# 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 & Related Models, 2021, 14 (3) : 407-427. doi: 10.3934/krm.2021010
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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)
]. 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 $T_+$">Figure 3.  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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