Magnetic field dependence of the in-plane hole $g$ factor in ZnSe- and CdTe-based quantum wells

The effective $g$ factor of holes is measured in modulation-doped ZnSe/(Zn,Mg)(S,Se) quantum wells and from surface-state $p$-doped CdTe/(Cd,Mg)Te quantum wells by time-resolved pump-probe Kerr rotation. The measurements are performed at a temperature of 1.7 K and in magnetic fields up to 5 T applied in the Voigt geometry with orientation perpendicular to the quantum-well growth axis. The absolute value of the in-plane hole $g$ factor increases with growing magnetic field in both studied heterostructures. A theoretical model is developed that considers the influence of magnetic field and interface mixing of heavy-hole and light-hole states on the $g$ factor. The model results are in good agreement with the experimental data.

Figure 1

8-nm-thick ZnSe/(Zn,Mg)(S,Se) QW (sample no. 1). (a) Photoluminescence spectrum. (b) Pump-probe Kerr rotation signal reflecting the coherent spin dynamics at various magnetic fields, measured at the trion resonance with the laser energy at $E_{\text{ex}}=2.809\mathrm{eV}$. $P_{\text{pump}}=1.4\mathrm{W}/{\mathrm{cm}}^2$ and $P_{\text{probe}}=0.4\mathrm{W}/{\mathrm{cm}}^2$. (c) Magnetic field dependence of the hole spin dephasing time (circles) and fit with Eq. (3) (line), which gives $\mathrm{\Delta }{g}_h=0.027$. (d) Kerr rotation traces at the transversal magnetic field of 2 T for different intensities of above-barrier illumination (3.06 eV). All data are at $T=1.7\mathrm{K}$.

Figure 2

8-nm-thick ZnSe/(Zn,Mg)(S,Se) QW (sample no. 1). (a) Magnetic field dependence of the hole Larmor precession frequency (circles) in comparison to linear dependence (black line) assuming $|g_h|=0.078$. (b) Magnetic field dependence of in-plane $|g_h|$ evaluated from the measured Larmor frequency (red circles) and calculated $|g_h|$ (black line) (See Sec. 4b). $E_{\text{ex}}=2.809\mathrm{eV}, P_{\text{pump}}=1.4\mathrm{W}/{\mathrm{cm}}^2, P_{\text{probe}}=0.4\mathrm{W}/{\mathrm{cm}}^2$, and $T=1.7\mathrm{K}$.

Figure 3

10.5-nm-thick ZnSe/(Zn,Mg)(S,Se) QW (sample no. 2). (a) Photoluminescence spectrum. (b) Kerr rotation traces at $B=5\mathrm{T}$ without (black line) and with above-barrier illumination (3.06 eV, $P_{\text{ill}}=2$ mW). $E_{\text{ex}}=2.8145\mathrm{eV}, P_{\text{pump}}=1.4\mathrm{W}/{\mathrm{cm}}^2, P_{\text{probe}}=0.4\mathrm{W}/{\mathrm{cm}}^2, T=1.7\mathrm{K}$. (c) Magnetic field dependence of the hole spin dephasing time (red circles) and fit to the data using Eq. (3) (black line) with $\mathrm{\Delta }{g}_h=0.045$.

Figure 4

10.5-nm-thick ZnSe/(Zn,Mg)(S,Se) QW (sample no. 2). (a) Magnetic field dependence of the hole Larmor precession frequency in comparison to a linear dependence (black line) with $|g_h|=0.085$. (b) Magnetic field dependence of in-plane $|g_h|$ extracted from measured Larmor frequencies: experimental data (red circles) and calculated $|g_h|$ (black line); for details, see Sec. 4b). $E_{\text{ex}}=2.8145\mathrm{eV}, P_{\text{pump}}=1.4\mathrm{W}/{\mathrm{cm}}^2, P_{\text{probe}}=0.4\mathrm{W}/{\mathrm{cm}}^2$, and $T=1.7\mathrm{K}$.

Figure 5

20-nm-thick $\mathrm{CdTe}/{\mathrm{Cd}}_{0.78}{\mathrm{Mg}}_{0.22}\mathrm{Te}$ QW (sample no. 3). (a) Photoluminescence spectrum excited at 1.670 eV. The red arrow marks the energy of the TRKR measurements. (b) Kerr rotation signal measured at $B=0.5\mathrm{T}$ (black line) and its components obtained from decomposition (vertically shifted for clarity): hole (red lines) and electron (blue, magenta and green lines) components. (c) Magnetic field dependence of the spin dephasing time for electrons (blue, magenta and green symbols) and holes (red symbols) and fit to the hole data using Eq. (3) (red line). $P_{\text{pump}}=4.2\mathrm{W}/{\mathrm{cm}}^2, P_{\text{probe}}=0.7\mathrm{W}/{\mathrm{cm}}^2, E_{\text{ex}}=1.597\mathrm{eV}$, and $T=1.7\mathrm{K}$.

Figure 6

20-nm-thick $\mathrm{CdTe}/{\mathrm{Cd}}_{0.78}{\mathrm{Mg}}_{0.22}\mathrm{Te}$ QW (sample no. 3). Magnetic field dependence of in-plane $|g_h|$: experimental data (red symbols) and calculated values (black line); for details, see Sec. 4b). $E_{\text{ex}}=1.597\mathrm{eV}, P_{\text{pump}}=4.2\mathrm{W}/{\mathrm{cm}}^2, P_{\text{probe}}=0.7\mathrm{W}/{\mathrm{cm}}^2$, and $T=1.7\mathrm{K}$.

Figure 7

Dependence of the heavy-hole and light-hole probability density in the ZnSe-based QWs corresponding to the highest valence-band energy level on the coordinate $z$ along the QW growth axis (the coordinate of the QW center is the zero on the horizontal axis) for different values of the magnetic field. The solid lines correspond to $B=0.5\mathrm{T}$ and the dashed lines demonstrate the results calculated for $B=5\mathrm{T}$. Panel (a) shows the calculations performed for sample no. 1 (QW width 8 nm) and panel (b) shows the results for sample no. 2 (QW width 10.5 nm). The solid black and dashed red lines depict the heavy-hole probability density $|{\chi }_{\pm 3/2}{\left(z\right)|}^2$; the solid blue and dashed green lines depict that of the light hole $|{\chi }_{\pm 1/2}{\left(z\right)|}^2$. The dashed vertical lines show the interfaces of the QWs.

Figure 8

Dependence of the heavy-hole and light-hole probability density in the 20-nm-thick CdTe-based QW corresponding to the highest valence-band energy level on the coordinate $z$ along the QW growth axis (the coordinate of the QW center is the zero on the horizontal axis) for different values of the magnetic field. The solid lines correspond to a magnetic field of $B=0.5\mathrm{T}$ and the dashed lines demonstrate the results calculated for a magnetic field of $B=5\mathrm{T}$. The solid black and dashed red lines depict the heavy-hole probability density $|{\chi }_{\pm 3/2}{\left(z\right)|}^2$; the solid blue and dashed green lines depict that of the light holes $|{\chi }_{\pm 1/2}{\left(z\right)|}^2$. The dashed vertical lines show the interfaces of the QWs.

Figure 9

Dependence of the heavy-hole probability density $|{\chi }_{\pm 3/2}{\left(z\right)|}^2$ in the ZnSe-based QWs corresponding to the highest valence-band energy level on the coordinate $z$ along the QW growth axis (the coordinate of the center of the QW is the zero on the horizontal axis). Panel (a) shows the calculations for the 10.5 nm width QW for different values of the mixing parameter $t_{l-h}$. Panel (b) shows the results for different widths of the QW using the fixed value of the mixing parameter $t_{l-h}=1.5$. All the curves are obtained for the external magnetic field $B=0.5\mathrm{T}$. The dashed vertical lines show the interfaces of the QWs.