Abstract
It is widely recognized that the effect of doping into a Mott insulator is complicated and unpredictable, as can be seen by examining the Hall coefficient in high T_{c} cuprates. The doping effect, including the electron–hole doping asymmetry, may be more straightforward in doped organic Mott insulators owing to their simple electronic structures. Here we investigate the doping asymmetry of an organic Mott insulator by carrying out electricdoublelayer transistor measurements and using cluster perturbation theory. The calculations predict that strongly anisotropic suppression of the spectral weight results in the Fermi arc state under hole doping, while a relatively uniform spectral weight results in the emergence of a noninteractinglike Fermi surface (FS) in the electrondoped state. In accordance with the calculations, the experimentally observed Hall coefficients and resistivity anisotropy correspond to the pocket formed by the Fermi arcs under hole doping and to the noninteracting FS under electron doping.
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Introduction
Electron–hole doping asymmetry in Mott insulators has been one of the key questions related to the origin of superconductivity in the proximity of the insulating state^{1,2,3}. To investigate this, one should prepare an exactly halffilled (zerodoped) Mott insulator and then inject or extract electrons, preferably by an electrostatic method, in the same sample. However, it is difficult to perform such measurements on highT_{c} cuprates for the following three reasons. First, they are mostly either only holedoped or electrondoped, and the crystallographic structures of the parent materials are often different. Second, they require a strong electric field to tune their band filling owing to their high halffilled carrier density (in the order of 10^{15} cm^{−2}). Third, since they are chargetransfertype insulators, electrons and holes are doped into different electronic orbitals, which may obscure the pure doping asymmetry. In practice, the inverse of the Hall coefficient monotonically increases with increasing doping concentration with the sign changing across the zerodoping point^{4}. This is a feature of band insulators rather than Mott insulators.
In contrast to the highT_{c} cuprates, the organic Mott insulator κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Cl (ref. 5) (abbreviated to κCl hereafter), where BEDTTTF represents bis(ethylenedithio)tetrathiafulvalene, serves as an appropriate material for examining the electron–hole asymmetry of doped Mott insulators because of its relatively low carrier density (∼1.8 × 10^{14} cm^{−2}) and its single electronic orbital nature. Indeed, electrons and holes are doped into the same π electron band and thus κCl can be modelled as a singleband Hubbard model on an anisotropic triangular lattice (Fig. 1). Because of its high controllability of electronic states, κCl has been studied intensively in terms of the bandwidthcontrolled Mott transition between Mott insulating and superconducting states at halffilling^{6,7}. The electrostatic method of carrier doping into organic Mott insulators with fieldeffect transistor (FET) structure has been implemented by the authors^{8,9}. However, the limited doping concentration with the FET still forbids the observation of electron–hole asymmetry.
Here, by fabricating electricdoublelayer transistors (EDLTs) based on thin single crystals of κCl, we realized for the first time both electron and hole doping into the organic Mott insulator. We measured the field effect on transport properties including the sheet resistivity, Hall coefficient and resistivity anisotropy to elucidate the doping anisotropy of the electronic state in the doped organic Mott insulator. To provide further insights into the Fermi surface (FS), we calculated the singleparticle spectral functions for an effective model of κCl by using cluster perturbation theory (CPT)^{10,11}. The calculations predict that strongly anisotropic suppression of the spectral weight results in the Fermi arc state under hole doping, while a relatively uniform spectral weight results in the emergence of a noninteractinglike FS in the electrondoped state. In accordance with the calculations, the experimentally observed Hall coefficients and resistivity anisotropy correspond to the pocket formed by the Fermi arcs under hole doping and to the noninteracting FS under electron doping.
Results
Resistivity
We fabricated the EDLT devices by mounting an ion gel on a Hallbarshaped thin single crystal of κCl and an Auside gate electrode (Fig. 2a,b). First, we obtained the resistivity curve for the sheet resistivity ρ at the temperature T=220 K (Fig. 2c). As expected for a Mott insulator, a clear ambipolar field effect with a resistance peak at the gate voltage V_{g}≈−0.2 V was observed. The hysteresis and the leakage current remained small without any signature of a chemical reaction between the electrolyte and κCl. Note that the field effect was ∼3% at 220 K, because thermally excited carriers in the bulk (about 100 molecular layers in a typical sample) dominated the electrical conduction owing to the low charge excitation gap (∼20 meV). We also confirmed that the carrier tunability of the κCl EDLT greatly exceeded that of SiO_{2}based FETs by comparing these devices using the same κCl crystal (Supplementary Fig. 1). Provided that the mobilities of the FET and EDLT are equivalent, a gate voltage of +1 V in the EDLT corresponded to ∼20% electron doping (see Supplementary Note 1). The resistivity curves at 220 K and the temperature dependences of the resistivity were reproducible after multiple temperature cycles (see Supplementary Note 2 and Supplementary Fig. 2).
With decreasing temperature, the ambipolar field effect became more distinct. Figure 2d,e show the temperature dependence of the sheet resistivity under a negative gate voltage (hole doping) and positive gate voltage (electron doping), respectively. Under hole doping, the activation energy significantly decreased to <1 meV at V_{g}=−1.35 V. On the other hand, the doping of electrons tended to reduce the resistivity more effectively and metallic conductivity was observed at V_{g}>1 V. In a highconductivity sample, we observed negative magnetoresistance accompanied with the upturn of resistivity at low temperature, which suggested a weak localization effect (see Supplementary Note 3 and Supplementary Fig. 3). Analysis of the weak localization indicated that the mean free path and dephasing length considerably exceeded the distance between BEDTTTF dimers indicating coherent transport.
Hall effect
To observe a more significant difference between the electron and holedoped states, we measured the Hall coefficient R_{H}, which indicates the FS topology. The magnetic field B dependence of the Hall resistance R_{xy} at 30 K is shown in Fig. 3a (R_{H} is given by the slope). Despite electron doping, the Hall coefficients were clearly positive and a simple estimation of the carrier density, 1/eR_{H}, gave ∼0.8 holes per BEDTTTF dimer. Since the injected carriers were electrons (∼0.2 electrons/dimer), the concentration of which was much lower than that of the observed holes, it suggests that dense hole carriers are delocalized by electron doping. This is considered to be the dopinginduced Mott transition previously observed in FET devices^{12}.
If we assume doping symmetry, a similar Hall effect is expected for the holedoped side. Namely, hole doping immediately collapses the Mott–Hubbard gap and 1/eR_{H} corresponding to 1−δ (δ: electron doping concentration, which is negative for hole doping) holes per BEDTTTF dimer should be observed. However, as shown in Fig. 3a, the Hall coefficient on the holedoped side was about three times larger than that on the other side, and 1/eR_{H} gave ∼0.3 holes per BEDTTTF dimer. Although 1/eR_{H} increased with increasing absolute value of the gate voltage, the rate of increase was low (only 7% difference between V_{g}=−0.8 V and −1.2 V). Surprisingly, under hole doping, 1/eR_{H} per dimer appears to be neither 1−δ (delocalized hole carriers+injected hole carriers) nor−δ (only injected hole carriers), indicating that the localized carriers were partially delocalized by excess hole doping as shown in Fig. 3c. These results indicate that the FS topology is significantly different between the electron and holedoped states. For the gate voltage dependence of the Hall mobility, see Supplementary Note 4 and Supplementary Fig. 4. The temperature dependence of R_{H} (Fig. 3b) shows that the R_{H} asymmetry was maintained up to ∼50 K, above which the Hall effect at the doped surface was obscured by the thermally excited carriers in the bulk. Note that increase in R_{H} at 30 K upon cooling is due to the decrease of the thermally excited carriers in the bulk (see Supplementary Note 5 and Supplementary Fig. 5). The data below 25 K contain ambiguity owing to the nonohmic behaviour (see Supplementary Note 6 and Supplementary Fig. 6).
What is the origin of R_{H} under hole doping? As shown in Fig. 1, κCl forms essentially an anisotropic triangular lattice resulting in an ellipsoidal noninteracting FS which is folded by the zone boundary. In a Fermi liquid with a single type of carrier, 1/eR_{H} denotes the carrier density corresponding to the volume enclosed by the FS (Luttinger’s theorem^{13}). Under electron doping, 1/eR_{H} is close to the carrier density corresponding to the noninteracting ellipsoidal FS (known as the βorbit in the study of Shubnikovde Haas oscillation^{14}). By contrast, 1/eR_{H} under hole doping somehow appears to correspond to the volume bounded by the lenslike closed portion of the FS (known as the αorbit). The doping dependences of the carrier density enclosed by the α and βorbits assigned for a κBEDTTTF salt^{15} are plotted as reference in Fig. 3c.
Singleparticle spectral functions
To verify this R_{H} asymmetry, we calculated the singleparticle spectral functions of the Hubbard model on an anisotropic triangular lattice (t′/t=−0.8, U/t=7) at 30 K using CPT and the results are shown in Fig. 4. First, the Mott insulating state was reproduced at halffilling, where the energy gap opened at all the kpoints (Fig. 4b,e). When electrons were doped (δ=+0.17), the FS appeared as shown in Fig. 4c,f. The FS topology was the same as that of the noninteracting case, although the shape and the spectral weight were not exactly the same. In this state, the value of R_{H} should be close to that of the Fermi liquid state, as suggested by the experiments. On the other hand, the FS under hole doping (δ=−0.17) was very different from the noninteracting FS (Fig. 4a,d). The spectral weight near the Z point was strongly suppressed. Namely, arcs around the X point and pseudogaps near the Z point appeared. The partial disappearance of FS by the pseudogap is similar to the Fermi arc state observed in lightly holedoped cuprates^{16,17,18}. In this state, Luttinger’s theorem is seemingly violated and the value of R_{H} is no longer simply estimated. However, considering the first Brillouin zone (BZ) of κCl, one can see that the Fermi arcs are folded and form closed lenslike orbits. These orbits originate from the αorbit mentioned above and correspond to the observed R_{H} in the holedoped states. These results imply that R_{H} is predominantly governed by quasiparticles with relatively long lifetime (bright points of the spectral function in the reciprocal space in Fig. 4). Therefore, the asymmetry of the FS topology was clearly observed via R_{H}.
Anisotropy of the resistivity
The partial disappearance of the FS can also be confirmed by the anisotropy in the carrier conduction. The strongly suppressed parts of the FS under hole doping should affect strongly the carrier conduction along the c axis. Indeed, our CPT calculations predict that the resistivity is more anisotropic in the holedoped state than in the electrondoped state (see Supplementary Note 7 and Supplementary Fig. 7). We measured the resistivity anisotropy as shown in Fig. 5. The crystallographic axes were determined by the crystal shape and the sign of the Seebeck coefficient at room temperature. At 220 K, where the thermally excited carriers in the nondoped bulk dominate the conductivity, the c axis resistivity ρ_{c} was slightly higher than the a axis resistivity ρ_{a} in both electron and hole doping. With decreasing temperature ρ_{c}/ρ_{a} increased under hole doping, while it decreased and remained near 1 in the nondoped and electrondoped states as shown in Fig. 5b. The contour plot of ρ_{c}/ρ_{a} under various gate voltage and temperature (Fig. 5c) clearly exhibited that the resistivity in the holedoped state was particularly anisotropic, in good qualitative agreement with our CPT calculations.
Discussion
The doping asymmetry is expected from the particle–hole asymmetric noninteracting band structure, as shown in Fig. 1c. The energy bands along the Z–M axis are very flat and the van Hove singularity (vHs) lies below the FS at halffilling. With the doping of holes, the FS approaches the vHs and the effect of the interaction is expected to be enhanced. Indeed, substantial suppression of the spectral weight of the FS along the Z–M axis, on which the van Hove critical points lie, is observed (Fig. 4a). By contrast, the FS departs from the vHs with the doping of electrons, resulting in a weaker interaction effect and a more noninteractinglike FS. It is also indicated that the spin fluctuation is stronger in the holedoped state because of the vHs. If a superconducting state could be induced by further hole doping, the transition temperature and pairing symmetry are of great interest since the holedoped state is substantially different from the metallic state created by controlling the bandwidth at halffilling.
To summarize, without gate voltage, κCl is a Mott insulator due to electron interaction and high commensurability between the hole density and periodic potential (one hole/one site). When holes or electrons are doped, the commensurability is reduced and the effect of electron interaction is weakened, resulting in carrier delocalization. However, the effect of interaction remains strong at specific kpoints where the energy dispersion is relatively flat. Owing to the particle–hole asymmetric band structure, the pseudogaps appeared only near the van Hove critical points under hole doping. In contrast to the calculations, our measurements did not observe the true metallic state (dρ/dT>0) at low temperature. This is probably due to the effect of localization caused by finite random potential at the surface^{19,20}, as the negative magnetoresistance indicated. Improvement of the surface roughness and cleanness may realize metallic phases under both dopings.
Thus, the doping asymmetry in a Mott insulator was demonstrated by a simple process in an organic Mott EDLT. The above results show that organic Mott insulators are good model materials for Mott physics even in the case of bandfilling control, and the electronic state in a simple Mott insulator may be predictable in the framework of CPT. Since molecular conductors consist of various molecules with different arrangements, the effect of doping into materials with different FS topologies is of interest in future experiments. Furthermore, the presence/absence of the superconducting phase under doping and its doping asymmetry are also intriguing.
Methods
Sample preparation
The source, drain and gate electrodes (18 nm thick Au) were patterned on a polyethylene naphthalate substrate (Teonex Q65FA, Teijin DuPont Films Japan Limited) using photolithography. A thin (∼100 nm) single crystal of κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Cl (abbreviated to κCl hereafter) was electrochemically synthesized by oxidizing BEDTTTF (20 mg) dissolved in 50 ml of 1,1,2trichloroethane (10% v/v ethanol) in the presence of TPP[N(CN)_{2}] (TPP, tetraphenylphosphonium; 190 mg), CuCl (60 mg) and TPPCl (100 mg). After applying current of 8 μA for 20 h, a thin crystal was transferred into 2propanol with a pipette and was guided on top of the PEN substrate. After the substrate was removed from the 2propanol and dried, the κCl crystal was shaped into a Hallbar using a pulsed laser beam with a wavelength of 532 nm. Typical dimensions of the Hall bar sample were approximately 15 μm (width) × 40 μm (length) × 100 nm (thickness). The EDLT devices were completed by mounting an ion gel (or ionic liquid) on the Hallbarshaped thin single crystal of κCl and an Auside gate electrode. The basic techniques, such as the electrochemical synthesis of thin single crystals and the laser shaping, are common to our previous FET devices^{12}. As gate electrolytes, we employed poly(vinylidene fluoridecohexafluoropropylene) (PVDFHFP) with 58% w/w 1butyl3methylimidazolium tetrafluoroborate (BMIMBF_{4}) (samples #1∼4), N,NdiethylNmethylN(2methoxyethyl) ammonium bis(trifluoromethanesulfonyl)imide (DEMETFSI) (sample #5 in Supplementary Fig. 1), polymethyl methacrylate (PMMA) with 57% w/w DEMETFSI (sample #6 in Supplementary Fig. 3). The PVDFHFP ion gel^{21} was spincoated from an acetone solution in which the weight ratio between the polymer, ionic liquid and solvent was 1:1.3:7.7. Although the ionic conductivity was the greatest in the liquid state, the large thermal stress usually broke the κCl crystal at a low temperature (T<150 K). Thinning of the gate electrolytes by gelation reduced the thermal stress at low temperatures, allowing the successful cooling of the samples down to 2 K. The area of the gate electrode was more than 100 times larger than that of the κCl channel in order to apply the gate voltage effectively.
Transport measurements
The transport measurements were performed using a Physical Property Measurement System (Quantum Design). The temperature and magnetic field normal to the substrate were controlled in the range of 2–300 K (cooling rate: 2 K min^{−1}) and ±8 T, respectively. When the gate voltage was varied, the samples were warmed to 220 K. In the Hall measurements, the magnetic field was swept in the range of ±8 T at a constant temperature and the forward and backward data were averaged to eliminate the small linear voltage drift.
Although it is difficult to experimentally determine the distribution of fieldinduced charge along the outofplane direction, we consider that the fieldinduced carriers predominantly exist in a single conducting molecular sheet^{12}. Recently, the quantum oscillations in the doped surface of the Dirac fermion system α(BEDTTTF)_{2}I_{3} (ref. 22) indicated that only two conducting layers from the surface were doped, in which the carrier density in the second layer was only 1/7 of that in the first layer. The above assumption is also likely to apply to κtype BEDTTTF salts, where a shorter screening length is expected.
Hubbard model on an anisotropic triangular lattice
To examine the electron correlation effects in the organic doped Mott insulator, we study the singleband Hubbard model defined on an anisotropic triangular lattice, which is known as the simplest model for κtype BEDTTTF salts^{23,24,25,26}. The model is defined by the Hamiltonian
where creates an electron on site i with spin σ(=, ) and . t_{ij} is the transfer integral between the neighbouring sites i and j, U is the onsite Coulomb repulsion and μ is the chemical potential. Here, a single site consists of a single dimer and a unit cell contains two dimers with different orientation (Supplementary Fig. 8a). The transfer integral between the different (same) dimers is given as t_{ij}=t (t′).
The series of κtype BEDTTTF salts serves as an ideal realization of the Hubbard model of equation (1). The anisotropy t′/t can be changed depending on substituted anions^{26,27,28,29}. The Coulomb repulsion U is estimated to be comparable to^{29,30} or larger than^{26} the band width, implying that the strongly correlated electronic state is realized at ambient pressure. Applied pressure^{28,31,32} or bending strain on thin films^{33} can control the electron correlation parameter t/U. Moreover, the chemical potential μ can be tuned efficiently by applied gate voltage in the EDLT, allowing for investigation of the electronic structure of electron and holedoped Mott insulators on equal footing. The series of organic charge transfer salts is indeed the highly controllable counterpart of strongly correlated transitionmetal compounds^{34}. In the main text, we have focused on a typical parameter set for κtype BEDTTTF salts of t′/t=−0.8, U/t=7 and t=55 meV. We have also considered the parameter set relevant for κCl of t′/t=−0.44, U/t=5.5 and t=65 meV (ref. 29), and found that the Fermi arc appears for holedoped case also in this parameter set (Supplementary Fig. 9). The noninteracting tightbinding band structure, the density of states, and the FS at halffilling for this parameter set are shown in Supplementary Fig. 8c,d.
The strongly correlated Hubbard model at halffilling or the Heisenberg model on the anisotropic triangular lattice have been investigated theoretically and the groundstate phase diagrams with various phases including metal, antiferromagnetic insulator, spiral, and spin liquid ground states have been obtained^{35,36,37,38,39,40,41,42}. Most of these previous calculations on the Hubbard model at halffilling have been carried out with t′/t>0. However, the sign of t′ is crucial when it comes to the electron–hole asymmetry of carrier doping because the sign determines whether the vHs appears in holedoped side or electrondoped side. The correct sign is t′/t<0 (ref. 26) and the vHs appears in the holedoped side (see Fig. 1c and Supplementary Fig. 8c).
Unit cell and Brillouin zone
In Fig. 4 and Supplementary Fig. 9, we show the singleparticle spectral functions and the FSs in the BZ of the unit cell consisting of two sites in the real space (see blue shaded region in Supplementary Fig. 10a). The BZ is relevant for the realistic structure of the κtype BEDTTTF salts and therefore the results are directly compared with those obtained by firstprinciples bandstructure calculations^{26,29}. However, the primitive unit cell of the singleband Hubbard model in equation (1) on the anisotropic triangular lattice consists of one site (see red shaded region in Supplementary Fig. 10a), unless a spontaneous symmetry breaking such as the Néel order occurs. Therefore the theoretical calculations for the Hubbard model on the anisotropic triangular lattice are usually performed with this primitive unit cell^{36,37,38,39,40,41,42,43}. In order to avoid possible confusions caused by the different choice of the unit cell, we describe the relation between the two BZs of the onesite unit cell and the twosite unit cell.
Supplementary Figure 10a shows the anisotropic triangular lattice with regulartriangular plaquettes. The primitive translational vectors for the one (two)site unit cell are given by {e_{1}, e_{2}} ({a, c}), and the corresponding BZ is diamond (rectangular) shaped, as shown in Supplementary Fig. 10c.
We also show in Supplementary Fig. 10b the lattice which is topologically equivalent to the triangular lattice in Supplementary Fig. 10a but forms the square plaquettes with diagonal lines. Notice that this square topology of the lattice is very often used for study of the Hubbard model on the anisotropic triangular lattice^{36,37,38,39,40,43}. The primitive translational vectors for the one (two)site unit cell are again given by {e_{1}, e_{2}} ({a, c}). The BZ of the one (two)site unit cell is now square (diamond) shaped, as shown in Supplementary Fig. 10d.
Let us now consider the basistransformation formula between the bases {a, c} and {e_{1}, e_{2}}, where the former is a set of the primitive translational vectors of the realistic structure of the κtype BEDTTTF salts, while the latter is that of the singleband Hubbard model. It is easily found from Supplementary Fig. 10b that they are related by the ±π/4 rotation and the scale transformation. To be concrete, we consider a vector r=r_{a}a+r_{c}c=r_{1}e_{1}+r_{2}e_{2} in the real space. It is easily shown that the coordinate (r_{a},r_{c}) is related to the coordinate (r_{1},r_{2}) as
Similarly, for a wave vector k=k_{a}g_{a}+k_{c}g_{c}=k_{1}g_{1}+k_{2}g_{2}, where {2π g_{a}, 2π g_{c}} ({2π g_{1}, 2π g_{2}}) is the set of the reciprocal vectors of {a, c} ({e_{1}, e_{2}}), (k_{a}, k_{c}) is related to (k_{1}, k_{2}) as
For example, (k_{a}, k_{c})=(0, 0),(0, π),(π, π) and (π, 0) in the {g_{a}, g_{c}} basis (that is, Γ, Z, M and X points in Supplementary Fig. 10c) correspond to (k_{1}, k_{2})=(0, 0),(π/2, π/2), (π, 0) and (π/2, −π/2) in the {g_{1}, g_{2}} basis (Supplementary Fig. 10d), respectively. Finally, we note that the rectangular BZ in Supplementary Fig. 10c corresponds to the antiferromagnetic BZ of the square lattice in Supplementary Fig. 10d (blue shaded regions).
Cluster perturbation theory
To examine the electron–hole asymmetry of the FS, the calculation of singleparticle Green’s functions is required. For this purpose, we employ the CPT^{11}. The CPT is a theory of strong coupling expansion for the singleparticle Green’s function and allows us to examine the singleparticle excitations including quasiparticle excitations, Mott gap, and even pseudogap phenomena in strongly correlated electron systems^{43,44,45}.
As shown in the following, the CPT requires the fully interacting singleparticle Green’s function (or the selfenergy^{46,47}) of openboundary clusters to obtain the singleparticle Green’s function of the lattice. The basic notion of the scheme was introduced in the earlier work by Gros and Valentí^{10}.
Let us outline the method of CPT. First, we divide the whole lattice on which the model Hamiltonian H is defined into identical finitesize clusters, each of which consists of L_{c} sites. We shall denote the Hamiltonian of a cluster as and the intercluster hopping as . Here, is given in the righthand side of equation (1) but it is defined on a L_{c}site cluster with openboundary conditions. Next, the singleparticle Green’s function G_{c} of the cluster Hamiltonian is calculated by the exact diagonalization method. Applying the strong coupling expansion with respect to the intercluster hopping , the singleparticle Green’s function is given as , where is the matrix representation of (ref. 48). By restoring the translational invariance, which is broken in due to the partitioning of the lattice, we obtain the singleparticle Green’s function G of the whole lattice for momentum k and complex frequency z as
where r_{i} is the position of the ith site within the cluster^{48}.
The cluster Green’s function matrix G_{c} is given as
with
and
where β=1/k_{B}T with T being the temperature, Ψ_{s}〉 is the sth eigenstate of with its eigenvalue E_{s} in ascending order, and is the partition function of the cluster. The sum over eigenstates is truncated by summing up to the s_{max}th eigenstate. The upper bound s_{max} of the sum is determined so as to satisfy that , where E_{0} is the groundstate energy and is a small real number. For the finite temperature calculations at T=100 K, we set , in which for example the lowest s_{max}∼1,400 eigenstates among 4^{12} eigenstates in total are included for the holedoped case with L_{c}=12.
The intercluster hopping between different clusters in equation (4) is represented as
where t_{ij} is given in equation (1), X_{1} and X_{2} are the translational vectors of the clusters, and m_{1}X_{1}+m_{2}X_{2} (m_{1} and m_{2}: integer) indicates a relative position of two different clusters, site i being located in one cluster and site j in the other cluster. For the 12site cluster, the translational vectors are X_{1}=2.5a+0.5c and X_{2}=0.5a+2.5c, and t_{ij} in equation (8) is finite only when (m_{1},m_{2})=(±1, 0), (0, ±1) and (±1, ±1), indicating the intercluster hopping between a cluster and its neighbouring six clusters (see Supplementary Fig. 8a). For the 16site cluster, the translational vectors are X_{1}=2a+2c and X_{2}=−2a+2c, and t_{ij} in equation (8) is finite only when (m_{1}, m_{2})=(±1, 0), (0, ±1), and (±1, ±1) (see Supplementary Fig. 8b).
Note that in principle a cluster with a larger size gives better results in the CPT because the electron correlation effects including spatial fluctuations within a cluster are fully taken into account. We have shown in the main text and Supplementary Fig. 9 the singleparticle spectral functions using the 12site cluster at finite temperatures. We also show in Supplementary Figs 11 and 12 the singleparticle spectral functions for the 16site cluster at zero temperature (see also Supplementary Note 8), where s_{max}=0 in equations (6) and (7). The results clearly demonstrate that the electron–hole asymmetric reconstruction of the FS is observed irrespectively of the cluster sizes used. As a technical point, we have made use of the C_{2v} symmetry of the 12 and 16site clusters to reduce computational costs for solving the eigenvalue problem of and for calculating the cluster singleparticle Green’s functions.
Singleparticle spectral function
As shown in Supplementary Fig. 8a, the partitioning of the triangular lattice into the 12site clusters does not respect the C_{2v} symmetry (for lack of the σ_{v} reflection symmetry) and the twosite unit cell structure (as X_{1} and X_{2} connect differently oriented dimers) of the dimer model on the anisotropic triangular lattice. However, these can be restored by averaging the singleparticle Green’s functions over four different momenta, that is,
This is because the average of the Green’s functions between k=(k_{a}, k_{c}) and (−k_{a}, k_{c}) restores the σ_{v} reflection symmetry and the addition of the Green’s functions with wave vectors shifted by a primitive reciprocal vector, that is, k=(±k_{a}, k_{c}+2π/c), recovers the twosite unit cell structure in the BZ. Therefore, given in equation (9) provides the singleparticle Green’s function relevant for the Hubbard model defined on the anisotropic triangular lattice even with the 12site cluster.
On the other hand, partitioning of the lattice into the 16site clusters respects both the C_{2v} symmetry and the twosite unit cell structure (see Supplementary Fig. 8b). Therefore, the singleparticle Green’s function with 16site cluster is calculated in a standard way of the CPT for multiband models described in ref. 48.
Using , the singleparticle spectral function is calculated as
where a small imaginary part iη of the complex frequency gives the Lorentzian broadening of the spectra. The FS is determined with the singleparticle spectral function at zero energy, A(k,0). We set the Lorentzian broadening factor η/t=0.2 for the singleparticle spectral functions and η/t=0.15 for the FS calculation. The singleparticle spectral functions and FS are shown in Fig. 4 and Supplementary Figs 11 and 12 (t′/t=−0.8, U/t=7 and t=55 meV), and in Supplementary Fig. 9 (t′/t=−0.44, U/t=5.5 and t=65 meV).
Data Availability
The data that support the findings of this study are available from the corresponding authors upon request.
Additional information
How to cite this article: Kawasugi, Y. et al. Electron–hole doping asymmetry of Fermi surface reconstructed in a simple Mott insulator. Nat. Commun. 7:12356 doi: 10.1038/ncomms12356 (2016).
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Acknowledgements
We acknowledge Teijin DuPont Films Japan Limited for providing the PEN films. Computations have been done using HOKUSAI facility of Advanced Center for Computing and Communication at RIKEN. This work was supported by MEXT and JSPS KAKENHI (Grant numbers JP16H06346, 15K17714, 26102012 and 25000003), JST ERATO and MEXT Nanotechnology Platform Program (Molecule and Material Synthesis).
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Y.K. and K.S. contributed equally to this work. Y.K. performed the planning, sample fabrication, cryogenic transport measurements and the data analyses. K.S. and S.Y. performed all CPT calculations and data analyses. Y.E., J.P. and T.T. developed the ion gel for κCl EDLT, performed transfer curve analyses and improved the device performance. Y.S. grew κCl single crystals of high quality and performed magnetoresistance data analyses. Y.K., K.S. and H.M.Y. wrote the manuscript. T.T., S.Y., H.M.Y. and R.K. supervised the investigation. All authors commented on the manuscript.
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Kawasugi, Y., Seki, K., Edagawa, Y. et al. Electron–hole doping asymmetry of Fermi surface reconstructed in a simple Mott insulator. Nat Commun 7, 12356 (2016). https://doi.org/10.1038/ncomms12356
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DOI: https://doi.org/10.1038/ncomms12356
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