Geometric and compositional influences on spin-orbit induced circulating currents in nanostructures

Circulating orbital currents, originating from the spin-orbit interaction, are calculated for semiconductor nanostructures in the shape of spheres, disks, spherical shells, and rings for the electron ground state with spin oriented along a symmetry axis. The currents and resulting orbital and spin magnetic moments, which combine to yield the effective electron $g$ factor, are calculated using a recently introduced formalism that allows the relative contributions of different regions of the nanostructure to be identified at zero magnetic field. For all these spherically or cylindrically symmetric hollow or solid nanostructures, independent of material composition and whether the boundary conditions are hard or soft, the dominant orbital current originates from intermixing of valence-band states in the electron ground state, circulates within the nanostructure, and peaks approximately halfway between the center and edge of the nanostructure in the plane perpendicular to the spin orientation. For a specific material composition and confinement character, the confinement energy and orbital moment are determined by a single size-dependent parameter for spherically symmetrical nanostructures, whereas they can be independently tuned for cylindrically symmetric nanostructures.

Figure 1

The orbital current within a unit cell can be split into an itinerant contribution ${\langle \mathbf{j}\rangle }_s$, and a localized contribution $\mathbf{j}\left(\mathbf{r}\right)-{\langle \mathbf{j}\rangle }_s$. Vector ${\mathbf{r}}_s$ points to the center of unit cell $s$.

Figure 2

Radius dependence of the confinement energy $\lambda$ and character of the electron ground state of an InAs sphere. The character is given in terms of the conduction-band ${\Gamma }_6^c$ (blue) and valence-bands ${\Gamma }_8^v$ (red) and ${\Gamma }_7^v$ (green) contributions.

Figure 3

The actual and expected [see Eq. (35)] radius $R_{\text{min}}$ where the electron ground state of spheres of different materials has the smallest conduction-band contribution.

Figure 4

The minimum contribution of the conduction band of spheres of different materials as a function of the parameter $\delta$, along with Eq. (36).

Figure 5

The radius dependence of the peak current densities ${\langle \mathbf{j}\rangle }^{\text{BV}}$ (red) and ${\langle \mathbf{j}\rangle }^{\text{EV}}$ (green), and their ratio (black) of an InAs sphere. The Bloch velocity related current is $\ge 10$ times the envelope velocity related current for radii $R\ge 1$ nm.

Figure 6

(a) The spatial distribution of the normalized magnitude of the ${\mathbf{e}}_y$ component of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ of a sphere. This current density peaks roughly at $R/2$ and resembles a current loop. (b)–(f) The magnetic moment density of the different components contributing to the orbital moment [(c)–(f)] and the spin moment (b) of a sphere. It can clearly be observed that ${\mu }_{\text{spin}}$ and ${\mu }_{\text{LC-EV}}$ have an even spatial symmetry, whereas the other orbital moments have an odd spatial symmetry. All figures are $xz$ cross sections; the white/black circles mark the boundary of the sphere.

Figure 7

The radius dependence of the different integrated orbital and spin moments and current $I_{\text{IC-BV}}$ of an InAs sphere.

Figure 8

The radius dependence of ${\mu }_{\text{IC-BV}}/{\mu }_{\text{Roth}}$ of spheres of various semiconducting materials (continuous lines). The same quantity is also plotted against the normalized radius $R/R^{*}$ (dotted lines).

Figure 9

The radius dependence of the character and integrated orbital moment ${\mu }_{\text{IC-BV}}$ of a HgTe sphere. The calculation is only valid for $R<2.7$ nm, hence the lines are dotted for $R>2.7$ nm.

Figure 10

The integrated orbital moment ${\mu }_{\text{IC-BV}}$ of InAs shells for various ratios $R_{\text{in}}/R_{\text{out}}$ as a function of the outer radius $R_{\text{out}}$ (continuous lines) and confinement energy $\lambda$ (dotted lines).

Figure 11

(a) The spatial distribution of the normalized magnitude of the ${\mathbf{e}}_y$ component of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ of a spherical shell. Due to the inner surface, an additional current loop is created, circulating oppositely to the outer current loop. (b), (c) The magnetic moment density of the most dominant orbital moment (c) and the spin moment (b) of a spherical shell. It can clearly be observed that ${\mu }_{\text{spin}}$ has an even spatial symmetry, whereas ${\mu }_{\text{IC-BV}}$ has an odd spatial symmetry. All figures are $xz$ cross sections, the white/black circles mark the boundaries of the spherical shell, and we choose $R_{\text{in}}/R_{\text{out}}=\frac{1}{2}$. Similar to the spheres, we have neglected the very small $x$ component of the spin moment density (see the discussion in Sec. 3a).

Figure 12

The height dependence of the character and confinement energy $\lambda$ of an InAs quantum well, where only $v_{4,7}$ intermix, in the cylindrical approximation (continuous lines) and in the spherical approximation (dotted lines).

Figure 13

The radius dependence of the character and confinement energy $\lambda$ of an InAs nanowire, where only $v_{3,5,8}$ intermix, in the cylindrical approximation (continuous lines) and in the spherical approximation (dotted lines).

Figure 14

The height and radius dependence of the character of the electron ground state of an InAs disk. The white lines are contours constant ${\left|N\right|}^2{\left|v_i\right|}^2$. The electron ground state $F_z=+\frac{1}{2}$ is dominated by the $v_1 (\mathrm{CB}\uparrow )$ state; note that (a) has a different color scale than the other plots. In the quantum wire limit (large $H$) only $v_{3,5,8}$ ($\mathrm{HH}\uparrow ,\mathrm{LH}\downarrow ,\mathrm{SO}\downarrow$) mix into the ground state, while in the quantum well limit (large $R$) only $v_{4,7}$ ($\mathrm{LH}\uparrow ,\mathrm{SO}\uparrow$) mix into the ground state. These coefficients are bounded by an imaginary $R=H$ line through their dependence on the confinement energy (see text). The contribution of $v_2 (\mathrm{CB}\downarrow )$ is absent and of $v_6 (\mathrm{HH}\downarrow )$ negligible (enhanced by ${10}^5$ for visibility).

Figure 15

(a), (b) The spatial distribution of the normalized magnitude of the ${\mathbf{e}}_y$ component of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ (a) and ${\langle \mathbf{j}\rangle }^{\text{EV}}$ (b) of a disk, with $v_6=0$. The current density peaks at roughly $R/2$ and $z=0$ and resembles a current loop in the plane of the disk. (c) The height and radius dependence of the ratio between the peak current densities ${\langle \mathbf{j}\rangle }_{\text{max}}^{\text{BV}}$ and ${\langle \mathbf{j}\rangle }_{\text{max}}^{\text{EV}}$ for an InAs disk. The color scale is logarithmic and the leftmost contour indicates a ratio of 10. As long as $R\ge 1$ nm, ${\langle \mathbf{j}\rangle }^{\text{BV}}\gg {\langle \mathbf{j}\rangle }^{\text{EV}}$. (d)–(h) The height and radius dependence of the different integrated orbital momenta (e)–(h) and the integrated spin moment (d) for an InAs disk, all in units of Bohr magnetons. The color scale is logarithmic and the white lines are contours of constant moment with a power of 10. The dashed black lines in (e) are contours of constant confinement energy.

Figure 16

The integrated orbital moment ${\mu }_{\text{IC-BV}}$ of an InAs quantum well/nanowire as function of height/radius (continuous lines) and confinement energy (dotted lines).

Figure 17

(a) Gates (red) define, by electrostatic means, a quantum dot in a quantum well (blue). We take hard-wall confinement in the $z$ direction and a harmonic potential in the lateral direction. (b) Gates (red) define, by electrostatic means, a quantum dot in a nanowire (transparent). We take hard-wall confinement at the nanowire surface and a harmonic confinement potential in the axial direction. (c) The normalized magnitude of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ in the ${\mathbf{e}}_y$ direction of an InAs quantum well with $H=10$ nm and $L_{\text{har}}^{\text{rad}}=10$ nm [45]. (d) The normalized magnitude of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ in the ${\mathbf{e}}_y$ direction of an InAs nanowire with $R=40$ nm and $L_{\text{har}}^{\text{ax}}=10$ nm [46]. The continuous white lines indicate hard-wall boundaries, the dashed ones indicate the harmonic confinement length.

Figure 18

The integrated orbital moment ${\mu }_{\text{IC-BV}}$ as a function of the harmonic confinement length $L_{\text{har}}^{\text{rad}}$ for gate-defined quantum dots in quantum wells (continuous lines) and as a function of radius $R$ for disk (dotted lines), with $H=10$ nm.

Figure 19

The integrated orbital moment ${\mu }_{\text{IC-BV}}$ as a function of the harmonic confinement length $L_{\text{har}}^{\text{ax}}$ for gate-defined quantum dots in nanowires (continuous lines) and as a function of height $H$ for disks (dotted lines), with $R=40$ nm [46, 48].

Figure 20

(a) The $xz$ cross section of the spatial distribution of the normalized magnitude of the ${\mathbf{e}}_y$ component of ${\langle \mathbf{j}\rangle }^{\text{BV}}$ of a ring. Similar to the spherical shells, there are two oppositely circulating current loops. For the plot we choose $R_{\text{in}}/R_{\text{out}}=\frac{1}{3}$ and set $v_6=0$. (b) The dependence of ${\mu }_{\text{IC-BV}}$ (in ${\mu }_B$) on the ring thickness $R_{\text{out}}-R_{\text{in}}$ and outer radius $R_{\text{out}}$, for an InAs ring with $H=100$ nm. Similar to the spherical shells, the orbital moment depends only on the ring thickness, as can be seen from the white contour lines. We have set $v_6=0$ to avoid numerical artifacts in the calculation.