Quantum transport simulation of exciton condensate transport physics in a double-layer graphene system

Spatially indirect electron-hole exciton condensates stabilized by interlayer Fock exchange interactions have been predicted in systems containing a pair of two-dimensional semiconductor or semimetal layers separated by a thin tunnel dielectric. The layer degree of freedom in these systems can be described as a pseudospin. Condensation is then analogous to ferromagnetism, and the interplay between collective and quasiparticle contributions to transport is analogous to phenomena that are heavily studied in spintronics. These phenomena are the basis for pseudospintronic device proposals based on possible low-voltage switching between high (nearly shorted) and low interlayer conductance states and on near-perfect Coulomb drag-counterflow current along the layers. In this work, a quantum transport simulator incorporating a nonlocal Fock exchange interaction is presented, and used to model the essential transport physics in, for specificity, a graphene-dielectric-graphene system. Finite-size effects, Coulomb drag-counterflow current, critical interlayer currents beyond which interlayer dc conductance collapses at subthermal voltages, nonlocal coupling between interlayer critical currents in multiple lead devices, and an Andreev-like reflection process are illustrated.

Figure 1

Simulated structure. Two graphene layers (infinite in the $y$ direction) are coupled via bare and Fock exchange interactions in a channel of length $L$ and with interlayer separation $d$ across a tunnel dielectric. The effective dielectric constant ${\epsilon }_{\text{r}}$ represents the dielectric environment between and, still more importantly, above and below the graphene layers as a whole. The periodicity in the $y$ direction is represented by the shaded stripes along the transport direction encompassing one armchair chain of graphene atoms per layer. Atoms within the armchair chain are labeled by consecutive integers, starting from 1 on the left of the channel and ending with $N$ on the right. Two consecutive atomic sites within one armchair chain of atoms and within the same layer correspond to a primitive unit cell of monolayer graphene. Four semi-infinite, perfect absorbing/injecting leads are modeled, and labeled TL (top left), BL (bottom left), TR (top right), and BR (bottom right), which conceptually (they do appear only implicitly in the boundary conditions) incorporate atoms 0, $-1,-2$, ... on the left and $N+1,N+2,N+3$, ... on the right within a layer.

Figure 2

Local density of states (LDOS) plotted in the center of the channel. The top-layer LDOS is plotted with solid lines and bottom-layer LDOS with dashed lines, respectively, with solid and open circles representing the condensate-free LDOS for comparison. An anticrossing band gap, which also is centered about the chemical potential used as the zero-energy reference, starts to form and saturates to its bulk value with increasing channel length. For stronger condensates (low ${\epsilon }_{\text{r}})$, the (incomplete) band gap saturates earlier, and is larger for the same channel length.

Figure 3

LDOS vs energy and position within the channel for ${\epsilon }_{\text{r}}=2.2,V_{\text{b}}=0.5$ meV, $d=1.0$ nm, $L=15$ nm, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$. The lack of LDOS is demonstrated by the dark regions.

Figure 4

The value of the transmission coefficients $T\left(E\right)$, including that for reflection to the lead of incidence and averaged over all incident modes, for injections from leads BL and TL. Transmission coefficients without the condensate are plotted in (a)–(d), and those with the condensate in (e)–(h). Simulation parameters are dielectric constant ${\epsilon }_{\text{r}}=2.2$, layer separation $d=1.0$ nm, channel length $L=15$ nm, interlayer bare coupling $V_{\text{b}}=1.0$ meV, and carrier concentrations $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$.

Figure 5

Illustrative samples for distributions of intralayer current per width $\left(I_{\text{intra}}\right)$ and interlayer current per area $\left(J_{\text{il}}\right)$. (a), (c) BiSFET-like biasing with $V_{\text{il}}=5$ mV and $V_{\text{b}}=0.5$ meV, and (b), (d): drag-counterflow current biasing with $V_{\text{al}}=40$ mV and $V_{\text{b}}=0.5$ meV. Other parameters are ${\epsilon }_{\text{r}}=2.2,d=1.0$ nm, $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$, and $L=15$ nm. The positive directions for intralayer and interlayer currents are as shown in the plots.

Figure 6

Bottom-layer $\left(I_{\text{B}}\right)$, top-layer $\left(I_{\text{T}}\right)$ current per width and interlayer $\left(J_{\text{il}}\right)$ current per area distributions for (a)–(c) BiSFET biasing, and (d)–(f) drag-counterflow biasing, using the same system parameters, biasing conditions and rules for signs of current flow as for Fig. 5. These current distributions are also visualized with arrows that originate at the positions they represent for (g) BiSFET-like and (h) drag-counterflow biasing, with vertical and horizontal components proportional to the interlayer and intralayer currents, respectively. However, with the interlayer and intralayer current having different units, the shown relative lengths of the horizontal and vertical components are chosen for illustration clarity only. The currents above $E_{\text{co}}$ correspond to electron and hole injection from the leads into or extraction to the leads from the channel region, while those below $E_{\text{co}}$ correspond to coherent exciton flow within the channel.

Figure 7

(a) Collapse of the near-perfect Coulomb drag-counterflow current in the presence of the condensate in a nanoscale channel with parameters provided in the figure. The bottom layer is biased, and thus the drag efficiency $\eta$ is defined as the top-to-bottom ratio between corresponding current amplitudes averaged across the channel. (b) LDOS in the center of the channel for representative voltages that do and do not support the near-perfect Coulomb drag-counterflow current, as indicated in (a). The collapse of the near-perfect Coulomb drag-counterflow current is accompanied by the collapse of the condensate.

Figure 8

(a) Amplitude of the Fock exchange interactions (proportional to the pseudospin amplitude) between bare-coupled pairs of top and bottom atoms as a function of position in a 15-nm channel. Other parameters are $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2},V_{\text{b}}=0.5$ meV, $d=1.2$ nm, and ${\epsilon }_{\text{r}}=2.2$. (b) Total pseudospin phase between the same pairs of atoms as a function of position with the same system parameters. Apparently, the pseudospin amplitude is position dependent yet voltage independent, while the pseudospin phase is voltage dependent yet largely position independent.

Figure 9

Linear $sin\left(\theta \right)$ vs $V_{\text{il}}$ relation for BiSFET biasing conditions, with ${\epsilon }_{\text{r}}=2.2,d=1.0$ nm, $V_{\text{b}}=0.75$ meV, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$, for varying channel lengths $L$.

Figure 10

Linear dependence of critical voltage $V_{\text{c}}$ on interlayer bare coupling energy $V_{\text{b}}$. System parameters are $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2},{\epsilon }_{\text{r}}=2.2$, and $d=1.0$ nm.

Figure 11

Effective interlayer critical voltage on the “drive” (drv) end of the channel at which the critical interlayer current is reached as a function of the interlayer voltage applied to the “control” (ctrl) end of the channel. The dashed line with slope of $-1$ corresponds to perfect critical voltage conservation such that sum of the two voltages at which the critical current is reached would be fixed. System parameters are ${\epsilon }_{\text{r}}=2.2,V_{\text{b}}=0.5$ meV, $L=15$ nm, $d=1.0$ nm, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$.

Figure 12

Interlayer conductance $G_{\text{il}}$ normalized to Laudauer-Büttiker limit for the leads at the chemical potential (0 eV), interlayer current density $I_{\text{c}}$ (per unit width), and critical voltage $V_{\text{c}}$ with varying effective dielectric constant ${\epsilon }_{\text{r}}=2.2$, 2.5, and 3.0 and corresponding bulk critical temperatures $T_{\text{c}}\approx 655$, 524, and 377 K, respectively. [For ${\epsilon }_{\text{r}}=3.0$ $(T_{\text{c}}\approx 377$ K) the condensate is only partially formed within the 25-nm channel, and therefore the transition length cannot be determined by the definition in the text. Therefore, $\lambda$ is taken to be simply less than the Fermi wavelength divided by half the maximum channel length (12.5 nm), which is 1.2.] The other parameters are $V_{\text{b}}=0.5$ meV, $d=1.0$ nm, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$.

Figure 13

Interlayer conductance $G_{\text{il}}$ normalized to Laudauer-Büttiker limit for the leads, interlayer critical current density $I_{\text{c}}$ per unit width, and (nonequilibrium) reflection probabilities, for incident electrons in the bottom-left lead BL $R_{\text{BL}}$ for varying interlayer separation $d$. The other parameters are ${\epsilon }_{\text{r}}=2.2,V_{\text{b}}=0.5$ meV, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$. The interlayer separations $d$ are 0.8, 1.0, and 1.2 nm, with associated bulk critical temperatures of $T_{\text{c}}\approx 747$, 655, and 585 K, respectively.

Figure 14

Interlayer conductance $G_{\text{il}}$ normalized to Laudauer-Büttiker limit for the leads, interlayer critical current density $I_{\text{c}}$ per unit width, and associated critical voltage $V_{\text{c}}$ for varying interlayer hopping energies $V_{\text{b}}=0.5$, 0.75, and 1.0 meV. The other parameters are ${\epsilon }_{\text{r}}=2.2,d=1.0$ nm, and $n=p\approx 6.0\times {10}^{12}{\text{cm}}^{-2}$. $T_{\text{c}}\sim 655$ K and $\lambda \sim 2.25$ for all three cases.

Figure 15

Interlayer conductance $G_{\text{il}}$ normalized to Laudauer-Büttiker limit for the leads, and interlayer critical current density, $I_{\text{c}}$ per unit width, with varying carrier concentrations $n=p\approx 3.8\times {10}^{12},6.0\times {10}^{12}$, and $8.6\times {10}^{12}{\text{cm}}^{-2}$ $(V_{\text{diff}}=0.3$, 0.5, and 0.7 eV, respectively) and corresponding bulk critical temperatures $T_{\text{c}}\approx 598$, 655, and 702 K, respectively.