Hiding the interior region of core-shell nanoparticles with quantum invisible cloaks

Based on the scattering cancellation, we provide a method not only making a nanoparticle nearly invisible, but also hiding its interior region from the outside probing matter wave. By applying the interplay among the nodal points of partial waves along with the concept of streamline in fluid dynamics for probability flux, a quantum invisible cloak to the electron transport in a host semiconductor is demonstrated by simultaneously guiding the probability flux outside a hidden region and keeping the total scattering cross section negligible. As the probability flux vanishes in the interior region, one can embed any materials inside a multiple core-shell nanoparticle without affecting physical observables from the outside. Our results reveal the possibility to design a protection shield layer for fragile interior parts from the impact of transport electrons.

Figure 1

A nanoparticle with the core-shell structure is illustrated as our quantum invisible cloak, where a different effective mass and potential energy are defined in each region, respectively. The quantum matter plane wave for the transport electron ${\Psi }_i$ is assumed to propagate along the $z$ axis.

Figure 2

(a) The parameter set $\{m_s,V_s\}$ to support a total internal reflection for the wave function in the shell region, ${\Psi }_s$; and (b) the corresponding parameter set $\{m_c,V_c\}$, marked in numbers, to simultaneously generate a negligible total normalized scattering cross section, as small as ${10}^{-4}$. (c) The probability amplitude of wave functions inside the barrier-well potential, as a function of the radial axis $r$, are shown for $|{\Psi }_c|$ and $|{\Psi }_s|$ in black and red colors, respectively. Decomposed channel functions $f_l$, $l=0,1,2$, are also shown as a comparison. Moreover, dashed lines in the barrier region show the extension of wave functions from the shell (potential well) region, all of which have a common nodal point. The parameter set $\{m_s,V_s,m_c,V_c\}$ used in (c) is $\{0.182m_e$, $-1.91$ eV, $0.1901m_e$, $6.368$ eV$\}$, with the geometric size parameters of our nanoparticle: $a=2$ and $a_c=1.7$ nm. The energy and effective mass for the transport electron are $E=0.01$ eV and $m_0=0.8m_e$ (in unit of the electron mass $m_e$), respectively.

Figure 3

(a) Probability density ${\left|\Psi \right|}^2$, and (b) the corresponding probability flux $\vec{J}$ of the wave function in the plane $x=0$ for a quantum invisible cloak, with the parameters reported in Ref. [13]. Even though this nanoparticle is invisible as shown in (a), but from (b) it can be seen that the probability flux goes through the entire object. Instead, (c) and (d) show the probability density and the corresponding probability flux, respectively, for hiding a nontransparent object, with the same parameter set used in Fig. 2. In this simulation, the probability inside the core region is calculated as small as ${10}^{-7}$, while near the interface of the core-shell, a probability flux is generated to satisfy the conservation of probability flux. Moreover, a three-layer nanoparticle as a quantum invisible cloak is demonstrated in (e) and (f), with the corresponding probability density and flux, respectively. Here, the parameters used are $a=2$, $a_c=1.7$, $a_h=0.68$ nm, $m_0=0.8m_e$, $m_s=0.182m_e$, $m_c=0.19m_e$, $m_h=0.5m_e$, $V_s=-1.91$, $V_c=6.368$, $V_h=-20$, and $E=0.01$ eV, respectively.