Coherent electronic transport in a multimode quantum channel with Gaussian-type scatterers

Coherent electron transport through a quantum channel in the presence of a general extended scattering potential is investigated using a $T$-matrix Lippmann–Schwinger approach. The formalism is applied to a quantum wire with Gaussian type scattering potentials, which can be used to model a single impurity, a quantum dot or more complicated structures in the wire. The well known dips in the conductance in the presence of attractive impurities are reproduced. A resonant transmission peak in the conductance is seen as the energy of the incident electron coincides with an energy level in the quantum dot. The conductance through a quantum wire in the presence of an asymmetric potential is also shown. In the case of a narrow potential parallel to the wire we find that two dips appear in the same subband which we ascribe to two quasi bound states originating from the next evanescent mode.

Figure 1

Schematic view of the system. An incoming wave is partly transmitted and partly reflected by the finite range scattering potential $V$.

Figure 2

(Color online). The energy levels of a quantum wire with a hard wall confinement. For a given energy $E$ (horizontal line) only a finite number of propagating modes are active (red markers), but higher lying evanescent modes can serve as intermediate states. This specific case is for a wire of width $L=4a_0$. Energy is scaled in Rydbergs (Ry) and lengths in Bohr radii $\left(a_0\right)$, cf. Sec. 3.

Figure 3

(Color online). Conductance of a hard wall quantum wire in units of $G_0=2e^2∕h$ as a function of energy in the presence of a single repulsive (dashed, $V_0=3\mathrm{Ry}$, $\alpha =2a_0^{-2}$) and attractive (solid, $V_0=-4\mathrm{Ry}$, $\alpha =2a_0^{-2}$) Gaussian scattering potential. The inset shows the scattering potentials in a cross section along the middle of the wire $(x=L∕2)$. The total number of modes is $N=8$. The width of the wire is $L=4a_0$.

Figure 4

(Color online). Conductance of a parabolic quantum wire in units of $G_0=2e^2∕h$ as a function of energy in the presence of a single repulsive (dashed, $V_0=1.35\mathrm{Ry}$, $\alpha =0.96a_0^{-2}$) and attractive (solid, $V_0=-1.35\mathrm{Ry}$, $\alpha =0.96a_0^{-2}$) Gaussian scattering potential. The inset shows the scattering potentials in a cross section along the middle of the wire $(x=0)$ and the parabolic confinement potential $(\hslash \omega =1.01\mathrm{Ry})$. The total number of modes is $N=9$. The arrows point at the energies at which the probability density is calculated in Figs. 5, 6

Figure 5

(Color online). The probability density of the scattering states ${\psi }_{nE}^{+}$ in the parabolic quantum wire in the presence of the single attractive Gaussian scattering potential of Fig. 4. The total energy of the incident particle is $E=2.02\mathrm{Ry}$ (cf. the left arrow in Fig. 4). The incoming wave is in mode $n=0$ (above) and $n=1$ (below).

Figure 6

(Color online). The probability density of the scattering states ${\psi }_{nE}^{+}$ in the parabolic quantum wire in the presence of the single attractive Gaussian scattering potential of Fig. 4. The total energy of the incident particle is $E=2.50\mathrm{Ry}$, coinciding with a dip in the conductance (cf. the right arrow in Fig. 4). The incoming wave is in mode $n=0$ (above) and $n=1$ (below).

Figure 7

(Color online). Conductance of a hard wall quantum wire in units of $G_0=2e^2∕h$ as a function of energy in the presence of a double Gaussian scattering potential with a varying depth of the well. The inset shows the scattering potentials, whose parameters in the solid (dashed) case are $V_1=5.21\left(6.16\right)\mathrm{Ry}$, $V_2=7.49\left(9.70\right)\mathrm{Ry}$, ${\alpha }_1=0.52\left(0.60\right)a_0^{-2}$ and ${\alpha }_2=1.52\left(1.60\right)a_0^{-2}$ in the cross section $x=L∕2$. The total number of modes is $N=8$. The width of the wire is $L=4a_0$.

Figure 8

(Color online). The solid line represents the double Gaussian potential of Fig. 7 along with its first two effective potentials.

Figure 9

(Color online). Conductance of a parabolic wall quantum wire in units of $G_0=2e^2∕h$ as a function of energy in the presence of a double Gaussian scattering potential. The inset shows the scattering potential, whose parameters are $V_1=2.02\mathrm{Ry}$, $V_2=3.71\mathrm{Ry}$, ${\alpha }_1=0.29a_0^{-2}$ and ${\alpha }_2=0.96a_0^{-2}$, in the cross section $x=0$. The parabolic confinement potential $(\hslash \omega =1.01\mathrm{Ry})$ is also shown. The total number of modes is $N=9$. The arrow points at the value of the energy at which the probability density in Fig. 10 is calculated.

Figure 10

(Color online). The probability density of the scattering states ${\psi }_{nE}^{+}$ in the parabolic quantum wire in the presence of the double Gaussian scattering potential of Fig. 9. The total energy of the incident particle is $E=0.64\mathrm{Ry}$, coinciding with a resonance in the conductance (cf. the arrow in Fig. 9). The incoming wave is in mode $n=0$.

Figure 11

(Color online). Conductance of a hard wall quantum wire in units of $G_0=2e^2∕h$, as a function of energy. A single contour of the scattering potential, an asymmetric single Gaussian, is shown in the inset. The contour is chosen to be at the value of $V_0{e}^{-1}=0.368V_0$. In the upper panel the potentials are repulsive $(V_0=3\mathrm{Ry})$ but attractive $(V_0=-3\mathrm{Ry})$ in the lower panel. In both cases the circularly symmetric potential has $\alpha =2a_0^{-2}$, the transverse one has ${\alpha }_x=0.10a_0^{-2}$ and ${\alpha }_z=6.73a_0^{-2}$, and the longitudinal has ${\alpha }_x=9.10a_0^{-2}$ and ${\alpha }_z=0.11a_0^{-2}$. The parameters are chosen such that the volume between the potential and the zero energy plane are equal. The width of the wire is $L=6a_0$ and the total number of modes is $N=8$. The symmetric and longitudinal potentials have been shifted by $1G_0$ and $2G_0$, respectively, for clarity.

Figure 12

(Color online). Conductance of a hard wall quantum wire in units of $G_0=2e^2∕h$, as a function of energy. The scattering potential has the same parameters as the attractive potential in Fig. 11, but the center is moved from the middle of the wire $x_i=L∕2$ to $x_i=5L∕8$ and $x_i=3L∕4$. The width of the wire is $L=6a_0$ and the total number of modes is $N=8$ as before. The inset shows the $V_0{e}^{-1}$ contour of the scattering potentials. The potentials in the center of the wire and the one with $x_i=5L∕8$ have been shifted by $1G_0$ and $2G_0$, respectively, for clarity.

Figure 13

The energy levels of the effective potential $V_{22}\left(z\right)$ of the longitudinal attractive potential of Fig. 12 with $x_i=3L∕4$. The levels are obtained in a box of length $L_{\text{box}}=150a_0$.

Figure 14

(Color online). The probability density of the scattering states ${\psi }_{nE}^{+}$ in the hard wall quantum wire in the presence of the elongated longitudinal Gaussian potential shown in the inset of Fig. 12 shifted to $x_i=3L∕4$. The incident wave is in mode $n=1$ and has total energy $E=0.865$ $\left(1.084\right)\mathrm{Ry}$ in the upper (lower) panel. These energies correspond to the two dips in the conductance of Fig. 12.