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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:
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Müller, Overcoming the shortcomings of the Nernst-Planck model, Physical Chemistry Chemical Physics, 15 (2013), 7075-7086. 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), 24-34. 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),143-160. Google Scholar [17] P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips, Solid-State Electronics, 52 (2008), 1364-1373. 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), 45-59. doi: 10.1007/PL00001469.  Google Scholar [19] D. Gillespie, W. Nonner and R. S. 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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), 387-390. 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), 1634-1654. 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), 2421-2440. 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, Heidelberg, New York, 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), 38-42. 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), 229-258. 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), 3318-3322. 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), 1845-1853. 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), 223-231. 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), 17-23. 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), 1893-1900. 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), 1805-1815. Google Scholar [36] A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction, Physical Review E, 55 (1997), 1779-1782. 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), 88-98. 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), 31-62. 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), 2025-2057. 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), 209-222. 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), 56-62. 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), 483-500. 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), 693-716. 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), 033703. doi: 10.1063/1.1847730.  Google Scholar [7] S. Birner, Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces, Ph.D thesis, TU München, 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), 111-124. 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), 012002. 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), 960-989. 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), 961-990. 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), 4745-4748. 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), 533-561. 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), 7075-7086. 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), 24-34. 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),143-160. Google Scholar [17] P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips, Solid-State Electronics, 52 (2008), 1364-1373. 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), 45-59. 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), 12129-12145. 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), 1-24. 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), 2515-2539. 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), 012004. 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), 387-390. 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), 1634-1654. 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), 2421-2440. 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, Heidelberg, New York, 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), 38-42. 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), 229-258. 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), 3318-3322. 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), 1845-1853. 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), 223-231. 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), 17-23. 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), 1893-1900. 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), 1805-1815. Google Scholar [36] A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction, Physical Review E, 55 (1997), 1779-1782. 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), 88-98. 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), 31-62. 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), 2025-2057. Google Scholar
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