Resonances in electronic transport through a quantum wire with impurities and variable cross-sectional shape

Electronic transport through a finite range Pöschl-Teller attractive impurity potential in a quantum wire is investigated using a coupled-channel theory. The impurity has a decay length along the propagation direction but is an arbitrary function of the lateral coordinate. In the presence of an impurity, the transmission of a quantum wire may exhibit resonances of the Fano type. The Fano line shape resulting from the interference of direct transmission and transmission via a quasibound state created in the impurity, is shown to be sensitive to the strength of the impurity. In particular, we show that by continuously increasing the strength of the impurity an asymmetric Fano resonance may evolve into a symmetric Breit-Wigner dip and subsequently into an “inverted” Fano resonance. Accordingly the asymmetry parameter $q$ oscillates between positive and negative values. The evolution of the bound states with increasing values of the decay length of the impurity is also examined. Furthermore, we consider the effects of the cross-sectional shape of the wire on the resonance line shape. It is shown that the Fano resonance structure may or may not be observed in electronic transport through three-dimensional quantum wires depending on the degree of anisotropy of their cross sections. We compare these results with the results for an impurity that has lateral extension but is a $\delta$ function along the propagation direction.

Figure 1

Impurity potential profile $V_{11}$ vs $x$ for scattering of the first channel mode $\nu =1$, for three values of the depth, i.e., $U_{11}=1.6$, 5.78, and 8.5 while $\alpha =1.7$ common for all three depths (Ref. 31). The units of $V_{11}$ and the units of $x$ are chosen as described in Ref. 30.

Figure 2

Transmission coefficient $T$ vs Fermi energy $E$ (in units as described in Ref. 30) through an attractive potential $V(x,y,z)=({\hslash }^2\gamma ∕2m){\mathrm{sech}}^2\left(\alpha x\right)v(y,z)$ in a 3D uniform quantum wire, for three values of $U_{11}$. We have chosen $U_{22}=1.1$, $v_{12}=0.08$, and $\alpha =1.7$ common for all three curves. (a) An asymmetric Fano resonance of the $0\to 1$ type. (b) A symmetric Breit-Wigner antiresonance for $U_{11}=5.78$ corresponding to one of the special values given in the text, at which the resonance energy coincides with the energy of the transmission zero. (c) An asymmetric Fano resonance of the $1\to 0$ type for which inversion of the resonance level occurred.

Figure 3

Fano asymmetry parameter $q$ plotted vs $U_{11}$, for three values of $\alpha$. For a description of the units see Ref. 30. For the three solid lines $U_{22}=1.1$, so that $0<U_{22}∕{\alpha }^2<2$, corresponding to the asymmetry parameter $q_0$ associated with the resonance originating from the first $(j=0)$ bound state. For the dashed line $\alpha =1$ and $U_{22}=3$, so that $U_{22}∕{\alpha }^2=3$, corresponding to the asymmetry parameter $q_1$ associated with the resonance originating from the second $(j=1)$ bound state. Resonance of the $0\to 1$ $(1\to 0)$ type results when $q>0$ $(q<0)$. For larger sizes of the impurity (i.e., smaller values of $\alpha$), inversion of the resonance level occurs more often while the resonance asymmetry progressively decreases. Note also that both $q_0$ (for $\alpha =1$) and $q_1$ cross the horizontal axis simultaneously giving rise to symmetric Breit-Wigner dips for exactly the same values of $U_{11}$.

Figure 4

Bound-state energies in the continuum of the first subband (Ref. 30) plotted vs the inverse decay length $\alpha$ of the impurity, for three values of the depth $U_{22}$. The lower three lines correspond to the first bound state, while the upper three lines correspond to the second bound state. Note that as the depth increases the bound state levels move lower in energy. The upper and lower horizontal double arrows show the ranges of $\alpha$ for which two bound states exist in the first subband, when $U_{22}=1.1$ and 1.4, respectively. The dots identify the points for which symmetric Breit-Wigner dips occur [as in Fig. 2b]. Note that these symmetric dips occur for both resonances at the same values of $\alpha$.

Figure 5

Transmission coefficients $T$ vs Fermi energy $E$ through a finite range PT (solid lines) or through a short range (dashed lines) impurity in 3D uniform quantum wires (Ref. 30) with asymmetric transverse cross sections (specified by the parameter $A$, where $A=l∕w$ and $w$, $l$ are the transverse dimensions of the cross section along $y$ and $z$). For all three shapes and for both types of an impurity $U_{11}=1.6$, $U_{22}=1.1$, and $v_{12}=0.08$ while $\alpha =1.45$. The cross-sectional area is kept constant to the value $wl=36$ as the shape of the cross section is varied. Note that increasing the anisotropy $A$ results to a shift of the Fano resonance toward the propagation threshold, regardless of the type of impurity, and finally when $A=A_c$ the resonance moves out of the energy window.

Figure 6

(a) Transverse energy levels $E_1$, $E_2$ vs the cross-sectional shape anisotropy $A$. The position of the transmission zero $E_0$ moves below $E_1$ at $A=A_c$ while $E_{\max }$ denotes the position of the transmission one (Ref. 30). (b) Critical value $A_c$ vs $\alpha$ for which the transmission zero (dashed) and transmission one (solid) move below $E_1$. Note that larger impurity size results to larger binding energy and thus smaller value of $A_c$ is required. (c) Fano asymmetry parameter $q$ vs $A$ for three values of $\alpha$ (corresponding to the three solid lines) for a PT impurity. The dashed line corresponds to a short range impurity. The solid line for $\alpha =1.45$ as well as the dashed line are plotted for the values used in Fig. 5. Note that as $A\to A_c$, $q$ grows rapidly resulting to a gradually sharper resonance peak, as shown in Fig. 5.