We present a symmetry analysis of electronic bandstructure including spin-orbit interaction close to the insulating gap edge in monolayer black phosphorus ('phosphorene'). Expressions for energy dispersion relation and spin-dependent eigenstates for electrons and holes are found via simplification of a perturbative expansion in wavevector $k$ away from the zone center using elementary group theory. Importantly, we expose the underlying symmetries giving rise to substantial anisotropy in optical absorption, charge and spin transport properties, and reveal the mechanism responsible for valence band distortion and possible lack of a true direct gap

We investigate the intrinsic spin relaxation of conduction electrons in germanium due to electronphonon scattering. We derive intravalley and intervalley spin-flip matrix elements for a general spin orientation and quantify the resulting anisotropy in spin relaxation. The form of the intravalley spinflip matrix element is derived from the eigenstates of a compact spin-dependent k·p Hamiltonian in the vicinity of the L point (where thermal electrons are populated in Ge). Spin lifetimes from analytical integrations of the intravalley and intervalley matrix elements show excellent agreement with independent results from elaborate numerical methods.

We derive a spin-dependent Hamiltonian that captures the symmetry of the zone edge states in silicon. We present analytical expressions of the spin-dependent states and of spin relaxation due to electron-phonon interactions in the multivalley conduction band. We find excellent agreement with experimental results. Similar to the usage of the Kane Hamiltonian in direct band-gap semiconductors, the new Hamiltonian can be used to study spin properties of electrons in silicon.PACS numbers: 78.60.Fi, 71.70.Ej Silicon is an ideal material choice for spintronics due to its relatively long spin relaxation time and central role in semiconductor technology. These characteristics are the reason for the wide interest in recent spin injection experiments [1][2][3][4]. To date, however, modeling of basic spin properties in silicon required elaborate numerical methods [5]. Notably, the availability of transparent spin-dependent theories in direct gap semiconductors have spurred the field of semiconductor spintronics [6]. The importance of a lucid theory that accurately describes spin properties of conduction electrons in silicon with relatively simple means is thus clear.In the first part of this letter we derive a Hamiltonian that captures spin properties of conduction electrons in silicon. The Hamiltonian is constructed by its invariance to the symmetry operations of the space group, G 2 32 , which describes the symmetry of the X-point at the edge of the Brillouin zone [7,8]. In silicon the X-point is closer to the absolute conduction band minimum than all other high symmetry points. While k·p and tight-binding models have been available for many decades [9][10][11][12][13][14][15][16][17], spin has heretofore been ignored since spin-orbit coupling in Si is weak [18][19][20][21][22] and lattice inversion symmetry causes spin degeneracy. The present work is motivated by the emergence of experimental work on spin-polarized electron transport in silicon [1][2][3][4].In the second part, this Hamiltonian is used to elucidate the nature of intravalley and intervalley spin relaxation processes in silicon due to electron-phonon interactions. Our approach unravels the underlying physics, structure and symmetries of dominant spin-flip mechanisms. These insights cannot be shown by state-of-the art numerical studies in which only the magnitude and temperature dependence are calculated [5]. We derive analytical forms and selection rules of the dominant spin-flip matrix elements and explain the subtle distinction between spin and momentum scattering processes. Importantly, it is shown that spin relaxation due to intravalley scattering is caused by coupling of the lower and upper conduction bands (whereas intravalley momentum relaxation is governed by dilation and uniaxial deformation potentials of the lower conduction band). The accepted

A unique spin depolarization mechanism, induced by the presence of g-factor anisotropy and intervalley scattering, is revealed by spin-transport measurements on long-distance germanium devices in a magnetic field longitudinal to the initial spin orientation. The confluence of electron-phonon scattering (leading to Elliott-Yafet spin flips) and this previously unobserved physics enables the extraction of spin lifetime solely from spin-valve measurements, without spin precession, and in a regime of substantial electric-field-generated carrier heating. We find spin lifetimes in Ge up to several hundreds of nanoseconds at low temperature, far beyond any other available experimental results.