Activation study of collective excitations of the soliton-lattice phase in the $\nu =1$ double-layer quantum Hall state

We investigate thermal excitations of the pseudospin soliton lattice in the double-layer quantum Hall (QH) state at total Landau-level filling factor $\nu =1$ by detailed measurements of the activation energy gap, where the pseudospin represents the layer degree of freedom. In a tilted magnetic field, the activation energy gap of the double-layer $\nu =1$ QH state shows a minimum near the phase transition point between the commensurate and incommensurate states. From a comparison of the in-plane field dependence of the activation energy gap with theories, we suggest that the minimum in the activation energy gap is due to collective modes of the soliton lattice.

Figure 1

(a) $R_{xx}$ and (b) $R_{xy}$ around $\nu =1$ at $\sigma =0$ as a function of $B_{\perp }$ for several values of the tilting angle. Tilting angles are $\theta =50.8°$, $52.0°$, $52.8°$, $53.2°$, $53.6°$, $54.0°$, $54.6°$, $55.0°$, and $57.0°$ from the bottom to top traces. Total electron density is $n_{\text{tot}}=0.9\times {10}^{11}{\text{cm}}^{-2}$. Temperature is 250 mK. Traces are vertically offset for clarity. Arrows in (a) indicate the positions of the peak in $R_{xx}$. In the inset of (a), $R_{xx}$ for another sample (see the text for details) is shown. $n_{\text{tot}}=1.0\times {10}^{11}{\text{cm}}^{-2}$ and $\theta =61°$ are set at the transition point. Temperature is 355 mK. Inset of (b) shows the definition of $\theta$.

Figure 2

Activation energy gap $\Delta$ of the double-layer $\nu =1$ QH state at $\sigma =0$ for several values of $n_{\text{tot}}$ as a function of $B_{\parallel }$. Traces are vertically offset by 2 K for clarity.

Figure 3

(a) Activation energy gap $\Delta$ of the double-layer $\nu =1$ QH state at $\sigma =0$ as a function of $B_{\parallel }$. Total electron density is $n_{\text{tot}}=0.9\times {10}^{11}{\text{cm}}^{-2}$. (b) $R_{xx}$ at $\nu =1$ as a function of $B_{\parallel }$ at 130 mK. The dashed lines in (a) and (b) are guides to the eye. Inset of (b) shows illustration of the number density of the pseudospin soliton $Q_s$ as a function of $B_{\parallel }$. The dashed line represents $B_{\parallel }d/{\varphi }_0$, where ${\varphi }_0$ is a flux quantum. (c) A schematic phase diagram of $n_{\text{tot}}$ and $B_{\parallel }$ for double-layer $\nu =1$ QH state at $\sigma =0$ (Ref. 16). $d/l$, where $l$ represents the magnetic length, is also indicated on the right axis. Data in (a) and (b) are taken along the arrow in (c).

Figure 4

(Color online) (a) Color-scale plot of $R_{xx}$ as a function of $\sigma$ and $B_{\text{tot}}$. Total density $n_{\text{tot}}=0.9\times {10}^{11}{\text{cm}}^{-2}$ and tilting angle $\theta =52.7°$ are fixed and the temperature is 60 mK. The yellow (light) and black (dark) regions represent small and large $R_{xx}$, respectively. Dashed line indicates a constant filling factor $\nu =1$. (b) Activation energy gap $\Delta$ of the $\nu =1$ QH state as a function of $\sigma$ at $n_{\text{tot}}=0.8\times {10}^{11}{\text{cm}}^{-2}$ and $\theta =53.6°$. In the inset, an Arrehnius plot at $\sigma =0$ (indicated by an arrow) is presented.