Boundary conditions for electron tunneling in complex two- and three-dimensional structures

We present boundary conditions given in integro-differential form for the single-particle two-dimensional (2D) or 3D Schrödinger equation, which allows for a treatment of nontrivial geometries, and an arbitrary number of input and output channels. The formalism is easy to implement using standard finite element packages. We consider a resonant dot structure and transport through a ringlike waveguide without barriers. The current in the dot is focused on an ellipsoid dot via a tunneling tip. The current-voltage characteristic is calculated for this system at the temperature $4.2\mathrm{K}$. Our results show that the current maxima appear close to the eigenstates of the quantum dot. We show, however, that only those modes which obey certain symmetry properties give rise to resonance in the dot, and current maxima are absent for antisymmetric modes at low temperatures. The current in the waveguide is shown to be a resonant function of the voltage, and the system exhibits current feedback and turbulence. Finally, we extend the formalism to other types of channels and equations other than the Schrödinger equation and we discuss some possible applications for these systems.

Figure 1

(Color online) The rectangular cylinder is the input channel (of infinite length) and is described by a wave function ${\Psi }_{r1}$ and wave numbers $k_j$. The circular cylinder is the output channel in where we have a wave function ${\Psi }_{t1}$ and wave numbers ${\kappa }_j$. The input- and output-boundary surfaces of the complex structure (the conical volume in this case) are denoted as $u_1(x,y)$ and $u_2(x,y)$.

Figure 2

(Color online) The upper figure shows the structure with incoming and outgoing current $i_0$. The $x$ and the $y$ axis are oriented horizontally, respectively, vertically in the figure. The arrows labeled $i_a$ and $i_b$ indicate currents in the lower and the upper branch, respectively. The lower figure shows the real part of the wave function $\Psi$ for a low voltage bias $(V=0.2)$. The grayscale ranges from black $[\mathrm{Re}\left(\Psi \right)=-1]$ to white $[\mathrm{Re}\left(\Psi \right)=+1]$. $\mathrm{Re}\left(\Psi \right)=0$ corresponds to “medium” gray, also valid for the boundary where $\Psi =0$.

Figure 3

(Color online) Total current ($i_0$, solid line) and gain ($G$, dash-dotted line) vs applied voltage. The current is a resonant function of the voltage. The gain is changing sign abruptly at $V\approx 10.8$, where $i_0\cong 0$. For $G<0$ and $G>1$, the current circulates clockwise or counter clockwise, respectively (see Fig. 2).

Figure 4

(Color online) (a) Current density shown for a low voltage bias $(V=0.2)$ where the current in the lower branch ($i_a$, see Fig. 2) is larger than the incoming current (i.e., positive feedback $G>1$). Arrow lengths are proportional to current density. (b) Current density shown for $V=4.8$, where the incoming current is divided equally between $i_a$ and $i_b$.

Figure 5

(Color online) Current density (arrows) and streamlines (thick lines) obtained by the equation system $dx\left(t\right)∕dt=j_x\mathbf{(}x\left(t\right),y\left(t\right)\mathbf{)}$ and $dy\left(t\right)∕dt=j_y\mathbf{(}x\left(t\right),y\left(t\right)\mathbf{)}$. In the figure the streamlines have finite lengths (of numerical reasons); they end at different places depending on where they start. The thin lines correspond to the isolevels of $j_x=0$ and $j_y=0$. At several places we obtain so-called critical points $(j_x=j_y=0)$ around where streamlines circulate. The flow goes from the input channel to the output channel via the upper branch. The lines starting from the middle of the upper branch circulate in clockwise direction.

Figure 6

(Color online) Upper figure: $j_y\left(x\right)$ shown for $y=1$ solid line, $y=0.3$ dash-dotted line, $y=0$ dashed line, and $y=2$ dotted line in the input channel (see Fig. 2 for information about coordinates). Lower figure: Current density shown for $y=1$ solid line, $y=1.2$ dash-dotted line, $y=1.6$ dashed line, and $y=2$ dotted line. $j_y\left(x\right)$ takes the ground state shape for large positive values of $y$ (see the dotted lines in both figures). Currents above the boundaries $(y>1)$ are obtained from the analytic expressions of the wave functions; meanwhile the profiles below the boundaries are interpolated form the numerical solution. It is instructive to compare $j_y\left(x\right)$ in the upper figure at $y=0$ with the current arrows and flow lines in Fig. 5 at the same position.

Figure 7

(Color online) (a) The structure specifying different materials and their effective barrier potentials in the structure (barriers are gray and channels are white). The $x$ and the $y$ axes are oriented horizontally, respectively, vertically in the figure. The shortest distance between the tip and the ellipsoid dot is $2.5\mathrm{nm}$. (b) Wave function ${\mid \Psi \mid }^{1∕5}$ (grayscale), with corresponding values at the bar to the right. The potential barrier is $113\mathrm{meV}$ and the figure shows the first resonant state (see Fig. 9, $V=143.1$). (c) Streamlines for the current flow (lines) for the same case as in (b) also including current probability density $j_x\left(y\right)$ at the input and the output, boundaries (Gaussian-like functions). (d) Current density (arrows) and input and output $j_x\left(y\right)$ for $V=298.3$ between the first and the second resonant state (see Fig. 9). The flow lines are focused onto the dot via the tunnel tip, but are then divided inside the dot. (e) Laminar and turbulent (circular) current streamlines and $j_x\left(y\right)$ at the input and the output boundaries for $V=313.8$. The laminar flow lines are focused on the dot via the sharp tunnel tip but splits into three beams. The turbulent (circular) current flow is located between the tip and the right side of the dot. (f) Current streamlines and $j_x\left(y\right)$ at the antisymmetric state $2p_y$ (see Fig. 10, $V=215.8$). The current at $y=0$ is zero and due to symmetry reasons the current must go a longer distance within the barrier. A focusing effect exists in the dot.

Figure 8

I-V characteristic for tunneling through the ellipsoid dot. A zoom of the first peak shows a negative differential resistance of the order of $500\mathrm{M}\Omega$. The first state corresponds to the ground state in the ellipsoid dot. The second peak with a smaller maximum current corresponds to a symmetric eigenstate, with three $\mid \Psi \mid$ maxima along the $y$ direction $\left(3p_y\right)$. The third peak corresponds to an eigenstate with two $\mid \Psi \mid$ maxima along the $x$ axis $\left(2p_x\right)$.

Figure 9

(Color online) Single-electron tunnel current and relative probability to find the electron within the dot (both dimensionless) shown as a function of the voltage at a constant energy. The peaks of the resonant current and probability appear at the same kind of resonant states in the dot. The incoming wave is taken to be in the ground state of the input boundary $\left({\lambda }_1\right)$. The first peak corresponds to the ground state “$1s$” and the second peaks to the excited state $3p_y$. The resonant state $2p_x$ appears close to $V=500$.

Figure 10

(Color online) Tunnel current and relative probability to find the electron within the dot (both dimensionless) shown as a function of the voltage at a constant energy. The same calculation as in Fig. 9 but the incoming wave is taken to be in the first excited $\left({\lambda }_2\right)$ antisymmetric state (around the $x$ axis) of the input boundary. The first peak corresponds to the antisymmetric state $2p_y$ (2 maxima of $\mid \Psi \mid$ in the dot) and the second little peak to the excited state $4p_y$ (4 maxima in the dot).

Figure 11

(Color online) In the upper part of the figure, a numerical grid is shown close to the boundary surface for an output channel (the output channel extends above the upper line, which is the boundary surface). The variable $u$ corresponds to the wave function in the structure (i.e., $\Psi$) and is on discrete form given by $u(x_n,y_m)=u_{n,m}$. The lower figures show the eigenmodes ${\varphi }_1\left(x\right)$ and ${\varphi }_2\left(x\right)$ with their corresponding wave numbers ${\kappa }_1$ and ${\kappa }_2$. $h$ is the step size of the numerical grid.