Finite-temperature pseudospin torque effect in graphene bilayers

We use self-consistent quantum transport theory to investigate the influence of layer Fermi energy and temperature on the interlayer transport properties of bilayer graphene in the regime of excitonic superfluidity. We conclude that at low temperature the critical tunneling currents and quasiparticle penetration depths are well explained by the existing zero-temperature pseudospin torque model. However, when the thermal broadening associated with finite-temperature transport is included, we find that modes injected above the gap lead to increased critical interlayer currents and quasiparticle penetration in to the gapped regime beyond what current analytical models predict. We find that the increases in the critical tunneling current are due to thermal smearing smoothing out the quantum interference effects present at low temperature leading to a slower pseudospin precession about the pseudospin Bloch sphere.

Figure 1

(Color online) Generic schematic of the simulated device geometry. Contacts are attached to the edges of each of the layers to inject and extract current. In this work we hold the layer widths, the length of the layers, and the thicknesses of the different oxide regions fixed.

Figure 2

Plot of the value of the order parameter plotted down the centerline of the system in equilibrium at $T=0\text{K}$. We see that the order parameter has a nearly constant, nonzero value across the system which is indicative of the presence of a condensate.

Figure 3

(Color online) Surface plots of the directional pseudospin fields (a) $⟨m_{ex}^x⟩$ and (b) $⟨m_{ex}^y⟩$ computed from the self-consistent density matrix for $V_{tg}=-V_{bg}=0.4\text{V}$.

Figure 4

(Color online) Plot of the interlayer current versus the applied interlayer voltage for gate voltages of $V_{tg}=-V_{bg}=200\text{mV}$ (solid black), 400 mV (dotted red), and 600 mV (dashed blue). The inset shows the differential interlayer conductance as a function of the applied interlayer voltage for $V_{tg}=-V_{bg}=200\text{mV}$ (solid black), 400 mV (dotted red), and 600 mV (dashed blue). This gives a much clearer picture of the location of the critical currents in our simulations.

Figure 5

(Color online) Surface plots of the interlayer current for interlayer voltages of (a) $V_{interlayer}=1\text{mV}$ and (b) $V_{interlayer}=1500\text{mV}$ the scales of both plots are in amperes. Note that the current flow between the two layers before the critical current is within $L_c$ of the contacts but past the critical current we see charge transfer over a much greater region.

Figure 6

(Color online) Surface plots of the directional pseudospin fields (a) $⟨m_{ex}^x⟩$ and (b) $⟨m_{ex}^y⟩$ computed from the self-consistent density matrix with $V_{tg}=-V_{bg}=400\text{mV}$ for interlayer voltage of $V_{interlayer}=1\text{mV}$. As we are still in the regime dominated by exciton condensation, the pseudospin fields are appreciable only within $L_c$ of the contacts. The scale on both plots is electron volt.

Figure 7

(Color online) Surface plots of the directional pseudospin fields (a) $⟨m_{ex}^x⟩$ and (b) $⟨m_{ex}^y⟩$ computed from the self-consistent density matrix for a 40-nm-wide graphene bilayer with $V_{tg}=-V_{bg}=400\text{mV}$ for interlayer voltage of $V_{interlayer}=1500\text{mV}$. We are beyond the critical current and the pseudospin fields are reduced in value and penetrate throughout the system. The scales for the figures are in electron volt.

Figure 8

(Color online) Plot of the value of the order parameter through the centerline of the system with $V_{tg}=-V_{bg}=400\text{mV}$ for interlayer voltages of $V_{interlayer}=1\text{mV}$ (solid black) and $V_{interlayer}=1500\text{mV}$ (dotted red).

Figure 9

Plot of the pseudospin torque term, $⟨m_{ex}^z⟩$, through the centerline of the system with $V_{tg}=-V_{bg}=400\text{mV}$ for interlayer voltages of (a) $V_{interlayer}=1\text{mV}$ and (b) $V_{interlayer}=1500\text{mV}$.

Figure 10

Plot of the value of the order parameter for 40-nm-wide layers through the centerline of the system with $V_{tg}=-V_{bg}=400\text{mV}$ at room temperature while the system is in equilibrium.

Figure 11

(Color online) Surface plots of the directional pseudospin fields (a) $⟨m_{ex}^x⟩$ and (b) $⟨m_{ex}^y⟩$ computed from the self-consistent density matrix for a 40-nm-wide graphene bilayer with $V_{tg}=-V_{bg}=400\text{mV}$ in equilibrium at $T=300\text{K}$. The scales for the figures are in electron volt.

Figure 12

(Color online) (a) Plot of the interlayer current versus the applied interlayer voltage for gate voltages of $V_{tg}=-V_{bg}=200\text{mV}$ (solid black), 400 mV (dotted red), and 600 mV (dashed blue) for a system temperature of 300 K. (b) Plot of the interlayer current for gate voltages of $V_{tg}=-V_{bg}=400\text{mV}$ for system temperatures of 0 K (dotted red) and 300 K (solid black). The inset shows the differential conductance as a function of the applied interlayer voltage for $V_{tg}=-V_{bg}=400\text{mV}$ for system temperatures of 0 K (dotted red) and 300 K (solid black). Note that the 300 K case allows for higher critical currents as is evidenced by the extended flat region of the differential conductance as compared to that of the low-temperature case.

Figure 13

(Color online) Plot of the differential tunneling conductance as a function of the applied interlayer voltage for $V_{tg}=-V_{bg}=400\text{mV}$ for system temperatures of 0 K (dotted red) and 300 K (solid black). Note that the 300 K case allows for higher critical currents as is evidenced by the extended flat region of the differential conductance as compared to that of the low-temperature case.

Figure 14

(Color online) Plot of the value of the order parameter for 40-nm-wide layers at a width of 20 nm down the length of the system with $V_{tg}=-V_{bg}=400\text{mV}$ at an interlayer bias of $V_{interlayer}=350\text{mV}$ for a system temperature of $T=300\text{K}$ (solid black) and $T=0\text{K}$ (dotted red).

Figure 15

(Color online) (a) Plot of the value of the interlayer current for temperatures of 0 K (dashed red), 150 K (dotted blue), and 300 K (solid black) with $V_{tg}=-V_{bg}=400\text{mV}$ showing the extension of the bias region for which the system is spontaneously coherent as the temperature is increased. (b) Plot of the pseudospin torque term, $⟨m_{ex}^z⟩$, plotted down the centerline of the bilayer system for temperatures of 0 K (dashed red), 150 K (dotted blue), and 300 K (solid black) with $V_{tg}=-V_{bg}=400\text{mV}$ showing the smoothing of variations in the precession of the pseudospins about the Bloch sphere as the temperature is increased.