Zeros in single-channel transmission through double quantum dots

By using a simple model we consider single-channel transmission through a double quantum dot that consists of two single dots coupled by a wire of finite length $L$. Each of the two single dots is characterized by a few energy levels only, and the wire is assumed to have only one level whose energy depends on the length $L$. The transmission is described by using $S$ matrix theory and the effective non-Hermitian Hamilton operator $H_{\mathrm{eff}}$ of the system. The decay widths of the eigenstates of $H_{\mathrm{eff}}$ depend strongly on energy. The model explains the origin of the transmission zeros of the double dot that is considered by us. Mostly, they are caused by (destructive) interferences between neighboring levels and are of first order. When, however, both single dots are identical and their transmission zeros are of first order, those of the double dot are of second order. First-order transmission zeros cause phase jumps of the transmission amplitude by $\pi$, while there are no phase jumps related to second-order transmission zeros. In this latter case, a phase jump occurs due to the fact that the width of one of the states vanishes when crossing the energy of the transmission zero. The parameter dependence of the widths of the resonance states is determined by the spectral properties of the two single dots.

Figure 1

The double dot system is connected to the reservoirs by the coupling constants $v$. The single dots are coupled to the wire by the coupling constants $u$.

Figure 2

(a) The transmission through a double QD with two identical single QDs that are connected by a wire according to Fig. 1. The eigenvalues of $H_B$ are shown by full lines. ${\epsilon }_1=-1.7$, ${\epsilon }_2=-1.4$, and $ϵ\left(L\right)=-1-L∕5$ (dashed line), $v=0.3$, $u=0.1$. (b) The modules of the transmission amplitude $\mid t(E,L)\mid$ for the same double QD as in (a) for fixed lengths $L=2.75$ (solid line) and $L=4$ (dashed line). The energies of the two single QDs are shown by circles. The real part (c) and imaginary part (d) of the five eigenvalues $z_k$ of the effective Hamiltonian as a function of $L$ for $E=-1.5$. Thin solid line: $z_1$; dashed line: $z_2$; thick solid line: $z_3$; dotted line: $z_4$; and dash-dotted line: $z_5$. At $L=2.75$ the imaginary part of the third eigenvalue is equal to zero at all energies $E$.

Figure 3

(a) The modules $\mid t\left(E\right)\mid$ (lower curve) and the phase $\arg \left(t\right)∕\pi$ (upper curve) of the transmission amplitude for a double QD with two identical single QDs. The parameters are the same as in Fig. 2, and $L=4$ as in Fig. 2b, dashed line. (b) The modules $\mid t_1\left(E\right)\mid$ (lower curve) and the phase $\arg \left(t_1\right)∕\pi$ (upper curve) of the transmission amplitude for one of the single QDs that is part of the double QD considered in (a). The energy positions of the single QD levels are shown by circles in Fig. 2(b). (c) Evolution of imaginary and real parts of the transmission amplitude $t$ of the double QD with energy. At the upper right corner, a zoomed fragment of the evolution is shown which demonstrates that the evolution has cusplike behavior in the vicinity of the transmission zero. (d) The same as (c) but for the single QD. Here, the evolution is of standard type.

Figure 4

(a) The transmission through a double QD consisting of two identical single QDs with $d=5$ states that are connected by a wire, versus energy and length of the wire. The eigenvalues of $H_B$ are shown by thin lines while the energy $ϵ\left(L\right)=5∕2-2L∕3$ of the mode in the wire is shown by the dashed line. (b) Energy dependence of the modules (solid line) and of the phase $\arg \left(t\right)∕\pi$ (dashed line) of the transmission amplitude for $L=4$. ${\epsilon }_n=-1.75$ $\cos (\pi n∕6)$, $n=1,2,\dots ,5$, $v=0.5$, $u=0.2$. There are 11 resonance states in the double dot, 4 transmission zeros of second order, and no phase jumps.

Figure 5

The energy dependence of the modules $\mid t\left(E\right)\mid$ (bottom) and of the phase $\arg \left[t\left(E\right)\right]∕\pi$ (top) of the transmission amplitude for $L=2.25,2.5,2.75,3.0,3.25$. The other parameters are the same as those in Fig. 2. The transmission zeros are denoted by stars. They are of second order. The ordinate is shifted every time by 0.1 when $L$ is changed by 0.25. All phases are shifted by $\pi$.

Figure 6

The same as Fig. 2, but with two different single QDs: ${\epsilon }_{1_L}=-1.7$, ${\epsilon }_{2_L}=-1.4$, ${\epsilon }_{1_R}=-1.6$, ${\epsilon }_{2_R}=-1.3$.

Figure 7

(a) The transmission $\ln \mid T(E,L)\mid$ through a double QD with different single QDs connected by a wire, and (b) the energy dependence of the modules (solid line) and of the phase $\arg \left(t\right)∕\pi$ (dashed line) of the transmission amplitude for this double QD system at a fixed length $L=4$ of the wire. The energy levels of the left single QD are ${\epsilon }_{k_L}=-1.6,-1.1,-0.6$, while those of the right single QD are ${\epsilon }_{k_R}=-1.4,-0.9,-0.4$. Further, $ϵ\left(L\right)=1-L∕2$, $v=0.3$, $u=0.2$. The full lines in (a) are the eigenvalues of $H_B$ while the dashed line is the energy $ϵ\left(L\right)$. There are seven resonance states of the double dot, four transmission zeros of first order, and the corresponding four phase jumps.