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Electron-phonon interaction in a spin-orbit coupled quantum wire with a gap

Interaction between electron and acoustic phonon in an in-plane magnetic field induced gapped quantum wire with Rashba spin-orbit interaction is studied. We calculate acoustic phonon limited resistivity ($ρ$) and phonon-drag thermopower ($S_g$) due to two well known mechanisms of electron-phonon interaction namely, deformation potential (DP) and piezoelectric (PE) scattering. In the so called Bloch-Gruneisen temperature limit both $ρ$ and $S_g$ depend on temperature ($T$) in a power law fashion i.e. $ρ$ or $S_g\sim T^{ν_T}$. For resistivity, $ν_T$ takes the value $5$ and $3$ due to DP and PE scattering respectively. On the other hand, $ν_T$ is $4$ and $2$ due to DP and PE scattering, respectively for phonon-drag thermopower. Additionally, we find numerically that $ν_T$ depends on Rashba parameter ($α$) and electron density ($n$). The dependence of $ν_T$ on $α$ becomes more prominent at lower density. We also study the variations of $ρ$ and $S_g$ with carrier density in the Bloch-Gruneisen regime. Through a numerical analysis a similar power law dependence $ρ$ or $S_g\sim n^{-ν_n}$ is established in which the effective exponent $ν_n$ undergoes a smooth transition from a low density behavior to a high density behavior. At a higher density regime, $ν_n$ matches excellently with the value obtained from theoretical arguments. Approximate analytical expressions for both resistivity and phonon-drag thermopower in the Bloch-Gruneisen regime are given.