NANO LETTERS
Nanofluidic Diode and Bipolar Transistor
2005 Vol. 5, No. 11 2274-2280
Hirofumi Daiguji,* Yukiko Oka, and Katsuhiro Shirono Institute of EnVironmental Studies, Graduate School of Frontier Sciences, The UniVersity of Tokyo, Tokyo 113-0033, Japan Received August 19, 2005; Revised Manuscript Received September 16, 2005
ABSTRACT Theoretical modeling of ionic distribution and transport in a nanochannel containing a surface charge on its wall, 30 nm high and 5 µm long, suggests that ionic current can be controlled by locally modifying the surface charge density through a gate electrode, even if the electrical double layers are not overlapped. When the surface charge densities at the right and left halves of a channel are the same absolute value but of different signs, this could form the basis of a nanofluidic diode. When the surface charge density at the middle part of a channel is modified, this could form the basis of a nanofluidic bipolar transistor.
Introduction. It is well known that surface charges induce electrostatic ion screening1,2 and electrokinetic effects3 such as electroosmosis, electrophoresis, streaming potentials, and streaming currents in micro/nanofluidic channels. Analogous to metal-oxide-semiconductor field-effect transistors (MOSFETs), Schasfoort et al.4 demonstrated that electrokinetically pumped liquid flow rates can be controlled by modifying the surface potential inside a microfluidic channel using a gate electrode. In nanofluidic channels, in contrast, the surface charges may have larger effects on the flow of ions as opposed to the flow of liquids.5,6 This difference between micro- and nanofluidics arises from a characteristic length scale for ion screening, the Debye length, λD ()1/κ), which generally ranges between 1 and 100 nm depending on the bulk ion concentration. At high bulk concentration of ion, where the channel height, h, is much larger than the Debye length, (κh . 1), the ion concentration inside the channel is the same as the bulk concentration. At low bulk concentration, where κh , 1, a unipolar solution of counterions is created inside the channel and the average concentration between two charged surfaces of surface charge density σ separated a distance h is given by n ) 2|σ|/eh from the requirement of electroneutrality. We7 have proposed theoretically that ionic flow can be regulated by locally modifying the surface charge density, σ, through a gate electrode when the electrical double layers are overlapped. Recently, Karnik et al.8 demonstrated the efficiency of field-effect control in nanofluidics experimentally, which could have broad implications on integrated nanofluidic circuits for the manipulation of ions and biomolecules in subfemtoliter volumes. However, when the * Corresponding author. E-mail:
[email protected]. Tel: +81-35841-8587. Fax: +81-3-3818-0835. 10.1021/nl051646y CCC: $30.25 Published on Web 10/04/2005
© 2005 American Chemical Society
electrical double layers are overlapped in a nanofluidic channel, it is possible to control the flow of only the counterions by modifying the surface charge density. If the electrical double layers are not overlapped, then both the counterion and coion flows can be controlled at the same time and the fluidic channels may have various currentpotential characteristics similar to those of semiconductor diodes and transistors. The objective of this study is to confirm theoretically that the ionic current can be controlled by locally modifying the surface charge density when the electrical double layers are not overlapped and to clarify the current-poential characteristics for two nanofluidic devices: (a) a nanofluidic diode and (b) a nanofluidic bipolar transistor. Governing Equations. In the analysis of ion transport in nanofluidic channels, the Poisson-Nernst-Planck (PNP) and the Navier-Stokes (NS) equations are employed.7,9 The governing equations are as follows ∇2φ ) -
1
∑a za ena
(1)
0
∇(na u + Ja) ) 0
(2)
∇u ) 0
(3)
1 u∇u ) [-∇p + µ∇2u - ( F
∑a za ena)∇φ]
(4)
where φ, na, p, and u are the electrostatic potential, concentration of ion species a, pressure, and velocity vector, respectively. Ja is the particle flux of ion species a due to a concentration gradient and electric potential gradient, which
Figure 1. (a) Calculation system of a nanochannel with a height of 30 nm and length of 5 µm with 1 × 1 µm2 reservoirs on either side and schematics of two different types of nanofluidic devices: (b-I) forward and (b-II) reverse biasing nanofluidic diodes; (c) nanofluidic bipolar transistor. The calculation system and schematics are not the same scale in the x and y directions.
is given by Ja ) -Da(∇na + (zaena/kT )∇φ), where Da and za are the diffusivity and valence of ion species a, 0 is the permittivity of vacuum, is the dielectric constant of medium, F is the fluid density, and µ is the viscosity. The boundary conditions at the channel walls, the walls face reservoirs, and the ends of reservoirs are given as follows ∇⊥φ ) -
σ 0
Ja⊥ ) 0 ∇⊥ p ) µ∇2⊥u - (
∑a za ena)∇⊥ φ
u ) 0 (at channel walls) (5) ∇φ ) 0 ∇nK+ ) ∇nCl- ) 0 ∇p ) 0 u ) 0 (at walls face reservoirs) (6) φ ) φb nK+ ) nCl- ) nb p ) pb ∇u ) 0 (at ends of reservoirs) (7) where ⊥ and b denote the wall-normal component and bulk value, respectively. σ is the surface charge density. Equations 1-4 are solved under boundary conditions 5-7 using a finite difference algorithm, yielding the electric potential, ionic concentration, pressure, and velocity in the system. If these are known, then the current density of ion species a is obtained by the following equation: ia ) ia,adv + ia,dif ) za e(na u + Ja) Nano Lett., Vol. 5, No. 11, 2005
(8)
The dielectric constant of KCl aqueous solution, , is 80, the diffusivities of K+ and Cl-, DK+ and DCl-, are 1.96 × 10-9 and 2.03 × 10-9 m2/s, respectively.10 The density and viscosity of the solution are assumed to be 103 kg/m3 and 10-3 Pa‚s, respectively. Calculation System. Figure 1a shows the schematic diagram of the 2D domain for the calculation, where the total length of the channel, Lx, is 5 µm and the height, Ly, is 30 nm. Reservoirs 1 × 1 µm2 in size are considered on either side of the channel. For a diode, the surface charge densities at the right and left halves are modified to be the same absolute value but of different signs. For a transistor, the gate electrode is located at the center of the channel with length, Lgx, of 2 µm and the surface charge density under the gate is modified. The surface charge density ranges from -5 × 10-3 to 5 × 10-3 C/m2. In the numerical simulations, the grid spacing outside the channel is 50 nm, whereas the minimum grid spacing of 10 nm is assumed in the x direction near the inlets and outlets of the channel. The minimum grid spacing in the y direction is inside the channel and the value is 2 nm. The temperature is 300 K and the bulk concentrations of the KCl aqueous solutions are 10-3, 5 × 10-3, and 10-2 M. Before showing calculation results, we briefly explain the principles of the two types of nanofluidic devices. Figures 1b-I and 1b-II show the schematics of forward and reverse biasing nanofluidic diodes, respectively. Under forward bias, counterions inside the electrical double layers 2275
Figure 2. Potential, concentration and pressure profiles (top), and velocity vectors (bottom) for (a) a forward bias of 5 V and (b) a reverse bias of 5 V. The electric potential and pressure profiles are along y ) 0, and the concentration profile is the profile of concentration averaged over the y direction. The surface charge densities at the left and right halves of the channel are assumed to be 2 and -2 mC/m2, respectively. The bulk concentration is 5 mM. The maximum velocities under forward and reverse biases are 5.42 × 10-3 and 4.12 × 10-4 m/s, respectively.
approach to the junction of positively and negatively charged walls. As a result, ions are accumulated near the junction and a continuous current can be maintained. Under reverse bias, counterions inside the electrical double layers apart from the junction. As a result, ions are depleted from the junction and the current ceases. Figure 1c shows the schematics of a nanofluidic bipolar transistor. When the surface charge density in the middle of a channel is modified from negative to positive, a nanofluidic bipolar transistor consists of two back-to-back nanofluidic diodes. In this condition, the current ceases because there is a depletion region. Calculation Results. Diode. The calibration method of the 2D PNP code was mentioned in refs 7 and 9. Consider the concentration of KCl aqueous solution to be 5 mM and the surface charge densities of left and right halves to be σ+ ) 2 mC/m2 and σ- ) -2 mC/m2, respectively. Figure 2 shows the electric potential, concentration and pressure profiles, and the plot of velocity vectors for the (a) forward and (b) reverse bias of 5 V. The electric potential and pressure profiles are along y ) 0 and the concentration profile is the profile of concentration averaged over the y direction. The concentration profiles show that under forward bias, ions are accumulated around x ) 0. However, under reverse bias the ions are depleted from x ) 0. The potential profiles show that under forward bias the potential changes largely at both ends of the channel, whereas under reverse bias it changes in the middle of the channel. Because the mass flux is 2276
constant at any cross-sectional area at steady state, the electric potential changes largely in the low-density region. In comparison to semiconductor diodes, under forward bias the charge carriers of electrons and holes in semiconductor diodes are combined with each other when they move across the junction. Therefore the concentrations of electrons and holes are restricted by the concentrations of doped acceptors and donors, respectively. Whereas in nanofluidic diodes, the charge carriers of positive and negative ions are not combined and accumulated near the junction. As a result, the ion concentrations at the junction become much higher than the bulk concentrations and the potential drop is all over across the two regions. Because of the requirement of neutrality, the concentration difference between the K+ ion and the Clion in the y direction, ∆n(x) ) nK+(x) - nCl-(x), is equal to -nw(x) ) -2σ(x)/eLy except around the middle (x ) 0) and at both ends (x ) (2.5 µm) of the channel. Assuming that the charge on the wall in the channel region is a fixed charge, of which the charge density is nw(x) ) 2σ(x)/eLy, the depletion width, xd, in the full depletion approximation is given by xd ) x(20Ly/|σ|)|φi-∆φ|, where |σ| is the absolute value for the surface charge density of the wall and φi is the built-in potential. The built-in potential is an internal potential at thermal equilibrium with no applied external potential, and is given by φi ) (2kT/e)sinh-1(|σ|/enb Ly), Nano Lett., Vol. 5, No. 11, 2005
Figure 3. Current-potential (I-∆φ) curves for three different surface charge densities |σ| ) 1, 2, and 5 mC/m2. The bulk concentrations are 1, 5, and 10 mM.
Figure 4. Potential, concentration and pressure profiles (top), and velocity vectors (bottom) for three different surface charge densities at the gate (-1.0 µm e x e 1.0 µm): (a) σg ) -2 mC/m2; (b) σg ) -1 mC/m2; (c) σg ) 2 mC/m2. The electric potential and pressure profiles are along y ) 0, and the concentration profile is the profile of concentration averaged over the y direction. The surface charge density except for the gate region is assumed to be -2 mC/m2. The bulk concentration and potential bias are 5 mM and 5 V, respectively. The maximum velocities at σg ) -2 and 2 mC/m2 are 7.54 × 10-3 and 5.79 × 10-4 m/s, respectively.
where nb is the bulk concentration. At nb ) 5.0 mM and |σ| ) 2.0 mC/m2, the depletion width, xd, is calculated to be 0.326 µm, which is in good agreement with the numerical calculation results. As shown in Figure 2, under forward bias, the fluid flows along the wall (y ) (Ly /2), from the two ends of the channel toward its center and the flows out from the center to two ends along the axis of symmetry (y ) 0). However, under reverse bias the flow direction is opposite. It is noted that the maximum velocities under forward and reverse biases are 5.42 × 10-3 and 4.12 × 10-4 m/s, Nano Lett., Vol. 5, No. 11, 2005
respectively. The flow velocity under reverse bias is much smaller than that under forward bias. Figure 3 shows the current-potential (I-∆φ) curves for three different surface charge densities, |σ| ) 1, 2, and 5 mC/m2 at the bulk concentrations, nb ) 1, 5, and 10 mM. At any bulk concentration, the ionic current increases with increasing potential bias under forward bias, and it is almost zero under reverse bias. But the current-potential curves for nb ) 1.0 mM have a different trend, that is, the gradient -dI/d(∆φ) decreases with increasing the potential bias. These 2277
Figure 5. Current-potential (I-∆φ) curves for five different surface charge densities at the gate, σg ) -2, -1, 0, 1, and 2 mC/m2. The bulk concentrations are 1, 5, and 10 mM. The surface charge density except for the gate region is assumed to be -2 mC/m2.
current-potential characteristics are discussed in the following section. Transistor. Figure 4 shows the potential, concentration, and pressure profiles and velocity vectors for three different surface charge densities at the gate (-1.0 µm e x e 1.0 µm): (a) σg ) -2 mC/m2; (b) σg ) -1 mC/m2; (c) σg ) 2 mC/m2. The electric potential and pressure profiles are along y ) 0 and the concentration profile is the profile of concentration averaged over the y direction. The surface charge density, except for the gate region, σ0, is assumed to be -2 mC/m2. The bulk concentration and potential bias are 5 mM and 5 V, respectively. When the surface charge density of the whole inner wall is -2 mC/m2 (case a), the concentrations of the K+ ion and the Cl- ion are substantially constant along the channel and the potential changes linearly along the channel direction. However, when the surface under the gate is σg ) 2 mC/m2 (case c), the concentration profile shows that ions are accumulated at x ) 1.0 µm and depleted from x ) -1.0 µm. This concentration profile is similar to the combination of two concentration profiles of diodes under forward and reverse biases, and the potential changes only in the depletion region. For σg ) -1 mC/m2 (case b), the potential and concentration change nonlinearly. The pressure profiles are similar to the concentration ones. For the velocity vectors, the maximum velocities at σg ) -2 and 2 mC/m2 are 7.54 × 10-3 and 5.79 × 10-4 m/s, respectively. As σg increases from -2 to 0 mC/m2, the direction of the electrostatic force, that is, the third term of square brackets in eq 4 under the gate is the same to the flow direction but its magnitude decreases. And when σg becomes positive, the direction of the electrostatic force changes and it suppresses the fluid flow. Figure 5 shows the current-potential curves for five different surface charge densities at the gate, σg ) -2, -1, 0, 1, and 2 mC/m2. The bulk concentrations, nb, are 1, 5, and 10 mM. When the surface charge density at gate is the same as it is everywhere else along the channel, σg ) σ0 ) -2 mC/m2, the current is proportional to the potential bias at any bulk concentration. But as -σg decreases, the current does not increase proportionally and saturates at a certain potential bias. For the bulk concentration nb ) 1 mM, the 2278
current is almost zero at -σg < 0, that is, the current can be suppressed completely. However, for the bulk concentration nb ) 5 and 10 mM, the current does not become zero at -σg < 0 and a leakage current is observed. But as the bulk concentration increases, the current becomes more sensitive to σg and ∆φ. These calculation results suggest that as the bulk concentration increases, both the current and the leakage current increase. Discussion. To understand the characteristics of the nanofluidic diode and bipolar transistor, we solved the potential and concentration profiles analytically in the channel direction in 1D approximation. When the channel height is in the 10-nm order, the wall-normal components of the current density vectors are much smaller than their wall tangential components. Therefore, the potential and the ion concentrations averaged over the wall-normal direction can be solved in the channel direction using the 1D approximation. Assuming that dnK+/dx ) dnCl-/dx and DK+ ) DCl-, the reduction and summation of the Nernst-Plank equations of two ions are given as follows 1 + r JK+ e dφ )n kT dx 2 DK+
( )
( )( )
dn e σ dφ 1 - r J K+ + )dx kT eLy dx 2 DK+
(9)
(10)
where n ) (nK+ + nCl-)/2 and r ) - JCl-/JK+. From the requirement of electroneutrality, (nK+ - nCl-)/2 ) -σ/eLy. Diode. By solving eqs 9 and 10 at r ) 1, we can give the potential and average concentration at x > 0 in the following equations φ(x) ) φ(0) +
n(x) )
x
kTLy (n(x) - n(0)) σ-
n2(0) -
2σ-JK+ x eLyDK+
(11)
(12)
Nano Lett., Vol. 5, No. 11, 2005
where σ- is the surface charge density of the wall in the right half (at x > 0) and the value is negative. From eq 11, n(0) is calculated to be 71.9 mM at n(Lx /2) ) 5 mM, φ(Lx /2) ) 5 V, φ(0) ) 0.5φ(Lx /2), T ) 300 K, Ly ) 30 nm, and σ- ) -2 mC/m2. This value is in good agreement with the former calculation result. By eliminating n(0) from eqs 11 and 12, the current under forward bias is given by the following equations
15 are satisfied in the following three regions (a) -2.5 µm e x e -1.0 µm, (b) -1.0 µm e x e 1.0 µm (gate region), and (c) 1.0 µm e x e 2.5 µm, then the total of six equations can determine the two unknown parameters, JK+ and r, and the four boundary values, φ(x ) (1.0 µm) and n(x ) (1.0 µm). However, these equations cannot be solved analytically. By using two parameters, ∆φg and φ0, the current is expressed as the functions of ∆φ, s()σg/σ0) and nb.
DK+σ- e 2 [(∆φ - φ0)2 - φ02] (13) 2Lx kT
-I ) -(1 + r)eJK+Ly ) 2DK+eLy nb e [∆φ - (1 - s)∆φg + φ0], (16) Lx kT
-I ) -2eJK+Ly ) -
( )
where ∆φ is the potential bias between two ends of a channel and φ0 ) 2kTLynb/σ-. The current is quadratic with respect to ∆φ and proportional to σ- and Lx-1. In semiconductor diodes, the current changes exponentially with respect to the potential bias, whereas in nanofluidic diodes the current changes quadratically. However, it is noted that as the bulk concentration decreases and the surface charge density increases, the current-potential curves do not remain quadratic, as is evident for nb ) 1 mM in Figure 3. In the 2D nanofluidic channel with reservoirs on either side, when the relation of |∆n| ) 2|σ|/eLy > nb is satisfied, at the entrance of a nanofluidic channel, the concentration of counterions is larger than that of the coions, that is, polarization occurs. As a result, the potential bias between the two ends of a channel becomes smaller than that between the two ends of reservoirs. However, in the 1D approximation the potential and ion concentrations at the two ends of a channel are given by the bulk values. Therefore, the current given by eq 13 is overestimated. As the bulk concentration decreases further and the solution inside each half of the channel becomes a unipolar solution of counterions, the current becomes zero at any potential bias because all ions can be located in only one-half of the channel. Besides, in nanofluidic diodes, because the fluid circulates symmetrically with respect to x ) 0 as shown in Figure 2, the number of ions passing through the channel due to the electroosmotic flow is much smaller than that due to electrophoretic flow. Therefore, the effect of advection of ions on the current is negligible. Transistor. By solving eqs 9 and 10 at r * 1, we can give the potential and average concentration profiles in the following equations x - x0 ) -
[
(
n(x) + nr 2 DK + (n(x) - n(x0)) - nr ln 1 - r J K+ n(x0) + nr
φ(x) - φ(x0) ) -
()(
n(x) + nr 1 + r kT ln 1-r e n(x0) + nr
)
)]
(14) (15)
where nr ) (1 + r)/(1 - r)(σ/eLy). The average concentration, n, is given by the inverse function of the linear plus logarithm functions of x. The functions, φ(x) and n(x), are continuous in the channel region if the depletion region does not appear. When eqs 14 and Nano Lett., Vol. 5, No. 11, 2005
( )
where ∆φgis the difference in potential between two ends of the gate region and φ0 ) (kTLx Ly/σ0)(dn/dx)x)-Lx/2. The ranges of these two parameters are (Lgx /Lx)∆φ e ∆φg < ∆φ and (kTLx Ly/σ0)(- nb/L0x) < φ0 e 0, where Lgx and 2L0x are the lengths of the gate region and except for the gate region, respectively. Equation 16 includes two unknown parameters, ∆φg and φ0, but it expresses that the current is suppressed by decreasing s because, in general, ∆φg increases and φ0 decreases with decreasing s. If |φ0| , ∆φ, then the third term of square brackets in eq 16 is negligible. At ∆φ ) 5 V, nb ) 1 mM, σ0 ) -2 mC/m2, Lx ) 5 µm, L0x ) 1.5 µm, and Ly ) 30 nm, the lower limit of φ0 is calculated to be (kTLx Ly/σ0)(- nb/L0x) ) -0.125 V, which is much smaller than the potential bias of ∆φ ) 5 V. However, as the bulk concentration decreases and surface charge density increases further, the concentrations of counterions and coions are not the same even near the axis of symmetry (y ) 0) because the solution inside the channel approaches to the unipolar solution of counterions. If so, then the current given by eq 16 is underestimated, besides, the current due to electroosmotic flow is not negligible. Conclusions. The Poisson-Nernst-Planck (PNP) equations and the Navier-Stokes (NS) equations were employed to calculate ionic current along a nanofluidic channel 5 µm long and 30 nm high, with a locally modified surface charge density. The surface charge density ranged from 0 to -5 × 10-3 C/m2, the bulk concentration of KCl aqueous solution ranged from 10-3 to 10-2 M, and the potential bias ranged from -5 to 5 V. The following conclusions can be drawn from this study: 1. In a nanofluidic channel for which the surface charge densities at the right and left halves are the same absolute value but of different signs, the currentpotential (I-φ) characteristics are similar to those of semiconductor diodes as long as the electrical double layer of the two surfaces do not overlap. The calculation results show that under forward bias the current is quadratic with respect to the potential bias. 2. By controlling the surface charge density in a region along the length of the channel, the ion current can be modulated even if the electrical double layer is not overlapped. At low bulk concentration, the current is small but it is suppressed completely by modifying the surface charge density through a gate electrode. As the bulk concentration increases, both the current and leakage current increase. 2279
Acknowledgment. We thank Professor Arun Majumdar for his valuable comments on this paper. This research was partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B), 16760149, 2005. References (1) Nishizawa, M.; Menon, V. P.; Martin C. R. Science 1995, 268, 700702. (2) Kang, M.-S.; Martin, C. R. Langmuir 2001, 17, 2753-2759. (3) Gad-el-Hak, M. The MEMS Handbook; CRC Press: Boca Raton, FL, 2002.
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(4) Schasfoort, R. B. M.; Schlautmann, S.; Hendrikse, J.; van den Berg, A. Science 1999, 286, 942-945. (5) Pu, Q. S.; Yun, J. S.; Temkin, H.; Liu S. R. Nano Lett. 2004, 4, 1099-1103. (6) Stein, D.; Kruithof, M.; Dekker, C. Phys. ReV. Lett. 2004, 93, 035901. (7) Daiguji, H.; Yang, P.; Majumdar, A. Nano Lett. 2004, 4, 137-142. (8) Karnik, R.; Fan, R.; Yue, M.; Li, D.; Yang, P.; Majumdar, A. Nano Lett. 2005, 5, 943-948. (9) Daiguji, H.; Yang, P.; Szeri, J. A.; Majumdar, A. Nano Lett. 2004, 4, 2315-2321. (10) Hille, B. Ion Channels of Excitable Membrane; 3rd ed.; Sinauer Associates Inc.: Sunderland, MA, 2001.
NL051646Y
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