Interband State Transfer in Double-Gated Bilayer Graphene at High Electric Field

TL;DR

This study demonstrates that double ionic gating can drive Bernal-stacked bilayer graphene into the large-$Δ$ regime, where the interlayer potential difference $Δ$ exceeds the interlayer hopping $t_\perp$. Transport measurements reveal a knee, peak splitting, and multiple sign reversals in the Hall response at high $Δ$, which the authors attribute to in-gap bound states bound to ions near the BLG interface crossing the mid-gap as $Δ$ grows. A minimal bound-state model shows that the bound-state energies $E_i^{±}$ move nonlinearly with $Δ$ and cross $E=0$ at Δ0 ≈ 1.3$t_\perp$ (≈ 0.45 eV), reshaping the in-gap density of states and the chemical potential. The work validates theoretical predictions for large-$Δ$ BLG, highlights the role of ion-induced in-gap states, and suggests future directions to suppress disorder (e.g., thin hBN spacers) to study intrinsic band-structure changes and possible excitonic effects in gapped BLG.

Abstract

The band structure of Bernal-stacked bilayer graphene can be tuned using double-gated transistors to apply a perpendicular electric field that generates an interlayer potential energy difference $Δ$. Dielectric breakdown limits the operation of conventional devices to the $Δ\ll t_\perp \simeq 360$ meV regime. We employ double ionic gating to reach fields past $ 1$ V/nm, for which $Δ> t_\perp$. We find that for $Δ\simeq t_\perp$, the evolution of the longitudinal resistance ($R_{xx}$) peak as a function of applied gate voltages undergoes a sharp change in slope, exhibiting a pronounced "knee". Increasing $Δ$ past the "knee" results in an unusual evolution transport properties: the peak in $R_{xx}$ decreases in magnitude, it exhibits a splitting concomitant with multiple sign reversals of the Hall resistance, and hysteresis in the peak position emerges. We explain the observed phenomenology in terms of in-gap bound states, whose energy strongly depends on the perpendicular electric field, and crosses the mid-gap level for sufficiently large $Δ> t_\perp$. The phenomenon causes large changes in the electronic density of in-gap states that profoundly affect the evolution of the chemical potential. Our experimental results and their interpretation reveal unique aspects of the physics of in-gap states in Bernal bilayer graphene and demonstrate that double ionic gating enables investigating the large-$Δ$ regime, which has remained experimentally inaccessible so far.

Figures (5)

• Figure 1: (a) Band structure of BLG for different values of interlayer energy difference $\Delta = 0$, $0.1$, and $0.4$ eV (the color scale shows how the layer-resolved weight of the electronic states varies on the different bands, as a function of energy). (b) Schematic cross-section (not to scale) of a BLG transistor equipped with an IL top gate and a Li-ion glass-ceramic back gate, together with their corresponding reference electrodes (RT, RB). The SiO$_2$/Al/Al$_2$O$_3$ trilayer is used to electrostatically decouple the top and bottom electrolytes away from the BLG. (c) Zoom-in on the device channel area, showing the Pt electrodes contacting the BLG. (d) Optical image of a device patterned in a Hall bar geometry on a LICGC substrate, prior to the deposition of the ionic liquid (IL). (e) Longitudinal ($R_{xx\square}$, black curve) and transverse ($R_{xy}$, red curve) resistance measured as a function of gate voltage on a Li-ion–gated BLG device, prior to the deposition of the ionic liquid ($R_{xy}$ is measured by applying a perpendicular magnetic field $B=1$ T). The top axis shows the value of the reference potential measured while sweeping the voltage applied to the backgate. (f) Mobility $\mu$ and carrier density $n$ extracted from the measurements shown in panel (e).
• Figure 2: (a) Color plot of the longitudinal square resistance $R_{xx\square}$ as a function of top and bottom reference potentials ($\delta V_R$) relative to the reference potential measured at zero perpendicular electric field (see main text). In this range of applied electric field, the data exhibit the characteristic evolution expected for a double-gated BLG regime: the maximum of $R_{xx}$ shifts linearly when $\delta V_R^T$ and $\delta V_R^B$ are varied with opposite polarity, and the peak resistance increases with perpendicular electric field (proportional to $\delta V_R^T - \delta V_R^B$). (b) Selected cuts of the color map in panel (a) for different fixed values of $\delta V_R^T$ (the blue curve of minimum peak resistance corresponds to the trace used to define the zero–electric-field point, $V_{R_0}^T$ and $V_{R_0}^B$ at the position of the peak). (c) Corresponding evolution of the transverse resistance $R_{xy}$ measured in the presence of a $B=1$ T perpendicular magnetic field.
• Figure 3: Color plot of $R_{xx\square}$ as a function of $\delta V_R^B$ and $\delta V_R^T$ extending to the high electric-field regime. The evolution of the peak position (marked by the white squares) strongly deviates from linearity: after an initial linear trend, it exhibits a clear "knee", and eventually flattens at large $\delta V_R^T$. (b) The evolution shown in panel (a) is reproducible across multiple devices and using different pairs of contacts. The panel contains seven distinct data sets measured using multiple pairs of contacts on two different devices (symbols of different colors represent data taken on the two different devices). (c) Selected line cuts extracted from panel (a), showing $R_{xx\square}$ as a function of $\delta V_R^B$ for fixed different values of $\delta V_R^T$. The height of the peak in $R_{xx\square}$ decreases once $\delta V_R^T$ is increased past the "knee", for both polarities of perpendicular electric field. (d) measurements of $R_{xx\square}$ as in panel (b) performed using a different pair of voltage probes as compared to those used to take the data in (a). Irrespective of the probes, the peak in $R_{xx\square}$ broadens substantially for negative values of $\delta V_R^B$ past the "knee", whereas for positive $\delta V_R^T$ the peak remains narrower, and --when using probes for which the peak is narrowest-- a splitting is detected. (e,f) $R_{xx\square}$ and Hall resistance $R_{xy}$ as a function of $\delta V_R^B$ in the low-and high–electric-field regimes, respectively ($R_{xy}$ is measured in the presence of a $B=1$ T perpendicular magnetic field). In the low-field regime the Hall effect behaves as expected, with the zero crossing occurring at the position of the maximum in $R_{xx\square}$. In the high-field regime, when a splitting in the peak of $R_{xx\square}$ is seen, $R_{xy}$ changes sign multiple times.
• Figure 4: (a,b) $R_{xx\square}$ measured for different values of $\delta V_R^T$ [(a) positive and (b) negative; in both panels the curves are offset vertically for clarity], while sweeping the back-gate voltage up (from negative to positive values, red curve) and down (from positive to negative values, blue curve). For small $\delta V_R^T$ (i.e., when $\delta V_R^T$ is below the “knee”), the peak position of $R_{xx\square}$ measured in the up- and down-sweeps coincides. For large magnitudes of $\delta V_R^T$ (above the “knee”), a sizable hysteresis in position of the $R_{xx\square}$ peak is observed. (c) Position of the peak in $R_{xx\square}$ extracted from up- (red symbols) and down- (blue symbols) sweeps, as a function of $\delta V_R^T$ and $\delta V_R^B$. The onset of hysteresis in the peak position coincides with the "knee" in the evolution of $R_{xx\square}$.
• Figure 5: (a) Model calculation of the energy of an individual in-gap state as a function of the interlayer energy difference $\Delta$. Red and blue lines indicate the energy of electron and hole in-gap bound states created respectively by positive and negative charged ions in the ionic liquid electrolyte. The energy of the in-gap state exhibits a non-monotonic evolution with $\Delta$ and for a certain value $\Delta = \Delta_0$ it crosses through $E=0$ (the mid-gap level). Black lines indicate the band edges. (b) Solution of the equation $\mu(V_B,V_T)=0$ calculated with the model discussed in the main text. The resulting curve can be compared to the evolution of the position of the peak in $R_{xx\square}$ as a function of $\delta V_R^B$ and $\delta V_R^T$ (see Fig.\ref{['fig:Figure3']}a,b). The "knees" in the curve occur when the edge of the impurity band passes through the mid-gap energy $E = 0$. (c) Non-monotonic dependence of the chemical potential as a function of $V_B$ for at a value of $V_T$ fixed in the vicinity of the knee (see red dashed line in (b)). All calculations correspond to a temperature $k_B T = 0.025 t_\perp$ and use a dielectric constant $\varepsilon = 3$ and a net impurity concentration $n_\textrm{imp} = 100 (t_\perp/\hbar v)^2$, where $v$ is the velocity of Dirac fermions in monolayer graphene. The symbols $C_B$ and $C_T$ denote the top gate and back gate capacitances, respectively.