = al, a2, cl, and 2c) forAg0 15TiS2 are much smaller than those for stage‑1 Fe0 15TiS2 andFeO.333TiS2'^ ^c2 is necessary to obtain the stage‑2 structure in Ago 15TiS2, as described in
Chapter 3. Such differences are attributable to the differences in the bond strengths between theguest atoms and host atoms; the bondings between the Ag‑S atoms and Ag‑Ti ones seem to bemuch weaker than the strong bondings between the Fe‑S and Fe‑Ti atoms through the hybridiza‑tion of the Fe 3d, S 3p and Ti 3d orbitals [38‑42].
Furthermore, it is instructive to recall that stage‑ 1 structure appears for other G^TiS2 withthe guests of Li and transition‑metal atom M (= Ti, V, Cr, Mn, Co and Ni), while the stage‑2 does51

Table VI The valence, spin state, and crystal radius for guest atom G withthe coordination number CN = 6 at the octahedral site in GxTiS2, as well As
the stage number n for GxTiS2 with x = 0.15.

Guest valence spinstate radius(A)
low

spin

stage‑n0.79

1high spin 0.92

1

1.29

2

1

2
2

for Na, and K guests with rather large ionic radius. In Table VI are listed the crystal radius of theguest atom with coordination number of 6, together with their valence, spin‑state, and stage num‑
ber for G = Co, Fe, Ag, and alkaline metal Li, Na and K; here we employ the crystal radius rc
reported by R.D. Shannon [68], which can be converted into effective ionic radius rv by r‑x = rc ‑00.14 A. From Table VI, it is clear that the small size of guest atom with rc less than 1.00 A (0.79
A for Co2+, 0.92 A for Fe2+, 0.90 A for Li+) produces the stage‑1 structure, while large size of0

guest atomwith rc larger than 1.0 A (1.29 A forAg+, 1.16 A forNa十1.52 A for K+) produces thestage‑2 compounds. These results indicates that the local lattice deformation around the guest
atoms caused upon intercalation plays a key role to determine the pair‑interaction strengths in thestage‑2 or stage‑ 1 structure.
Schematic model for local lattice deformation upon guest atoms and its effect to the neigh‑boring guest sites is depicted in Fig. 4. 13, where (100) plane cross‑sections are illustrated. For°

small guest atom with rc < 1.0 A, an induced deformation is rather small and it affects on lst NNs
both in the plane and along the c‑axis as repulsive interactions Val and Vcl, while it gives nodirect effect on 2nd NN sites in the plane separated by寸3a. On the other hand, for large guestO

atom with rc > 1.0 A, the van der Waals gap is remarkably expanded, and lattice deformation
becomes larger, and it affects on the c2 sites between the 2nd NNs in the next layers as some
repulsive interaction Vc2, as well as repulsive interactions Val and Vc l , since the superposition of
the deformations at c2 sites become too large to neglect. Thus, the size of the guest atom deter‑
mines whether the pair‑interaction Vc2 may be neglected or not. However, it is difficult to explain
the origin of Va2 and V^c between 2nd NN sites in the a‑axis plane and those in the next nearest
neighboring layers, because they may be determined as the result of the superposition of thevarious interactions, such as bond strengths between the guest and host atoms (Ti‑S, G‑S, and G‑
Ti bonds), van der Waals force, and local lattice deformation, each of which is strongly dependent
on the valence and electronic configuration of the intercalated guest atom, as well as it's concen‑
tration x.

For further understanding of the origin of the repulsive or attractive pair‑interactions andtheir magnitudes, more也eoretical and experimental studies will be needed, such as bond orders
based on the electronic structures [40], precise X‑ray diffraction and X‑ray absorption using52

synchrotron radiation source, and scanning tunneling microscopy, as done for Fe/T^ [69] andthe surface atomic arrangements of the J3a x J3a structure in Ag/Si(lll) 【70,71】, where thedisplacements of surface Ag atoms and the Si atoms in the top layer of the substrate are essential.

(a)

Smal一

guest

(b) Large guest atom

atom

Fig. 4.13 Schematic model for local lattice deformation in GxTiS2 upon intercalation. The (100)plane cross‑sections for (a) small guest atom (rc < 1.0 A), and (b) large guest atom (rc > 1.0 A). The
arrows show repulsive paiトinteractions between guest atoms. For the large arom, the van der Waals
gap is remarkably expanded, and lattice deformation becomes larger, and it affects on the c2 sites
between the 2nd NNs in the next layers.

53

Chapter 5
C onclusion
Monte Carlo simulations of atomic distribution of guest atoms G in intercalation com‑pound O^82 have been carried out in the triangular lattices in the a‑axis plane stacked with 6
layers along the c‑axis (lattice size: 18 × 18 × 6), by taking into account attractive or repulsive
pair‑interactions between the neighboring guest atoms in the a‑axis plane Vaj and Va2, and thosealong the c‑axis, Vcl, V2c an(* ^c2‑ From these calculations, we have made clear the followings.
(1) The atomic distribution of the guest atoms in the a‑axis planes is primarily determinedby the pair‑interactions between the lst NNs V&¥ and the 2nd NN pair‑interaction V^ as well. The
pair‑interaction Vcl between the nearest layers separated by c along the c‑axis governs the atomic
distribution along this direction, and in particular, the attractive interaction V2c between the next
nearest layers separated by 2c can account for the observed 2c‑periodicity in the superlattices

2寸3a x 2a x 2c for FeJCTiS2 (jc = 1/4) and J3a x J3a x 2c for Fe^TiSj (x = 1!3) and stage‑2
structure of Ago 1 5TIS2.

(2) The X‑ray diffraction patterns for Fe^T^ are calculated from the atomic distributions
obtained using four kinds of pair‑interactions, in qualitative agreement with the experimental
data of the 2a x 2a x 2c short‑range ordered structure for x ‑ 0. 15, and superlattices of 2寸3a x 2a
x 2c for x = 0.25 and J3a x J3a x 2c for jc = 0.333 with fractional site occupancy.
(3) A set of averaged numbers of the neighboring guest atoms, z^ (k = al, a2, a3, cl, and2c), specify a given atomic distribution. The comparison of the averaged numbers for the calcu‑
lated distributions with those for the superlattices with the fractional occupancies in FeA‑TiS2
confirms that the latter is not enough to reproduce the details of the atomic distributions, because
we have used the averaged fractional values.

(4) Based on the calculated Fe atomic distributions in Fe^T^, we have discussed on theformation of various types of clusters and the percolation cluster. We have found that one‑ andtwo‑dimensional percolation clusters are formed by the 3rd NNs in the a‑axis plane for x = 0. 15
and 0.25, respectively, and a two‑dimensional one is formed by the 2nd NNs for x = 0.333, whichcorresponds

to

the

magnetic

phase

diagram[spin‑glass

(x

=

0.

15)

and

cluster一glass

(x

=

0.25

and

0.333)]. The difference in the type of 2D‑percolation cluster is responsible for the difference inthe dynamical relaxation behaviors in the thermoremanent magnetization.
(5) Furthermore, using the probability of the number of the Fe atoms near the host S atoms,p(n), evaluated from the Fe atom distributions obtained by the Monte Carlo simulation, as well as
EXAFS data, we have calculated the interatomic distance /?(Ti‑S) for MxTiS2. In the case of
Fe;cTiS2 the change in the S atom position along the c‑axis is almost linear to the number of thenearest Fe atoms n, whereas for Co^T^ it is nonlinear. The atomic distributions of the guest
atoms around their host S atoms affect the local structures.
(6) Taking account of the repulsive pair‑interaction strength Vc2, in addition to four kindsof pair‑interactions Val , V&, Vci , and V^c usec* for AfxTiS2 atomic distributions in stage‑2 GxTiS2

54

with the in‑plane J3a x J3a structure are obtained. The domain size of the stage‑2 structure
depends on the strength Vc2. From the temperature dependence of the atomic distributions, thephase transition from the V3a x y3a to random arrangement is confirmed.
(7) The X‑ray diffraction patterns of Ago 1 5TiS2 with stage‑2 structure are calculated fromthe obtained distributions, in good agreement with the experimental data of diffuse rod elongatedalong the c ‑axis. The in‑plane order‑disorder phase transition of the Ag atoms are discussed
based on the calculated distributions, together with experimental results of the thermal displace‑ments of Ti, S, and Ag atoms and entropy change due to the phase transition.
(8) Difference between FeA‑TiS2 and Ago. i5TiS2 is discussed from the viewpoint of local
lattice deformation caused upon intercalation of guest atoms. The size of the guest atom plays acrucial role to form stage‑2 intercalation compound. Stage‑2 compound is formed only for largeO

guest atom with crystal radius of rc > 1.0 A, due to a large lattice deformation, which affects on
the 2nd NN sites in the next layers to exert some repulsive interaction Vc2.
Thus, Monte Carlo simulation method with the pair‑interactions is powerful and useful to
get atomic distribution of guest atoms in GxTiS2, as shown in this work. This work will be ex‑tended further to the pressure‑induced stage transition in Ago 35TiS2 from stage‑2 at ambientpressure to stage‑1 at high pressure [72], and to GxTiS2 with higher guest concentration x ≧ 0.40,
where the effect of the interaction between the 1 st NNs could be dominant and some complicatedpatterns could be appeared.

55

Acknowledgements
The author would like to express her sincere thanks to Prof. Masafumi Sera for his interest
and encouragement through this work. Furthermore, she would like to express her sincere grati‑

tude to Emeritus Prof. Masasi Inoue for his patient guidance, constant encouragement and inter‑
est, and Prof. M. Sasaki for his valuable discussion, useful suggestions and kind advice. She isdeeply grateful to Dr. H. Negishi for his beneficial discussions and continuous support. She ap‑
preciate Prof. Y. Kuroiwa at Okayama University for giving her research group the detailed ex‑penmental data of the neutron and X‑ray diffractions on FeJTiS2 and AgxTiS2, and valuable
discussion.

She would like to thank Emeritus Prof. H. Fujiwara, Prof. M. Nomura at Ube National
College of Technology, the late Prof. H. Kadomatsu, and Dr. K. Umeo, who helped her to con‑tinue her research works with their encouragements. Sincere thanks are due to Prof. S. Keitoku
and Dr. K. Nishioka at Hiroshima Women's University for their collaboration, who taught infinite
possibility to research under restricted conditions to the author. She is grateful to Ms. R. Tsukamoto,Ms. J. Nakagawa, Ms. J. Koyama, Ms. Y. Nishigaki, Ms. Y. Okuda, and Ms. S. Tokumitsu for
their supports on office routine and heartful encouragements. She also acknowledges Drs. S.
Ohara, K. Takase, Y. Hara, Mrs. A. Yamasaki, H. Yamada, S. Kakita, and all members of Inoue
Laboratory for their collaborations, exciting discussions and friendly help.
Finally, I acknowledge my parents, children, and all persons who have helped me丘om theoutside of this work for their understanding on my work and heartful supports.

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