Fictitious gauge fields in bilayer graphene

We discuss the effect of elastic deformations on the electronic properties of bilayer graphene membranes. Distortions of the lattice translate into fictitious gauge fields in the electronic Dirac Hamiltonian that are explicitly derived here for arbitrary elastic deformations, including in-plane as well as flexural (out-of-plane) distortions. We include gauge fields associated to intra- as well as interlayer hopping terms and discuss their effects on the electronic band structure and on the transport properties of suspended bilayer membranes. In particular, we consider the electron-phonon coupling induced by the fictitious gauge fields and analyze its contribution to the electrical resistivity. Of special interest is the appearance of a linear coupling for flexural modes, in stark contrast to the case of monolayer graphene. This new coupling channel is shown to dominate the temperature-dependent resistivity in suspended samples with low tension.

Figure 1

Atomic structure of perfect bilayer graphene in Bernal stacking (top view). The honeycomb lattice with full (dashed) lines corresponds to the upper (lower) layer $l=2$ ($l=1$). The atomic sites A2 and B1 coincide once projected on the plane. The three vectors ${\mathbf{e}}_j^{}$, ($j=1,2,3$) connecting A and B sites are indicated (see text for more details).

Figure 2

Schematic description of the effect of deformations on bond lengths. This side view, taken along the direction ${\mathbf{e}}_j^{}$, shows the atoms involved in intra and interlayer hopping processes in a unit cell. (a) Bilayer under flexural deformations $h_{}^{\left(S\right)}$ and $-h_{}^{\left(S\right)}$. The difference between the hopping lengths ${\mathbf{3}}^{\prime }$ and ${\mathbf{3}}^{\prime \prime }$ (as well as between ${\mathbf{4}}^{\prime }$ and ${\mathbf{4}}^{\prime \prime }$) breaks the symmetry with respect to the plane and is responsible for the appearance of a linear coupling with $h_{}^{\left(S\right)}$ in the gauge field $F_3^{\left(\tau \right)}$ (and $F_4^{\left(\tau \right)}$). The fact that ${\mathbf{3}}^{\prime }={\mathbf{4}}^{\prime \prime }$ leads to the symmetry $h_{}^{\left(S\right)}\to -h_{}^{\left(S\right)}$ between $F_3^{\left(\tau \right)}$ and $F_4^{\left(\tau \right)}$. (b) Same as in (a), but for in-plane antisymmetric deformations ${\mathbf{u}}_{}^{\left(A\right)}$ and $-{\mathbf{u}}_{}^{\left(A\right)}$. The equality ${\mathbf{4}}^{\prime }={\mathbf{3}}^{\prime \prime }$ accounts for the symmetry ${\mathbf{u}}_{}^{\left(A\right)}\to -{\mathbf{u}}_{}^{\left(A\right)}$ between $F_3^{\left(\tau \right)}$ and $F_4^{\left(\tau \right)}$.

Figure 3

Electronic band structure of bilayer graphene without deformations. All energy scales $\epsilon$ are in $\mathrm{meV}$. (a) The low-energy spectrum neglecting terms in $v_4^{}$. Four massless cones touch at zero energy. (b) Equipotential lines for (a). Dashed lines correspond to $\epsilon <{\epsilon }_{}^{*}$, yielding four disconnected electron pockets. The thick line corresponds to $\epsilon ={\epsilon }_{}^{*}$ where the LT occurs. All other continuous lines are at $\epsilon >{\epsilon }_{}^{*}$, yielding a single connected electron pocket. Dark areas correspond to states close to zero energy, while light ones are for higher energies. (c) Band structure including $v_4^{}$. The central Dirac cones touch at zero energy, while the other three touch at $\tilde{\epsilon }$. The boxes $[N_e^{},N_h^{}]$ in different energy windows indicate that the FS is made out of $N_e^{}$ electron pockets and $N_h^{}$ hole pockets. (d) Same as (c), but with asymmetric intralayer velocities, corresponding to $t_2^{}=t_1^{}/4=1\mathrm{eV}$. This large asymmetry is used to stress the formation of the minigaps $\tilde{\Delta }$. (e) Magnification of the low energy spectrum in (c), but with a small interlayer gap $\Delta <\tilde{\epsilon }$. (f) Same as in (e), but with a larger interlayer gap $\Delta >\tilde{\epsilon }$.

Figure 4

Electronic band structure in the wave-vector space and equipotential lines for different values of strain $\beta$ along $\theta =0$, see text. Energy ($\epsilon$) is expressed in milli electron volt and the plots are taken for $t_1^{}=t_2^{}=2.47\mathrm{eV}$ and ${\eta }_{\mathrm{A}1,\mathrm{B}2}^{}={\eta }_{\mathrm{A}1,\mathrm{A}2}^{}={\eta }_{\mathrm{B}1,\mathrm{B}2}^{}=1$. For these parameters, we get ${\beta }_{c1}^{}\simeq 2.3\times {10}_{}^{-3}$ and ${\beta }_{c2}^{}\simeq 2\times {10}_{}^{-2}$. (a1) Band structure for $\beta =1.5\times {10}_{}^{-3}$. (b1) Equipotential lines for (a1), showing two LT at ${\epsilon }_{\mathrm{L}1}^{}$ and ${\epsilon }_{\mathrm{L}2}^{}$ (thick lines). For $\epsilon >{\epsilon }_{\mathrm{L}1}^{}$ the FS is of type [1, 0] for this and all other panels. Dashed lines show the [3, 0] FS at an energy between the two LT, while the thin lines exemplify a [4, 0] FS at $\epsilon <{\epsilon }_{\mathrm{L}2}^{}$. (a2) Band structure with critical strain ${\beta }_{c1}^{}$. (b2) Equipotential lines for (a2). One LT occurs at ${\epsilon }_{\mathrm{L}1}^{}$, below which the Fermi surface is of type [3, 0]. (a3) Band structure for $\beta =4\times {10}_{}^{-3}$. The corresponding equipotential lines are shown in (b3). The dashed line shows the [3, 0] Fermi surface at ${\epsilon }_{\mathrm{m}}^{}<\epsilon <{\epsilon }_{\mathrm{L}1}^{}$, while thin lines show the [2, 0] FS at $0<\epsilon <{\epsilon }_{\mathrm{m}}^{}$. (a4) Band structure for $\beta =3\times {10}_{}^{-2}$. The local minimum disappears. The corresponding equipotential lines are shown in panel (b4). One LT occurs at ${\epsilon }_{\mathrm{L}1}^{}$. The dashed line shows a [2, 0] FS for $0<\epsilon <{\epsilon }_{\mathrm{L}1}^{}$.

Figure 5

The electronic DOS and Fermi surface for different values of strain $\beta$ along $\theta =0$. Here, we choose ${\eta }_{\mathrm{A}1,\mathrm{B}2}^{}=1$ and ${\gamma }_4^{}=0$. (a) Electronic DOS as a function of energy for $\beta =0$ (thick line), $\beta ={10}_{}^{-3}$ (dashed line), $\beta =2\times {10}_{}^{-3}$ (dotted line), and $\beta =5\times {10}_{}^{-3}$ (dot-dashed line). The peaks in the DOS at the various LT are clearly visible, as well as the linear dependence on energy in the low-energy regime due to the massless Dirac cones. The dot-dashed line shows a steplike feature at ${\epsilon }_{\mathrm{m}}^{}$ associated to the local parabolic minimum in the dispersion [see Fig. 4(a3)]. (b) Fermi surface at electron doping corresponding to ${\epsilon }_{\mathrm{F}}^{}=0.8\mathrm{meV}<{\epsilon }_{}^{*}$. The thin line is for $\beta =0$, yielding a [4, 0] FS. The thick line shows the FS at the LT (${\epsilon }_{\mathrm{F}}^{}={\epsilon }_{\mathrm{L}2}^{}$) for $\beta \simeq 1.2\times {10}_{}^{-3}$, while the dashed line shows the [3, 0] FS at $\beta \simeq 3\times {10}_{}^{-3}$. (c) Fermi surface at electron doping corresponding to ${\epsilon }_{\mathrm{F}}^{}=2.2\mathrm{meV}>{\epsilon }_{}^{*}$. The thin, thick, and dashed lines correspond to $\beta =0$, $\beta \simeq 1.6\times {10}_{}^{-3}$, and $\beta \simeq 3\times {10}_{}^{-3}$, respectively. The LT here occurs at ${\epsilon }_{\mathrm{F}}^{}={\epsilon }_{\mathrm{L}1}^{}$.