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Quantum theory provides a powerful theoretical tool to deeply understand various physical phenomena in emerging semiconducting crystals. The resulting electronic band structure and wave functions can be employed to theoretically calculate observable physical quantities, e.g., the spontaneous emission rate with the aid of Fermi's golden rule52, which in turn accounts for the light absorption and emission induced by optical interband transitions of electrons. Several effective approaches have been introduced to quantitatively understand the physical properties of strained 2D materials, including DFT and symmetry-allowed low-energy effective fields based on group theory ideas53, 54. In the absence of an external strain, graphene and most TMDC monolayers share the same hexagonal Bravais lattice characterized by primitive unit cells plus a set of basis atoms. Mathematically, the two primitive lattice vectors, designated a1 and a2 can be expressed by a linear combination of bond vectors between adjacent atoms, δn, such as three and six nearest-neighbor vectors for graphene and TMDCs, respectively55. However, when the 2D materials are strained, the Bravais lattices are deformed. At the same time, the modified bond vectors can be exploited to parameterize the valence force models to describe the elastic energy of strained 2D materials56. On the other hand, from a macroscopic point of view, the 2D materials deformed under external applied strains can be described by continuum elasticity theory, provided that the length scale of the deformation is large compared with the lattice constant54. The elastic energy stored in 2D materials can be divided into stretching and bending energies. In such a configuration, a generic atom in the deformed crystalline structure of a 2D material whose original position is r = (x, y) experiences a displacement $ {\boldsymbol{u}}\left( {\boldsymbol{r}} \right) + \hat zh\left( {\boldsymbol{r}} \right), $ where $ {\boldsymbol{u}}\left( {\boldsymbol{r}} \right) = \left( {u_x\left( {\boldsymbol{r}} \right), u_y\left( {\boldsymbol{r}} \right)} \right)$ is a 2D vector field describing the in-plane deformation, and h(r) is a scalar field accounting for out-of-plane deformations. The strain is a second-rank tensor field
$$ \varepsilon _{ij} = \frac{1}{2}\left( {\partial _iu_j + \partial _ju_i + \partial _ih\partial _jh} \right) $$ (1) where i, j ∈ {x, y}. The elastic energy density Fel (a contribution to the potential energy of the Hamiltonian) of anisotropic membrane can be written as a sum of the stretching energy density Fst (energy cost due to in-plane relative distance changes) and the bending energy density Fb (resulting from deviations from the flat configuration), as described in the following equations:
$$ F_{{\mathrm{el}}} = F_{{\mathrm{st}}} + F_{\mathrm{b}}, {\mathrm{with}}\, F_{{\mathrm{st}}} = \frac{1}{2}\left( {\lambda \varepsilon _{ii}^2 + 2\mu \varepsilon _{ij}\varepsilon _{ij}} \right)\, {\mathrm{and}}\, F_{\mathrm{b}} = \frac{1}{2}\kappa \left( {\nabla ^2h} \right)^2 $$ (2) where λ and μ are the (in-plane) Lamé constants of the material and κ is the bending rigidity. The stretching energy density can be explicitly written as55
$$ F_{{\mathrm{st}}} = \frac{{Y_{2d}}}{{2\left( {1 - \nu ^2} \right)}}\left[ {\varepsilon _{xx}^2 + \varepsilon _{yy}^2 + 2\nu \varepsilon _{xx}\varepsilon _{yy} + \left( {1 - \nu } \right)\varepsilon _{xy}^2} \right] $$ (3) where Y2d and v are the 2D Young's modulus and Poisson ratio, respectively, which can be expressed in terms of the Lamé constants as
$$ Y_{2d} = \left( {\lambda + 2\mu } \right)\left( {1 - \nu ^2} \right)\, {\mathrm{and}}\, \nu = \frac{\lambda }{{\lambda + 2\mu }} $$ (4) Here, it is worth noting that the Lamé constants and bending rigidity can be related to the microscopic parameters of 2D materials through the valence force field55. Thus, Eqs. (2)-(4) allow the development of a bridge between macroscopic strains and microscopic parameters of the materials. We apply these equations to determine the electronic band structure by constructing microscopic Hamiltonians in the tight-binding (TB) and k · p models through the strain-displacement relations obtained by minimizing the elastic energy Fel for a given strain (i.e., mechanical equilibrium). More specifically, for the non-Bravais lattices of strained materials, the strain-dependent first-neighbor vector $ {{{\rm{ \mathsf{ δ} }}}}^{\prime}_n = \left( {{\boldsymbol{I}} + \varepsilon } \right) \cdot {{{\rm{ \mathsf{ δ} }}}}_n$, where ε is the strain tensor and relates to the displacement via u(r) = ε · r, breaks down within the Cauchy-Born approximation53 and should be corrected as $ {{{\rm{ \mathsf{ δ} }}}}^{\prime}_n = \left( {{\boldsymbol{I}} + \varepsilon } \right) \cdot {{{\rm{ \mathsf{ δ} }}}}_n + \Delta$, where the additional displacement Δ accounts for the additional degrees of freedom introduced by multiple basis atoms in the primitive unit cell. The vector Δ is determined by minimizing the deformation energy of the material. In this way, the TB Hamiltonian incorporating first-neighbor vectors can be used to deduce strain-related effects in a graphene lattice53
$$ H = - \mathop {\sum}\limits_{{\boldsymbol{r}}^\prime} \mathop{\sum}\limits_{n = 1}^3t_{{\boldsymbol{r}}^\prime, {{{\rm{ \mathsf{ δ} }}}}_{\boldsymbol{n}}^\prime }a_{{\boldsymbol{r}}^\prime}^\dagger b_{{\boldsymbol{r}}^\prime + {{{\rm{ \mathsf{ δ} }}}}_{\boldsymbol{n}}^\prime} + {\mathrm{H}}.{\mathrm{c}}. $$ (5) where H.c. is the Hermitian conjugate, $ t_{{\boldsymbol{r}}^\prime , {\boldsymbol{\delta }}_{\boldsymbol{n}}^\prime }$ indicates the hopping integral in the deformed lattice, and $ a_{{\boldsymbol{r}}^\prime }^\dagger$ and $ b_{{\boldsymbol{r}}^\prime + {\boldsymbol{\delta }}_{\boldsymbol{n}}^\prime }$ are electron creation and annihilation operators on the A sublattice (at position r′) and the B sublattice (at position $ {\boldsymbol{r}}^\prime + {\boldsymbol{\delta }}_{\boldsymbol{n}}^\prime$), respectively.
In practice, the TB model discussed above is widely applied to explore the strain-induced energy landscape of graphene57, 58 (Fig. 5a), while the DFT method is used to determine the effect of strain on the band structure of 2D TMDCs (Fig. 2a). However, the application of DFT methods is computationally prohibitive when the elastic strains are inhomogeneous. In such situations, the TB model provides a useful approach to evaluate the effects of strain on the band structure of 2D materials under inhomogeneous strains (Fig. 3a).
Recently, other effective Hamiltonians with coupling terms related to external fields (e.g., strain and electromagnetic fields), have been proposed to gain a better understanding of the physics behind novel effects in 2D materials (e.g., those associated with strains and valley Hall effects)59. One way to construct such effective Hamiltonians that account for strain and electron momentum is through the crystal symmetry group of 2D materials, and their corresponding irreducible representations. An alternative way of deriving effective Hamiltonians is based on direct expansion of the TB Hamiltonians. For example, in the presence of nonuniform lattice deformations, the low-energy effective Hamiltonian of graphene is given by60
$$ H = H_0 + \mathop {\sum}\limits_{m = 1}^6 {a_mH_m} + \mathop {\sum}\limits_{m = 1}^6 {\tilde a_m\tilde H_m} $$ (6) where $ H_0 = \nu _F\left( { \pm \sigma _xk_x + \sigma _yk_y} \right)$ is the unstrained standard Dirac contribution (vF and σi, i ∈ {x, y}, are the Fermi velocity and Pauli matrices). The strained terms amHm and $ \tilde a_m\tilde H_m$ relate to different effects induced by nonuniform strains54, such as the Dirac cone shift and tilt in momentum space and the gap-opening mechanism. As another example, the TB Hamiltonian of a uniformly strained TMDC (ignoring spin-orbit interactions) can be approximated by a reduced two-band model near the K points consisting of the highest valence band and the lowest conduction band59. The total Hamiltonian is H = H0 + H1, where H0 is the effective Hamiltonian of the unstrained TMDC and the contribution of strains can be expressed by
$$ H_1 = f_3\mathop {\sum}\limits_i {\varepsilon _{ii} + f_4} \mathop {\sum}\limits_i {\varepsilon _{ii}\sigma _z + f_5} \left[ {\left( {\varepsilon _{xx} - \varepsilon _{yy}} \right)\sigma _x - 2\varepsilon _{xy}\sigma _y} \right] $$ (7) where f3, f4, and f5 represent the modification of the mid-gap position and pseudo-gauge field terms. Moreover, the spin-orbit couplings (1 ± σz)sz and out-of-plane deformations can be accounted for by corrections to these Hamiltonians.
The macroscopic and microscopic strain models, introduced above, can assist in understanding and predicting a wide range of optical and electronic properties of strained 2D materials, such as their PL and Raman spectra, as a result of interactions between photons, electrons, and phonons as a function of the (tensor) strain fields.
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The intriguing optical properties of 2D materials are related to their complex electronic band structures73. As noted in the previous section, the dependence of the Hamiltonians of 2D materials on external strains can be used to identify strain-dependent effects in the electronic band structure. For example, both DFT and GW calculations predict a decrease in the bandgap in monolayer MoS2 with increasing tensile strain32, and suggest that a semiconductor-to-metal phase transition can occur under very large strains43. As illustrated in Fig. 2a, the impact of mechanical deformation on the electronic band structure is especially noticeable at the K point in the BZ, where the intrinsic direct bandgap decreases with increasing strain. Meanwhile, the top of the valence band shifts slightly upwards at the Γ point, stemming from the decrease in the orbital overlap between the metal and sulfur atoms associated with strain-induced changes in the bond distance. At sufficiently large strain, the valence band maximum moves from the K to Γ point, leading to a direct-to-indirect bandgap transition32. In addition to these uniform strain effects, wrinkling in MoS2 creates a nonuniform strain distribution that can reduce the direct bandgap (consistent with TB calculations)41.
Fig. 2 Band structures and PL and absorption spectra of strained TMDCs under different strains.
a Band structures of monolayer MoS2 at lattice constant of 3.160, 3.190, and 3.225 Å corresponding to 0%, 1%, and 3% tensile strains calculated by DFT, G0W0, and SCGW0, respectively32. b, c Evolution of PL and absorption spectra (b) and peak positions (c) of monolayer MoS2 under various uniaxial tensile strains. Note that the dashed black line overlaid on the PL spectra in b indicates the PL peak of PMMA rather than that of the B exciton. Lines in c are linear fits to the data points extracted from b29. d PL spectra of bilayer MoS2 under uniaxial tensile strain increasing from 0 to 0.6%. The insert shows the indirect bandgap of bilayer MoS2 (ref. 28). e PL spectra of trilayer MoS2 under various biaxial compressive strains76. f PL spectra of bilayer WSe2 (solid lines) under various uniaxial tensile strains compared with that of unstrained monolayer WSe2 (dashed line)78. Reprinted a with permission from ref. 32. Copyright (2013) by the American Physical Society. Adapted b, c with permission from ref. 29. Copyright (2013) American Chemical Society. Adapted d with permission from ref. 28. Copyright (2013) American Chemical Society. Adapted e with permission from ref. 76. Copyright (2013) American Chemical Society. Reprinted f with permission from ref. 78. Copyright (2014) American Chemical Society -
As noted above, low-symmetry 2D TMDCs provide an excellent platform to continuously tune the electronic band structures and optical properties via external strains, as seen in many spectroscopy experiments28, 74, 75. For example, the peak positions in both the PL and absorption spectra of 2D TMDCs show a linear redshift for both the A and B excitons, when homogeneous tensile strain is applied (Fig. 2b). The experimental results shown in Fig. 2c reveal that the redshift rates in the absorption spectrum of monolayer MoS2 are -64 ± 5 meV/% (/% means per percent of tensile strain) for the A exciton and -68 ± 5 meV/% for the B exciton29. In addition, redshift rates of -54 ± 2 and -50 ± 3 meV/% are obtained for the A and B excitons in monolayer WSe2, respectively74. In addition to shifts in absorption peaks, PL spectral peaks also exhibit an approximately linear dependence on tensile strain75; for example, Fig. 2d shows redshift rates of -45 ± 7 meV/% for the A exciton in MoS2 monolayers, and -53 ± 10 meV/% for the A exciton and 129 ± 20 meV/% for the indirect bandgap (marked as "I" in the insert) in MoS2 bilayers28. Figure 2e shows a linear blueshift of the peak in trilayer MoS2 on a piezoelectric substrate with the application of a compressive biaxial strain76.
In addition to the redshift, when the strain is sufficiently large, a direct-to-indirect bandgap transition or an indirect-to-direct bandgap transition can be induced in TMDCs. For example, an indirect peak emerges when the strain is larger than 2.5% in strained WS2 monolayers77, while an indirect-to-direct transition occurs in strained WSe2 bilayers with an obvious enhancement of the PL intensity at 0.73% uniaxial tensile strain (Fig. 2f)78. Motivated by the aforementioned controllable optical properties, the variation in the measurable strain-related parameters with temperature79, input laser intensity79, and voltage76 can also be used to quantify the strains on TMDCs; this further broadens the potential optoelectronic, as well as strain sensor applications.
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Apart from homogenous in-plane strains, local inhomogeneous strains can also be induced by local strain engineering, providing other exciting avenues for tailoring distinctive optical properties of 2D materials on the nanoscale. As a concrete example, the electronic band structures of a zigzag MoS2 ribbon calculated using the TB model with periodic boundary conditions are plotted in Fig. 3a. The corresponding bandgap decreases as the strain increases, modifying the electronic band structure on the nanoscale. As a result, the excitons drift hundreds of nanometers to the lower bandgap regions on the top of the wrinkles before recombination, as depicted in Fig. 3b; this is referred to as the "funnel effect". The PL spectrum of the A exciton on top of four different wrinkles (Fig. 3c) demonstrates that the A exciton can be confined by local strains41. The funnelling of photogenerated excitons toward regions of higher strain has potential for many applications in diverse fields, such as single-photon sources80 for quantum networks and communications and solar cells44 for photovoltaic devices.
Fig. 3 Band structure, exciton funnel effect, spontaneous emission enhancement, SHG, and phase transitions in TMDCs under local strain.
a Calculated band structures for nonuniformly strained monolayer MoS2 under 0% (left panel), 2% (middle panel), and 4% (right panel) strian41. b Schematic of the exciton funnel effect induced by the inhomogeneous strain in a wrinkled MoS2 region41. c Spatial distribution of the A exciton wavelength in a four-layer thick MoS2 flake with four wrinkles41. d SEM micrograph of a WSe2 monolayer conformally coated on a rough silver surface, where red dotted circles denote strained regions on silver nanoparticles80. e Spatially resolved and wavelength integrated (750-850 nm) PL intensity map of the sample in c80. f PL spectra taken at the locations circled in e. The spectra labeled U and B are for the pristine monolayer and bare substrate80. g Schematic of the SHG process in strained MoS2 (ref. 81). h SHG intensity maps for monolayer (1L), bilayer (2L), and trilayer (3L) MoS2 on a Si/SiO2 wafer81. i Polarization-resolved SHG intensity patterns for a strained (red) and unstrained (blue) MoS2 monolayer81. j Schematic of the strain-induced reversible phase transition in MoTe2 (ref. 86). k Raman spectra taken at the suspended (red), peripheral (black), and supported (blue) areas in a cavity-supported MoTe2 thin film with an initial phase of 2H (ref. 86). l Raman intensity maps at the 140 cm-1 (1T′ phase, top panel) and 230 cm-1 (2H phase, bottom panel) peaks86. Reprinted a-c with permission from ref. 41. Copyright (2013) American Chemical Society. Reprinted d-f with permission from ref. 80. Copyright (2018) American Chemical Society. Reprinted g-i with permission from ref. 81. Copyright (2018) Springer Nature. Reprinted j-l with permission from ref. 86. Copyright (2016) American Chemical SocietyWhen a homogeneous strain is applied to two- or three-layer TMDCs, a rapid increase in the PL intensity can be observed (Fig. 2f, e), which can be attributed to the transition from an indirect to direct band structure. In addition, TMDCs can be designed as ultracompact quantum light emitters under local strain (due to tight exciton localization) to produce large spontaneous emission enhancements and very sharp spectral lines at several wavelengths. For example, depositing monolayer WSe2 on a rough metal (Ag) surface creates strain-induced quantum emitters at Ag islands and nanoparticles (as shown in Fig. 3d), and their emission is largely enhanced by the localized surface plasmon resonances (LSPRs) of the Ag nanostructures through the Purcell effect80. A finite-difference time-domain simulation showed an ~30-fold field enhancement at the tip of a Ag nanocone excited by a dipole source, which induces large decay rate enhancements. The emission lines (Fig. 3f) corresponding to selected positions on the WSe2 on the rough Ag surface (Fig. 3e) show very sharp PL peaks (linewidths much narrower than those of the unstrained material). This phenomenon motivated the rapid growth of research on TMDC-based single-photon emission on metallic substrates.
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Second-harmonic generation (SHG) is a second-order nonlinear optical process, in which two photons with the same frequency are combined into a single photon in a noncentrosymmetric medium (within the electrostatic approximation); see monolayer MoS2 in Fig. 3g. An example of SHG in TMDCs is illustrated in Fig. 3h (ref. 81), where a MoS2 monolayer exhibits giant second-order nonlinear polarizability due to the breaking of spatial inversion symmetry; this makes the MoS2 monolayer a good platform for studying SHG. Inspired by the close relationship between the SHG signal and the crystal lattice, SHG mapping can be exploited to quickly and nondestructively identify the strains in 2D TMDCs82.
We now discuss how the SHG signal can be used to sense strain in TMDC monolayers. The second-order nonlinear susceptibility tensor $ {{{\rm{ \mathsf{ χ} }}}}_{klm}^{(2)}$ of 2D materials, dependent on strain, can be written (to lowest order) as81, 83
$$ {{{{\rm{ \mathsf{ χ} }}}}}_{klm}^{\left( 2 \right)} = {{{{\rm{ \mathsf{ χ} }}}}}_{klm}^{\left( {2, 0} \right)} + {\bf{p}}_{klmij}{{{{\rm{ \mathsf{ ε} }}}} }_{ij} $$ (8) where $ {\bf{p}}_{klmij} = \frac{{\partial {{{\rm{ \mathsf{ χ} }}}}_{klm}^{\left( {2, 0} \right)}}}{{\partial {\boldsymbol{\varepsilon }}_{ij}}}$ and $ {{{\rm{ \mathsf{ χ} }}}}_{klm}^{\left( {2, 0} \right)}$ describes the second-order nonlinear susceptibility of the unstrained crystal. pklmij is the fifth-rank photoelastic tensor, which relates strain εij to the nonlinear susceptibility. Given the symmetric strain tensor (εij = εji) and dispersion-free SHG process, the photoelastic tensor of monolayer TMDCs should possess the following symmetries: pklmij = pkmlij = pklmji = pkmlji. Since hexagonal TMDC monolayers have D3h lattice symmetry, considering these symmetries, the 2D TMDC monolayer photoelastic tensor should have 12 nonzero elements that are functions of (only) two parameters, P1 and P2 (ref. 81). The induced second-order nonlinear polarization can be described by $ {\bf{P}}_k^{\left( 2 \right)}\left( {2\omega } \right) \propto$ $ {{{{{\rm{ \mathsf{ χ} }}}} }}_{klm}^{\left( 2 \right)}E_l(\omega )E_m(\omega )$, where E refers to the incident electric field. For linearly polarized light, incident at angle ϕ, analysis of the SHG signal with the same polarization yields the SHG intensity of the form81:
$$ I_\parallel ^{\left( 2 \right)} \propto \frac{1}{4}\left( {A\cos \left( {3\phi } \right) + B\cos \left( {2\theta + \phi } \right)} \right)^2 $$ (9) where $ {\mathrm{A}} = \left( {1 - {{{{\rm{ \mathsf{ ν} }}}}}} \right)\left( {P_1 + P_2} \right)\left( {\varepsilon _{xx} + \varepsilon _{yy}} \right) + 2\chi ^0$ and $ {\mathrm{B}} = (1 + {{{{\rm{ \mathsf{ ν} }}}}})\left( {P_1 - P_2} \right)\left( {\varepsilon _{xx} - \varepsilon _{yy}} \right)$. Note that χ0 is the nonlinear susceptibility parameter of unstrained monolayer TMDC crystals, v is the Poisson's ratio, εxx and εyy denote the principal strains, and θ is the direction of the principal strain. Consequently, once the parameters P1 and P2 are deduced from the polarization-resolved SHG intensity at different strain levels, these two parameters can be utilized to spatially map the strain field81. The proof-of-concept polarization-resolved SHG intensity study demonstrated in Fig. 3i opened up a window for imaging the strain distribution below the optical diffraction limit.
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Many TMDCs exhibit different crystal structures (polymorphs) for various deposition or postprocessing conditions. These polymorphs retain the general MX2 (X-M-X trilayer) structure, but exhibit different chalcogen coordination structures around the transition metal atoms. The most common structures are the 2H (trigonal prismatic coordination) phase, 1T (octahedral coordination) phase, and 1T′ (distorted octahedral coordination) phase84. For group-VI TMDC monolayers, theoretical predictions showed that the most stable phase under ambient conditions is 2H (except for in WTe285)—these materials are semiconductors with bandgaps in the 1-2 eV range. On the other hand, 1T and 1T′ tend to be metallic. The coexistence of metallic and semiconducting TMDC monolayer polymorphs has spurred studies of phase transitions in these systems. The existence of metal-insulator transitions under or close to ambient conditions suggests possible nonvolatile information storage applications85.
A strain-driven semiconductor-to-metal (2H-1T′) transition of MoTe2 under room temperature was reported in 2016 (ref. 86; such a reversible phase transition is shown in Fig. 3j). Given that the 2H and 1T′ phases of monolayer MoTe2 present different Raman signatures and peaks (140 cm-1 for the 1T′ phase and 230 cm-1 for the 2H phase), the Raman spectra could be used to identify the occurrence of phase transitions (Fig. 3k). In this experiment, a thin film of MoTe2 was transferred onto a substrate patterned with cavities of different diameters. The MoTe2 monolayers suspended over the cavities were then subjected to external tensile strains using AFM tips. Evidence of a phase transition from 2H to 1T′ was only observed in the suspended regions. These suspended areas showed 1T′ Raman signals (140 cm-1 in the top panel of Fig. 3l), while only 2H MoTe2 signals were found in the supported (unstrained) areas (230 cm-1 in the bottom panel of Fig. 3l). The application of strain reduced the phase transition temperature from 855 ℃ to room temperature in this experiment. Thus, phase-change-related TMDC electrical and optical property manipulation at room temperature may find applications in extremely sensitive optical and electrical sensors86.
Electronic band structures of strained 2D TMDCs
Strain tuning of the photoluminescence, absorption, and bandgap transition
Exciton funnel effect and spontaneous emission enhancement
Strain-induced second-harmonic generation in TMDCs
Phase transition of TMDCs engineered by strain
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In this section, we briefly review various techniques for applying strains to 2D materials. This section is divided into three parts based on the nature of the applied strain: homogeneous uniaxial strain, homogeneous biaxial strain, and inhomogeneous local strain.
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Controllable homogeneous uniaxial strain can be applied to a 2D material via bending. This is usually accomplished by transferring the 2D material to a flexible substrate (PMMA29, PC74, PDMS115, etc.) followed by bending the substrate, as depicted in Fig. 7a, b. One surface of the bent, flexible substrate will be in compression, and the other in tension, resulting in a uniform, uniaxial strain (the uniaxial strain is perpendicular to the axis of bending) on the 2D material on the substrate surface. We can divide such techniques into two main categories based on the apparatus used: two-point bending systems (Fig. 7a) and cantilever bending systems (Fig. 7b). The uniaxial strain induced on the surface of a flexible substrate in two-point bending is115
Fig. 7 Bending and elongation for generating homogenous uniaxial strain.
a Experimental setup for two-point bending. The insert shows a PC-MoSe2-PDMS stack created by placing a viscoelastic (PDMS) stamp carrying exfoliated MoSe2 upside down onto a polycarbonate (PC) substrate115. b Schematic diagram of cantilever bending where the (1L or 2L) MoS2 sample is placed near the fixed edge of the flexible PMMA substrate (see the inset)29. c Schematic of rolling two transparent oxide layers stacked on a Ge substrate118. d Schematic of a MEMS device for generating large strains in graphene (upper panel) and its SEM micrograph (lower panel)119. Reprinted a from ref. 115. Published by The Royal Society of Chemistry. Reprinted b with permission from ref. 29. Copyright (2013) American Chemical Society. Reprinted c from ref. 118 with permission from Wiley. Adapted d with permission from ref. 119. Copyright (2014) American Chemical Society$$ \varepsilon = \frac{t}{{2R}} $$ (10) where t is the substrate thickness and R is the radius of curvature (assuming R $ \gg $ t)109. For a cantilever of length L, the uniaxial strain on the substrate surface is29
$$ \varepsilon = \frac{{3t\delta }}{{2L^2}}\left( {1 - \frac{x}{L}} \right) $$ (11) where x is the distance from the fixed edge of the substrate and δ is the deflection of the bendable edge, assuming that δ is small such that the maximum slope is $ \ll $ 1 (ref. 116).
Normally, two-point bending systems are commonly employed for large 2D sheets, while the cantilever approach is applied when the 2D sheet is small. Commonly, large means that the lateral extent is ≥100 µm (Fig. 7a), whereas small means ≤10 µm (Fig. 7b). These methods have been widely applied in studies of graphene and TMDCs28, 29, 74, 75, 78, 109-111, 114, 115, 117. Depending on the strength of the 2D sheet and the equipment, the maximum uniaxial strain achieved by these methods is 0.5-3.8%.
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Because of their high strength, 2D materials can be readily rolled up. For example, many nanofabrication processes for graphene employ rolling techniques, which result in a uniaxial strain perpendicular to the rolling axis, as shown in Fig. 7c (ref. 118). This approach can be used to apply tensile or compressive strains depending on the pre-straining. Typical maximum compressive strains are on the order of -0.3% (ref. 118; a larger magnitude than most other techniques).
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Conceptionally, elongation is the simplest approach for applying a uniaxial homogeneous strain to a 2D material77, 112, 120. The 2D material is placed on the surface of a substrate, which is elongated in a tensile test device. The tensile strain in this case is easily controllable via the application of a load to the substrate. This method can routinely achieve a much higher maximum strain level (~ 4%)77 than those achieved via the above techniques. A MEMS device was recently119 fabricated to achieve uniaxial strains in graphene in excess of 10% (Fig. 7d). In this experiment, two suspended shuttle beams are bridged by graphene, one of which is thermally actuated, while the other is affixed to springs to measure the pulling force. When an external power is applied to the contact pad, current flows through the thermal shuttle beam to induce Joule heating, which expands the beam, and thereby actuates the shuttle to uniformly apply uniaxial strain to the 2D material sample119.
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A biaxial strain can be applied using the idea of differential thermal expansion (Fig. 8a (ref. 121)). This requires a large difference in the thermal expansion coefficients of the 2D materials and the substrate. In the example, in Fig. 8a, the thermal expansion coefficient of the 2D material exceeds that of the substrate. When the 2D material is strongly adhered to the substrate and the system if heated or cooled, the expansion difference between these materials generates a homogeneous tensile biaxial strain79, 121. While the thermal expansion coefficient is a symmetric second-rank tensor, the thermal strain will be (balanced) equal in two orthogonal directions if the substrate is amorphous, cubic, or has a (0001) surface in a hexagonal crystal or a (001) surface in a tetragonal crystal. There are two obvious deficiencies of this method: (1) the modest magnitude of the induced strain (a few tenths of a percent) and (2) the difficulty in studying the temperature dependence of a property that depends on strain.
Fig. 8 Thermal expansion and piezoelectric effect for generating homogeneous biaxial strain.
a Schematic of the differential thermal expansion induced homogeneous biaxial tensile strain in monolayer WSe2, whose thermal expansion coefficient is larger than that of the substrate121. b Schematic diagram of MoS2 sandwiched between a piezoelectric PMN-PT substrate and a graphene layer serving as an electrode76. c Schematics of scotch-tape-exfoliated graphene on a PMN-PT substrate (upper panel) and an electromechanical device applying an in-plane biaxial strain to graphene (lower panel)97. Reprinted a with permission from ref. 121. Copyright (2017) Springer Nature. Reprinted b with permission from ref. 76. Copyright (2013) American Chemical Society. Reprinted c with permission from ref. 97. Copyright (2010) American Chemical Society -
Piezoelectric materials are strained under the application of an applied external electric field. They can serve as substrates suitable for applying both tensile and compressive strains on attached 2D materials76, 97, 104. The working principle, together with a schematic, is shown in Fig. 8b, c, where a 2D material is deposited on a hybrid substrate incorporating a PMN-PT layer, whose thickness changes upon application of an external applied electric field. Under an appropriate electrical bias, the substrate elongates in the vertical direction and compresses in the horizontal direction, which in turn applies a homogeneous in-plane biaxial compressive strain to the 2D materials (Fig. 8b)76. When the bias direction is reversed, tensile strain can also be applied (Fig. 8c)97. In this way, application of an electric field can continuously tune the magnitude of the strain (and its direction) in the -0.2% to 0.1% range76, 97.
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In this section, we focus on techniques for applying locally inhomogeneous strains to 2D materials, i.e., there is a spatial variation in the strain tensor. It is also of technological interest to apply strains in very small areas, e.g., for miniaturization of conventional devices suitable for higher integrity and lower power consumption.
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The laser illumination method follows a similar principle to the thermal expansion method discussed in the previous section. A high intensity, focused laser is used to efficiently heat a local region of a 2D material, resulting in a nonuniform temperature distribution (e.g., a higher temperature in the center and a lower temperature at the edges), which results in an inhomogeneous local strain in the sample via differential thermal expansion79.
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Transferring 2D materials onto prestrained flexible elastomeric substrates (a relatively simple and broadly used method) provides a means of introducing inhomogeneous local strains into any type of 2D materials41, 122-124. The working principle is illustrated in Fig. 9a, where an elastomeric substrate is initially stretched, 2D material sheets (e.g., exfoliated MoS2) are subsequently deposited on the prestrained substrate, and the load that strains the substrate is released. The large strain energy in the 2D material sample is released via localized buckling and debonding, creating a distribution of wrinkles in the 2D material (Fig. 9b). In this technique, the maximum strain accumulates on the top of the wrinkles and can be defined as41
Fig. 9 Wrinkling and nanostructure support for local strain generation.
a Schematic of the formation of wrinkled 2D MoS2 (ref. 41), and b optical microscopy image of a wrinkled MoS2 sheet41. c, d SEM micrograph of an array of silica nanopillars on a silicon substrate and d fabrication procedures for a layered material (LM) on the nanopillars126. e AFM image of monolayer WSe2 on a silica nanopillar (upper panel), and height profiles of the nanopillar before (blue area) and after (pink line) LM deposition (lower panel)126. Reprinted a and b with permission from ref. 41. Copyright (2013) American Chemical Society. Reprinted c-e with permission from ref. 126. Copyright (2017) Springer Nature$$ \varepsilon = \pi ^2h\delta /\left( {1 - \nu ^2} \right)\lambda ^2 $$ (12) where ε is the uniaxial strain, h is the sample thickness, δ is the wrinkle height (distance between the top of the sample and the substrate), λ is the wrinkle width, and v is the Poisson's ratio125. Of course, if the prestrain in the substrate is not uniaxial, then a more complex distribution of wrinkles and final strain states may be induced.
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In this class of techniques, the morphology of the substrate must be patterned prior to transferring the 2D material. This can be accomplished by positioning nanopillars126-129, nanorods130, etched nanoholes131, or other nanostructures on a flat surface, by producing rough substrates80, 113 or by substrate patterning via nanolithography methods before transferring the 2D materials. In all of these cases, the non-flatness of the substrate and the conformality of the 2D materials induce strains in the 2D material via bending/stretching. These types of approaches have been applied in recent years due to the rising interest in single-photon emission, which is a key element in optical quantum computation.
For the sake of clarity, two typical examples are introduced here, in which nanopillars and rough metallic surfaces are used. In the first example126, exfoliated monolayer WSe2 flakes are deposited on a substrate with a square lattice of Si nanopillars. Figure 9d shows a schematic of the formation of the inhomogeneous local strain, while Fig. 9c, e show an SEM micrograph of the Si nanopillar array and an AFM image of monolayer WSe2 on a Si nanopillar, respectively. In the second example80, a sapphire substrate coated by a thin silver film (200 ± 10 nm) followed by deposition of a 3 nm Al2O3 layer to protect the silver film from oxidation is used as a rough metallic surface to apply local strains to the later transferred WSe2 (ref. 80).
Homogeneous uniaxial strain
Bending
Rolling
Elongation
Homogeneous biaxial strain
Thermal expansion
Piezoelectric straining
Inhomogeneous local strain
Laser illumination
Wrinkling
Nanostructure support
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We first provide a brief summary of the optical effects induced by various types of strain (homogeneous uniaxial strain, homogeneous biaxial strain, and local inhomogeneous strain). Table 1 summarizes the spectral shift coefficients and optical properties of TMDCs and graphene under various strains. Among them, homogeneous uniaxial strain is viewed as the easiest-to-access strain technique with the highest achievable reliability, associated with the simplicity of the equipment and the ease of manipulation of the environment. Such an approach is simple to implement through bending28, 74, 75, 115 or elongation77. Moreover, the induced uniaxial strains can be easily quantified and thus controlled. Alternatively, homogeneous biaxial strains in 2D materials can be applied in situations where harsher conditions can be sustained (e.g., high temperature79, 121, high electrical voltage76, 104, and high laser intensity79). However, such methods typically require more complex experimental instruments and more control. We note that in experiments that produce macroscopic strains, viscoelastic stamps can act as a mechanical clamp to improve the stability, and prevent interfacial sliding between the 2D material and the substrate during straining, when the interfacial adhesion is low (Figs. 7a, c and 8b). However, such an approach requires special attention to avoid the build-up of strain when the carried 2D material is transferred from the viscoelastic stamp to the stretchable substrate.
Type of strain Materials (max strain) Optical effects Refs. Homogeneous uniaxial strain 1–2L MoS2 (2.2%) -120 meV/% for 2L,
-45 meV/% for 1L (PL)a]28 1–2L MoS2 (0.8%) -48 meV/% for 1L,
-46 meV/% for 2L DBb,
-86 meV/% for 2L IBc (PL)75 1L WSe2 (1.4%) -54 meV/% for A exciton,
-50 meV/% for B exciton (Abs)d]74 2–4L WSe2 (2.0%) Indirect-to-direct bandgap transition 78 1L MoSe2 (1.1%) -27 ± 2 meV/% (PL) 115 Graphene (0.8%) -10.8 cm-1/% for G+ band,
-31.7 cm-1/% for G- band,
-64 cm-1/% for 2D band109 Graphene (0.62%) -31.4 ± 2.8 cm-1/% for G+ band,
-9.6 ± 1.4 cm-1/% for G- band111 Graphene (-0.62%)e] 22.3 ± 1.2 cm-1/% for G+ band,
5.5 ± 1.9 cm-1/% for G band111 1–2L MoS2 (0.52%) -64 ± 5 meV/% for 1L A exciton,
-68 ± 5 meV/% for 1L B exciton,
-71 ± 5 meV/% for 2L A exciton,
-67 ± 5 meV/% for 2L B exciton (Abs)
-48 ± 5 meV/% for 2L A exciton,
-77 ± 5 meV/% for 2L IB (PL)29 1L WS2 (4%) -11.3 meV/% (PL), direct-to-indirect bandgap transition at 2.5% 77 Homogeneous biaxial strain 3L MoS2 (-0.2%) -300 meV/% (PL) 76 Graphene (-0.15%) 160.3 cm-1/% for G′ band,
-57.3 cm-1/% for G band97 1L WSe2 (-0.2% to 1.0%) N/A 121 1L MoS2 (0.2%) -0.42 meV K-1 (PL) 79 Inhomogeneous local strain 1L MoS2 (N/A) -0.114 meV kW-1 cm-2 (PL) 79 3–5L MoS2 (2.5%) Total -90 meV change for DB (PL) 41 2–4L WS2, 2–4L WSe2 (1% for 3L, 2% for 4L) Locally enhanced PL intensity with bandgap patterning 122 1L WSe2 on Ag film (N/A) Strong emission enhancement and single-photon emitter 80 1–2L WSe2 on substrate (N/A) Strong emission enhancement and single-photon emitter 127 Graphene on Ag film (N/A) Strong emission enhancement and splitting of G band 113 Au deposited on graphene (N/A) Strong emission enhancement and splitting of G band 108 aPhotoluminescence.
bDirect bandgap.
cIndirect bandgap.
dAbsorption.
e"-" indicates compressive strain.Table 1. Summary of different strains and corresponding optical effects.
Apart from the techniques for generating homogeneous strain, new approaches for inducing spatially inhomogeneous strains are gaining popularity for application on scales down to tens of nanometers, e.g., wrinkling 2D materials or depositing 2D materials on nanostructures. These approaches include the use of gels as an elastomeric substrate to realize buckling-induced delamination41, where a viscoelastic stamp is used as a sample carrier (to transfer the 2D material to the nanostructure) that can later be washed away (or peeled off)80, 108, 113, 122, 127. The local strains induced by these methods are usually much stronger, leading to a significant enhancement of the local photon emission and long lifetimes of excited electrons. Because of these characteristics, such methods are finding increasing uses in the field of 2D-material-based quantum emitters80, 127. On the other hand, these methods suffer several disadvantages, including the costly and challenging nanofabrication process, and the lack of an effective model to quantify the local strains.