# Characteristics of Resonant Tunneling in Nanostructures with Spacer Layers
## 2 Materials and Methods  
In the present study, we considered two-barrier resonant tunneling nanostructures. Schematically, the potential profile of the bottom of the conduction band of such a structure is shown in Figure 1. Electrons distributed by energies according to the Fermi-Dirac statistics are injected into the structure from contacts located to the left and right of the active region. We assumed that the transport of charge carriers is ballistic. We considered the model that takes into account the presence of highly doped contact regions (in Figure 1, these regions are highlighted in green and have a thickness of \(L_{c}\) and a donor concentration of \(n_{D}\)), as well as undoped spacer layers that prevent the penetration of impurities into the active quantum region. The thickness of the emitter spacer is denoted by \(L_{SE}\), and the thickness of the collector spacer is \(L_{SC}\). The active region of the resonant tunneling nanostructure is formed by the two potential barriers with a thickness of \(L_{b}\) and a height of \(\Delta E_{c}\), between which the quantum well with the thickness of \(L_{QW}\) is located. Calculations of the electronic transport through the resonant tunneling nanostructures within the framework of the numerical solution of the stationary system of Schrodinger-Poisson equations taking into account spacer layers and strongly doped contact regions face divergence problems partially described in [23]. Our study showed that all the methods proposed so far allow us to achieve the convergence of the solution only in a limited range of structure parameters. For structures with small values of the resonance-level width (for example, as for the structure considered in [23; 24]), the divergence of the numerical solution is observed at emitter spacer thicknesses less than 6 nm. For structures with large values of the resonance-level width (for example, the structures from [25; 26]), the minimum thickness of the emitter spacer, at which there is no divergence of the numerical solution of the stationary problem, increases. Thus, the thicknesses of spacer layers less than 8-10 nm are inaccessible for analysis in the framework of a stationary formulation of the problem, although such thicknesses are used in experimental structures [25; 26]. A possible reason for the divergence of the numerical solution of the stationary problem may be the poor fulfillment of the electroneutrality condition. To calculate the static current-voltage characteristics, we developed a technique based on the numerical solution of the open time-dependent Schrodinger-Poisson equations:  
\[i\hbar\frac{\partial\psi_{k}(x,t)}{\partial t} = -\frac{\hbar^{2}}{2}\frac{\partial}{\partial x}\bigg{(}\frac{1}{ m^{*}(x)}\frac{\partial\psi_{k}(x,t)}{\partial x}\bigg{)}+(V(x,t)+V_{0}(x)+ \varphi(x,t))\psi_{k}(x,t),\] \[V(x,t) = -V_{0}(t)\frac{x}{L},\] \[\frac{\partial^{2}\varphi}{\partial x^{2}} = -\frac{e^{2}}{\varepsilon\varepsilon_{0}}(n(x,t)-n_{D}),\] \[\varphi(0,t) = \varphi(L,t)=0,\] \[n(x,t) = \int|\psi_{k}(x,t)|^{2}g(k)dk,\] \[g(k) = \frac{mk_{B}T}{2\pi^{2}\hbar^{2}}\ln\left(1+\exp\left(\frac{E_{F} -\frac{\hbar^{2}k^{2}}{2m^{*}}}{k_{B}T}\right)\right);\]  
a set of Schrodinger equations with different values of the wave vectors \(k\) is solved; \(V(x,t)\) is the potential of the time-dependent electric field; \(\varphi(x,t)\) is the self-consistent potential in the Hartree approximation; \(n(x,t)\) is the electron density; \(g(k)\) is the Fermi-Dirac distribution integrated along the transverse wave vector; \(T\) is the temperature (all calculations were carried out for \(T=300\) K); \(m^{*}\) is the electron effective mass; \(E_{F}\) is the Fermi energy. A detailed description of the difference scheme for solving the time-dependent Schrodinger-Poisson equations is given in [23]. Exact discrete open boundary conditions for the Schrodinger equation are used to prevent the non-physical reflections from the boundaries of the computational domain [23; 27; 28; 29].