Phonon Drag in InSb: Theory and ``spin''-motive force

The phonon number operator $\hat n \to \sin^2 \frac{\theta}{2}$ defines the Euler angle $\theta$ and with the phase $\phi$ this maps to a precessing spin. Defined are a ``spin" Berry phase and a ``spin''-motive force (smf)[1]. Unlike an emf, an smf can act upon neutral phonons. Tradition[2] has sub-thermal phonons as central to the thermopower of semi-conductors. The momentum given to these phonons, by the temperature gradient, is transferred to the electrons by ``drag'' where it cancels a Seebeck effect electric field $\vec E$. Here, for InSb at low temperatures, thermal phonons actually relax momentum via boundary and umklapp scattering and energy conservation involves sub-thermal phonons, created by anharmonic effects, with a frequency $\hbar \omega_{\vec q} \sim k_B (dT/dx) \ell$ where $\ell$ is the phonon mean-free-path (mfp). The resulting smf acting upon the thermal phonons produces a ``spin'' voltage $\sim (k_B/e) \Delta T \sim 100\mu$V/K. Via the electron-phonon interaction, the smf, multiplied by the ratio $\ell_{ep}/\ell$, where $\ell_{ep}$ is the electron-phonon mfp, are detected, but not created by the few electrons in our InSb samples. [1] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. {\bf 98}, 246601 (2007) [2] C. Herring, Phys. Rev. {\bf 95}, 954 (1954).