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December  2016, 11(4): 603-625. doi: 10.3934/nhm.2016011

A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion

1. 

Department of Information Engineering, Computer Sciences and Mathematics, University of L'Aquila - via Vetoio, loc. Coppito, I-67100 L'Aquila, Italy, Italy, Italy

Received  December 2014 Revised  May 2016 Published  October 2016

In this paper we present a suitable mathematical model to describe the behaviour of a hybrid electrolyte-oxide-semiconductor (EOS) device, that could be considered for application to neuro-prothesis and bio-devices. In particular, we discuss the existence and uniqueness of solutions also including the effects of the size exclusion in narrow structures such as ionic channels or nanopores. The result is proved using a fixed point argument on the whole domain.
    Our results provide information about the charge distribution and the potential behaviour on the device domain, and can represent a suitable framework for the development of stable numerical tools for innovative nanodevice modelling.
Citation: Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011
References:
[1]

U. Ascher, J. Christiansen and R. D. Russell, Collocation software for boundary-value ODEs,, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209. doi: 10.1145/355945.355950. Google Scholar

[2]

J. N. Y. Aziz, R. Genov, B. L. Bardakjian, M. Derchansky and P. L. Carlen, Brain-silicon interface for high-resolution in vitro neural recording,, IEEE Transactions on Biomedical Circuits and Systems, 1 (2007), 56. Google Scholar

[3]

G. Bader and U. Ascher, A new basis implementation for a mixed order boundary value ODE solver,, SIAM J. Sci. Stat. Comput., 8 (1987), 483. doi: 10.1137/0908047. Google Scholar

[4]

R. Baronas, F. Ivanauskas and J. Kulys, Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians,, Springer Science & Business Media, (2010). doi: 10.1007/978-90-481-3243-0_5. Google Scholar

[5]

S. Baumgartner and C. Heitzinger, Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors,, Commun. Math. Sci, 10 (2012), 693. doi: 10.4310/CMS.2012.v10.n2.a13. Google Scholar

[6]

M. Bayer, C. Uhl and P. Vogl, Theoretical study of electrolyte gate AlGaN/GaN field effect transistors,, Journal of Applied Physics, 97 (2005). doi: 10.1063/1.1847730. Google Scholar

[7]

S. Birner, Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces,, Ph.D thesis, (2011). Google Scholar

[8]

S. Birner, S. Hackenbuchner, M. Sabathil, G. Zandler, J.A. Majewski, T. Andlauer, T. Zibold, R. Morschl, A. Trellakis and P. Vogl, Modeling of Semiconductor Nanostructures with nextnano3,, Acta Physica Polonica A, 110 (2006), 111. doi: 10.12693/APhysPolA.110.111. Google Scholar

[9]

S. Birner, C. Uhl, M. Bayer and P. Vogl, Theoretical model for the detection of charged proteins with a silicon-on-insulator sensor,, Journal of Physics: Conference Series, 107 (2008). doi: 10.1088/1742-6596/107/1/012002. Google Scholar

[10]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM Journal on Applied Mathematics, 67 (2007), 960. doi: 10.1137/060664689. Google Scholar

[11]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961. Google Scholar

[12]

E. Cianci, S. Lattanzio, G. Seguini, S. Vassanelli and M. Fanciulli, Atomic layer deposited $TiO_2$ for implantable brain-chip interfacing devices,, Thin Solid Films, 520 (2012), 4745. Google Scholar

[13]

C. De Falco, E. Gatti, A. L. Lacaita and R. Sacco, Quantum-corrected drift-diffusion models for transport in semiconductor devices,, Journal of Computational Physics, 204 (2005), 533. doi: 10.1016/j.jcp.2004.10.029. Google Scholar

[14]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model,, Physical Chemistry Chemical Physics, 15 (2013), 7075. doi: 10.1039/c3cp44390f. Google Scholar

[15]

P. Fromherz, Semiconductor chips with ion channels, nerve cells and brain,, Physica E: Low-dimensional Systems and Nanostructures, 16 (2003), 24. doi: 10.1016/S1386-9477(02)00578-7. Google Scholar

[16]

P. Fromherz, Three levels of neuroelectronic interfacing,, Annals of the New York Academy of Sciences, 1093 (2006), 143. Google Scholar

[17]

P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips,, Solid-State Electronics, 52 (2008), 1364. Google Scholar

[18]

I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 48 (1997), 45. doi: 10.1007/PL00001469. Google Scholar

[19]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, Journal of Physics: Condensed Matter, 14 (2002), 12129. doi: 10.1088/0953-8984/14/46/317. Google Scholar

[20]

W. M. Grill, S. E. Norman and R. V. Bellamkonda, Implanted neural interfaces: Biochallenges and engineered solutions,, Annual Review of Biomedical Engineering, 11 (2009), 1. doi: 10.1146/annurev-bioeng-061008-124927. Google Scholar

[21]

Y. He, I. Gamba, H.-C. Lee and K. Ren, On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells,, SIAM Journal on Applied Mathematics, 75 (2015), 2515. doi: 10.1137/130935148. Google Scholar

[22]

C. Heitzinger, R. Kennell, G. Klimeck, N. Mauser, M. McLennan and C. Ringhofer, Modeling and simulation of field-effect biosensors (BioFETs) and their deployment on the nanoHUB,, Journal of Physics: Conference Series, 107 (2008). doi: 10.1088/1742-6596/107/1/012004. Google Scholar

[23]

C. Heitzinger and G. Klimeck, Computational aspects of the three-dimensional feature-scale simulation of silicon-nanowire field-effect sensors for DNA detection,, Journal of Computational Electronics, 6 (2007), 387. doi: 10.1007/s10825-006-0139-x. Google Scholar

[24]

C. Heitzinger, N. J. Mauser and C. Ringhofer, Multiscale modeling of planar and nanowire field-effect biosensors,, SIAM Journal on Applied Mathematics, 70 (2010), 1634. doi: 10.1137/080725027. Google Scholar

[25]

A. Jüngel and I. V. Stelzer, Existence Analysis of Maxwell-Stefan Systems for Multicomponent Mixtures,, SIAM Journal on Mathematical Analysis, 45 (2013), 2421. doi: 10.1137/120898164. Google Scholar

[26]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Springer Science & Business Media, (1986). doi: 10.1007/978-3-7091-3678-2. Google Scholar

[27]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag: Berlin, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[28]

M. Mojarradi, D. Binkley, B. Blalock, R. Andersen, N. Ulshoefer, T. Johnson and L. Del Castillo, A miniaturized neuroprosthesis suitable for implantation into the brain,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11 (2003), 38. doi: 10.1109/TNSRE.2003.810431. Google Scholar

[29]

X. Navarro, T.B Krueger, N. Lago, S. Micera, T. Stieglitz and P. Dario, A critical review of interfaces with the peripheral nervous system for the control of neuroprostheses and hybrid bionic systems,, Journal of the Peripheral Nervous System, 10 (2005), 229. doi: 10.1111/j.1085-9489.2005.10303.x. Google Scholar

[30]

Y. Ohno, K. Maehashi, Y. Yamashiro and K. Matsumoto, Electrolyte-gated graphene field-effect transistors for detecting pH and protein adsorption,, Nano Letters, 9 (2009), 3318. doi: 10.1021/nl901596m. Google Scholar

[31]

W. R. Patterson, Y. Song, C. W. Bull, I. Ozden, A. P. Deangellis, C. Lay, J. L. McKay, A. V. Nurmikko, J. D. Donoghue and B. W. Connors, A microelectrode/microelectronic hybrid device for brain implantable neuroprosthesis applications,, IEEE Transactions on Biomedical Engineering, 51 (2004), 1845. doi: 10.1109/TBME.2004.831521. Google Scholar

[32]

I. Peitz and P. Fromherz, Electrical interfacing of neurotransmitter receptor and field effect transistor,, The European Physical Journal E: Soft Matter and Biological Physics, 30 (2009), 223. doi: 10.1140/epje/i2009-10461-3. Google Scholar

[33]

R. Popovtzer, A. Natan and Y. Shacham-Diamand, Mathematical model of whole cell based bio-chip: An electrochemical biosensor for water toxicity detection,, Journal of Electroanalytical Chemistry, 602 (2007), 17. doi: 10.1016/j.jelechem.2006.11.022. Google Scholar

[34]

M.J. Schöning and A. Poghossian, Bio FEDs (Field-Effect Devices): State-of-the-Art and New Directions,, Electroanalysis, 18 (2006), 1893. Google Scholar

[35]

W. M. Siu and R. S. C. Cobbold, Basic properties of the electrolyte-SiO2-Si system: Physical and theoretical aspects,, IEEE Transactions on Electron Devices, 26 (1979), 1805. Google Scholar

[36]

A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction,, Physical Review E, 55 (1997), 1779. doi: 10.1103/PhysRevE.55.1779. Google Scholar

[37]

T. Tokuda, Y. L. Pan, A. Uehara, K. Kagawa, M. Nunoshita and J. Ohta, Flexible and extendible neural interface device based on cooperative multi-chip CMOS LSI architecture,, Sensors and Actuators A: Physical, 122 (2005), 88. doi: 10.1016/j.sna.2005.03.065. Google Scholar

[38]

R. E. G. van Hal, J. C. T. Eijkel and P. Bergveld, A general model to describe the electrostatic potential at electrolyte oxide interfaces,, Advances in Colloid and Interface Science, 69 (1996), 31. Google Scholar

[39]

M. W. Shinwari, M. J. Deen and D. Landheer, Study of the electrolyte-insulator-semiconductor field-effect transistor (EISFET) with applications in biosensor design,, Microelectronics Reliability, 47 (2007), 2025. Google Scholar

show all references

References:
[1]

U. Ascher, J. Christiansen and R. D. Russell, Collocation software for boundary-value ODEs,, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209. doi: 10.1145/355945.355950. Google Scholar

[2]

J. N. Y. Aziz, R. Genov, B. L. Bardakjian, M. Derchansky and P. L. Carlen, Brain-silicon interface for high-resolution in vitro neural recording,, IEEE Transactions on Biomedical Circuits and Systems, 1 (2007), 56. Google Scholar

[3]

G. Bader and U. Ascher, A new basis implementation for a mixed order boundary value ODE solver,, SIAM J. Sci. Stat. Comput., 8 (1987), 483. doi: 10.1137/0908047. Google Scholar

[4]

R. Baronas, F. Ivanauskas and J. Kulys, Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians,, Springer Science & Business Media, (2010). doi: 10.1007/978-90-481-3243-0_5. Google Scholar

[5]

S. Baumgartner and C. Heitzinger, Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors,, Commun. Math. Sci, 10 (2012), 693. doi: 10.4310/CMS.2012.v10.n2.a13. Google Scholar

[6]

M. Bayer, C. Uhl and P. Vogl, Theoretical study of electrolyte gate AlGaN/GaN field effect transistors,, Journal of Applied Physics, 97 (2005). doi: 10.1063/1.1847730. Google Scholar

[7]

S. Birner, Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces,, Ph.D thesis, (2011). Google Scholar

[8]

S. Birner, S. Hackenbuchner, M. Sabathil, G. Zandler, J.A. Majewski, T. Andlauer, T. Zibold, R. Morschl, A. Trellakis and P. Vogl, Modeling of Semiconductor Nanostructures with nextnano3,, Acta Physica Polonica A, 110 (2006), 111. doi: 10.12693/APhysPolA.110.111. Google Scholar

[9]

S. Birner, C. Uhl, M. Bayer and P. Vogl, Theoretical model for the detection of charged proteins with a silicon-on-insulator sensor,, Journal of Physics: Conference Series, 107 (2008). doi: 10.1088/1742-6596/107/1/012002. Google Scholar

[10]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM Journal on Applied Mathematics, 67 (2007), 960. doi: 10.1137/060664689. Google Scholar

[11]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961. Google Scholar

[12]

E. Cianci, S. Lattanzio, G. Seguini, S. Vassanelli and M. Fanciulli, Atomic layer deposited $TiO_2$ for implantable brain-chip interfacing devices,, Thin Solid Films, 520 (2012), 4745. Google Scholar

[13]

C. De Falco, E. Gatti, A. L. Lacaita and R. Sacco, Quantum-corrected drift-diffusion models for transport in semiconductor devices,, Journal of Computational Physics, 204 (2005), 533. doi: 10.1016/j.jcp.2004.10.029. Google Scholar

[14]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model,, Physical Chemistry Chemical Physics, 15 (2013), 7075. doi: 10.1039/c3cp44390f. Google Scholar

[15]

P. Fromherz, Semiconductor chips with ion channels, nerve cells and brain,, Physica E: Low-dimensional Systems and Nanostructures, 16 (2003), 24. doi: 10.1016/S1386-9477(02)00578-7. Google Scholar

[16]

P. Fromherz, Three levels of neuroelectronic interfacing,, Annals of the New York Academy of Sciences, 1093 (2006), 143. Google Scholar

[17]

P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips,, Solid-State Electronics, 52 (2008), 1364. Google Scholar

[18]

I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 48 (1997), 45. doi: 10.1007/PL00001469. Google Scholar

[19]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, Journal of Physics: Condensed Matter, 14 (2002), 12129. doi: 10.1088/0953-8984/14/46/317. Google Scholar

[20]

W. M. Grill, S. E. Norman and R. V. Bellamkonda, Implanted neural interfaces: Biochallenges and engineered solutions,, Annual Review of Biomedical Engineering, 11 (2009), 1. doi: 10.1146/annurev-bioeng-061008-124927. Google Scholar

[21]

Y. He, I. Gamba, H.-C. Lee and K. Ren, On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells,, SIAM Journal on Applied Mathematics, 75 (2015), 2515. doi: 10.1137/130935148. Google Scholar

[22]

C. Heitzinger, R. Kennell, G. Klimeck, N. Mauser, M. McLennan and C. Ringhofer, Modeling and simulation of field-effect biosensors (BioFETs) and their deployment on the nanoHUB,, Journal of Physics: Conference Series, 107 (2008). doi: 10.1088/1742-6596/107/1/012004. Google Scholar

[23]

C. Heitzinger and G. Klimeck, Computational aspects of the three-dimensional feature-scale simulation of silicon-nanowire field-effect sensors for DNA detection,, Journal of Computational Electronics, 6 (2007), 387. doi: 10.1007/s10825-006-0139-x. Google Scholar

[24]

C. Heitzinger, N. J. Mauser and C. Ringhofer, Multiscale modeling of planar and nanowire field-effect biosensors,, SIAM Journal on Applied Mathematics, 70 (2010), 1634. doi: 10.1137/080725027. Google Scholar

[25]

A. Jüngel and I. V. Stelzer, Existence Analysis of Maxwell-Stefan Systems for Multicomponent Mixtures,, SIAM Journal on Mathematical Analysis, 45 (2013), 2421. doi: 10.1137/120898164. Google Scholar

[26]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Springer Science & Business Media, (1986). doi: 10.1007/978-3-7091-3678-2. Google Scholar

[27]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag: Berlin, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[28]

M. Mojarradi, D. Binkley, B. Blalock, R. Andersen, N. Ulshoefer, T. Johnson and L. Del Castillo, A miniaturized neuroprosthesis suitable for implantation into the brain,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11 (2003), 38. doi: 10.1109/TNSRE.2003.810431. Google Scholar

[29]

X. Navarro, T.B Krueger, N. Lago, S. Micera, T. Stieglitz and P. Dario, A critical review of interfaces with the peripheral nervous system for the control of neuroprostheses and hybrid bionic systems,, Journal of the Peripheral Nervous System, 10 (2005), 229. doi: 10.1111/j.1085-9489.2005.10303.x. Google Scholar

[30]

Y. Ohno, K. Maehashi, Y. Yamashiro and K. Matsumoto, Electrolyte-gated graphene field-effect transistors for detecting pH and protein adsorption,, Nano Letters, 9 (2009), 3318. doi: 10.1021/nl901596m. Google Scholar

[31]

W. R. Patterson, Y. Song, C. W. Bull, I. Ozden, A. P. Deangellis, C. Lay, J. L. McKay, A. V. Nurmikko, J. D. Donoghue and B. W. Connors, A microelectrode/microelectronic hybrid device for brain implantable neuroprosthesis applications,, IEEE Transactions on Biomedical Engineering, 51 (2004), 1845. doi: 10.1109/TBME.2004.831521. Google Scholar

[32]

I. Peitz and P. Fromherz, Electrical interfacing of neurotransmitter receptor and field effect transistor,, The European Physical Journal E: Soft Matter and Biological Physics, 30 (2009), 223. doi: 10.1140/epje/i2009-10461-3. Google Scholar

[33]

R. Popovtzer, A. Natan and Y. Shacham-Diamand, Mathematical model of whole cell based bio-chip: An electrochemical biosensor for water toxicity detection,, Journal of Electroanalytical Chemistry, 602 (2007), 17. doi: 10.1016/j.jelechem.2006.11.022. Google Scholar

[34]

M.J. Schöning and A. Poghossian, Bio FEDs (Field-Effect Devices): State-of-the-Art and New Directions,, Electroanalysis, 18 (2006), 1893. Google Scholar

[35]

W. M. Siu and R. S. C. Cobbold, Basic properties of the electrolyte-SiO2-Si system: Physical and theoretical aspects,, IEEE Transactions on Electron Devices, 26 (1979), 1805. Google Scholar

[36]

A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction,, Physical Review E, 55 (1997), 1779. doi: 10.1103/PhysRevE.55.1779. Google Scholar

[37]

T. Tokuda, Y. L. Pan, A. Uehara, K. Kagawa, M. Nunoshita and J. Ohta, Flexible and extendible neural interface device based on cooperative multi-chip CMOS LSI architecture,, Sensors and Actuators A: Physical, 122 (2005), 88. doi: 10.1016/j.sna.2005.03.065. Google Scholar

[38]

R. E. G. van Hal, J. C. T. Eijkel and P. Bergveld, A general model to describe the electrostatic potential at electrolyte oxide interfaces,, Advances in Colloid and Interface Science, 69 (1996), 31. Google Scholar

[39]

M. W. Shinwari, M. J. Deen and D. Landheer, Study of the electrolyte-insulator-semiconductor field-effect transistor (EISFET) with applications in biosensor design,, Microelectronics Reliability, 47 (2007), 2025. Google Scholar

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