Upper limit on shift current generation in extended systems

Despite a long history of research into nonlinear response theory, there has been no systematic investigation into the maximum amount of nonlinear optical response attainable in solid-state materials. In this work, we present an upper bound on the second-order response functions of materials, which controls the shift current response. We show that this bound depends on the band gap, bandwidth, and geometrical properties of the material in question. We find that delocalized systems generally have larger responses than more localized or isolated ones. As a proof of principle, we perform first-principles calculations of the response tensors of a wide variety of materials, finding that the materials in our database do not yet saturate the upper bound. This suggests that new large shift current materials will likely be discovered by future materials research guided by the factors mentioned in this work.

Figure 1

Geometrical factor $\mathrm{\Xi }\left(\xi \right)$ for the upper bound on nonlinear optical response, as a function of the hopping range $\xi$, defined in Eq. (10) of the text. The geometrical factor is shown for different lattices and for different measurement directions. Anisotropic lattices show the highest potential for large nonlinear responses. Here, the tetragonal lattice has $c/a=2.0$ ratio, and has largest nonlinear response upper limit for light polarization and current measurement directions along the $c$ axis.

Figure 2

Integrated nonlinear response for a test set of semiconductors and semimetals. The largest tensor component of the integrated nonlinear response for each material is plotted against the band gap. Dashed lines indicate the value of the nonlinear response upper bound [Eq. (9)] as a function of the band gap, for different values of hopping strength ($A$) and geometrical factor ($\mathrm{\Xi }$). Select materials with large responses (${\mathrm{TaSe}}_2, {\mathrm{TaS}}_2, {\mathrm{Ca}}_3{\mathrm{Bi}}_2{\mathrm{O}}_7$, LaAlGe), or which deviate from the overall trend (SrGaSiH, SrAlSiH, CaAlSiH, NaSnP) are indicated on the plot. Shown in purple are the integrated nonlinear response of BaGaSiH and the theoretical bound constructed using the $A$ and $\mathrm{\Xi }$ values of BaGaSiH.

Figure 3

Wannier functions constructed from frontier (conduction and valence) orbitals of (a) BaGaSiH and (b) ${\mathrm{BaTiO}}_3$. BaGaSiH is a large second harmonic generation and shift current material, with integrated response tensor (see text) $\int \sigma d\omega =2.7\times {10}^{-6}\text{A}/\text{V}$. In contrast, ${\mathrm{BaTiO}}_3$ has relatively low nonlinear response, with $\int \sigma d\omega =1.6\times {10}^{-7}\text{A}/\text{V}$. These differences are explained in terms of the bonding character between the two materials. Isosurfaces of the Wannier functions of the two materials are plotted, with the isolevel chosen at 20% of the maximum value of the Wannier function. The Wannier orbitals of BaGaSiH ($\xi =0.80, \mathrm{\Xi }=10.12$) are more diffuse than those of ${\mathrm{BaTiO}}_3$ ($\xi =0.61, \mathrm{\Xi }=0.64$), giving rise to longer range hopping in BaGaSiH.