Conductance of a quantum wire with a Gaussian impurity potential and variable cross-sectional shape

We calculate the conductance through a Gaussian impurity potential in a quantum wire using the Lippmann-Schwinger equation. The impurity has a decay length $d$ along the propagation direction while it is localized along the transverse direction. In the case of a repulsive Gaussian impurity it is shown that the conductance quantization is strongly affected by the decay length. In particular, increasing $d$ causes gradual suppression of backscattering and smaller contribution of evanescent modes, leading to progressively sharper conductance steps. The dependence of the conductance on the impurity position is also examined. In the case of an attractive Gaussian impurity it is shown that multiple quasibound states are formed due to the finite size of the impurity. By varying the size of the impurity these quasibound states may evolve into highly localized states with greatly enhanced lifetime. It is also shown (for a model impurity potential very similar to the Gaussian) that the transmission exhibits asymmetric Fano line shape. Under certain circumstances the Fano line shape may appear “inverted” or evolve into a Breit-Wigner dip. We consider also the effects of the cross-sectional shape of the wire on the quantum transmission. It is shown that varying the cross-sectional shape causes shifting of the positions of the conductance steps (which is due to the rearrangement of the transverse energy levels) and influences the character of conductance quantization.

Figure 1

(a) Conductance $G$ (in units of $2e^2∕h$) as a function of Fermi energy $E$ (over the first subband) through a Gaussian impurity of strength $\gamma =2.5\times {10}^6{\mathrm{cm}}^{-1}$ in a 2D rectilinear quantum wire of width $w$. $E$ is given in units of $E_1$ (where $E_1$ is the bottom of the first subband in the clean wire). Results are shown for various sizes of the impurity (specified by the parameter $d∕w$). Note that increasing $d$ results to sharper rise of the conductance. (b) Tunneling probability vs $d∕w$, for two values of the Fermi energy. The solid line corresponds to $E=0.9E_{1,s}$ while the dashed line to $E=0.88E_{1,s}$, where $E_{1,s}$ is the bottom of the first subband in the scattering region.

Figure 2

Conductance $G$ (in units of $2e^2∕h$) vs Fermi energy $E$ through a Gaussian impurity of strength $\gamma =2.5\times {10}^6{\mathrm{cm}}^{-1}$ in a 2D rectilinear quantum wire of width $w$, for three values of the parameter $d∕w$. Note the sharper rise of the second conductance step (compared to the first step) which is due to the smaller distance between $E_{2,s}$ and $E_2$, where $E_{2,s}$ is the bottom of the second subband in the scattering region and $E_2$ is the corresponding subband in the clean part of the wire.

Figure 3

(a) Conductance $G$ (in units of $2e^2∕h$) vs Fermi energy $E$ through a Gaussian impurity of strength $\gamma =2.5\times {10}^6{\mathrm{cm}}^{-1}$ and $d=0.7w$ in a 2D quantum wire of width $w$, for three different impurity positions (specified by the parameter $y_i∕w$). Note the shifting of the positions of the conductance steps which is due to the shifting of the subbands $E_{n,s}$ in the scattering region. (b) The bottoms of the first two subbands vs impurity position $y_i∕w$. The two straight horizontal lines represent the subbands $E_1$, $E_2$ of the clean wire while $E_{1,s}$, $E_{2,s}$ are the subbands in the scattering region which follow the structure of the transverse wave functions ${\mid {\varphi }_n\left(y\right)\mid }^2$.

Figure 4

(a) Conductance $G$ (in units of $2e^2∕h$) vs Fermi energy $E$ through an attractive Gaussian impurity of strength $\gamma =-2\times {10}^6{\mathrm{cm}}^{-1}$ in a 2D quantum wire of width $w$, for three values of the parameter $d∕w$. The multiple dips in the conductance are due to the formation of quasibound states. The upper two curves (corresponding to $d∕w=0.48$ and 0.56) are vertically offset by an amount $1.1(2e^2∕h)$ and $2.2(2e^2∕h)$ respectively. (b) Location of the first dip $E_{\mathit{res}}$ (in units of $E_1$) in the first subband vs the size of the impurity (specified by the parameter $d∕w$). (c) Half width $\Gamma$ (in units of $E_1$) vs $d∕w$. The two minima in the HW correspond to highly localized states with enhanced lifetime.

Figure 5

Three-dimensional plots of the squared modulus of the wave function ${\mid {\psi }_{\mathbf{p}}^{(+)}(x,y)\mid }^2$ vs $x$ and $y$. (a) corresponds to the conductance minimum in the subband of the lower curve in Fig. 4a which occurs at $E_{\mathit{res}}=3.7363E_1$ while (b) corresponds to the minimum in the first subband of the middle curve in Fig. 4a and occurs at $E_{\mathit{res}}=3.5805E_1$. (c) corresponds to the stable state with infinitesimally small half width [i.e., the first minimum in Fig. 4c] which corresponds to $d=0.4352w$ and $E_{\mathit{res}}=3.6263E_1$.

Figure 6

Impurity potential profiles versus $x∕w$. The solid line represents the Gaussian impurity potential with decay length $d$ while the dashed line represents the impurity potential $V\left(x\right)={\mathrm{sech}}^2\left(\alpha x\right)$ with decay length ${\alpha }^{-1}=3^{1∕3}d∕\sqrt{\pi }$ (as explained in the text).

Figure 7

Transmission coefficient $T$ vs Fermi energy $E$ through an attractive impurity potential $V_i(x,y)=({\hslash }^2\gamma ∕2m^{*}){\mathrm{sech}}^2\left(\alpha x\right)\upsilon \left(y\right)$ in a 2D quantum wire, for three values of the inverse decay length $\alpha$ while ${\upsilon }_{11}$ is kept fixed at the value ${\upsilon }_{11}=-1.5$. (a) The solid line, which corresponds to $\alpha =1.7$ with ${\upsilon }_{22}=-1.4$ and ${\upsilon }_{12}=0.08$, shows an asymmetric Fano line shape while the dashed line represents the direct transmission. (b) Corresponds to $\alpha =1.4$, ${\upsilon }_{22}=-0.8$, ${\upsilon }_{12}=0.08$ and shows a distortion of the Fano resonance. (c) Corresponds to $\alpha =1$, ${\upsilon }_{22}=-0.5$, ${\upsilon }_{12}=0.12$ and the transmission (which is generally enhanced) exhibits only a dip.

Figure 8

Transmission coefficient $T$ vs Fermi energy $E$ through an attractive impurity potential $V_i(x,y)=({\hslash }^2\gamma ∕2m^{*}){\mathrm{sech}}^2\left(\alpha x\right)\upsilon \left(y\right)$ in a 2D quantum wire. (a) Both solid and dashed lines correspond to ${\upsilon }_{11}=-2$ and $\alpha =1$ so that $E_R-E_0=0$. The solid line corresponds to ${\upsilon }_{22}=-0.8$ and ${\upsilon }_{12}=0.1$ while the dashed line corresponds to${\upsilon }_{22}=-0.95$ and ${\upsilon }_{12}=0.14$. Notice that for the larger value of $\mid {\upsilon }_{22}\mid$ the antiresonance moves lower in energy while its width increases (due to the larger value of ${\upsilon }_{12}$). (b) Corresponds to appropriate values (given in the text) such that $E_R<E_0$ (i.e., the resonance level is “inverted” with respect to Fig. 7).

Figure 9

(a) Conductance $G$ (in units of $2e^2∕h$) vs Fermi energy $E$ through a Gaussian impurity of strength $\gamma =2.3{\mathrm{cm}}^{-2}$ and decay length $d=20\mathrm{nm}$ in 3D rectilinear quantum wires with asymmetric cross sections (specified by the parameter $s$, where $s=l∕w$ and $w$, $l$ are the transverse dimensions of the cross section along $y$ and $z$). $E$ is given in units of $E_{11}$ (where $E_{11}$ is the bottom of the first subband in the clean wire). The cross-sectional area is kept constant to the value $wl=600{\mathrm{nm}}^2$ as the shape of the cross section is varied. (b) Transverse energy levels vs the cross-sectional shape anisotropy $s$. Solid lines correspond to the levels $E_{\mathit{nm}}$ of the unperturbed wire while the dashed lines correspond to the levels $E_{\mathit{nm},s}$ in the impurity region.