Spin-polarized two-dimensional electron gas embedded in a semimagnetic quantum well: Ground state, spin responses, spin excitations, and Raman spectrum

We present theoretical aspects of spin-polarized two-dimensional electron gas (SP2DEG) which can be achieved in doped semimagnetic quantum wells. This original model system has been recently studied by magneto-Raman-scattering experiments, which has given access to spin-resolved excitations and spectrum of the SP2DEG. Starting from the diluted magnetic semiconductor (DMS) Hamiltonian in the presence of the Coulomb interaction between conduction electrons, we define the conditions to reach such a SP2DEG. The equilibrium state is studied at low temperature; in particular, a theory for the degree of spin polarization is derived. Dynamical spin susceptibilities are further calculated in the framework of a spin-density-functional formalism already developed in the past. We then derive spin-conserving and spin-flip excitation dispersions using a recent determination of the SP2DEG correlation energy corrected from the thickness of the well. The SP2DEG presents two key features: the spin-flip wave, whose existence is a direct consequence of the Coulomb interaction between the spin-polarized electrons with a dispersion and energy range typical of the SP2DEG obtained in DMS, and the spin-density fluctuations exhibiting a specific collective behavior when the spin polarization is increased. The dissipation spectrum through these excitations is studied in detail. Particular attention is given to the spectrum determined by resonant Raman scattering. We show, indeed, that the latter gives unique access to the spin-fluctuation spectrum of the SP2DEG.

Figure 1

(Color online) Zeeman energy of Eq. (10) for ${\text{Cd}}_{0.99}{\text{Mn}}_{0.01}\text{Te}$ and various temperatures.

Figure 2

(Color online) Map of the maximum equilibrium spin-polarization degree $\zeta =\left(n_{\uparrow }-n_{\downarrow }\right)/\left(n_{\uparrow }+n_{\downarrow }\right)$ as a function of the electron sheet density $n_{2\text{D}}$ and nominal Mn concentration $x$ for a typical ${\text{Cd}}_{1-x}{\text{Mn}}_x\text{Te}/\text{CdMgTe}$ quantum well of width $w=150\text{Å}$. The Zeeman energy is taken at its maximum value [maximization of Eq. (10)] for temperature $T=1.5\text{K}$. (a) Bare spin-polarization degree ${\zeta }_0$ of Eq. (15). (b) Calculation of $\zeta$ with interaction enhancement of Eq. (16).

Figure 3

(Color online) Map of the SPE continua for $\zeta =-40\%$ and $T=0\text{K}$. (a) The spin-conserving SPE (SC-SPE) continuum is the overlap of majority and minority SC-SPE continua. The high-energy boundary of the former (the latter) is indicated by line 4 (line 2). These two lines correspond to the particular SPEs indicated by arrows 4 and 2 in the inset. For SPEs having an initial state in between 1 and 2 (3 and 4), no phase-space reduction due to the final-state occupancy does occur. SPEs with energies below line 1 (line 3) have in contrary a reduced final-state phase space. Hence, lines 1 and 3 indicate the peak position in the joint density of excitations [peak in $\text{Im}{\Pi }_{\uparrow \uparrow }\left(q,\omega \right)$ and $\text{Im}{\Pi }_{\downarrow \downarrow }\left(q,\omega \right)$, respectively]. (b) Spin-flip SPE continuum (SF-SPE): lines 1 and 4 mark the boundaries of the ${\downarrow }_{\to }\uparrow$ SPE continuum corresponding to excitations of arrows 1 and 4 in figure (c). The low-energy boundary 1 reaches 0 for $q=q_0=k_{F,\downarrow }-k_{F,\uparrow }$. For $q_0\le q\le k_{F,\downarrow }+k_{F,\uparrow }$, it is always possible to find a zero-energy SPE. Lines 2 and 3 indicate the peak position of the joint density of excitations [peaks in $\text{Im}{\Pi }_{\downarrow \uparrow }\left(q,\omega \right)$] due to the spin-up phase space that is reduced (arrows 2 and 3). Line 5 is the boundary of the ${\uparrow }_{\to }\downarrow$ SPE continuum corresponding to excitation of arrow 5 in figure (d). Line 6 indicates the peak position of the excitation count occurring because of the spin-down states occupancy (arrow 5). ${\uparrow }_{\to }\downarrow$ SPE require a minimum wave vector $q_0$ to exist.

Figure 4

(Color online) Single-particle Raman dissipation spectrum as a function of wave vector $q$ for a given spin-polarization degree $\zeta$. Calculations have been performed for $T=1.5\text{K}$ and homogeneous broadening $\hslash \eta =0.02E_F$. (a) Longitudinal spectrum of Eq. (73) with ${\beta }_2=1$. (b) Transverse spectrum of Eq. (74). Arrows 1–4 correspond to excitations pointed in Fig. 3.

Figure 5

(Color online) Plasmon dispersions: (a) Effect of corrections: without exchange and $\text{correlations}=\text{RPA}$ (straight line), with exchange only (dotted line), and with exchange and correlations (dashed line). (b) Effect of density: calculations are carried out with exchange and correlations for $r_s=0.4,1,2,5,10$. (c) Effect of spin-polarization degree: variation in the plasmon dispersions for $\zeta$ ranging from 0 to −1, the shaded area are the intermediate values. In (a) and (b) the ${\text{SPE}}_{\uparrow \uparrow }$ and ${\text{SPE}}_{\downarrow \downarrow }$ boundaries are plotted for $T=0\text{K}$ (sparse domains). In (c) only the ${\text{SPE}}_{\downarrow \downarrow }$ boundary is plotted for $\zeta =0$ and $\zeta =-1$.

Figure 6

(Color online) Dissipation spectrum of spin-conserving response functions as a function of the spin-polarization degree $\zeta$ and for $r_s=2$, wave vector $\tilde{q}=0.1$, homogenous broadening $\tilde{\eta }=\hslash \eta /E_F=0.02$, temperature $\tilde{T}=kT/E_F=0.02$, and $\tilde{\omega }=\hslash \omega /E_F$. (a) Charge-charge density response function with its plasmon peak. Inset: illustration of the SPE screening, the region of SPE has been magnified. (b) Spin-spin density response function exhibiting a low-energy spin-fluctuation part with a plasmon peak appearing at high $\zeta$. Inset: illustration of the shape difference between the SPE dissipation and spin-fluctuations part: this part is a collective effect. (c) Charge-spin density response functions (extra diagonal term in the longitudinal response). Both spin fluctuations and plasmon contribute with growing weight as the spin-polarization degree increases.

Figure 7

(Color online) Collective behavior of the spin-fluctuation part in the spin-spin density response function. Calculations have been made at $\tilde{q}=0.1$.(a) Line-shape deformation and peak shifting of the spin fluctuations as a function of $r_s$ for $\zeta =0$, $\tilde{\eta }=0$, and $\tilde{T}=0$. In this reduced unity scale, the SPE line in $\text{Im}\Pi$ remains constant. [(b)–(d)] Origin of the spin-fluctuations line-shape deformation. $\text{Im}{\chi }_{zz}$ is calculated at $r_s=2$ as a function of spin degree $\zeta$ by modifying the dynamical screening: (b) all the dynamical screenings are set to zero, (c) screening due to spin-density $\left(G_{L,\sigma }^{-}\right)$ set to zero, (d) no modification.

Figure 8

(Color online) Behavior of the longitudinal Raman dissipation spectrum. (a) Case with $\left({\gamma }_{\uparrow \downarrow }-{\gamma }_{\downarrow \downarrow }\right)/\left({\gamma }_{\uparrow \downarrow }-{\gamma }_{\downarrow \downarrow }\right)\le 0$. (b) Case with $\left({\gamma }_{\uparrow \downarrow }-{\gamma }_{\downarrow \downarrow }\right)/\left({\gamma }_{\uparrow \downarrow }-{\gamma }_{\downarrow \downarrow }\right)\ge 0$.

Figure 9

(Color online) Dispersion of spin-flip excitations calculated for a zero thickness quantum well at $T=0\text{K}$, $\eta =0$, and (a) $\zeta =-0.4$ or (b) $\zeta =-1$. Lines are the SFW mode corresponding to the indicated $r_s$ values. Domains are boundaries of the SF-SPE continua (which does not depend on $r_s$ in the reduced unity frame) and ${\tilde{q}}_0=\sqrt{1-\zeta }-\sqrt{1+\zeta }$.

Figure 10

(Color online) Spin-flip dissipation spectrum $\text{Im}{\chi }_{+-}$ calculated for $r_s=2$, $\tilde{\eta }=0.02$, and $\tilde{T}=0.02$: (a) as a function of the wave vector for $\zeta =-0.4$; (b) as a function of $\zeta$ for $q=0$. The inset of (b) plots the weight of the SFW peak deduced from the spectrum as a function of $\zeta$.