![UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 1
Investigation of Steady-State Carrier Distribution
in CNT Porins in Neuronal Membrane
Kyle Poe, University of the Pacific
Abstract—In this work, the carrier distribution of a carbon
nanotube inserted into the spinal ganglion neuronal membrane
is examined. After primary characterization based on previous
work, the nanotube is approximated as a one-dimensional
system, and the Poisson and Schr¨odinger equations are solved
using an iterative finite-difference scheme. It was found that
carriers aggregate near the center of the tube, with a negative
carrier density of ρn = 7.89 × 1013
cm−3
and positive car-
rier density of ρp = 3.85 × 1013
cm−3
. In future work, the
erratic behavior of convergence will be investigated.
I. INTRODUCTION
IN this work, we consider the equilibrium carrier distri-
bution of a semiconducting carbon nanotube porin (CNT
porin) in the cellular membrane of the spinal ganglion
neuron. It has recently been demonstrated that nanotubes
of inner diameter 1.51 ± 0.21 nm and lengths comparable
to or slightly greater than the thickness of the membrane
self-insert into DOPC membranes, of thickness 4.6 ± 0.2
nm with a low angular deviation [1]. Furthermore, it was
shown that these CNTs could conduct ions, hence the name
porin. The variously metallic, conducting, or semiconducting
nature of CNTs in addition to their porin-esque proper-
ties provide an intriguing foothold for the investigation of
biomedical and pharmaceutical technologies which exploit
CNT properties [2], [3]. Due to the electrically intriguing
properties of this material and its self-inserting behavior,
it is worthwhile to investigate how they respond to the
neuronal membrane environment to better understand their
response to various other stimulus in future work. Recent
studies have characterized the electrical potential across
spinal ganglion neuron membranes in resting and action
potential states [4]. Having characterized this potential, we
may then look to model the response of the CNT porin
based on equilibrium considerations. As a first step in the
investigation of potential applications in pharmaceuticals and
biomedical devices which could employ CNT porins, we first
set out to characterize the steady-state carrier distribution of
CNT porins. By understanding the carrier distribution, other
electronic properties of the nanotube may be studied.
II. METHODS
A. CNT Characterization
The CNT model considered in this work is identical to
those examined by Geng et al [1]. They have diameter 1.51
K. Poe was mentored by Rahim Khoie in this work.
± 0.21 nm. The chiral indices for such a tube may be found
by considering the equation for tube diameter, where a =
0.246 nm is the separation between carbon atoms [5]
d =
a
π
(n2 + nm + m2) (1)
B. Membrane Characterization
While the electrical environment of the cellular membrane
has been modeled in some detail by Pinto et al [4], as
a first approach, we consider a potential constant within
the cytoplasm and outside the cell. Since the phospholipid
bilayer of the membrane is very hydrophobic [6], we assume
a linear gradient from the inside to outside with a potential
difference equal to −77 mV (see Fig. 1).
Fig. 1. Cross-section of the cellular membrane-nanotube environment
Due to the relative hydrophobicity of the bilayer, any
deviation from a linear potential gradient will arise from
carriers in the CNT (see figure 6).
C. Potential
Considering that the charge distribution and electrical po-
tential are interdependent, the resultant potential and charge
distribution must satisfy both the Poisson and Schr¨odinger
equations. Since here we consider the CNT to be a quasi-one
dimensional system, Poisson’s equation reduces to (2)
d2
V
dz2
=
−Q(z)
εt
(2)
where the dielectric constant of the tube is taken to be 1 [7],
and with the charge distribution given by
Q(z) = q(p(z) − n(z)) (3)](https://image.slidesharecdn.com/researchwriteupv2-190107202512/85/Investigation-of-Steady-State-Carrier-Distribution-in-CNT-Porins-in-Neuronal-Membrane-1-320.jpg)
![UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 2
where q is the elementary charge and p(z), n(z) are the
carrier densities for holes and electrons respectively.
In the present work, the voltage is taken to be fixed within
the cell and outside of the cell, but since carriers will exist
within the bilayer due to the CNT, the voltage may vary
non-linearly. It should be noted that this constant voltage
outside the bilayer is only an approximation, and does not
claim to be a rigorous assumption.
Here we assume that due to the abundance of electrolyte,
the potential is defined by equation (4), with a constant
potential of Vins = −77 mV in the cytosol and Vout = 0 mV
in the extracellular matrix (ECM).
V : z →
Vins, z ∈ Cytosol
Vout, z ∈ ECM
VB(z), z ∈ Bilayer
(4)
Since the potential is free to vary within the bilayer, we must
solve the Poisson equation with boundary conditions fixed by
the potential in the cytosol and ECM. This is accomplished
by constructing the Laplacian for the bilayer using a finite-
difference method and rewriting the Poisson equation as a
matrix-vector equation [8]
LB =
∆zB
NB
2
−2 1 0 · · · 0
1 −2 1 · · · 0
0 1 −2 · · · 0
...
...
...
...
...
0 0 0 · · · −2
(5)
where ∆zB denotes the thickness of the bilayer, and NB is
the number of points in out finite difference scheme within
the bilayer. Poisson’s equation for the bilayer then becomes
LBVB =
−Q
εt
− ρBC (6)
where ρBC = (∆zB/NB)2
[Vins, 0, 0, · · · ]T
is the vector
enforcing the Dirichlet boundary conditions. To solve for the
potential VB, the system is subjected to numerical inversion.
D. Carrier Distribution
To compute the steady-state electron concentration distri-
bution n(z) and hole distribution p(z) as well as the allowed
energies it is necessary to consider solutions to the time-
independent Schr¨odinger equation (7)
Eψ =
− 2
2m∗
e
d2
ψ
dz2
+ Up/n(z)ψ (7)
Where m∗
e = is the effective mass of the electron. The
effective mass in the tube is approximately m∗
e = 0.35me,
where me is the electron rest mass [9].
Assuming that the local electrostatic potential rigidly
shifts the local band structure [10], the potential function
that the electrons and holes see is found by
Un(z) = −qV (z) − χCN (8)
Up(z) = Eg − Un(z) (9)
Where χCN denotes electron affinity and Eg denotes the
bandgap energy. To find n(z) and p(z), we must consider
the contribution of each possible energy state for the carriers
in the first band. After using finite differences to discretize
the domain by zn = Lt
n
N , n ∈ {0, 1, · · · , N} we solve the
eigenvalue problem for a given ψ(En) = ψn to numeri-
cally approximate the allowed energy states, using a central
difference quotient for the Laplacian of the system
LS =
Lt
N
2
−2 1 0 · · · 0
1 −2 1 · · · 0
0 1 −2 · · · 0
...
...
...
...
...
0 0 0 · · · −2
(10)
we may express the Hamiltonian of the system as a matrix
Hp/n =
− 2
2m∗
e
LS + diag(Up/n) (11)
for which we may solve the eigenvalue problem to obtain
the allowed energies for holes and electrons
Enψn = Hp/nψn (12)
with the typical vanishing condition ψ(−Lt/2) =
ψ(Lt/2) = 0 built in. Once the wavefunctions have been
found, they are normalized to |ψn|2
= 1. Each of these
wavefunctions represent the spatial probability function for
a carrier with a given energy En. For a state density function
g(E) and state probability function f(E) the total carrier
count (shown for electrons, similar for holes) is given by
equation (13) [11]
ntot =
∞
EC
fn(E)gn(E) dE (13)
For a CNT, the state density function is given in [12] by
equation (14)
gn(E) = 2
All Bands
i
4
πVppπa
√
3
E
E2 − E2
cmini
(14)
where Vppπ = 2.5 eV is the nearest neighbor overlap integral
and Ecmini
is the lowest energy level of the current energy
band. The state probability function may be approximated
by considering the state probability to be given by the Fermi-
Dirac distribution, given that the membrane voltage is small
and does not significantly shift the band structure
fn(E) =
1
e(E−EF )/(kT ) + 1
(15)
where the Fermi Energy is approximated as constant, the
average of the vacuum energy level imposed by the electrical
potential minus the work function
EF,n =
−q
Lt
Lt/2
−Lt/2
V (z) dz − φCN (16)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Investigation of Steady-State Carrier Distribution in CNT Porins in Neuronal Membrane
- 1. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 1 Investigation of Steady-State Carrier Distribution in CNT Porins in Neuronal Membrane Kyle Poe, University of the Pacific Abstract—In this work, the carrier distribution of a carbon nanotube inserted into the spinal ganglion neuronal membrane is examined. After primary characterization based on previous work, the nanotube is approximated as a one-dimensional system, and the Poisson and Schr¨odinger equations are solved using an iterative finite-difference scheme. It was found that carriers aggregate near the center of the tube, with a negative carrier density of ρn = 7.89 × 1013 cm−3 and positive car- rier density of ρp = 3.85 × 1013 cm−3 . In future work, the erratic behavior of convergence will be investigated. I. INTRODUCTION IN this work, we consider the equilibrium carrier distri- bution of a semiconducting carbon nanotube porin (CNT porin) in the cellular membrane of the spinal ganglion neuron. It has recently been demonstrated that nanotubes of inner diameter 1.51 ± 0.21 nm and lengths comparable to or slightly greater than the thickness of the membrane self-insert into DOPC membranes, of thickness 4.6 ± 0.2 nm with a low angular deviation [1]. Furthermore, it was shown that these CNTs could conduct ions, hence the name porin. The variously metallic, conducting, or semiconducting nature of CNTs in addition to their porin-esque proper- ties provide an intriguing foothold for the investigation of biomedical and pharmaceutical technologies which exploit CNT properties [2], [3]. Due to the electrically intriguing properties of this material and its self-inserting behavior, it is worthwhile to investigate how they respond to the neuronal membrane environment to better understand their response to various other stimulus in future work. Recent studies have characterized the electrical potential across spinal ganglion neuron membranes in resting and action potential states [4]. Having characterized this potential, we may then look to model the response of the CNT porin based on equilibrium considerations. As a first step in the investigation of potential applications in pharmaceuticals and biomedical devices which could employ CNT porins, we first set out to characterize the steady-state carrier distribution of CNT porins. By understanding the carrier distribution, other electronic properties of the nanotube may be studied. II. METHODS A. CNT Characterization The CNT model considered in this work is identical to those examined by Geng et al [1]. They have diameter 1.51 K. Poe was mentored by Rahim Khoie in this work. ± 0.21 nm. The chiral indices for such a tube may be found by considering the equation for tube diameter, where a = 0.246 nm is the separation between carbon atoms [5] d = a π (n2 + nm + m2) (1) B. Membrane Characterization While the electrical environment of the cellular membrane has been modeled in some detail by Pinto et al [4], as a first approach, we consider a potential constant within the cytoplasm and outside the cell. Since the phospholipid bilayer of the membrane is very hydrophobic [6], we assume a linear gradient from the inside to outside with a potential difference equal to −77 mV (see Fig. 1). Fig. 1. Cross-section of the cellular membrane-nanotube environment Due to the relative hydrophobicity of the bilayer, any deviation from a linear potential gradient will arise from carriers in the CNT (see figure 6). C. Potential Considering that the charge distribution and electrical po- tential are interdependent, the resultant potential and charge distribution must satisfy both the Poisson and Schr¨odinger equations. Since here we consider the CNT to be a quasi-one dimensional system, Poisson’s equation reduces to (2) d2 V dz2 = −Q(z) εt (2) where the dielectric constant of the tube is taken to be 1 [7], and with the charge distribution given by Q(z) = q(p(z) − n(z)) (3)
- 2. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 2 where q is the elementary charge and p(z), n(z) are the carrier densities for holes and electrons respectively. In the present work, the voltage is taken to be fixed within the cell and outside of the cell, but since carriers will exist within the bilayer due to the CNT, the voltage may vary non-linearly. It should be noted that this constant voltage outside the bilayer is only an approximation, and does not claim to be a rigorous assumption. Here we assume that due to the abundance of electrolyte, the potential is defined by equation (4), with a constant potential of Vins = −77 mV in the cytosol and Vout = 0 mV in the extracellular matrix (ECM). V : z → Vins, z ∈ Cytosol Vout, z ∈ ECM VB(z), z ∈ Bilayer (4) Since the potential is free to vary within the bilayer, we must solve the Poisson equation with boundary conditions fixed by the potential in the cytosol and ECM. This is accomplished by constructing the Laplacian for the bilayer using a finite- difference method and rewriting the Poisson equation as a matrix-vector equation [8] LB = ∆zB NB 2 −2 1 0 · · · 0 1 −2 1 · · · 0 0 1 −2 · · · 0 ... ... ... ... ... 0 0 0 · · · −2 (5) where ∆zB denotes the thickness of the bilayer, and NB is the number of points in out finite difference scheme within the bilayer. Poisson’s equation for the bilayer then becomes LBVB = −Q εt − ρBC (6) where ρBC = (∆zB/NB)2 [Vins, 0, 0, · · · ]T is the vector enforcing the Dirichlet boundary conditions. To solve for the potential VB, the system is subjected to numerical inversion. D. Carrier Distribution To compute the steady-state electron concentration distri- bution n(z) and hole distribution p(z) as well as the allowed energies it is necessary to consider solutions to the time- independent Schr¨odinger equation (7) Eψ = − 2 2m∗ e d2 ψ dz2 + Up/n(z)ψ (7) Where m∗ e = is the effective mass of the electron. The effective mass in the tube is approximately m∗ e = 0.35me, where me is the electron rest mass [9]. Assuming that the local electrostatic potential rigidly shifts the local band structure [10], the potential function that the electrons and holes see is found by Un(z) = −qV (z) − χCN (8) Up(z) = Eg − Un(z) (9) Where χCN denotes electron affinity and Eg denotes the bandgap energy. To find n(z) and p(z), we must consider the contribution of each possible energy state for the carriers in the first band. After using finite differences to discretize the domain by zn = Lt n N , n ∈ {0, 1, · · · , N} we solve the eigenvalue problem for a given ψ(En) = ψn to numeri- cally approximate the allowed energy states, using a central difference quotient for the Laplacian of the system LS = Lt N 2 −2 1 0 · · · 0 1 −2 1 · · · 0 0 1 −2 · · · 0 ... ... ... ... ... 0 0 0 · · · −2 (10) we may express the Hamiltonian of the system as a matrix Hp/n = − 2 2m∗ e LS + diag(Up/n) (11) for which we may solve the eigenvalue problem to obtain the allowed energies for holes and electrons Enψn = Hp/nψn (12) with the typical vanishing condition ψ(−Lt/2) = ψ(Lt/2) = 0 built in. Once the wavefunctions have been found, they are normalized to |ψn|2 = 1. Each of these wavefunctions represent the spatial probability function for a carrier with a given energy En. For a state density function g(E) and state probability function f(E) the total carrier count (shown for electrons, similar for holes) is given by equation (13) [11] ntot = ∞ EC fn(E)gn(E) dE (13) For a CNT, the state density function is given in [12] by equation (14) gn(E) = 2 All Bands i 4 πVppπa √ 3 E E2 − E2 cmini (14) where Vppπ = 2.5 eV is the nearest neighbor overlap integral and Ecmini is the lowest energy level of the current energy band. The state probability function may be approximated by considering the state probability to be given by the Fermi- Dirac distribution, given that the membrane voltage is small and does not significantly shift the band structure fn(E) = 1 e(E−EF )/(kT ) + 1 (15) where the Fermi Energy is approximated as constant, the average of the vacuum energy level imposed by the electrical potential minus the work function EF,n = −q Lt Lt/2 −Lt/2 V (z) dz − φCN (16)
- 3. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 3 where φCN is the work function of the CNT. Considering that the energy space is naturally partitioned by the eigen- spectrum of the Hamiltonian, we may furthermore assert that ni = Ei+1 Ei f(E)g(E) dE (17) where ni is the carrier density in the ith state. Since the probability density for a carrier in a given state is given by |ψi(z)|2 , we may find the (discretized) carrier distribution to be n(z) = N i=1 ni|ψi(z)|2 (18) For holes, the same treatment is given, except from the perspective of the energy that the holes “see”. While the energy levels and the distributions are still given by the same functions, it is important to note that the approximate Fermi energy as “seen” by holes is EF,p = −EF,n. This “negative energy” treatment comes from the view that holes behave in a way similar to antiparticles, in that they annihilate with electrons. E. Numerical Scheme To generate a convergent solution for the system of equations that has been constructed, we employ an itera- tive Picard scheme, as has been previously employed for Schr¨odinger-Poisson solvers [10] Vk+1 = Vk − αP−1 (rk), rk = Q(Vk) − P(Vk) where P(V ) = Q is equation (6) written such that the charge is a function of potential and α ≤ 1 is a damping factor. From some initial guess, we may calculate the carrier distribution, and using equation (3) we calculate the residual of the iteration. Measuring the magnitude of the residual gives an idea of the agreement between the potential as seen by Schr¨odinger’s and by Poisson’s equation. III. RESULTS In this study we examine a zigzag nanotube with ideal geometry to self-insert into the spinal ganglion neuron. Considering that the most ideal diameter identified in [1] was 1.51 ± 0.21, we treat nanotubes with the chiral vector specified by (n, 0) = (1.51 ± 0.21)π 0.246 , 0 ≈ (19 ± 3, 0) per equation (1). Assuming a constant proportionality be- tween bilayer thickness and optimal tube length, the optimal length for the CNT for spinal ganglion neuron insertion is assumed to approximately be LSGN = TSGN TDOP C LDOP C = 16 nm (19) Conductivity is not a strong function of length for CNTs, so this is a safe assumption [3]. −8 −6 −4 −2 0 2 4 6 8 z (nm) −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0.00 V0(V) Fig. 2. Initial guess for potential V0 Since the solution to Poisson without any charges in the bilayer is linear, our initial guess VB for the bilayer potential is that it is linear over the bilayer (see (4)) VB(z) = Vins 1 − z − z0 ∆zB (20) where z0 denotes the first part of the bilayer as measured from the interior of the cell, and positive z is closer to the outside of the cell (see fig 2). For χCN = 4.2 eV, Eg = 0.62 eV, and φCN = 4.5 eV (approximated from [10]), and body temperature, we run the solver for k = 331 iterations. In the course of this solver, the Schr¨odinger equation was solved for both holes and electrons (see fig 3). After computing the wavefunctions, z (nm)<latexit sha1_base64="HYe1xDb0F4S14OLh2R2Wg1cPN5U=">AAAB9XicbVBNSwMxEM3Wr1q/qh69BItQQcquCHosevFYwX5Au5Zsmm1Dk+ySzKp16f/w4kERr/4Xb/4b03YP2vpg4PHeDDPzglhwA6777eSWlldW1/LrhY3Nre2d4u5ew0SJpqxOIxHpVkAME1yxOnAQrBVrRmQgWDMYXk385j3ThkfqFkYx8yXpKx5ySsBKd0+dk3IH2COkSo6Pu8WSW3GnwIvEy0gJZah1i1+dXkQTyRRQQYxpe24Mfko0cCrYuNBJDIsJHZI+a1uqiGTGT6dXj/GRVXo4jLQtBXiq/p5IiTRmJAPbKQkMzLw3Ef/z2gmEF37KVZwAU3S2KEwEhghPIsA9rhkFMbKEUM3trZgOiCYUbFAFG4I3//IiaZxWPLfi3ZyVqpdZHHl0gA5RGXnoHFXRNaqhOqJIo2f0it6cB+fFeXc+Zq05J5vZR3/gfP4AMfGSTQ==</latexit><latexit 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Wavefunctions for holes and electrons superimposed over energy band diagram at final potential solution. Normalized, squared wavefunctions are shown for each of the first three energy levels, E0 = −4.18 eV, E1 = −4.15 eV, E2 = −4.12 eV. Wavefunctions for holes are inverted (higher concentrations are down) for readability. the number of carriers were computed for each energy level 133 was not decided upon purposefully. It was simply decided that the solver was not going to improve past that point by inspection.
- 4. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 4 −8 −6 −4 −2 0 2 4 6 8 z (nm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Concentration(carriers/cm−3) 1e14 n(z) p(z) ⟨ρn⟩ ⟨ρp⟩ Fig. 4. Electron and hole concentrations n(z) and p(z) over the CNT, given in carriers per cubic centimeter. Average carrier densities are shown, with ρn = 7.89 × 1013 cm−3 and ρp = 3.85 × 1013 cm−3 using numerical quadrature to solve (13). Using equation (18), the resulting carrier concentrations were computed. The resulting carrier concentrations were found to aggregate near the middle of the tube (fig 4). Further, it was found that there was a far greater concentration of electrons on the nanotube than holes, with an average concentration of ρn = 7.89 × 1013 cm−3 for electrons and ρp = 3.85 × 1013 cm−3 for holes. After 33 iterations, there is no evidence of convergence at α = 0.2, with the residual oscillating chaotically such that r ∈ [1.4E − 23, 2.2E − 23], where denotes the 2-norm as a measure of magnitude (see fig 5). This was tested across a range of N and α values, with no significant changes. This may suggest that either the model is inaccurate, or something is computationally incorrect. As articulated in the problems section, it is more likely that the problem lies with the model. The final potential distribution is very similar to the initial 0 5 10 15 20 25 30 Nth residual 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 Magnitude 1e−23 Residual Magnitude Fig. 5. Magnitude of the residual as a function of scheme iteration guess, with an approximately Gaussian difference near the center of the tube (fig 6). −8 −6 −4 −2 0 2 4 6 8 z (nm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 V−V0(V) 1e−16 Fig. 6. Difference between the the final V (z) and initial guess V0(z) IV. DISCUSSION AND FUTURE WORK In this work, strides were made toward the characteriza- tion of CNTs in the neuronal membrane. While it may not be an exact result, it is nontrivial that carrier concentrations were determined in the expected range. Before any more work is done with this problem, the problems mentioned in appendix A must be addressed. In the future, we hope to examine the transient response to the neuron action potential, after improving results for this model. Progress in understanding the behavior of short CNT porins in vivo is crucial to the development of biotechnology which exploits the unique properties of carbon nanotubes in the cellular membrane. REFERENCES [1] J. Geng, K. Kim, J. Zhang, A. Escalada, R. Tunuguntla, L. R. Comolli, F. I. Allen, A. V. Shnyrova, K. R. Cho, D. Munoz, Y. M. Wang, C. P. Grigoropoulos, C. M. Ajo-Franklin, V. A. Frolov, and A. Noy, “Stochastic transport through carbon nanotubes in lipid bilayers and live cell membranes,” Nature, vol. 514, p. 612, oct 2014. [Online]. Available: http://dx.doi.org/10.1038/nature13817http: //10.0.4.14/nature13817 [2] R. Alshehri, A. M. Ilyas, A. Hasan, A. Arnaout, F. Ahmed, and A. Memic, “Carbon Nanotubes in Biomedical Applications: Factors, Mechanisms, and Remedies of Toxicity,” Journal of Medicinal Chemistry, vol. 59, no. 18, pp. 8149–8167, sep 2016. [Online]. Available: https://doi.org/10.1021/acs.jmedchem.5b01770 [3] H. He, L. A. Pham-Huy, P. Dramou, D. Xiao, P. Zuo, and C. Pham- Huy, “Carbon nanotubes: Applications in pharmacy and medicine,” BioMed Research International, vol. 2013, no. November 2014, 2013. [4] T. M. Pinto, R. S. Wedemann, and C. M. Cortez, “Modeling the electric potential across neuronal membranes: The effect of fixed charges on spinal ganglion neurons and neuroblastoma cells,” PLoS ONE, vol. 9, no. 5, 2014. [5] R. A. Bell, Conduction in Carbon Nanotube Networks, 2015. [Online]. Available: http://link.springer.com/10.1007/978-3-319-19965-8 [6] J. F. Nagle and S. Tristram-Nagle, “Structure of lipid bilayers,” Biochimica et Biophysica Acta (BBA) - Reviews on Biomembranes, vol. 1469, no. 3, pp. 159–195, 2000. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0304415700000162
- 5. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 5 [7] F. L´eonard and J. Tersoff, “Dielectric response of semiconducting carbon nanotubes,” Applied Physics Letters, vol. 81, no. 25, pp. 4835–4837, dec 2002. [Online]. Available: https://doi.org/10.1063/1. 1530373 [8] P. J. Olver, “Finite Differences BT - Introduction to Partial Differential Equations,” P. J. Olver, Ed. Cham: Springer International Publishing, 2014, pp. 181–214. [Online]. Available: https://doi.org/ 10.1007/978-3-319-02099-0{ }5 [9] K. El Shabrawy, K. Maharatna, D. Bagnall, and B. M. Al-Hashimi, “Modeling SWCNT bandgap and effective mass variation using a monte carlo approach,” IEEE Transactions on Nanotechnology, vol. 9, no. 2, pp. 184–193, 2010. [10] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, “A Schr¨odinger-Poisson Solver for Modeling Carbon Nanotube FETs,” Proc NSTI Nanotech, vol. 3, no. 1, pp. 2–5, 2004. [Online]. Available: http: //scholar.google.com/scholar?hl=en{&}btnG=Search{&}q=intitle: A+Poisson+Solver+for+Modeling+Carbon+Nanotube+FETs{#}0 [11] R. F. Pierret, Advanced semiconductor fundamentals / R.F. Pierret., dec 2018. [12] H. Usmani, R. Singh, and N. Gupta, “Calculation of carrier density and effective mass of carbon nanotube,” International Journal of Advanced Research in Electronics and Communication Engineering, vol. 1, no. 6, pp. 36–39, 2012. APPENDIX A ASSUMPTIONS AND VALIDITY OF THE MODEL Ultimately, it is difficult to draw rigorous conclusions, as the solutions did not converge. It is possible that an invalid assumption was made about the nature of the system that caused something to fail. Here, the various difficulties that were faced throughout this investigation will be discussed. A. The Cellular Membrane Environment In [4], the membrane environment is carefully character- ized and a mathematical model is assembled to describe the various influences of the cellular environment on the membrane potential. To produce this work, there was not enough time to consider the development of a model that would complement the mathematical structure of Pinto et al.’s paper. Possibly incorrect assumptions may include the cytoplasmic/ECM charge distribution having a non- negligible effect on the tube, or capacitative effects of the charge distribution on the membrane. B. Boundary Conditions of Poisson’s Equation Perhaps the least rigorous assumption made in this work was that V (z) was constant where z was not within the cellular membrane. This decision was largely motivated by inadequate understanding of the effects of the electrolyte- CNT junction and implications of the 2-D representation. When considering the carrier distribution per the results of the Schr¨odinger equation, it was evident that the charge distribution was nonzero outside of the membrane. Future work should address this by: • Changing the boundary conditions such that only the very ends of the CNT are fixed at Vins and Vext, and in- cluding some ρ(z) term that contains information about the influence of the environment on charge distribution • Altering the topology of the model to respect the permeable nature of the CNT (more on this later) • Modifying the boundary conditions for the Schr¨odinger equation to agree with a constant external potential (not ideal) C. Topological Incorrectness At the outset of this project, one of the primary motivators for dealing with CNTs is that they may often be modeled as one-dimensional systems. This is very nice when considering conductance through the nanotube where the tube itself is the conductor, and the system has azimuthal symmetry. Aiding this is the fact that carbon nanotubes are nanoscopically thin, with the “walls” being on the order of Angstroms thick. In effect, this grants the nanotube an intrinsic two-dimensional topology, where attributes of the system may be described as a function of z (length) and ϕ (azimuth). In this problem, we are faced with the tricky circumstance of a tube which maintains a steady flow of ions due to the existence of an electrochemical gradient across the membrane. There is little reason to suspect that this effect is insignificant or identical to the current of electrons within the nanotube lattice itself. In reality, the system is then a function of radius and length—however, there is no reason to suspect that azimuthal symmetry is not still preserved, assuming that the CNT insertion has negligible angular aberration and that the local electrochemical gradient is locally perpendicular to the membrane. While the model could have been modified to 2D, it is not clear how exactly the internal ionic flow would have been accounted for. Future approaches to better tackle the problem include • Choosing a 2D instead of 1D model • Modeling the immediate environment outside the CNT rather than only the CNT itself • Investigating the properties of interfaces between fluids and solids on the quantum scale D. CNT Properties There were several proposed properties of CNTs used in this work. Among them were • m∗ e - The effective mass • χCN - Electron affinity • Eg - Bandgap energy • gn(E) - The state density function • EF - The Fermi level The effective mass of a carrier within a nanotube is a strong function of the chiral vector. The value used in this work was taken from analysis of a plot for zigzag nanotubes in [9]. It was assumed to be equal for electrons and holes in this work. Further, the electron affinity and bandgap energies used were taken from [10]. It is not certain that these values were absolutely correct. Although these certainly are potentially offensive, these are all constant properties. What may be more cause for alarm are those which are not. The state density function gn(E) was derived in the cited work from an expression for the energy states involving wavevectors. Notably, this
- 6. UNDERGRADUATE RESEARCH. WRITTEN DECEMBER 2018 6 function has very different properties from the one developed in Semiconductor Device Fundamnetals [11], which gave the density of states as proportional to the square root of the difference between the energy and the bottom of the conductance band. Perhaps the most problematic of all was the Fermi level EF and the attached Fermi-Dirac distribution. Per the au- thor’s understanding, the Fermi level is only a well-defined concept for systems at equilibrium, which this is not. In certain cases, a quasi-Fermi level may be used, but it was unclear whether or not it was an applicable concept in this circumstance. Ideally, another description should be used for non-equilibrium state probability densities. E. Numerical Implementation While there is little doubt in the general efficacy of the numerical approach employed, there is a possibility that the scheme implemented was not robust enough for the problem at hand. It was evident that the scheme did not converge, and it is possible that this was partially due to selecting a range of damping parameters that were simply too large. In a future implementation, an adaptive scheme should be used that can better respond to a poorly-converging system.