Strain-engineered growth of two-dimensional materials

The application of strain to semiconductors allows for controlled modification of their band structure. This principle is employed for the manufacturing of devices ranging from high-performance transistors to solid-state lasers. Traditionally, strain is typically achieved via growth on lattice-mismatched substrates. For two-dimensional (2D) semiconductors, this is not feasible as they typically do not interact epitaxially with the substrate. Here, we demonstrate controlled strain engineering of 2D semiconductors during synthesis by utilizing the thermal coefficient of expansion mismatch between the substrate and semiconductor. Using WSe2 as a model system, we demonstrate stable built-in strains ranging from 1% tensile to 0.2% compressive on substrates with different thermal coefficient of expansion. Consequently, we observe a dramatic modulation of the band structure, manifested by a strain-driven indirect-to-direct bandgap transition and brightening of the dark exciton in bilayer and monolayer WSe2, respectively. The growth method developed here should enable flexibility in design of more sophisticated devices based on 2D materials.

However, the amorphous substrate still represents a corrugated potential, albeit without long-range crystalline order, for the deposited film. It is interesting, then, to consider the amplitude of the corrugation necessary to sustain the thermal-expansion mismatch strain within the films. In order to explore the amplitude of the surface corrugation required to retain the strain arising due to thermal expansion mismatch, we explored a simple 2D Frenkel-Kontorova model 16 . Though the model is simple, it provides substantial insight into the considered problem, and enables exploration of the large systems. Though the model is best suited for epitaxy, it does allow exploration of the relationship between the corrugation of the potential and the retention of thermal mismatch strain.
The 2D Frenkel-Kontorova model considered here consists of a single plane of atoms connected by springs interacting with a substrate potential that has the same triangular symmetry In this model, the initial state corresponds to the growth temperature of the material. It is assumed that at the growth temperature, the materials are lattice matched. This is, in general, not true. However, for the case of the amorphous substrate, this is probably a reasonable assumption, as the growing film is not likely strained at the growth temperature. The reduction in the equilibrium spring length, then, is meant to model the thermal contraction of the WSe2 film relative to the substrate. A reduction of 1% in the spring length is comparable to the thermal expansion mismatch between WSe2 and SiO2.
The model is then used to explore the amplitude of the substrate corrugation required to retain the thermal mismatch strain. The amplitude of the substrate potential, o , is reduced to the point where it can no longer sustain a 1% strain in the film. It is important to note that this strain (1) is sample size dependent, with larger samples requiring smaller, on average, corrugations to sustain the strain. Accordingly, we have considered an equilateral triangular sample containing 45,451 "atoms," with an edge length of approximately 100 nm (this corresponds to minimizing a function of roughly 91,000 variables, exploration of larger sizes is computationally more difficult). We considered a range of potential energy corrugation amplitudes, o . For o << 1, an example of the pattern of relaxation that we observe is shown in Supplementary Figure 1 (b). Here, the imposed thermal mismatch strain is 0.8% biaxial. The strain plotted is defined to be ( Through trial and error, we discovered that a potential corrugation amplitude of o ≈ 5 × 10 −4 was sufficient to retain the substrate lattice parameter over most of the sample (the results are shown in Supplementary Figure 1 (b)). Recall that this amplitude is measured in units of To compute the strain energy of the WSe2 monolayer, we employ linear elasticity theory.
The lattice parameter and elastic constants for WSe2 are computed as discussed below, arriving at the values 11 = 120 J m −2 and 12 = 23 J m −2 , and o = 3.361 Å. Note that since the elastic constants represent the elastic properties of a 2D material, they have dimension of energy per area.
Noting that the elastic energy of the film computed under a biaxial strain of is given by ( 11 + 12 ) 2 , and equating the two strain energies we find: This required corrugation, then, sets a scale for the retention of thermal mismatch strain.
It is much less than a typical covalent bond strength (of the order of 1 eV). Interestingly, this corrugation is very near to the strength of a typical van der Waals bond (of the order of 20-40 meV). We conclude that a covalent bond between the atoms of the growing film and that of the substrate is not necessary to enable strain tuning of the film via thermal expansion mismatch.
The elastic constants of a monolayer of WSe2 were calculated using the plane-wave density functional theory (DFT) program VASP 1 . Projector augmented wave potentials were used for the ion-electron interactions, 2 and the PBE generalized gradient approximation was used for the exchange-correlation functional 3 . The system modeled was a WSe2 monolayer, which was represented by a 3 atom cell with a 15.44 Å vacuum layer, normal to the monolayer. The plane wave energy cutoff was 600 eV, and the tetrahedron smearing method was used. The electronic (3) self-consistent loop's convergence criterion was set to 1 x 10 -8 , and the system was relaxed until the Hellmann-Feynman forces on each atom were below 0.001 eV Å -1 . A gamma centered 29 × 29 × 1 Monkhorst pack grid was used to sample k-space. First, the system was relaxed allowing the cell and atomic positions to change. The lattice parameter for monolayer WSe2 was found to be 3.316 Å. Using the relaxed system, the elastic constants were found using the method presented by de Jong et. al 4 . The 2D elastic constants for monolayer WSe2 were calculated to be C11 = 120 J m -2 and C12 = 23 J m -2 .

Supplementary Note 2: Expected strain from lattice misfit.
Due to the fact that the WSe2 is not covalently bound to the growth substrate we would not expect the strain to be transferred from the mismatch between the lattice constant of the substrate and the 2D material. However, to verify this we calculated the expected strain due to lattice misfit.
The lattice misfit equation for heteroepitaxial strain is given by 5 : where aSub, a2D, and T are the lattice parameter of the substrate, the lattice parameter of the 2D material, and temperature respectively. In the case where f > 0 the film is expected to be under tensile strain, while for the case where f < 0 the film is expected to be under compressive strain. In our case we compute the expected lattice misfit both at room temperature as well as the growth

Supplementary Methods
The furnace temperature profile as a function of the set point temperature for both zones was characterized to determine the actual sample temperature as well as the cross talk between furnace zones. This was used to adjust the set point of the upstream zone such that the selenium powder would reach the target temperature from excess heat provided by the downstream zone as Ar/H2 ratio, growth time, and gas pressure are shown in Supplementary Figures 9, 10, 11, and 12 respectively. Additionally, photoluminescence imaging and atomic force microscopy were performed to verify sample uniformity and is shown in Supplementary Figure 3.

Growth on AlN:
NaBr is employed as a promotor for the growth of WSe2 instead of KBr. NaBr is mixed with WO3 at 1:2 ratio. All the other growth parameters are kept the same as the method described in the manuscript.

Growth on sapphire (Al 2 O 3 ):
NaCl or KBr is used as a promotor for the growth of WSe2 on sapphire. The promotor is mixed with WO3 at the ratio of 1:2. All the other growth parameters are kept the same as the method described in the manuscript.

Growth on STO:
NaBr is mixed with WO3 at the ratio of 1:2. As the downstream furnace temperature reaches 875°C, the synthesis of the WSe2 is initiated by introducing hydrogen. All the other growth parameters are kept the same as the method described in the manuscript.  Fig. 3 (c) and Supplementary Fig. 5), the WO3/promoter boat and the target substrates were placed in separate furnace zones (shown schematically in Supplementary   Fig. 7 (b)).
Device fabrication: Back-gated devices were fabricated on WSe2 samples grown on fused silica and transferred to Si/SiO2 (50 nm thick oxide) substrates as well as WSe2 directly exfoliated on Si/SiO2 (50 nm thick oxide) substrates to further verify material quality. All patterning was performed using electron beam lithography with PMMA C4 as the resist. Samples were first etched using XeF2, and subsequently 40 nm thick Ni was deposited by thermal evaporation as the contact electrode. Both CVD and exfoliated WSe2 devices show ambipolar behavior with similar on-currents as shown in Supplementary Figure 13.