I have been using the Deformation Potential theory to evaluate the electron scattering rates in nanomaterials. In each case, I was under the impression that the Conduction Band Minimum (CBM) and the Valence Band Maximum (VBM) should be expected to vary linearly with applied strain. For example, refer to Figure 2 of Reference 1.

However, Reference 2 (Figure 5) shows the CBM and VBM varying in a non-linear form.

Is there an explanation for such a variation? Was it wrong to assume that the band edges of semiconductors would vary linearly with applied strain?

Note: I have read a significant number of articles (for various materials and nano-structures) where the band edges vary linearly.


  1. Chen, X.; Zhang, X.; Gao, J.; Li, Q.; Shao, Z.; Lin, H.; Pan, M. Computational Search for Better Thermoelectric Performance in Nickel-Based Half-Heusler Compounds. ACS Omega, 2021, 6, 18269–18280. https://doi.org/10.1021/acsomega.1c02172.
  2. Ma, W.; Record, M.-C.; Tian, J.; Boulet, P. Strain Effects on the Electronic and Thermoelectric Properties of n(PbTe)-m(Bi2Te3) System Compounds. Materials, 2021, 14, 4086. https://doi.org/10.3390/ma14154086.
  • $\begingroup$ I would never compare ACS to MDPI unless you are certain the MDPI authors are of high scientific profiles. $\endgroup$
    – Sha
    Jun 29, 2023 at 19:18
  • 1
    $\begingroup$ @Sha all comparisons aside, Is it possible for the band edges to vary non-linearly with applied strain? $\endgroup$
    – PBH
    Jun 30, 2023 at 7:05
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    $\begingroup$ Not commenting on the two papers you cited, but it would not surprise me if there can be a nonlinear behavior at high strain, as in doi.org/10.1016/j.commatsci.2018.05.047 $\endgroup$
    – Anyon
    Jul 1, 2023 at 15:28
  • $\begingroup$ Could this by any chance be related to the anisotropy in the valence band induced by the strain as mentioned in this article? pubs.aip.org/aip/jap/article-abstract/113/18/183718/1015663/… $\endgroup$
    – PBH
    Jul 2, 2023 at 4:50

1 Answer 1


Generally speaking, one would expect that the band edges should vary linearly with applied strain, especially for small strains. This linear relationship is often observed in many materials and nanostructures, as you mentioned in your question. However, there are cases where the band edges deviate from linearity. Take a look at those examples https://doi.org/10.1103/PhysRevB.91.085423 and https://doi.org/10.1021/acs.jpcc.7b12408.

There are several reasons that could lead to this non-linear variation of band edges with strain and here I will mention some of them:

  1. Anisotropy: as you have mentioned in your comment, anisotropy can lead to non-linear variations in the band edges, where materials with anisotropic properties could show different behavior based on the direction of the applied strain.
  2. electron-electron interaction: for some materials with complex electronic structure, such as high electron-electron interactions, it is possible that the relationship between band edges and strain may not be linear.
  3. Piezoelectric contribution, the coupling between strain and polarization, can contribute to non-linear variations in the band edges
  4. Large strain: as mentioned in the comments non-linear strain regime is expected at higher values of strain where the material reaches the elastic limit or for instance undergoing geometrical changes.

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