Spontaneous Layer-Pseudospin Domain Walls in Bilayer Graphene

Bilayer graphene is susceptible to a family of unusual broken symmetry states with spin and valley dependent layer polarization. We report on a microscopic study of the domain walls in these systems, demonstrating that they have interesting microscopic structure related to the topological character of the ordered states. We use our results to show that the metal-insulator transition temperature in bilayer graphene is reduced from mean-field estimates by thermal excitation of domain walls.

Figure 1

(a) Schematic summary of our domain wall calculations. Two domain walls are oriented along the $y$ direction and the mass changes sign along the $x$ direction. (b),(c) Typical mean-field solutions for $m_z(x)$ and $m_x(x)$ variation across a domain wall. Note the different scales in (b) and (c). (d) Energy spectrum of a model with sharp domain walls. The gray area is the bulk continuum. Black and gray colors are used to distinguish chiral states localized at the domain walls which propagate in opposite directions, while solid and dashed lines are used to distinguish states with $⟨{\sigma }_x⟩<(>)0$. The two black dots identify the states with $E=\pm |m_0|/\sqrt{2}$.

Figure 2

Microscopic domain wall properties for square simulation cells with side $L$ and a uniform energy gap $2m_0$. In these figures, red dots are numerical data, while the thin solid lines are power-law fits. (a) Condensation energy $E_{\mathrm{c}}$ of bilayer graphene (in units ${\gamma }_1/\mu {\mathrm{m}}^2$) as a function of $m_0$. The dashed line is obtained from microscopic calculations. (b) Domain wall width $\xi$ as a function of $m_0$. (c) Domain wall energy $E_{\mathrm{DW}}$ as a function of $L$. (d) Domain wall surface tension $J\equiv E_{\mathrm{DW}}/L$ (in units of ${\gamma }_1/0.1\mu {\mathrm{m}}^2$) as a function of $m_0$. The dashed line is the Ginzburg-Landau theory prediction for the domain wall surface tension. (e) and (f) Comparison of the collective ($T_c^{\mathrm{DW}}$) and mean-field ($T_c^{\mathrm{MF}}$) critical temperatures.

Figure 3

Tight-binding calculation of distinct domain wall zero-mode patterns in gapped bilayer graphene samples with spin-rotational invariance. The red lines denote the zero modes localized at domain walls between (a) two QAH regions with opposite total Hall conductance, (b) two QVH regions with opposite layer polarization, and (c) a QVH and a QAH region. The gray lines represent the edge states on the outermost zigzag boundaries.