Strain engineering in two-dimensional materials has emerged as a powerful tool to modulate their electronic, optical, and mechanical properties. Hexagonal boron nitride (hBN), with its atomically smooth surface, high thermal stability, and insulating nature, serves as an ideal substrate for strain engineering in 2D materials such as graphene, transition metal dichalcogenides (TMDCs), and black phosphorus. The ability to controllably introduce strain in these materials enables precise tuning of their band structures, leading to novel functionalities for flexible electronics, optoelectronics, and quantum devices.

One of the primary methods to induce controlled strain in 2D materials on hBN substrates is through mechanical deformation. This can be achieved by transferring the 2D material onto a pre-stretched elastomeric substrate, such as polydimethylsiloxane (PDMS), followed by relaxation to create compressive or tensile strain. The hBN substrate acts as an intermediate layer, ensuring minimal charge scattering and preserving the intrinsic electronic properties of the 2D material. Strain levels up to 10% have been demonstrated using this method, with precise control achievable by adjusting the pre-stretch parameters.

Another approach involves the use of patterned substrates. By depositing hBN on nanostructured surfaces, such as arrays of nanopillars or trenches, non-uniform strain fields can be generated in the overlying 2D material. The lattice mismatch between the 2D material and the hBN substrate further enhances strain effects, leading to localized modifications in the electronic structure. For instance, in monolayer MoS2 on hBN, strain gradients of 0.5% per micrometer have been reported, resulting in spatially varying bandgap modulation.

Thermal mismatch strain is also a viable technique. Due to differences in the thermal expansion coefficients of hBN and the 2D material, heating or cooling the heterostructure induces strain. For example, cooling a graphene-hBN stack from room temperature to cryogenic temperatures can generate tensile strain in graphene due to the higher thermal contraction of hBN. This method allows for reversible strain tuning, with strain values typically ranging from 0.1% to 1%.

The effects of strain on the electronic band structure of 2D materials are profound. In graphene, uniaxial strain breaks the hexagonal symmetry, leading to a shift in the Dirac cone and the emergence of pseudo-magnetic fields exceeding 100 Tesla for strains around 5%. This pseudo-magnetic field quantizes the electronic states into Landau levels, enabling zero-field quantum Hall effects. For TMDCs like WS2 or MoSe2, tensile strain reduces the bandgap by approximately 50 meV per percent strain, while compressive strain increases it. The strain-induced bandgap modulation is accompanied by changes in the effective masses of charge carriers, altering transport properties.

In black phosphorus, anisotropic strain along different crystal directions results in dramatic changes in its electronic properties. Applying tensile strain along the armchair direction reduces the bandgap, while strain along the zigzag direction can induce a transition from direct to indirect bandgap. The anisotropic response makes black phosphorus highly suitable for strain-sensitive devices, with carrier mobility variations exceeding 20% for strains below 2%.

The optical properties of 2D materials are equally sensitive to strain. Photoluminescence (PL) spectroscopy reveals strain-induced shifts in excitonic peaks, with red shifts observed under tensile strain and blue shifts under compressive strain. In MoS2, the PL peak position shifts at a rate of 30 meV per percent strain, providing a direct measure of the applied strain. Strain also affects exciton binding energies and trion populations, offering additional knobs for optoelectronic device engineering.

Applications of strain-engineered 2D materials on hBN substrates span multiple domains. In flexible electronics, strain-tunable transistors exhibit enhanced performance metrics. For example, strained graphene transistors on hBN demonstrate increased carrier mobility due to reduced phonon scattering, with values exceeding 100,000 cm²/Vs at room temperature. The combination of strain and hBN encapsulation also suppresses charge inhomogeneities, leading to more uniform device characteristics.

Strain-engineered photodetectors benefit from the tunable bandgap and enhanced light-matter interaction. A strained MoS2 photodetector on hBN shows a 40% increase in photoresponsivity compared to its unstrained counterpart, attributed to the strain-induced reduction in bandgap and improved carrier separation. The mechanical flexibility of these devices enables integration into wearable systems, where strain can be dynamically adjusted to optimize performance.

Quantum devices also leverage strain engineering for novel functionalities. Strain-induced pseudo-magnetic fields in graphene create confined states for quantum dots without external magnetic fields. These quantum dots exhibit discrete energy levels with tunable spacing, making them suitable for spin qubits. Similarly, strained TMDCs on hBN show valley polarization effects, where the degeneracy between K and K’ valleys is lifted, enabling valleytronic applications.

The mechanical robustness of hBN substrates ensures that strain-engineered devices maintain their performance under repeated bending or stretching cycles. Studies have shown that graphene-hBN heterostructures withstand over 1,000 bending cycles with minimal degradation in electrical properties, highlighting their potential for durable flexible electronics.

Challenges remain in achieving uniform strain distribution over large areas and in understanding the long-term stability of strained interfaces. Advances in transfer techniques and in-situ characterization methods are addressing these issues, paving the way for scalable fabrication of strain-engineered devices.

In summary, strain engineering using hBN substrates provides a versatile platform for tailoring the properties of 2D materials. The ability to precisely control strain enables unprecedented modulation of electronic and optical behaviors, driving innovations in flexible electronics, optoelectronics, and quantum technologies. As research progresses, the integration of strain engineering with other modulation techniques will further expand the functional landscape of 2D materials.