Metallic Phase and Temperature Dependence of the $\nu =0$ Quantum Hall State in Bilayer Graphene

The $\nu =0$ quantum Hall state of bilayer graphene is a fertile playground to realize many-body ground states with various broken symmetries. Here we report the experimental observations of a previously unreported metallic phase. The metallic phase resides in the phase space between the previously identified layer polarized state at large transverse electric field and the canted antiferromagnetic state at small transverse electric field. We also report temperature dependence studies of the quantum spin Hall state of $\nu =0$. Complex nonmonotonic behavior reveals concomitant bulk and edge conductions and excitations. These results provide a timely experimental update to understand the rich physics of the $\nu =0$ state.

Figure 1

(a) The optical micrograph of device N01-A. The thin black line outlines the BLG sheet. The local bottom gate (BG) is outlined in dashed purple. The top gate (TG) overlaps the BG over the BLG sheet. The two side contacts are as labeled. (b) A schematic side view of the device. The TG and side contacts are made of $\mathrm{Cr}/\mathrm{Au}$ and the BG is made of multilayer graphite. (c) Two terminal conductance $G$ ($V_{\mathrm{TG}}$) in devices N01-A (red trace and left and bottom axes) and S24-A (blue trace and top and right axes) at $B_{\perp }=2\mathrm{T}$. $V_{\mathrm{BG}}=0\mathrm{V}$, $V_{\mathrm{Si}}=40\mathrm{V}$, and $T=0.3\mathrm{K}$ for both traces. The filling factors are marked in the figure.

Figure 2

(a) Comparison of $G$ ($D$) in device N01-A (red traces) and S24-A (blue traces) at $B_{\perp }=3$, 8, and 14 T. Thin vertical lines and symbols mark $D^{*}$ in S24-A and $D_{\text{in}/\text{out}}^{*}$ in N01-A. $D_{\text{in}/\text{out}}^{*}$ follow a prominent peak consistently as $B_{\perp }$ evolves. $T=20\mathrm{mK}$ for the red traces and $T=1.4\mathrm{K}$ for the blue traces. (b) ($D^{*}$, $B_{\perp }$) phase diagram of N01-A constructed from data in (a) and additional data given in Figs. S3 and S4 [32]. Below 2 T, boundary conductance peaks (open gray circles) are present although the conductance of the CAF phase does not drop to zero (See Fig. S3 [32]). Gray lines are guide to the eye. (c) The ($D^{*}$, $B_{\text{tot}}$) phase diagram of N01-A constructed from data in Fig. S7. Dashed lines are guide to the eye. The three “$+$” symbols mark the spots where $T$-dependent conductance $G(T)$ measurements were taken and plotted in (d). $D^{*}=0$, 38.2, and $80.1\mathrm{mV}/\mathrm{nm}$, respectively, for the black, green, and red spots. $B_{\perp }=2\mathrm{T}$ and $B_{\text{tot}}=25\mathrm{T}$ for all three.

Figure 3

(a) $G$ ($D$) of $\nu =0$ in device S24-A at $B_{\perp }=2\mathrm{T}$ and selected $B_{\text{tot}}$’s. From blue to red: $B_{\text{tot}}=2$, 10, 15, 18, 22, 26, and 31 T. (b) ($D^{*}$, $B_{\text{tot}}$) phase diagram constructed by reading transition $D^{*}$, as marked in (a). $B_{\perp }=2(3)\mathrm{T}$ for blue (red) symbols. The arrows point to intersections of low-$D$ and high-$D$ trend lines, which are $(B_{\text{tot}}^{*},D^{*})=(8.3\mathrm{T},26\mathrm{mV}/\mathrm{nm})$ and (11.2 T, $34\mathrm{mV}/\mathrm{nm}$), respectively, for $B_{\perp }=2$ and 3 T. (c) $G$ ($D$) of the same device at selected temperatures as indicated in the plot. $B_{\perp }=3\mathrm{T}$ and $B_{\text{tot}}=31\mathrm{T}$. (d) $G$ ($T$) at $D=0$. From top to bottom $B_{\text{tot}}=31$, 25, 17, 12, 3 T. $B_{\perp }=3\mathrm{T}$ for all traces. Inset: data in the low-$T$ range. The blue solid and dashed traces correspond to $(B_{\text{tot}},B_{\perp })=(31,3)\mathrm{T}$ in two different cooldowns. The black trace corresponds to $(B_{\text{tot}},B_{\perp })=(31,2)\mathrm{T}$. They are all taken on S24-A. The orange data are from device N01-A at $(B_{\text{tot}},B_{\perp })=(25,2)\mathrm{T}$. The sign of $dG/dT$ at low temperatures varies among the traces shown. (e) Remaining $G_e$ ($T$) of top three traces in (d) after subtracting “bulk contribution” as described in the text. The subtracted data are given in Fig. S10 [32]. The dashed line marks the estimated temperatures, below which $G_e$ ($T$) rises rapidly.