Magnetic Structures of Itinerant Electron Systems on the Extended Spatially Completely Anisotropic Triangular Lattice

概要

ത࢜࿦จ

Magnetic Structures of Itinerant
Electron Systems on the Extended
Spatially Completely Anisotropic
Triangular Lattice
⎫
⎧
֦ுۭؒҟํతࡾ֯֨ࢠ্ͷ ⎪
⎪
⎪
⎪
⎭
⎩
࣓‫ߏؾ‬଄ͷ࣓‫ߏؾ‬଄

޿ౡେֶେֶӃઌ୺෺࣭Պֶ‫ڀݚ‬Պ
Տ໺ɹ༎‫ل‬

2022 ೥ 3 ݄

໨࣍
1. ओ࿦จ
Magnetic Structures of Itinerant Electron Systems on the Extended
Spatially Completely Anisotropic Triangular Lattice
(֦ுۭؒҟํతࡾ֯֨ࢠ্ͷวྺిࢠ‫ܥ‬ͷ࣓‫ߏؾ‬଄) ɹ
Տ໺ɹ༎‫ل‬
2. ެද࿦จ
(1) On Scaling Relations of Organic Antiferromagnets with Magnetic
Anions
Hiroshi Shimahara and Yuki Kono
Journal of the Physical Society of Japan, 86, 043704-1 - 043704-5
(2017).
(2) Magnetic Structures of Electron Systems on the Extended
Spatially Completely Anisotropic Triangular Lattice near Quantum Critical Points
Yuki Kono and Hiroshi Shimahara
Journal of the Physical Society of Japan, 90, 024708-1 - 124711-8
(2021).

ओ࿦จ

Acknowledgments
I would like to thank Professor Hiroshi Shimahara for his careful guidance
and many advices. I am indebted to Professor Yutaka Nishio for useful
discussions, information, and experimental data. I am indebted to Professors
Sinya Uji, Yugo Oshima, Takaaki Minamidate, Shuhei Fukuoka, and Takuya
Kobayashi for useful discussions.

3

Contents
1 Introduction and Purpose
6
1.1 Extended Spatially Completely Anisotropic Triangular Lattice
6
1.2 Organic Compound λ-(BETS)2 FeCl4 . . . . . . . . . . . . . . 11
1.3 Purpose of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 13
2 Scaling Relations in Mixed Crystal Systems
2.1 Scaling Relations in Organic Compound . . . . . . . . . . .
2.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Analysis Based on the Random Phase Approximation
2.2.2 Analysis Based on the Mean Field Theory . . . . . .

.
.
.
.

16
16
18
20
21

3 Magnetic Structures of Electron Systems on the ESCATL
3.1 Energy Dispersion . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Parameter Sets and Fermi Surface . . . . . . . . . . . . . . . .
3.3 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Magnetic Structure of λ-Fe . . . . . . . . . . . . . . . . . . .
3.5 Phase Diagram and Effect of Imbalance of Spatial Anisotropy
3.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . .

24
24
26
28
33
36
36

4 Summary of Thesis and Conclusion

45

A Trapezoidal Formula

47

B Linear Response Theory

52

C Derivation of Eq. (3.50)

56

D Derivation of Eq. (2.15)

65

E Derivation of Eq. (2.19)

68

4

F Derivation of Eqs. (2.28) and (2.29)

5

74

Chapter 1
Introduction and Purpose
1.1

Extended Spatially Completely Anisotropic
Triangular Lattice

Electron systems on triangular lattices have been extensively researched because they exhibit interesting phenomena, such as quantum spin liquids,
magnetic plateaus, and spiral magnetic structures [1–3]. They originate from
the geometrical frustration of the spin alignment. The frustration is maximum when the magnitudes of antiferromagnetic exchange interactions on the
bonds are all equal, and the spatial anisotropy of the antiferromagnetic exchange interactions reduces the geometrical frustration. However, the effect
of spatial anisotropy can be significant when real compounds are examined.
Some compounds contain spatially anisotropic triangular lattices that consist of two types of triangles of the bonds. These lattices are called extended
spatially completely anisotropic triangular lattice (ESCATL). The localized
spin model on the ESCATL has six kinds of exchange interactions with the
coupling constants Jl and Jl as shown in Fig. 1.1, whereas the itinerant electron model on the ESCATL has six kinds of transfer integrals tl and tl as
shown in Fig. 1.2, where l=1, 2, and 3.
When we consider special cases, the ESCATL is reduced to some frustrated lattices. For example, when Jl = Jl for all l and J2 = J3 , it reduces
to the spatially anisotropic triangular lattice (SATL) [2,4–9]. The SATL has
been studied as the lattice that is composed of π-electron systems in
β-Me4−n Etn X [Pd(dmit)2 ]2 (X = P, Cs, N, Sb, and As) and κ-(BEDTTTF)2 Cu(CN)3 , where dimt and BEDT-TTF stand for 1,3-dithiol-2-thione4,5-dithiolate and bis(ethylenedithio)tetrathiafulvalene. Hereafter, we abbreviate β-Me4−n Etn X [Pd(dmit)2 ]2 and κ-(BEDT-TTF)2 Cu(CN)3 as X-n and
κ-ET, respectively. In κ-ET, it has been suggested from the experimen6

J2

J3
J1

J2′

J2

J3′
J1′
J3
J1

y

x

Figure 1.1: Extended spatially completely anisotropic triangular lattice and
definition of exchange coupling constants.

t2

t3
t1

t2′

t2

t3′
t1′
t3
t1

y

x

Figure 1.2: Definitions of the transfer integrals on the ESCATL.

7

tal results of susceptibility and nuclear magnetic resonance (NMR) that the
ground state is quantum spin liquid [10]. The SATL has also been studied
in the spin system of Cs2 CuBr4 and Cs2 CuCl4 . In the compound Cs2 CuBr4 ,
a magnetization plateau was observed in the magnetization process [11, 12].
When Jl = Jl for all l, it is reduced to the spatially completely anisotropic
triangular lattice (SCATL) [4]. Hauke examined it for X-n (X= P, Cs, N,
Sb and As) and showed that the ground states of As-2 and Sb-0 are the
N´eel-(π, π) states shown in Fig. 1.3(a) [4].
When J3 = 0, the ESCATL reduces to the trellis lattice [13]. The trellis
lattice is contained the compounds SrCu2 O3 , NaV2 O5 , and Agx V2 O5 [14–16].
When J3 = J1 = J1 = 0, the ESCATL reduces to the honeycomb lattice [17]. The honeycomb lattice is contained the compounds InCu2/3 V1/3 O3
and Na3 T2 SbO6 (T = Cu, Ni, and Co) [18]. Compared to these lattices, the
ESCATL has the unique feature of the imbalance of the spatial anisotropies
in two types of triangles.
In this thesis, we introduce the itinerant electron model on the ESCATL
to examine the compound λ-(BETS)2 X Cl4 (X=Fe, Ga, Fex Ga1−x ), where
BETS stands for bis(ethylenedithio)tetraselenafulvalene. Hereinafter, we abbreviate λ-(BETS)2 X Cl4 as λ-X. We extend the knowledge of the ESCATL
antiferromagnets and examine the magnetic structure of the λ-X system.
The ESCATL in π-electron system corresponds to each BETS molecule as
shown in Fig. 1.2. The λ-Fe system exhibits an antiferromagnetic long-range
order (AF LRO) [19, 20]. In the λ-Fe system, as the temperature decreases,
the magnetization m of the itinerant π-electrons saturate first, and the 3dspins follow a constant exchange field created by the π electrons [21, 22]. We
consider a realistic situation in which the spiral state is suppressed [4]. We
assume the N´eel and up-up-down-down (uudd) phases defined in Fig. 1.3
and 1.4 as collinear spin structures [28]. In this thesis, we examine the magnetic structure of ESCATL antiferromagnets in the ground state within a
mean-field approximation. In particular, we reveal the effect of the imbalance of the spatial anisotropies in two types of triangles. We define the
parameter rimb that represents the imbalance of spatial anisotropies as
rimb ≡

t3 /t2 − t3 /t2
t3 /t2

(1.1)

because t1 = t1 is satisfied in the λ-Fe system [23].
The organic compound λ-(BEDT-STF)2 X Cl4 (X=Fe,Ga) [24, 25], where
BEDT-STF stands for bis(ethylenedithio)diselenadithiafulvalene, is a material in which Se atoms of BETS molecules are replaced with S atoms. This
replacement is effectively a negative pressure applied to the λ-X system. In
8

(b)

(a)
B’

A’

B’

A’

A

B

A

B

1
2’
B’

3’
A’

2’
B’

3’

A’
1’

1’
2
A

1

3

B

2

B

A

3
1

1
(c)
B’

B

B’

A’

A

A’

3’
B’

B

B’

A

A’

B
A
2’
B’

1
1’

2
A

3

A’

1

Figure 1.3: Magnetic structures which is examined. (a) N´eel-(π, π) state, (b)
N´eel-(π, 0) state, (c) N´eel-(0, π) state.

9

(b)

(a)
B’

A’

B’

A’

A’

B

A

B

A

B

3’
B’

A’

2’
B’

B’

B’

A

B

A

3’
A’

B’

A’

A

B

1’

1’
2
A

A’

1

1
2’
A’

B’

3

B

A

2
A

B

3

B

1

1

Figure 1.4: Magnetic structures which are examined. (a) uudd-2 state and
(b) uudd-2 state.

the λ-(BEDT-STF)2 FeCl4 system, the metal-insulator transition accompanied by the paramagnetic-antiferromagnetic transition has been observed at
TN = 16 K [26, 27]. In organic compound λ-D2 A, where D and A represent
a doner and an anion, respectively, the dimerized doners form the ESCATL.
We examine wide ranges of parameters so that we can consider compounds
that have not yet been discovered.
For the localized spin model on the ESCATL, the classical phase diagram
includes five different collinear antiferromagnetic phases and spiral phase [28].
It was shown that the imblance of the spatial anisotropies, which are parameterized by J3 /J2 and J3 /J2 , stabilizes the uudd phase shown in Fig. 1.4(a)
and (b).
A uudd phase has been studied in several materials [29–34]. It has been
suggested that in the solid 3 He the ground state is the uudd state [29].
Roger examined the magnetism of 3 He by a two-parameter model based
on three-spin exchange and planar four-spin exchange [30]. In the insulating perovskite HoMnO3 , which is a frustrated spin system having ferromagnetic nearest-neighbor and antefferomagnetic next-nearest-neighbor interactions within a MnO2 , the uudd phase has been experimentally suggested [31, 32]. Kaplan proposed a possible mechanism for this state by the
frustrated classical Heisenberg model in one dimension with nearest-neighbor
biquadratic exchange [33]. Zou et al. found the uudd structure in the spinel
conpound GeCu2 O4 by the effective classical spin Hamiltonian containing
10

nearest-neighbor biquadratic exchange interaction [34].

1.2

Organic Compound λ-(BETS)2FeCl4

In the compound λ-(BETS)2 FeCl4 , the BETS layers and FeCl4 anion layers
are stacked alternately, and the BETS layers and FeCl4 anion layers have π
electrons and 3d spins, respectively. The λ-Fe system is highly conductive
in the direction parallel to the ac plane, whereas it has low conductivity in
the b-axis direction because the electron transfer is blocked by the anion
layer. The crystal system is triclinic and the space group in paramagnetic
phase is P¯1 [35]. ...