Pressure-induced phase switching of Shubnikov de Haas oscillations in the molecular Dirac fermion system $\alpha-$(BETS)₂I₃

Figure 1(b) shows the temperature dependence of the resistivity at ambient pressure in sample #1 ($\alpha-$(BETS)₂I₃/polyimide/PET). Compared with a bulk crystal Inokuchi1995 , the sample exhibits slightly lower metal-insulator crossover temperatures and more moderate resistivity increases at lower temperatures. The former is ascribable to the fact that the thermal contraction of the PET substrate applies compressive strain to $\alpha-$(BETS)₂I₃ Kawasugi2008 (therefore, this sample at ambient pressure corresponds to a bulk crystal under weak pressure), and the latter is attributable to the doping effect of the polyimide layer. As the single crystal consists of several tens of conducting (BETS) layers and the doped carriers are confined at the interface, the doping effect appears only at low temperatures where the bulk is insulating. At 1.7 K, the sheet resistivity (resistance $\times$ width $/$ length) is 8.4 $\times$ 10³ $\Omega$.

Figure 1(c) shows the magnetoresistance (upper) and the Hall resistance (lower) at 1.7 K. The SdH oscillations along with negative magnetoresistance are visible. The magnetoresistance is complicated in detail. It is slightly negative up to 0.4 T, turns positive up to 1.5 T, and then becomes negative again by a further magnetic field. Such a negative magnetoresistance has not been observed in $\alpha-$(BEDT-TTF)₂I₃ under low pressures Unozawa2020 . The magnetoresistance can be simply explained by neither the weak localization nor weak-antilocalization. It is reminiscent of the negative longitudinal magnetoresistance in topological semimetals Schumann2017 due to charge carrier density or mobility fluctuations. However, the magnetic field direction is different in this study (current is perpendicular to the magnetic field). Its origin cannot be clarified at this moment. We observe the negative magnetoresistance (without oscillations) in a bulk crystal Supplemental . Although we cannot see whether the doped interface also shows the negative magnetoresistance or not, the oscillations originate from the interface. We focus on the oscillation signals in this study.

The low-field Hall resistances are positive and proportional to the magnetic field. The sign becomes negative at around 23 K with increasing temperature Supplemental . By contrast, a bulk crystal shows negative Hall resistance at low temperatures Supplemental . Therefore, the doped carriers are holes and the concentration is $\sim$10¹² cm^-2 by ignoring electrons in bulk (Hall mobility $\sim 1080$ cm²/Vs).

The quantum oscillations originate from the quantization condition for the energy levels of the electron Onsager1952 ; Lifshitz1956 :

$$S_{n}=\frac{2\pi e}{\hbar}B(n+\gamma),$$

(1)

where $S_{n}$ is the area of the cyclotron orbit in k-space, $n$ is an integer, $B$ is the magnetic field, and $\gamma$ is the phase factor. The oscillation signal is periodic against 1/$B$, and the area can be estimated using the measurement $B_{F}\equiv(\frac{1}{B_{n+1}}-\frac{1}{B_{n}})^{-1}$. Assuming that the spin and valley degeneracies are both 2, the carrier density $N$ is

$$N=\frac{4e}{h}B_{\rm F}.$$

(2)

The SdH oscillation is given by

$$\Delta R_{xx}=R(B,T)\cos 2\pi(\frac{B_{\rm F}}{B}-\gamma),$$

(3)

where $R_{xx}$ and $R(B,T)$ are the longitudinal resistance and the oscillation amplitude, respectively Sharapov2004 ; Lukyanchuk2004 ; Zhang2005 . The phase factor $\gamma$ is associated with the Berry phase $\phi_{\rm B}$ as

$$\gamma-\frac{1}{2}=-\frac{\phi_{\rm B}}{2\pi}.$$

(4)

In a conventional electron system with isolated bands, $\phi_{\rm B}=0$ and $\gamma=1/2$. However, if the cyclotron orbit surrounds the contact point of the bands and the energy dispersions are linear in ${\bf k}$ in the vicinity of the contact point, the $\pi$ Berry phase emerges and $\gamma$ becomes zero Mikitik1999 . Accordingly, when we plot $1/B$ corresponding to the peaks against Landau level index (Landau fan diagram), the intercept $-\gamma=$ 0 or $-$1/2 for 2D Dirac or normal electrons.

Nevertheless, we have to be careful about the phase analysis of the SdH oscillations. Eq. (3) assumes the condition $R_{xx}<<|R_{xy}|$ (graphene, $\alpha-$(BEDT-TTF)₂I₃, and many low-carrier-density semiconductors meet this condition), and the minima in $R_{xx}$ coincide with those in the conductance $G_{xx}=R_{xx}/(R_{xx}^{2}+R_{xy}^{2})$. The condition may be violated due to low mobility or the presence of a highly conducting bulk transport channel. In the case that $R_{xx}>>|R_{xy}|$, the minima in $R_{xx}$ correspond to the maxima in $G_{xx}$. The difficulty of the phase analysis using resistance data has been pointed out for topological insulators Xiong2012 ; Ando2013 . However, the Hall response is usually weak and the resistance oscillation is more apparent in many cases. One may still use the reversed resistance data when $R_{xx}>>|R_{xy}|$, as in the case of graphite Lukyanchuk2004 . Here, we analyze the oscillations of both $R_{xx}$ and $G_{xx}$ in $\alpha-$(BETS)₂I₃/polyimide/PET at ambient pressure. To eliminate the background, we show the second derivative of the data with respect to the magnetic field.

Figure 1(d) shows the $1/B$ dependences of $d^{2}R_{xx}/dB^{2}$ and $-d^{2}G_{xx}/dB^{2}$ derived from Fig. 1(c). They correspond to $-\Delta R_{xx}$ and $\Delta G_{xx}$, respectively. The oscillation signal is more apparent in the upper curve because $\Delta R_{xx}/R>\Delta G_{xx}/\sigma$. However, both curves show almost the same periods and phases, indicating $-\Delta R_{xx}\propto\Delta G_{xx}$. $B_{\rm F}$ and $N$ estimated from the upper curve are 12.1 T and $1.17\times 10^{12}$ cm^-2, respectively, and correspond to 0.6% hole doping provided that the doped carriers are confined within one conducting layer. The $N$ value provides a realistic Hall scattering factor $\gamma_{\rm H}$ of 1.70 ($N=\gamma_{\rm H}/eR_{\rm H}$). $\gamma$ is almost $1/2$, indicating that the carriers are not Dirac fermions at this doping level. The same analysis of sample #2 is described in Fig. S4 Supplemental . The transport properties of sample #1, such as the temperature dependence of the resistance, the magnetoresistance, the sign and magnitude of the Hall effect, the relationship between resistance and conductance oscillations, and the phase factor reproduced in sample #2.

The leftmost peak in Fig. 1(d) (denoted by yellow diamonds) deviates from the position predicted from the fitting line in Fig. 1(e). Provided that this is a split peak as a result of the Zeeman effect, we estimate the effective mass $m^{*}\sim 0.43m_{e}$ from the relation $(n_{\rm LL}+\frac{1}{2})\frac{\hbar e}{m^{*}}B_{\rm LL}=(n_{\rm LL}+\frac{1}{2})\frac{\hbar e}{m^{*}}B_{\rm Z}+\mu_{\rm B}B_{\rm Z}$, where $B_{\rm LL}$ and $B_{\rm Z}$ are the predicted and observed peaks, respectively.

With increasing pressure, the entire sample is compressed. The bandwidths of the bulk and surface are enhanced, and their resistances decrease. The metal-insulator crossover gradually diminishes and disappears at around 0.6 GPa, as shown in Fig. 2(a). The dip at around 35 K and the upturn below 5 K of the resistance have also been observed in bulk crystals Inokuchi1993 , but the detailed mechanisms are still unclear. Above 0.6 GPa, the resistivity is almost constant down to approximately 15 K, below which the metallic behavior of the doped holes appears. The sheet resistivity per conducting layer is close to the quantum resistance $h/e^{2}$, as in the case of $\alpha-$(BEDT-TTF)₂I₃. The negative magnetoresistance observed at ambient pressure diminishes and becomes positive (Fig. 2(b)). The Hall resistance also decreases and becomes nonlinear, probably due to the emergence of a conducting bulk transport channel (Fig. 2(c)). As stated above, the bulk crystal of $\alpha-$(BETS)₂I₃ does not show the SdH oscillations even under pressure. We investigate the doped surface by the analysis of the oscillations.

Figure 3(a) shows the pressure dependence of $-d^{2}G_{xx}/dB^{2}$. At 0.35 and 0.4 GPa, the minima give $\gamma\sim 1/2$; this tendency is also observed at ambient pressure. $m^{*}$ values are estimated to be $\sim 0.37m_{e}$ and $0.35m_{e}$, respectively, showing a decreasing trend with pressure. At 0.45 GPa, we cannot construct a convincing fan diagram because of ambiguous oscillation signals and large background signals. However, we can see a half-period oscillation (up to $1/B\sim 0.4$ T^-1), probably indicating the coexistence of anti-phase oscillations. One possible scenario is the phase separation between the regions with $\gamma$=1/2 and 0. Above 0.5 GPa, $\gamma$ becomes almost zero, implying the emergence of Dirac fermions, although the oscillations at low $1/B$ are not clear. The Landau fan diagrams and the pressure dependence of $B_{\rm F}$ and $\gamma$ are summarized in Fig. 3(b) and (c). The conductance oscillations and Landau fan diagrams of sample #3 at ambient pressure and 1.2 GPa are also shown in Fig. S5 Supplemental .

We cannot observe Dirac fermions unless the metal-insulator crossover is sufficiently suppressed by pressure. These results are in contrast to those for $\alpha-$(BEDT-TTF)₂I₃, in which the Dirac fermions and the semiconducting behavior are simultaneously observed. Besides, we cannot confirm the coexistence of Dirac and normal electrons (which has been reported for $\alpha-$(BEDT-TTF)₂I₃ Monteverde2013 ) in this study. The SdH oscillations survive beyond the pressure-induced transition and $B_{\rm F}$ does not significantly vary with pressure. If large Fermi surfaces emerge by applying pressure, as predicted by band calculations in early reports Kondo2009 ; Alemany2012 , the doping effect (and accordingly the quantum oscillations) should be obscured by the dense carriers.

The most straightforward interpretation of the phase switching is that the pressure-induced resistive transition is a semiconductor-Dirac fermion system transition. Another possible scenario is the pressure-induced merging of the Dirac cones. In that case, the number and area of the Fermi surface generally change at the transition. However, Fig. 3 shows that the $B_{\rm F}$ does not significantly change during the transition. To consider the merging of the Dirac cone as the origin of the transition, we need a model and conditions consistent with these measurements.

The SdH oscillations become more evident as pressure increases. Figure 4 shows the magnetotransport properties of sample #4 at 1.8 GPa and 0.5 K. Here, the minima of $R_{xx}$ coincide with those of $G_{xx}$ Supplemental . The minima indicate $\gamma=0$ and the interval of the Zeeman splitting peaks (yellow diamonds in Fig. 4(b)) gives effective Fermi velocity $v_{\rm F}\sim$3.6 $\times$ 10⁴ m/s, which is approximately 20% lower than that from the same analysis for $\alpha-$(BEDT-TTF)₂I₃ Unozawa2020 ($v_{\rm F}$ is estimated from the relation $\sqrt{2e\hbar v_{\rm F}^{2}|n|B_{\rm Zh}}-\mu_{\rm B}B_{\rm Zh}=\sqrt{2e\hbar v_{\rm F}^{2}|n|B_{\rm Zl}}+\mu_{\rm B}B_{\rm Zl}$, where $B_{\rm Zh}$ and $B_{\rm Zl}$ are the peak fields). The low $v_{\rm F}$ indicates that the Dirac cone is more tilted or more blunted than $\alpha-$(BEDT-TTF)₂I₃. However, we cannot determine the central origin because the estimated $v_{\rm F}$ is average over the orbit in the reciprocal space Tajima2007 . The peak around $1/B\sim 0.15$ further separates into two peaks (green squares in Fig. 4(b)). We assign the bottom between these peaks to the Zeeman splitting peak because a similar $v_{\rm F}$ of 3.6$\times$ 10⁴ m/s is estimated from the bottoms denoted by red circles. Therefore, the small separation is considered a valley splitting. We roughly estimate the valley-splitting energy $\Delta_{\rm v}/k_{\rm B}\sim 0.92B$ K using a similar analysis to $v_{\rm F}$, assuming that $\Delta_{\rm v}$ is proportional to the magnetic field. A relative permittivity $\epsilon$ of $\sim 350$ is derived from the relation $\Delta_{\rm v}=\frac{e^{3}}{\epsilon\epsilon_{0}K\hbar}B$, where $K$ is the distance between the Dirac cones in k-space (approximated by the inverse lattice constant). In bulk $\alpha-$(BEDT-TTF)₂I₃, $\epsilon$ near the Dirac point is estimated to be $\sim 190$ from the interlayer magnetoresistance Tajima2010 . As the permittivity decreases near the Dirac point, a comparable value is expected at the Dirac point in $\alpha-$(BETS)₂I₃.

We have investigated the pressure dependence of the magnetoresistance and the Hall effect in slightly hole-doped thin single crystals of $\alpha-$(BETS)₂I₃ laminated on polyimide films, and verified that the material is in the Dirac fermion phase under pressure. We found a phase switching of the SdH oscillation near the pressure-induced metal-insulator crossover, unlike in $\alpha-$(BEDT-TTF)₂I₃ in the vicinity of the phase transition. At ambient pressure, the system exhibits a metal-insulator crossover below 50 K, and the phase of the SdH oscillation at 1.7 K indicates $\gamma=1/2$. Under pressure, $\gamma$ becomes zero above 0.5 GPa, whereas the metal-insulator crossover disappears at approximately 0.6 GPa. A half-period oscillation appears at the boundary ($\sim 0.45$ GPa), although the oscillation signal is ambiguous. It may originate from the coexistence of the regions with normal and Dirac fermions. The pressure-induced phase switching of the SdH oscillation indicates the presence of a semiconducting phase with normal electrons next to a Dirac fermion phase in $\alpha-$(BETS)₂I₃. In $\alpha-$(BEDT-TTF)₂I₃, the $\pi$ Berry phase appears even in the highly resistive states, and such a trivial insulating phase has not been observed Unozawa2020 . Recently, Kitou et al. reported that $\alpha-$(BETS)₂I₃ maintains the inversion symmetry below the metal-insulator crossover Kitou2020 , implying a different insulating mechanism from $\alpha-$(BEDT-TTF)₂I₃. Ohki et al. suggested that the insulating phase is a spin-ordered massive Dirac electron phase where time-reversal symmetry is broken but spatial inversion and translational symmetries are conserved Ohki2020 . Tsumuraya et al. explained that the system is in the massless Dirac state but a gap opens as a result of the spin-orbit interaction Kitou2020 ; Tsumuraya2020 . However, we cannot confirm the presence of Dirac fermions at ambient pressure in this study. Under high pressure (1.8 GPa), $\alpha-$(BETS)₂I₃ is a Dirac fermion system with $v_{\rm F}$ of $\sim$3.6 $\times$ 10⁴ m/s. At high magnetic fields, the valley splitting is observed in the SdH oscillation. The valley splitting energy $\Delta_{\rm v}/k_{\rm B}$ is estimated to be $\sim 0.92B$ K. Further study is required to clarify the electronic states of $\alpha-$(BETS)₂I₃ which may provide a unique Dirac fermion system different from $\alpha-$(BEDT-TTF)₂I₃.