Bioinspired carbon nanotube–based nanofluidic ionic transistor with ultrahigh switching capabilities for logic circuits

Abstract

The voltage-gated ion channels, also known as ionic transistors, play substantial roles in biological systems and ion-ion selective separation. However, implementing the ultrafast switchable capabilities and polarity switching of ionic transistors remains a challenge. Here, we report a nanofluidic ionic transistor based on carbon nanotubes, which exhibits an on/off ratio of 10⁴ at operational gate voltage as low as 1 V. By controlling the morphology of carbon nanotubes, both unipolar and ambipolar ionic transistors are realized, and their on/off ratio can be further improved by introducing an Al₂O₃ dielectric layer. Meanwhile, this ionic transistor enables the polarity switching between p-type and n-type by controlled surface properties of carbon nanotubes. The implementation of constructing ionic circuits based on ionic transistors is demonstrated, which enables the creation of NOT, NAND, and NOR logic gates. The ionic transistors are expected to have profound implications for low-energy consumption computing devices and brain-machine interfacing.

The nanofluidic ionic transistor with an on/off ratio of 10⁴ enables the polarity switching for ionic circuits.

INTRODUCTION

In biological systems, the transmembrane protein channels, which open and close in response to cell membrane potential to regulate the transport of ion or molecular species, act as a gatekeeper for information integration and transmission (Fig. 1A) (1–4). These voltage-gated ion channels, also known as ionic transistors, have a comparable mechanism to the three terminal electronic semiconductor field-effect transistors (SFETs), which manipulate the charge carriers (electrons and holes) of the semiconductor channel under a gate voltage (5–8). The underlying mechanisms of ionic transistors are spatiotemporally electrophoretic modulation of the transport of ions and charged molecules across the nanochannel and membranes by surface charge–governed electrical double layers (EDLs) and steric hindrance–induced dehydration effect (9–15). Taking SFETs as a reference, the primary necessity for bioinspired nanofluidic ionic transistors is to have fundamental properties, specifically high switching ratios, high switching sensitivity, and robust switching performance, which underscores the importance of constructing a system that precisely controls ion transmission. Meanwhile, these fundamental properties determine the application scope of nanofluidic ionic transistors, for example, in performing high-throughput single-molecule level DNA sequencing, efficient metal ion sieving (13, 14, 16, 17), and other potential applications in chemical logical circuits for revolutionary next-generation, affordable, and unconventional information processing units.

Fig. 1. Bioinspired design of nanofluidic ionic transistor and its on-off states.

(A) Illustration of biological synapse (left) and schematic of the on-off states of life’s transistor (right). (B) Schematic of nanofluidic ionic transistor based on carbon nanotubes (left) and schematic of the on-off states of nanofluidic ionic transistor (right). (C) SEM image of cross section of ACNT. The output curves (D) and transfer curves (E) of nanofluidic ionic transistor based on ACNT as a function of V_G with V_SD.

Recently, a handful of ionic transistors were proposed on the basis of different materials (10, 11, 13, 14, 16, 18–23). First-order nonconductive nanochannels based on one-dimensional (1D) silicon or alumina, with a minimum channel size typically greater than 20 nm, are the most extensively studied of these (10, 11, 16, 23). There is limited controllable gating ion transport when the gate voltage is applied to the silicon- or alumina-based nanochannel due to the fairly weak EDL effect caused by the surface charge. A low on-off ratio (≤10) is the outcome of this. In addition, the ionic transistors based on silicon or alumina nanochannels require a considerably high voltage for operation, substantially greater than the water-splitting threshold (1.23 V equivalent to a standard hydrogen electrode), which invariably results in additional adverse chemical reactions and also a clear and pressing critical issue of energy consumption (24). In the past decade, ionic transistors based on 2D-layered channel materials, such as conductive graphene or MXene, were proposed to realize ultrafast and highly selective ion transport in the subnanometer channel via the gate voltage (12–14). Nevertheless, the voltage-gating effect of long-term stability is considerably constrained by the low swelling-resistant and corrugated membrane with a wide channel size distribution (14). In addition, the existing examples of previously reported state-of-the-art nanofluidic ionic transistors only have unipolar switching capabilities. Despite tremendous efforts, realizing advanced and feasible nanofluidic ionic transistors with high on/off ratio and polarity switching capabilities at lower operating voltages to overcome the abovementioned bottlenecks is still a long-standing challenge.

Here, we report the unipolar and ambipolar nanofluidic ionic transistors based on two configurations of asymmetric carbon nanotubes (ACNTs) and symmetrical carbon nanotubes (CNTs), respectively. The unipolar ionic transistor exhibits an on/off ratio of 10⁴ in 0.1 mM KCl solution and 10³ even at various salt concentrations by applying a gate voltage as low as 1 V, which is capable of surpassing the switching performance threshold that is met by conventional nanofluidic materials (10, 11, 14, 22, 25). Subsequently, it was successfully implemented to alter the polarity switching between p-type and n-type ambipolar ionic transistors using CNT, which also contrasts with the switching properties of previously reported unipolar nanofluidic ionic transistors (13, 26). The ambipolar transistor characteristics would enable the creation of complementary ionic logic circuits and substantially raise the level of sophistication of bioinspired nanofluidic iontronic devices. In addition, the introduction of an Al₂O₃ dielectric layer promotes a higher level of electrostatic controllability and volumetric capacitance, consequently obtaining a prominent on/off ratio of ionic transistor. Ultimately, the ionic logic systems consisting of NOT, NAND, and NOR logic gates based on nanofluidic ionic transistors are realized. The nanofluidic ionic transistors described here may provide a route to build ionic logic system for biomimetic computations.

RESULTS

Device design of nanofluidic ionic transistor

The schematic of the carbon nanotube–based nanofluidic transistor device structure is a three-terminal configuration (Fig. 1B, left side). The source and drain electrodes (Ag/AgCl) were inserted into electrolyte reservoirs to detect the current across the carbon nanotubes, whereas the third terminal (Pt electrode) was directionally attached to the anodic aluminum oxide (AAO) membrane, with carbon nanotubes serving as the gate electrode (Fig. 1C and figs. S1 and S2).

The electric field–driven ion transport behaviors under the applied gate potential (V_G) are studied by effectively modulating the surface charge of carbon nanotubes, so as to realize the switching behavior of the nanofluidic ion transistor, as shown in Fig. 1B. In the case of positive V_G, fewer ions flow into the carbon nanotubes since the positive V_G neutralizes the original negative charges on the nanochannel wall of carbon nanotubes, which weakens the surface charge–governed EDL effect and decreases the inherent carrier concentration (14). Under such a circumstance, the depletion region will form at the center of the carbon nanotubes, resulting in the “off” state. Conversely, more ions flow into the nanochannel when the carbon nanotubes are subjected to negative V_G. In this case, more negative surface charges are formed in the inner surface wall of carbon nanotubes (23, 27), resulting in the “on” state. The on/off performance calculated by the ratio of maximum to minimum current is about 16,452 at source-drain bias (V_SD) of 1 V (Fig. 1D), which outperforms other reported nanofluidic ionic transistors (10, 11, 22, 25, 28, 29). In addition, the transfer characteristics of nanofluidic ionic transistor are similar to the p-type behavior seen in the electronic counterparts (1). Therefore, controlling the surface charge of carbon nanotubes is able to alter the inherent carrier types of nanofluidic ionic transistor in a manner similar to how electronic equivalents’ doping levels are regulated.

Nanofluidic ionic transistor with different morphology features

To study the effect of morphology, two configurations of CNTs and ACNTs were prepared using AAO as a template by a chemical vapor deposition (CVD) process (figs. S3 and S4) (30). Electronegativity doping of N atoms is responsible for the negative charges observed in carbon nanotubes, as indicated by zeta potential and x-ray photoelectron spectroscopy (XPS) (figs. S5 and S6). From the morphology characterization, the hollow ACNT formed inside the pore channel of the AAO membrane, the generated ACNT with an asymmetric pore diameter gradient from 5 to 50 nm can be clearly seen from the top and bottom views of the scanning electron microscope (SEM) (Fig. 2A). The fabricated device presents typical transistor behaviors by monitoring the representative transfer and output characteristics. The output current (I_SD) demonstrates well-defined nonlinear dependence as a function of V_G that varied from −1 to 1 V, with V_SD changing from 0 to +1 V (Fig. 2B). By sweeping the positive V_G (from 0 to 1 V), the ionic current is essentially undetectable, corresponding to the “off” state of ionic transistor. By contrast, in negative V_G varying from 0 to −1 V, a substantial ohmic-like “on” state current can be rapidly modulated from 0.3 to 243.8 nA, and the maximum on/off ratio is estimated to be 10³ (fig. S7A). Differently from the aforementioned phenomenon, the field-effect modulation output curves of I_SD from forward-sweep to reverse-sweep (V_SD = −1 to 0 V) always display the “off” state. The asymmetric behavior for both circumstances between forward-sweep and reverse-sweep arises from the highly rectifying nature of the asymmetric ACNT (30), which causes the nanofluidic ion transistor to behave in a unipolar p-type manner (Fig. 2C).

Fig. 2. Electrical characterizations of nanofluidic ionic transistor with different morphology features.

(A) Schematic and SEM image of top view and bottom side of ACNT. The output curves (B) and transfer curves (C) of nanofluidic ionic transistor based on ACNT as a function of V_G with V_SD. (D) Schematic and SEM image of top view and bottom side of CNT. The output curves (E) and transfer curves (F) of nanofluidic ionic transistor based on CNT as a function of V_G with V_SD. (G) TEM image and elemental maps of CNT by introduction of an Al₂O₃ dielectric layer. The output curves (H) and transfer curves (I) of nanofluidic ionic transistor based on Al₂O₃/CNT as a function of V_G with V_SD.

The asymmetric unipolar nanofluidic ionic transistor was further characterized in different conditions. The gate-dependent electrical characteristics with a high on/off ratio up to 10³ (V_SD > 0) are maintained pronouncedly in various electrolyte concentrations (fig. S8). In 0.1 mM KCl solution, the unipolar ionic transistor exhibits an on/off ratio of 10⁴ by applying a gate voltage as low as 1 V, this high switching capabilities are more dominated by the primary operation mechanism of conventional nanofluidic devices, which are supported by surface charge–governed EDL effect (12, 31, 32). Traditionally, the EDLs that arise from the electrostatic interactions between mobile ions of electrolyte solution and surface charge of confined channels are responsible for the ion selectivity (33). EDL thickness is characterized by the Debye length, which is defined as the following equation

$${\mathrm{\lambda }}_D=\sqrt{\frac{{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_r\mathit{RT}}{F^2{\mathrm{\sum }}_i{z_i}^2{c}_i^{\text{bulk}}}}$$ (1)

where ε₀ is the permittivity of vacuum, ε_r is the relative permittivity of the electrolyte, R is the gas constant, T is the absolute temperature, F is the Faraday constant, z_i is the valence of ion species i, and c_i is the concentration of ion species i in the bulk solution. As can be seen from the equation above, it indicates that the thickness of EDL is higher for low-concentration electrolytes. Accordingly, this confirms that the electrostatic force generated by the EDL promotes the high switching capabilities in 0.1 mM KCl. In addition, the nanofluidic ionic transistor exhibits excellent stability (fig. S9), which holds potential applications in water-based complex systems with switching and amplifying requirements (22), and also demonstrates a highly desirable maneuverability in chemical circuits (31, 34).

By contrast, the ambipolar nanofluidic ionic transistors based on symmetrical CNT are demonstrated in contrast to unipolar behavior of ACNT. Figure 2D shows SEM images of CNT with a symmetrical internal pore diameter of 50 nm. On the basis of the same analysis as the one mentioned above, the output characteristics disclose a symmetrical nonlinear ohmic-like state behavior (Fig. 2E), which follows the on/off switchable law of nanofluidic ionic transistor based on ACNT (Fig. 2B). The maximum on/off ratio is estimated to be 479 (fig. S7B). In addition, the symmetrical typical p-type and n-type ambipolar behavior of nanofluidic ionic transistor is observed (Fig. 2F). The change of the polarity switching of the ionic transistor is due to the reversal of the surface charge of CNT, which enables bipolar ion transport.

In SFETs, an ultrathin gate dielectric layer with high internal capacitance would strongly influence the electrical characteristics of transistors, such as the charge carrier density, on/off ratio, and low-voltage operation (35). Here, an Al₂O₃ dielectric layer was further introduced in the aforementioned fabricated nanofluidic ionic transistor by atom layer deposition (ALD). The transmission electron microscopy (TEM) image proves the existence of an Al₂O₃ dielectric layer with a thickness of about 2 nm, and the energy-dispersive x-ray analysis (EDX) provides additional evidence of the successful doping of N element in the carbon nanotubes (Fig. 2G and fig. S10). It can be observed that the introduction of the dielectric layer does not alter the original geometric structure, but only alters the surface properties of CNT. The output characteristics of nanofluidic ionic transistor showcased similar nonlinear behavior based on CNT and ACNT (Fig. 2H and fig. S11A), by contrast, the leakage current of the nanofluidic ionic transistor decreases from 8.54 to 1.43 nA for CNT and from 11.7 to 2.7 nA for ACNT by introducing an Al₂O₃ dielectric layer (V_SD = 0 V, V_G = −1 V). In addition, the on/off ratio increases by about five times (V_SD = 1 V, V_G = −1 V) and reaches 1855 for CNT at the condition of V_SD = −1 V, V_G = 1 V (fig. S7, B and C), whereas the corresponding on/off ratio of ACNT increases from 940 to 1313 (figs. S7A and S11B). By sweeping V_G from −1 to 1 V, the typical p-type behavior at V_SD = 1 V switches to n-type ambipolar switching characteristics for ACNT at V_SD = −1 V (fig. S11C). The improved performance by introducing an Al₂O₃ dielectric layer suggests that the dielectric layer can cause the accumulation of surface charges and produce greater EDL capacitance (35, 36).

Mechanism verification of nanofluidic ionic transistor

To explain how the polarity reversal of the surface charge contributed to the polarity switching of the nanofluidic ionic transistor, the surface charges of ACNT and CNT were monitored by Kelvin probe force microscopy (KPFM). As expected, we directly observed the gate voltage–governed surface charge reversal and thus demonstrated that polarity switching of the nanofluidic ionic transistor between p-type and n-type ambipolar was achieved (Fig. 3, A and B, and figs. S12 and S13).

Fig. 3. Working principle and theoretically predicted electrical characteristics of the nanofluidic ionic transistors based on two carbon configurations (ACNT or CNT).

(A) Contact potential differences (CPD) evolution of ACNT as a function of V_G. (B) On-off state responses of surface charge into ACNT as a function of V_G. Theoretically predicted output characteristics (C), on/off ratios (D), and transfer curves (E) of the nanofluidic ionic transistor based on asymmetric ACNT as a function of V_G with V_SD from −1 to 1 V. Theoretically predicted output characteristics (F), on/off ratios (G), and transfer curves (H) of the nanofluidic ionic transistor based on CNT as a function of V_G with V_SD from −1 to 1 V.

A model combining the coupled Poisson and Nernst-Planck (PNP) equations was performed to give a theoretical prediction of ionic transistors’ electrical characteristics (fig. S14A). In the asymmetric model, asymmetric electrical characteristic curves of nanofluidic ionic transistor are realized because of the establishment of asymmetric electric field induced by the hierarchical pore gradient of ACNT (Fig. 3, C to E). Conversely, the electrical properties of nanofluidic ionic transistor are virtually symmetrical using a symmetric model (Fig. 3, F to H). The switch characteristics based on the calculation are comparable with that obtained by experimental data (Fig. 3, C and F). In addition, a typical concentration polarization phenomenon (ion accumulation and depletion) is observed, and the total ion concentration that has accumulated in the small pore ACNT gradually depletes in the entrance regions of large pore ACNT at V_SD = 1 V as a function of V_G varied from −1 to 1 V (fig. S14B). This phenomenon is ascribed to an asymmetric genuinely electrostatic interaction due to the substantial confinement effect caused by the asymmetric gradient structure. By contrast, an opposite tendency of concentration polarization is shown at V_SD = −1 V (fig. S14C). The simulation’s outcomes of electric field–driven ion transport behaviors under the applied gate potential exhibit a high degree of agreement with the experimental part (fig. S15).

Logic gate construction by nanofluidic ionic transistor

To prove the broad application prospect of the nanofluidic ionic transistor in ionic logic circuits, the standard logic gates, including NOT, NAND, and NOR gates, were successfully constructed (37, 38). This ionic logic circuit stitched the characterization of electronic chip and neuron system (Fig. 4A). Its equivalent circuit diagram and corresponding features are shown in the lower part of Fig. 4. The NOT gate was fabricated by connecting a resistor in series with drain electrode of nanofluidic ionic transistor (Fig. 4B), which can be used to invert its input signal. When the input voltage of A (V_inA) is 0 V (logic state “0”), meaning the “off” state of nanofluidic ionic transistor, its output voltage (V_out) of 0.19 V approaches the supply voltage of 0.4 V (logic state “1”) (Fig. 4C, green line). On the contrary, V_out decreases to 0.03 V, corresponding to the logic state “0”, if the nanofluidic ionic transistor switches to the “on” state with V_inA of −1 V (Fig. 4C, orange line). This reversible signal switching of NOT gates is a necessary consequence of the nanofluidic ionic transistors’ exceptional electrical characterization, especially owing to the prominent on/off ratio.

Fig. 4. Logic gates assembled from nanofluidic ionic transistors.

(A) The ionic logic circuit stitched the characterization of electronic chip and neuron system. Circuit diagrams of NOT (B), NAND (D), and NOR (F). Output performance of NOT (C), NAND (E), and NOR (G).

On the combination of two logically input voltage (V_inA,B), the complicated NAND and NOR gates were also realized by constructing the ionic circuit with an ionic transistor, two diodes, and external resistors (Fig. 4, D and F). The NAND gate produces an output of logic state “0” only if all its inputs are in the logic “1” (Fig. 4E, orange line), otherwise, the output is in the state of logic “1” (Fig. 4E, green and gray line). For the NOR gate, a high output is generated (logic state “1”) only if both the inputs are low (logic state “0”) (Fig. 4G, green line), if one or both input is high (logic state “1”), a low output (logic state “0”) is generated (Fig. 4G, orange and gray line).

We execute the logic gates with a single nanofluidic ionic transistor, in contrast to standard monopolar NAND/AND gates that use at least two transistors (39). By adjusting the external gate voltages, the nanofluidic ionic transistor can operate in a variety of logical states because of the ultrafast switching behavior. Thus, the successful construction of multifunctional logic gates based on nanofluidic ionic transistor configuration represents a feasible approach to be extended to complex functional integrated ionic logical circuits for temporal manipulation applicable (40).

DISCUSSION

In summary, the ambipolar and unipolar nanofluidic ionic transistors have been successfully fabricated on the basis of carbon nanotube, the polarity switching of which depends on the geometric structure and charge characteristics of carbon nanotubes. With a high on/off ratio of 10⁴ that can be further improved by introducing a thin Al₂O₃ dielectric layer, the advanced and feasible nanofluidic ionic transistors can overcome the bottlenecks that previously reported nanofluidic devices meet. However, it has not yet surpassed the electronic counterparts and life’s transistor, meaning that the development of nanofluidic ionic transistors is still in the toddler stage. One possible solution is to effectively design and optimize artificial ion channel structure with atomic-scale dimensions, plentiful surface charge characteristics, and intricate conformational essential for faithfully duplicating the multifunctional capabilities of life’s transistors. In addition, despite the nanofluidic ionic transistor being used in the ionic logic circuit, its complexity and computational performance still lag far behind complex biological neuronal network. More advanced all-ionic circuit consisting of ionic rectifier diode, ionic resistance, and ionic transistor deserves to be explored to approach biological intelligence infinitely in the near future.

MATERIALS AND METHODS

Fabrication of AAO-grown ACNT and CNT

The target substance was fabricated using asymmetric porous AAO (FlexiPor, 0.03- to 0.2-μm pore size) as the growth template and acetonitrile as the carbon source precursor by following a published procedure previously. Typically, the AAO substrate was placed into the horizontal growth chamber of a standard atmospheric pressure CVD system. Subsequently, the temperature of chamber was heated up to 1090°C and reduction treatment for 0.5 hours with a gas mixture [Ar, 100 standard cubic centimeter per minute (sccm) and H₂, 50 sccm]. Then, the growth step started to be implemented for 1 hour by introducing the reaction carbon precursor (humidified with acetonitrile bubbler) into the system, followed by cooling down to room temperature under Ar protection. The as-obtained sample was denoted as ACNT. Furthermore, the procedure for resultant CNT was identical to that of ACNT using symmetric porous AAO (Puyuan Nano, 0.1-μm pore size), except that the target temperature (1000°C) and time (1 hour) with a gas mixture (Ar, 50 sccm and H₂, 50 sccm) for growth are different.

Preparation of Al₂O₃ dielectric layer

Al₂O₃ was coated on the inner surface of CNT or ACNT in a self-made ALD system. N₂ was used as a carrier gas and delivered into the chamber at a flow rate of 1000 sccm. First, CNT or ACNT was loaded in the ALD reactor. Then, trimethylaluminum (TMA) and H₂O were introduced into the deposition chamber and maintained at 120°C for 30 min in vacuum. Subsequently, TMA and H₂O were maintained at room temperature. One ALD deposition cycles consisted of pulse (TMA)/exposure (TMA)/purge (N₂)/pulse (H₂O)/exposure (H₂O)/purge (N₂), corresponding to 0.5, 2, 30, 0.5, 0.5, and 45 s, respectively. The ALD reactor was cooled to room temperature with N₂ flow after growth.

Fabrication and measurement of devices

First, the voltage-gated ionic transistor devices based on ACNT and CNT were packaged by silicon chips with holes and polydimethylsiloxane (PDMS). Subsequently, the devices were mounted between two reservoirs of a homemade electrochemical cell, which stored electrolyte solutions with an equivalent volume. Electrolyte solutions were symmetrically positioned in contact with the AAO membrane containing CNTs, which had an effective testing area of 0.0289 mm². One side of the membrane was attached to a Pt electrode as gate, and two Ag/AgCl electrodes were inserted into reservoirs to collect the current signals through the membranes. Electrical characteristics (output curve and transfer curve) of the fabricated ionic transistor were measured using Keithley 2634 B (Keithley Instruments, Cleveland, OH). The output curves (I_SD − V_SD) of the ionic transistors were recorded when scanning V_G from −1 to 1 V in the step of 0.1 V, and V_SD was set in the range of −1 to 1 V. In addition, the transfer curve (I_SD − V_G) was recorded when V_SD was fixed at −1 or 1 V and V_G was changed from −1 to 1 V with 0.1-V steps. Under each condition, we repeated the measurement at least three times.

KPFM measurements

KPFM measurements were performed using a commercial atomic force microscopy system (Bruker Dimension Icon). Platinum/iridium-coated silicon tips (Bruker SCM-PIT) were used for the measurements. To monitor the gate voltage–induced surface charge, the sample surface was attached to an external potentiostat via Cu wire to apply gate voltage during the KPFM measurements. To estimate the net surface charge, the electrostatic potential signals at the sample surface were obtained via a peak force KPFM-HV mode.

Material characterization

SEM characterized the morphologies of ACNT and CNT at an accelerating voltage of 10 kV by Hitachi-SU8220. TEM, EDX, and elemental mapping were performed using an FEI Talos F200X with an acceleration voltage of 300 kV. The Raman spectrometer (Horiba, LabRAM HR Evolution) of ACNT was characterized by an excitation wavelength of 532 nm. The XPS was conducted using Al Kα x-ray source by PHI 5000 VersaProbe (ULVAC-PHI Inc.). The binding energies were standardized to the C1s peak at 284.8 eV from adventitious carbon. Surface photovoltaic spectroscopy was conducted by a surface photovoltaic spectrometer (CEL-TPV1000). The zeta potential of as-prepared samples was measured with SurPASS 3, Anton Paar.

PNP simulation

The trans-nanotube potential was systematically analyzed using a theoretical model based on PNP equations with proper boundary conditions by the following equation

$${\nabla }^{2}U=-\frac{F}{\mathrm{ϵ}}\sum z_i{c}_i$$ (2)

$${\mathrm{j}}_i=D_i\nabla c_i+\frac{z_{i}F}{\mathit{RT}}{D}_i{c}_i\nabla \mathrm{U}$$ (3)

$$\nabla · j_i=0$$ (4)

where U, c_i, D_i, j_i, and z_i are the electrical potential, ion concentration, diffusion constant, ionic flux, and charge of species i, respectively. We use a constant ϵ to describe the dielectric properties of electrolyte solution.

The individual ionic current (I_i) through the junction region is then integrated by Eq. 5

$${\mathrm{I}}_{\mathrm{i}}={\mathrm{\int }}_{\text{LX}}{z}_{i}e{D}_{\text{eff}}(\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }x}+\frac{\frac{\mathrm{\partial }U}{\mathrm{\partial }x}{z}_{i}e{c}_i}{k_{B}T})\mathit{dy}$$ (5)

where LX is a cut line inside the junction region (the line x = 110 is used in our calculations); e is elementary charge (e = 1.60 × 10⁻¹⁹ C); T is the absolute temperature (T = 300 K); c_sol = 0.01 M; D_eff is charge-specific effective diffusion coefficient for the K⁺ and Cl⁻ ions, which is defined as the following equation

$$D_{\text{eff}}=D_{\text{bulk}}*\frac{1}{\mathit{LK}}{\mathrm{\int }}_{\mathit{LK}}{e}^{\mp \frac{\mathit{eU}(x)}{k_{B}T}}\mathit{dx}$$ (6)

where LK is a cut line along the tunnel; U(x) is the electrical potential along the line LK; the bulk diffusion coefficients (D_bulk) for K⁺ and Cl⁻ ions are 1.96 × 10⁻⁵ cm² s⁻¹ and 2.03 × 10⁻⁵ cm² s⁻¹, respectively.

The boundary condition for potential U on the nanotube wall is

$$\overrightarrow{n}·\nabla \mathrm{U}=-\frac{\mathrm{\sigma }}{\mathrm{ϵ}}$$ (7)

where σ is the surface charge density, which is always subject to variation of our experiment condition. The ion flux has no normal components on the borders

$$\overrightarrow{n}· \mathrm{j}=0$$ (8)

A 2D structure was constructed for PNP model according to the geometry size as 1615 nm × 800 nm (fig. S14A). The commercial finite-element stationary solver was generally used (COMSOL Multiphysics 5.4). But when it fails, the parametric solver was applied. For all the calculations, the accuracy is set to be less than 10⁻³.

Supplementary Materials

This PDF file includes:

Figs. S1 to S15