Wave packet dynamics in semiconductor quantum rings of finite width

The time evolution of a wave packet injected into a semiconductor quantum ring is investigated in order to obtain the transmission and reflection probabilities. Within the effective-mass approximation, the time-dependent Schrödinger equation is solved for a system with nonzero width of the ring and leads and finite potential-barrier heights, where we include smooth lead-ring connections. In the absence of a magnetic field, an analysis of the projection of the wave function over the different subband states shows that when the injected wave packet is within a single subband, the junction can scatter this wave packet into different subbands but remarkably at the second junction the wave packet is scattered back into the subband state of the incoming wave packet. If a magnetic field is applied perpendicularly to the ring plane, transmission and reflection probabilities exhibit Aharonov-Bohm (AB) oscillations and the outgoing electrons may end up in different subband states from those of the incoming electrons. Localized impurities, placed in the ring arms, influence the AB oscillation period and amplitude. For a single impurity or potential barrier of sufficiently strong strength, the period of the AB oscillations is halved while for two impurities localized in diametrically opposite points of the ring, the original AB period is recovered. A theoretical investigation of the confined states and time evolution of wave packets in T wires is also made, where a comparison between this system and the lead-ring junction is drawn.

Figure 1

(Color online) Potential profiles for (a) rings and (b) T wires, considering (top) right-angle and (bottom) smooth connections. The smooth connections are described by circles of radius $R_s=300\text{Å}$. The width in both systems is $W=100\text{Å}$ and the average radius is $R_{\text{av}}=600\text{Å}$. The potential is defined as $V\left(x,y\right)=0$ inside ring and channel regions (blue) and $V_e$ outside (white).

Figure 2

(Color online) (a) Ground-state squared wave function (ground and first-excited states are degenerate states with symmetric and antisymmetric wave functions), for sharp channel-ring connections. (b) Ground and (c) second-excited-states squared wave functions, for smooth connections. Black lines illustrate the limits of the channel-ring profiles.

Figure 3

(Color online) Energy versus wave-vector $k_x$ diagram for a quantum wire (solid curves). The average energies of the considered three wave packets, ${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$ (thick dotted lines) are indicated. The widths of $k_x$ and $E$ distributions of the initial wave packet ${\epsilon }_3$ are represented by horizontal and vertical Gaussians, respectively. The wave vector $k_3^{\left(1\right)}$ is related to the energy ${\epsilon }_3$ of the ground-state subband whereas $k_3^{\left(2\right)}$ is for the same energy but for the first-excited subband.

Figure 4

(Color online) Time evolution of the projection of the wave function on the ground (black) and first-excited (red) subbands of a quantum well of width $W=100\text{Å}$, calculated at three different locations: left channel, upper arm of the ring, and right channel. The injected wave packet is in the lowest subband and has average energy (a) ${\epsilon }_1$, (b) ${\epsilon }_2$, and (c) ${\epsilon }_3$. [see Fig. 3]

Figure 5

(Color online) The same as Fig. 4c but now we consider the wave packet with energy ${\epsilon }_3$ to be in the second subband, with wave vector $k_3^{\left(2\right)}$.

Figure 6

(Color online) Time evolution of the wave-function projections on the ground (black), first-excited (red), and second-excited (blue) subband states of a quantum well of width $W=100\text{Å}$, calculated at the upper arm of the ring, for a wave packet in the second subband with energy 430 meV.

Figure 7

(Color online) Contour plots of the time evolution of the squared wave function, for right-angle (left panels) and smooth (right panels) lead-ring connections. The initial wave packet is in the lowest subband with width $\sigma =200\text{Å}$ and energy (a) ${\epsilon }_1$ and (b) ${\epsilon }_2$.

Figure 8

(Color online) Contour plots of the time evolution of the squared wave function, for right-angle (left panels) and smooth (right panels) lead-ring connections. The initial wave packet has width $\sigma =200\text{Å}$, energy ${\epsilon }_3$, and is in the (a) lowest subband with wave vector $k_3^{\left(1\right)}$, and in the (b) second subband with wave vector $k_3^{\left(2\right)}$.

Figure 9

(Color online) Time-dependent probability current, calculated at three points: left channel, upper arm of the ring, and right channel, for wave packets with energies ${\epsilon }_1$ (black, solid), ${\epsilon }_2$ (red, dashed), and ${\epsilon }_3$ (blue, dotted) in the case of sharp lead-ring connections.

Figure 10

(Color online) Transmission probabilities $T$ for wave packets with energies ${\epsilon }_1$ (black, dashed) and ${\epsilon }_2$ (red, dotted) in the first subband, and with energy ${\epsilon }_3$ in the first (blue, solid) and second (green, dashed-dotted) subbands, as functions of the magnetic field, for (a) right-angle and (b) smooth lead-ring connections.

Figure 11

(Color online) Transmission probabilities $T$ for wave packets injected in the lowest subband with different energies $\epsilon =140$ (black, dashed), 160 (red, dotted), and 170 meV (blue, solid), as functions of the magnetic field, considering (a) right-angle and (b) smooth with $R_s=300\text{Å}$ lead-ring connections.

Figure 12

(Color online) Contour plots of the time evolution of the electron density, for right-angle (left panels) and smooth (right panels) lead-ring connections, in the presence of an applied magnetic field $B=0.18\text{T}$, perpendicular to the ring plane. The initial wave packet has width $\sigma =200\text{Å}$, energy ${\epsilon }_3$ and is in the (a) lowest subband with wave vector $k_3^{\left(1\right)}$, and in the (b) second subband with wave vector $k_3^{\left(2\right)}$.

Figure 13

(Color online) Contributions $⟨J_x^{\left(j\right)}⟩$ of the ground (black, dashed), first-excited (red, dotted), and second-excited (green, dashed-dotted) subband states to the transmission current, as functions of the magnetic field $B$, calculated at the right channel, considering wave packets with energy ${\epsilon }_3$ in the (a) first and (b) second subband, and (c) with energy $\epsilon =430\text{meV}$ in the second subband. The blue solid line is the sum of the contributions due to the three subband states, $⟨J_x^{\left(1\right)}⟩+⟨J_x^{\left(2\right)}⟩+⟨J_x^{\left(3\right)}⟩$.

Figure 14

(Color online) [(a),(d)] Transmission probability as a function of the depth $V_G$ of the Gaussian well, for wave packets with energy ${\epsilon }_3$ in the first subband $\left(k_3^{\left(1\right)}\right)$ and in the second subband $\left(k_3^{\left(2\right)}\right)$, considering quantum rings with right-angle (left panels) and smooth, with $R_s=300\text{Å}$, (right panels) lead-ring connections. Two values of the magnetic field $B$ are considered: 0 T (solid) and 0.18 T (dotted). [(b),(e)] Time-averaged currents $⟨J_x^{\left(1\right)}⟩$ and $⟨J_x^{\left(2\right)}⟩$, of the first and second subbands, respectively, considering a wave packet initially in the first subband, with $k_3^{\left(1\right)}$. [(c),(f)] Time-averaged currents $⟨J_x^{\left(1\right)}⟩$ and $⟨J_x^{\left(2\right)}⟩$ for a wave packet initially in the second subband, with $k_3^{\left(2\right)}$. The lines $⟨J_x^{\left(1\right)}⟩$ and $⟨J_x^{\left(2\right)}⟩$ for $B=0.18\text{T}$ (dotted) were shifted by $+0.5$, for clarity [except in (c), where they were shifted by $+1.0$].

Figure 15

(Color online) Transmission probability as a function of the magnetic field for wave packets with energy (a) ${\epsilon }_2$ and (b) ${\epsilon }_3$ in the first subband $\left(k_3^{\left(1\right)}\right)$, considering quantum rings with smooth $R_s=300\text{Å}$ lead-ring connections, for several values of depth $V_G$ of the Gaussian well: 0 (black, solid), 10 (red, dashed), 20 (blue, dotted), 30 (green, dashed-dotted), and 40 V (yellow, dashed-dotted-dotted). The latter four curves are shifted in order to help visualization and the amount of shift is indicated on top of the curves.

Figure 16

(Color online) Transmission probabilities for a wave packet with energy ${\epsilon }_2$, as functions of the magnetic field, considering smooth channel-ring connections, in the presence of (a) one impurity, localized at three different distances $z_{\text{imp}1}$ from the ring plane: $1\text{Å}$ (black, solid), $100\text{Å}$ (red, dashed), and $400\text{Å}$ (blue, dotted), and (b) two impurities, each one localized symmetrically in one arm of the ring, at distances $z_{\text{imp}1}=1\text{Å}$ and $z_{\text{imp}2}=1\text{Å}$ (black, solid) or $z_{\text{imp}2}=100\text{Å}$ (red, dashed).

Figure 17

(Color online) Energies of four low-lying bound states of a T-wire junction [see Fig. 1b], as functions of the radius of the circle describing the smooth junction. The insets show the probability density of the four bound states in case of $R_s/W=4$.

Figure 18

(Color online) Time-dependent probability current, calculated at the vertical (black, dotted) and horizontal (red, solid) leads of a $W=100\text{Å}$ T-shaped wire, at a distance $1400\text{Å}$ from the center of the junction, considering right-angle and smooth, with $R_s=300\text{Å}$, connections. In (a), two values of wave packet energy were considered, ${\epsilon }_1$ and ${\epsilon }_2$ while in (b), we consider ${\epsilon }_3$ in subbands $k_3^{\left(1\right)}$ and $k_3^{\left(2\right)}$.

Figure 19

(Color online) (a) Calculated transmission times for a classical particle traveling in a T-shaped wire (black circles) and the time where the maximum value of $J_T$ occur in the vertical lead (red triangles), as functions of the wave packet energy $\epsilon$, for leads of $W=100\text{Å}$ width, considering right-angle connections. (b) Time-averaged currents in the first ($⟨J_x^{\left(1\right)}⟩$, black) and second ($⟨J_x^{\left(2\right)}⟩$, red) subbands.

Figure 20

(Color online) (a) Transmission probability as a function of the wave packet energy $\epsilon$ (in the lowest subband) for a T-shaped wire with leads of $W=100\text{Å}$, considering several values of radius $R_s$ for smooth connections. (b) Square root of the energies of the peaks and valleys for each curve shown in (a). The straight lines are linear fits to the symbols. Similar types of curves in (a) and (b) correspond to the same $R_s$.