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We investigate the effects of strain on superconductivity with particularreference to SrTiO$_3$. Assuming that a ferroelectric mode that softens undertensile strain is responsible for the coupling, an increase in the criticaltemperature and range of carrier densities for superconductivity is predicted,while the peak of the superconducting dome shifts towards lower carrierdensities. Using a Ginzburg-Landau approach in 2D, we find a linear dependenceof the critical temperature on strain: if the couplings between the orderparameter and strains in different directions differ while their sum is fixed,different behaviours under uniaxial and biaxial (uniform) strain can beunderstood.

Strain and ferroelectric soft mode induced superconductivity in strontium titanate