Enhancement of valley susceptibility upon complete spin polarization

Measurements on a two-dimensional electron system confined to an AlAs quantum well reveal that for a given electron density the valley susceptibility, defined as the change in valley population difference per unit strain, is enhanced as the system makes a transition from partial to full spin polarization. This observation is reminiscent of earlier studies in which the spin susceptibility of AlAs electrons was observed to be higher in a single-valley system than its two-valley counterpart.

Figure 1

(Color online) (a) Schematic showing the strain-induced transfer of electrons from one valley to another. (b) Orientation of the sample with respect to applied magnetic field. (c) (Top) Strain-induced valley splitting in two-spin and single-spin systems. (Bottom) Spin splitting caused by a magnetic field in two-valley and single-valley systems.

Figure 2

(Color online) (a) Coincidence method of measuring ${\chi }_v^{\ast }$ in the partially spin-polarized regime shown for $n=3.7\times {10}^{11}{\text{cm}}^{-2}$ at $\theta =0°$. Corresponding energy diagram is shown in the top panel. The bottom panel shows the piezoresistance trace for $\nu =10$ where the maxima correspond to coincidences in the energy diagram. The inset shows energy levels for the $i=1$ peak which is also marked by the brown (open) circle in the top panel. (b) Similar measurements for the fully spin-polarized case taken at a high angle of $\theta =74°$. (c) Index plot of data shown in the left and middle panels combined with similar measurements at other $\nu$’s. The blue (open) and red (closed) symbols represent partial and complete spin polarizations. The slopes give ${\chi }_v^{\ast }$.

Figure 3

(Color online) ${\chi }_v^{\ast }$, normalized to its band value, as a function of $P_v$ and $P_s$ for two densities. The blue (open) and red (closed) symbols correspond to partially and completely spin-polarized regimes, respectively. Panels (a) and (b) are for $n=3.7\times {10}^{11}{\text{cm}}^{-2}$, while (c) and (d) correspond to $n=5.1\times {10}^{11}{\text{cm}}^{-2}$. Each data point is an average of values obtained for positive and negative values of $\Delta ϵ$ and for different tilt angles and fillings.

Figure 4

(Color online) Summary of susceptibility data. The left panel shows ${\chi }_v^{\ast }$ for partial [blue (open) circles] and complete [red (closed) squares] spin polarization. $P_v$ is held constant within a narrow range to facilitate comparison. The right panel shows ${\chi }_s^{\ast }$ measurements for zero [blue (open) circles] and complete [red (closed) squares] valley polarizations. Again, $P_s$ is held constant within a narrow range to minimize its influence on ${\chi }_s^{\ast }$. Both ${\chi }_v^{\ast }$ and ${\chi }_s^{\ast }$ are normalized to their respective band values.