# American Institute of Mathematical Sciences

June  2021, 14(3): 407-427. doi: 10.3934/krm.2021010

## Mathematical modelling of charge transport in graphene heterojunctions

 1 Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italia 2 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale Andrea Doria 6, 95125 Catania 3 Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse, France

* Corresponding author: Luigi Barletti

Received  May 2020 Revised  December 2020 Published  June 2021 Early access  March 2021

A typical graphene heterojunction device can be divided into two classical zones, where the transport is basically diffusive, separated by a "quantum active region" (e.g., a locally gated region), where the charge carriers are scattered according to the laws of quantum mechanics. In this paper we derive a mathematical model of such a device, where the classical regions are described by drift-diffusion equations and the quantum zone is seen as an interface where suitable transmission conditions are imposed that take into account the quantum scattering process. Numerical simulations show good agreement with experimental data.

Citation: Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic and Related Models, 2021, 14 (3) : 407-427. doi: 10.3934/krm.2021010
##### References:
 [1] C. Bardos, R. Santos and R. Sentis, Diffusion approximation and the computation of the critical size, T. Am. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0. [2] L. Barletti, Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle, J. Math. Phys., 55 (2014), 083303, 21 pp. doi: 10.1063/1.4886698. [3] L. Barletti and C. Cintolesi, Derivation of isothermal quantum fluid equations with Fermi-Dirac and Bose-Einstein statistics, J. Stat. Phys., 148 (2012), 353-386.  doi: 10.1007/s10955-012-0535-5. [4] L. Barletti and C. Negulescu, Quantum transmission conditions for diffusive transport in graphene with steep potentials, J. Stat. Phys., 171 (2018), 696-726.  doi: 10.1007/s10955-018-2032-y. [5] N. Ben Abdallah, A hybrid kinetic-quantum model for stationary electron transport, J. Stat. Phys., 90 (1998), 627-662.  doi: 10.1023/A:1023216701688. [6] N. Ben Abdallah, P. Degond and I. Gamba, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), 1-24.  doi: 10.1063/1.1421635. [7] V. D. Camiola, G. Mascali and V. Romano, Charge Transport in Low Dimensional Semiconductor Structures, Springer, 2020. doi: 10.1007/978-3-030-35993-5. [8] A. Castro Neto, F. Guinea, N. Peres, K. Novoselov and A. Geim, The electronic properties of graphene, Rev. Mod. Phys., 81 (2009), 109-162. [9] V. Cheianov and V. Fal'ko, Selective transmission of Dirac electrons and ballistic magnetoresistance of n-p junctions in graphene, Phys. Rev. B, 74 (2006), 041403(R). [10] V. Cheianov, V. Fal'ko and B Altshuler, The focusing of electron flow and a Veselago lens in graphene, Science, 315 (2007), 1252-1255. [11] M. Coco, A. Majorana, G. Nastasi and V. Romano, High-field mobility in graphene on substrate with a proper inclusion of the Pauli exclusion principle, Atti della Accademia Peloritana dei Pericolanti, 97 (2019), A6, 15 pp. doi: 10.1478/AAPP.97S1A6. [12] P. Degond and A. El Ayyadi, A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181 (2002), 222-259.  doi: 10.1006/jcph.2002.7122. [13] P. Degond and C. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), 4634-4663.  doi: 10.1063/1.532528. [14] T. Fang, et al., Carrier statistics and quantum capacitance of graphene sheets and ribbons, Appl. Phys. Lett., 91 (2007), 092109. [15] F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems, J. Stat. Phys., 80 (1995), 1033-1061.  doi: 10.1007/BF02179863. [16] B. Huard, et al., Transport measurements across a tunable potential barrier in graphene, Phys. Rev, Lett., 98 (2007), 236803. [17] M. I. Katsnelson, K. S. Novoselov and A. K Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys., 2 (2006), 620-625.  doi: 10.1038/nphys384. [18] G. M. Landauer, D. Jimènez and J. L. Gonzàlez, An accurate and Verilog-A compatible compact model for graphene Field-Effect Transistors, IEEE Transactions on Nanotechnology, 13 (2014), 895-904. [19] G. Lee, G. Park and H. Lee, Observation of negative refraction of Dirac fermions in graphene, Nat. Phys., 11 (2015), 925-929.  doi: 10.1038/nphys3460. [20] P. Lichtenberger, O. Morandi and F. Schürrer, High-field transport and optical phonon scattering in graphene, Phys. Rev.. B, 84 (2011), 045406. doi: 10.1103/PhysRevB.84.045406. [21] L. Luca and V. Romano, Quantum corrected hydrodynamic models for charge transport in graphene, Annals of Physics, 406 (2019), 30-53.  doi: 10.1016/j.aop.2019.03.018. [22] A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys.: Condens. Matter, 30 (2018), 053001. [23] A. Majorana, G. Nastasi and V. Romano., Simulation of bipolar charge transport in graphene by using a discontinuous Galerkin method, Commun. Comput. Phys., 26 (2019), 114-134.  doi: 10.4208/cicp.OA-2018-0052. [24] G. Nastasi and V. Romano, Improved mobility models for charge transport in graphene, Communications in Applied and Industrial Mathematics, 10 (2019), 41-52.  doi: 10.1515/caim-2019-0011. [25] G. Nastasi and V. Romano, Simulation of graphene field effect transistors, in Scientific Computing in Electrical Engineering - SCEE 2018 (eds. G. Nicosia and V. Romano), Springer, 32 (2018), 171–178. doi: 10.1007/978-3-030-44101-2_16. [26] G. Nastasi and V. Romano, A full coupled drift-diffusion-Poisson simulation of a GFET, Communications in Nonlinear Science and Numerical Simulations, 87 (2020), 105300, 16 pp. doi: 10.1016/j.cnsns.2020.105300. [27] K. Novoselov, Electric field effect in atomically thin carbon films, Science, 306 (2004), 666-669.  doi: 10.1126/science.1102896. [28] B. Özyilmaz, et al., Electronic transport and quantum Hall effect in bipolar graphene p-n-p junctions, Phys. Rev, Lett., 99 (2007), 166804. [29] V. Romano, Quantum corrections to the semiclassical hydrodynamical model of semiconductors based on the maximum entropy principle, J. Math. Phys., 48 (2007), 123504, 24 pp. doi: 10.1063/1.2819600. [30] S. A. Thiele, J. A. Schaefer and F. Schwierz, Modeling of graphene metal-oxide-semiconductor field-effect transistors with gapless large-area graphene channels, J. Appl. Phys. 107 (2010), 094505. doi: 10.1063/1.3357398. [31] A. F. Young and P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nat. Phys., 5 (2009), 222-226.  doi: 10.1038/nphys1198. [32] N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamical approach, Math. Methods Appl. Sci., 34 (2011), 807-818.  doi: 10.1002/mma.1403.

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##### References:
 [1] C. Bardos, R. Santos and R. Sentis, Diffusion approximation and the computation of the critical size, T. Am. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0. [2] L. Barletti, Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle, J. Math. Phys., 55 (2014), 083303, 21 pp. doi: 10.1063/1.4886698. [3] L. Barletti and C. Cintolesi, Derivation of isothermal quantum fluid equations with Fermi-Dirac and Bose-Einstein statistics, J. Stat. Phys., 148 (2012), 353-386.  doi: 10.1007/s10955-012-0535-5. [4] L. Barletti and C. Negulescu, Quantum transmission conditions for diffusive transport in graphene with steep potentials, J. Stat. Phys., 171 (2018), 696-726.  doi: 10.1007/s10955-018-2032-y. [5] N. Ben Abdallah, A hybrid kinetic-quantum model for stationary electron transport, J. Stat. Phys., 90 (1998), 627-662.  doi: 10.1023/A:1023216701688. [6] N. Ben Abdallah, P. Degond and I. Gamba, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), 1-24.  doi: 10.1063/1.1421635. [7] V. D. Camiola, G. Mascali and V. Romano, Charge Transport in Low Dimensional Semiconductor Structures, Springer, 2020. doi: 10.1007/978-3-030-35993-5. [8] A. Castro Neto, F. Guinea, N. Peres, K. Novoselov and A. Geim, The electronic properties of graphene, Rev. Mod. Phys., 81 (2009), 109-162. [9] V. Cheianov and V. Fal'ko, Selective transmission of Dirac electrons and ballistic magnetoresistance of n-p junctions in graphene, Phys. Rev. B, 74 (2006), 041403(R). [10] V. Cheianov, V. Fal'ko and B Altshuler, The focusing of electron flow and a Veselago lens in graphene, Science, 315 (2007), 1252-1255. [11] M. Coco, A. Majorana, G. Nastasi and V. Romano, High-field mobility in graphene on substrate with a proper inclusion of the Pauli exclusion principle, Atti della Accademia Peloritana dei Pericolanti, 97 (2019), A6, 15 pp. doi: 10.1478/AAPP.97S1A6. [12] P. Degond and A. El Ayyadi, A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181 (2002), 222-259.  doi: 10.1006/jcph.2002.7122. [13] P. Degond and C. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), 4634-4663.  doi: 10.1063/1.532528. [14] T. Fang, et al., Carrier statistics and quantum capacitance of graphene sheets and ribbons, Appl. Phys. Lett., 91 (2007), 092109. [15] F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems, J. Stat. Phys., 80 (1995), 1033-1061.  doi: 10.1007/BF02179863. [16] B. Huard, et al., Transport measurements across a tunable potential barrier in graphene, Phys. Rev, Lett., 98 (2007), 236803. [17] M. I. Katsnelson, K. S. Novoselov and A. K Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys., 2 (2006), 620-625.  doi: 10.1038/nphys384. [18] G. M. Landauer, D. Jimènez and J. L. Gonzàlez, An accurate and Verilog-A compatible compact model for graphene Field-Effect Transistors, IEEE Transactions on Nanotechnology, 13 (2014), 895-904. [19] G. Lee, G. Park and H. Lee, Observation of negative refraction of Dirac fermions in graphene, Nat. Phys., 11 (2015), 925-929.  doi: 10.1038/nphys3460. [20] P. Lichtenberger, O. Morandi and F. Schürrer, High-field transport and optical phonon scattering in graphene, Phys. Rev.. B, 84 (2011), 045406. doi: 10.1103/PhysRevB.84.045406. [21] L. Luca and V. Romano, Quantum corrected hydrodynamic models for charge transport in graphene, Annals of Physics, 406 (2019), 30-53.  doi: 10.1016/j.aop.2019.03.018. [22] A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys.: Condens. Matter, 30 (2018), 053001. [23] A. Majorana, G. Nastasi and V. Romano., Simulation of bipolar charge transport in graphene by using a discontinuous Galerkin method, Commun. Comput. Phys., 26 (2019), 114-134.  doi: 10.4208/cicp.OA-2018-0052. [24] G. Nastasi and V. Romano, Improved mobility models for charge transport in graphene, Communications in Applied and Industrial Mathematics, 10 (2019), 41-52.  doi: 10.1515/caim-2019-0011. [25] G. Nastasi and V. Romano, Simulation of graphene field effect transistors, in Scientific Computing in Electrical Engineering - SCEE 2018 (eds. G. Nicosia and V. Romano), Springer, 32 (2018), 171–178. doi: 10.1007/978-3-030-44101-2_16. [26] G. Nastasi and V. Romano, A full coupled drift-diffusion-Poisson simulation of a GFET, Communications in Nonlinear Science and Numerical Simulations, 87 (2020), 105300, 16 pp. doi: 10.1016/j.cnsns.2020.105300. [27] K. Novoselov, Electric field effect in atomically thin carbon films, Science, 306 (2004), 666-669.  doi: 10.1126/science.1102896. [28] B. Özyilmaz, et al., Electronic transport and quantum Hall effect in bipolar graphene p-n-p junctions, Phys. Rev, Lett., 99 (2007), 166804. [29] V. Romano, Quantum corrections to the semiclassical hydrodynamical model of semiconductors based on the maximum entropy principle, J. Math. Phys., 48 (2007), 123504, 24 pp. doi: 10.1063/1.2819600. [30] S. A. Thiele, J. A. Schaefer and F. Schwierz, Modeling of graphene metal-oxide-semiconductor field-effect transistors with gapless large-area graphene channels, J. Appl. Phys. 107 (2010), 094505. doi: 10.1063/1.3357398. [31] A. F. Young and P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nat. Phys., 5 (2009), 222-226.  doi: 10.1038/nphys1198. [32] N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamical approach, Math. Methods Appl. Sci., 34 (2011), 807-818.  doi: 10.1002/mma.1403.
Schematic geometry of our model: the rectangle represents the graphene sheet and the central strip represents the quantum active region, i.e. the zone where the variations of $V = V(x)$ are localized. Outside the strip, in the two classical regions, the potential $V$ has constant values $V_0$ and $V_0+{\delta V}$
A schematic picture of a n-p-n graphene device: the graphene sheet is represented as the black honeycomb (not in scale), the grey regions represent gates and contacts, and the blue box represents some substrate layer (typically an oxide)
Gray-scale plots of $T_+( {\boldsymbol{p}})$, as a function of the energy $E = {v_F}{\vert {{ {\boldsymbol{p}}}} \vert}$ and of the incidence angle $\phi$, for different values of the energy height $E_h$. White corresponds to perfect transmission ($T_+ = 1$) and black to total reflection ($T_+ = 0$). Note that for $\phi = 0$ the barrier is always completely transparent, regardless to $E_h$, which is the so-called Klein paradox [17]. The dashed red line is a contour line of $M( {\boldsymbol{p}}) \cos\phi$, corresponding to a region that encompasses approximately 90% of its integral; such region is therefore where the main contribution to the integrals in (53) comes from (the same region for $M( {\boldsymbol{p}}) \cos^2\phi$ is just slightly narrower). In this figure the barrier width is $50\,\mathrm{nm}$ and the temperature is $40\,\mathrm{K}$. For lower values of the temperature, the Maxwellian will be narrower, resulting in a higher sensitivity to the variations of $T_+$
Conductance as a function of the top gate voltage $V_\mathit{tg}$ for different values of the back gate (left column) and for different values of the temperature (right column). In the left plots, the temperature is fixed at $T = 10\,\mathrm{K}$ while, in the right plots, the back gate voltage is fixed at $V_\mathit{bg} = 23\, \mathrm{V}$
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