*Article* **Optical Conductivity Spectra of Charge-Crystal and Charge-Glass States in a Series of** *θ***-Type BEDT-TTF Compounds**

**Kenichiro Hashimoto 1,2,\*, Ryota Kobayashi 2, Satoshi Ohkura 2, Satoru Sasaki 2, Naoki Yoneyama 3, Masayuki Suda 4,5, Hiroshi M. Yamamoto <sup>5</sup> and Takahiko Sasaki <sup>2</sup>**


**Abstract:** In the 3/4-filled band system *θ*-(BEDT-TTF)2*X* with a two-dimensional triangular lattice, charge ordering (CO) often occurs due to strong inter-site Coulomb repulsion. However, the strong geometrical frustration of the triangular lattice can prohibit long-range CO, resulting in a chargeglass state in which the charge configurations are randomly distributed. Here, we investigate the charge-glass states of orthorhombic and monoclinic *θ*-type BEDT-TTF salts by measuring the electrical resistivity and optical conductivity spectra. We find a substantial difference between the charge-glass states of the orthorhombic and monoclinic systems. The charge-glass state in the orthorhombic system with an isotropic triangular lattice exhibits larger low-energy excitations than that in the monoclinic one with an anisotropic triangular lattice and becomes more metallic as the isotropy of the triangular lattice increases. These results can be understood by the different charge-glass formation mechanisms in the two systems: in the orthorhombic system, the charge-glass state originates from geometric frustration due to the equilateral triangular lattice, leading to metallic 3-fold COs, whereas in the monoclinic system, the charge-glass formation originates from geometric frustration of the isosceles triangular lattice, in which the charge-glass state is described by the superposition of insulating 2-fold stripe COs.

**Keywords:** strongly correlated electrons; metal-insulator transition; charge order; charge glass; charge crystal; geometrical frustration; organics; optical conductivity

#### **1. Introduction**

Charge ordering (CO), in which electrons self-organize into an alternating pattern of charge-rich and charge-poor sites owing to strong Coulomb interactions, often emerges in strongly correlated electron systems [1,2]. Inter-site Coulomb interactions play an essential role in the formation of CO, mostly leading to a long-range order. In geometrically frustrated systems, however, disordered ground states without long-range CO have been reported [3–10]. This is considered due to competition among various types of CO patterns coming from geometrical frustration [11], which prevents a specific charge configuration, similar to geometrically frustrated spin systems such as a quantum spin liquid and spin glass. Thus, geometrical frustration can suppress the tendency toward long-range CO, leading to exotic electronic states such as a charge-glass state.

**Citation:** Hashimoto, K.; Kobayashi, R.; Ohkura, S.; Sasaki, S.; Yoneyama, N.; Suda, M.; Yamamoto, H.M.; Sasaki, T. Optical Conductivity Spectra of Charge-Crystal and Charge-Glass States in a Series of *θ*-Type BEDT-TTF Compounds. *Crystals* **2022**, *12*, 831. https:// doi.org/10.3390/cryst12060831

Academic Editor: Andrej Pustogow

Received: 16 May 2022 Accepted: 9 June 2022 Published: 12 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Quasi-two-dimensional (quasi-2D) organic compounds with a triangular lattice, *θ*- (BEDT-TTF)2*MM*- (SCN)4 (*M* = Tl, Rb, Cs, *M*- = Zn, Co) (where BEDT-TTF denotes the donor molecule bis(ethylenedithio)tetrathiafulvalene and *MM*- (SCN)4 represents a monovalent anion), have been extensively studied as a platform of the CO metal-insulator transition system [12–31]. In *θ*-(BEDT-TTF)2*MM*- (SCN)4, there are two crystal forms with orthorhombic (*I*222) and monoclinic (*C*2) symmetries [10,12] (see Figure 1a–d). The both crystal structures consist of an alternating stack of BEDT-TTF and anion layers, and the charge transfer between these layers leads to a quarter-filled hole band system. Figure 1b,d show the 2D molecular arrangement of each BEDT-TTF layer for the orthorhombic *θ*o-type and monoclinic *θ*m-type systems, respectively. The nearest-neighbor Coulomb interactions are given by *V*<sup>1</sup> and *V*<sup>2</sup> in the *θ*o-type system and by *V*1, *V*2, and *V*- <sup>2</sup> in the *θ*m-type system. Note that *<sup>V</sup>*<sup>2</sup> <sup>≈</sup> *<sup>V</sup>*- <sup>2</sup> for the *θ*m-type system [10,12,25]. While the *θ*o-type system exhibits a horizontal CO pattern (Figure 1b), the *θ*m-type system shows a diagonal CO pattern (Figure 1d) [20]. The ground states of *θ*o-type salts, ranging from charge ordered insulating states to a metallic state, can be tuned using the anisotropy of the nearest-neighbor Coulomb interactions on the triangular lattice, *V*2/*V*1, which depends on the anions [8] (see Figure 1e). Indeed, the orthorhombic *θ*-(BEDT-TTF)2TlZn(SCN)4 (hereafter, abbreviated as *θ*o-TlZn) and *θ*-(BEDT-TTF)2RbZn(SCN)4 (*θ*o-RbZn) with relatively anisotropic *V*2/*V*<sup>1</sup> values are well-known long-range CO compounds with the transition temperature *T*CO ≈ 240 K [20] and 200 K [3], respectively. Such a periodic CO state can be regarded as a charge-crystal state. Importantly, this charge-crystal state can be kinetically avoided when the sample is cooled faster than a critical cooling rate, leading to a charge-glass state where the charge configurations are randomly quenched. For instance, the charge-crystal state in *θ*o-RbZn can be suppressed for the critical cooling rate of ∼30 K/min, resulting in a charge-glass state (see Figures 1f and 2c). In *θ*o-CsZn, which has a more isotropic triangular lattice, the critical cooling rate becomes much slower. As a result, the charge-glass state can be realized even upon very slow cooling (<0.1 K/min) (see Figures 1f and 2a). These experimental facts imply that geometrical charge frustration between *V*<sup>1</sup> and *V*<sup>2</sup> plays an important role for the charge-glass formation in the *θ*o-type salts.

**Figure 1.** Crystal structure and phase diagram of *θ*–type (BEDT–TTF) salts. (**a**) Crystal structure of the *θ*o-type salts viewed along the *c*-axis direction. The rectangle indicates the unit cell. (**b**) 2D conducting BEDT-TTF layers within the *a*-*c* plane for the *θ*o-type salts. (**c**) Crystal structure of the *θ*m-type system viewed along the *b*-axis direction. The parallelogram indicates the unit cell. (**d**) 2D conducting BEDT-TTF layers within the *b*-*c* plane for the *θ*m-type system. (**e**) Phase diagram of the *θ*o-type system a function of the anisotropy parameter, *V*2/*V*1. (**f**) Critical cooling rate for charge-glass formation for various *θ*-type salts as a function of *V*2/*V*1.

The monoclinic *θ*-(BEDT-TTF)2TlZn(SCN)4 (*θ*m-TlZn) shows a diagonal CO at *T*CO = 170 K [10] (see Figure 2e). In *θ*m-TlZn, the long-range CO can be suppressed by rapid cooling (>∼50 K/min), and the charge-glass state can be realized. Although the triangular lattice for *θ*m-TlZn is more anisotropic than that for *θ*o-TlZn, the critical cooling rate of *θ*m-TlZn is much slower than that of *θ*o-TlZn (see Figure 1f). This suggests different mechanisms of charge-glass formation between these two systems. In this study, in order to clarify the different charge-glass formation mechanisms between the orthorhombic and monoclinic *θ*-type salts, we measured electrical resistivity and optical conductivity spectra of *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn.

**Figure 2.** Resistivity and Arrhenius plot in the *θ*–type salts. (**a**) *ρ*(*T*) curve of *θ*o-CsZn measured during cooling. (**b**) Arrhenius plot of the same data in (**a**). (**c**) *ρ*(*T*) curve of *θ*o-RbZn measured during rapid cooling of 30 K/min (blue) and slow cooling of 0.1 K/min (red). (**d**) Arrhenius plot of the same data in (**c**). (**e**) *ρ*(*T*) curve of *θ*m-TlZn measured during rapid cooling of 100 K/min (blue) and slow heating after slow cooling of 0.1 K/min (red). (**f**) Arrhenius plot of the same data in (**e**). The black lines in (**b**,**d**,**f**) represent the fits to *ρ* ∝ exp (Δ/(*k*B*T*)).

#### **2. Materials and Methods**

Single crystals of *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn were grown by the electrochemical oxidation method [17]. The typical sample size used for the resistivity and optical conductivity measurements was ∼0.1 mm × 1 mm × 3 mm. The in-plane dc resistivity was measured by the 4-terminal method in the linear *I*-*V* region. The polarized optical conductivity measurements were carried out with a Fourier transform microscope spectrometer in the range of 600–8000 cm<sup>−</sup>1. For *θ*o-CsZn, the optical conductivity measurements in the farinfrared region (100–650 cm<sup>−</sup>1) were performed using a synchrotron radiation light source at BL43IR in SPring-8. The optical conductivity was calculated through a Kramers–Kronig (KK) transformation from the optical reflectivity determined by comparison with a gold thin film evaporated on the sample surface.

#### **3. Results**

#### *3.1. Electrical Resistivity in θo-CsZn, θo-RbZn, and θm-TlZn*

Figure 1a,c,e show the temperature dependence of resistivity *ρ*(*T*)for *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn, respectively. *θ*o-CsZn with the most isotropic triangular lattice shows no longrange CO and enters the charge-glass state below ∼100 K. Regardless of the cooling rate of the sample, *θ*o-CsZn always shows the charge-glass state. In contrast, *θ*o-RbZn and *θ*m-TlZn show the long-range CO transition at 200 K and 170 K, respectively, which can be suppressed by rapid cooling, resulting in the charge-glass state. Figure 1b,d,f show the Arrhenius plot

of the resistivity for *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn, respectively. Clear activation-type behaviors can be seen both for the charge-crystal and charge-glass states in all the salts. By fitting the data to *ρ*(*T*) = *ρ*<sup>0</sup> exp (Δ/*k*B*T*), we obtained the activation energies for the charge-crystal and charge-glass states as shown in Figure 2b,d,f. The activation gap for *θ*o-CsZn is very small (approximately 18 K), consistent with the fact that this material is located near the phase boundary. As for *θ*o-RbZn, the gap sizes of the charge-crystal and charge-glass states show a large difference. Previous X-ray diffuse scattering experiments in *θ*o-CsZn have revealed a short-range 3 × 3 CO [5,13,14]. Since the 3-fold CO in *θ*o-RbZn is expected to be metallic, the gap size of the charge-glass state (∼400 K) becomes much smaller than that of the charge-crystal state (∼1950 K). In contrast, in *θ*m-TlZn, there is little difference in the activation gaps between the charge-crystal and charge-glass states. These differences are considered to originate from the different mechanisms of the charge-glass formation in the *θ*o- and *θ*m-type systems, as will be discussed later.

#### *3.2. Optical Conductivity Spectra in θo-CsZn, θo-RbZn, and θm-TlZn*

For a comprehensive understanding of charge-glass formation in *θ*-type salts, we measured the optical conductivity spectra in the series of *θ*-type salts. Figure 3a shows the optical conductivity spectra *<sup>σ</sup>*1(*ω*) of *<sup>θ</sup>*o-CsZn for *E a* at various temperatures. At room temperature, there are two characteristic broad bands at around 1000 and 2000 cm<sup>−</sup>1. In addition, the antisymmetric or antiresonance features of the vibrational modes of the BEDT-TTF molecule around 400, 900, and 1300 cm−<sup>1</sup> can be seen, which become more noticeable at low temperatures [7]. Moreover, as lowering the temperature, *σ*1(*ω*) in the low-energy region below ∼500 cm−<sup>1</sup> is strongly enhanced. The enhancement of *<sup>σ</sup>*1(*ω*) is different from a Drude response since the dc conductivity *σ*dc in *θ*o-CsZn decreases to ∼<sup>1</sup> <sup>Ω</sup>−1cm−<sup>1</sup> at 4 K. The absence of a Drude peak at low temperatures is also expected from the fact that the resistivity obeys the Arrhenius law as shown in Figure 2b. Thus, the optical spectra are mainly composed of three characteristic structures with center frequencies of 100–300, 800–1000, and 2000–2500 cm−<sup>1</sup> (referred to as *L*low, *L*middle, and *L*high, respectively). As discussed in Ref. [7], the low-energy peak can be attributed to the short-range CO with a relatively long-period 3 × 3 CO. As for the broad bands *L*middle and *L*high, very similar features have been observed in other quarter-filled organic conductors close to a CO phase [32–34]. A transition between Hubbard-like bands induced by the intersite Coulomb repulsion *V* gives rise to a broad band in the mid-infrared region of the order of *V*, which corresponds to *L*high. The other band *L*middle is a charge-fluctuation band originating from short-range CO fluctuations.

Figure 3b shows the temperature dependence of *<sup>σ</sup>*1(*ω*) of *<sup>θ</sup>*o-RbZn for *E a* measured during slow cooling (charge-crystal state) and heating after rapid cooling (charge-glass state). The optical conductivity spectra at room temperature are similar to that of *θ*o-CsZn, having two broad structures at around 1000 and 2500 cm−1. When the sample is slowly cooled, the CO transition occurs at 200 K, below which the optical conductivity spectra show a drastic change. The spectral weight below 3000 cm−<sup>1</sup> disappears, and a clear optical gap is observed below ∼2000 cm<sup>−</sup>1, which is comparable to <sup>Δ</sup>/*k*<sup>B</sup> obtained from the Arrhenius plot of the resistivity in the charge-crystal state. In contrast, when the sample is quenched, the CO transition is suppressed and a charge-glass state is realized. The optical conductivity spectra in the charge-glass state share a similar shape with that above the CO transition. Thus, the optical conductivity spectra between the charge-crystal and charge-glass states show a large difference, indicating that the charge configurations are very different between these two states.

**Figure 3.** Optical conductivity spectra in the *θ*–type salts. (**a**) Optical conductivity spectra *σ*1(*ω*) in *θ*o-CsZn measured at several temperatures during slow cooling of 1 K/min. (**b**) Optical conductivity spectra *σ*1(*ω*) in *θ*o-RbZn measured at several temperatures during slow cooling of 1 K/min (solid line) and slow heating after rapid cooling of more than 50 K/min when passing through *T*CO (dashed line). (**c**) Optical conductivity spectra *σ*1(*ω*) in *θ*m-TlZn measured at several temperatures during slow cooling of 1 K/min (solid line) and slow heating after rapid cooling of more than 50 K/min when passing through *T*CO (dashed line). For clarity, the data are shifted vertically. Note that the sharp peak at approximately 2100 cm−<sup>1</sup> is the CN stretching mode of SCN in the anion layer.

Figure 3c shows the temperature dependence of *<sup>σ</sup>*1(*ω*) of *<sup>θ</sup>*m-TlZn for *E c* measured in the charge-crystal and charge-glass states. Although the optical conductivity spectra are slightly different in the charge-crystal and charge-glass states, the optical gaps are almost identical in the whole temperature range, which are consistent with the dc resistivity data in which the activation energies for the charge-crystal and charge-glass states are close to each other (see Figure 2f). The obtained optical gaps in the charge-crystal and charge-glass states are about 1200–1300 cm−<sup>1</sup> at low temperatures, which are comparable to Δ/*k*<sup>B</sup> obtained from the Arrhenius plot of the resistivity. Importantly, in the charge-crystal state, in addition to a broad peak structure around 3500 cm−1, a shoulder-like feature around 2000 cm−<sup>1</sup> emerges as lowering the temperature, whereas in the charge-glass state, the growth of the 2000 cm−<sup>1</sup> feature seems to be frozen (see the hatched area in Figure 3). The optical spectra obtained for the charge-crystal state can be well understood by previous theoretical calculations performed for the diagonal CO phase [31], in which the first low-energy peak and the high-energy broad feature are well reproduced.

#### **4. Discussion**

We compare the optical conductivity spectra of the three salts in the charge-glass state. Figure 4 shows the optical conductivity spectra of the charge-glass states in *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn. It can be clearly seen that the optical conductivity of *θ*o-CsZn with the most isotropic triangular lattice shows a significant low-energy peak, and as the anisotropy of the triangular lattice increases, the spectral weight in the low-energy region shifts to a higher-energy region. This systematic evolution of the optical conductivity spectra is considered to reflect the different charge configurations in the charge-glass states in the three salts.

To discuss the difference of the optical conductivity spectra in the three salts, we consider the extended Hubbard model. The ground-state properties of *θ*-(BEDT-TTF)2*X* have been extensively studied by the extended Hubbard model on an anisotropic triangular lattice [25,27,28,30,31,35–38]. The Hamiltonian of the extended Hubbard model is given by

$$\mathcal{H}\_{\rm EHM} = \sum\_{\langle i,j \rangle \sigma} \left( -t\_{\bar{i}\bar{j}} c\_{i\sigma}^{\dagger} c\_{j\sigma} + \text{h.c.} \right) + \mathcal{U} \sum\_{i} n\_{i\uparrow} n\_{i\downarrow} + \sum\_{\langle i,j \rangle} V\_{i\bar{j}} n\_{i} n\_{j\uparrow} \tag{S1}$$

where *c*† *<sup>i</sup><sup>σ</sup>* (*ciσ*) is the creation (annihilation) operator for a hole at the *i*-th site with spin *<sup>σ</sup>* (↑ or ↓), *ni* (≡ <sup>∑</sup>*<sup>σ</sup> ni<sup>σ</sup>* ≡ <sup>∑</sup>*<sup>σ</sup> <sup>c</sup>*† *<sup>i</sup><sup>σ</sup>ciσ*) is the number operator, *tij* and *Vij* are the transfer integrals and the intersite Coulomb interactions between the *i*-th and *j*-th sites, respectively, and *U* is the on-site Coulomb repulsion.

**Figure 4.** Comparison of optical conductivity spectra of the charge–glass states in the *θ*–type salts. Optical conductivity spectra of the charge-glass states in *θ*o-CsZn (red), *θ*o-RbZn (blue), and *θ*m-TlZn (green) measured at 4 K, 50 K, and 50 K, respectively.

In Ref. [31], the polarization dependence of the optical conductivity spectra has been calculated for various charge ordering patterns, based on the extended Hubbard model. When *V*<sup>1</sup> and *V*<sup>2</sup> are close, the optical conductivity spectra for two polarization directions become isotropic, except for the difference in the magnitude of the optical conductivity spectra (which reflects the difference between the intermolecular distances in the *V*<sup>1</sup> and *V*<sup>2</sup> directions [31]), indicating the strong geometric frustration of the isotropic triangular lattice. In contrast, when *V*<sup>1</sup> is larger than *V*<sup>2</sup> (that is, in the case of diagonal CO), the optical conductivity spectra for two polarization directions become anisotropic: the optical spectra for the polarization parallel to the *V*<sup>1</sup> direction (*b*-axis direction in *θ*m-TlZn) have only a low-energy peak, whereas the optical spectra for the polarization perpendicular to the *V*<sup>1</sup> direction (*c*-axis direction in *θ*m-TlZn) show a step-like increase in the lowenergy region, followed by a broad structure in the high-energy region. Indeed, very similar behaviors have been observed in our experimental results (see Figure 5). Such a polarization dependence in the optical spectra has also been observed in other 1D chargeordered organic materials [39], where the polarization dependence corresponding to the stripe charge ordering pattern has been reported.

Next, we discuss the charge configurations of the charge-glass states in the *θ*o-type and *θ*m-type systems. When *U tij*, the extended Hubbard model is compatible to the spinless fermion model (*t*-*V* model) that neglects the spin degrees of freedom. The Hamiltonian of the *t*-*V* model is given by

$$\mathcal{H}\_{t \cdot V} = \sum\_{\langle i, j \rangle} \left( -t\_{ij} f\_i^{\dagger} f\_j + \text{h.c.} + V\_{ij} \vec{n}\_i \vec{n}\_j \right), \tag{S2}$$

where *f* † *<sup>i</sup>* (*fi*) is the creation (annihilation) operator for a spinless fermion at the *i*-th site and *n*˜*<sup>i</sup>* = *f* † *<sup>i</sup> fi* is the number operator. It has been well established that the classical ground states of the *t*-*V* model (*tij* = 0) on an isosceles triangle lattice as shown in Figure 6b are disordered owing to geometric frustration when *V*<sup>1</sup> ≥ *V*2, whereas the vertical CO becomes a unique ground state at *V*<sup>1</sup> < *V*<sup>2</sup> [29,40,41]. At *V*<sup>1</sup> > *V*2, the chain-striped states such as the horizontal and diagonal COs emerge owing to the geometric frustration between the two diagonal Coulomb interactions *V*2, which is the case for *θ*m-TlZn.

**Figure 5.** Polarization dependence of optical conductivity spectra in the charge–glass/crystal states of the *<sup>θ</sup>*–type salts. (**a**–**c**) Optical conductivity spectra of (**a**) the charge-glass state in *<sup>θ</sup>*o-CsZn for *E a* (blue) and *E c* (red) measured at 4 K, (**b**) the charge-crystal state in *<sup>θ</sup>*o-RbZn for *E a* (blue) and *E c* (red) measured at 50 K, and (**c**) the charge-crystal state in *<sup>θ</sup>*m-TlZn for *E c* (blue) and *E b* (red) measured at 50 K.

When *V*<sup>1</sup> = *V*2, the ground states include a vertical-striped state, horizontal-striped state, diagonal-striped state, and three-sublattice state, all of which are degenerate (see Figure 6a,). The three-sublattice state has been discussed in terms of a pin-ball liquid [29]. The *θ*o-type compounds can be categorized in this regime. Indeed, the X-ray diffuse scattering experiments have revealed that for *<sup>θ</sup>*o-CsZn, diffuse rods with *<sup>q</sup>*<sup>d</sup> = (2/3, *<sup>k</sup>*, 1/3) corresponding to a 3 × 3 CO are observed [5,13,14], while for *θ*o-RbZn above *T*CO, diffuse rods associated with a short-range 3 <sup>×</sup> 4 CO are observed at *<sup>q</sup>*<sup>d</sup> = (±1/3, *<sup>k</sup>*, <sup>±</sup>1/4) [4,15,16]. Such three-fold diffuse rods are different from that of *θ*m-TlZn, where diffuse lines at *<sup>q</sup>*<sup>d</sup> = (1/2, *<sup>l</sup>*) corresponding to the superposition of the chain striped COs as shown in Figure 6b have been observed [10].

Based on the above calculations, we discuss the charge-glass formation mechanisms in the *θ*o-type and *θ*m-type systems. In the charge-glass state of the *θ*o-type salts with an isotropic triangular lattice, the short-range 3-fold periodic CO patterns have been observed. Although the *q* vectors of the short-range COs are slightly different from the 3-fold CO shown in Figure 6a, the presence of the 3-fold COs makes the system metallic. As a result, the optical gap and Arrhenius gap in the charge-glass state become smaller than that in the charge-crystal state. On the other hand, in the charge-glass state of the *θ*m-type system with an isosceles triangular lattices, the short-range 2-fold COs have been reported. In this case, the charge-glass state is described by the superposition of the insulating 2-fold stripe COs [10]. Therefore, there is no significant difference in the magnitude of charge separation between the charge-crystal and the charge-glass states. Thus, the sizes of the optical gap and Arrhenius gap become almost the same in the charge-crystal and chargeglass states. From these facts, we conclude that the charge-glass states in the *θ*o-type and *θ*m-type systems originate from the geometrical frustration of the equilateral and isosceles triangular lattices, respectively. Since recent thermal expansion and noise spectroscopy measurements have pointed out that the lattice degrees of freedom in addition to the electron degrees of freedom play an important role for the charge-glass formation in the *θ*-type BEDT-TTF compounds [42,43], elucidating the effect of lattice degrees of freedom on the charge-glass formation needs to be addressed in the future.

**Figure 6.** Schematic charge configurations on triangular lattices. (**a**) Charge configurations on the isosceles triangular lattice. Vertical, horizontal, diagonal, and three-sublattice COs are described. In the three-sublattice structure, the sublattice A is filled by one hole (pin), the sublattice B is empty, and the sublattice C is randomly occupied by the remaining holes (ball). The green hexagon stands for the unit cell. (**b**) Chain striped CO patterns on the isosceles triangular lattice, such as horizontal and diagonal COs. *V*<sup>1</sup> and *V*<sup>2</sup> (*V*<sup>1</sup> > *V*2) are the nearest-neighbor Coulomb interactions. Since all these states are degenerate in the classical limit of the *t*-*V* model, the classical ground state can be described by the superposition of these states. The magenta and white circles represent the charge-rich and charge-poor sites, respectively.

#### **5. Conclusions**

We investigated the charge-glass states of *θ*o-CsZn, *θ*o-RbZn, and *θ*m-TlZn by measuring the electrical resistivity and optical conductivity spectra. We find that there is a fundamental difference between the charge-glass formation mechanisms in the *θ*o-type and *θ*m-type systems. The charge-glass state in *θ*o-CsZn exhibits large low-energy excitations, consistent with the fact that the material is located near the CO phase boundary. In *θ*o-RbZn, the optical gaps between the charge-crystal and charge-glass states show a large difference, indicating that the charge configurations are very different between the two states. In contrast, the optical gap of the charge-glass state in *θ*m-TlZn does not differ from that in the charge-crystal state. These results can be understood by the different charge-glass formation mechanisms in the *θ*o-type and *θ*m-type systems: in the *θ*o-type system, the charge-glass state originates from geometric frustration due to the equilateral triangular lattice, leading to metallic 3-fold COs, whereas in the *θ*m-type system, the charge-glass formation originates from geometric frustration of the isosceles triangular lattice, in which the charge-glass state can be described by the superposition of insulating 2-fold stripe COs.

**Author Contributions:** K.H. and T.S. conceived the project. K.H., S.O. and S.S. carried out electrical resistivity measurements. K.H., R.K., S.O., S.S. and T.S. performed optical measurements. N.Y., M.S., H.M.Y. and T.S. carried out sample preparation. K.H. wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) (Nos. JP21H01793, JP20H05144, JP19K22123, JP19H01833, JP18KK0375, JP18H01853), and Grantin-Aid for Scientific Research for Transformative Research Areas (A) "Condensed Conjugation" (No. JP21H05471, JP20H05869, JP20H05870) from Japan Society for the Promotion of Science (JSPS).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Acknowledgments:** We thank M. Naka, K. Yoshimi, C. Hotta, T. Thomas, and J. Mu¨ller for fruitful discussion and Y. Ikemoto and T. Moriwaki for technical assistance. Optical measurements using a synchrotron radiation light source were performed at SPring-8 with the approvals of the Japan Synchrotron Radiation Research Institute (Grant Nos. 2016A0073, 2020A0639, 2020A1065).

**Conflicts of Interest:** The authors declare no competing interests.

#### **References**

