Onset temperature for the Kosterlitz-Thouless transition in the ${\nu }_t=1$ bilayer quantum Hall state

We investigate the onset temperature $T^{*}$ of the bilayer quantum Hall state at the total Landau level filling factor ${\nu }_{\mathrm{t}}=1$ together with the activation energy gap $\Delta$. The onset temperature is the temperature at which the quantum Hall state starts to form. We make two remarkable experimental observations: (1) As the total electron density decreases, $T^{*}$ increases while $\Delta$ decreases on account of the deterioration of electron mobility. (2) As the in-plane magnetic field increases, $T^{*}$ exhibits a rise at the point where $\Delta$ becomes a minimum. The minimum of $\Delta$ occurs at the transition point from the commensurate phase to the recently discovered soliton lattice phase. We discuss the relation between $T^{*}$ and the Kosterlitz-Thouless transition on the basis of the $\mathit{XY}$ symmetry possessed by the ${\nu }_{\mathrm{t}}=1$ bilayer system and discuss the experimentally observed behavior of $T^{*}$.

Figure 1

Graphical representation of parameters $T^{*}$ and $\Delta$. The temperature at which the activated behavior terminates is marked by $T^{*}$.

Figure 2

Plots of $\Delta$ (left axis) and $T^{*}$ (right axis) as a function of $n_{\mathrm{t}}$. Inset: The zero-temperature phase diagram of the $\nu =1$ bilayer system at $B_{\parallel }=0$. $E_{\mathrm{C}}$ represents the Coulomb energy [$=e^2/\left(\epsilon {\ell }_{\mathrm{B}}\right)$, $e$ denotes the elementary charge and $\epsilon$ denotes the dielectric constant]. The circles indicate the phase positions corresponding to the data and the arrow indicates the direction in which $n_{\mathrm{t}}$ increases.

Figure 3

Arrhenius plot ($\ln R_{xx}$ versus $1/T$) for several values of $\theta$ at $n_{\mathrm{t}}=0.8\times {10}^{11}$ cm${}^{-2}$.

Figure 4

(a) $\Delta$ and (b) $T^{*}$ as a function of $\theta$ for $n_{\mathrm{t}}=0.6$ and $0.8\times {10}^{11}$ cm${}^{-2}$. Inset: The zero-temperature phase diagram at the ${\nu }_{\mathrm{t}}=1$ bilayer QHS in the $n_{\mathrm{t}}$ versus $B_{\parallel }$ plane. The points shown correspond to the experimental regions to which $B_{\parallel }$ is applied.

Figure 5

$R_{xx}$ at $T\sim 110$ mK as a function of $\theta$ for $n_{\mathrm{t}}=0.6$ and $0.8\times {10}^{11}$ cm${}^{-2}$. The arrows indicate the transition points.

Figure 6

Plot of $T^{*}$ as a function of $d/{\ell }_{\mathrm{B}}$ along with the curve for $T_{\mathrm{KT}}$ that follows the equation $T_{\mathrm{KT}}=\pi {\rho }_{\mathrm{ps}}/2$.

Figure 7

(a) Theoretically predicted values of $K_{11}$, $K_{22}$, and $T_{\mathrm{KT}}$ as a function of $B_{\parallel }$. (b) and (c) The fitting results of $T^{*}$ with the theoretical $T_{\mathrm{KT}}$ curve of the SL phase as obtained by Read[11] for $n_{\mathrm{t}}=0.6$ and $0.8\times {10}^{11}$ cm${}^{-2}$, respectively.