Multiple negative differential conductance regions and inelastic phonon assisted tunneling in $\text{graphene}/h-\mathrm{BN}/\text{graphene}$ structures

In this paper we study in detail the effect of the rotational alignment between a hexagonal boron nitride ($h$-BN) slab and the graphene layers in the vertical current of a a $\text{graphene}/h-\mathrm{BN}/\text{graphene}$ device. We show how for small rotational angles, the transference of momentum by the $h-\mathrm{BN}$ crystal lattice leads to multiple peaks in the $I-V$ curve of the device, giving origin to multiple regions displaying negative differential conductance. We also study the effect of scattering by phonons in the vertical current and see how the opening up of inelastic tunneling events allowed by spontaneous emission of optical phonons leads to sharp peaks in the second derivative of the current.

Figure 1

(a) Schematic of a typical graphene/$h$-BN/graphene vdW structure with four boron-nitride layers, with applied gate $V_{\mathrm{gate}}$ and bias $V_{\mathrm{bias}}$ voltages. (b) Representation of crystalline structure shared by a graphene/boron-nitride monolayer, showing the lattice basis $\left\{{\mathit{a}}_1,{\mathit{a}}_2\right\}$, the nearest-neighbor vectors ${\mathbf{\tau }}_i, i=1,2,3$, and the sublattice $A/B$ sites. (c) Representation of the first Brillouin zone of the rotated bottom and top graphene layers, showing the $K$ points of both layers and the reciprocal lattice basis vectors $\left\{{\mathit{b}}_{1,\mathrm{bg},\mathrm{tg}},{\mathit{b}}_{2,\mathrm{bg}/\mathrm{tg}}\right\}$.

Figure 2

Band diagram representing the constraints imposed by energy-momentum conservation and Pauli's exclusion principle in the vertical current of a graphene/$h$-BN/graphene device. The two cones represent the dispersion relation for electrons of the bottom and top graphene layers. The shadowed blue regions represent the occupation of electronic states in both graphene layers. Energy-momentum conservation is only satisfied when the two shifted Dirac cones intersect and the energy windows where this occurs are represented by the dashed arrows. The following cases are represented: (a) Only intrabands are possible, ${\varepsilon }_{n,m}<1$; these are, however, Pauli blocked or there are no states available, therefore in the low-temperature limit, no vertical current flows. (b) Threshold bias voltage above which intraband processes which satisfy energy-momentum conservation appear in the energy window where tunneling is allowed by the electronic occupation factors. (c) The condition which corresponds to the occurrence of a peak in the current, when ${\varepsilon }_{n,m}=1$, when both intraband and interband (conduction-to-valence and valence-to-conduction) processes are allowed. (d) If one further increases the bias voltage, only interband tunneling ${\varepsilon }_{n,m}>1$ becomes possible and the current diminishes.

Figure 3

Plot of the quantity ${\mathrm{TDoS}}_{m,n}(\omega +{\varepsilon }_{m,n}{v}_F\hslash |{\mathcal{Q}}_{n,m}|/2,\omega -{\varepsilon }_{n,m}{v}_F\hslash |{\mathcal{Q}}_{n,m}|/2)$ for different values of ${\varepsilon }_{m,n}$ as a function of the energy at zero magnetic field and for rotation angles of ${\theta }_{\mathrm{TG}}=1^{\circ }$ and ${\theta }_{h-\mathrm{BN}}=1.5^{\circ }$. The solid red line shows the tunneling density of states if the wave-function overlap factors ${\mathrm{{\rm Y}}}_{\mathit{k},\lambda }^{B/T,n}$ in Eq. (40) are set to one. A constant broadening factor of $\gamma =2.5\times v_F\hslash \left|{\mathcal{Q}}_{n,m}\right|\times {10}^{-3}$ was used in all plots.

Figure 4

$I-V$ curves for vertical current in a graphene/$h$-BN/graphene device with four layers of $h$-BN for rotation angles of ${\theta }_{\mathrm{tg}}=1^{\circ }$ and ${\theta }_{h-\mathrm{BN}}=1.5^{\circ }$ at gate voltage $V_{\mathrm{gate}}=0$ for two different temperatures. The solid red line indicates the current due to all the nine processes coupling both graphene layers, for graphene electrons, while the dashed black lines represents the total current for scalar electrons (by setting the wave-function factors ${\mathrm{{\rm Y}}}_{\mathit{k},\lambda }^{\mathrm{bg}/\mathrm{tg},n}$ to 1). The remaining lines represent the contributions to the current arising from processes involving different ${\mathcal{Q}}_{n,m}$ [taking into account the relations imposed by threefold rotational invariance, Eq. (12)]. The dashed vertical lines labeled by ${(n,m)}^{\pm }$ mark the bias voltages when the condition ${\epsilon }_{\mathrm{F},\mathrm{tg}}+e{V}_{\mathrm{bias}}-{\epsilon }_{\mathrm{F},\mathrm{bg}}=\pm v_F\hslash \left|{\mathcal{Q}}_{n,m}\right|$ is satisfied. Notice that while for scalar electrons all the expected peaks in the current are present, for Dirac electrons some of them are absent. Is is due to the suppression by the ${\mathrm{{\rm Y}}}_{\mathit{k},\lambda }^{\mathrm{bg}/\mathrm{tg},n}$ factors. A constant broadening factor of $\gamma =2.5$ meV was used.

Figure 5

$I-V$ curves at constant $V_{\mathrm{gate}}=0$ in a graphene/$h$-BN/graphene device with four layers of $h$-BN, at $V_{\mathrm{gate}}=0$ and $T=300$ K, for different rotation angles between the top and bottom graphene layers, and the $h$-BN slab and the bottom graphene layer. The black dashed line marks the bias voltage when ${\varepsilon }_{0,0}=\pm 1$ (a condition that is independent of ${\theta }_{h-\mathrm{BN}}$). The remaining vertical lines mark the bias voltages when ${\varepsilon }_{n,m}=\pm 1$ for $n\ne m$ for different values ${\theta }_{h-\mathrm{BN}}$ (the color and type of line match the ones used in the plots).

Figure 6

Density plot of current $I$ and its second derivative with respect to the applied bias voltage $d^{2}I/d{V}_{\mathrm{bias}}^2$, as a function of the applied bias and gate voltages at $T=10$ K. In the current plot, it is also shown the lines defined by the following conditions: ${\epsilon }_{\mathrm{F},\mathrm{bg}}=0$ and ${\epsilon }_{\mathrm{F},\mathrm{tg}}=0$, represented by the solid lines in red and purple; ${\epsilon }_{\mathrm{F},\mathrm{tg}}+e{V}_{\mathrm{bias}}-{\epsilon }_{\mathrm{F},\mathrm{bg}}=\pm v_F\hslash \left|{\mathit{Q}}_{0,m}\right|$ [Eq. (41)] for $m=0$, 1, and 2, represented by the solid lines in blue, green, and yellow, respectively; ${\epsilon }_{\mathrm{F},\mathrm{tg}}\pm e{V}_{\mathrm{bias}}+{\epsilon }_{\mathrm{F},\mathrm{bg}}=\pm \frac{1}{2}{v}_F\hslash \left|{\mathit{Q}}_{0,m}\right|$ for $m=0$, 1, and 2 [Eq. (42)] represented by the dashed lines in blue, green, and yellow. Notice now the guide lines shown in the current plot match perfectly the sharp features shown in the $d^{2}I/d{V}_{\mathrm{bias}}^2$ plot. Also, the peaks expected to occur through channels ${(0,1)}^{+}$ and ${(0,2)}^{-}$ are absent. A constant broadening factor of $\gamma =2.5$ meV was assumed for both layers.

Figure 7

$I-V$ curves for a graphene/$h$-BN/graphene device with four layers of $h$-BN, with rotation angles of ${\theta }_{\mathrm{tg}}=1^{\circ }$ and ${\theta }_{\mathrm{hBN}}=1.5^{\circ }$ at constant $V_{\mathrm{gate}}=0$ and $T=300$ K, for different values and orientation of the in-plane magnetic field and electronic broadening factor. The vertical lines, labeled by ${(n,m)}^{\pm }$, mark the bias voltages for which ${\varepsilon }_{n,m}=\pm 1$. Notice how the applied magnetic field leads to a splitting of the peaks that occur at zero magnetic field. As the broadening factor is increased, the peaks become less resolved.

Figure 8

Diagram representing the possible effect of the reconstruction of the graphene Dirac spectrum, due to the presence of $h$-BN, in the vertical current of a graphene/$h$-BN/graphene device when ${\varepsilon }_{n,m}=\pm 1$. The red bars represent the position in energy of the regions, of width $\mathrm{\Delta }$, where graphene's spectrum reconstruction is significant. Provided $e{V}_{\mathrm{bias}}/\mathrm{\Delta }\gg 1$, the peaks in the current will still be present.

Figure 9

Diagrammatic representation of contributions to the current involving phonon scattering. The dots represent the level width functions due to the bottom and top external metallic contacts, the squares represent the graphene/$h$-BN coupling, solid lines represent graphene electronic propagators, and dashed lines represent $h$-BN propagators. The wiggly lines represent phonon propagators. (a) Ladder diagrams that are resummed in Eq. (63). To lowest order in the graphene/$h$-BN coupling, these are all the contributions due to electron-phonon interaction in the graphene layers. (b) Diagram contributing to the current in higher order in the graphene/$h$-BN coupling, including the renormalization of the top graphene layer Green's function by phonons. This kind of diagram can be captured in Eq. (63), provided the effect of coupling to the graphene layers is included into ${\mathit{G}}_{h-\mathrm{BN}}$. (c) Higher-order diagrams in the graphene/$h$-BN coupling, including electron-phonon interaction in the graphene layers, that is not included in Eq. (63) and more generically cannot be captured when evaluating the current following approach (A).

Figure 10

$I-V$ curves at constant $V_{\mathrm{gate}}=0$ in a graphene/$h$-BN/graphene device with rotations angles ${\theta }_{\mathrm{tg}}=1^{\circ }$ and ${\theta }_{h-\mathrm{BN}}=1.5^{\circ }$, considering different sources of scattering in the graphene layers: (RT) constant relaxation time of $\gamma =3$ meV; (Imp) scattering by resonant scatterers treated within the SCBA with an impurity concentration of $n_{\mathrm{imp}}={10}^{-4}$ impurities per graphene unit cell; (Imp+RT) scattering by resonant scatterers and graphene in-plane optical phonons also with $n_{\mathrm{imp}}={10}^{-4}$.

Figure 11

$I-V$ curve and $d^{2}I/d{V}_{\mathrm{bias}}^2$ as a function of bias voltages at a constant $V_{\mathrm{gate}}=10$ V for different temperatures and for rotation angles ${\theta }_{\mathrm{tg}}=2^{\circ }$ and ${\theta }_{h-\mathrm{BN}}=3^{\circ }$, including effects of scattering by out-of-plane breathing phonons of $h$-BN, ${\omega }_{\mathrm{ZB}}^{h-\mathrm{BN}}=15$ meV [58], and of the in-plane graphene phonons, ${\omega }_{\mathrm{\Gamma }\mathrm{O}}^{\mathrm{g}}=196$ meV [59] (represented by the vertical dashed lines). Processes involving spontaneous emission of phonons open up new tunneling channels that appear as peaks in $d^{2}I/d{V}_{\mathrm{bias}}^2$ at low temperature. The solid lines are the curves computed with the electron-phonon coupling parameters estimated using Eqs. (71) and (E2): $g_{\mathrm{ZB}}^{h-\mathrm{BN}}\simeq 0.06$ eV and $g_{\mathrm{\Gamma }\mathrm{O}}^{\mathrm{g}}\simeq 0.6$ eV for $h$-BN and graphene, respectively. The dashed lines show the curves computed with a $g_{\mathrm{ZB}}^{h-\mathrm{BN}}$ that is 10 times larger. The inset zooms in the small peak due to the $h$-BN out-of-plane breathing phonon. The feature that occurs around $V_{\mathrm{bias}}\sim 0.1$ V is not due to phonons, but due to the tunneling density of states structure.

Figure 12

Computed Fermi levels for the bottom and top graphene layers as a function of bias voltage for different gate voltages obtained by solving Eqs. (A9) and (A10). We assume that the following parameters $d_{{\mathrm{SiO}}_2}=285$ nm, $d_{h-\mathrm{BN}}=40$ nm for the thickness of the back-gate dielectric, with out-of-plane dielectric constants ${\overline{\epsilon }}_{{\mathrm{SiO}}_2}=3.9$ and ${\overline{\epsilon }}_{h-\mathrm{BN}}=5.09$ [35]. We assumed that the distance between the two graphene layers is separated by four monolayers of $h-\mathrm{BN}$, which corresponds to a distance between the graphene layers of $d\simeq 1.6$ nm.

Figure 13

Real and (minus) imaginary parts of the retarded self-energy for graphene electrons due to resonant impurities treated within the SCBA, for two different impurity concentrations (number of impurities per graphene unit cell).

Figure 14

Real and (minus) imaginary parts of the self-energy for graphene electrons due to scattering by in-plane optical phonons for two different temperatures for doped graphene with ${\epsilon }_{\mathrm{F}}=0.3$ eV. The zero of energy corresponds to the Dirac point. The dashed vertical line marks $\omega ={\epsilon }_{\mathrm{F}}$ and the dotted lines mark $\omega ={\epsilon }_{\mathrm{F}}\pm {\omega }_{\mathrm{\Gamma }\mathrm{O}}^{\mathrm{g}}$.

Figure 15

Diagrammatic representation of the Bethe-Salpeter equation (F6).