From Spin Flip Excitations to the Spin Susceptibility Enhancement of a Two-Dimensional Electron Gas

The $g$-factor enhancement of the spin-polarized two-dimensional electron gas was measured directly over a wide range of spin polarizations, using spin flip resonant Raman scattering spectroscopy on two-dimensional electron gases embedded in ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ semimagnetic quantum wells. At zero Raman transferred momentum, the single-particle spin flip excitation, energy $Z^{*}$, coexists in the Raman spectrum with the spin flip wave of energy $Z$, the bare giant Zeeman splitting. We compare the measured $g$-factor enhancement with recent spin-susceptibility enhancement theories and deduce the spin-polarization dependence of the mass renormalization.

Figure 1

(a) Typical depolarized Raman spectra of one of the four samples studied (sample A), measured at $E_i=1.621\mathrm{eV}$, showing the spin flip wave (SFW) and the spin flip single-particle excitation (SF-SPE). The background at high Raman shift is from luminescence. The Voigt backscattering geometry [$x(\sigma \pi )\overline{x}$] is parallel to the growth axis $x$ and perpendicular to the in-plane field. (b) Magnetic field dependence of energies $Z$ and $Z^{*}$ deduced from (a). $Z(B)$ has been fitted with the Brillouin function.

Figure 2

(a) Typical theoretical dispersions of spin flip excitations of the SP2DEG calculated for $\zeta =-0.4$ and $r_s=2.0$. Full circles are the SFW [20] and the hatched area is the SF-SPE continuum. Lines numbered 1, 2, 3, 4 correspond, respectively, to the excitations 1, 2, 3, 4 in (d): line 4 (1) is the excitation of a spin down electron with initial wave vector $k=k_{F\downarrow }$ ($k=-k_{F\downarrow }$). Lines 2 and 3 are limits where the number of excitations is restricted due to filling of the up spin subband. Overlaid are the calculated Raman spectra for $q/k_F=0$ and 0.2. (b) Experimental spectra obtained on sample A for $B=4.8\mathrm{T}$ and $q=0$ and $9.5\mu {\mathrm{m}}^{-1}$ ($q/k_F\approx 0.1$). (c), (d) Schematic of spin-split subbands indicating representative SF-SPE excitations for (c) $q=0$ and (d) nonzero $q$.

Figure 3

(a) $Z^{*}$ ($Z$) for all samples. (b) Comparison of $Z^{*}/Z$ (symbols) deduced from (a) and theoretical $\zeta /{\zeta }_0$ (lines) given by Eq. (5) as a function of the bare spin polarization ${\zeta }_0$. Line colors match with symbol colors for the different samples. The black dotted lines around the solid black line for sample A are limits of $\zeta /{\zeta }_0$ when including the error on $r_s^0$ and varying the thickness of the square well from $\pm 10\%$ (a variation which overestimates the error of the growth technique and any error from neglecting the wave-function penetration into the barrier). (c) Symbols indicate the average renormalized mass ${\overline{m}}^{*}(\zeta )/m_b$ extracted from (b) as a function of the spin polarization $z$, determined from ${\zeta }_0$ using $\zeta /{\zeta }_0$. Lines are Z&D prediction (Ref. 13 ) for $r_s=1$, 2, and 3. We have added $m_F^{*}(\zeta =0)$ from Ref. 23. Sample C has been removed for clarity.