Photoluminescence of $p$-doped quantum wells with strong spin splitting

The spectroscopic properties of a spin-polarized two-dimensional hole gas are studied in modulation doped ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ quantum wells with variable carrier density up to $5\times {10}^{11}{\mathrm{cm}}^{-2}$. The giant Zeeman effect, which is characteristic of diluted magnetic semiconductors, induces a significant spin splitting even at very small values of the applied field. Several methods of measuring the carrier density (Hall effect, filling factors of the Landau levels at high field, various manifestations of Moss-Burstein shifts) are described and calibrated. The value of the spin splitting needed to fully polarize the hole gas shows a strong enhancement of the spin susceptibility of the hole gas due to the carrier-carrier interaction. At small values of the spin splitting, whatever the carrier density (nonzero) is, photoluminescence lines are due to the formation of charged excitons in the singlet state. Spectral shifts in photoluminescence and in transmission (including an “excitonic Moss-Bustein shift”) are observed and discussed in terms of excitations of the partially or fully polarized hole gas. At large spin splitting, and without changing the carrier density, the singlet state of the charged exciton is destabilized in favor of a triplet state configuration of holes. The binding energy of the singlet state is thus measured and found to be independent of the carrier density (in contrast to the splitting between the charged exciton and the neutral exciton lines). The state stable at large spin splitting is close to the neutral exciton at low carrier density, and close to an uncorrelated electron-hole pair at the largest values of the carrier density achieved. The triplet state gives rise to a characteristic double-line structure with an indirect transition to the ground state (with a strong phonon replica) and a direct transition to an excited state of the hole gas.

Figure 1

(a) Typical transmission spectra and (b) magnetic field dependence of the transition energies, observed at $1.9\mathrm{K}$ on sample N0, an $8\mathrm{nm}$ broad CdTe QW with ${\mathrm{Cd}}_{0.69}{\mathrm{Zn}}_{0.08}{\mathrm{Mg}}_{0.23}\mathrm{Te}$ barriers, moderately doped $p$-type (two symmetrical N-doped layers at $50\mathrm{nm}$). In (b), the right part (positive fields) displays the ${\sigma }^{-}$ polarization data and the left part (negative fields) the ${\sigma }^{+}$ ones. Dashed lines are quadratic fits; the solid line through the charged-exciton positions is obtained when taking into account the dependence of the $X∕X^{+}$ splitting on the hole polarization.

Figure 2

Energy of the transmission line, as a function of applied field, at $1.7\mathrm{K}$, for sample N5, a $p$-doped, $8\mathrm{nm}$ wide, ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ QW. Ar-ion laser illumination strongly reduces the carrier density to the low ${10}^{10}{\mathrm{cm}}^{-2}$ range. Solid lines are drawn with the same parameters as in Fig. 1 to describe the normal Zeeman effect and diamagnetic shift, and a Brillouin function with $x_{\mathit{eff}}=0.0048$ (and $x=0.0052$, $T_0=0.17\mathrm{K}$) to describe the giant Zeeman effect, and adjustable values for the zero-field energies of $X$ and $X^{+}$.

Figure 3

(a) Optical density spectra obtained from transmission, at $1.7\mathrm{K}$ and magnetic fields from $0\text{to}20\mathrm{T}$, in ${\sigma }^{+}$ polarization, for a ${\mathrm{Cd}}_{0.9982}{\mathrm{Mn}}_{0.0018}\mathrm{Te}$ QW (sample N2); spectra have been shifted in both directions to enhance the principal features; (b) same as (a), in ${\sigma }^{-}$ polarization, from $0\text{to}10\mathrm{T}$; shifts are different in order to enhance the features of interest.

Figure 4

Position of the absorption lines vs applied magnetic field. (a) ${\mathrm{Cd}}_{0.9982}{\mathrm{Mn}}_{0.0018}\mathrm{Te}$ QW (sample N2); open symbols are for experimental data in ${\sigma }^{-}$ polarization, full symbols for ${\sigma }^{+}$ polarization; dotted lines assume excitonic-like variations (normal and giant splitting and diamagnetic shift), with the zero-field energy as only adjustable parameter; solid lines assume Landau levels with the theoretical masses and the normal and giant Zeeman effects, with the zero-field energy as only adjustable parameter. The dotted line with a higher slope is shifted from the exciton in ${\sigma }^{-}$ polarization by the Landau fan splitting (sum of cyclotron resonances). The dashed line shows the position expected for the absorption threshold at low field (it includes the normal and giant Zeeman effects and the Moss-Burstein shift added to the zero-field energy of the Landau fan, with no adjustable parameter). The lower part shows the intensity of each line in ${\sigma }^{+}$ polarization. The vertical dashed lines show the magnetic field values corresponding to integer filling factors. (b) same as (a), for sample N5, with a ${\mathrm{Cd}}_{0.9941}{\mathrm{Mn}}_{0.0053}\mathrm{Te}$ QW.

Figure 5

(a) PL spectra, at $1.7\mathrm{K}$, with various values of the applied field, from a ${\mathrm{Cd}}_{0.9963}{\mathrm{Mn}}_{0.0037}\mathrm{Te}$ QW (sample N3); the PL was excited using an Ar-ion laser, which almost completely depletes the QW. Solid lines are for ${\sigma }^{+}$ spectra and dotted line for ${\sigma }^{-}$; (b) same as (a), but with excitation with a Ti-sapphire laser with photon energy below the barrier gap, which has no effect on the density of the carrier gas.

Figure 6

(a) Three-carrier diagram of optical transitions. Single arrows show the electron spin and double arrows the hole spin; the dash-dotted line marks the level crossing. (b) Relative intensities of the $X$ and $X^{+}$ photoluminescence lines measured in ${\sigma }^{+}$ polarization for sample N3, ${\mathrm{Cd}}_{0.9963}{\mathrm{Mn}}_{0.0037}\mathrm{Te}$ QW, under illumination by an Ar-ion laser. (c) Comparison of the valence band Zeeman splitting (points and solid line) and the $X∕X^{+}$ splitting (dashed horizontal line) for sample N3. Dot-dashed vertical lines mark the field at which the $X∕X^{+}$ crossing occurs.

Figure 7

Time-resolved photoluminescence of sample N4 (${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$ QW) measured in ${\sigma }^{+}$ polarization in a magnetic field of $0.5\mathrm{T}$ close to the crossing of the excited levels. Symbols give the temporal profiles after excitation by the laser pulse (solid line), detected at different energies corresponding to different components of the PL (marked $D_{\mathit{\text{low}}}$, $X_s{}^{+}$, $D_{\mathit{\text{high}}}$ on the spectrum in the inset).

Figure 8

Sample N4, ${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$ QW (a) integrated PL decay time measured in ${\sigma }^{+}$ polarization, as a function of the magnetic field; open squares denote the experimental values, stars represent the average of two times: ${\tau }_{X^{+}}=80\mathrm{ps}$ and ${\tau }_{D_{\mathit{\text{low}}}}={\tau }_{D_{\mathit{\text{high}}}}$ $140\mathrm{ps}$ (arrows), weighted by the intensities of each of the two components of the PL spectra, plotted in (b). (b) Relative time-integrated intensities of the two components of photoluminescence spectra: double line $D_{\mathit{\text{low}}}∕D_{\mathit{\text{high}}}$ (squares), and excitonic line (circles) measured in ${\sigma }^{+}$ polarization, as a function of the magnetic field.

Figure 9

Comparison of the transition energy in PL (down triangles) vs transmission (lines showing the fits used in Fig. 4). Note that the PL and transmission scales have been shifted (see text). (a) Sample N2, ${\mathrm{Cd}}_{0.9982}{\mathrm{Mn}}_{0.0018}\mathrm{Te}$ QW and (b) sample N5, ${\mathrm{Cd}}_{0.9947}{\mathrm{Mn}}_{0.0053}\mathrm{Te}$ QW, as in Fig. 4.

Figure 10

Example of the transmission and PL spectra at zero magnetic field (a and b) and magnetic field $0.5\mathrm{T}$ (c) for sample N4, (${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$ QW). In (a) the sample was illuminated by a strong blue light, which almost completely depleted the QW. In (b) and (c) there was no above barrier illumination so there was an equilibrium density of carriers. Solid lines are for ${\sigma }^{+}$ spectra and dashed lines for ${\sigma }^{-}$. The data from (b) and (c) were used to draw Fig. 11.

Figure 11

(a) Positions of the transmission lines (open circles) and PL lines (full circles) vs the applied magnetic field, in the low-field range, for sample N4, ${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$ QW. The ${\sigma }^{-}$ data are plotted at negative fields. The dotted line gives the Brillouin function determined at high field. (b) Schematic diagram of the initial and final states and of the optical transitions. (c) Experimentally determined energies (see text) of the final states of optical transitions, obtained from data in (a); diamonds are for the double line $(D_{\mathit{\text{low}}},D_{\mathit{\text{high}}})$, circles for the excitonic line, and the solid lines are drawn using the energies of initial and final state plotted in (b). In all 3 panels, dashed vertical lines indicate the field values for complete hole polarization, and the dash-dotted vertical line indicates the destabilization of $X^{+}$.

Figure 12

Schematic diagram of the initial and final states in optical transitions from Fig. 11.

Figure 13

(a) cw PL spectra of sample N4 at level crossing (linear scale); the solid line gives the experimental PL spectrum in ${\sigma }^{+}$ polarization and the dashed line the experimental spectrum in ${\sigma }^{-}$ polarization; the dotted lines give the proposed decomposition of the ${\sigma }^{+}$ line into the double line $\left(D_{\mathit{\text{high}}}\text{−}{D}_{\mathit{\text{low}}}\right)$ and the excitonic one. (b) Typical spectra in ${\sigma }^{+}$ polarization for samples N4 (at $1\mathrm{T}$, solid line), and S7 (QW doped from surface states, at $0.5\mathrm{T}$, dotted line); the energy scale is with respect to the position of the upper component of the PL double line, $D_{\mathit{\text{high}}}$, and the intensity scale is a log scale.

Figure 14

(a) The hole gas density determined from filling factors in the Landau fan, or from Hall resistance, versus the polarized Moss-Burstein shift between the absorption line and the lower component of the double PL line, $D_{\mathit{\text{low}}}$, in ${\sigma }^{+}$ polarization; the solid line is the calculated slope using the electron and hole masses as described in the text. (b) “Excitonic mb shift” (measured in ${\sigma }^{+}$ polarization between the absorption line and the excitonic PL line, at a field such that the hole gas is completely polarized, but which is lower than the value where the level crossing occurs. The solid line is a linear fit. (c) Full symbols represent the splitting of the double line $D_{\mathit{\text{high}}}\text{−}{D}_{\mathit{\text{low}}}$—the energy of the final state involved in the $D_{\mathit{\text{high}}}$ component—vs the carrier density. Note that two samples have been measured at different carrier densities, changed either optically (sample N4, triangles) or through an electrostatic gate (sample D3, circles and sample D50, diamonds). Open diamonds represent the valence band Zeeman splitting at complete hole gas polarization; gray area is the Fermi energy of the polarized hole gas.

Figure 15

Dissociation energy of the charged exciton, determined from the level crossing (squares) vs the carrier density; the solid line shows the $X\text{−}{X}^{+}$ splitting determined from absorption spectra in Ref. 23.