Structural semiconductor-to-semimetal phase transition in two-dimensional materials induced by electrostatic gating

Dynamic control of conductivity and optical properties via atomic structure changes is of technological importance in information storage. Energy consumption considerations provide a driving force towards employing thin materials in devices. Monolayer transition metal dichalcogenides are nearly atomically thin materials that can exist in multiple crystal structures, each with distinct electrical properties. By developing new density functional-based methods, we discover that electrostatic gating device configurations have the potential to drive structural semiconductor-to-semimetal phase transitions in some monolayer transition metal dichalcogenides. Here we show that the semiconductor-to-semimetal phase transition in monolayer MoTe2 can be driven by a gate voltage of several volts with appropriate choice of dielectric. We find that the transition gate voltage can be reduced arbitrarily by alloying, for example, for MoxW1−xTe2 monolayers. Our findings identify a new physical mechanism, not existing in bulk materials, to dynamically control structural phase transitions in two-dimensional materials, enabling potential applications in phase-change electronic devices.

between the simulation and model prediction, estimated by dividing the difference between the simulation and model prediction for the energy difference of 2H and 1T' phases by the excess charge transferred to 2H-MoTe 2 . In panels c and d, the Cs substrate consists of four layers for the red curves and six layers for the blue curves. Therefore, monolayer TMDs with small energy difference between charge-neutral 2H and 1T' phases should be selected to display the phase transition driven by electrostatic gating.
where is the ground-state energy of an electrically neutral monolayer, the integral is the energy of moving charge from the reference plane to the Fermi level of the monolayer, and ∆ , is the electronic potential difference between the reference plane and the surface of the material defined as . We take to be the plane at which the plane-averaged Kohn-Sham potential is equal to Fermi level, as shown in Supplementary Fig. 1. This position changes with the charge in the monolayer, i.e. = ′ , defined via: The potential difference ∆ , in equation (1) is, where is the electrostatic potential generated by the uniform compensating background charge − and is the area of the monolayer.
The background potential in equation (3) is given by where is the size of the computational cell along c, interlayer axis. Equation (4) assumes that the monolayer is placed at = 0.
The third and fourth terms in equation (3)  Secondly, a reference plane = is chosen. For monolayer TMDs, this distance is chosen to be 17.25 Å. Thirdly, for each value of , the potential difference between the reference plane at and the surface of the material ∆ , can be calculated using equation (3). Fourth, first-degree polynomial fitting is performed using a linear least squares regression method to get an analytical expression for ∆ , as a function of , which can be written as: where and ′ are fitting coefficients. Supplementary Fig. 1 shows that at a large value of = = 17.25 Å, the potential difference between and (blue dashed line) is primarily determined by the uniform electric field, which is in direct proportion to . Therefore, linear fitting is an appropriate first-order approximation.
Supplementary Fig. 2 shows the fitting results of Equation (5) for 2H-and 1T'-MoTe 2 , and the root-mean-squared error is smaller than 0.08 V. The discontinuity at ′ = 0 in semiconducting 2H-MoTe 2 is due to the band gap.
Fifth, the potential difference at any other reference plane , ∆ , , can be computed from equation (5) using the uniform electric field: Last, the full energy of the charged monolayer, , , can be calculated using equation (1) and (6):

Supplementary Note 2: Test case of a monolayer MoTe 2 on top of a Cs substrate.
In the main text, we have discussed how to calculate the energy To compute the energy difference between phases, we first define some notation. Let the energy of the Cs substrate computed at the H lattice constants E(Cs H ), and the total energy of the system consisting of 2H-MoTe 2 on Cs substrate is E(2H-MoTe 2 + Cs H ). We adopt similar notation for 1T'-MoTe 2 . Because the Cs substrate is strained to fit into the computational cell, it has nonzero strain energy, which is defined as the energy difference between the strained Cs substrate and a zero-stress Cs substrate. The energy of a zero-stress Cs substrate is denoted as . We take the energy difference where the strain energy of the Cs substrate has been subtracted.
Plotted in Supplementary Fig. 3, panel b is the averaged electrostatic potential along the vacuum direction obtained from DFT simulation. A uniform electric field between the monolayer and the substrate is generated by the charge transferred between them, where is monolayer area.
Using the approach we have proposed to make model predictions for such a system  (7): where is the distance from the center of MoTe 2 to the surface of the uniform electric field region and is the distance from the surface Cs atom to the other surface of the uniform electric field region, as shown in Fig. 2 of main text. Therefore, the separation between the center of MoTe 2 and the surface Cs atoms can be written as The total energy of the system , can be computed using Equation (1) of the main text: Using equations (10) and (11), we can further compute: As discussed in the main text, the equilibrium charge transferred, , can be computed through minimization of the total energy , (equivalent to the grand potential since no gate voltage is applied). After which, we can further compute the total energy at equilibrium: The model prediction results from equation (14) and DFT simulation results are compared in Supplementary Fig. 3, panels c and d. Plotted in Supplementary Fig. 3

Supplementary Note 3: Distance parameters of TMDs.
The distance between a monolayer TMDs and the substrate, , may vary over a few Angstroms 10,11 . DFT-based calculations of the average separation between the Mo layer of monolayer MoS 2 and the top layer of substrate has been reported to vary, e.g. from 3.57 Å in the case of Ti-substrate to 4.21 Å in the case of Au-substrate 12 . When identifying appropriate parameters for monolayer TMDs, one can refer to distance parameters in bulk TMDs. Some distance parameters in bulk TMDs in the 2H phase are labeled in Supplementary Fig. 5. Parameter is the distance between the centers of two neighboring layers. Parameter is the distance from the transition metal atom centers to the chalcogenide atom centers. Parameter is the distance from the transition metal atom centers of one layer to the chalcogenide atom surfaces of the neighboring layer. Parameter represents the distance from the chalcogenide atom centers to the chalcogenide atom surfaces, and is estimated from empirically determined atomic radii 5 . The values used for these parameters are listed in Supplementary Table 1. The parameters for 1T'/1T phases are taken to be the same as those for 2H phase. For the capacitor structure shown in Fig. 4(a) in main text, is chosen to be = − − (see values in Supplementary Table 1).
The dependence of transition voltages on the separation is explored in Supplementary   Fig. 6 ( = ). Supplementary Fig. 6 shows that the magnitudes of transition voltages increase modestly with the separation between monolayer TMDs and substrate. The All DFT calculations are performed at 0 K ionic and electronic temperature.
In the stress-free case, both 2H and 1T' phases are structurally relaxed at a condition of constant in-plane stress, and = 0. Therefore, Gibbs free energy is the relevant potential, and is written as: where is internal energy.
In the scenario of constant area, 1T' is constrained to the computational cell of the relaxed 2H phase. Therefore, Helmholtz free energy should be computed, and is written as: Because free energy is the same as internal energy in either case, they are denoted with the same symbol in main text, although it represents different thermodynamic potentials in different conditions as discussed here.

Supplementary Note 5: Mechanism for charge-induced structural phase transition.
Despite the earlier theoretical studies on structural phase transitions in monolayer TMDs driven by strain 1,7 and heating 8 , the mechanism for such a structural phase transition induced by charge is not understood. Here, we explain the mechanism for such a charge-induced phase transition using energy band diagrams 9 . Supplementary   Fig. 8 shows the electronic Kohn-Sham density of states for monolayer MoTe 2 and TaSe 2 with Kohn-Sham energies (y axis) shifted so that zero corresponds to the vacuum level in each case. 2H phase has a lower free energy when the monolayer is electrically neutral in both cases. Supplementary Fig. 8(a) shows that semiconducting 2H-MoTe 2 has a Kohn-Sham band gap while 1T'-phase has no band gap. When monolayer MoTe 2 is negatively charged, the excess electrons will occupy the lowest energy states above the Supplementary Fig. 8(b) shows that the case of TaSe 2 is a metal-to-metal transition which gives rise to a phase boundary at only one sign of the charge. When TaSe 2 is sufficiently positively charged, the 1T phase will be favored because the Fermi level in 1T-TaSe 2 is higher than that in the 2H phase. However, in contrast to MoTe 2 , 2H- TaSe 2 always has a lower free energy than the 1T phase when the monolayer is negatively charged.
The simplistic picture presented in Supplementary Fig. 8 suggests that other monolayers that exhibit metal-metal transitions are most likely to exhibit a phase boundary at one sign (positive or negative) while a gap in one of the phases can lead to transitions at both positive and negative charges.
Note that this charging mechanism is distinct from the application of an electric field to an electrically isolated monolayer, where the charge on the monolayer will remain neutral. Charge must be allowed to flow on and off the monolayer to achieve the transition described in this work. When excess electrons are assigned to the computational cell, a homogeneous positive background charge is automatically introduced in the vacuum space in order to compensate the excess charge. When the number of excess electrons is increased to some value and the vacuum separation in the direction perpendicular to the monolayer surface is bigger than some corresponding threshold, the Kohn-Sham states will begin to occupy low-lying vacuum electronic states in the center of the vacuum region, at the boundary of the computational cell. Special attention has been given to avoid the formation of these vacuum electronic states by not adding too many excess electrons.