Quantum solid phase and Coulomb drag in two-dimensional electron-electron bilayers of MoS2

Coulomb drag experiments can give us information about the interaction state of double-layer systems. Here, we demonstrate anomalous Coulomb drag behaviours in a two-dimensional electron-electron bilayer system constructed by stacking atomically thin MoS2 on opposite sides of thin dielectric layers of boron nitride. In the low temperature regime, the measured drag resistance does not follow the behaviour predicted by the Coulomb drag models of exchanging momenta and energies with the particles in Fermi-liquid bilayer systems. Instead, it shows an upturn to higher and higher values. We investigate quantum solid/fluid phases and the Kosterlitz-Thouless/Wigner two-dimensional quantum melting transition in this bilayer system and describe this interesting phenomenon based on thermally activated carriers of quantum defects from the formation of the correlation-induced electron solid phases with enhanced stabilization by the potential due to the boron nitride dielectric layers.


Introduction
Coulomb drag experiments can give us information about the effect of the long-range Coulomb interaction between charge particles in two closely spaced but electrically isolated lowdimensional electronic systems. [1] The transport characteristics of Coulomb drag are generally reflected by the drag resistance Rdrag, defined as the ratio of the drag voltage Vdrag (in the passive layer) to the drive current Idrive (in the active layer) as illustrated in Figure 1a. Coulomb drag measurements can directly show the interactions of charged particles both between the two coupled electronic systems and inside them, [1] and have become part of the standard toolbox in condensed matter physics widely used to investigate various properties in low dimensional electronic systems such as quantum metal-insulator transition, [2][3][4][5] exciton effects [6][7][8][9][10][11][12] and quantum coherence. [13,14] In two-dimensional Fermi liquid (FL) systems [15][16][17][18] (e.g., semiconductor heterostructures of GaAs double-quantum wells and graphene double layers), the dominant drag mechanism is the momentum transfer between carriers in the two layers by mutual Coulomb scattering: Rdrag is usually proportional to the electron-electron momentum relaxation time [16] and should vanish as T 2 when the temperature T goes to zero. However, in semiconductor bilayers with larger interlayer spacing d, only very small momentum can be transferred. In this condition, as much as 30% of the measured Rdrag was ascribed to phonon-mediated interactions, where both energy and momentum transfer takes place. [1] In graphene bilayers (separated by an atomically thin boron nitride (BN) dielectric layer), long-range Coulomb interaction between charge particles is largely enhanced and the friction drag dominates over the phonon drag, resulting in giant momentum transfer between the two layers. [13,18,19] However, due to the high Fermi velocity and small effective mass, the Bohr radius = 90 Å for the graphene system is comparable to that in GaAs. The dimensionless parameter for distance, the Wigner Seitz radius that is measured relative to , is extremely small for experimentally accessible electron densities in graphene. [20] The physics is dominated by the quantum kinetic energy that is proportional to 1/ 2 . It is therefore difficult to study the physics of the effect of the Coulomb interaction which is proportional to 1/ and becomes important at low densities. Transition metal dichalcogenide (TMDC) semiconductors have geometries similar to that of graphene and possess many valuable properties (e.g., large effective masses and strong electronelectron interactions with = 6 Å for the experimental TMDC systems). TMDCs process a of the order of 10 for the experimentally accessible electron densities ( 10 12 /cm 2 ), [21] one order of magnitude larger than that in monolayer graphene, thus opening up new dimensionless density regimes to explore Coulomb drag properties experimentally. In this work, we demonstrate anomalous drag behaviours in MoS2/BN/MoS2 electron-electron bilayers. At cryogenic temperatures, the measured drag resistance Rdrag neither vanishes nor follows the T 2 behaviour.
Instead, it shows an upturn to higher and higher values. We attribute the unexpected Coulomb drag behaviour to the formation of bilayer quantum solid phases of electrons and their possible stabilization by the potential due to the BN dielectric material.

Experimental Section
Atomically thin bilayer devices, as shown in Figure 1b and 1c, were constructed through the mechanical exfoliation and dry transferring method developed previously. [22] Few-layer MoS2 flakes (4L ~ 8L) were exfoliated onto SiO2/Si substrates, separated by a thin potential barrier layer of BN (5nm thick) and sandwiched by thicker BN (1020nm thick) layers. Transport measurements were performed in a cryogenic system which provided stable temperatures ranging from 1.5 to 300 K. The top and bottom gates, Vtg and Vbg, controlled the carrier densities in the two layers (dual-gate tunable layer resistance can be found in Figure S1, we have set the chemical potential of the fluid/solid phases to zero as estimated from the gate voltages and the corresponding electron densities). An electrical current Idrive was injected into the bottom active layer, and the Coulomb drag voltages Vdrag were measured from the top passive layer. The Vdrag-Idrive characteristic ( Figure S2) and the interlayer leakage current ( Figure S3) are checked to confirm the validity of the drag measurements. We first measure the drag resistance Rdrag versus Vtg (or equivalently the carrier density in the layer) at a fixed Vbg. Figure 1d shows a typical Rdrag obtained at 1.5K. Strikingly, Rdrag reaches a value of ~2.5MΩ at the 1:1 ratio of the carrier densities top = bottom = = 3 × 10 12 cm −2 , which is three orders of magnitude larger than ~1kΩ when the carrier densities of two layers mismatch (as shown in inset of Figure 1d). One possible explanation for this Rdrag peak is the formation of the bilayer quantum solid phase. Previous studies show that when the Coulomb interaction between electrons dominates their kinetic energy, electrons can condense into a close-packed lattice at a low enough temperature and result in a quantum solid phase. [23][24][25] In our case, when the layer densities are about 1:1 matched, electrons in two layers can form solids that are commensurate and most effectively and completely coupled. The charge carriers are frozen out and resulting in an extremely high Rdrag. The carrier density and temperature dependence of the drag signal are then performed to study the details of the drag behaviour.  (Figure 2b). At the low carrier density region (1-2×10 12 /cm 2 ), Rdrag increases exponentially with decreasing carrier density. We explained this based on the picture of thermally activated quantum defects in the quantum solid state. In the solid state, the free electrons are frozen, and the carriers can come from thermally activated quantum defects. [26] The effect of such defects was first emphasized by Kosterlitz and Thouless, who proposed that in two dimensions the solid phase can be characterized by a finite shear modulus, even though there is no long-range order. [27] The zero shear modulus in the fluid phase occurs when dislocation pairs in the solid phase become unbound. A different idea of the stabilization of the solid came from Wigner, who pointed out that the potential energy gained is more than the quantum kinetic energy lost for the solid at low densities. [28] We found that defects such as dislocations in electron solids are quantum and form waves with an effective mass of the order of the electron mass. [26] The energy of a dislocation in the electron solid can be lowered by coupling to the zero point motion of the phonons. [29] This lowering becomes big at low densities, thus combining the Wigner idea and the Kosterlitz Thouless picture. An example of a defect responsible for the transport is a dislocation pair, [26,29] with an energy approximately given by ∆≅ 2 1.9 [0.048 (1 − 0 ) + 0.021] for a constant 0 . The exponential increase of Rdrag can be a result of the term of ∆ that is proportional to 1/ 2 ; as the density n is decreased, the solid defect energy is strengthened due to the decrease of the quantum kinetic energy from the zero point motion. The carrier density decreases and Rdrag is increased.   Figure   3c). We explained this anomalous Rdrag drag on the formation of the correlation-induced solid phases as shown below. Second, when the temperature is higher than the melting temperature (50~175K), the system transits into a quantum fluid phase. In this regime, our experimental results can be well fitted by the diffusive model, [1] demonstrating a robust drag signal from momentum transfer between the MoS2 layers. Further increasing the temperature (175~300K), the approximation that the temperature is much less than the Fermi temperature becomes invalid.
Thermally excited quasiparticles and plasmons coexist and give additional contributions to the drag signal, thus the observed Rdrag is slightly higher than the value predicted by the diffusive model.

Discussion and Modelling
The anomalous upturn of the drag resistance has been observed in electron-hole bilayer systems such as GaAs-AlGaAs [6] and graphene-GaAs [30] where electron-hole pairing and possible exciton condensation play important roles. However, the mechanism of this anomalous behavior in our system should be totally different from theirs, as electron-hole pairing cannot happen in our electron-electron system. We attribute the anomalous Coulomb drag behavior at low temperatures to the formation of MoS2 two-dimensional bilayer crystal phases of electrons. The low-temperature dependence in Figure 3b can be understood as follows. As we explained, we expect the transport to be carried by thermally activated defects of a density that is proportional to −∆/ . This results in a Rdrag~∆ / . As shown in Figure 4, our experimental results can be well fitted by this formula, demonstrating the formation of the inter-layer solid phase in our sample. To improve our understanding, we have performed numerical studies of our system with no adjustable parameters as described below. We have calculated the quantum solid and fluid energies and their transition in this bilayer system with fixed node diffusion Monte Carlo method, [31,32] taking into account the realistic experimental structure including the screening effect [33][34][35] of the BN spacer, the encapsulation and the substrate. [36] As mentioned previously, the smaller Bohr radius for the TMDC systems favors the formation of the crystal electronic phases at a much higher electron density. Previous numerical [37][38][39][40] and analytical studies [32] showed that the interlayer Coulomb interaction enhances the formation of the Wigner crystal phases and extends the maximum solidification density. Those numerical studies were motivated by the study of GaAs heterostructures and did not correspond to the experimentally available parameter range of the TMDC systems. However, compared to the formation of intra-layer Wigner crystals, a coupled bilayer system has an enhanced Coulomb potential energy which is favorable to the formation of inter-layer crystal phases. Here, we consider one more favourable factor, the BN spacer with thickness d stacked between the two MoS2 layers which is critical for the formation of the coupled bilayer device structure.
The energies of the solid and the fluid phases for a system with 30 particles on each side of the structure for different densities and BN spacer thicknesses d are calculated. In Figure 5a, we show the difference between the solid and the fluid energies as a function of rs that is defined in terms of the density of each layer for d = 5nm. We have also carried out calculations with the effect of the BN potential by considering static Coulomb potential from the periodic array of boron and nitrogen ions (details of the calculation process can be found in our previous paper [41] ). The energy difference between the solid and the fluid phases is shown in Figure 5b. From where the energy difference is equal to zero, we obtain the phase boundary of the quantum melting transition of the present system and the critical Wigner Seitz radius rsc. The critical rsc for three different cases are shown in Figure 5c. As expected, the bilayer system tends to stabilize the Wigner solid phase, resulting in a transition at a dimensionless distance rsc ≈ 8.4, much smaller than the value of 30 for the single-layer TMDC case. In addition, the consideration of the BN potential results in a rsc ≈ 7.4, the smallest value in these cases. The stabilization of the Wigner solid phase here is partly due to the interlayer coupling effect.

Conclusion
In conclusion, we demonstrate that the Coulomb drag resistance measured from atomically thin MoS2 bilayers shows an upturn when decreasing the temperature, very different from the Coulomb drag models of exchanging momenta and energies with the particles in Fermi-liquid bilayer systems. We describe the anomalous Coulomb drag behaviours based on the properties of the quantum defects of the electron solid and their possible stabilization by the potential caused by the BN dielectric layers.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.