## 2 Theoretical Model Background and Simulation Details  
The potential \(V=V(x)\) is a function of height \(w\) and is coupled to a rectangular-like barrier of height \(w_{0}\). However, the following condition is required: the potential \(V=V(x)\) must approach zero potential when the height of the potential barrier \(w\) tends to zero. \[\lim_{w\to 0}(V(w))=0 \tag{1}\]  
Figure 1 represents the approximation of the Kronig-Penney model, the potential to which the other potentials studied will be approximated, and those that meet the condition of Equation (1). The solutions of the potentials displayed in Figure 1 of the associated time-independent SE are commonly special functions [22; 23; 26; 31], while the zero-like potential has its solution as a linear combination of complex exponentials [3]. To treat these periodic potentials, the same method proposed for potentials approximating the Dirac-delta potential was used [12; 14], but with the assumption that the potential \(V=V(x)\) must now satisfy Equation (1). For this, the well-known Bloch theorem, widely used in periodic potentials, was applied [32]. In other words, \(a\) is the width of the rectangular-like potential, \(2b\) is the width of the potential \(V=V(x)\), and \(T=2b+a\) is the period of the periodic potential (see Figure S1), while \(\Psi_{I}(x),\Psi_{II}(x)\), and \(\Psi_{0}(x)\) represent the wave functions and \(\Psi_{I}^{\prime}(x),\Psi_{II}^{\prime}(x)\), and \(\Psi_{0}^{\prime}(x)\) are the derivatives of the wave functions. The applied contour and continuity functions have the same form as in [30]. Furthermore, the application of these is shown in the Supplementary Materials. It has been reported in the literature [3; 9; 11; 12] that Equations (2) and (3) have solutions for the cases of the rectangular-like potential and the Dirac-delta potential of positive intensity given, respectively, by  
\[\cos\left(k(a+2b)\right) =\left(\frac{\beta^{2}-\alpha^{2}}{2\beta\alpha}\right)\sinh \left(2b\beta\right)\sin\left(\alpha a\right)+\cosh\left(2b\beta\right)\cos \left(\alpha a\right) \tag{2}\] \[\cos\left(ka\right) =\frac{p}{\alpha a}\sin\left(\alpha a\right)+\cos\left(\alpha a\right) \tag{3}\]  
Equation (2) tends to Equation (3), resulting in an infinitely thin barrier width and an infinitely long potential barrier height. In Equations (2) and (3), the \(a\) and \(b\) parameters are defined in Figure 1. \(P\) is a positive intensity of the Dirac-delta potential, and the parameters that define the energies are \(\alpha=\sqrt{2 mE}/\hbar\) and \(\beta=\sqrt{2m(w_{0}-E)}/\hbar\). Then,  
Figure 1: Schemes of the quantum potentials. The dark blue lines represent the Kronig–Penney potential. The are due to the rectangular-like potential of negative intensity (magenta color). The red and green lines represent the linear-like potentials, while the cyan and yellow lines are the triangular-like potentials. in Figure S1, the numerical simulation of the Kronig-Penney potential energy is plotted, starting from the analytical solution of Equation (2). For the simulations in the figures below, an approximation \(\hbar^{2}/2m=1\) is made in order to reduce the number of physical constants of the problem (it can be easily recovered at the end if necessary using this transformation and comparing the dimensions of the physical quantities [10]). In other words, this approximation does not affect the physics of the simulations. However, with this approximation, it is not necessary to place units in the simulations. In Equation (2), \(k=1/\lambda\), where \(\lambda\) is the wavelength of the electrons. This is also important because the wave vector is usually defined as \(k=2\pi/\lambda\), which takes discrete values from \(-\pi/T\) to \(\pi/T\) and thus defines the Brillouin zone. In this case, the trigonometric functions in Equations (4)-(7) are periodic with period \(k.T\). It is worth mentioning that the units of \(k\) are the opposite of those of the period \(T\).