Effects of domain walls in bilayer graphene in an external magnetic field

We investigate bilayer graphene systems with layer switching domain walls separating the two energetically equivalent Bernal stackings in the presence of an external magnetic field. To this end, we calculate quantum transport and local densities of three microscopic models for a single domain wall: a hard wall, a defect due to shear, and a defect due to tension. The quantum transport calculations are performed with a recursive Green's function method. Technically, we discuss an explicit algorithm for the separation of a system into subsystems for the recursion and we present an optimization of the well known iteration scheme for lead self-energies for sparse chain couplings. We find strong physical differences for the three different types of domain walls in the integer quantum Hall regime. For a domain wall due to shearing of the upper graphene layer, there is a plateau formation in the magnetoconductance for sufficiently wide defect regions. For wide domain walls due to tension in the upper graphene layer, there is only an approximate plateau formation with fluctuations of the order of the elementary conductance quantum ${\sigma }_0$. A direct transition between stacking regions like for the hard wall domain wall shows no plateau formation and is therefore not a good model for either of the previously mentioned extended domain walls.

Figure 1

Illustration of Bernal stacked BLG for a nearest-neighbour model. The green sheet is on top and the magenta sheet is in the bottom. The blue and red circles indicate the A and B sublattice respectively for both sheets.

Figure 2

Homogeneous and HLSW system. The lower layer is in blue and the upper layer is striped or dotted in red and green. On the left is a homogeneous AB stacked BLG system, which will be used as a reference system. On the right is the hard wall model for a LSW at the zigzag nanoribbon. The upper layer is decoupled, but any effective interaction may be of interest, in particular a fully coupled upper layer.

Figure 3

Heterogeneous systems with LSWs containing BLG under shear (left) and tension (right). The lower layer is the solid blue lattice. The upper layer is in both cases either dotted or striped in green, orange and red respectively. Only the nearest-neighbour hoppings are shown.

Figure 4

Hall bar system with TLSW representing a general LSW. The lower graphene layer is solid blue and the upper layer is striped green. This is the generic system which will be used to discuss the IQHE in inhomogenous or homogeneous BLG systems. It is a six-terminal structure with a lead connected to the system at each of the labels $L$ (left), $R$ (right), $\mathit{BL}$ (bottom left), $\mathit{BR}$ (bottom right), $\mathit{TL}$ (top left) and $\mathit{TR}$ (top right). The electron propagation in a homogeneous medium is indicated by blue arrows. The electron injection electrodes are also indicated with arrows.

Figure 5

Schematic depiction of a partition of a quite complicated geometry into appropriate subsystems such that the resulting Hamiltonian matrix is block tridiagonal with small blocks. Leads are shown in red and the dotted lines in grey show the lines that define the subsystems.

Figure 6

Hall bar plot for a homogeneous AB stacked Hall bar system. On the left is a logarithmic plot of the LED in units of $1/t$ due to electron injection from the lower left and right lead with a cutoff at $1\times {10}^{-4}$. On the right are the elements of the conductance tensor. The different conductances have different colors and marker styles. The indices for the conductances are as indicated in Fig. 4. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 5\times {10}^5$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 1.5\times {10}^5$. The width of the Hallbar in the LED calculation is $222l_m$ and the height is $55l_m$.

Figure 7

Plot of a slab geometry for a system with HLSW. Shown is a logarithmic plot of the LED $\rho$ in units of $1/t$ due to electron injection from $L$ with a cutoff at $1\times {10}^{-4}$. The LSW is located precisely in the middle of the geometry. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=34.5{\varphi }_0^{-1}$. The lower layer is fully coupled $t_{\mathrm{LSW}}^{\left(1\right)}=t$. The upper layer is decoupled $t_{\mathrm{LSW}}^{\left(2\right)}=0$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 6.5\times {10}^5$. The width of the slab in the LED calculation is $284l_m$ and the height is $141l_m$. This plot shows the system in a configuration at the edge of a plateau, where bulk modes are not gapped.

Figure 8

Hall bar plot, see Fig. 6, for a Hall bar system with HLSW. The LED calculation is performed at an external magnetic ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The lower layer is fully coupled $t_{\mathrm{LSW}}^{\left(1\right)}=t$. The upper layer is decoupled $t_{\mathrm{LSW}}^{\left(2\right)}=0$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 5\times {10}^5$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 1.5\times {10}^5$. The width of the Hallbar in the LED calculation is $242l_m$ and the height is $60l_m$.

Figure 9

Parameter study of the HLSW interaction strength $t_{\mathrm{LSW}}^{\left(U\right)}$ for different external magnetic fields ${\varphi }^{-1}$ for a HLSW system. On the left is ${\sigma }_{\mathrm{BL}\to \mathrm{BR}}$ and on the right is ${\sigma }_{\mathrm{BL}\to \mathrm{TL}}$. The two plots show the conductance in color with the HLSW interaction strength $t_{\mathrm{LSW}}^{\left(U\right)}$ on the $y$ axis and the magnetic field on the $x$ axis. The number of sites in the system used for the transmission calculations is $N_{\text{C}}\approx 1.6\times {10}^5$.

Figure 10

Hall bar plot, see Fig. 6, for a Hall bar system with SLSW. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The cutoff between the two layers in the LSW is long, $DC=4a_0$. The shear width parameter is $L_S=44$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 1.2\times {10}^6$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.3\times {10}^5$. The width of the Hallbar in the LED calculation is $294l_m$ and the height is $61l_m$.

Figure 11

Parameter study of the SLSW width $L_s$ for different external magnetic fields ${\varphi }^{-1}$ for a SLSW system. On the left is ${\sigma }_{\mathrm{BL}\to \mathrm{BR}}$ and on the right is ${\sigma }_{\mathrm{BL}\to \mathrm{TL}}$. The two plots show the conductance in color with the SLSW width $L_s$ on the $y$ axis and the magnetic field on the $x$ axis. The number of sites in the system used for the transmission calculations is $N_{\mathrm{C}}\approx 2.3\times {10}^5$. The cutoff between the two layers in the LSW is long, $D_C=4a_0$.

Figure 12

Transmission calculation, see Fig. 6, for a Hall bar system with SLSW. The cutoff between the two layers in the LSW is long, $D_C=4a_0$. The shear width parameter is $L_S=150$. For both calculations The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.3\times {10}^5$.

Figure 13

Hall bar plot, see Fig. 6, for a tensed system. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The cutoff between the two layers in the LSW is short, $D\_C=1.3a_0$. The tension length parameter is $L_T=60$. The number of sites contained in the system used for the LED calculation is. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.0\times {10}^5$. The width of the Hallbar in the LED calculation is $180l_m$ and the height is $60l_m$.

Figure 14

Parameter study of the TLSW width Lt for different external magnetic fields ${\varphi }^{-1}$ for a TLSW system. On the left is ${\sigma }_{\mathrm{BL}\to \mathrm{BR}}$ and on the right is ${\sigma }_{\mathrm{BL}\to \mathrm{TL}}$. The two plots show the conductance in color with the TLSW width $L_t$ on the $y$ axis and the magnetic field on the $x$ axis. The number of sites in the system used for the transmission calculations is $N_{\mathrm{C}}\approx 2.0\times {10}^5$. The cutoff between the two layers in the LSW is long, $DC=4a_{i}0$.

Figure 15

Transmission calculation, see Fig. 6, for a tensed Hall bar system. The cutoff between the two layers in the LSW is long, $D_{\mathrm{C}}=4a_0$. The tension length parameter is $L_{\mathrm{T}}=150$. For both calculations The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.0\times {10}^5$.

Figure 16

Hall bar system finite-size analysis for transmission diagrams, see Fig. 6 for a description of transmission diagrams. The lower layer is fully coupled $t_{\mathrm{LSW}}^{\left(1\right)}=t$. The upper layer is decoupled $t_{\mathrm{LSW}}^{\left(2\right)}=0$.

Figure 17

SLSW Hall bar system finite-size analysis for transmission diagrams, see Fig. 6 for a description of transmission diagrams. The cutoff between the two layers in the LSW is long, $D_{\mathrm{C}}=4a_0$. The shear width parameter is $L_S=44$.

Figure 18

TLSW Hall bar system finite-size analysis for transmission diagrams, see Fig. 6 for a description of transmission diagrams. The cutoff between the two layers in the LSW is long, $D_{\mathrm{C}}=4a_0$. The tension length parameter is $L_T=60$.

Figure 19

Parameter study of the cutoff distance $D_c$ for different external magnetic fields ${\varphi }^{-1}$ for a SLSW system. On the left is ${\sigma }_{\mathrm{BL}\to \mathrm{BR}}$ and on the right is ${\sigma }_{\mathrm{BL}\to \mathrm{TL}}$. The two plots show the conductance in color with the cutoff distance $D_c$ on the $y$ axis and the magnetic field on the $x$ axis. The number of sites in the system used for the transmission calculations is $N_{\mathrm{C}}\approx 2.3\times {10}^5$. The shear width parameter is $L_S=44$.

Figure 20

Parameter study of the cutoff distance $D_c$ for different external magnetic fields ${\varphi }^{-1}$ for a SLSW system. On the left is ${\sigma }_{\mathrm{BL}\to \mathrm{BR}}$ and on the right is ${\sigma }_{\mathrm{BL}\to \mathrm{TL}}$. The two plots show the conductance in color with the cutoff distance $D_c$ on the $y$ axis and the magnetic field on the $x$ axis. The number of sites in the system used for the transmission calculations is $N_{\mathrm{C}}\approx 2.0\times {10}^5$. The tension length parameter is $L_T=60$.

Figure 21

A study of magnetoconductance for three different Fermi energies. Transmission for HLSW $N\approx 1.6\times {10}^5$ (left), SLSW $N\approx 1.7\times {10}^5$ (middle), TLSW $N\approx 1.8\times {10}^5$ (right) systems. The energies are $E_{\mathrm{F}}=0.85\mathrm{eV}$ (top), $E_{\mathrm{F}}=0.9\mathrm{eV}$ (middle), EF = 0:95 eV (bottom).

Figure 22

Hall bar plot for a homogeneous AB stacked Hall bar system. On the left and right are logarithmic plots of the LED in units of $t$ due to electron injection from the lower left and right lead with a cutoff at $1\times {10}^{-4}$. The LED is split by layer with the lower layer on the left and the upper layer on the right. The different conductances have different colors and marker styles. The indices for the conductances are as indicated in Fig. 4. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 5\times {10}^5$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 1.5\times {10}^5$. The width of the Hallbar in the LED calculation is $222l_m$ and the height is $55l_m$.

Figure 23

Hall bar plot for a Hall bar system with HLSW. On the left and right are logarithmic plots of the LED in units of $t$ due to electron injection from the lower left and right lead with a cutoff at $1\times {10}^{-4}$. The LED is split by layer with the lower layer on the left and the upper layer on the right. The different conductances have different colors and marker styles. The indices for the conductances are as indicated in Fig. 4. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The lower layer is fully coupled $t_{\mathrm{LSW}}^{\left(1\right)}=t$. The upper layer is decoupled $t_{\mathrm{LSW}}^{\left(2\right)}=0$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 5\times {10}^5$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 1.5\times {10}^5$. The width of the Hallbar in the LED calculation is $242l_m$ and the height is $60l_m$.

Figure 24

Hall bar plot for a Hall bar system with SLSW. On the left and right are logarithmic plots of the LED in units of $t$ due to electron injection from the lower left and right lead with a cutoff at $1\times {10}^{-4}$. The LED is split by layer with the lower layer on the left and the upper layer on the right. The different conductances have different colors and marker styles. The indices for the conductances are as indicated in Fig. 4. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The cutoff between the two layers in the LSW is long, $D_{\mathrm{C}}=4a_0$. The shear width parameter is $L_S=44$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 1.2\times {10}^6$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.3\times {10}^5$. The width of the Hallbar in the LED calculation is $294l_m$ and the height is $61l_m$.

Figure 25

Hall bar plot for a tensed system. On the left and right are logarithmic plots of the LED in units of $t$ due to electron injection from the lower left and right lead with a cutoff at $1\times {10}^{-4}$. The LED is split by layer with the lower layer on the left and the upper layer on the right. The different conductances have different colors and marker styles. The indices for the conductances are as indicated in Fig. 4. The LED calculation is performed at an external magnetic field ${\varphi }^{-1}=10.5{\varphi }_0^{-1}$. The cutoff between the two layers in the LSW is short, $D_{\mathrm{C}}=1.3a_0$. The tension length parameter is $L_T=60$. The number of sites contained in the system used for the LED calculation is $N_{\mathrm{L}}\approx 8.1\times {10}^5$. The number of sites contained in the system used for the transmission calculation is $N_{\mathrm{C}}\approx 2.0\times {10}^5$. The width of the Hallbar in the LED calculation is $180l_m$ and the height is $60l_m$.