Geometry and magnetic-field-induced vortices and antivortices in mesoscopic two-dimensional systems

A two-dimensional mesoscopic system of arbitrary shape placed in a perpendicular homogeneous magnetic field is investigated for arbitrary boundary conditions. The energy spectrum of quantum dots and the phase boundary of mesoscopic superconductors are obtained. The wave function for quantum dots and the complex order parameter for mesoscopic superconductors exhibit the simultaneous stabilization of vortices and antivortices at specific ranges of the magnetic field for noncircular symmetric samples. When the vorticity is changed, we found that antivortices can be introduced into the system in order to preserve the symmetry of the sample in the vortex pattern.

Figure 1

(a) Energy spectrum of an electron in an infinite high square well $(\text{lines}+\text{symbol})$ and in an infinite high circular well (dashed lines), placed in a magnetic field $B$ perpendicular to the surface, as function of the flux $\varphi =BS$ through the surface $S$ of the square. ${\varphi }_0$ denotes the elementary flux quantum $hc∕\mid e\mid$ and $E_0={\hslash }^2∕2mS$. The numbers indicated on the spectrum are the $l$ quantum numbers for the circular disk. (b) Expectation value of the angular momentum operator $L_z$ as function of the flux (magnetic field). (c) Expectation value $⟨x^2+y^2⟩$ as function of the flux (magnetic field). The same type of symbols are used in all three figures for curves corresponding to the same energy level.

Figure 2

Vortex patterns in (a)–(l) for the infinite square well. A clockwise current flow (i.e., a vortex) corresponds to $l=+1$, while $l=-1$ corresponds to a counterclockwise current flow (i.e. an antivortex).

Figure 3

Zoom of the energy spectrum of the square well (full lines) and the circular well (dashed lines). The current flow patterns (i.e., the vortex structure) in the points (a)–(l) are shown in Fig. 2. The gray circle highlights a crossing in the spectrum that will be discussed further in combination with Figs. 7, 8, 9.

Figure 4

Position of a vortex–antivortex pair as function of the flux (magnetic field). The inset schematically shows the vortex pattern and the set of vortices (circled pair) that was used to determine the position in the $x$ direction (the $y$ coordinate does not change in that direction). $W$ is the side length of the square.

Figure 5

Energy spectrum for an electron in a finite height square well $(V_0∕E_0=100)$ as function of the flux $\varphi =BS$ through the surface $S$ of the well. Note that $E_0={\hslash }^2∕2mS$ and ${\varphi }_0=hc∕\mid e\mid$.

Figure 6

Vortex patterns in (a)–(d) for the finite height square well. The wave functions that correspond to the points indicated in Fig. 5 were used to make these plots. The square indicates the boundary of the well.

Figure 7

Energy difference between a pair of energy levels for different values of $\delta$. Inset: detail of the studied pair of energy levels at different values of $\delta$. The complete spectrum for $\delta =0$ and $\delta =0.3$ can be found in Figs. 3, 8, respectively, with a gray circle indicating the studied crossing/anticrossing.

Figure 8

Energy spectrum for an electron confined in a rectangular well $(\delta =0.3)$. The gray circle indicates the anticrossing that was studied in Fig. 7.

Figure 9

Probability of the electron corresponding to points (a)–(d), which are indicated in the inset of Fig. 7. High (low) density is given by the dark (light) regions.

Figure 10

(a) Energy spectrum of an electron in an infinite high triangular well placed in a magnetic field $B$ perpendicular to the surface, as function of the flux $\varphi =BS$ through the surface $S$ of the triangle. ${\varphi }_0$ denotes the elementary flux quantum $hc∕\mid e\mid$ and $E_0={\hslash }^2∕2mS$. (b) Expectation value of the angular momentum operator $L_z$ as function of the flux (magnetic field). (c) Expectation value $⟨x^2+y^2⟩$ as function of the flux (magnetic field). $W^{\prime }=2∕\sqrt[4]{3}W$, where $W$ is the length of a side of the triangle. The same type of symbols are used in all three figures for curves corresponding to the same energy level.

Figure 11

Vortex patterns in points (a)–(f) indicated in Fig. 10a for the triangular well.

Figure 12

The energy levels as function of $\beta$ (the parameter that determines the boundary condition).

Figure 13

Wave functions of the ground (a), the first (b), and the second (c) excited state for different values of $\beta$. $W$ is the width of the well.

Figure 14

Comparison of the effective critical temperature [for 1D (i.e., film) and 2D (i.e., square) confinement] with the critical temperature $T_c$ of the bulk superconductor as function of the boundary conditions, indicated by the parameter $\beta$.

Figure 15

Spectrum as function of the parameter $\beta$ for a square sample. Note that $E_0={\hslash }^2∕2mS$.

Figure 16

Spectrum as function of the flux $\varphi$ through the surface $S$ of the square well for (a) $\beta =-1.25$ and (b) $\beta =1$. The units are $E_0={\hslash }^2∕2mS$ and ${\varphi }_0=hc∕\mid e\mid$.

Figure 17

Superconducting—normal state phase boundary for $\beta =-1.25$ (left curve) and $\beta =1$ (right curve) for the case of a square with width $W∕\xi \left(0\right)=5$.

Figure 18

Phase diagram for the vorticity of a mesoscopic superconductor at the S/N boundary. $\beta$ is the parameter determining the boundary conditions, $\varphi =BS$ and ${\varphi }_0$ is the elementary flux quantum. The dashed line separates the superconducting phase (S) with the normal phase (N) and is drawn for $W∕\xi \left(0\right)=5$.

Figure 19

Schematic vortex patterns for different values of $l$. Only a small part of the square with $W∕\xi \left(0\right)=5$ near the vortex pattern is shown. A counterclockwise rotation corresponds to $l=1$, a clockwise rotation to $l=-1$ (Ref. 29). The double circle in the case of $l=2$ and $l=6$ indicates a giant vortex which caries two flux quanta. These patterns can be found in areas with corresponding $l$-value in Fig. 18.