Linear polarization of the photoluminescence of quantum wells subject to in-plane magnetic fields

The degree and orientation of the magnetic-field induced linear polarization of the photoluminescence from a wide range of heterostructures containing (001) (Cd,Mn)Te quantum wells between (Cd,Mn,Mg)Te barriers has been studied as a function of detection photon energy and of the strength and direction of a magnetic field applied in the plane of the quantum well. Three field-induced contributions to the linear polarization of the photoluminescence are observed which differ in their dependence on the angle, $\phi$, between the magnetic field and the [110] direction, being, respectively, independent of $\phi$, varying as $\cos \left(2\phi \right)$, and as $\cos \left(4\phi \right)$. A theoretical description of each of these contributions in terms of an in-plane deformation acting on the valence band states is presented and verified by comparison with the experimental data. In our model, we account for the possibility that the in-plane deformations are distributed in both magnitude and direction. We conclude that it is possible to account for the magnetic-field induced linear polarization of the photoluminescence via in-plane deformations and without invoking terms in the valence band spin Hamiltonian which are cubic in $J$. The models developed in the present paper apply in full measure to nonmagnetic quantum wells as well as ensembles of disklike quantum dots with shape and/or strain anisotropy.

Figure 1

Photoluminescence (solid lines) and linear polarization degree ${\mathcal{R}}_0$ (circles connected by a spline) spectra taken from different QWs in the magnetic field $\mathit{H}\perp z$: (a) sample 2, QW $60\mathrm{\AA }$, $H=2.5\mathrm{T}$; (b) sample 1, QW $40\mathrm{\AA }$, $H=0.4\mathrm{T}$; (c) sample 4, QW $20\mathrm{\AA }$, $H=1.5\mathrm{T}$. Excitation by He-Ne laser $\left(1.96\mathrm{eV}\right)$, $T=2\mathrm{K}$.

Figure 2

In-plane angular scans of polarization parameters ${\mathcal{R}}_0$ (open circles) and ${\mathcal{R}}_{45}$ (closed circles) taken in the following conditions: (a) sample 2, QW $60\mathrm{\AA }$, $H=2.5\mathrm{T}$ $(\mathit{H}\perp z)$; (b) sample 6, QW $45\mathrm{\AA }$, $H=0$. Angle $\phi =0°$ corresponds to $\mathit{H}\Vert \left[110\right]$. Lines show the pure second harmonic fits: ${\mathcal{R}}_0\propto \cos 2\phi$, and ${\mathcal{R}}_{45}\propto \sin 2\phi$.

Figure 3

In-plane angular scans of polarization parameters ${\mathcal{R}}_0$ (open circles) and ${\mathcal{R}}_{45}$ (closed circles) taken for: (a) sample 2, QW $20\mathrm{\AA }$, $H=2.5\mathrm{T}$; (b) sample 2, luminescence from the barrier layer, $H=1\mathrm{T}$. Points are fitted by a sum of zeroth, second, and fourth harmonics (in case of ${\mathcal{R}}_0$) and by that of second and fourth harmonics (in case of ${\mathcal{R}}_{45}$).

Figure 4

In-plane angular scans of polarization parameter ${\mathcal{R}}_0$ (open circles) taken for: (a) sample 4, QW $20\mathrm{\AA }$, $H=1.5\mathrm{T}$; (b) sample 4, luminescence from the barrier layer, $H=0.3\mathrm{T}$.

Figure 5

In-plane angular scan of polarization parameter ${\mathcal{R}}_0$ (shown in polar coordinates) taken for: (a) sample 6, QW $45\mathrm{\AA }$, $H=1.25\mathrm{T}$; (b) sample 5, QW $30\mathrm{\AA }$, $H=2.6\mathrm{T}$.

Figure 6

Magnetic field dependences of the polarization degree ${\mathcal{R}}_0$ taken for: (a) sample 2, QW $40\mathrm{\AA }$, two different orientations of the crystal with respect to the field; (b) sample 1, QW $40\mathrm{\AA }$. Solid lines are guides for the eye. The dotted line in panel (a) shows the quadratic fit.

Figure 7

Magnetic field dependences of the amplitudes of zeroth, second, and fourth angular harmonics of ${\mathcal{R}}_0$, taken for sample 6, QW $16\mathrm{\AA }$. Solid lines show the quadratic fits of the two former dependences. Since for the fourth harmonic the quadratic fit does not work in this magnetic field range, a straight line is shown instead (dotted).

Figure 8

Resonant spin-flip Raman scattering in sample 2, QW $60\mathrm{\AA }$ subject to tilted magnetic field ($\theta =60°$ to the growth axis) at different field strengths. The spectral position of the laser is seen in the figure, $T=1.6\mathrm{K}$; spectra are consecutively upshifted for convenience. Spectral features corresponding to spin flips of manganese, electron, hole, and exciton are marked as described in the text.

Figure 9

Raman shifts of the ESF, HSF, and XSF features in sample 2, QW $60\mathrm{\AA }$ (at $\mathit{H}$ along $\theta =60°$) plotted versus the value of the field (a) and the magnetization of the manganese spin system (b). Lines in (a) show the Brillouin fits.

Figure 10

Out-of-plane angular dependences ($\theta$ dependences) of the Raman shifts of ESF, HSF, and XSF features corresponding to spin flips of the electron, the hole, and of the exciton, respectively. $H=1\mathrm{T}$, $T=1.6\mathrm{K}$. Lines show the results of calculation as described in Sec. 3B.