Thermoelectric Effect at Quantum Limit in Two-Dimensional Organic Dirac Fermion System with Zeeman Splitting

The thermoelectric effect in a two-dimensional (2D) massless Dirac fermion (DF) system at the quantum limit is discussed to verify the prediction of high-performance thermopower in an organic conductor \alpha-(BEDT-TTF)2I3. Because of relatively large Zeeman splitting in \alpha-(BEDT-TTF)2I3, the boundless increase of thermopower at high magnetic fields, predicted without the Zeeman effect, is hardly expected, whereas there appears to be a broad local maximum. This is characteristic of 2D DF systems with Zeeman splitting and is recognized in the previous experiment. In contrast to 3D Dirac/Weyl semimetals with robust gapless features, it might be difficult to realize high-performance thermopower in real 2D DF systems under high magnetic fields.


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The thermoelectric effects of topological semimetals have recently attracted a great deal of attention. For three-dimensional (3D) Dirac or Weyl semimetals at the highmagnetic-field quantum limit, it has been shown that the thermoelectric Hall conductivity (3D) xy  takes a constant value that is proportional to temperature but independent of the magnetic field and/or carrier density, which is owing to the constant density of states of the 0 N = chiral Landau subband [1,2]. Correspondingly, the Seebeck coefficient xx S increases linearly with no saturation as the magnetic field increases, resulting in highperformance thermopower. This feature has been observed in a 3D Dirac semimetal Pb1-xSnxSe with a small spin-orbit gap [3] and was recently confirmed in a 3D Dirac semimetal ZrTe5 [4].
In 2019, Liang Fu predicted a similar high-performance thermopower in twodimensional (2D) Dirac fermion (DF) systems at the quantum limit and cited a layered organic conductor -(BEDT-TTF)2I3 as a candidate material [5]. The thermoelectric characteristics of 2D DFs under magnetic fields have already been investigated theoretically [6] to explain graphene experiments [7]. Fu particularly focused on the DF system with a small carrier density near charge neutrality. At the quantum limit, the Fermi level exists at the 0 N = Landau level (LL) at zero energy. It has four-fold spin and valley degeneracy when the interaction and Zeeman effect can be neglected. In this situation, the 2D thermoelectric Hall conductivity xy  acquires a universal quantized value B 4(log 2) / k e h , which is independent of temperature as long as the LL width  is much smaller than B kT (the dissipationless limit). This feature leads to unprecedented growth in the thermopower xx S and thermoelectric figure of merit under high magnetic 3 fields even at low temperatures, where no realistic thermoelectric material is known, similar to the 3D Dirac/Weyl semimetals [5]. The thermoelectric features were also discussed for another 2D Dirac-like system, where a pair of Dirac cones merged [8].
In this study, we consider how the thermoelectric features predicted by Fu appear in real -(BEDT-TTF)2I3 at the dissipationless limit is usually regarded as a 2D system owing to its weak interlayer coupling. A 2D massless DF system with a pair of tilted Dirac cones (valleys) is realized in the high-pressure (>1 .3 GPa) metallic phase [9]. This fact was originally found theoretically [10] and later experimentally confirmed using interlayer magnetotransport [11][12][13], specific heat [14], and nuclear magnetic resonance (NMR) [15].
The thermoelectric effect of the DF state in -(BEDT-TTF)2I3 under magnetic fields was experimentally investigated by Konoike et al. [16]. They found that the The LLs of 2D massless DFs with Zeeman splitting are given by where N is the LL index ( 0, 1, 2, N =   ), and 1, 1

)
indicates electron spin. F v and B  are the constant velocity of DFs and Bohr magneton, respectively. We assumed that the g-factor was two. Each LL has two-fold valley degeneracy.
The magnetic field dependence of LLs is shown in Fig. 1 Following the G-J theory and Fu's argument, we consider a ribbon-shaped DF system with the left and right edges shown in Fig. 1 is the magnetic length. The current carried by the hole-like edge state, Here, factor 2 comes from valley degeneracy. The chemical potential  included in 0 , is determined by the following condition for the imbalance between the 2D electron density (n) and 2D hole density (p): The magnetic field dependence of the chemical potential,  is shown in Fig.   1(a) for several temperatures. Here, we assumed a small carrier imbalance    for a small carrier density in a field range wider than that in Fig. 1(a) This causes the periodic oscillation of .
The corresponding field dependence of the Seebeck coefficient xx S is illustrated for the case with (solid curves) and without (dashed curves) Zeeman splitting in Fig. 3(b). 8 The yellow shaded region corresponds to the quantum limit. As the magnetic field increases, xx S exhibits quantum oscillations at low temperatures before the quantum limit.
In the quantum limit region, Finally, it should be noted that the present results are obtained for the dissipationless limit B kT  , which was assumed in the 3D Dirac/Weyl semimetals 9 argument. Therefore, the present model cannot provide any results on the zerotemperature limit and dissipative scattering effects. At zero temperature, xy  and xx S must be zero. In fact, theory that considers scattering concluded that xy  decreases to zero with decreasing temperature [6]. For the dissipative effect, the Nernst coefficient xy S , which we ignored because of the assumption of 0 xx  = and 0 xx  = , becomes finite in a disordered system [6,17,19]. The observed peak in the temperature dependence of xx S and xy S , which originates from the thermal distribution to 1,   , is reproduced by considering the disorder [17].
In conclusion, we investigated the thermoelectric effect at the high-magneticfield quantum limit in a 2D massless DF system to verify the high-performance thermopower proposed by Fu in an organic DF system -(BEDT-TTF)2I3. We assumed the dissipationless limit as in the 3D topological semimetal arguments. The boundless increase in thermopower, predicted without the Zeeman effect, is hardly expected because of the relatively large Zeeman splitting in -(BEDT-TTF)2I3, whereas the local maximum structure characteristic of the DF system appears in the field dependence of the thermopower. This feature is recognized in the previous experimental data. In contrast to 3D Dirac/Weyl semimetals with robust gapless features, it might be difficult to realize high-performance thermopower in real 2D DF systems under high magnetic fields.