Zero- and one-dimensional magnetic traps for quasiparticles in diluted magnetic semiconductors

We investigate the possibility of trapping quasiparticles in hybrid structures composed of a magnetic quantum well placed a few nanometers below a ferromagnetic micromagnet. We are interested in two different micromagnet shapes: cylindrical (microdisk) and rectangular geometry. We show that in the case of a microdisk, the quasiparticles are localized in all three directions and therefore zero-dimensional states are created. In the case of an elongated rectangular micromagnet the quasiparticles can move freely in one direction, hence one-dimensional states are formed. After calculating the magnetic field profiles produced by the micromagnets, we analyze in detail the light absorption spectrum for different micromagnet thicknesses and different distances between the micromagnet and the semimagnetic quantum well. We argue that the discrete spectrum of the localized states can be detected via spatially resolved low-temperature optical measurement.

Figure 1

(Color online) In the absence of a micromagnet we expect the main absorption edge (MAE) between heavy hole $\left(H{H}_1\right)$ and electron $\left(E_1\right)$ energy states (full lines). Both energy states show twofold spin degeneracy without external magnetic field. After the deposition of the micromagnet, new states appear below the $E_1$ state and above the $H{H}_1$ state (dashed lines). The spin degeneracy is lifted in each new energy state. In a QW structure the heavy hole states and the light hole states (e.g., $H{H}_1$ and $L{H}_1$) are split even without external magnetic field.

Figure 2

(Color online) Top panel: splitting of the heavy hole $\left(H{H}_1\right)$ and light hole $\left(L{H}_1\right)$ edges at the $\Gamma$ point in a QW vs $B_x$ (we use the electron representation). $B_y$ and $B_z$ are set to zero. Bottom panel: splitting of the heavy hole and light hole edges in a QW vs $B_z$ ($B_x=0$ and $B_y=0$). Solid and dashed lines represent states with opposite spin. Heavy hole states $\mp 3∕2$ do not split when the magnetic field is applied in the plane of the QW as seen on the upper panel.

Figure 3

(Color online) Three-dimensional image of the magnetic field $B(x,y,z=d)$ on the $XY$ surface at a distance $d$ above Fe microdisk. It is assumed that the micromagnet is in the vortex state. The diameter of the region where the $z$ component of the magnetization $M_z$ is nonzero (the core region) is of the order of $R_c=60\mathrm{nm}$, whereas the diameter of the microdisk $R$ is $1\mu \mathrm{m}$, and the thickness of the disk $D_z$ is $50\mathrm{nm}$. $M_z$, depicted by vertical arrows in the core region, is very well approximated by a parabolic profile.

Figure 4

(Color online) Absorption coefficient for three distances $d$ between the microdisk and the QW with $D_z=50\mathrm{nm}$. The number $\left(n_c{n}_v\right)$ corresponds to transitions between the $n_c^{\mathit{th}}$ electron state and the $n_v^{\mathit{th}}$ hole state.

Figure 5

(Color online) Absorption coefficient for two thicknesses $D_z$ of the ferromagnetic microdisk. In magnetostatic language, $D_z$ is the distance between magnetic charges. The separation between the microdisk and the QW was set to $d=10\mathrm{nm}$.

Figure 6

$Z$ component of the magnetic field $\vec{B}$ produced by a rectangular micromagnet at a distance $d$ below its narrow edge. We show results for two distances, $d=10\mathrm{nm}$ and $d=60\mathrm{nm}$, and for two micromagnet thicknesses, $D_z=150\mathrm{nm}$ and $D_z=500\mathrm{nm}$. Contour plots on the $xy$ planes indicate localization and “strength” of the $z$ component of the magnetic field.

Figure 7

(Color online) Absorption coefficient for distances $d=10$, 20, and $30\mathrm{nm}$ (from top to bottom) between the rectangular micromagnet ($D_z=150\mathrm{nm}$, $D_x=6\mu \mathrm{m}$, and $D_y=2\mu \mathrm{m}$) and the QW. The binding energy as well as the separation between the peaks decreases almost linearly with increasing $d$. In the inset we show the arrangement of the rectangular micromagnet in a single domain state. Magnetization is pointing in the $x$ direction.

Figure 8

(Color online) Absorption coefficient for different thicknesses $D_z$ of a rectangular micromagnet. In this case the distance between the micromagnet and the QW is constant, $d=10\mathrm{nm}$. Separation between peaks is barely distinguishable for all three $D_z$ values. Note that the absorption peak (11) shifts almost linearly with $D_z$.

Figure 9

(Color online) Comparison of the absorption spectrum in two models: the isotropic and the anisotropic $g$ factor of the hole. The electron $g$ factor is isotropic. The top and bottom spectra are very similar. Calculations done for rectangular Fe micromagnet with dimensions of $D_x=6\mu \mathrm{m}$, $D_y=2\mu \mathrm{m}$, $D_z=0.15\mu \mathrm{m}$ on the top of a QW structure at a distance $d=10\mathrm{nm}$ apart.