**2. Method**

## *2.1. Formulation*

In *α*-(BEDT-TTF)2I3, there are two band-crossing points called Dirac points being assigned as the right (R) and left (L) valleys. The effective Hamiltonian describing the Dirac electron system in *α*-(BEDT-TTF)2I3 is given by [10,31]

$$H = H\_0 + H',\tag{1}$$

where kinetic energy term *H*0 is

$$H\_0 = \sum\_{\mathbf{q}\gamma\gamma'\tau s} a\_{\mathbf{r}\gamma,\tau s}^{\dagger} [\hat{H}\_0^{\tau s}]\_{\gamma\gamma'} a\_{\mathbf{r}\gamma',\tau s'} \tag{2}$$

$$
\hat{H}\_0^{\pi s} = -\text{i}\tau \hbar \upsilon \left( \begin{array}{cc} \eta \partial\_{\overline{x}} & \partial\_{\overline{x}} - \text{i}\tau \partial\_{\overline{y}} \\ \partial\_{\overline{x}} + \text{i}\tau \partial\_{\overline{y}} & \eta \partial\_{\overline{x}} \end{array} \right), \tag{3}
$$

where *a*† **<sup>r</sup>***γ*,*τ<sup>s</sup>* and *<sup>a</sup>***r***γ*,*τ<sup>s</sup>* represent creation and annihilation operators, respectively, with two dimensional space **r** = (*<sup>x</sup>*, *y*), Luttinger–Kohn base *γ* [10,37], valley *τ* = ± (R, L), and spin *s* = ± (↑, ↓). The degree of tilt *η* is defined by *η* = *v*0/*v* with cone velocity *v* and tilt velocity *v*0, where the anisotropy of the cone velocity is ignored here for simplicity [31]. The energy eigenvalue of *H*0 is given by

$$E = \hbar v \left( \eta k\_{\text{x}} \pm \sqrt{k\_{\text{x}}^2 + k\_{\text{y}}^2} \right),\tag{4}$$

with two-dimensional momentum *kx* and *ky*. The left inset in Figure 1 shows the tilted Dirac cone, where *v* ± *v*0 are velocities in the ±*kx* directions, respectively. Interaction term *H*ˆ is given by

$$
\hat{H}' = \frac{1}{2} \int \mathbf{dr} \int \mathbf{dr'} V(\mathbf{r} - \mathbf{r'}) n(\mathbf{r}) n(\mathbf{r'}) \tag{5}
$$

with long-range Coulomb interaction *<sup>V</sup>*(**r**) = *e*2/*εr* and electron density operator *<sup>n</sup>*(**r**).

**Figure 1.** Theoretical results on |*B* cos *θ*| dependences of *V*HS for *η* = 0 (circle), 0.5 (triangle), 0.8 (inverted triangle), and 0.9 (diamond). The right inset shows the *η* dependence of *γ*. The left inset indicates the tilted Dirac cone, where *v* and *v*0 are cone velocity and tilt velocity, respectively.

The energy spectrum of the two-dimensional electronic system is quantized under a tilted magnetic field, **B** = ∇ × **A** = (*Bx*, *By*, *Bz*), given by the vector potential with the Landau gauge, **A** = (*Byz* − *Bzy*, <sup>−</sup>*Bxz*, <sup>0</sup>). The Hamiltonian under the magnetic field is obtained by Peierls substitution −*i*∇→−*i*∇ + (*e*/¯*h*)**<sup>A</sup>** and Zeeman energy *E*Z = *gμ*B*B*/2 with *B* = |**B**|. The Landau levels are given by *ENs* = sgn(*N*)(*hv*¯ /*l*) 2*λ*<sup>3</sup>|*N*| − *sE*Z with *λ* = 1 − *η*2 and magnetic length *l* = *<sup>h</sup>*¯ /*e*|*Bz*|. The wave function **Ψ***τ <sup>N</sup>*,*<sup>k</sup>*(**r**) for the *N* = 0 Landau level is given by References [11,27,38]

$$\Psi\_{N=0,k}^{\mathbf{r}=+}(\mathbf{r}) = \frac{1}{\sqrt{2(1+\lambda)}} \begin{pmatrix} -\eta \\ 1+\lambda \\ \vdots \end{pmatrix} \phi\_{N=0,k}(\mathbf{r}),\tag{6}$$

$$\Psi\_{N=0,k}^{\tau=-}(\mathbf{r}) = \frac{1}{\sqrt{2(1+\lambda)}} \begin{pmatrix} -1-\lambda\\ \eta \end{pmatrix} \phi\_{N=0,k}(\mathbf{r}),\tag{7}$$

where

$$\phi\_{N=0,k}(\mathbf{r}) = \frac{\lambda^{1/4}}{\sqrt{2\pi}} \frac{1}{\left(\sqrt{\pi}l\right)^{1/2}} e^{i\mathbf{k}\cdot\mathbf{r}} e^{-\hat{Y}^2/2} \tag{8}$$

with *Y*˜ = √*λ l* (*y* − *Y*) and center coordinate *Y* = *l* <sup>2</sup>*k*. When *H* = 5 T, *E*1 ∼= 3.5 meV using values of parameters for *α*-(BEDT-TTF)2I3, where velocity *v* = 1.0 × 10<sup>5</sup> m/s and *v*0 = 0.8 × 10<sup>5</sup> m/s are given by the band calculation [10,23,31], 2*E*Z ∼= 0.5 meV with *g* = 2, *l* ∼= 14 nm.

In the present study, we consider the *N* = 0 Landau levels in order to study interlayer magnetoresistivity in the quantum limit and, for the case of *E*1 *E*Z. The effective *g*-factor, *g*<sup>∗</sup>, is treated by the mean field theory. The Green function *Gs*(*k*, <sup>i</sup>*εn*) is defined by

$$\text{iG}\_{\text{s}}(k, \text{i}\varepsilon\_{\text{n}}) = \frac{1}{\text{i}\varepsilon\_{\text{n}} + \mu + s\text{E}\_{\text{Z}} - \Sigma\_{\text{s}}},\tag{9}$$

where  *n* = (<sup>2</sup>*n* + <sup>1</sup>)*π*/*β* is fermion Matsubara frequency with *β* = 1/*k*B*T* and integer *n*, and *μ* is the chemical potential, determined so that the Fermi energy coincides with the Dirac point. The self-energy Σ*s* is given by self-consistent equation

$$
\Sigma\_{\uparrow} - \Sigma\_{\downarrow} = -V\_{\text{HS}}m\_{\Xi \prime} \tag{10}
$$

with effective Coulomb interaction *V*HS = ∑*τ*[*V*(0)]*<sup>τ</sup>*,*τ*,*τ*,*τ* , magnetization *mz* = *<sup>n</sup>*↑ − *<sup>n</sup>*↓, and electron density for spin *s*, *ns*, where

$$\Sigma\_{s} = \frac{1}{D} \sum\_{k'} \frac{1}{\beta} \sum\_{\mathbf{r}'\_n} \sum\_{\mathbf{r}'} [V(0)]\_{\mathbf{r}, \mathbf{r}, \mathbf{r}', \mathbf{r}'} G\_{-s}(k', \mathbf{i} \varepsilon'\_n) \tag{11}$$

and the Coulomb interaction matrix [*V*(*q*)]*<sup>τ</sup>*1*τ*2,*τ*3,*τ*4 for the *N* = 0 Landau levels is given by

$$\begin{split} [V(q)]\_{\mathbf{r}\_{1}\mathbf{r}\_{2},\mathbf{r}\_{3},\mathbf{r}\_{4}} &= \frac{1}{2} \int \mathrm{d}\mathbf{r} \int \mathrm{d}\mathbf{r}' V(\mathbf{r} - \mathbf{r}') \times \\ [\![\![\mathbf{Y}]\!]\_{N=0,k+q}^{\mathbf{T}}(\mathbf{r})\!]^{\dagger} \cdot \mathbf{Y}\_{N=0,k}^{\mathbf{r}\_{1}}(\mathbf{r}) [\![\![\mathbf{Y}]\!]^{\dagger} \cdot \mathbf{Y}\_{N=0,k'}^{\mathbf{T}}(\mathbf{r}') \!] , \end{split} \tag{12}$$

where the degeneracy of center coordinate *D* = *LxLy*/2*πl*<sup>2</sup> with length *Lx* and *Ly* in the *x* and *y* directions, respectively. The effective Coulomb interaction depends on magnetic length *l*, so effective Coulomb interaction depends on angle *θ*. Effective spin splitting *E*∗Z is given by

$$E\_{\mathbb{Z}}^{\*} = \mathbb{g}^{\*} \mu\_{\mathbb{B}} B/2 = E\_{\mathbb{Z}} + \Delta \tag{13}$$

with Δ = <sup>Σ</sup>↓ = −Σ<sup>↑</sup> ≥ 0 for the charge neutral system *μ* = 0. Thus, effective g-factor *g*∗ is given by

$$
\mathbb{g}^\* = \mathbb{g} + 2\Delta / \mu\_\mathbb{B} \mathbb{B}.\tag{14}
$$

Thus, the energy eigenvalues of *N* = 0 Landau levels are modified as follows:

$$E\_{N=0,\sharp} = -sE\_{\mathbb{Z}}^{\ast} = -s(E\_{\mathbb{Z}} + \Delta). \tag{15}$$

Interlayer conductivity is given by interlayer coupling as a perturbation [25]. The perturbation Hamiltonian *H* is given by

$$
\hat{H}' = -2t\_{\hat{c}} \cos \left( -ic \frac{\partial}{\partial z} \right) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \tag{16}
$$

where *tc* and *c* represent interlayer transfer energy and interlayer spacing, respectively. In the quantum limit, *N* = 0 Landau levels are dominant in magnetotransport. The effective transfer energy between *N* = 0 Landau levels in neighboring layers is given by [25]:

$$\begin{split} \left| t\_{\varepsilon}(\mathbf{Y}', z\_i'; \mathbf{Y}, z\_i) \right| &= \left| t\_{\varepsilon} \times \exp \left[ -\frac{1}{4} \frac{\varepsilon^2 \varepsilon \left( B\_x^2 + B\_y^2 \right)}{\hbar |B\_z|} \right] \right| \\ &\times \exp \left[ i \frac{\varepsilon B\_x}{\hbar} (z\_i' - z\_i) \frac{\mathbf{Y} + \mathbf{Y}'}{2} \right], \end{split} \tag{17}$$

where *zi* is the layer position, and the neighboring layers are *zi* = *zi* ± *c*. The center coordinate of initial state *Y* on one layer *z* = *zi*, and that of final state *Y* on neighboring layer *z* = *zi* ± *c* satisfy the condition

$$Y' = Y \pm \frac{B\_y}{B\_z}c.\tag{18}$$

The complex interlayer conductivity *<sup>σ</sup>*˜*zz*(*ω*) is given by [25]:

$$\begin{split} \left| \mathcal{T}\_{zz}(\omega) \right| &= -\frac{i\hbar}{L^2} \sum\_{\substack{\mathbf{Y}, z\_i, \mathbf{r}, \mathbf{s}^\prime, z\_i^\prime, z\_i^\prime, \mathbf{r}^\prime, \mathbf{s}^\prime}} \left< N = 0, Y, z\_i, \mathbf{r}, \mathbf{s} \mid \bigwedge\_{z} N = 0, \mathbf{Y}^\prime, z\_i^\prime, \mathbf{r}^\prime, \mathbf{s}^\prime \right> \\ & \quad \times \left< N = 0, Y^\prime, z\_i^\prime, \mathbf{r}^\prime, \mathbf{s}^\prime \mid \bigwedge\_{z} N = 0, Y, z\_i, \mathbf{r}, \mathbf{s} \right> \\ & \quad \times \frac{f(E\_{N=0,s}) - f(E\_{N=0,s^\prime})}{E\_{N=0,s^\prime} - E\_{N=0,s}} \frac{1}{E\_{N=0,s^\prime} - E\_{N=0,s} - \hbar \omega - i\hbar/\pi}, \end{split} \tag{19}$$

where *τ*˜ and ˆ*jz* represent relaxation time and interlayer current density, respectively. The center coordinate *Y* is associated with wavenumber *k* by relation expression *Y* = *l* <sup>2</sup>*k*. Interlayer current density is given by ˆ*jz* = ( −*<sup>e</sup>*)(1/*ih*¯)[*z*ˆ, *H*ˆ ].

The leading term of real conductivity is given by [25]:

$$\sigma\_{zz} = \frac{2Ct\_c^2 c e^3 \tilde{\tau} |B\_{\tilde{z}}|}{\pi \hbar^3} \exp\left[-\frac{c c^2 (B\_x^2 + B\_y^2)}{2\hbar |B\_z|}\right] \tag{20}$$

with

$$\mathcal{C} = \sum\_{s\tau} \int D\_{s\tau}(E)^2 \left( -\frac{\mathbf{d}f}{\mathbf{d}E} \right) \mathbf{d}E \tag{21}$$

where *Dsτ*(*E*), *tc* and *c* denote density of states for spin *s* and valley *τ*, interlayer transfer energy, and interlayer spacing, respectively. Interlayer resistivity *ρzz* is given by *ρzz* 1/*<sup>σ</sup>zz*, since interlayer Hall conductivities *σxz* and *<sup>σ</sup>yz* are negligibly smaller than other components [39,40].

#### *2.2. Experimental Method*

#### 2.2.1. Crystal Growth

Either the electrolysis method or the diffusion method are generally used in the crystal growth of organic conductors. We synthesized organic conductor *α*-(BEDT-TTF)2I3 by the electrolysis method using an H-type cell. BEDT-TTF molecules and I− 3 anions are desolved in a supporting electrolyte (THF, benzonitrile, chlorobenzene, etc.). Then, we supply electrical current (1 ∼5 μA) between platinum electrodes. After about 1 ∼2 weeks, small single crystals appear on the positively based platinum electrode. The typical size of a crystal is 1 × 0.5 × 0.05 mm3.

#### 2.2.2. Experiments of Interlayer Magnetoresistance Under Pressure

A sample with a dimension of approximately 0.7 × 0.5 × 0.05 mm3, on which four electrical leads (gold wire with a diameter of 15 μm) are attached by carbon paste, is put in a Teflon capsule filled with the pressure medium (Idemitsu DN-oil 7373); then, the capsule is set in a NiCrAl clamp cell, and hydrostatic pressure of up to 1.7 GPa is applied. The hydrostatic pressure is determined at room temperature by a Manganine resistance gauge in the pressure cell. The interlayer resistance of a crystal is measured by a conventional DC method with an electrical current of 1 μA along the *c*-crystal axis, which is normal to the two-dimensional plane. In the investigation, interlayer magnetoresistance is measured as functions of *B* and *θ* which is the angle between the magnetic-field direction and *c*-crystal axis at *T* = 0.5, 1.7, 2.5, 3.0, 3.5, and 4.2 K. As mentioned in the introduction, interlayer magnetoresistance is a useful tool to detect the effects of *N* = 0 Landau levels, including its Zeeman splittings.

## **3. Results**

Figure 1 shows the theoretical results on |*B* cos *θ*| (= |*Bz*|) dependences of effective Coulomb interaction *V*HS for tilt parameter *η* = 0, 0.5, 0.8, and 0.9. It is numerically shown that *V*HS ∝ |*B* cos *θ*| *γ* approximately for |*B* cos *θ*| > 3. Effective Coulomb interaction depends on magnetic length *l*, which is a function of |*Bz*|, as shown in Equation (12). Thus, effective Coulomb interaction depends on angle *θ*. The left inset shows the tilting Dirac cone, where *v* and *v*0 represent cone velocity and tilt velocity, respectively. The right inset shows the *η* dependence of *γ*, where *γ* increases as *η* increases. Thus, we use a relation, *V*HS = *u*|*<sup>B</sup>* cos *θ*| *γ*, with *γ* = 0.87 for the tilted Dirac cones in *α*-(BEDT-TTF)2I3 with *η* = 0.8 [23] hereafter. Parameter *u* = 0.3 is chosen to fit with the experimental results.

Figure 2a,b shows the theoretical results on the *B*-dependences of *g*∗ and *E*∗ Z, respectively, for *θ* = 0, 20, 40, and 60◦ at *T* = 1.7 K. It is found that both *g*∗ and *E*∗ Z enhance by *V*HS depend on *θ*. When *θ* = 90◦, *g*∗ = *g* and *E*∗Z = *E*Z, since *V*HS = 0. Although *E*∗Z increases monotonically as *B* increases, *g*∗ has a maximum since Δ divided by *B* contributes to *g*<sup>∗</sup>.

Figure 3a shows the theoretical results on *B*-dependences of interlayer resistivity *ρzz* for *θ* = 0, 20, 40, and 60◦ at *T* = 1.7 K. *ρzz* has a minimum at *B*0. It is found that *B*0 increases as *θ* increases due to *θ*-dependence of *<sup>E</sup>*<sup>∗</sup>Z. Figure 3b shows the experimental results on *B*-dependences of interlayer resistance *Rzz* for *θ* = 0, 20, 40, and 60◦ at *T* = 1.7 K. *B*0 obtained in the experimental results also increases as *θ* increases.

In a general two-dimensional system, *E*∗Z dose not depend on *θ*. The agreemen<sup>t</sup> between the theory and experiment of interlayer magnetoresistance shown in Figure 3 indicates that effective Coulomb interaction plays an important role to the Zeeman effects. In the following, peculiar Zeeman effects in this system are examined.

The first step is to investigate *g*∗ for *θ* = 0 from the interlayer-magnetoresistance minimum, where *g*<sup>∗</sup>*μ*B*B*0 ∼ 2*h*¯ /*τ*˜. Here, the rough of broadening energy *h*¯ /*τ*˜ of the Landau levels in this system is 3 K at low temperatures [32]. *g*∗ for *θ* = 0 is roughly estimated to be 5 experimentally at 1.7 K; this is close to the theoretical value, which is approximately 5.3 at *B*0 ∼ 1.8 T, as shown in Figure 2a.

**Figure 2.** Numerical results on *B*-dependences of *g*∗ (**a**) and *E*∗Z (**b**), respectively, at *T* = 1.7 K for *θ* = 0 (solid line), 20 (dotted line), 40 (dashed line), and 60◦ (dot-dashed line). *g* and *E*Z in the absence of interaction are drawn by the thin solid line.

**Figure 3.** *Cont*.

**Figure 3.** (**a**) theoretical results on *B*-dependences of *ρzz* for *θ* = 0 (solid line), 20 (dotted line), 40 (dashed line), and 60◦ (dot-dashed line) at *T* = 1.7 K. (**b**) experimental results on *B*-dependences of *Rzz* for *θ* = 0, 20, 40, and 60◦ at *T* = 1.7 K.

Lastly, the peculiar Zeeman effects on the *θ*-dependence of the interlayer magnetoresistance minimum are detected. In Figure 4a, the theoretical results on *θ* dependences of *B*−<sup>1</sup> 0 are drawn as functions of cos *θ* for *T* = 0.5, 1.7, 2.5, 3.0, 3.5, and 4.2 K. It is numerically shown that 1/*B*0 ∼= *a* cos *θ* + *b* when *θ* is close to 0◦. Note that this numerical calculation is not applicable when *θ* is close to 90◦, where contributions of higher Landau levels in interlayer resistivity are not negligible. Figure 4b shows the experimental results on cos *θ* dependences of *B*−<sup>1</sup> 0 for *T* = 0.5, 1.7, 2.5, 3.0, 3.5, and 4.2 K. The experimental results also show the same approximate relation on *B*−<sup>1</sup> 0 as a linear function of cos *θ*.

**Figure 4.** *Cont*.

**Figure 4.** (**a**) theoretical results on cos *θ* dependences of *B*−<sup>1</sup> 0 for *T* = 0.5 (open square), 1.7 (open circle), 2.5 (open triangle), 3.0 (inverted open triangle), 3.5 (open diamond), and 4.2 K (open pentagon). When cos *θ* is close to 1, numerical results show a relation 1/*B*0 ∼= *a* cos *θ* + *b* approximately, where *a* and *b* are independent of *θ*. (**b**) experimental results on cos *θ* dependences of *B*−<sup>1</sup> 0 for *T* = 0.5 (filled square), 1.7 (filled circle), 2.5 (filled triangle), 3.0 (inverted filled triangle), 3.5 (filled diamond), and 4.2 K (filled pentagon). As with the theoretical results, the experimental results show relation 1/*B*0 ∼= *a* cos *θ* + *b* approximately when cos *θ* is close to 1.

Figure 5 shows the theoretical and experimental results on *T* dependencies of coefficient *a*. It is found that coefficient *a* increases as *T* decreases, indicating remarkable *θ* dependence of *B*−<sup>1</sup> 0 at very low temperatures *T* ≤ 1.7 K. Those results show qualitative agreemen<sup>t</sup> between theory and experimental results for the field-angle dependence of interlayer magnetoresistance.

**Figure 5.** Theoretical results (filled square) and experimental results (filled circle) of coefficient *a* as a function of *T* in relation 1/*B*0 ∼= *a* cos *θ* + *b*. Coefficient *a* increases as *T* decreases. Coefficient *a* especially rapidly increases at low temperatures. Theoretical and experimental results coincide.

## **4. Discussion**

Those theoretical curves in Figures 3a and 4a qualitatively reproduce the experimental results including the characteristic features of the resistance minimum as shown in Figures 3b and 4b. Here, the theoretical results do not coincide with the experimental results near *B* = 0 T, since only *N* = 0 Landau levels are examined in the present theoretical calculation. This theoretical approach, however, is valid because the special Landau-level structure in this system realizes the quantum limit in the magnetic field above 0.07 T for a perpendicular component to the two-dimensional plane at 1.7 K [32,41].

In the present study, the value of *u* for *V*HS is chosen to fit with the experimental results. The value of *u* will be evaluated by taking the Thomas–Fermi screening effect into account. The effects of the Coulomb interaction in *N* = 0 Landau levels is also going to be studied in the mean field calculation. In addition, interlayer magnetoresistance is also going to be studied, taking *N* = 0 Landau levels into account, which leads to the maximum of the interlayer resistance under weak magnetic fields [26,28].
