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A zero-gap state with a Dirac cone type energy dispersion was discovered in an organic conductor α-(BEDT-TTF)_{2}I_{3} under high hydrostatic pressures. This is the first two-dimensional (2D) zero-gap state discovered in bulk crystals with a layered structure. In contrast to the case of graphene, the Dirac cone in this system is highly anisotropic. The present system, therefore, provides a new type of massless Dirac fermion system with anisotropic Fermi velocity. This system exhibits remarkable transport phenomena characteristic to electrons on the Dirac cone type energy structure.

_{2}I

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Since Novoselov _{2}I_{3} (BEDT-TTF = bis(ethylenedithio) tetrathiafulvalene) was found to be a new type of massless Dirac fermion state under high pressures [_{2}I_{3} under high pressure.

The organic conductor α-(BEDT-TTF)_{2}I_{3} is a member of the (BEDT-TTF)_{2}I_{3} family [_{3}^{−} anions as shown in _{C} values of several Kelvin [_{2}I_{3} is different from the other crystals. According to the band calculation, this system is a semimetal with two small Fermi surfaces; one with electron character and the other with hole character [

(_{3}^{−} anion; (_{2}I_{3} viewed from

When cooled, it behaves as a metal above 135 K, where it undergoes a phase transition to an insulator as shown in

Temperature dependence of the resistance under several hydrostatic pressures [

When the crystal is placed under a high hydrostatic pressure of above 1.5 GPa, the metal-insulator transition is suppressed and the metallic region expands to low temperatures as shown in ^{5} cm/V·s exhibit extremely large magnetoresistance at low temperatures [

To clarify the mechanism of these apparently contradictory phenomena, the Hall effect was first examined by Mishima ^{14} cm^{−3} and a high mobility of approximately 3 × 10^{5} cm^{2}/V·s [

Temperature dependence of the carrier density and the mobility for P= 1.8 GPa. The data plotted by close circle is the effective carrier density _{eff} and the mobility _{eff}. The magnetoresistance mobility _{M} and the density _{M}, on the other hand, is shown by open square from 77 K to 2 K. The results of two experiments agree well and the density obeys

The mechanism of such anomalous phenomena, however, was not clarified until Kobayashi _{2}I_{3} under uniaxial strain by Kondo and Kagoshima [_{F}(_{0}) where _{F}(_{0} are the positions of the two contact points. As shown in _{F}(_{0}), which is denoted by

The picture of the zero-gap system grow understanding of the remarkable transport phenomena of α-(BEDT-TTF)_{2}I_{3}. In this paper, we describe the interpretation of transport phenomena in this system based on the zero-gap picture.

(

First, let us examine the temperature dependence of the carrier density shown in _{F} is located at the contact point and does not move with temperature, this relationship is derived as

where _{F}(^{4} cm/s. This value corresponds to the result obtained from realistic theories [

Carrier mobility, on the other hand, is determined as follows. According to Mott's argument [^{−2} in the 2D zero-gap system. Consequently, the Boltzmann transport equation gives the temperature independent quantum conductivity as

where _{xx}

In order to examine Equation 2, we refer to _{s}) for seven samples is plotted. Note that a conductive layer of this material is sandwiched by insulating layers as shown in ^{2} = 25.8 kΩ. It varies from a value approximately equal to the quantum resistance at 100 K to about 1/5 of it at 7 K. The reproducibility of data was checked using six samples. Many realistic theories predict that the sheet resistance of intrinsic zero-gap systems is given as _{s} = ^{2}, where

(_{s} for seven samples under a pressure of 1.8 GPa. The inset _{s} at temperatures below 10 K; (_{s} for Sample 6. It was examined down to 80 mK. Reproduced with permission from [

Here, we mention the resistivity below 7 K in _{3}^{−} anions appears strongly. A rise of _{s} may be a symptom of localization because it is proportional to log _{3}^{−} gives rise to a partially incommensurate structure in the BEDT-TTF layers. As for Sample 6, the log _{s} was examined down to 80 mK. The log _{2}I_{3}, on the other hand, Kanoda _{s}. This answer, however, will answer the question as to why the sample with the higher carrier density (lower _{H} saturation value in _{s} at low temperature. Further investigation should lead us to interesting phenomena.

Temperature dependence of the Hall coefficient for (_{H} is plotted. Thus, the dips in (

Here we adduce other examples for the effect of unstable I_{3}^{−} anions. It was also seen in the superconducting transition of the organic superconductors β-(BEDT-TTF)_{2}I_{3} [_{2}I_{3} [

In conclusion of this section, α-(BEDT-TTF)_{2}I_{3} under high hydrostatic pressure is an intrinsic zero-gap system with Dirac type energy band _{F}(_{0}). In the graphite systems, a monolayer sample (graphene) is inevitable to realize a zero-gap state. On the other hand, in the present system, bulk crystals can be two dimensional zero-gap systems. The carrier density depends on temperature as ^{2} = 25.8 kΩ within a factor of 5.

A magnetic field gives us characteristic phenomena. According to the theory of Fukuyama, the vector potential plays an important role in inter-band excitation in electronic systems with a vanishing or narrow energy gap [_{2}I_{3}. In this section, we demonstrate that these effects give rise to anomalous Hall conductivity in α-(BEDT-TTF)_{2}I_{3}. Bismuth and graphite are the most well-known materials that serve as testing ground for the interband effects of magnetic fields. To our knowledge, however, α-(BEDT-TTF)_{2}I_{3} is the first organic material in which the inter-band effects of the magnetic field have been detected.

Realistic theory predicts that the interband effects of the magnetic field are detected by measuring the Hall conductivity _{xy}

We find two types of samples in which electrons or holes are slightly doped by unstable I_{3}^{−} anions. The doping gives rise to strong sample dependences of the resistivity or the Hall coefficient at low temperatures (_{H} is intense. In the hole-doped sample as shown in the inset of the lower part of _{H} is positive over the whole temperature range (_{H} is understood as follows. In contrast to graphene, the present electron-hole symmetry is not good except at the vicinity of the Dirac points [_{H} = 0 [_{H} at the vicinity of _{H} = 0 for electron-doped samples must be determined to detect the inter-band effects of the magnetic field. The saturation value of _{H} at the lowest temperature, on the other hand, depends on the doping density _{d}, as _{d} = _{s}/_{H}), where _{s} is the sheet density. _{d} = _{s}/_{H}).

Note that the carrier density and the mobility in

In this section, thus, to detect the inter-band effects of the magnetic field, we focused on the behavior of _{H} in which the polarity is changed (

The first step is to examine the temperature dependence of _{0}, shown as _{H} = 0 [_{s}, on the other hand, is approximately proportional to _{0}^{2} as shown in _{B} = _{F}/_{B}− _{0} approximately because _{F} versus _{0} curve in _{F} is estimated from the relationship _{s} = _{F}^{2}/(4π^{2}_{F}^{2}), where the averaged Fermi velocity _{F}, is estimated to be approximately 3.3 × 10^{4} m/s from the temperature dependence of the carrier density. Note that the weak sample dependence of both _{s} and _{H} at temperatures above 7 K (_{F} values of all samples are almost the same. When we assume that _{F}, _{B} = _{F}/_{B}−

(_{s} for five samples plotted against temperature at _{H}. (_{F} was estimated from the relationship _{s} = _{F}^{2}/(4π^{2}_{F}^{2}) with _{F} ~ 3.3 × 10^{4} m/s. From this curve, the temperature dependence of _{B} = _{F}/_{B}− _{F}. (_{F} = 0. Our experimental formula is quantitatively consistent with the theoretical curve of Kobayashi

The second step is to calculate the Hall conductivity as _{xy}_{yx}_{xx}ρ_{yy}_{yx}^{2}). In this calculation, we assume _{xx}_{yy}

Based on this assumption, we show the temperature dependence of _{xy}_{xy}_{xy}

In the last step, we redraw _{xy}_{B} = _{F}/_{B} − _{xy}

(

Lastly, we briefly mention the zero-gap structure in this material. The smooth change in the polarity of _{xy}_{xy}

In conclusion of this section, we succeeded in detecting the interband effects of the magnetic field on the Hall conductivity when

One of the characteristic features in Dirac fermion system is clearly seen in the magnetic field. In this section, the transport phenomena in the magnetic field are described. Note that we interpret the transport phenomena based on the assumption which _{F} locates at the Dirac point because _{F} is lower than broadening energy of Landau levels.

In the magnetic field, the energy of Landau levels (LLs) in zero-gap systems is expressed as _{F} irrespective of the field strength, the Fermi distribution function is always 1/2. It means that half of the Landau states in the zero mode are occupied. Note that in each LL, there are states whose density is proportional to

For _{B}T < _{1LL}, most of the mobile carriers are in the zero-mode. Such a situation is called the quantum limit. The carrier density per valley and per spin direction in the quantum limit is given by _{0}, where _{0} = _{F}. In moderately strong magnetic fields, the density of carriers induced by the magnetic field can be very high. At 3 T, for example, the density of zero-mode carriers will be 10^{15} m^{−2}. This value is by about 2 orders of magnitude higher than the density of thermally excited carriers at 4 K, and, in the absence of the magnetic field, it is 10^{13} m^{−2}. Therefore, carrier density in the magnetic field is expressed as _{0} except for very low fields.

This effect is detected in the inter-layer resistance, _{zz}

(

Recently, Osada gave an analytical formula for interlayer magnetoresistance in a multilayer Dirac fermion system as follows:

where ^{3}/2_{c}^{2}^{3} is a parameter that is considered to be independent of the magnetic field if the system is clean. _{0} is a fitting parameter that depends on the quality of the crystal, _{0}(

Except for narrow regions around _{zz}_{0}). Using this formula and assuming _{0} = 0.7 T, we tried to fit the curves in _{2}I_{3} at high pressures.

Here, we briefly mention the origin of positive magnetoresistance around

Angle dependence of interlayer Hall resistivity _{zx}_{zx}_{zx}^{2} = 25.8 kΩ and is independent of the magnetic field. This is simply understood as follows. In general, Hall voltage is proportional to cos_{zx}_{0} cos_{0} sin_{zx}^{2} cot

cot_{zx}

Here, let us return to _{p}, shifts to a lower field with decreasing temperature, where it almost saturates at about 0.04 K, as shown in

(_{zz}_{zz}_{zz}_{p} (solid triangles) and _{min} (solid circles). The solid line is the curve of _{1LL} with _{F} ~ 4 ×10^{4} m/s. Reproduced with permission from [

The overlap between the zero-mode and other LLs, primarily the _{p} and as a result, the negative magnetoresistance is observed there. Then, we have a tentative relationship: _{1LL} ~ 2_{B}_{p} at _{p} [_{1LL} with _{F} ~ 4 × 10^{4} m/s is reproduced well except in the temperature region below 2 K. This Fermi velocity corresponds to that estimated from the temperature dependence of the carrier density [_{1LL} below 2 K, on the other hand, suggests that thermal energy is sufficiently lower than the scattering broadening energy г_{0}. Thus, г_{0} is roughly estimated to be approximately 2 K from the constant value of _{p} as _{0} was estimated to be about 30 K.

The deviation from Equation 3 of data in the high field region of

In the above discussion, we did not consider the Zeeman effect. The Zeeman effect, however, should be taken into consideration because it has a significant influence on the transport phenomena at low temperatures. In the presence of a magnetic field, each LL is split into two levels with energies _{nLL} = ±Δ_{B}_{F}) = 1/2 to _{B}_{B}_{B}_{0} at the magnetoresistance minimum [

The Zeeman energy when

Recently, Osada pointed out the possibility of _{xx}_{xy}

Edge state Spector in

In conclusion of this section, we succeeded in detecting the zero-mode Landau level. The characteristic feature of zero-mode Landau carriers including the Zeeman effect was clearly seen in the inter-layer transport. The experimental data suggest that with increasing magnetic field or decreasing temperature, the system changes from a “Dirac fermion” state to a “quantum limit state”, then to a “spin-splitting” state and then to a “

Schematic diagram of boundaries in

(

α-(BEDT-TTF)_{2}I_{3} under high hydrostatic pressure is an intrinsic zero-gap conductor with a Dirac type energy band. The carrier density, expressed as ^{2} as _{s} = ^{2}, where _{s} at low temperature below 7 K gives us the impression that the zero-gap structure is unstable. The detection of zero-mode Landau carriers including its spin splitting down to 0.07 K, however, strongly suggests that the zero-gap structure is robust. Further investigation for the anomalous phenomena at low temperature should lead us to interesting phenomena.

This system offers a testing ground for a new type of particles, namely, massless Dirac fermions with a layered structure and anisotropic Fermi velocity.

We are grateful to M. Sato, S. Sugawara, M. Tamura, R. Kato and Y. Iye for fruitful collaborations which helped us understand the subjects discussed in this paper. We thank T. Osada, A. Kobayashi, S. Katayama, Y. Suzumura, R. Kondo, T. Morinari, T. Tohyama and H. Fukuyama for valuable discussions. This work was supported by Grant-in-Aid for Scientific Research (No. 22540379 and No. 22224006) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

_{2}I

_{3}Salt

_{2}I

_{3}: Ultra narrow gap semiconductor, superconductor, metal, and charge-ordered insulator

_{2}I

_{3}under pressure

_{2}I

_{3}under High Pressures

_{2}I

_{3}under high pressure: Discovery of a novel type of conductor

_{2}I

_{3}and discovery of superconductivity

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_{2}I

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_{2}I

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_{2}I

_{3}by synchrotron X-ray diffraction

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_{2}I

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_{3}at ambient pressure and with uniaxial strain

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_{2}I

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_{2}I

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_{2}I

_{3}and θT-(BEDT-TTFET)

_{2}I

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_{2}I

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_{2}I

_{3}under high pressure

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_{3}using interlayer magnetoresistance

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_{3}: Positive versus negative contributions in a tilted dirac cone system