Fig. 3: Phonon-mediated spin polarization.
From: Phonon-mediated spin transport in quantum paraelectric metals
a Schematic illustration of the vector calculations involved in computing phonon-mediated spin polarization. The outcome, \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\cdot {{{{\boldsymbol{P}}}}}_{\perp }\), represents the value of the desired spin polarization. b–d Plots of \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\cdot {{{{\boldsymbol{P}}}}}_{\perp }\) as a function of the azimuthal (ϕk) and polar (θk) angles of \(\hat{{{{\boldsymbol{k}}}}}=(\sin {\theta }_{k}\cos {\phi }_{k},\sin {\theta }_{k}\sin {\phi }_{k},\cos {\theta }_{k})\). b Plot for \({{{\boldsymbol{P}}}}\parallel {\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\), where \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }=\hat{z}\) and \({{{\boldsymbol{P}}}}=| {{{\boldsymbol{P}}}}| \hat{z}\). (c) Plot for \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }=\hat{z}\) and \({{{\boldsymbol{P}}}}=| {{{\boldsymbol{P}}}}| (\sqrt{2}/2,0,\sqrt{2}/2)\). (d) Plot for \({{{\boldsymbol{P}}}}\perp {\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\), where \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }=\hat{z}\) and \({{{\boldsymbol{P}}}}=| {{{\boldsymbol{P}}}}| \hat{x}\). The unit vectors \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\) and \(\hat{{{{\boldsymbol{k}}}}}\) denote the directions of the spin polarization and the phonon wave vector, respectively. P = q × (p + k/2) denotes the vector product of q and p + k/2. \({{{{\boldsymbol{P}}}}}_{\perp }={{{\boldsymbol{P}}}}-\hat{{{{\boldsymbol{k}}}}}(\hat{{{{\boldsymbol{k}}}}}\cdot {{{\boldsymbol{P}}}})\) represents the vector rejection of P from \(\hat{{{{\boldsymbol{k}}}}}\). (b-d) The color scale denotes the value of \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\cdot {{{{\boldsymbol{P}}}}}_{\perp }\). Specific relative orientations between \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\) and P are utilized in each plot. The cyan and magenta lines indicate the conditions under which \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\cdot {{{{\boldsymbol{P}}}}}_{\perp }\) vanishes, corresponding to \(\hat{{{{\boldsymbol{k}}}}}| | \pm {\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\) and \({\hat{{{{\boldsymbol{e}}}}}}_{\alpha }\perp {{{{\boldsymbol{P}}}}}_{\perp }\), respectively.
