Absence of inter-layer tunnel coupling of $K$-valley electrons in bilayer MoS$_{2}$

TL;DR

This study uses magnetotransport experiments to show that in high-quality bilayer MoS$_{2}$, the two layers act as independent electronic systems with negligible interlayer tunnel coupling, highlighting the dominance of electrostatic interactions.

Contribution

It provides direct experimental evidence that bilayer MoS$_{2}$ exhibits weak interlayer tunnel coupling, contrasting with behaviors seen in bilayer graphene, and emphasizes the role of electron-electron interactions.

Findings

Layers behave as independent electronic systems.

Large negative compressibility at bottom layer population.

Weak interlayer tunnel coupling observed.

Abstract

In Bernal stacked bilayer graphene interlayer coupling significantly affects the electronic bandstructure compared to monolayer graphene. Here we present magnetotransport experiments on high-quality $n$-doped bilayer MoS$_{2}$. By measuring the evolution of the Landau levels as a function of electron density and applied magnetic field we are able to investigate the occupation of conduction band states, the interlayer coupling in pristine bilayer MoS$_{2}$, and how these effects are governed by electron-electron interactions. We find that the two layers of the bilayer MoS$_{2}$ behave as two independent electronic systems where a two-fold Landau level's degeneracy is observed for each MoS$_{2}$ layer. At the onset of the population of the bottom MoS$_{2}$ layer we observe a large negative compressibility caused by the exchange interaction. These observations, enabled by the high electronic quality of our samples, demonstrate weak interlayer tunnel coupling but strong interlayer electrostatic coupling in pristine bilayer MoS$_{2}$. The conclusions from the experiments may be relevant also to other transition metal dichalcogenide materials.

Figures

Equations

$${\frac{d{D}_{\mathrm{BL}}}{d{D}_{\mathrm{B}}}|}_{V_{\mathrm{TG}}}=\frac{C_{\mathrm{BL}}}{C_{\mathrm{B}}}\times$$

$$\times \frac{C_{\mathrm{B}}\left(C_{\mathrm{T}}+e{\frac{d{n}_{\mathrm{t}}}{d{V}_{\mathrm{BG}}}|}_{V_{\mathrm{TG}}}\right)-C_{\mathrm{T}}e{\frac{d{n}_{\mathrm{b}}}{d{V}_{\mathrm{BG}}}|}_{V_{\mathrm{TG}}}}{C_{\mathrm{BL}}\left(C_{\mathrm{T}}+e{\frac{d(n_{\mathrm{t}}+n_{\mathrm{b}})}{d{V}_{\mathrm{BG}}}|}_{V_{\mathrm{TG}}}\right)+C_{\mathrm{T}}e{\frac{d{n}_{\mathrm{b}}}{d{V}_{\mathrm{BG}}}|}_{V_{\mathrm{TG}}}},$$

Full text

Of the multitude of two-dimensional (2D) host materials, transition metal dichalcogenides (TMDs) are promising candidates for exploring quantum correlated electronic phases and electron-electron interaction effects due to their intrinsic 2D nature, large spin-orbit interaction and large effective mass carriers. Molybdenum disulfide (MoS2) is one of the most widely studied TMDs and still most of its fundamental quantum electronic properties have thus far been elusive. Contrary to monolayer MoS2, in pristine bilayer MoS2 inversion symmetry is restored Xiao et al. (2012); Mak et al. (2012); Cao et al. (2012). As a result, the orbital magnetic moment and the valley-contrasting optical dichroism vanish Yao et al. (2008); Xiao et al. (2012). A potential difference between the two layers breaks the inversion symmetry  Wu et al. (2013); Lee et al. (2016). The influence of a perpendicular electric field on bilayer MoS2 has been extensively probed by optical excitation Wu et al. (2013); Lee et al. (2016). Very little is known about the electronic transport properties of bilayer MoS2 when electric and magnetic fields are both applied perpendicular to the sample plane Lin et al. (2018). Magnetotransport studies of 2D holes have been recently performed in bilayer WSe2 revealing the presence of two subbands, each localized in the top and bottom layer, and demonstrating an upper bound of the interlayer tunnel coupling of $19\,\mathrm{meV}$ Fallahazad et al. (2016); Movva et al. (2017). A thorough study of the interlayer coupling in the conduction band of bilayer transition metal dichalcogenides is still missing Lin et al. (2018); Larentis et al. (2018).

Here we report a magnetotransport study of electrons in the conduction band of dual-gated bilayer MoS2. All studied bilayer samples exhibit Shubnikov-de Haas (SdH) oscillations with a twofold Landau level degeneracy at $T=1.5\,\mathrm{K}$. At lower temperature the valley degeneracy is lifted and spin-valley coupled Landau levels are resolved. The evolution of the Landau level spectrum as a function of density indicates that electrons occupy states of the $K$ and $K^{\prime}$ valley in each layer. By tuning the Fermi energy in each layer individually we are able to populate lower and upper spin-orbit split bands in both layers. The exchange interaction in a single layer yields a pronounced negative compressibility visible in occupation of the states detected via the Landau fan diagram. In addition, we observe an intricate interplay between spin- and valley-polarized Landau levels originating from the two decoupled MoS2 layers. We do not observe any obvious signature in the Landau level spectrum when the electrostatic potential difference between the two layers vanishes and the structural inversion symmetry is expected to be restored.

Figure 1(a) shows the schematic cross-section of the dual-gated bilayer MoS2 device under study. The MoS2 is encapsulated between two hBN dielectrics with graphite layers as top and bottom gates. We fabricate pre-patterned Au bottom contacts below the bilayer MoS2. Ohmic behavior of these contacts is achieved by applying a sufficiently positive top gate voltage ($V_{\mathrm{TG}}$). The heterostructure is assembled using a dry pick-up and transfer method Pisoni et al. (2017, 2018a, 2018b). We fabricated and measured three bilayer MoS2 samples, labeled A, B, and C, which show the same behavior. We will mainly discuss samples A and B here.

The samples use commercial bulk MoS2 crystal (SPI supplies) mechanically exfoliated on SiO2/Si substrates. Using a combination of optical contrast, photoluminescence spectroscopy and atomic force microscopy bilayer MoS2 flakes are identified. Figure 1(b) shows the optical micrograph of sample A. The bilayer MoS2 flake is outlined with a white dashed line. Top and bottom graphite gates are outlined in purple and top and bottom hBN are outlined in blue and cyan, respectively. In the inset of Fig. 1 (b) we sketch the contact geometry where contacts 1 and 4 are used for current injection and extraction and contacts 2 and 3 serve as voltage probes.

Figure 1(c) shows the four-terminal resistance, $R_{\mathrm{14,23}}$, as a function of $V_{\mathrm{BG}}$ and $V_{\mathrm{TG}}$ at $T=1.5\,\mathrm{K}$. Green solid lines denote specific resistance values as a function of $V_{\mathrm{TG}}$ and $V_{\mathrm{BG}}$. At fixed $V_{\mathrm{TG}}$ we observe a sudden increase in $R_{\mathrm{14,23}}$ at $V_{\mathrm{BG}}\approx 1\,\mathrm{V}$ (white dashed line) that we attribute to the population of the bottom MoS2 layer. As a result, for the $V_{\mathrm{BG}}$ values on the left side of the white dashed line we probe the electron transport only through the top MoS2 layer. At $V_{\mathrm{BG}}\approx 3.4\,\mathrm{V}$ (white dotted line) and for $V_{\mathrm{TG}}$ and $V_{\mathrm{BG}}$ values along the white solid line, we observe additional resistance kinks that we attribute to the occupation of the upper spin-orbit split bands in the bottom and top MoS2 layers, respectively.

We investigate magnetotransport phenomena in bilayer MoS2 using lock-in techniques at $31.4\,\mathrm{Hz}$. Figure 1(d) shows $\Delta R_{\mathrm{14,23}}$, the four-terminal linear resistance with a smooth background subtracted, as a function of magnetic field $B$ at various temperatures ranging from $1.5\,\mathrm{K}$ to $4.5\,\mathrm{K}$ and a density $n=3.8\times 10^{12}\,\mathrm{cm}^{-2}$. For $T=1.5\,\mathrm{K}$ SdH oscillations start at $\approx 3\,\mathrm{T}$. At $T=100\,\mathrm{mK}$ the onset of SdH oscillations moves to yet lower magnetic fields yielding a lower bound for the quantum mobility of $\approx 5000\,\mathrm{cm}^{2}/\mathrm{V}\mathrm{s}$ (see Fig. 4). The electron density is calculated from the period of the SdH oscillations in $1/B$ considering valley degenerate Landau levels at the $K$ and $K^{\prime}$ conduction band minima Pisoni et al. (2018b). At $T=1.5\,\mathrm{K}$ and $n=3.8\times 10^{12}\,\mathrm{cm}^{-2}$ we observe the sequence of odd filling factors $\nu=21,23,25,...$. The twofold Landau level’s valley degeneracy is lifted at lower temperatures (see below).

We determine the electron effective mass $\,m^{\mathrm{\ast}}$ from the temperature dependence of the SdH oscillations by fitting $\Delta R_{\mathrm{14,23}}$ to $\varepsilon/\sinh(\varepsilon)$, where $\varepsilon=2\pi^{\mathrm{2}}\,k_{\mathrm{B}}T/\,\hbar\omega_{\mathrm{c}}$ and $\,\hbar\omega_{\mathrm{c}}=eB/\,m^{\mathrm{\ast}}$ is the cyclotron frequency Ando et al. (1982); Isihara and Smrcka (1986); Pudalov et al. (2014). The inset of Fig. 1 (c) shows the extracted $\,m^{\mathrm{\ast}}$ for the three different samples. We obtain a density-averaged mass of $\,m^{\mathrm{\ast}}\approx 0.62\,m_{\mathrm{e}}$ which does not show any obvious dependence neither on $n$ nor on $B$ Zhang and Das Sarma (2005); Attaccalite et al. (2002).

In Fig. 1 (d) we extract the $\,m^{\mathrm{\ast}}$ of the $K$ and $K^{\prime}$ electrons localized in the top MoS2 layer, thus effectively calculating the effective mass of a monolayer MoS2 Lin et al. (2018). The effective masses we extract in our bilayer samples are systematically $10-20\%$ lower compared to the ones measured in monolayer MoS2 Pisoni et al. (2018b). In bilayer MoS2 the top MoS2 layer is encapsulated between hBN and the bottom MoS2 layer, which is devoid of electrons. We speculate that the higher dielectric constant ($\epsilon\approx 6.4$) reported for monolayer MoS2 Cheiwchanchamnangij and Lambrecht (2012); Ramasubramaniam (2012); Molina-Sánchez and Wirtz (2011); Laturia et al. (2018) compared to hBN ($\epsilon\approx 3.5$) causes a weakening of electron-electron interaction effects thus affecting the $\,m^{\mathrm{\ast}}$ value.

The interaction strength is characterized by the dimensionless Wigner-Seitz radius $r_{\mathrm{s}}=1/(\sqrt{\pi\,n}a^{\ast}_{\mathrm{B}})$, where $a^{\ast}_{\mathrm{B}}=a_{\mathrm{B}}(\kappa m_{\mathrm{e}}/m^{\ast})$ is the effective Bohr radius and $\kappa$ the dielectric constant. For the considered electron density range we estimate $r_{\mathrm{s}}=1.9-10$, placing the system in a regime where interaction effects are important  Okamoto et al. (1999); Shashkin et al. (2001); Vakili et al. (2004).

Figure 2(a) shows $\Delta R_{\mathrm{14,23}}$ (color scale) as a function of $B$ applied perpendicular to the sample and $V_{\mathrm{BG}}$, at $V_{\mathrm{TG}}=9\,\mathrm{V}$ and $T=1.5\,\mathrm{K}$. For $V_{\mathrm{BG}}<1.3\,\mathrm{V}$ (black arrow in Fig. 2 (a)) the Landau level’s evolution as a function of $V_{\mathrm{BG}}$ resembles the one of monolayer MoS2 Pisoni et al. (2018b). For $V_{\mathrm{BG}}<-1.5\,\mathrm{V}$ only the Landau levels of the lower spin-orbit split $K_{\uparrow}$ and $K^{\prime}_{\downarrow}$ bands are seen. As the electron density increases in this regime, we observe an alternating parity of the filling factor sequence [see filling factor sequences in Fig. 2 (a)]. These results can be explained in an extended single-particle picture where the valley $g$ factor is density dependent, following the interpretation of previous works Lin et al. (2018); Fallahazad et al. (2016); Movva et al. (2017); Gustafsson et al. (2017); Larentis et al. (2018).

At $V_{\mathrm{BG}}=-1.5\,\mathrm{V}$ there is a sudden change in the slope of the Landau fan related to the occupation of the higher spin-orbit split $K_{\downarrow}$ and $K^{\prime}_{\uparrow}$ valleys. Where the slope changes, the electron density is $n=3.6\times 10^{12}\,\mathrm{cm}^{-2}$. Assuming two-fold valley-degeneracy and using the experimentally determined electron effective mass, we calculate the Fermi energy to be $E_{\mathrm{F}}=14\,\mathrm{meV}$, in good agreement with the intrinsic spin-orbit interaction measured previously for $K$-valley electrons in monolayer MoS2 Pisoni et al. (2018b). We would like to note that our results justify the assumptions in Lin et al. (2019) that bilayer MoS2 investigated in the right regime behaves as single-layer MoS2 with the caveat that the effective mass is different because of the dielectric environment.

The measured Landau level structure for $V_{\mathrm{BG}}<1.2\,\mathrm{V}$ fully agrees with our previous monolayer MoS2 results Pisoni et al. (2018b). For $V_{\mathrm{BG}}$ crossing the voltage $1.2\,\mathrm{V}$ from below, we observe two important changes in the SdH oscillations compared to the monolayer system. First, the slope of the SdH oscillation minima vs $V_{\mathrm{BG}}$ that existed below this threshold changes sign from positive to negative. Second, an additional set of Landau levels appears [blue dashed lines in Fig. 2 (a)]. At $V_{\mathrm{BG}}=3.4\,\mathrm{V}$ the slope of these secondary Landau levels changes by about a factor of 2, as indicated with white dashed lines in Fig. 2 (a).

To interpret these observations, we extract how the electron densities change as we tune $V_{\mathrm{BG}}$. To determine the electron density of the individual layers and bands from the Landau fan diagram we generate a Fourier transform map of $\Delta R_{\mathrm{14,23}}$ vs. $1/B$ for each $V_{\mathrm{BG}}$ value in Fig. 2 (a) (see supplemental information). The Fourier transform of the SdH oscillations shows multiple peaks in the amplitude spectrum as we increase $V_{\mathrm{BG}}$. From these peaks we extract the electron density of the various spin-orbit split bands in bilayer MoS2 using $n=(g_{\mathrm{v}}e/h)\times f$, where $f$ is the frequency of the Fourier transform peaks and $g_{\mathrm{v}}=2$ accounts for the valley degeneracy. The results of this procedure are shown in Fig. 2 (b). For $V_{\mathrm{BG}}<1.2\,\mathrm{V}$ electrons populate only the top MoS2 layer where they occupy the lower (green dashed line) and upper (orange dashed line) spin-orbit split bands as we increase $V_{\mathrm{BG}}$. At $V_{\mathrm{BG}}=1.2\,\mathrm{V}$ the bottom MoS2 layer starts to be populated (blue dashed line). The secondary Landau fan that appears at $V_{\mathrm{BG}}=1.2\,\mathrm{V}$ in Fig. 2 (a) originates from the Landau levels of the electrons populating the bottom MoS2 layer. Beyond $V_{\mathrm{BG}}=1.2\,\mathrm{V}$ we see an increasing density in the bottom layer, whereas the top layer density starts to decrease. This density decrease is direct experimental evidence for the negative compressibility of the bottom layer at low densities Eisenstein et al. (1992); Bello et al. (1981); Tanatar and Ceperley (1989); Kravchenko et al. (1990). At $V_{\mathrm{BG}}>3.4\,\mathrm{V}$ the two valleys of the upper spin-orbit split bands in the bottom layer start to be populated (cyan dashed line).

In the following, we will quantify this negative compressibility effect in MoS2 based on the data in Fig. 2. To this end, we consider the electrostatic model schematically displayed in the inset of Fig. 3 consisting of three layers of different dielectric constants, in which electric displacement fields exist due to the applied voltages $V_{\mathrm{TG}}$ and $V_{\mathrm{BG}}$. The MoS2 bilayer is modeled as two grounded conducting planes of finite density of states with a geometric capacitance ${C_{\mathrm{BL}}}$ and a displacement field ${D_{\mathrm{BL}}}$ between them. It is our goal to express $d{D_{\mathrm{BL}}}/d{D_{\mathrm{B}}}$, i.e., the change in ${D_{\mathrm{BL}}}$ upon a change in the displacement field ${D_{\mathrm{B}}}$ between back gate and MoS2, at constant top gate voltage in terms of the measured $V_{\mathrm{BG}}$-dependent changes of the layer densities. This quantity allows us to directly compare the strength of the effect with the results obtained by Eisenstein et al Eisenstein et al. (1992) in the case of a GaAs double quantum well, and with the numerical results of Tanatar and Ceperley Tanatar and Ceperley (1989).

The model (see supplemental material for details) results in

[TABLE]

[TABLE]

where ${C_{\mathrm{T}}}$ and ${C_{\mathrm{B}}}$ are the geometric capacitances per unit area between MoS2 and top- and bottom-gate, respectively. The quantities ${n_{\mathrm{t}}}$ and ${n_{\mathrm{b}}}$ are the measured total electron densities in the two layers shown in Fig. 2 (b). In the case of $V_{\mathrm{BG}}<1.2\,\mathrm{V}$, where ${n_{\mathrm{b}}}$ and its $V_{\mathrm{BG}}$-derivative are zero, the displacement field ratio in eq. (1) is exactly one. Negative compressibility in the region ${V_{\mathrm{BG}}}>1.2\,\mathrm{V}$ will manifest itself by $d{D_{\mathrm{BL}}}/d{D_{\mathrm{B}}}<0$.

Figure 3 displays the result of applying eq. (1) to the data in Fig. 2 (b). Two curves are shown, in which the quite uncertain value of ${C_{\mathrm{BL}}}$ takes on two plausible extreme values (see supplemental information for details). This shows that the result depends very little on the exact value of this parameter. A strong negative compressibility with $d{D_{\mathrm{BL}}}/{D_{\mathrm{B}}}\approx-0.5$ is seen at $V_{\mathrm{BG}}=-2\,\mathrm{V}$ where ${n_{\mathrm{b}}}=1\times 10^{12}\,\mathrm{cm^{-2}}$, roughly ten times stronger than the effect observed in Ref. Eisenstein et al. (1992). To compare the value to the numerical results of Ref. Tanatar and Ceperley (1989), we have added an estimated scale bar of $r_{\mathrm{s}}$-values to the top axis in Fig. 3. The negative compressibility values measured in our sample agree fairly well with the predictions of the numerical calculations at these $r_{\mathrm{s}}$-values.

Resolving individual layer electron densities in Fig. 2 indicates that the two MoS2 layers are weakly coupled. This observation is in contrast to Bernal stacked bilayer graphene, where the interlayer coupling of $\approx 0.4\,\mathrm{eV}$ strongly affects the energy momentum dispersion compared to monolayer graphene Zhang et al. (2008). Previous work probing hole transport in bilayer WSe2 reported an upper bound for the interlayer tunnel coupling of $\approx 19\,\mathrm{meV}$ Fallahazad et al. (2016). In our results the interlayer coupling in the conduction band of bilayer MoS2 is not observable. We achieve same electron densities in both layers (red circle in Fig. 2 (b)) for three different samples with no experimental evidence for interlayer coupling. Band structure calculations Kormányos et al. reveal that strong interlayer hybridization in the conduction band of MoS2 occurs predominantly from orbitals which are responsible for the minima at the $Q$-point, which are not occupied in our samples. Conversely, weak interlayer hybridization is expected from the orbitals forming the $K$-valleys, which is consistent with our experimental observations.

At lower temperatures finer details of the Landau level structure are resolved. In Fig. 4 (a) we show $\Delta R_{\mathrm{14,23}}$ as a function of $B$ and $V_{\mathrm{BG}}$ at $V_{\mathrm{TG}}=13.5\,\mathrm{V}$ and $T=100\,\mathrm{mK}$ for sample B. For $V_{\mathrm{BG}}<2\,\mathrm{V}$ (white dashed line) the bottom MoS2 layer is devoid of electrons and we only observe the Shubnikov-de Haas oscillations of the top layer. At $T=100\,\mathrm{mK}$ we are able to resolve valley-spin polarized Landau levels originating from the lowest conduction band minima in the top layer. The Landau level structure of the spin-valley coupled bands in the bottom layer appears for $V_{\mathrm{BG}}\geq 2\,\mathrm{V}$. Figures 4(b-c) show an enlargement of Fig. 4 (a). When only the top layer is populated [see Fig. 4 (b)] we observe a pattern of avoided crossings, a signature of the coupling between the spin-valley polarized Landau levels of the lower and upper spin-orbit split bands Pisoni et al. (2018b). Figure 4(c) shows that Landau levels of electrons populating the two different layers cross each other, indicating weak coupling between the two electronic systems below our measurement resolution.

In conclusion, we observe SdH oscillations at magnetic fields as low as $2\,\mathrm{T}$ at $T\approx 100\,\mathrm{mK}$, testifying to the high-mobility of our dual-gated bilayer MoS2 devices. We are able to measure spin-valley polarized LLs originating from the lower and upper spin-orbit split bands of $K$-valley electrons populating the top and bottom MoS2 layers. Our observations demonstrate that electrons in bilayer MoS2 behave like two independent electronic systems. The exchange interaction at the turn on of the two-dimensional electron gas in the bottom layer leads to the observation of a large negative compressibility. Our work demonstrates fundamental electronic transport properties as well as the importance of interaction effects in pristine bilayer MoS2. These results bear relevance for understanding electronic transport in twisted bilayer TMDs.