Introduction

Organic small-molecular semiconductors are intrinsic semiconductors composed of π-conjugated charge neutral molecules, whose electronic states are weakly connected solely via van der Waals bonds. Recent advances in molecular synthesis and the fabrication of single crystalline thin films have demonstrated that band-transport, based on coherent electronic states, can be formed in these molecular systems1,2,3. This progress has significantly accelerated applications in organic electronics, particularly in the development of high-speed integrated circuits for flexible, ultralight sensors4,5,6,7 and radio frequency devices8,9,10. Concurrently, the emergence of these new classes of coherent electronic systems, characterized by narrow bands within a soft-lattice environment, has posed fundamental challenges in elucidating quantum mechanical electronic phases as a function of carrier concentration at low temperatures.

A technical challenge in achieving sufficient doping levels for metallic ground states in these systems is to prevent the soft surfaces of organic crystals from being damaged under high-concentration doping. Recently, we employed a specific quantum-well molecule, 3,11-dioctyldinaphtho[2,3-d:2’,3’-d’]benzo[1,2-b:4,5-b’]dithiophene (C8-DNBDT)11. In this approach, the molecules crystallize to spontaneously form a two-dimensional (2D) π-electron layer, which is protected against the random potentials of dopants by alkyl-chain layers with higher band gap energy, as shown in Fig. 1a. We achieved a metallic phase in C8-DNBDT single crystals at the interface with an ionic liquid, at a hole concentration of 0.5 × 1014 cm2, forming an electric double layer (EDL)12.

Fig. 1: Illustrations of electric-double-layer gated transistor using quantum-well molecules.
figure 1

a Schematic picture of a quantum-well molecular compound. b Schematic image of an electric-double-layer transistor. Here, S, D, BG, SG, and VP stand for the source, drain, bottom-gate, side-gate, and voltage probe, respectively. c Structural formula of a C8-DNBDT molecule (Upper) and the herringbone structure in the bc plane of the single crystal (Lower). Alkyl chains are omitted for visibility. d Cross section across ionic liquid/bilayer C8-DNBDT/parylene, and the hole distribution (red line, upper horizontal axis) and potential height (blue line) against the depth direction (vertical axis). The hole distribution is calculated by the Poisson–Schrödinger equation.

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In this study, we reveal exotic properties of the metallic electronic phases obtained by further doping carriers up to concentrations of 1014 cm−2 (about 1/4 per molecule), which is a range previously unexplored in van der Waals crystals. Specifically, we examined whether, and at what doping level, strong electronic correlations become apparent in such narrow-band systems, moving beyond the simple one-electron rigid band model. Indeed, various exotic electronic phases have been observed in both inorganic13,14,15,16 and organic compounds17,18,19,20 when the doping level approach half-filled bands. A fundamental question is whether the electronic phase of such exotic metals abruptly or gradually manifests as a function of carrier concentration.

Our findings indicate that electronic correlation dominates even at concentrations of holes less than 1/4 per molecule, evolving from weak-localizing behaviour with the correction of electron–electron interaction to a regime of considerably stronger electron correlation, potentially with charge ordering instability at increasing doping levels. Consequently, an unusual temperature dependence of the Hall coefficient is observed, contradictory to the rigid-band model. This observation is notable given that band-structure calculations predict a simple 2D cylindrical Fermi surface without strong correlation effects.

Results and discussion

Device fabrication and measurement setup

We first fabricated bimolecular-layer thin films of single-crystalline C8-DNBDT on a parylene-coated polyethylene naphthalate (PEN) substrate, following a one-shot printing process reported earlier6,21. To construct a Hall-bar geometry of the EDL transistors, gold and chromium were deposited through a shadow mask for source, drain, four voltage-probing probes, and a side gate. A thin film of ionic-liquid gel, comprising an ionic liquid of N, N-diethyl-N-methyl-N-(2-methoxyethyl) ammonium bis (trifluoromethanesulfonyl) imide ([DEME][TFSI]) and poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) copolymer film, were laminated (Fig. 1b). Detailed fabrication procedures are provided in the Supplementary Fig. 1 and ref. 12. In the semiconductor crystals, a two-dimensionally ordered herringbone structure of the conjugated molecular cores is formed within the bc-plane (Fig. 1c). Magneto-transport measurements were conducted after charging the samples by applying a side-gate voltage (VSG) slowly at temperatures ranging from 220 K to 250 K, continuously monitoring the conductivity of the samples to confirm the absence of abrupt change due to obvious damage to the semiconductors. Details on this process are available in the Supplementary Figs. 2–5. Fig. 1d presents the calculated hole carrier distribution caused by the electric field along the a-axis (perpendicular to the substrate) at 300 K with a hole concentration of 0.5 × 1014 cm−2, based on the Poisson–Schrödinger equation. This calculation suggests that hole carriers are present exclusively in the first quantum well in contact with the anion layer under a high electrical field. The computational methods used are detailed in the Methods section.

Anomalous temperature dependence of Hall coefficients

Figure 2a, b depicts the temperature dependence of sheet resistivities ρsheet and Hall coefficients RH of C8-DNBDT for various side-gate voltages VSG. We present results from two devices, labelled samples #1 and #2, to confirm reproducibility. The resistivities exhibit typical metallic behaviour at higher temperatures, falling below the quantum resistance (h/e2 ~ 25.8 kΩ) and decreasing monotonically with the decrease of temperature to 10–20 K for both samples. We estimate the hole carrier concentration, measuring the Hall coefficient RH, as long as (eRH)−1 is temperature-independent and is considered to be equivalent to the electrostatic doping concentration p in normal 2D metals. In the low doping region with |VSG| <2.8 V, the attempt of the estimation looks successful because the Hall coefficient is nearly temperature-independent at least above 100 K. Plotted in Fig. 2c is (eRH)−1 at 180 K as a function of VSG, exhibiting a linear relationship between (eRH)−1 and VSG in the low |VSG| region. The observation indicates that the EDL capacitance C is 8.6 μF cm-2 in this region, which is consistent with our previous studies12 and the values specified in literature22. The electrostatic doping concentration p is thus defined properly for |VSG| <2.8 V, reaching the level of 1014 cm−2 (~1/4 hole per molecule, Fig. 2d), which has not been explored in small molecular organic semiconductors. However, the values of (eRH)−1 at 180 K deviates from the linear relation with the application of |VSG| higher than 2.8 V. Also in the high doping region, significant temperature dependence of RH appears even around 180 K, which are contradictory to the model of Fermi-degenerated normal metallic states, violating the relation of (eRH)−1 = p. We also tried another attempt using gate current charging analysis, which is proposed as an independent method to estimate carrier density in EDL transistors23, however, the analysis did not give proper values for the present ion-gel gated devices (see Supplementary Fig. 6). Therefore, we were not able to precisely estimate the electrostatic doping concentration and carrier density in the region of |VSG| > 2.8 V. We paid particular caution on the experimental conditions of charging rate and temperature sweep so that the samples preserve healthy nature in the boundary between the semiconductor crystal and ionic liquid monitoring stability of the devices during the whole measurement including the region of |VSG| > 2.8 V. At around 240 K, VSG was carefully raised with the condition of less than 0.04 V/min, continuously measuring four terminal conductivity which did not show any jump, indicating absence of significant damage upon doping (See Supplementary Table 1). Furthermore, no abrupt change in the ρsheet vs T data is observed, either. We changed the temperature at the rate of 0.2 K/min, which we previously confirmed to be sufficiently slow in our preliminary experiment, without hysteresis between the data taken with decreasing and increasing temperatures (See Supplementary Fig. 7).

Fig. 2: Hall effects of quantum-well molecular compound continuously doped up to a density of 1014 cm−2.
figure 2

Temperature dependence of (a) sheet resistivity and (b) Hall coefficient RH at various side-gate voltage VSG. #1 and #2 represents the sample #1 and #2, respectively. c Hall carrier concentration (eRH)−1 at 180 K for various VSG. The broken line shows the electrostatic doping concentration of hole p as a function of VSG derived from the capacitance of EDL, which is valid at lower |VSG|(<2.8 V). d Calculated valence band, density of states, and number of holes per unit cell, where the top of valence band is set to 0 eV. The coloured horizontal lines represent the position of Fermi energy EF for several p values corresponding to (c). For reference, the EF value when the system is half-filled is shown by the broken line. e Temperature dependence of RH normalized by RH at 180 K. f RH(T)/RH(180 K) as a function of VSG for several temperatures. Source data are provided as a Source Data file.

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To discuss the temperature dependence of RH, Fig. 2e, f plot the normalized Hall coefficient, defined by RH(T)/RH(180 K), as a function of the temperature and VSG, respectively. When metallic phases emerge upon doping holes of 10% per molecule, both conductivity and the Hall coefficient exhibit logarithmic T dependence below 10 K, explainable as weak localization including electron–electron interactions24,25,26 (see Supplementary Fig. 8). To consider Coulomb interaction screening effects in the 2D hole gas of C8-DNBDT, we estimate the average hole distance as \(r={r}_{s}\cdot {a}_{B}^{*}\) based on the effective Bohr radius \({a}_{B}^{*}=4\pi \varepsilon {\hslash }^{2}/\left({m}^{*}{e}^{2}\right)\), where the dimensionless interaction parameter \({r}_{s}\), indicating Coulomb interaction significance, is given by the ratio of the potential (Coulomb, EC) to kinetic (Fermi, EF) energy: \({r}_{s}={E}_{C}/{E}_{F}={\left({a}_{B}^{*}\sqrt{\pi p}\right)}^{-1}\). In general, when \({r}_{s}\) > 1, the Coulomb interaction is not negligible and causes logarithmic divergence of RH under cooling. Given m∗ = 1.51 m0 (m0 is the mass of free holes) as an average effective mass in the bc-plane of C8-DNBDT, the typical dielectric constant of OSCs ε = 3ε0, and the hole concentration of p = 0.4 × 1014cm−2, \({r}_{s}\) is calculated to be ~8.5, which is considerably larger than unity and comparable to values previously obtained for inorganic 2D weakly localized systems with electron–electron interaction27. \({r}_{s}\) decreases due to substantial screening, reducing the electron–electron interaction effect within the weak-localization model. Nonetheless, as shown in Fig. 2e and f, an unusual temperature dependence of RH emerges over a wide temperature range at higher |VSG|; while the T dependence of normalized RH remains logarithmic at relatively lower |VSG|(<2.8 V), additional T dependence emerges at higher temperatures around 180 K, at higher |VSG|. As the Fermi surface is expected to be simply cylindrical, originating from a single band at these doping concentrations (Fig. 2d), this highly temperature-dependent RH contradicts the rigid band model. Thus, strong electronic correlation is suspected in the highly doped organic 2D hole gas28.

Various exotic electronic phases are observed in both inorganic13,14,15,16 and organic compounds17,18,19,20 with the doping level close to that of half-filled bands. However, our present case is far from half-filling, and to our knowledge, electronic correlation effects have been seldom addressed theoretically at such low doping levels29. Since the wave function of the molecular orbital in organic semiconductors is rather extended in comparison with the d orbitals of inorganic superconductors, the on-site Coulomb repulsion is expected to be suppressed so that it can be comparable to neighbouring off-site Coulomb repulsion. Furthermore, the screening of off-site Coulomb repulsion cannot be expected for the valence band which is separated from the other bands30. It has been recognized that the off-site Coulomb repulsion induces the charge-ordered states in several materials30,31. To evaluate spin and charge ordering instabilities originating from the electronic correlation, we calculated spin and charge susceptibilities at 60 and 180 K using the fluctuation exchange approximation32,33 based on the extended Hubbard Hamiltonian including on-site and off-site Coulomb repulsions with C8-DNBDT parameters (See Supplementary Fig. 9). Charge density wave (CDW) and spin density wave (SDW) phases are defined by the divergence of charge and spin susceptibilities, respectively. These calculations suggest possible states in the phase diagram close to the CDW regime at 180 K, even at hole concentrations far from the half-filled condition, i.e., ~\(4\times {10}^{14}\) cm-2, as shown in Supplementary Fig. 10. Therefore, we can speculate that the experimentally observed anomalous features result from strongly correlated electronic states on the verge of showing a charge ordering instability. As compared to most previous studies for inorganic and organic strongly correlated electron systems, impurities are doped to introduce charge carriers, so that the dopants often induce lattice distortion, significant disorders or even other crystalline structures, making it difficult to experimentally separate the charge concentration or the lattice distortion as the driving mechanism of the charge ordering instability. On the other hand, the advantage of this study using the EDL transistors is that we widely vary the carrier concentration of the same sample, minimizing extrinsic effects. Since less than 10-nm thick C8-DNBDT is strongly laminated on the substrate, it is not likely to cause major structural change due to lattice deformation in the present devices, though the possibility of electronically driven minor lattice distortion is not completely excluded because of softness in the alkyl-chain layers.

Figure 3 illustrates the temperature dependence of Hall mobility μHall, defined by the standard definition μHall = σsheetRH. Despite the unconventional T-dependent RH, a semi-quantitative discussion of μHall is presented; it appears to increase with the decrease of temperature from approximately 7 cm2 V−1 s−1 at 180 K to 20–30 cm2 V−1 s−1 at 5 K. This pattern resembles the temperature dependence in typical degenerate 2D electronic systems such as GaAs/AlGaAs devices34,35 and transition metal dichalcogenide nanosheets27, where mobility increases with temperature reduction due to inelastic scattering suppression (see Supplementary Note 1) and saturates at low temperatures dominated by elastic scattering, such as remote ionized impurity scattering. At lower |VSG|(<2.8 V) as well as higher temperature (>100 K), where p = (eRH)-1, plots of μHall vs T for sample #1 show almost p-independent (VSG-independent) behaviour. The result implies that μHall is identical to the drift mobility and the charge transport properties are dominated by the electron-phonon scatterings, which can be seen in normal metals. However, as decreasing temperature and/or increasing |VSG|, it seems that the electronic states are gradually deviating from the normal metallic states due to the electronic correlations, resulting in the complexed VSG and T dependence of μHall. Below 20 K, where negative dρsheet/dT is observed, μHall does not decrease upon cooling. Since the conductivity is usually proportional to the density of states and the mobility, the result may imply the reduction of density of states at the Fermi energy by pseudo-gap opening at a lower temperature due to evolution of electron correlations36. Further research about the correlation effects is needed to fully elucidate the detailed electronic structures of the system. In addition, the mobilities tend to slightly decrease with increasing |VSG|, suggesting a possibility that the holes attracted to the interface with an ionic liquid by applying VSG are scattered by the electronic static disorder potential induced by the ionic liquid. Estimating the static mean-free path \({l}_{e}=v{\tau }_{e}\) from μHall at the low-temperature limit (~20 cm2 V−1 s−1), \({l}_{e}\) is ~65 Å, corresponding to 10 unit cells of C8-DNBDT, thus satisfying the Mott–Ioffe–Regel criterion37.

Fig. 3: Temperature dependence of Hall mobilities μHall.
figure 3

The mobilities are plotted for various VSG. Source data to calculate the Hall mobilities are provided as a Source Data file.

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Mean free paths and phase coherence length

Figure 4a presents the magnetoconductivity (MC) \(\Delta {\sigma }_{{sheet}}\left(B\right)\equiv {\sigma }_{{sheet}}\left(B\right)-{\sigma }_{{sheet}}\left(0\right)\) in the strongly correlated 2D organic hole gas at VSG = −3.2 V (p = 1.1 × 1014 cm−2) as a function of the magnetic field B perpendicular to the conducting plane from 5 to 10 K for sample #1. We observed a significant positive MC around zero magnetic field atop a parabolic negative MC, attributed to semiclassical Lorentz contribution, possibly enhanced due to complications in momentum-dependent scattering times resulting from electronic correlation38. The positive MC aligns with the weak localization model, Hikami–Larkin–Nagaoka (HLN) formula39 (see Methods section and Supplementary Fig. 11). Assuming weak spin-orbit interaction, Bϕ, the characteristic field for phase-backscattering loss, is the sole fitting parameter, reproducing both the magnitude and curvature of the positive MC. Bϕ is related to the phase coherence length Lϕ by \({B}_{\phi }=\hslash /\left(4e{L}_{\phi }^{2}\right)\). The phase coherence length Lϕ from 5 to 10 K, represented by red circles in Fig. 4b, increases as T-0.5 with temperature decreases, exceeding 10 nm at 5 K. This temperature dependence is reported in inorganic 2D electron gas systems where electronic interactions are significant40,41,42,43.

Fig. 4: Magnetoconductivity and characteristic lengths such as mean free paths and phase coherence length.
figure 4

a Magnetoconductivity as a function of the magnetic field B perpendicular to the two-dimensional plane at VSG = −3.2 V for at several different temperature. b Temperature dependence of the phase coherence length, inelastic mean free path, and elastic mean free path at VSG = −3.2 V. Source data are provided as a Source Data file.

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The blue circles in Fig. 4b represent the inelastic mean free path \({l}_{{ie}}\equiv v{\tau }_{{ie}}\) below 10 K, calculated using the relations \({L}_{\phi }\equiv \sqrt{D{\tau }_{{ie}}}\). Additionally, the mean free path can be estimated from the Hall mobility data (Fig. 3) across the entire temperature range by assuming the Hall mobility equals to \(e\tau /{m}^{*}\). The calculation procedure is detailed in the Methods section. The resulting mean free path \(l\), shown by black circles in Fig. 4b, increases with decreasing temperature from 180 K due to inelastic scattering. The temperature dependence of lie below 10 K, derived from magnetoconductivity, smoothly connects to the lie above 50 K, obtained from Hall mobility, supporting the validity of estimating elastic and inelastic scattering times using relatively temperature-independent μHall below 10 K.

In summary, we employed the method of EDL doping to achieve a high carrier density up to 1014 cm-2 in C8-DNBDT single-crystal semiconductors with molecular layers which is characterized by the defect-free quantum-well structure. The resultant electronic system turned out to be clean as the result of the magnetoconductivity measurements so that the phase coherence length can exceed 10 nm at 5 K. This study has revealed the presence of strong electronic correlation in a simple band-insulating small-molecular organic semiconductor by employing electric double layer transistors, which enabled doping a concentration of 1014 cm-2 (~1/8 filled band), a range previously unexplored. Despite the hole density being significantly lower than that of a half-filled band, the observed conductivities and Hall coefficients indicate a substantial deviation of the electronic states from Fermi-degenerated metallic states, likely due to potential charge order instability. The unusual T dependence of RH at several doping levels implies dominant electronic correlation even at hole concentrations below 1/4 per molecule, evolving from weak-localizing behaviour with electron–electron interaction to strong electron correlation with potential charge ordering instability at increased doping levels. The narrow band characteristic of small molecular organic semiconductors suggests that even off-site Coulomb energy might compete effectively with the screening effects typically observed in Fermi-degenerated systems. Therefore, the exotic electronic phase can manifest at hole concentrations substantially different from those found in Mott insulators.

Methods

Hole distribution calculations using the Poisson–Schrödinger equation

To calculate the hole distribution in a bilayer C8-DNBDT sandwiched between an ionic liquid and a Parylene substrate, as shown in Fig. 1d, we employed a one-dimensional quantum-well model along the a-axis. In this model, the quantum well represents the DNBDT core, with a length of 16.4 Å, while a potential barrier of 5.7 eV is formed by the alkyl chain, measuring 9.6 Å in length. The thicknesses of the ionic liquid and Parylene layers are fixed at 10 Å each. The relative dielectric constants for these materials are as follows: 10.0 for the ionic liquid, 4.0 for C8-DNBDT, and 3.34 for Parylene. Upon the application of a side-gate voltage that induces a hole concentration of 0.5×1014 cm−2 in the quantum wells at 300 K, we compute the spatial distribution of holes via the shooting method based on the Poisson–Schrödinger equation.

Evaluation of mean free paths

By fitting our magnetoconductivity data (Fig. 4a) with \(\Delta {\sigma }_{{sheet}}\left(B\right)=\Delta {\sigma }_{{WL}}\left(B\right)+\Delta {\sigma }_{{Lorentz}}\left(B\right)\), we employed the HLN model as follows:

$$\Delta {\sigma }_{{WL}}\left(B\right)=-\frac{{e}^{2}}{\pi h}\left\{{ln}\left(\frac{{B}_{\phi }}{B}\right)-\psi \left(\frac{1}{2}+\frac{{B}_{\phi }}{B}\right)\right\}$$
(1)

where \(\psi\) is the digamma function. Note that this HLN formula is valid only when \({B}_{0}=\hslash /\left(4e{l}_{e}^{2}\right)\) is larger than B37. In this study, we calculated B0 from the mean free path le at low temperatures and found that it meets the criterion \({B}_{0}/B\gg 1\). Conversely, at a relatively small hole concentration of 0.62×1014 cm−2, we observed negative MC, which is explicable by the Lorentz-type behaviour; \({\sigma }_{{Lorentz}}\left(B\right)={\sigma }_{0}/\left\{1+{\left(\mu \cdot B\right)}^{2}\right\}\) (Supplementary Fig. 11). In this equation, σ0 represents the Drude conductivity. The observed Lorentz term is notably large compared with normal metals, which may be attributed to the complexity in the scattering rate of charge carriers around the Fermi surface.

Subsequently, we calculated the inelastic mean free path defined by \({l}_{{ie}}\equiv v{\tau }_{{ie}}\) using the relations \({L}_{\phi }\equiv \sqrt{D{\tau }_{{ie}}}\), where the diffusion coefficient \(D\equiv {v}^{2}{\tau }_{e}/2\), where \(v\) and \({\tau }_{e}\) are the Fermi velocity and elastic scattering time, respectively. We obtain the Fermi velocity and effective mass from the derivative of valence band structure, i.e., \(v=\partial E\left(k\right)/\hslash \partial k\) and \(1/{m}^{*}=\left(1/{\hslash }^{2}\right)\left({\partial }^{2}E\left(k\right)/\partial {k}^{2}\right)\), using the density functional theory calculations. Assuming that the observed Hall mobility is equal to \(e\tau /{m}^{*}\), the scattering time τ can be evaluated from the data shown in Fig. 3. The obtained mean free path \(l\equiv v\tau\) is shown by black circles in Fig. 4b. l is saturated below 10 K, since the elastic scattering is predominant, resulting in the identification of \({\tau }_{e}\) with \(\tau\) at 5 K. Accordingly, we can evaluate the diffusion coefficient \(D\equiv {v}^{2}{\tau }_{e}/2\). Finally, the inelastic mean free path \({l}_{{ie}}\) is obtained, as shown by the blue circles in Fig. 4b.