Band Structure and Physical Properties of α-STF₂I₃: Dirac Electrons in Disordered Conduction Sheets

Abstract

The compound being investigated is an organic charge-transfer complex of the unsymmetrical donor STF with I₃ [STF = bis(ethylenedithio)diselenadithiafulvalene], which is isostructural with α-ET₂I₃ and α-BETS₂I₃ [ET = bis(ethylenedithio)tetrathiafulvalene, BETS = bis(ethylenedithio)tetraselenafulvalene]. According to recent studies, the calculated band structure should represent a zero-gap semiconductor at 1 bar that is similar to α-ET₂I₃ under high pressure (>15 kbar). Such materials have attracted extensive interest because the electrons at the Fermi level can be massless Dirac fermions (MDFs), with relativistic behaviors like those seen in graphene. In fact, α-STF₂I₃ exhibited nearly temperature-independent resistivity, ρ, (~100–300 K), a phenomenon that is widely observed in zero-gap semiconductors. The non-Arrhenius-type increase in ρ (<~100 K) was consistent with the characteristics of interacting MDFs. The paramagnetic susceptibility, χ, (2–300 K)—as well as the reflectivity, R and optical conductivity, σ, (25–300 K; 400–25,000 cm⁻¹)—were also almost temperature independent. Furthermore, σ was practically independent of wavenumber at ~6000–15,000 cm⁻¹. There was no structural transition based on X-ray studies (90–300 K). Considering all the electrical, magnetic, optical and structural properties of α-STF₂I₃ at 1 bar, it was concluded that the salt possesses a band structure characterized with Dirac cones, which was consistent with the calculation.

1. Introduction

Recently, topological materials have attracted increased research interest in the field of solid-state chemistry and physics [1,2,3]. Various descriptors are used to refer to these materials, such as topological insulators, Dirac fermions (Dirac electrons), Weyl semimetals, Berry phases, time reversal symmetry, non-Abelian anyons and Majorana particles. These materials are considered to contain or connect fundamental problems in nature concerning the states of matter, nuclear physics and mathematics. The traditional topological materials are diverse, such as graphene [4], quasicrystals [5] and the surfaces of particular materials [1], all of which should be of high-quality with well-defined structures and chemical formulae. Among them, bulk samples, crystalline materials in particular, are valuable and advantageous because they facilitate various types of physical measurements, which enable a detailed study of their properties and structures. An advantage is that the materials are readily prepared for most experimental purposes by standard methods of chemical synthesis as single crystals of sufficient dimensions, qualities and quantities that are stable enough to be handled under normal atmospheric conditions. Examples include α-ET₂I₃ (ET = bis(ethylenedithio)tetrathiafulvalene, also abbreviated as BEDT-TTF (Scheme 1); “α-” denotes one of the polymorphs reported on ET₂I₃.) [6,7,8,9,10,11,12,13,14,15,16], which is an organic charge-transfer complex with two-dimensional conduction sheets consisting of a self-aggregated donor molecule (D) sublattice (Figure 1a). The α-ET₂I₃ is now known as a zero-gap semiconductor with Dirac cones under high pressure (>15 kbar) [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. However, many of the experiments are very challenging or impossible to perform under such high pressures. Therefore, the observation and characterization of Dirac electrons require a sample with Dirac Cones at ambient pressure. In this sense, α-STF₂I₃ (STF = bis(ethylenedithio)diselenadithiafulvelene, also abbreviated as BEDT-STF [33,34,35,36,37,38]) is the sample of choice, which is an isostructural complex with α-ET₂I₃ and has been recently suggested to be a zero-gap semiconductor at 1 bar by a band calculation (Figure 2) [37]. The STF, together with BETS (BETS = bis(ethylenedithio)tetraselenafulvelene) [38,39,40,41,42,43], is an electron donor, which is obtained by the partial replacement of sulfur atoms with selenium atoms in ET (Scheme 1). The substitution of chalcogen atoms generally results in two major effects in the physical properties of the STF complexes. First, in addition to the single crystals of STF itself, all of the STF complexes currently known contain disorder at all of the STF sites (Figure 1b). The unit cell parameters of α-STF₂I₃ are almost identical to those of α-ET₂I₃. Hence, the α-STF₂I₃ contains disorder at every STF site due to the asymmetric molecular structure about the central C=C bond, which could influence its electrical and magnetic properties. Second, the intermolecular interactions in the STF sublattice are enhanced based on the closer interatomic overlaps involving Se atoms, which are often termed chemical pressure effects. This effect makes the salt correspond to an approximately higher-pressure state of the isostructural ET salts. Accordingly, α-STF₂I₃ exhibits qualitatively different physical properties from those of α-ET₂I₃ under ambient pressure. For example, α-ET₂I₃ exhibits a metal-insulator (M-I) transition at T_M-I originating from charge ordering in the ET sublattice at a pressure (P) less than ~12–15 kbar (T_M-I = 135 K at 1 bar). However, α-STF₂I₃ exhibits a gradual increase in resistivity (ρ) at a temperature (T) lower than T_min(ρ) ≈ 130 K, without phase transitions [34,35,36,37]. Interestingly, in all the known solids containing STF molecules, the differences in electron densities between S and Se atoms are almost completely averaged out by disorder. As a result, the STF complexes typically behave as though they consist of ET-type symmetrical donor molecules containing imaginary atoms between S and Se. In fact, in α-STF₂I₃, even diffuse streaks or scatterings have never been observed in the X-ray oscillation photographs. In this way, α-STF₂I₃ may uniquely provide an overview of or insight into the Dirac electrons in disordered lattices by various physical property measurements under ambient pressure. However, some basic physical properties of α-STF₂I₃ remain unknown because of limited sample availability and limited research on the salt as a candidate of the zero-gap semiconductors. In order to provide a basis for discussion on this salt, it is important to present various aspects of basic physical properties. In this article, we report electrical, structural, magnetic and optical properties of α-STF₂I₃.

2. Materials and Methods

2.1. Sample Preparation

Neutral STF molecules were synthesized by following the reported procedure [34,36]. Single crystals of α-STF₂I₃ were prepared by a galvanostatic (1.4 μA) electrocrystallization of STF (6 mg, 1.25 × 10⁻² mmol) and (n-C₄H₉)₄N·I₃ (50.5 mg, 8.10 × 10⁻² mmol) in distilled C₆H₅Cl (10 mL) at 43 °C under an N₂ atmosphere for 2–3 weeks in the dark [34,36]. Prior to the physical measurements, all the single crystals were examined using X-ray oscillation photographs (R-AXIS RAPID-R, Rigaku, Tokyo, Japan) for identifying the lattice parameters and the conduction sheets (the crystallographic ab-planes) in addition to investigating the crystal quality.

2.2. Electrical Resistivity Measurements

Electrical resistivity was measured by a standard four-probe method using gold wires attached with carbon paste to the ab-planes of the single crystals. The morphology varied from crystal to crystal (from thick needles to irregular-shaped plates) and there was no particular relationship between the direction of the a- or b-axes and that of the most developed dimension in the ab-plane (L-axis). The L-axis made an angle of ~0 to ~90° with the a-axis, depending on the crystal. The four gold wires were always aligned along the L-axis. Owing to the highly isotropic nature in the conduction sheets and the error in the estimation of the crystal dimensions, the observed absolute values of resistivity and the temperature dependence agreed with each other semiquantitatively between different measurements. For the estimation of the anisotropy in the conduction plane (the ab-plane), the resistivity was measured along the a- and b-axes independently. The resistivity at 2–300 K was measured using a Physical Property Measurement System PPMS-9 with an EverCool II (Quantum Design Japan, Inc., Tokyo, Japan). Below 10 K, the resistance was too high to measure. Thus, only the data at 10–300 K were shown in the results below. The magnetic field (1 T) was applied below 18 K in zero-field-cooling (ZFC) processes, but no hysteresis or significant magnetoresistance was observed.

2.3. Magnetic Susceptibility Measurements

The magnetic susceptibility was measured using the polycrystalline samples and the DC mode of a SQUID susceptometer (MPMS-XL7minB×R3 with an EverCool, Quantum Design Japan, Inc., Tokyo, Japan). The sample was contained in a gelatin capsule with ventilation holes, which was set in the middle of a polystyrene straw (Quantum Design Japan, Inc.). The magnetic field of 1 T was applied, which was in the linear range between the applied magnetic field and the observed magnetization of the samples. Both FC and ZFC processes were measured at 2–300 K. The diamagnetic contributions from the STF and I₃⁻ species were estimated by the same measurements on the neutral STF and (n-C₄H₉)₄NI₃ at 300 K. The obtained diamagnetic susceptibilities, χ_dia, are −1.55 × 10⁻⁴ and −3.61 × 10⁻⁵ emu mol⁻¹G⁻¹ for STF and I₃⁻, respectively. The scatterings between observed magnetization for each measurement/compound were less than 10⁻⁶ emu (≤50 ppm of the observed signals).

2.4. X-ray Structural Studies

The X-ray single-crystal structural analyses and the measurements of temperature dependence of the lattice constants of α-STF₂I₃ were performed using a VariMax RAPID/α diffractometer (Cu-Kα radiation; λ = 1.54187 Å) (Rigaku, Tokyo, Japan) and a VariMax SaturnCCD724α diffractometer (Mo-Kα radiation; λ = 0.71075 Å) (Rigaku, Tokyo, Japan), each with a continuous N₂-gas-flow type temperature controller. X-ray structural analyses and magnetic susceptibility measurements were repeated several times using the samples prepared from independently synthesized neutral STF and (n-C₄H₉)₄NI₃. Sample-dependence was hardly observed in the measurements below.

2.5. Optical Studies

The reflectance spectra were measured on the ab-plane of the single crystal (450–25,000 cm⁻¹, 25–300 K). The optical conductivity spectra were obtained through the Kramers–Kronig transformation of the reflectance spectra. Details are described in our previous paper [35]. In this paper, only the reflectance and conductivity spectra of α-BETS₂I₃ were presented and those of α-STF₂I₃ were not, because they were nearly identical to those of α-BETS₂I₃ and were not required in the discussion then. However, as the actual spectra are indispensable in the discussion of this paper, they are present here with permission of the authors then.

2.6. Calculation of the Density of States

The density of states at the Fermi level of α-STF₂I₃ was estimated using the crystal structures of α-ET₂I₃ and α-BETS₂I₃ by assuming the identical atomic parameters with α-STF₂I₃, where extended Hückel tight-binding calculation was performed using Caesar vers.1 and 2 (PrimeColor Software, Inc., Raleigh, NC, USA) [44].

3. Results

3.1. Electrical Behavior

The observed electrical behavior (Figure 3a–c) quantitatively reproduced the previously reported temperature dependence of electrical resistivity [34,35,36,37]. The resistivity (25–40 mΩcm at 300 K) is higher than the bulk resistivity corresponding to the quantum sheet resistance (4.5 mΩcm in this case) by nearly an order of magnitude. This is consistent with the interpretation that the observed low temperature resistivity behavior is suggestive of a strong correlation between Dirac electrons (see below), as electron correlation deviates the resistivity from the quantum sheet resistance [22,23,24,28]. The resistivity slightly decreased but was almost temperature independent at 300–T_min(ρ) ~130 K and gradually increased at lower temperature (~1000 Ωcm at 10 K). At ~100–300 K the temperature dependence was continuous and monotonic (Figure A1a,b). In addition, since the temperature dependence of lattice constants at 90–300 K—a/Å, b/Å, c/Å, α/°, β/°, γ/° and V/ų (Figure 4a,b)—and that of magnetic susceptibility at 2–300 K (see below) exhibited no sudden change or hysteresis, there should be no first-order phase transitions in α-STF₂I₃. The anisotropy of the conductivity in the ab-plane (the ratio between the conductivities along the respective crystallographic axes) σ_b/σ_a is almost constant (3.3–3.9) at ~80–300 K. This is in agreement with our previous estimation by calculation (σ_b/σ_a ≈ 5) [37]. However, below ~80 K, the conductivities along both crystallographic axes decrease rapidly and cannot be reliably detected. As such, the σ_b/σ_a ratio is unreliable for use in quantitative assessment. The increase in resistivity at low temperature is neither of Arrhenius type (Figure 3c) nor logarithmic form (Figure 3a). For α-ET₂I₃, the temperature dependence of resistivity below T_min(ρ) (~10 K) at 18 kbar resembles the power-law or logarithmic form [22,24,28] with some sample-dependence [6,24,28,32]. Some of the samples exhibited closely related behavior to that of α-STF₂I₃. The insulating behavior toward the ground state originates from strong electron correlation between the Dirac fermions in α-ET₂I₃ [22,23,24,28]. Since such correlation originates from the band structure (Dirac cones), which is independent of disorder, the low-temperature resistivity upturn in α-STF₂I₃ can also originate from strong correlation.

3.2. Magnetic Behavior

The temperature dependence of molar magnetic susceptibility of α-STF₂I₃, χ, at 2–300 K is shown in Figure 5a,b. Both FC and ZFC processes produced identical results, which is also the case between independent samples/measurements (Figure A2). The good linearity between χT and T at 2–300 K indicates that χ is almost temperature independent, like the Pauli paramagnetism of metallic substances. At a closer examination, χ slightly decreased from 300 K to T₀(χ) ~120–130 K, and then more steeply decreased down to T_min(χ) ~40–50 K, where it gradually began to increase down to 2 K. The increase in χ at T ≤ 40 K is proportional to T^−0.2. If the increase originates from the lattice defects in the sample such as impurities and the disorder between Se and S at the STF sites, T⁻¹-dependence, i.e., the Curie law is expected. On the other hand, there are a number of papers on disordered systems such as Si:P [45] and FeSi_1−xAl_x [46] reporting their T^−α-dependences (0 < α < 1) of magnetic susceptibilities near the ground states. In order to clarify whether the increase in χ with the T^−α-dependence (0 < α < 1) at T ≤ 40 K originates from disorder, the same measurement on α-BETS₂I₃ is necessary, which we are currently investigating.

Now, we tentatively apply the relationship in the Fermi liquids between the Pauli paramagnetism, χ_Pauli and the density of states at the Fermi level N(E_F),

$${\chi }_{\mathrm{Pauli}}={\mu }_B^{2}N\left(E_F\right)$$

(1)

where, μ_B is the Bohr magneton. Our tight-binding calculation, where we assumed that α-ET₂I₃ and α-BETS₂I₃ should have the same atomic parameters with those of α-STF₂I₃, suggested that N(E_F) = 8.06–8.64 (eV⁻¹). Thus, χ_Pauli = (4.67–5.00) × 10⁻⁴ emu mol⁻¹G⁻¹, which is significantly smaller than the observed values at T ≥ T_min(χ) (~40 K) by a factor of ~3. By considering the low-temperature increase in the resistivity, the magnetic susceptibility can be consistently interpreted that the massless Dirac fermions (MDFs) in α-STF₂I₃ should have significant interaction with each other [22,23,24,28].

3.3. Optical Behavior

The reflectance and conductivity spectra of α-STF₂I₃ are shown in Figure 6a,b and Figure 7a,b, respectively [35]. General features of the reflectance spectra are closely related to those of usual two-dimensional molecular metals with Drude dispersions [47]. However, the spectral features can be explained by the coexistence of massive and massless fermions [19]. Such coexistence is theoretically predicted [7] and is actually observed in α-ET₂I₃ under high pressure [27]. The following analysis corroborates the coexistence of the two kinds of fermions in α-STF₂I₃. The plasma frequencies of the Drude dispersions along the a- and b-axes, ω_p,a and ω_p,b, were determined from the optical conductivity spectra at 300 K as 0.59 and 0.99 eV, respectively [35]. This gives anisotropy in conductivity as $\raisebox{1ex}{${\sigma }_b$}\!\left/ \!\raisebox{-1ex}{${\sigma }_a$}\right.={\left(\raisebox{1ex}{${\omega }_{p,b}$}\!\left/ \!\raisebox{-1ex}{${\omega }_{p,b}$}\right.\right)}^2\cong 2.8$, which agrees with the directly measured values of 3.3–3.9 shown in Figure 3d. Based on ω_p,a and ω_p,b, the effective mass along the a- and b-axes, (m*_opt)_a and (m*_opt)_b, are estimated as 4.60 m and 1.64 m, respectively, where m designates the electron rest mass. The larger effective mass generally means stronger correlation in the electronic system and leads to enhanced magnetic susceptibility. Since the Dirac electrons do not give a Drude dispersion [5], it originates from the semimetallic (massive) carriers. Accordingly, the effective masses are those of massive carriers and they are too small to account for the observed magnetic susceptibility. Therefore, the strong correlation originates from the interacting MDFs instead of the semimetallic carriers.

The overall features of the spectra and the temperature dependence clearly indicate that α-STF₂I₃ remains a metal-like highly conducting material without band gaps, exhibiting no phase transitions at 25–300 K. This apparently contradicts with the insulating resistivity behavior at T < ~130 K, which is, again, consistent with each other by assuming the coexistence of massless and massive carriers in α-STF₂I₃. The electrical properties at low temperatures are dominated by the massless carriers (interacting MDFs) [22,23,24,28], whereas the Drude dispersion originates from the (semimetallic) massive carriers as stated above. The most striking feature shared by reflectance and conductivity spectra is their negligible temperature dependence in a wide temperature range (25–300 K), which is shared by many Dirac electron systems [7,27,48] and is similar to those of α-BETS₂I₃ [35]. Although the reflectance spectra (Figure 6a,b) appear to be common for two-dimensional molecular metals, the optical conductivity spectra (Figure 7a–c) exhibit constant conductivity versus a wide range of wavenumbers (see below). Again, either of the constant values (~18 and ~30 Scm⁻¹ for E // a- and E // b-axes, respectively) is extremely different from the corresponding value of the quantum sheet conductance [(4.5 mΩcm)⁻¹ ≈ 220 Scm⁻¹ in this case]. The reflectance and conductivity spectra are almost quantitatively identical with each other between α-STF₂I₃ and α-BETS₂I₃ at respective temperatures. The latter (α-BETS₂I₃) is also considered to be a zero-gap semiconductor based on NMR measurements [38], theoretical studies [39], X-ray structural analyses with band calculation [40] and magnetotransport properties [41]. Similar temperature-independent features of the reflectivity and optical conductivity are observed in Cd₃As₂, which belongs to the three-dimensional Dirac electron systems [49].

Generally, the interband optical response of d-dimensional Dirac electrons is supposed to be universal [49,50,51]: the optical conductivity σ(ω) should follow a power-law frequency (ω) dependence,

$$\sigma \left(\omega \right)\propto {\omega }^{\raisebox{1ex}{$d-2$}\!\left/ \!\raisebox{-1ex}{$z$}\right.},$$

(2)

where, σ(ω) is the real part of the optical conductivity and z is the exponent in the band dispersion relation,

$$E\left(k\right)\propto {\left|k\right|}^z$$

(3)

In the case of α-STF₂I₃, d ≈ 2. Accordingly, Equation (2) indicates that σ(ω) should be independent of ω, irrespective of z. In fact, the obtained values of σ(ω) along both polarization directions are approximately ω-independent at ω ≈ 6000–15,000 cm⁻¹; σ(ω)/Scm⁻¹ ≈ 18 and 30 (independent of temperature) along the a- and b-axes, respectively (Figure 7b,c). The ω-independent range is possibly masked in part by the intense peak centered at 22,000–23,000 cm⁻¹ assigned to the local excitation of the I₃ anions, which hides the spectral features originating from the Dirac electrons.

4. Discussion

The optical results conclusively support that α-STF₂I₃ contains both massless and massive electron systems. All the rationalization on the resistivity and magnetic susceptibility data are established based on the conclusions of the optical results. Furthermore, considering the temperature dependencies of electrical, magnetic and structural properties, it is concluded that the band structure with Dirac cones remains in the ground state. In fact, including the T^−α-temperature dependence (0 < α < 1) of the resistivity below T_min, the observed electrical property is consistent with the typical behavior of interacting MDFs. The T^−α-temperature dependence (0 < α < 1) at low temperatures is also demonstrated by the magnetic behavior. These findings will aid in various successive experimental results at lower temperatures with smaller energy scales, as well as advancement in related theories. For the Dirac electron systems, the STF salt reveals similarities with the ET salt in the following points. (i) Anisotropic Dirac cone dispersion with accidental degeneracy at low symmetry, i.e., general k-points in the a*b*-plane. (ii) Almost identical lattice parameters with each other. (iii) The Fermi energy is located exactly on the Dirac point due to the 3/4-filled band. (iv) Electron correlation plays an important role in the physical properties.

In conclusion, all the experimental results consistently indicate the existence of the Dirac electrons in α-STF₂I₃, being consistent with the recent band calculation [37].

Finally, the Dirac electron systems of the α-STF₂I₃ are qualitatively different from those of α-ET₂I₃ in the following points. (i) The ET salt possesses substantially tilted the Dirac cones, whereas the STF salt possesses the almost vertical Dirac cones. The difference in the orientation of the Dirac cones will manifest influence on the magnetotransport properties and the orbital diamagnetism [31]. (ii) The ET salt is a clean system based on the well-defined crystal structure, whereas the STF salt is not based on such a system since it contains heavy disorder in the conduction sheets. The disorder of the STF salt could not be lowered to the ground state; thus, it would be interesting to investigate how the Dirac electron system in the salt would respond when cooled to 0 K. Since every perfect crystalline solid should have zero entropy—and since the STF salt could not be a perfect crystalline solid—it would be intriguing to see how the system reconciles itself with the third law of thermodynamics when T → 0. (iii) The phase transition is observed in the ET salt under low pressure. Accordingly, the band structure and the strength of electron correlation vary near the phase boundary [19,24,25]. In α-STF₂I₃, the aforementioned parameters are practically constant, without a phase transition. Therefore α-STF₂I₃ is anticipated to provide different aspects of the Dirac electron systems from those of α-ET₂I₃ or other related materials.