Valley Susceptibility of an Interacting Two-Dimensional Electron System

We report direct measurements of the valley susceptibility, the change of valley population in response to an applied symmetry-breaking strain, in an AlAs two-dimensional electron system. As the two-dimensional density is reduced, the valley susceptibility dramatically increases relative to its band value, reflecting the system’s strong electron-electron interaction. The increase has a remarkable resemblance to the enhancement of the spin susceptibility and establishes the analogy between the spin and valley degrees of freedom.

Figure 1

(a) The first Brillouin zone and conduction band constant energy surfaces for bulk AlAs. (b) Schematic diagram showing that electrons are transferred from the $X$ to the $Y$ valley upon the application of strain while the total 2DES density is independent of strain. (c) Experimental setup for the valley susceptibility measurements. (d) Energy diagram showing the spin ($E_Z$) and valley ($E_V$) subband splittings with applied magnetic field or strain, respectively. (e) Sample resistance, $R_{xx}$, vs the piezo bias, $V_P$ at various filling factors $\nu$. The traces are offset for clarity. The dashed and dotted lines, respectively, mark $V_P$ at which $X$ and $Y$ are equally populated, and $V_P$ beyond which all the electrons are transferred to the $Y$ valley. The top axis shows the strain as measured by the strain gauge. (f) Same data as in (e) but now plotted vs a normalized horizontal (top) axis $ϵ/B_{\nu }$, showing that the oscillations have the same period. The traces are offset for clarity and, for $\nu =8$, 10, and 12, a smooth, parabolic background is subtracted from the traces to highlight the oscillations. (g) Plot of the coincidence index, $i$ normalized by $\nu$, showing linearity with $V_P$. From the slopes of the lines fitted through the data points we obtain ${\chi }_v$, and the $i=0$ intercepts give the value of $V_P$ at which the $X$ and $Y$ valleys are equally populated ($ϵ=0$). Three data sets A, B, and C correspond to densities of: 2.8, 4.8, and $7.6\times {10}^{11}{\mathrm{cm}}^{-2}$, respectively. (h) Schematic energy fan diagram showing the relevant energies: the cyclotron ($E_C$), Zeeman ($E_Z$) and valley splitting ($E_V$).

Figure 2

Valley and spin susceptibilities, normalized to their band values, vs density. ${\chi }_v$ was measured for three samples, S1, S2, and S3 using various techniques, as indicated. ${\chi }_s$ data were measured in several AlAs 2DES samples, including S2, and are shown for the case when the $X$ and $Y$ valleys are equally occupied (open circles), and for when only one valley is occupied (open triangles). Data points are from measurements at $T=0.3\mathrm{K}$ except as indicated. Lines are guides to the eye.