Depletion Type Organic Electrochemical Transistors and the Gradual Channel Approximation

The gradual channel approximation forms the foundation for the analysis of field‐effect transistors. It has been used to discuss transistors that are not necessarily based on the field‐effect as well, such as the organic electrochemical transistor (OECT). Here, the applicability of the gradual channel approximation for OECTs is studied by a 2D drift‐diffusion model. It is found that OECT switching can be described by two separate effects—a doping/dedoping mechanism and the formation of an electrostatic double layer at the interface between the mixed conductor and the electrolyte. The balance between these two mechanisms is determined by the morphology of the mixed conductor, in particular the question if ions move in the same phase and electric potential as the holes, or if separate ion and hole phases are formed. It is argued that the gradual channel approximation can only be used to describe electrostatic switching at the mixed conductor/electrolyte interface (the two‐phase model), but cannot be employed to analyze devices operating on a doping/de‐doping mechanism (the one‐phase model).


Introduction
The gradual channel approximation forms the basis for the analysis of field-effect transistors.It was first proposed by Shockley in his seminal paper from 1952 [1] for a device that would nowadays be called a junction field effect transistor.Later on, the gradual channel was used to discuss the behavior of MOSFETs. [2]he gradual channel approximation greatly simplifies the 2D problem of a field-effect transistor, in particular when solving the In its original formulation, Shockley proposed that the curvature of the electric potential perpendicular to the channel (e.g., d 2  dy 2 , assuming that the channel extends along the x-axis) is small compared to the charge density (x,y)  , which is equivalent to [3,4] d 2  dx 2 ≪ d 2  dy 2 (2) With this approximation, Poisson's equation reduces to the 1D problem  .For a standard MOSFET geometry, it can be solved analytically, which results in a surface charge density induced at the insulator/semiconductor interface  p (x) that is proportional to the applied gate voltage, that is, Here, C G =  d ox is the gate capacitance per unit area, d ox and ϵ are the thickness and dielectric constant of the gate oxide, and ϕ(x) is the potential at position x within the channel.
Under the assumption of Equation (2),  p will be independent of the lateral field dx created by the drain potential V D .Therefore, Equation (3) can be used to calculate the drain current I D in a channel with width W as dx (: charge mobility), which leads, after integration along the transistor channel, to the known dependencies of the drain current on the gate and drain potential of fieldeffect transistors. [4]lthough MOSFETs have been the dominating transistor technology that is unrivaled in terms of integration density and computational complexity, other technologies have emerged for applications, where MOSFETs are not suitable.[14][15][16] Overall, these properties make OECTs highly suited for bio-electronic applications. [17,18]The geometry of an OECT is sketched in Figure 1.The conductance of a thin film of an organic semiconductor is controlled by the injection of cations from an electrolyte, which in turn is modulated by a gate electrode.Often, the organic semiconductor PE-DOT:PSS is used, which is highly p-doped and hence shows high conductivity in its initial state.During operation, cations are injected, which de-dope the transistor and turn it off.
Following the success of the gradual channel approximation, it was used to describe OECTs as well. [19,20]Similar to the treatment of MOSFETs, gate capacitance, that is, a capacitance formed between the electrolyte and the PEDOT:PSS channel, was defined as the ratio of the density of injected cations p ion to the potential drop between gate and channel (V G − ϕ(x)).In its initial formulation, a capacitance per unit area as for MOSFETs was proposed. [19]ater on, it was found that the gate capacitance C G scales with the whole volume of the semiconducting layer, leading to the introduction of the volumetric capacitance C*. [17] C The assumption of a fixed gate capacitance C* implicitly invokes the gradual channel approximation, that is, Equation (2)  has to be valid.However, by the definition of a gate capacitance, the OECT could be treated analytically and it was shown that the drain current I D depends on the gate voltage V G as: [17,19] where q is the elementary charge, T the channel thickness, L the channel length, and p 0 is the effective doping concentration (which is assumed as proportional to the density of PSS − ions in the PEDOT:PSS film).The term C * is usually referred to as the pinch-off voltage, that is the voltage that has to be applied between gate and drain electrode (i.e., V G − V D ) to completely de-dope the channel at the drain.Once the channel is de-doped at the drain (i.e., at V G − V D = V P ), the drain current saturates at leading to the transconductance of that is, the transconductance is supposed to scale linearly with the gate voltage.Although this approach was highly successful in discussing general trends in OECT behavior, several non-idealities were found.For example, Figure 2a plots the transfer characteristic and transconductance of a representative OECT (W = 50 μm, L = 200 μm).From Equation ( 8), a continuous decrease of the transconductance g m with an applied gate voltage V G is expected.In the experiment, however, the transconductance displays a distinct maximum at about V G = 0.4V.This peaking transconductance is often observed in depletion type OECTs [8,20] and has been discussed by us in terms of a contact limitation [21] or was proposed to be caused by disorder incurred by hopping transport in mixed semiconductors. [22]Similarly, an absence of an accumulation region in PEDOT:PSS based OECTs, [23,24] that is, the assumption that injection of negative ions into the PEDOT:PSS layer does not (significantly) increase the hole concentration, [25] would lead to a drop in transconductance at smaller positive or even negative gate voltages.
The complex shape of the transfer characteristic shown in Figure 2a leads to further challenges in the discussion of OECTs.In Figure 2b the square root of the drain current √ I D is plotted versus the gate voltage.Following Equation ( 7), the slope of this plot should be proportional to the charge mobility and the pinchoff voltage can be extracted as the crossing point of a linear fit of √ I D and the voltage axis.However, Figure 2b shows that although the plot of √ I D versus V G can be fitted locally by a linear function, the overall results will be highly influenced by the choice of fitting range.
These challenges of fitting the transfer characteristic of depletion type OECTs are well known, [26] and usually more complex, transient methods are employed to extract the charge mobility in OECTs. [27]Oftentimes, the mobility is not extracted individually, but the product of the mobility and the capacitance μC* is quoted as a figure of merit to compare materials performance in OECTs. [28]To extract the μC* product, the transconductance g m is plotted versus the geometric factor WT L or the geometric factor normalized by the applied overpotential WT L (V p − V G ).In line with Equation (8), a linear dependency is often observed [28] with a slope that is proportional to the μC* product.
However, even this analysis is facing challenges.Equation ( 8) shows that a direct proportionality is only observed if g m is analyzed at a constant overpotential V P − V G .Determining the pinchoff voltage is limited by the problems discussed above, that is, the fact that the transfer characteristic cannot be perfectly fitted by Equation (7) (i.e., √ I D,sat , cf. Figure 2b).Furthermore, the pinch-off voltage itself depends on the gate capacitance C*, making the extraction of μC* difficult.To somehow circumvent these problems, the maximum transconductance (i.e., the peak in transconductance as seen in Figure 2a) can be plotted instead of the transconductance at constant overpotential, which, however, can only be seen as a rough approximation.
Another challenge of this analysis is displayed by the experimental results plotted in Figure 3a.Here, a series of OECTs with channel lengths of 100 μm to 1 mm and widths of 50 to 600 μm were measured.Although the maximum transconductance scales linearly at low WT L , there is a slight saturation at larger WT L .This saturation is mainly caused by a saturation at shorter channel lengths L, that is, scaling with the channel width W is perfectly linear, as shown in Figure 3b.
This saturation in transconductance was explained by a parasitic resistance in series with the OECT channel, [20] either caused by a contact resistance at the source and drain electrode, [24] or by resistances of the contact line in vertical OECTs [29] that drive extremely large drain currents.Figure 4 plots a characteristic contact resistance measured at varying gate potentials V G .[32] However, neither has the origin of the contact resistance been clarified, nor is the significance of contact resistance for OECTs generally accepted. [20,22]hese non-idealities in the electric response of depletion type OECTs are well known and have already been discussed in the literature.Procedures have been found to account for the intricate behavior of OECTs and to discuss trends in their behavior. [26]Here, however, we consider the question of whether these challenges are not mere deviations from an ideal system, but are caused by the fact that the initial assumption of standard thin film theory-the gradual channel approximation-is not justified for OECTs.We will use a 2D simulation to discuss two limiting cases of OECT-a one-phase and a two-phase model, and will discuss the applicability of the gradual channel approximation for OECTs.We will argue that the gradual channel approximation is indeed not valid for a volumetric doping/de-doping mechanism, which is dominant in a onephase system, but that a two-phase system can be described by the approximation more safely.Distinguishing between the two regimes from experimental data is challenging, and it is argued that in real devices both regimes are in effect with varying weights.

Modeling Results
To test if the gradual channel approximation can be used for organic electrochemical transistors, a 2D drift-diffusion model of an OECT as presented previously is used. [33]In the model, the continuity equations of holes and cations are solved alongside Poisson's equation.Source and drain electrodes are modeled as ohmic for holes and as ideally reflecting for cations.The cation concentration at the gate electrode is set to its equilibrium condition, which effectively sets the voltage drop at the gate interface to zero.This is a good approximation for a macroscopic gate electrode that is much larger than the transistor channel.
We will focus the discussion on depletion type OECTs, such as PEDOT:PSS based OECTs, as this material is commercially available and one of the most heavily used materials. [7]Two different cases are compared.At first, it is assumed that the mixed conductor (the PEDOT:PSS layer), transports holes and ions in the same phase, that is, the mixed conductor is represented by a homogeneous film with a certain ion and hole mobility, and both species experience the same electric field (Section 2.1).
Later on, a two-phase model is considered, where ions and holes move in separate phases, for example, PSS as ion conductor and PEDOT as hole conductor (Section 2.2).The two phases will feature separate electric potentials.

The Gradual Channel Approximation in a One-Phase System
The setup of the one-phase OECT is shown in Figure 5.The model consists of two layers, the mixed conductor on the bottom (transporting both, holes and cations), and the electrolyte layer on the top (only conducting cations).The potential of the electrolyte is controlled by the gate electrode on top of the device, whereas holes are injected into the mixed semiconductor at the source electrode on the bottom left, and extracted at the drain on the bottom right.The interface between the two layers is permeable for ions, but ideally reflecting for holes.The source and drain electrodes are assumed to be ohmic for holes (the boundary condition for holes p inj is set to the effective doping concen-tration p 0 ,) and ideally reflecting for ions.The gate electrode is treated as ohmic for ions, that is, the concentration of cations is set to the concentration of anions.Only the movement of cations is considered, anions are approximated to be stationary.Previously, we were able to show that this assumption is well justified and leads to a straightforward explanation for the peak in transconductance. [24]lthough there are reports on a charge carrier dependent hole mobility, [22,34] a constant charge carrier mobility is assumed here, which was shown to lead to qualitatively correct results. [24]ll quantities are normalized to increase convergence of the calculation.In particular, the voltages are normalized to the thermal voltage V T = k B T q (k B : Boltzmann constant, T: temperature).We are neglecting structural inhomogeneity, which is known to exist in PEDOT:PSS based devices. [35]Furthermore, we assume that the diffusion constant of the ions is constant and only one ionic species is treated.Still, as we will show in the following, the model is sufficient for discussing qualitative trends.

Working Mechanism -Double Layer versus Bulk Doping/De-Doping
The transfer characteristics obtained by the one-phase model are shown in Figure 6.The parameters used to obtain this result are summarized in Table 1.The transfer characteristic shown in Figure 6a shows a pronounced depletion region at positive voltages marked by a sharp drop in drain current, and a constant current at negative voltages.This constant current at negative voltages can be explained by the absence of an accumulation region.At negative voltages, all cations are removed from the mixed layer and no de-doping occurs.However, anions are not accumulated inside the device (they are treated as immobile), which would otherwise increase the current further. [24]This mechanism explains the peaking transconductance curve shown in Figure 6c, that is, the peak marks the transition between (absent or weak) accumulation and depletion.Finally, the plot of √ I D versus the gate current shown in Figure 6b re-iterates the challenges of fitting the transfer curve by Equation ( 7), as already observed in the experimental results.Although the plot could be fitted by a piecewise linear function, this fit will be highly dependent on the chosen fitting ranges.
To discuss if the gradual channel approximation can nevertheless be used to describe the device, its working mechanism will be discussed with the help of a short channel device with model parameters as summarized in Table 1.The electric potential inside the device at a gate voltage of V G = 5.8V T and a drain potential of V D = −2V T is shown in Figure 7a.Whereas the potential in the electrolyte approximately equals the gate potential, the potential inside the mixed conductor is lower and closer to the source and drain potential.Due to the small drain potential, the potential is almost constant along the horizontal channel direction (i.e., the x-direction).Most importantly, however, the potential inside the mixed semiconductor seems to be almost constant along the vertical direction (along the y-axis) as well.
This observation is confirmed by Figure 7c, where the electric potential along the vertical direction in the middle of the channel ϕ(x = 0.5L, y) (marked by the red line in Figure 7a) is plotted for varying gate potentials.In this plot, the gate voltage is applied on the right side of the electrolyte region, while the region on the left side shows the potential within the PEDOT:PSS layer.
At the interface between the electrolyte and the PEDOT:PSS layer, marked by the dashed line, a double layer is formed, and the potential gradually transitions from the electrolyte potential to the potential inside the mixed conductor.Away from this double layer, the potential inside the mixed conductor is constant along the y-direction and the electric field along y vanishes (E y ≈ 0).
The origin of the constant potential in the mixed conductor can be found in Figure 7b,d, which plot the distribution of holes inside the device for the same potential as in Figure 7a (V G = 5.8V T , V D = −2V T ) and the variation of the hole   concentration in the middle of the channel along the y direction p(x = 0.5L, y) (red line in Figure 7b), respectively.In contrast to the potential, a small gradient in the hole concentration along the transistor channel can be observed (Figure 7b).For increasing positive gate voltages, more cations are entering the mixed conductor, which leads to a de-doping effect and hence a reduction in the density of free holes (Figure 7d).Depending on the polarity of the gate voltage, a small accumulation or depletion of holes is observed at the interface between mixed conductors and electrolyte, but the hole concentration remains constant along the y-direction in the bulk of the mixed conductor.
Overall, due to the significant mobility of ions inside the mixed conductors, a constant electrochemical potential is formed in the bulk of the film.The complete potential difference between the channel and the electrolyte drops across a thin double layer at the interface between the two layers, which in turn leads to a vanishing small gradient in the hole concentration and potential along the vertical direction, that is, d dy | x=L∕2 ≈ 0 and dp dy | x=L∕2 ≈ 0. To test if the gradual channel approximation can be used in the one-phase model, the second derivative of the potential with respect to the horizontal and vertical direction d 2  dx 2 (x = L 2 , y) and The second derivatives are plotted along a vertical line (indicated by a red line e.g., in Figure 7a).The PEDOT:PSS layer is seen on the left and the electrolyte on the right.The interface between the layers is indicated by a black line.
Figure 8 reflects the observations made above.There are two regimes visible.At the interface between the mixed conductor and the electrolyte, a double layer is formed, which is  dx 2 exceeds d 2  dx 2 closer to the interface with the electrolyte (marked by the black line).Overall, these results show that the gradual channel approximation cannot be used in the bulk of the PEDOT:PSS layer (i.e., where the layer is doped or de-doped), but only at its interface to the electrolyte.The current density along the x-direction j x is plotted as well.Although the current density increases slightly at the PEDOT:PSS/electrolyte interface, most of the current is carried by the bulk of the PEDOT:PSS layer.This indicates that the switching mechanism of the device is dominated by bulk (de)-doping, and not by a modulation of the double layer at the electrolyte/PEDOT:PSS interface.
marked by a larger curvature in the electric potential along the y-direction,( d 2  dy 2 ≫ d 2  dx 2 ), indicating the assumption taken by Schockley to treat field-effect transistors is indeed justified within the double layer.However, in the bulk of the mixed semiconductor, both curvatures almost vanish ( d 2  dy 2 ≈ d 2  dx 2 ≈ 0), or in other words, the gradual channel approximation cannot be used.
These two regimes represent two different operation modes of the OECTs.It seems that OECTs can operate by a modulation of the thickness of the electric double layer formed at the interface between the electrolyte and the mixed conductor, which is akin to a field-effect device that can be treated by the gradual channel approximation.Another separate operation regime is given by the injection of cations into the mixed semiconductor, which de-dope the film and modulate the conductivity in the bulk of the semiconductor.This mechanism is marked by a vanishing small electric field inside the PEDOT:PSS layer, and the gradual channel approximation cannot be used.
To quantify these two effects, and to decide which one is dominating, the electric current density flowing in the x-direction is plotted along the middle of the transistor channel (i.e., j x (x = L 2 , y)) in Figure 8 as well.It can be seen that most of the current flows along the bulk of the mixed conductor.Only for large negative voltages are additional holes accumulated in the double layer region (cf. Figure 7d), and a spike in current density j x at the interface is observed.This spike, however, is small, and the majority of the current seems to be caused by bulk conduction within the mixed semiconductor.
Therefore, it seems that the doping/de-doping mechanism is dominating switching in the one-phase model, which is in agree-ment with the observation that the transconductance of an OECT scales with the volume of the channel.For this bulk doping/dedoping mechanism, however, the potential along the vertical direction is approximately constant, and the gradual channel approximation cannot be used.

Contact Resistance in the One-Phase Model
The bulk switching mechanism observed to be dominating in the one-phase system leads to an accumulation of cations at the source and drain electrode, which is shown in Figure 9a.The accumulation of cations is caused by a lateral current of cations along the horizontal electric field of the drain potential, which only stops when an equilibrium of equal but opposite ion and diffusion currents is reached. [33]his accumulation of ions at the contacts leads to a screening of the applied source-drain potential V D , which is shown in Figure 9b.The amount of accumulated ions and hence the magnitude of the potential drop at the contacts is proportional to the applied gate potential V G .This additional potential drop at the contacts resembles the effect of contact resistances as observed for example, in OFETs.
In Figure 10a, the results of a numerical experiment mimicking the transmission line method used to measure the contact resistance of OFETs are shown.Here, the channel length of OECTs is varied and the resistance of the channel is measured at a low drain voltage V D , that is, in the linear regime of the OECT.
As expected, the resistance of the channel decreases linearly with the channel length.The y-intercept of the fit line represents  the total contact resistance consisting of the sum of resistance at the source and drain.
The contact resistance scales with the gate voltage V G , as shown in Figure 10b.Overall, the characteristic exponential shape as already observed in Figure 4 is reproduced, which is in line with other literature reports. [21,30,32]ith the help of Figure 9a, an origin for this contact resistance can be proposed.It seems that contact resistances can be caused by the accumulation of cations at the source and drain electrodes, which screen the applied potential and result in an additional potential drop.In equilibrium, the amount of accumulated ions should depend exponentially on the applied gate voltage, [33] resulting in the characteristic exponential dependency of contact resistance on the gate voltage.
This origin of contact resistance is different from a "conventional" contact resistance observed, for example, in OFETs, as it is not caused by energy barriers at the contacts and inefficient injection of holes.In fact, perfect injection is assumed in the current model here.Nevertheless, ion accumulation will weaken the potential at the source and drain leading to an additional potential drop and an apparent contact resistance.

The Gradual Channel Approximation in a Two-Phase System
The discussion of the one-phase system has shown that two switching mechanisms of OECTs are in effect: modulation of the hole density by the extent of the double layer at the PE-DOT:PSS/electrolyte interface and a doping/de-doping effect in the bulk of the mixed conductor.Whereas the gradual channel approximation is valid for the first mechanism, the second one cannot be described using the assumptions underlying the gradual channel approximation.
In the one-phase system, the majority of the current is carried by the bulk of the mixed conductor and hence the doping/dedoping switching mechanism dominates.However, the balance of the two mechanisms can be altered if a two phase system is used consisting of separate ion and hole conductive phases.
PEDOT:PSS as the archetypical mixed conductor used in OECTs was indeed shown to feature two phases -the PEDOT phase and the PSS phase. [36]Whereas the PEDOT conducts holes, ions are transported along the PSS phase.Hence, it is usually classified as a heterogeneous blend, [37] in which ion and hole transport is carried by separate phases.
A model that explicitly treats PEDOT:PSS as a two-phase system was proposed by Tybrandt et al. [38] In their and derived [39] models, two electrical potentials are used-one for the electronic and one for the ionic system.The potential difference between these two phases leads to a charging process at the interface, which is described by a volumetric capacitance, implicitly invoking the gradual channel approximation again.
Here, the 2D drift-diffusion simulation routine was adapted to model a two-phase system without the a-priory assumption of interfacial capacitance.The setup is shown in Figure 11.In reality, the ion and hole conductive phases form a complex interwoven network, that extends across the whole volume of the organic semiconductor, leading to the observed scaling of transistor characteristics with channel thickness.To simplify this problem, only one pair of two straight fibers or layers is modeled.The bottom layer represents the hole conductive phase (e.g., PEDOT), and the top one is the ion conductive phase (e.g., PSS).Ions cannot enter the hole conductive phase and viceversa, that is, the interface between the two phases is treated as ideally reflecting for both species.Due to these assumptions, the total current obtained by the model cannot be compared directly to experimental results but has to be scaled by the number of pairs of ion and hole conductive phases that form an interface.
The potential inside the ion conductive phase is controlled by the applied gate potential at the top of the device.In contrast to the one phase model, the source and drain potential is applied at the sides of the model.This change was implemented to model the fact that both, the ion and hole conductive phase, will be in contact with the source and drain electrode and will experience the applied voltage at these contacts.At the source and drain contact, holes can be injected (modeled by assuming a boundary condition of p inj at source and drain in the PEDOT phase), whereas source and drain represent an ideally reflecting contact for ions, that is, no ions can be injected (e.g., reduced or oxidized).Therefore, the drain current plotted in the following is a hole current only and does not include ionic currents.and an ion-conducting layer on top (e.g., PSS).Both layers are contacted by the source electrode on the left and the drain electrode on the right.At these electrodes, the hole concentration is set to a constant value p inj .The gate electrode is used to modulate the electrochemical potential inside the ion-conducting layer.The interface between the two layers is impermeable for both charge carriers.This is modeled by enforcing that the y-component of the current densities is zero (i.e., the interface is ideally reflecting).
Within the hole conductive phase on the bottom of the device, a small amount of doping is assumed (N A = 10 16 m −3 ), representing impurities inside the semiconductor.Most importantly, this doping level is much lower than assumed for the one-phase model.Dopants are assumed to be stationary, that is, they cannot move within the phase.
Inside the ion conductive phase, cations are treated to be mobile, whereas anions (e.g., the PSS − groups) are assumed to be stationary.At the gate electrode, the density of cations is set to the density of anions.

Working Mechanism of the Two Phase System
Figure 12a plots the transfer characteristic of the two phase model (cf.Table 1 for the modeling parameters).The transfer characteristic shows an almost exponential decrease in drain current at positive voltages, and a weaker dependency at negative voltages (cf. Figure 12a).The absolute current is strongly dependent on the injection condition at the source and drain electrodes p inj , but for high concentrations this dependency seems to weaken.Again, only a limited proportionality of √ I D on V G is observed  ).The device consists of an ion conducting (e.g., PSS) phase on top and a hole conducting phase (e.g., PEDOT) on the bottom.In contrast to the one-phase model, source and drain electrodes are at the side of the device (indicated by red boxes), which takes into account that both phases have to be in contact with the electrodes.The gate electrode on top (marked by a small red box as well) controls the potential inside the ion conducting phase.b) Potential (x = L 2 , y) along the red cross-section shown in (a) for increasing gate potential V G .For larger voltages, a double layer is formed at the interface between the two phases, which leads to the observed on and off switching.In comparison to the one-phase model, no injection of cations and no (de)-doping effect is observed.
Although the results are similar to the one-phase model, the two-phase model results in significantly lower currents.However, this is a result of the fact that the model shown in Figure 11 only represents one pair of ion and hole conducting layer.In reality, these two phases are forming a complex network with each one carrying a current that behaves as shown in Figure 12a.However, from the comparison to the one phase model one can see that a very large number of these pairs or fibers (in the order of 10 3 ) have to be combined to reach a current that is comparable to the one phase model, at otherwise identical parameters.
In addition, the transfer characteristic obtained by the twophase model is shifted by a constant voltage compared to the results of the one-phase model.This observation can be explained by the fact that the one-phase system works as depletion transistor, whereas in the two-phase system, a channel of holes has to be accumulated.However, one has to keep in mind that the effective gate voltage plotted in Figure 12a neglects any difference in work function at the electrodes, any additional energy needed to mix the different ionic species, [40,41] or different redox reactions at the source/drain and gate electrodes, each of which will shift the effective gate voltage, similar to a threshold voltage shift.
To understand the working mechanism of OECTs in the two-phase model in detail, the potential inside short channel devices is shown in Figure 13a (cf.Table 1 for detailed simulation parameters).From the 2D plot one can observe that the source and drain electrodes control the potential inside the hole-conductive phase, and a potential distribution similar to the triangular region of field-effect transistors is formed.
Again, vertical potential profiles in the middle of the channel (x = L 2 , y), for example, along the red cross section of Figure 13a, are shown for increasing gate voltages V G in Figure 13b.For increasingly negative gate voltages, a larger potential drops between the two phases (Figure 13b).
The difference in potential between the ion and hole conductive phases is reflected in the distribution of holes and cations inside the device.Figure 14a plots the hole concentration and Figure 14b cation concentration.Clearly, a thin channel of increased hole concentration is formed at the interface between ion and hole conductive phase.This increased positive charge is balanced by an increased depletion in the ion conductive phase, that is, a depletion of cations at the interface.Both effects are again proportional to the applied gate voltage (cf. Figure 14c,d).
Overall, a thin channel of holes is formed in the device, which is balanced by a depletion of cations in the ion conducting phase.In Figure 15 the curvature of the potential at the center of the device along the dashed red line in Figure 13a ( d 8V T , that is, when a channel is formed.The curvature along the y-direction is several orders of magnitude larger than the one in x-direction ( d 2  dy 2 (x = L 2 , y) ≫ d 2  dx 2 (x = L 2 , y), or, in other words, the gradual channel approximation can be safely used in the center of the device.
However, the devices show an anomaly at the source and drain contacts, which is visible in the plot of the hole distribution p(x, y) shown in Figure 14a.Although a channel of holes is formed at the interface between the two phases, the channel is depleted at the source and drain contacts and no holes are accumulated.This effect is independent of the applied drain potential, and therfore not caused by a regular channel pinch-off observed in field-effect devices.This unusual pinch-off is accompanied by a strong depletion of cations at the source and drain contacts, visible in Figure 14b.
The effect of an unusual pinch-off at the source/drain electrodes is caused by the influence of the potential applied by the source and drain electrodes.In Figure 16a the potential close to the drain contact ϕ(x = L, y) is plotted.Although a small variation is observed, the potential largely remains at the applied drain potential of V D = −2V T .
Figure 16b plots the hole concentration along the drain (p(x = L, y), p ion (x = L, y)).Indeed, the very small potential difference between the hole and ion conducting phases seen in Figure 16a leads to a very weak or even absent accumulation of holes, for example, the pinch-off of the transistor channel even in the linear regime observed in Figure 14a.
The observation of an unusual channel pinch-off is accompanied by the fact that the gradual channel approximation cannot be used at the source and drain contacts.dx 2 (x = L, y) ≪ d 2  dy 2 (x = L, y) cannot be taken, that is, the gradual channel approximation is violated at the source and drain contacts.

Contact Resistance in the Two-Phase Model
In the two phase model the source and drain contacts display a behavior that is markedly different from the center of the channel.In particular the close coupling of the source and drain potentials to both phases leads to an unusual pinch-off of the transistor channel.
This observation is reflected in the transmission line experiment shown in Figure 18a.As for the one phase model, the resistance increases linearly with the channel length.Extending the lines to the left results in a residual or contact resistance.This contact resistance is proportional to the boundary condition at the source and drain p 0 (cf. Figure 18b), that is, more efficient injection leads to a smaller contact resistance.

Conclusion
The working mechanism of OECTs has been described by a capacitive element that couples the applied gate potential to the density of ions injected into the mixed conductor (cf.Equation ( 4)).This definition of gate capacitance was highly successful and led to a closed analytical model that is often used to discuss trends in the behavior of OECTs.
However, this assumption of gate capacitance is closely tied to the gradual channel approximation as stated in Equation ( 2), that is, the change in the lateral electric field has to be smaller than the change in the vertical electric field.Only when this assumption is correct, the space charge created by the gate voltage can be approximated by a simple capacitive element.If, however, this assumption can be justified, the 2D transistor problem can be reduced into two (weakly coupled) 1D problems, that can be solved analytically.
Here, we discuss if the often implicit assumption of the gradual channel approximation is indeed justified in OECTs.We show that drain current modulation as observed in OECTs can be caused by two separate effects-a doping/de-doping mechanism by injection of ions into the mixed conductor, and the formation of an electrostatic double layer at the interface between the electrolyte and the organic conductor.
These two mechanisms are markedly different.In particular, whereas the doping/de-doping effect leads to vanishing small electric fields in the bulk of the organic conductor (cf. Figure 7c), large electric fields perpendicular to the OECT channel are formed at the interface.Consequently, the gradual channel approximation can not be used for the bulk doping/de-doping mechanism, but only for the electrostatic accumulation effect at the interface between the electrolyte and the mixed conductor.
The balance between these two switching mechanisms depends strongly on the microscopic structure of the organic conductor.When the organic conductor can be described by a single phase, that is, when ions and holes are transported within the same phase and experience the same electric potential, the doping/de-doping mechanism dominates (cf. Figure 8).However, if ions and holes are transported in separate phases, each one featuring an independent electric potential, the influence of the electrostatic double layer dominates.
From the experimental data shown above, it is difficult to distinguish between the two mechanisms and to decide if a one or a two-phase model is better suited to describe OECTs.Both models result in similar scaling laws, and for both models, it is observed that the transconductance g m scales with the geometric factor WT L .Both models result in a peaking transconductance curve (e.g., as shown in Figure 2a for experiments, and Figures 6c and 12c for the model calculations).However, whereas the peak in transconductance in the one-phase model is explained by a weak or absent accumulation, [23,24] an external contact resistance seems to dominate the effect for the two-phase model.The previous observation that the peak in transconductance can be shifted by the applied drain potential [24] can be explained easily by the one-phase model, but is more challenging to discuss in the two-phase model.Furthermore, the observation that the peak in transconductance is visible in OECTs with very low or even absent contact resistance hints as well towards a one-phase system. [22]inally, the characteristic exponential dependency of the contact resistance on the gate potential as observed experimentally is straightforwardly explained in the one-phase model, where the amount of ions accumulated at the drain electrode is expected to scale exponentially with the gate potential (cf. Figure 10b).For a two-phase model, the dependency of the contact resistance on the gate potential is much smaller, and mainly influenced by the boundary condition p inj , that is, the injection barrier for holes at the electrodes (cf. Figure 18b).
Although there are some indications that a homogeneous doping/de-doping effect is dominating in PEDOT:PSS based depletion type transistors, it has to be kept in mind that a mixture of both mechanisms is most likely in effect in real devices.The balance between the two mechanisms will depend on the microstructure of the organic layer.Not only is the formation of separate phases for holes and ions crucial, but so is the efficiency of ion transport within the ion-transporting phase.Only when ion transport is efficient enough to establish an independent electrical potential within the volume of the OECT, an electrostatic, two-phase mechanism can be reached.In contrast to the one-phase model (Figure 10b) the exponential dependency of the contact resistance on the gate potential often observed in experiments cannot be reproduced.

Experimental Section
Fabrication of Organic Electrochemical Transistors (OECTs): The transistors used in this work were fabricated on 100 mm silicon wafers.The wafers are coated with a 5 μm thick layer of polyimide to act as substrate.The polyimide was spin coated on the wafer and cured on a vacuum hotplate.A 200 nm layer of gold was deposited by sputtering for the source, drain, and gate electrodes.The electrodes were then structured by standard photolithography and iodine wet etching.Two more layers of polyimide were applied, one 300 nm layer to act as passivation for the contacts and one 5 μm layer to act a shadow mask for the channel material.Both of these layers were structured by standard photolithography and reactive ion etching to open the channels, gate, and contact pads.Finally, the organic semiconductor was spin coated over the surface and annealed.The organic semiconductor was 20 mL PEDOT:PSS (Clevios PH-1000 from Heraeus) mixed with 5 mL ethylene glycol, 50 μL dodecylbenzenesulfonic acid (DBSA), and 242 μL GOPTS ((3-glycidyloxypropyl)trimethoxysilane).The mixed solution was spincoated at 17 rps and annealed for 10 min at 110 °C resulting in a layer thickness of ≈ T = 200 nm.The top layer of polyimide was peeled such that the semiconductor layer only covered the channels and gate.
The channels were structured such that the widths vary from 50 to 600 μm and the lengths vary from 100 to 1000 μm, which resulted in 80 unique channel sizes.The 3.5 × 3.5 mm gate electrode lay in-plane with the channels, and was coated with PEDOT:PSS in the same step as the channels.There was a 10 μm overlap between the semiconductor layer and the source/drain electrodes to ensure a connection to the channel.
Measurement Procedure: A solution of 100 mm NaCl was used as electrolyte for all measurements.For the transfer characteristics, the drain voltage was held at −0.2 V (the source electrode is always grounded), while the gate electrode was swept from −0.1 to 1.1 V. To ensure steady-state during measurement, a delay of 0.1 s was used between each measurement point.For the output characteristics, the gate voltage was held constant (at values from 0 to 0.9 V), while the drain-source voltage was swept from 0 to −0.6 V.The resistance of the channels was extracted from the linear region of these curves.Then, the contact resistance was calculated by linear fitting, using a modified transmission line method. [42]

Figure 1 .
Figure 1.Sketch of an OECT including source, drain, and gate electrodes.The orientation of the x and y-axes used here is indicated.

Figure 2 .
Figure 2. Representative transfer characteristic of an OECT.These OECTs have a channel width of 50 μm with channel lengths varying from 100 μm to 1 mm.a) Absolute drain current I D versus gate voltage V G and transconductance g m (channel length L = 200 μm).The transconductance displays the characteristic bell-shaped form, which is not explained well by the current OECT model based on the gradual-channel approximation.b) Plot of √ I D versus gate voltage V G .Fitting this curve with a linear function to extract the threshold voltage and hole mobility is difficult and strongly depends on the assumed fitting range.

Figure 3 .
Figure 3. a) Scaling of transconductance with the geometric factor WT L , showing a linear trend for small, but a saturation at larger WT L .This saturation behavior is indicative of a contact resistance.b) Scaling of the maximum transconductance g m with the width W of the device, showing a linear trend without saturation.This behavior shows that the contact resistance is not dominated by lead resistance of the contact lines.

Figure 4 .
Figure 4. Contact resistance observed in OECTs.A characteristic exponential increase in contact resistance is observed.

Figure 5 .
Figure 5. Sketch of the one-phase model of OECTs.The transistor is formed by an electrolyte layer and a layer of a mixed conductor, for example, PEDOT:PSS.Drift and diffusion of cations are considered within the whole device area, whereas holes can only move within the mixed conductor.

Figure 6 .
Figure 6.Result of a model calculation obtained using the one-phase model as shown in Figure 5.The mixed conductor has a thickness of T = 120nm, a hole mobility of μ p = 1cm 2 V −1 s −1 and a doping concentration N A = 10 21 cm −3 .All results are obtained at a drain potential of V D = −0.1 V. a) Transfer characteristic for varying channel length, displaying the expected scaling with channel length.b) Plot of √ I D versus the gate potential, equivalent to the experimental results shown in Figure 2b.Again, fitting this curve by a linear function to extract charge mobility and threshold voltage is challenging.c) Transconductance versus gate potential, correctly reproducing the bell-shaped form of the experiment plotted in Figure 2a.

Figure 7 .
Figure 7. Results of the 1-phase model to describe OECTs.a) Potential and b) hole distribution within the device consisting of the electrolyte and PEDOT:PSS layer.The gate (top), source (bottom left) and drain (bottom right) are marked by red boxes.A double layer is formed at the interface between PEDOT:PSS and electrolyte.c) Plot of the electric potential ϕ(x = 0.5L, y) along the vertical red line in the center of the PEDOT:PSS layer, showing the influence of the gate potential on the extent of this double layer.Most importantly, the potential inside the PEDOT:PSS layer is constant.d) Hole concentration p(x = 0.5L, y) along the red cross-section, showing a slight hole accumulation at the electrolyte interface for negative gate voltages, and a modulation of the bulk hole concentration for positive gate voltages.

Figure 8 .
Figure 8. Second derivative of the electric potential with respect to x and y ( d 2  dx 2 (x = L 2 , y) and d 2  dx 2 (x = L 2 , y)) plotted along a vertical cross-section in the center of the transistor.Whereas in the bulk of the PEDOT:PSS layer (i.e., below y < 25nm), both derivatives vanish, d 2 dx 2 exceeds d 2  dx 2 closer to the interface with the electrolyte (marked by the black line).Overall, these results show that the gradual channel approximation cannot be used in the bulk of the PEDOT:PSS layer (i.e., where the layer is doped or de-doped), but only at its interface to the electrolyte.The current density along the x-direction j x is plotted as well.Although the current density increases slightly at the PEDOT:PSS/electrolyte interface, most of the current is carried by the bulk of the PEDOT:PSS layer.This indicates that the switching mechanism of the device is dominated by bulk (de)-doping, and not by a modulation of the double layer at the electrolyte/PEDOT:PSS interface.

Figure 9 .
Figure 9. a) Plot of the cation concentration p ion (x, y = 0) along the transistor channel.Due to the potential difference between the gate and source/drain, cations accumulate on top of the drain electrode.b) Electric potential ϕ(x, y = 0) along the transistor channel.The accumulation of cations on the source and drain electrode leads to a potential drop at the source and drain electrode, which can be described by a contact resistance.

Figure 10 .
Figure 10.a) Transmission line method.The resistance of the transistor in the linear regime is plotted versus the channel length for various gate voltages V G .The contact resistance is found as the intercept of a linear fit with the resistance axis.b) Extracted contact resistance versus gate voltage V G .An exponential increase in contact resistance is found, which can be explained by an increase in the density of accumulated ions at the source and drain contact for larger gate voltages.

Figure 11 .
Figure11.Two phase model of organic electrochemical transistors (OECTs).The device is formed by a hole-only layer on the bottom (e.g., PEDOT) and an ion-conducting layer on top (e.g., PSS).Both layers are contacted by the source electrode on the left and the drain electrode on the right.At these electrodes, the hole concentration is set to a constant value p inj .The gate electrode is used to modulate the electrochemical potential inside the ion-conducting layer.The interface between the two layers is impermeable for both charge carriers.This is modeled by enforcing that the y-component of the current densities is zero (i.e., the interface is ideally reflecting).

Figure 12 .
Figure 12.Result of a model calculation obtained using the two-phase model as shown in Figure 11.The hole conductive layer has a hole mobility of μ p = 1cm 2 V −1 s −1 , and the channel length is L = 125 μm.All characteristics are obtained at a drain potential of V D = −0.1 V. a) Transfer characteristic for varying channel length.Similar trends to the one-phase model are observed.The current values are lower as the model treats only one pair of PSS and PEDOT phase.b) Plot of √ I D versus the gate potential, equivalent to the experimental results shown in Figure 2b.c) Transconductance versus gate potential, correctly reproducing the bell-shaped characteristic observed in the experiment (Figure 2a).

Figure 13 .
Figure 13.a) Potential distribution ϕ(x, y) within the OECT for the two-phase model (p inj = 10 22 m −3).The device consists of an ion conducting (e.g., PSS) phase on top and a hole conducting phase (e.g., PEDOT) on the bottom.In contrast to the one-phase model, source and drain electrodes are at the side of the device (indicated by red boxes), which takes into account that both phases have to be in contact with the electrodes.The gate electrode on top (marked by a small red box as well) controls the potential inside the ion conducting phase.b) Potential (x = L 2 , y) along the red cross-section shown in (a) for increasing gate potential V G .For larger voltages, a double layer is formed at the interface between the two phases, which leads to the observed on and off switching.In comparison to the one-phase model, no injection of cations and no (de)-doping effect is observed.

Figure 14 .
Figure 14.a) Hole distribution p(x, y) within the OECT at V G = −15V T and V D = −2V T (p inj = 10 22 m −3 ).A channel of free holes is formed at the interface between the two phases.b) Distribution of cations p ion (x, y) at the same gate and drain potentials.c) Hole concentration p(x = L 2 , y) along the red line shown in (a).The density of holes accumulated at the interface is proportional to the applied gate potential.d) Ion concentration p ion (x = L 2 , y) along the red line shown in (b).Overall, these results confirm that switching in the two-phase model is dominated by the formation of an electrostatic double layer at the interface between the PSS and PEDOT phase.

Figure 15 .
Figure 15.Curvature in electric potential along the red line shown in Figure 13a.Due to the double layer formation, one finds that d 2  dy 2 (x = L 2 , y) ≫ d 2  dx 2 (x = L 2 , y), that is, the gradual channel approximation can be used in the center of the device.

Figure 17
plots the curvatures of the potential d 2  dx 2 (x = L, y) and d 2  dy 2 (x = L, y) at the drain.Clearly, the assumption d 2

Figure 16 .
Figure 16.a) Potential ϕ(x = L, y) along the drain electrode (p inj = 10 22 m −3).Although there is a drop in potential between the two phases, the difference is very small.b) Hole concentration p(x = L, y) along the drain electrode.The small variation in potential drop results in only a weak accumulation of holes at the drain.This result indicates that the formation of the double layer is disturbed at the drain contact.

Figure 17 .
Figure 17.Curvature in electric potential d 2  dx 2 (x = L, y) and d 2  dy 2 (x = L, y) at the drain electrode.The assumption of d 2  dx 2 (x = L, y) ≪ d 2  dy 2 (x = L, y) is not justified and the gradual channel approximation can not be used at the contacts.

Figure 18 .
Figure 18.a) Transmission line method for a boundary condition of p inj = 1 × 10 22 .b) Contact resistance versus gate voltage for different injection conditions.In contrast to the one-phase model (Figure10b) the exponential dependency of the contact resistance on the gate potential often observed in experiments cannot be reproduced.

Table 1 .
Simulation parameters used here.