Inelastic light scattering study of the $\nu =1$ quantum Hall ferromagnet

We report on an inelastic light scattering study in the two-dimensional electron system at filling factor $\nu =1$. The energy and the inelastic cross section of the cyclotron spin-flip mode are analyzed. The exchange enhanced electronic $g$ factor in the quantum Hall ferromagnet is measured. In addition to the magnetoplasmon and cyclotron spin-flip modes, inelastic scattering lines associated with spin-singlet and spin-triplet $D^{-}$ complexes are observed. The inelastic light scattering is shown to probe the thermodynamics of the quantum Hall ferromagnet and gives insight into the formation of ferromagnetic order in two dimensions. It is established that at temperatures above the bare Zeeman energy, the ferromagnet breaks up into domains, whose size and number change with temperature. The experimental data allow us to construct a stability diagram of the $\nu =1$ quantum Hall ferromagnet in the temperature versus magnetic field plane.

Figure 1

An example of an inelastic light scattering spectrum at filling factor $\nu =1$ recorded at $8.5\mathrm{T}$ on a $25\mathrm{nm}$ wide quantum well sample. The features denoted as MP and CSFM correspond to the magnetoplasmon and the cyclotron spin-flip modes. The inset shows a cartoon of the three lowest energy excitations for the $\nu =1$ quantum Hall ferromagnet.

Figure 2

Right axis: $\Delta (0,B)$ as a function of density (or equivalently $B$). Open circles are the values extracted from inelastic light scattering experiments. The dashed and solid lines are the results from a first order Hartree-Fock simulation for a 2D system with zero width (dashed line) and when the finite width of the confining potential is taken into account (solid line). Left axis: the exchange enhancement of the electron $g$ factor as a function of density (or equivalently $B$) calculated from the experimental data points (solid circles). The line is a guide to the eyes.

Figure 3

Inelastic light scattering spectrum at $9\mathrm{T}$ $(\nu =1)$ for a $25\mathrm{nm}$ wide quantum well. The cartoon at the top explains the origin of the $S$ and $T^{+}$ lines.

Figure 4

Measured dependence of the difference between the central energy of the threefold split spin-singlet line $\left(S\right)$ and the cyclotron energy (open symbols). The dotted line is a guide to the eyes. In the inset, the central energy of the spin singlet feature is plotted versus filling factor $\nu$.

Figure 5

(A) Changes as a function of temperature for the cyclotron spin-flip mode in the inelastic light scattering spectrum at $B=5.5\mathrm{T}$. (B) The scattering efficiency of the cyclotron spin-flip mode (open dots, left axis) and the Coulomb term $\Delta (0,B)$ (solid dots, right axis) as a function of temperature. The dashed line indicates $T_Z=\mu gB∕k_B$. The bulk $g$ factor in QWs is taken from Ref. 13.

Figure 6

(A) Temperature dependence of the inelastic light scattering efficiency for the cyclotron spin-flip mode (open circles) in comparison with the magnetization of the 2D electron system. The magnetization results were taken from nuclear magnetic resonance (Ref. 4) (solid circles) and optical absorption (Ref. 5) (triangulars) experiments in the literature. The solid lines are guides to the eyes. The dotted line shows the magnetization of noninteracting electrons obeying Fermi-Dirac statistics, while the dashed line presents the spin-exciton activated magnetization described by Eq. (2). The dash-dotted line is simulated within the $O\left(N\right)$ continuum quantum field theory of Heisenberg ferromagnets with adjusted nearest neighbor exchange interaction strength. The spin stiffness is calculated with the inclusion of finite width effects for the electron wave function in the growth direction. This wave function is obtained from a self-consistent solution of the one-dimensional Schrödinger and Poisson equations. (B) The stability diagram for the $\nu =1$ quantum Hall ferromagnet. $T_f$ (open circles) and $T_d$ (solid circles) are plotted as a function of the magnetic field. The regions with macroscopic ferromagnetic order and with only domains or clusters with ferromagnetic order are delineated by solid lines.