Coulomb Correlation Gap at Magnetic Tunneling between Graphene Layers

The strong suppression of equilibrium magnetic tunneling in a graphene/hBN/graphene heterostructure caused by the Coulomb correlation gap in the tunneling density of states has been found. Comparison has shown that the suppression of the equilibrium tunneling conductivity ${{G}_{0}}$ in a high magnetic field with a decrease in the temperature and the dependence of the observed gap width Δ on the filling factor of Landau levels ν are qualitatively similar to the respective results of similar experiments in GaAs heterostructures and has confirmed our hypothesis concerning the nature of the effect. However, the determined gap width Δ is much larger than those measured in all previous works in semiconductor heterostructures probably because the cyclotron energy in graphene is much higher than that in GaAs in the region of low Landau levels.

Great attention has been paid for a long time to multiparticle correlation effects in semiconductor tunneling heterostructures with two parallel layers of a two-dimensional electron gas. In particular, manifestations of a strongly correlated electron liquid in layers of the two-dimensional electron gas in a magnetic field in the formation of a Coulomb gap in the tunneling density of states and in the suppression of tunneling between parallel layers of the two-dimensional electron gas in semiconductor GaAs/AlAs heterostructures were detected experimentally more than 30 years ago [1–3]. Such a Coulomb gap at transitions of electrons between two strongly correlated layers in the two-dimensional electron gas is due to an additional energy for the extraction of an electron from one layer and its introduction in another layer. The total energy for these processes is determined by the Coulomb interaction between electrons in the two-dimensional electron gas and is equal to the formed energy gap Δ.

Multiparticle theories [4–6], which were developed at that time predict that this gap at a fixed filling factor $\nu < 1$ is $\Delta \approx 0.4{{e}^{2}}{\text{/}}4\pi \varepsilon {{l}_{B}}$, where ${{l}_{B}} = (\hbar {\text{/}}eB{{)}^{{1/2}}}$ is the magnetic length, or $\Delta \approx {{e}^{2}}{\text{/}}4\pi \varepsilon a$ if the electron liquid forms a lattice similar to a Wigner crystal [7, 8], where $a = 1{\text{/}}{{(n\pi )}^{{1/2}}}$ is the distance between electrons in the layer. The developed theories of the gap in low magnetic fields (at large filling factors $\nu \gg 1$) [9–12] also could not provide a satisfactory description of the experimental magnetic field dependences $\Delta (B)$ [2, 3]. The empirical dependence Δ ≈ Aℏω_c, where A is a constant and ω_c = eB/m* is the cyclotron frequency, obtained in the experiment [2] is consistent with predictions of various theories [9–12]. In this work, this relation is used to compare gaps in graphene and GaAs.

The discovery of graphene and the appearance of van der Waals heterostructures stimulated investigations of correlation effects. The observation of a tunneling gap due to the intralayer Coulomb correlation in graphene heterostructures was first mentioned in [13], but it was not proved likely because the authors of [13] were focused on the study of the giant enhancement of the tunneling conductivity caused by interlayer correlation. Later, the Coulomb correlation gap was observed and studied in [14] at the magnetic tunneling between graphene layers through a localized state in an hBN barrier, where the experiment was complicated by the possible effect of the charge on this localized state both on the process of tunneling and on the formation of the gap and its measurement. An additional difficulty in the interpretation of the experiment reported in [14] was the observation of the gap only at tunneling transitions between Landau levels with different indices due to features of the design of the structure.

In this work, we detect and study the Coulomb correlation gap in the direct (without intermediate states) tunneling between graphene layers in a graphene/hBN/graphene heterostructure with two gates, which allow us to avoid the problems of the experiment reported in [14] and to correctly examine the formation of the gap and to directly measure its width Δ. To implement tunneling transitions between graphene layers at relatively low energies without intermediate states, heterostructures with a high degree ~2° of angular matching between graphene layers were used in our experiment. The comparison of features of the temperature behavior of the detected effect such as the inversion of the temperature dependence of the equilibrium conductivity with a decrease in the filling factor ν of Landau levels and its activation character at low filling factors ν with the results of similar experiments in GaAs shows their qualitative similarity and confirms our interpretation of the effect. The observed increase in the gap width with decreasing filling factor ν is also consistent with previous results for semiconductor systems. However, the characteristic parameters and dependences of the gap width strongly differ from those measured in semiconductor systems. In particular, the parameter Δ is much larger than the values measured in all cited works, which is likely due both to higher cyclotron energy in graphene compared to GaAs in the relevant region of Landau levels to the possible effect of the interlayer Coulomb interaction between electrons [12, 15].

The sample was a vertical van der Waals heterostructure, which was fabricated by mechanical splitting and transfer of graphene and hBN layers and was studied in our previous works [16, 17]. The main layers of the heterostructure and the measuring circuit are shown schematically in Fig. 1, and the micrograph and the detailed description of the sample are given in Fig. S1 in the supplementary material. The graphene layers were crystallographically co-oriented so that the angular mismatch of the crystal lattices of the top and bottom graphene sheets was ~2° and the crossing area was ~25 µm². The application of the magnetic field perpendicular to the graphene layers led to the quantization of the in-plane electron motion and to the formation of nonequidistant Landau levels, which graphically are circles on the surface of Dirac cones (see Fig. 2), whose energy spectrum is described by the expression E_N = ±${{v}_{{\text{F}}}}$(2eℏ|N|B)^1/2, where N = 0, ±1, ±2, … and ${{v}_{{\text{F}}}}$ is the Fermi velocity. The voltages V_tg and V_bg applied to the top and bottom gates change the chemical potentials μ_t and μ_b and result in successive filling/emptying of Landau levels in the layers. The bias voltage V_b changes the mutual energy positions of the Dirac cones, and a channel of possible resonant-tunneling current is opened in the case of coincidence of filled and empty Landau levels. Strictly speaking, the condition of resonance between the Landau levels with the energy and momentum conservation corresponds to crossing of the filled and empty Landau levels in the k space, which cannot be represented in the shown simplified scheme of tunneling. Tunneling between the Landau levels in two graphene sheets separated by a tunnel barrier was described in detail in [18]. The variation of the voltage V_b and the control of V_tg and V_bg allowed us to observe the sequence of features in the current–voltage characteristics corresponding to tunneling resonances between individual Landau levels in the top and bottom layers. By keeping the value V_b = 0 and varying the values V_tg and V_bg, we plotted three-dimensional maps of the equilibrium (i.e., near V_b = 0) tunneling conductivity of these heterostructures (see the map for B = 12 T in Fig. S2 in the supplementary material). An ideal tool to study manifestations of collective effects in tunneling heterostructures is experimental measurement of the equilibrium interlayer tunneling conductivity in a magnetic field with keeping equal electron densities in the layers between which tunneling occurs by means of the top and bottom gates. To observe and correctly measure the Coulomb correlation gap in the tunneling density of states, as was previously shown in experiments with GaAs heterostructures [1, 2], where the Coulomb gap was observed, it is crucially important to ensure these conditions because the electric field in the barrier in the case of disbalance of the densities will strongly complicate the interpretation of the measured gap widths. This work is devoted to just such measurements of the interlayer tunneling conductivity with electron densities in graphene layers maintained equal by controlling gate voltages. Conductivity maps were plotted using the modulation technique with a modulating voltage amplitude of 0.1–1 mV at a frequency of 27 Hz. The measurements were carried out at temperatures of 1.8–50 K. In contrast to our previous works [16, 17], we also measured the dependence of the tunneling current I on the bias voltage V_b (or dI/dV_b as a function of V_b) at the equal electron densities in the top and bottom graphene layers near zero bias voltage, i.e., at n_top = n_bottom and V_b = 0.

Fig. 1. 

(Color online) (a) Schematic of equilibrium tunneling between graphene monolayers, where Gr_top and Gr_bottom are the top and bottom graphene monolayers, respectively; hBN-1, hBN-2, and hBN-3 are the bottom subgate, tunneling, and top BN layers, respectively; SiO₂ is the main dielectric layer of the bottom gate; μ_t and μ_b are the chemical potentials in the top and bottom layers, respectively; and ΔE_D is the energy shift of Dirac points depending on V_tg, V_bg, and V_b.

Fig. 2.

(Color online) Section of the equilibrium conductivity map at B = 12 T through the line of equal densities n_top = n_bottom. (Upper and lower right panels) Schematics of L3–L3 and L1–L1 tunneling transitions between Landau levels of the top and bottom graphene monolayers, respectively.

Figure 2 presents the dependence of the equilibrium tunneling conductivity on the filling factor ν = n_sh/(eB), where n_s is the density of the two-dimensional electron gas, obtained from the section of the experimental (V_tg, V_bg) map at B = 12 T by the equal density line n_top = n_bottom, which is indicated by the dashed line in Fig. S2 in the supplementary material and approximately corresponds to the relation V_tg = 0.4 + 0.083V_bg. Each feature corresponding to a transition between Landau levels with identical indices is split into four peaks due to the lifting of the spin and valley degeneracy of Landau levels in high magnetic fields [17, 19–21]. The splitting value for all measured features at B = 12 T is in the interval of 7–9 meV. The amplitude of resonance features increases with the filling factor ν in the graphene layers due both to the natural increase in the transparency of the tunnel barrier and to the forbiddenness of tunneling transitions between low Landau levels with the conservation of the longitudinal momentum component parallel to the layers because even a small misorientation of the graphene layers results in the separation of Dirac cones in the k space and they begin to intersect each other only at a certain energy. According to our estimates, this energy at a misorientation of the graphene layers of about 2° is approximately 150–200 meV and corresponds to L3–L3 transitions between Landau levels at B = 12 T. Figure 2 shows the schemes of possible L1–L1 and L3–L3 transitions between Landau levels with indices of 1 and 3, respectively, at B = 12 T. It is seen that the corresponding orbitals for L3–L3 Landau levels, which are represented as circles on the cones in Fig. 2, intersect in the k space and transitions between them with the conservation of both the energy and the longitudinal momentum component parallel to the layers, whereas the tunneling between lower Landau levels in a magnetic field of 12 T is possible only involving phonons [22] or due to the scattering from the graphene/hBN interface.

Figure 3 presents two I–V_b characteristics of our sample at B = 12 T, T = 2 K, ν = (а) 12 and (b) 8, and such V_tg and V_bg values that n_top = n_bottom ~ (a) 3.46 × 10¹² and (b) 2.3 × 10¹² cm^–2 at V_b = 0. The I–V_b characteristic in Fig. 3a has a shape conventional for resonant tunneling with maxima of the current at V_b ≈ ±7 mV and regions of a negative differential conductivity behind these maxima. By analogy with conventional resonant tunneling, a large peak-to-valley current ratio I_peak/I_valley ~ 2 indicates that the momentum in transitions between Landau levels is conserved with a high accuracy [16] and thereby confirms our estimate that the orbitals corresponding to the L3–L3 tunneling transition between Landau levels in a magnetic field of B = 12 T intersect in the k space at a misorientation of the graphene layers of about 2° (see the inset of Fig. 3a). A further increase in the bias voltage broadens the conduction window both tunneling through high Landau levels and tunneling involving phonons become possible simultaneously [18, 22], which leads to an increase in the current and a wide peak near V_b ~ 30 mV. However, the I–V_b characteristic in Fig. 3b corresponding to tunneling between L2‒L2 Landau levels at low bias voltages does not have the region of a negative differential conductivity, which indicates predominantly inelastic processes of transitions of electrons between these levels and also confirms our estimate. In view of this circumstance, to exclude the influence of inelastic processes and to simplify the interpretation of observed effects, we focus here on the L3–L3 transitions.

Fig. 3.

(Color online) I–V_b characteristics of the sample at B = 12 T, T = 2 K, ν = (а) 12 and (b) 8, and n_top = n_bottom ~ (a) 3.46 × 10¹² and (b) 2.3 × 10¹² cm^–2 at V_b = 0. The insets show schematics of the orbitals of Landau levels in the k space corresponding to tunneling transitions between Landau levels (а) L3–L3 and (b) L2–L2 at V_b = 0 in the magnetic field B = 12 T for the sample with the disoriented crystal lattices of the top and bottom graphene sheets by about 2°.

With a decrease in the temperature at tunneling between L3–L3 Landau levels in high magnetic fields, a dip near V_b = 0 is manifested in the dependences dI/dV_b(V_b). Figure 4a shows such dependences for ν = 12 and n_top = n_bottom ~ 3.46 × 10¹² cm^–2 in the magnetic field B = 12 T at T = (red line) 15 and (blue line) 2 K. The width of the dip at a temperature of 2 K in the magnetic field B = 12 T, which is measured as the distance between the conductivity maxima, was Δ ≈ 6 mV, and it disappeared completely as the temperature was increased to ~15 K. Any dip in the dependences dI/dV_b(V_b) at the same electron densities in low magnetic fields was absent (see, e.g., Fig. 4b). Furthermore, temperature-induced changes in the dependences in Figs. 4a and 4b are crucially different: an increase in the temperature increases the tunneling conductivity near V_b = 0 in the presence of the dip, whereas this increase decreases the conductivity in the absence of the dip and leads to the conventional te-mperature smearing of the resonance due to the broadening of Landau levels. Thus, it can be already concluded that the suppression of the equilibrium tunneling conductivity in the studied graphene heterostructure also has a magnetic-field nature, as in [1–3]. Moreover, as shown below, the observed suppression of magnetic tunneling is due to the Coulomb gap in the tunneling density of states because of the formation of correlated collective states in the two-dimensional electron gas in the graphene layers. Since the theory of such Coulomb gap for graphene heterostructures is currently absent, we sought the nature of the observed effect by comparing the data obtained in this work and the experimental results on the observation of the gap in GaAs heterostructures [1–3], which are closest in experimental implementations and methods and, correspondingly, allow the most direct comparison of the data. The measured gap width Δ ≈ 6 mV for ν = 12 and n_top = n_bottom ≈ 3.46 × 10¹² cm^–2 in the magnetic field B = 12 T is much larger than the values obtained at comparable ν values in [2] and is comparable only with the gap widths measured in [1–3] in the ultraquantum limit for ν < 1/2. Since all early works with in GaAs systems showed that the gap width is described by the expression Δ ≈ Aℏω_c, where A is a constant and ω_c is the cyclotron frequency [1, 2, 9–12] and under the assumption that the mechanism of the formation of the gap is the same in GaAs and graphene, the difference of the gap width Δ measured in this work from that in GaAs can be attributed to a significant difference between the cyclotron masses m_c [23] and, correspondingly, the cyclotron energies ℏω_c = ℏeB/m_c in GaAs and graphene in the region of several low Landau levels. The influence of the interlayer Coulomb interaction between electrons on the gap width Δ is also not excluded [12, 15].

Fig. 4.

(Color online) (a) Bias voltage dependence of dI/dV_b at n_top = n_bottom ~ 3.46 × 10¹² cm^–2 and T = (red line) 15 and (blue line) 2 K (a) for ν = 12 in the magnetic field B = 12 T and (b) for ν = 36 in the magnetic field B = 4 T. The inset shows the gaps for the L1–L1, L2–L2, and L3–L3 transitions.

The interpretation of the gap in previous works [1–3] with GaAs structures is based on the fact that the gap width Δ is the sum of energies spent on the extraction of an electron from one layer of the correlated two-dimensional electron gas and on its introduction in another layer and increases with decreasing $\nu$. Therefore, Δ for tunneling transitions between Landau levels with neighboring indices, e.g. ${{\Delta }_{{12}}}$, is intermediate between ${{\Delta }_{{11}}}$ and ${{\Delta }_{{22}}}$, i.e., ${{\Delta }_{{22}}} < {{\Delta }_{{12}}} < {{\Delta }_{{11}}}$, which is observed for measured ${{\Delta }_{{11}}}$ and ${{\Delta }_{{22}}}$ values (see the inset of Fig. 4a) and ${{\Delta }_{{12}}}$ from the experiment [14].

Similar to [1–3], we analyzed the temperature behavior of the equilibrium tunneling conductivity G₀ = dI/dV_b at V_b = 0, which makes it possible to unambiguously determine the presence/absence of the tunneling gap. The temperature dependences of G₀ = dI/dV_b at V_b = 0 for the L9–L9 and L3–L3 transitions at B = 4 and 12 T, respectively, presented in Fig. 5a demonstrate opposite behaviors at low temperatures up to T ~ 10 K: the conductivity in the case of the L9–L9 transition decreases smoothly with increasing T in the entire temperature range, reflecting the conventional broadening of Landau levels, whereas the equilibrium tunneling conductivity for the L3–L3 transition first increases with the temperature up to T ~ 10 K, reflecting the temperature-induced closure of the tunneling gap, and then decreases slowly, reflecting the conventional broadening of Landau levels. Figure 5b shows the same conductivity G₀(V_b = 0) at B = 12 T for tunneling between L3–L3 Landau levels but as a function of the inverse temperature 1/T. This dependence demonstrates the activation behavior of the equilibrium conductivity G₀ at low temperatures, as in [1–3], which clearly indicates the existence of the gap in the tunneling density of states. The determined activation energy E_a ~ 0.68 meV is approximately an order of magnitude smaller than the gap width Δ determined from the plot in Fig. 4a. A similar ratio Δ/E_a ≈ 14–20 was already observed both in GaAs heterostructures with double quantum wells [1–3] and in graphene structures [14] and was explained by the fact that the activation energy is determined not by the gap width Δ but by a much lower characteristic energy of collective excitations in the system [24, 25]. We observed a similar suppression of the equilibrium tunneling conductivity with decreasing temperature for the L2–L2 and L1–L1 temperatures (see Fig. S4 in the supplementary material). However, as mentioned above, we focus in this work on the L3–L3 transitions with the conservation of the momentum component parallel to the layers.

Fig. 5.

(Color online) (a) Temperature dependence of G₀ = dI/dV_b at V_b = 0 (red) for L9–L9 transitions at B = 4 T and (blue) for L3–L3 transitions at B = 12 T. (b) G₀ = dI/dV_b for the L3–L3 transition versus the inverse temperature 1/T at ν = 12 in the magnetic field B = 12 T.

To conclude, we emphasize that, although all L1–L1, L2–L2, and L3–L3 transitions demonstrate the splitting of Landau levels due to the lifting of degeneracy, as seen in Fig. 2, any section of the map in Fig. S3 along V_b shows also the dip of the conductivity near V_b = 0, which means the coexistence of real gaps in the density of states due to the splitting of Landau levels and the Coulomb “pseudogap” in the tunneling density of states.

To summarize, we have detected the strong magnetic-field-induced suppression of the conductivity near V_b = 0 at tunneling in graphene/hBN/graphene heterostructures and the activation temperature dependence of the equilibrium conductivity, which we have attributed to the Coulomb correlation gap in the tunneling density of states. the increase in the gap width Δ with decreasing $\nu$ is also in agreement with early results for semiconductor systems. However, the determined gap width $\Delta$ is much larger than those measured in all previous works probably because the cyclotron energy in graphene is much higher than that in GaAs in the region of low Landau levels. This result is consistent with the results reported in [14].