Main

Conducting polymers are an example of a wider class of disordered metals that include liquid alkali metals, metallic glasses and granular metals1. Electrons in these materials are localized but correlated due to their Coulomb interactions. As a consequence, a soft Coulomb gap, that is, a suppression of the density of states (DOS) around the Fermi level EF, is expected to form and manifest itself in the charge-transport properties2. Coulomb-gap formation in conducting polymers has been inferred from simulations3,4 or from the temperature dependence of conductivity5,6, but is hard to study directly in experiments as it tends to be masked by structural heterogeneity7. Another consequence of Coulomb interactions, reported in some disordered metals at low temperatures8,9, are electron glass signatures in the conductivity, which reflect long-lived, metastable conduction states that do not equilibrate on the timescale of transport measurements. Whether similar ‘glassy’ transport signatures can be observed in conducting polymers has not yet been studied.

Polymers offer two key advantages for a better fundamental understanding of this correlated electron-transport regime at high carrier density: the ability to tune the carrier concentration over a wide range and the ease of integration into devices. In electrolyte-gated organic electrochemical transistors (OECTs), defined doping states can be dialled in by applying an ion gate voltage, VIG; charge transport at this doping level can then be studied using different techniques10,11,12,13. There have been recent reports of p-type14 and n-type OECTs4 in which the valence or conduction bands (derived from the highest (lowest) (un)occupied molecular orbital, HOMO (LUMO)) can be completely emptied (filled). However, in most materials such extreme band filling is limited by electrochemical degradation15. Here we show that in some p-type, donor–acceptor polymer-based OECTs, band filling can stably be driven even further to access deeper, HOMO−1-derived bands that cannot be realized in covalently bonded inorganic materials without inducing structural collapse. We also demonstrate that by adding a second, field-effect gate electrode to the OECT ion gate, non-equilibrium states can be induced in the coupled system of electrons and ions, in which the counterions are unable to rearrange in response to field-effect-induced changes in electron density. This approach enables us to study the correlated electron–ion transport physics of conducting polymers over a very wide range of doping levels.

Extreme band filling in polymer OECTs

Our OECTs (Fig. 1a) comprise large side gates to induce ion injection from a solid, non-aqueous ion gel based on 1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide (BMP TFSI)/poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) (Fig. 1c)16 on top of the polymer channel (Supplementary Note 1) and multi-functional electrodes for accurate measurements of four-point-probe conductivity σ and the Seebeck coefficient S (Supplementary Note 2). We focus on three representative widely studied polymers: indacenodithiophene-co-benzothiadiazole (IDT-BT), a donor–acceptor copolymer with high carrier mobilities in field-effect transistors (FETs) due to a low degree of energetic disorder17, despite exhibiting a high paracrystallinity (g ≈ 25%) and only local chain alignment;18 poly(2,5-bis(3-dodecylthiophen-2-yl)thieno(3,2-b)thiophene) (PBTTT), a polythiophene-based semicrystalline polymer that can be doped to conductivities of σ > 1,000 S cm−1;19 and poly[[2,5-bis(2-octadecyl)-2,3,5,6-tetrahydro-3,6-diketopyrrolo[3,4-c]pyrrole-1,4-diyl]-alt-(2-octylnonyl)-2,1,3-benzotriazole] (DPP-BTz), a semicrystalline copolymer20 with σ = 200–300 S cm−1. We focus on IDT-BT in the main text and discuss the other polymers in Supplementary Notes 3–6.

Fig. 1: Observation of deep band-filling behaviour in IDT-BT OECTs.
figure 1

a, Device structure. b, Transfer curve for an IDT-BT OECT, taken at room temperature with a drain voltage VD of −0.1 V. The dashed grey line indicates the gate leakage current and the arrow indicates the voltage sweep direction. n represents the number of dopant ion per polymer repeat unit. c, Chemical structures of IDT-BT, BMP TFSI and PVDF-HFP. n, x  and y represent the number of IDT-BT, PVDF, and HFP repeat units, respectively. d, Temperature dependence of the Seebeck coefficient S of IDT-BT at various doping levels. These doping levels are shown schematically on an OECT transfer curve in the inset. e, Seebeck–conductivity plot at 200 K for the data shown in d. The arrows denote the direction of increasing doping level for a given regime. In d and e, the data are presented as the mean Seebeck coefficient ± standard error of the mean, originating from fitting uncertainties of the on-chip thermometer calibration and the thermovoltage versus temperature difference plots. Lines are included as a guide to the eye.

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IDT-BT shows unusual OECT behaviour not reported previously: a peak in the drain current ID around VIG = −0.6 V is followed by a drop with increasingly negative VIG. However, rather than reaching the insulating state seen in other polymers4,14,21, we observe a second current rise when VIG < −1 V (Fig. 1b). We refer to three different transport regimes (colour-coded in Fig. 1 and subsequent figures): Regime I at low doping levels up to the ID peak (blue); Regime II between the peak and the valley state (green); and Regime III beyond the valley (red). Despite the high FET mobilities of IDT-BT, room-temperature conductivities at the peak are only 5–10 S cm−1, but in Regime III they can exceed 30 S cm−1.

We measured the temperature dependence of σ and S as a function of VIG. The conductivity σ has a non-metallic temperature dependence in all three regimes and reduces strongly with decreasing temperature (Supplementary Note 3.2). However, S varies distinctly across the regimes, and provides more direct insight into the electronic structure. In Regime I, S is positive; when approaching the peak, it reduces in magnitude and approaches a linear temperature dependence characteristic of metallic transport (Fig. 1d). This is consistent with theoretical predictions of a divergence in the dielectric constant, leading to an insulator–metal transition at relatively low doping levels (n ≈ 0.1)22. In Regime II, S switches from p-type to n-type, increases in magnitude with increasing ∣VIG∣ and becomes nearly independent of temperature between 100 and 250 K. When sweeping VIG across the valley, S switches sign again from a large negative to a large positive value. In Regime III, S varies again linearly with temperature but is approximately threefold larger at a comparable p-type conductivity than in Regime I (Fig. 1e).

X-ray photoemission spectroscopy and gate current charging analysis show that the peak of conductivity corresponds to one dopant per polymer repeat unit (n = 1), that is, the HOMO-derived band is half-filled, whereas in Regime III the doping concentration exceeds n = 2 (Extended Data Fig. 1 and Supplementary Notes 3.4 and 3.5). Our observations are consistent with a simple band-filling interpretation, which assumes that in Regime III, holes are induced in deeper-lying states derived from HOMO−1: in Regimes I and III, S is determined by holes at the top of the HOMO- and HOMO−1-derived bands, respectively, whereas in Regime II, electrons remaining at the bottom of the HOMO-derived band determine the sign of S. At the transition between Regimes II and III, S is governed by the competition between electrons in the HOMO and holes in the HOMO−1-derived bands—causing S to be near zero in the valley but to take on a large value on either side. Complementary measurements presented in Extended Data Figs. 2–5 are fully consistent with this band-filling interpretation. We considered alternative mechanisms for the non-monotonic OECT and Seebeck behaviour, including the opening of a Coulomb gap around the Fermi level due to electron–electron repulsion4, changes in structural ordering upon doping and a previously reported mechanism that involves the coexistence of crystalline and amorphous domains;23 however, these did not provide a consistent explanation for our observations (Supplementary Notes 3 and 4).

Regime III operation was also observed in several other polymers (Extended Data Fig. 6a–c) and in IDT-BT OECTs gated with different ionic liquid anions (Extended Data Fig. 6d–f). We also observe consistent band-filling behaviour in PBTTT and DPP-BTz (Extended Data Fig. 7 and Supplementary Note 3.1), but with a smaller range of accessible doping levels: all three polymers exhibit a positive S in Regime I and approach a conductivity maximum near half-band filling (n = 1). In PBTTT, Regime II is not accessible. In DPP-BTz, we observe inversion to a negative S in Regime II, as in IDT-BT; however, a fully insulating state is reached, that is, Regime III is inaccessible.

Operando grazing-incidence wide-angle X-ray scattering (GIWAXS) (Supplementary Note 4) shows that microstructural changes upon ion incorporation play a major role in determining accessible doping ranges. In IDT-BT, doping generates only a minor reversible increase in crystallinity that is uncorrelated with conductivity. PBTTT, by contrast, forms a highly ordered co-crystal with one ion per monomer, the stability of which prevents the incorporation of further ions (Supplementary Note 4.3). DPP-BTz shows a partially irreversible reduction in crystallinity at high doping levels (Supplementary Note 4.2) that limits operation at very high doping levels. The absence of Regime III in DPP-BTz can also be rationalized via density functional theory (DFT) calculations performed on single, neutral polymer chains, which show that the second occupied band is expected to be energetically more accessible in IDT-BT than in DPP-BTz (Extended Data Fig. 8 and Supplementary Note 5). We warn, however, that these calculations neglect Coulomb interactions between charge carriers and with the counterions that are important in highly doped systems24,25. A more realistic electronic structure model, fully accounting for Coulomb interactions, will be provided below.

Probing non-equilibrium states via double gating

Whereas electrochemical doping enables EF to be shifted deep into the HOMO/HOMO−1 band, the potential formation of a Coulomb gap at these ultrahigh carrier concentrations cannot be inferred unambiguously from OECT characteristics as the temperature scaling of conductivity varies with the doping level (Supplementary Note 3.2). To probe DOS changes more directly, we introduce a new double-gating technique that incorporates both a field-effect gate and an ion gate (Fig. 2a). First, we set the bulk doping level by applying VIG and cooling the device with VIG held on. Below the glass transition temperature of the electrolyte (~200 K), VIG can be removed and the field-effect gate voltage (VFG) can be swept to record field-effect transfer curves without affecting the film doping level. This experiment probes the transport of additional electronic carriers injected into the DOS at a bulk doping state defined by VIG, without being charge compensated by extra counterions. By choosing a positive or negative VFG, we can inject either additional electrons or holes.

Fig. 2: Non-equilibrium transport signatures in IDT-BT double-gated transistors.
figure 2

a, Schematic of the double-gated transistor experiment. b, Electrical conductivity at VFG = 0 V of various doping states at 160 K. Insets show field-effect transfer curves for six representative doping states at 160 K. As the film conductivity changes by orders of magnitude as the doping level is increased, the drain current is normalized as (ID(VFG) − Imin)/Imin, where Imin is the minimum current in the field-effect transfer curve of the corresponding doping state. For each inset the maximum field-effect modulation is indicated as a percentage value. At the conductivity peak, the current decreases when changing VIG but increases when changing VFG. c, Field-effect transfer curves for an ex-situ-doped IDT-BT device between 300 and 220 K, exhibiting non-linear behaviour at low temperatures. The transfer curves are vertically shifted for clarity. The magnitude of the arrow represents 0.4 μS. The inset shows a schematic of the decomposition of the 220 K transfer curve into a linear antisymmetric component and a symmetric component. d, 19F nuclear spin–lattice relaxation time (T1) and spin–spin relaxation time (T2) versus temperature (T) obtained from 19F NMR saturation recovery and spin-echo delay experiments of IDT-BT doped with TFSI. The inset shows the extracted correlation times (τc) from a two-component fit described in Supplementary Note 7.

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An important practical requirement for such double-gating experiments is minimizing the polymer thickness. In undoped FETs, only a region of ~1 nm thick near the polymer/SiO2 interface is affected by VFG; in our highly doped devices, the expected doping-induced increase in dielectric constant22 should further limit the field effect to the first polymer monolayer near the interface. Even for the thinnest practically achievable polymer films (~10 nm), VFG only injects the equivalent of a bulk charge concentration of ~1019 cm−3, which is much smaller than the VIG-induced bulk doping levels (~1021 cm−3). The expected, VFG-induced current modulation is therefore at most a few per cent.

Figure 2b shows six selected field-effect transfer curves (insets) measured at different doping states along the IDT-BT OECT transfer curve (main panel; full dataset in Supplementary Fig. 26). The experimental data disclose several surprising observations. First, the field-effect modulation is much larger than expected, varying from 12% near the peak to 297% near the valley. Second, we had expected the field-effect response to follow the OECT ion gate transfer characteristics—to be unipolar p-type in Regimes I and III and n-type in Regime II. At the peak (doping state 4), we had expected a current reduction for both positive and negative VFG. Instead, we observe a current increase in both directions. At most of the other doping states, a similar ‘graphene like’ ambipolar current increase is observed, although the current rise for negative VFG is stronger than for positive VFG in Regimes I and III and weaker in Regime II. Note that at the highest doping level (state 14), the field-effect transfer characteristics resemble those at the first peak (state 4), suggesting that we may be approaching half-filling of the HOMO−1 band. This behaviour was found to be general and was also observed in DPP-BTz and PBTTT (Extended Data Fig. 9 and Supplementary Note 6).

To analyse the non-linear transfer curves, we decompose them into a linear component that is antisymmetric with respect to VFG = 0 (that is, an odd function: ID( − VFG) − ID(VFG = 0) = −(ID(VFG) − ID(VFG = 0))) and a non-linear component that is symmetric with respect to VFG = 0 (that is, an even function: ID( − VFG) − ID(VFG = 0) = ID(VFG) − ID(VFG = 0)) (inset of Fig. 2c). The linear, antisymmetric component follows the behaviour expected from the ion gate transfer characteristics and the Seebeck coefficient, and is believed to be determined by the slope of the DOS at EF: in Regimes I and III, the DOS rises as EF moves towards the centre of the HOMO/HOMO−1-derived band, resulting in a p-type linear component; and in Regime II, the DOS falls as EF is pushed beyond the centre of the HOMO-derived band, resulting in an n-type linear component. At half-filling of the HOMO-derived band (doping state 4) the slope of the DOS at EF vanishes, and the linear component disappears.

The symmetric component, on the other hand, will be shown below to probe a Coulomb gap at EF that is superimposed on the DOS. We can understand not only the ambipolar nature but also the unexpectedly strong field-effect response if we assume that the additional, VFG-induced carriers are able to access states at the edges of the Coulomb gap that have a higher DOS and/or higher degree of charge delocalization than states at the centre of the gap. This will allow the additional charge carriers, in essence, to move more quickly than the carriers induced by doping (Extended Data Fig. 10).

The observed ambipolar signatures are reminiscent of low-temperature (<4 K) field-effect gating experiments on other disordered metals, such as granular or oxide metals9,26,27. One explanation proposed for these is slow electron relaxation in the presence of a Coulomb gap caused by electron–electron interactions28. However, there are important differences—here, we observe these signatures at much higher temperatures and more stably, even when measuring very slowly or applying a field-effect bias stress for up to one day (Supplementary Note 6); we do not observe, for example, so-called double-dip signatures under similar measurement conditions26. This makes it unlikely that glassy electron relaxation, that is, the inability of the additional carriers to equilibrate with the doping-induced electronic carriers on the measurement timescale, is responsible in our system, as this would be expected to be fast at such high temperatures—that is, a soft Coulomb gap will remain tied to the shifting Fermi level when VFG is changed29,30.

To better understand the origin of the non-linear symmetric component, we investigated its temperature dependence. Non-linear field-effect transfer characteristics were observed over a wide temperature range below 190 K (Supplementary Note 6.1). Measurements at higher temperatures are not possible on OECTs (as ions move in and out of the film in response to VFG), so films were doped ex situ via ion-exchange doping or by washing off the ionic liquid after electrochemical doping. The change in conductivity Δσ = σ − σ0 (where σ0 = σ(VFG = 0)) is plotted versus VFG at different temperatures in Fig. 2c (raw data in Supplementary Fig. 28). Strikingly, at 300 K, the transfer curve of this mildly doped IDT-BT film in Regime I is unipolar p-type and perfectly linear. The non-linearity reproducibly appears only below ~240 K, which is close to the melting point of the ionic liquid (255 K)31 and the β-glass transition temperature of IDT-BT (249 K)32 associated with side-chain motion. This suggests that the appearance of the non-linear component is linked to the relaxation dynamics of the dopant counterions rather than the electrons.

To study the ion dynamics we measured the 19F nuclear spin–lattice and spin–spin relaxation times T1 and T2, respectively, of the TFSI ions in an IDT-BT film (Fig. 2d) using NMR. Nuclear spin relaxation is driven by fluctuating fields associated with molecular motions, enabling us to determine the correlation times of these motions. T1 decreases monotonically with increasing temperature, whereas T2 exhibits three different temperature regimes. Above 255 K and below 215 K, T2 increases with increasing temperature, indicating motional narrowing regimes corresponding to two different types of molecular motion that are fast enough to average effectively over local magnetic environments. The intermediate regime (215–255 K) reflects a slowing of the high-temperature motion, which causes motional narrowing to become ineffective. A fit to Redfield theory (Supplementary Note 7) yields correlation times for both motions. The faster, low-temperature-dominating motion with an activation energy (Ea) of ~16 meV is probably associated with an intra-ionic motion, such as functional group rotations. The slower, high-temperature-dominating motion has a much higher activation energy (574 meV), which is typical for translational ion motion in polymers33,34, and slows by more than five orders of magnitude in the temperature range where the non-linearity appears.

This suggests that the unusual field-effect response is caused by the slowing of translational ion motion as the polymer approaches its glass transition. At room temperature, ionic motion is fast enough for ions to adjust to changes in carrier concentration and find their lowest energy equilibrium configuration as VFG is swept. At 260 K, the timescale of ionic relaxation becomes comparable to the VFG sweep rate, which manifests itself as hysteresis in the transfer curves (Fig. 2c). Below 240 K the ions can no longer respond to VFG-induced changes in carrier concentration and remain frozen in configurations they adopted before cooling. If instead a non-zero VFG is applied to perturb this configuration during cooling, the system ‘memorizes’ this high-temperature equilibrium configuration, and the minimum of the ambipolar transfer characteristics at low temperature shifts to the VFG value applied during cooling (Supplementary Fig. 30).

Simulating non-equilibrium transport

Theoretical simulations were performed using a two-dimensional Hubbard-type model that we recently applied to study the interplay between Coulomb interactions and disorder19. The model has been generalized to enable the tuning of carrier concentrations with and without introducing charge-neutralizing counterions, thus mimicking electrochemical doping and the application of a field-effect bias, respectively. The simulations consider either equilibrium conditions, where ion positions are relaxed upon applying a gate bias, or non-equilibrium conditions, where ion positions are frozen (Supplementary Note 8). The field-effect transfer curves simulated under frozen-ion, non-equilibrium conditions are in good agreement with the experiments. For doping levels around half-filling (Fig. 3b, 1.2 ions per monomer) the conductivity, presented as its relative change Δσ/σ, is enhanced symmetrically when adding or removing additional carriers, Δn. When moving away from half-filling (Fig. 3a,c, for 0.4 and 1.6 ions per monomer, respectively), asymmetric curves are observed, agreeing with our experiments. This ambipolarity is absent in equilibrium simulations (Fig. 3a–c).

Fig. 3: Theoretical modelling of non-equilibrium transport.
figure 3

a–c, Transfer curves calculated in and out of equilibrium for systems at electrochemical doping levels of 0.4 (a), 1.2 (b) and 1.6 ions per monomer (c). The effect of the field-effect gate bias is modelled as a relative change in the charge density, Δn/n0, where n0 is the number of electrons in a full band. Data are presented as the mean of 400 realizations ± standard error of the mean. d, DOS in the absence of field-effect gating, with and without ionic relaxation at a doping level of 1.2 ions per monomer. e-e interaction, electron–electron interaction. e, DOS around EF for positive and negative gate bias under out-of-equilibrium conditions. The arrows mark the Fermi level for different Δn/n0. f,g, Relative variation of the DOS at EF (f) and relative variation of the participation ratio L(EF) (g), measuring the degree of delocalization of the carriers. h, Schematic of the free energy landscape in equilibrium (left) and out of equilibrium (right). Equilibrium cases refer to mobile ion and/or zero-gate bias conditions, while non-equilibrium refers to the case of frozen ions with non-zero gate bias. The contributions of disorder and e-e interaction (top), ionic relaxation (middle) and their sum (bottom) are shown schematically. In the non-equilibrium condition the gate bias alters the electronic energy landscape (red arrow), but the ionic relaxation contribution remains unchanged. This leads to a smaller energy barrier (green arrow) to escape from the energy minimum.

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The ambipolarity is rooted in the formation of a soft Coulomb gap in the DOS at EF, which originates from the interactions between localized electrons in a disordered medium2,19. However, crucially, ionic relaxation deepens the Coulomb gap by enhancing disorder and charge localization (Fig. 3d)35. Under non-equilibrium conditions, EF is shifted with respect to the centre of the gap upon field-effect gating (Δn ≠ 0; Fig. 3e) because the dominant contribution to the pseudogap from the ionic reaction field is fixed, that is, frozen at its original, zero-bias position. This shift increases the number of states available for transport (Fig. 3f) as well as the delocalization of these states (Fig. 3g), both effects co-operating to yield an ambipolar conductivity enhancement. Under equilibrium conditions, on the other hand, EF remains pinned at the DOS minimum (Supplementary Fig. 42), resulting in flat transfer curves.

To demonstrate the proposed mechanism Fig. 3h illustrates the free energy landscape in configuration space as determined by disorder and electron–electron interactions. The ion relaxation at zero gate bias deepens the potential wells and enhances state localization. The application of a gate bias with frozen ions alters the electronic contribution to the energy landscape but leaves the ionic contribution unchanged. This lowers the activation barriers (green arrows) for carriers to move, rationalizing the ambipolar conductivity increase.

Spectroscopic evidence for enhanced delocalization

To validate our theoretical prediction of enhanced state delocalization at EF upon field-effect gating (Fig. 3g), experimentally we used charge modulation spectroscopy (CMS), which measures the differential optical absorption (ΔAbs) as VFG is modulated. This provides direct spectroscopic characterization of the additional carriers;36 a redshifted infrared polaron-induced absorption (0.1–0.5 eV) would indicate increased polaron delocalization37.

At 270 K, (Fig. 4a) a negative VFG leads to increased polaronic absorption, whereas a positive VFG produces a corresponding bleaching signal without spectral changes (apart from an absorption tail below 0.1 eV from the CMS response of the silicon substrate). At 190 K (Fig. 4b) the CMS spectrum becomes more complex; the induced absorption and bleaching spectra at negative and positive VFG values, respectively, no longer have the same shape. To interpret these spectra, we mirror the analysis of the electrical characteristics and decompose the measured absorption A(ω) into component spectra c(ω) that are symmetric and antisymmetric with respect to VFG, that is A(ω) = ca(ω)VFG + cs(ω)∣VFG∣. A least-squares fit at each frequency provides a good match with the experimental data (Supplementary Note 9).

Fig. 4: Spectroscopic evidence for enhanced delocalization of non-equilibrium states.
figure 4

a,b, CMS spectra of an IDT-BT:TFSI film doped to Regime I at 270 K (a) and 190 K (b). The bottom left insets show the transfer curves at each temperature. The symbols show the experimental data, and the solid lines are fits from the decomposed symmetric and antisymmetric component spectra as described in the text. OD, optical density. c, Antisymmetric (top) and symmetric (bottom) component spectra used to fit the data. The antisymmetric component (blue line, 270 K data shown) matches closely the OECT moving difference spectrum (green line) calculated from the data in Extended Data Fig. 4. The symmetric component (red line, difference between 190 K and 270 K data shown) is nearly independent of the doping level and reflects an increase in carrier delocalization upon field-effect gating below the ionic glass transition temperature.

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At 270 K, the antisymmetric component dominates (blue line in Fig. 4c) and matches closely the corresponding moving difference spectrum obtained from the OECT data in Extended Data Fig. 4 (green line in Fig. 4c). This indicates that above the transition temperature the polarons induced by ion or field-effect gating are indistinguishable spectroscopically and have the same degree of polaron delocalization. This is consistent with our previous work, showing that at such high doping levels the ions act as a uniform background charge density19.

Below the transition temperature (190 K data), whereas the antisymmetric component remains qualitatively unchanged (Supplementary Note 9), a pronounced symmetric component emerges. The response of the silicon appears in the symmetric component as well, but this can be removed by subtracting the 270 K symmetric component. The resulting subtracted symmetric component (Fig. 4c, red line) is remarkably consistent in spectral shape and intensity across a range of doping levels (Supplementary Note 9) and shows a broad negative, bleaching feature at high energy and a positive, induced absorption at low energy, that integrate to near zero absorption. This signal therefore does not reflect a change in carrier density but instead indicates that the polaron bands of both additional electron and hole carriers are shifted to lower energy. This shift provides clear validation that the field-induced, non-equilibrium carriers are indeed more delocalized than the ion-gate-induced equilibrium carriers, in good agreement with our theoretical model and electrical characteristics.

Enhanced transport coefficients

The ability to shift EF away from the centre of the Coulomb gap provides a route to enhancing the Seebeck coefficient. Within the Boltzmann transport framework, S is proportional to the logarithmic slope of the energy-dependent conductivity function, which is smallest at the bottom of the gap but rises as more delocalized states with a larger DOS become accessible at the edges of the gap. As a proof of principle we attempted to detect changes in the thermopower upon the field-effect gating of an IDT-BT double-gated device (Fig. 5) operating in a highly conducting Regime III state, similar to doping state 14. Indeed, when the soft gap is frozen at 190 K, a 10% larger Seebeck coefficient of 66.0 μV K−1 is recorded at VFG = −40 V, compared with the ungated value of 60.5 μV K−1. This is remarkable, as field-effect gating only affects carriers near the dielectric interface. From a parallel conduction model, we estimate that a thermopower of up to an order of magnitude larger can be achieved if EF can be similarly shifted in the bulk. For VFG = +40 V, S is reduced compared with the ungated state, due to competition between the p-type bulk response and the expected n-type field-effect response (Supplementary Note 10). These observations validate our proposed model and demonstrate that thermoelectric transport coefficients σ and S can both be substantially enhanced in the non-equilibrium regime.

Fig. 5: Enhancement of the Seebeck coefficient due to the frozen Coulomb gap.
figure 5

a,b, Field-effect transfer curve (VD = −0.1 V) of a highly doped IDT-BT device measured at 190 K (in Regime III, see inset) (a), where changes in the magnitude of the Seebeck coefficient across the gap are indicated on the transfer curve, as inferred from thermovoltage measurements at various field-effect gate voltages (b).

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Our findings open up new strategies to considerably enhance the performance of conducting polymers for applications from thermoelectrics to bioelectronics. (1) In Regime III, markedly higher conductivities and Seebeck coefficients are achievable, possibly because polarons in the HOMO−1-derived band experience a higher dielectric constant/more efficient screening of Coulomb interactions22. The hitherto neglected HOMO−1-derived band should therefore be considered when designing high-performance polymers. (2) Although non-equilibrium transport phenomena in disordered metals have been confined to very low temperatures8,9, our new mechanism involves freezing a Coulomb gap in the electronic DOS by controlling the relaxation of dopant counterions that interact Coulombically with the electrons. This mechanism enables non-equilibrium phenomena much closer to room temperature. For practical applications, freezing of the gap above room temperature should become possible using polymers and ionic liquids/salts with higher glass transition/melting temperatures. Methods to induce highly delocalized non-equilibrium carriers, not just at interfaces but also in the bulk, are needed; this may be achievable through a two-step doping protocol, with ions introduced during the first step unable to relax during the second step. During the review process of this work we have become aware that potentially related non-equilibrium states in the system of polarons/bipolarons and counterions can be observed in poly(3,4-ethylenedioxythiophene)/poly(styrene sulfonate) (PEDOT/PSS) via X-ray photon correlation spectroscopy38. Our results suggest polymers could become powerful model systems for fundamental studies of non-equilibrium transport phenomena in disordered metals.

Methods

Sample preparation

Silicon/SiOx substrates were used for the double-gated devices, and all the other devices were fabricated on Corning Eagle XG glass substrates. Electrodes were fabricated using standard photolithography, metal deposition and a lift-off process. Conjugated polymer films were spin coated using glass pipettes to minimize contamination39. All spin coating and annealing was performed in a nitrogen glovebox (<1 ppm H2O, O2) equipped with a molecular sieve solvent trap. For ex-situ-doped samples, the polymer films were doped using either ion-exchange doping or electrochemical doping in a nitrogen glovebox using a PalmSens4 potentiostat. For OECTs and double-gated devices, the polymer channels were patterned using photolithography, and the ion gel layers were spin coated to cover both the ion gate electrodes and the polymer channels. Further details (for example, spin-coating parameters for different polymers and doping procedures) can be found in Supplementary Note 1.1.

Electrical, thermoelectric and electrochemical measurements

Room-temperature OECT measurements were carried out in a nitrogen glovebox. Low-temperature OECT and double-gated measurements were performed using a cryogenic probe station (a Desert Cryogenics probe station or a LakeShore Cryotronics CRX-4K closed-cycle cryogenic probe station). Seebeck measurements were performed in a LakeShore Cryotronics CRX-4K closed-cycle cryogenic probe station. For these measurements, a Keithley 2612B source measure unit and an Agilent 4155B semiconductor parameter analyser were used for the source voltage and to measure the current, and the thermovoltage signals were measured using a Keithley 2182A nanovoltmeter or a Keithley 6430 subfemtoamp source measure unit with a remote preamplifier. Cyclic voltammetry measurements were performed with a PalmSens4 potentiostat, a Pt counter electrode and an Ag quasi-reference electrode. Polymer films served as the working electrodes and ionic liquids as the electrolyte. To estimate the conductivity at different oxidation potentials, we used the chronoamperometry and linear sweep voltammetry functions of the potentiostat: a constant voltage was applied to the polymer film for 30 s, and then a current–voltage scan was quickly carried out between two electrodes in contact with the polymer film. Further details (for example, device geometries) can be found in Supplementary Note 1.2.

Magnetic resonance spectroscopy

Continuous-wave electron spin resonance (ESR) measurements were carried out using a Bruker E500 spectrometer with an X-band microwave source and a Bruker ER 4122SHQE cavity. All measurements were carried out using an Oxford Instruments ESR900 helium cryostat. More details on our ESR set-up and analysis routines can be found in ref. 40. ESR studies at various doping levels were performed on OECTs, enabling in situ adjustment of the doping level.

NMR measurements were carried out using a Bruker 400 MHz dynamic nuclear polarization spectrometer (~9.4 T). A 3.2 mm LT MAS (low-temperature magic-angle spinning) probe was used, which could be tuned to 1H or 19F. Samples were spun at a MAS frequency of 11 kHz at all temperatures. The spin–lattice (T1) relaxation time values were obtained from saturation recovery experiments with echo detection (such a recoupling detection pulse can considerably attenuate any probe background signals). The spin–spin (T2) relaxation time values were obtained from spin-echo decay experiments. Spectral fitting and lifetime extractions were performed using the ssNAKE program41. Rotors were packed with the sample of interest (~5.35 mg of doped IDT-BT) together with the KBr reference in an argon glovebox. The sample temperature could be determined accurately from the well-established temperature dependence of the T1 of 79Br nuclei42. Accounting for possible thermal drift over the prolonged duration of signal acquisition, each nominal temperature value has been taken as an average between two measurements at the beginning and at the end of the signal acquisition period.

Photoemission spectroscopy

X-ray photoemission spectroscopy was carried out using a JEOL JPS-9030 set-up with a monochromatic Al Kα source (photon energy, hν = 1,486.6 eV). The core-level binding energy was calibrated by setting the C 1s peak of adventitious carbon to 284.8 eV. Ultraviolet photoemission spectroscopy was performed using monochromatized light from a He discharge (hν = 21.22 eV; HIS 13, Focus) and a SPECS PHOIBOS 100 hemispherical analyser. A bias of −10 V was applied between the sample and the analyser for measurement of the secondary electrons to eliminate the influence of the analyser workfunction. All photoemission spectroscopy measurements were carried out under an ultrahigh vacuum of ~10−9 mbar.

Photoemission samples were ex situ electrochemically doped and then rinsed with acetonitrile in a nitrogen glovebox to remove any residual ionic liquid and then transferred to the spectrometer using vacuum suitcases in a nitrogen atmosphere to eliminate air exposure at any point after doping. Photoemission measurements were performed within 30 min of the samples being prepared, to minimize the possibility of sample dedoping and/or degradation.

Optical spectroscopy

Optical spectroscopy measurements were performed in Oxford Instruments Optistat CF-V cryostats under a high vacuum of ~10−6 mbar. For the ultraviolet-visible-near-infrared (UV-Vis-NIR) measurements, the cryostat was outfitted with crystalline quartz windows, whereas for the Fourier transform infrared (FTIR) measurements, thallium bromoiodide (KRS-5) windows were used. UV-Vis-NIR measurements were performed using an Agilent Cary 6000i UV-Vis-NIR spectrophotometer fitted with a tungsten halogen Vis-NIR lamp and a deuterium arc UV-lamp. The NIR and infrared portion of the spectrum (<1.1 eV) was smoothed using a Savitzky–Golay filter with a filter window of 50 points. The UV-Vis-NIR spectra shown are single scans collected with an integration time of 0.025 s per point. FTIR measurements were performed using a Bruker VERTEX 70v FTIR spectrometer with a Bruker DigiTect mid-infrared DLATGS (deuterated L-alanine-doped triglycine sulfate) detector. Measurements on OECTs used an interferometer scan rate of 1.6 kHz and a resolution of 4 cm−1. Data were measured under a constant gate current and are presented as a simple moving average of nine scans. For the infrared field-effect-gated CMS measurements, an interferometer scan rate of 5 kHz and a resolution of 16 cm−1 were used for all measurements. A total of 128 scans were collected per modulation step, and 64 modulation cycles of (OFF ON ON OFF) were collected per gate voltage (8,192 scans total). The Keithley 2612B source measure unit was used to electrically gate the devices during the spectral measurements.

Grazing-incidence wide-angle X-ray scattering

GIWAXS measurements were performed at the I07 Surface and Interface Diffraction beamline of Diamond Light Source using a photon energy of hν = 12.5 keV; images were collected using a PILATUS 2M detector. We used a smaller incidence angle of 0.15° for the operando OECT measurements to maximize the signal, whereas a slightly larger angle of 0.2° was used for film samples. Electrical measurements of the device were carried out using the Keithley 2612B source measure unit. The device was enclosed in a helium atmosphere to minimize sample degradation and scattering from the ambient gaseous species. Beam damage was characterized by changes in the source–drain current after each X-ray exposure. To minimize this degradation, we reduced the X-ray flux as much as practically possible while maintaining a sufficient signal-to-noise ratio.

Modelling

Periodic DFT simulations were performed to compute the band structure of pristine polymers using the all-electron atomic orbital formalism implemented in the CRYSTAL17 code43. Calculations were performed at the PBE0/def2-SVP level of theory44 for a single, periodic one-dimensional polymer chain with a 6 × 1 × 1 sampling of the Brillouin zone. Further details of the DFT simulations can be found in Supplementary Note 5.

The electronic structure of doped polymers was modelled with a two-dimensional extended Hubbard-type model with long-range Coulomb interactions, solved in the Hartree–Fock approximation19. Calculations were performed on large supercells of an anisotropic square lattice. Model parameters are representative of the IDT-BT polymer (see Supplementary Note 8 for details). The conductivity was calculated within the framework of transient localization theory19,45. Results were obtained as an average over 400 realizations of the energetic disorder. Error bars in the plots correspond to the standard error of the mean.