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We present a theoretical model of composite excitonic states in doped semiconductors. Many-body interactions between a photoexcited electron-hole pair and the electron gas are integrated into a computationally tractable few-body problem, solved by the variational method. We focus on electron-doped ML-MoSe$_2$ and ML-WSe$_2$ due to the contrasting character of their conduction bands. In both cases, the core of the composite is a tightly-bound trion (two electrons and valence-band hole), surrounded by a region depleted of electrons. The composite in ML-WSe$_2$ further includes a satellite electron with different quantum numbers. The theory is general and can be applied to semiconductors with various energy-band properties, allowing one to calculate their excitonic states and to quantify the interaction with the Fermi sea.

The monolayer transition metal dichalcogenides are an emergent semiconductor platform exhibiting rich excitonic physics with coupled spin-valley degree of freedom and optical addressability. Here, we report a new series of low energy excitonic emission lines in the photoluminescence spectrum of ultraclean monolayer WSe2. These excitonic satellites are composed of three major peaks with energy separations matching known phonons, and appear only with electron doping. They possess homogenous spatial and spectral distribution, strong power saturation, and anomalously long population (> 6 $��$s) and polarization lifetimes (> 100 ns). Resonant excitation of the free inter- and intra-valley bright trions leads to opposite optical orientation of the satellites, while excitation of the free dark trion resonance suppresses the satellites photoluminescence. Defect-controlled crystal synthesis and scanning tunneling microscopy measurements provide corroboration that these features are dark excitons bound to dilute donors, along with associated phonon replicas. Our work opens opportunities to engineer homogenous single emitters and explore collective quantum optical phenomena using intrinsic donor-bound excitons in ultraclean 2D semiconductors., to be appear in Nature Communications

The coupling between spin, charge, and lattice degrees of freedom plays an important role in a wide range of fundamental phenomena. Monolayer semiconducting transitional metal dichalcogenides have emerged as an outstanding platform for studying these coupling effects because they possess unique spin-valley locking physics for hosting rich excitonic species and the reduced screening for strong Coulomb interactions. Here, we report the observation of multiple valley phonons, phonons with momentum vectors pointing to the corners of the hexagonal Brillouin zone, and the resulting exciton complexes in the monolayer semiconductor WSe2. From Lande g-factor and polarization analyses of photoluminescence peaks, we find that these valley phonons lead to efficient intervalley scattering of quasi particles in both exciton formation and relaxation. This leads to a series of photoluminescence peaks as valley phonon replicas of dark trions. Using identified valley phonons, we also uncovered an intervalley exciton near charge neutrality, and extract its short-range electron-hole exchange interaction to be about 10 meV. Our work not only identifies a number of previously unknown 2D excitonic species, but also shows that monolayer WSe2 is a prime candidate for studying interactions between spin, pseudospin, and zone-edge phonons., to appear in Nature Communications

In the archetypal monolayer semiconductor WSe_{2}, the distinct ordering of spin-polarized valleys (low-energy pockets) in the conduction band allows for studies of not only simple neutral excitons and charged excitons (i.e., trions), but also more complex many-body states that are predicted at higher electron densities. We discuss magneto-optical measurements of electron-rich WSe_{2} monolayers and interpret the spectral lines that emerge at high electron doping as optical transitions of six-body exciton states ("hexcitons") and eight-body exciton states ("oxcitons"). These many-body states emerge when a photoexcited electron-hole pair interacts simultaneously with multiple Fermi seas, each having distinguishable spin and valley quantum numbers. In addition, we explain the relations between dark trions and satellite optical transitions of hexcitons in the photoluminescence spectrum.

Monolayers of transition metal dichalcogenides are ideal materials to control both spin and valley degrees of freedom either electrically or optically. Nevertheless, optical excitation mostly generates excitons species with inherently short lifetime and spin/valley relaxation time. Here we demonstrate a very efficient spin/valley optical pumping of resident electrons in n-doped WSe2 and WS2 monolayers. We observe that, using a continuous wave laser and appropriate doping and excitation densities, negative trion doublet lines exhibit circular polarization of opposite sign and the photoluminescence intensity of the triplet trion is more than four times larger with circular excitation than with linear excitation. We interpret our results as a consequence of a large dynamic polarization of resident electrons using circular light. Optical excitation of transition metal dichalcogenide monolayers mostly generates excitons species with inherently short lifetime and spin/valley relaxation time. Here, the authors demonstrate efficient spin/valley optical pumping of resident electrons in n-doped WSe2 and WS2 monolayers.

While materials design for many device applications usually relies on adding impurities, recent advances in scaling-down heterostructures with improved interfacial properties offer a different way to transform a large class of materials. A given material can be drastically changed by inheriting properties leaking from its neighboring regions, such as magnetism, superconductivity, or spin-orbit coupling. While these proximity effects often have a short range and are considered negligible, the situation is qualitatively different in atomically thin and two-dimensional materials where the extent of proximity effects can exceed their thickness. Consequently, proximitized materials have a potential to display novel properties and device opportunities, absent in any of the constituent region of the considered heterostructures. Such proximitized materials could provide platforms for a wide range of emerging applications: from seamless integration of memory and logic, to fault-tolerant topologically protected quantum computing.

Advances in scaling down heterostructures and having an improved interface quality together with atomically-thin two-dimensional materials suggest a novel approach to systematically design materials. A given material can be transformed through proximity effects whereby it acquires properties of its neighbors, for example, becoming superconducting, magnetic, topologically nontrivial, or with an enhanced spin-orbit coupling. Such proximity effects not only complement the conventional methods of designing materials by doping or functionalization, but can also overcome their various limitations. In proximitized materials it is possible to realize properties that are not present in any constituent region of the considered heterostructure. While the focus is on magnetic and spin-orbit proximity effects with their applications in spintronics, the outlined principles provide also a broader framework for employing other proximity effects to tailor materials and realize novel phenomena., Comment: Invited Review to appear in Materials Today, 28 pages, 22 figures

The electron valley and spin degree of freedom in monolayer transition-metal dichalcogenides can be manipulated in optical and transport measurements performed in magnetic fields. The key parameter for determining the Zeeman splitting, namely, the separate contribution of the electron and hole g factor, is inaccessible in most measurements. Here we present an original method that gives access to the respective contribution of the conduction and valence band to the measured Zeeman splitting. It exploits the optical selection rules of exciton complexes, in particular the ones involving intervalley phonons, avoiding strong renormalization effects that compromise single particle g-factor determination in transport experiments. These studies yield a direct determination of single band g factors. We measure g_{c1}=0.86±0.1, g_{c2}=3.84±0.1 for the bottom (top) conduction bands and g_{v}=6.1±0.1 for the valence band of monolayer WSe_{2}. These measurements are helpful for quantitative interpretation of optical and transport measurements performed in magnetic fields. In addition, the measured g factors are valuable input parameters for optimizing band structure calculations of these 2D materials.

The valley degree of freedom is a sought-after quantum number in monolayer transition-metal dichalcogenides. Similar to optical spin orientation in semiconductors, the helicity of absorbed photons can be relayed to the valley (pseudospin) quantum number of photoexcited electrons and holes. Also similar to the quantum-mechanical spin, the valley quantum number is not a conserved quantity. Valley depolarization of excitons in monolayer transition-metal dichalcogenides due to long-range electron-hole exchange typically takes a few ps at low temperatures. Exceptions to this behavior are monolayers MoSe$_2$ and MoTe$_2$ wherein the depolarization is much faster. We elucidate the enigmatic anomaly of these materials, finding that it originates from Rashba-induced coupling of the dark and bright exciton branches next to their degeneracy point. When photoexcited excitons scatter during their energy relaxation between states next to the degeneracy region, they reach the light cone after losing the initial helicity. The valley depolarization is not as fast in monolayers WSe$_2$, WS$_2$ and likely MoS$_2$ wherein the Rashba-induced coupling is negligible., We would appreciate any feedback

Monolayer transition-metal dichalcogenides (ML-TMDs) offer exciting opportunities to test the manifestations of many-body interactions through changes in the charge density. Tuning the charge density by a gate voltage leads to profound changes in the optical spectra of excitons in ML-TMDs. We review the band-gap renormalization and dynamical screening as a function of charge density, and then incorporate these effects through various approximations that model long-wavelength charge excitations in the Bethe-Salpeter Equation (BSE). We then show that coupling between excitons and shortwave charge excitations is essential to resolve several experimental puzzles. Unlike ubiquitous and well-studied plasmons, driven by collective oscillations of the background charge density in the long-wavelength limit, we discuss the emergence of shortwave plasmons that originate from the short-range Coulomb interaction through which electrons transition between the $\mathbf{K}$ and $-\mathbf{K}$ valleys. We study the coupling between the shortwave plasmons and the neutral exciton through the self-energy of the latter. We then elucidate how this coupling as well as the spin ordering in the conduction band give rise to an experimentally observed optical sideband in electron-doped W-based MLs, conspicuously absent in electron-doped Mo-based MLs or any hole-doped ML-TMDs. While the focus of this review is on the optical manifestations of many-body effects in ML-TMDs, a systematic description of the dynamical screening and its various approximations allow one to revisit other phenomena, such as nonequilibrium transport or superconducting pairing, where the use of the BSE or the emergence of shortwave plasmons can play an important role., Comment: This revised version is far more extensive. Any feedback is welcome

Photoluminescence experiments from monolayer transition-metal dichalcogenides often show that the binding energy of trions is conspicuously similar to the energy of optical phonons. This enigmatic coincidence calls into question whether phonons are involved in the radiative recombination process. We address this problem, unraveling an intriguing optical transition mechanism. Its initial state is a localized charge (electron or hole) and delocalized exciton. The final state is the localized charge, phonon and photon. In between, the intermediate state of the system is a virtual trion formed when the localized charge captures the exciton through emission of the phonon. We analyze the difference between radiative recombinations that involve real and virtual trions (i.e., with and without a phonon), providing useful ways to distinguish between the two in experiment., Any Feedback is welcome. Enjoy the reading

An upconversion from negatively charged to neutral excitons is observed in monolayer tungsten diselenide, which could provide a route for cooling two-dimensional semiconductors using lasers.

Uniaxial compressive strain along the [001] direction strongly suppresses the spin relaxation in silicon. When the strain level is large enough so that electrons are redistributed only in the two valleys along the strain axis, the dominant scattering mechanisms are quenched and electrons mainly experience intra-axis scattering processes (intravalley or intervalley scattering within valleys on the same crystal axis). We first derive the spin-flip matrix elements due to intra-axis electron scattering off impurities, and then provide a comprehensive model of the spin relaxation time due to all possible interactions of conduction-band electrons with impurities and phonons. We predict a nearly three orders of magnitude improvement in the spin relaxation time of $\ensuremath{\sim}{10}^{19}\phantom{\rule{4pt}{0ex}}{\text{cm}}^{\ensuremath{-}3}$ antimony-doped silicon (Si:Sb) at low temperatures.

Ge is emerging as an interesting material for spintronic applications. We demonstrate that in SiGe heterostructures strain and quantum confinement effects can be used to tailor spin related properties enhancing the spin polarization of injected carriers above the bulk limit. Moreover the fast Γ-L electron scattering and the strain induced removal of the heavy-hole, light-hole degeneracy, make SiGe heterostructures an ideal material platform for the study of hole dynamics.

Future development in spintronic devices will require an advanced control of spin currents, for example by an electric field. Here we demonstrate an approach that differs from previous proposals such as the Datta and Das modulator, and that is based on a van de Waals heterostructure of atomically thin graphene and semiconducting MoS2. Our device combines the superior spin transport properties of graphene with the strong spin–orbit coupling of MoS2 and allows switching of the spin current in the graphene channel between ON and OFF states by tuning the spin absorption into the MoS2 with a gate electrode. Our proposal holds potential for technologically relevant applications such as search engines or pattern recognition circuits, and opens possibilities towards electrical injection of spins into transition metal dichalcogenides and alike materials., By forming heterostructures of different layered two-dimensional materials, functional spintronic devices may be built by exploiting the materials' different spin-orbit coupling and spin transport properties. Here, the authors demonstrate a spin switch in a gated structure of graphene and MoS2.

This paper presents a 1Mb STT-MRAM consisting of novel planar memory cells with built-in differential voltage output. Each memory cell contains four planar ferromagnets, one PMOS and one NMOS to enable read operation, one NMOS to enable write operation and four transmission gates to pass write current in opposite directions. The output differential voltage of two read contacts in each memory cell is 23.66 mV for logic '1' and only 0.46 mV for logic '0', which renders high signal-to-noise ratio. Estimated area of each memory cell in 65-nm CMOS technology is 1.5 um2 (355 F2). The proposed 1Mb STT-MRAM consumes dynamic write power of 12.8 mW, static leakage power of 4mW and dynamic read power of 11.10 mW to achieve 100MHz operation.

Exciton optical transitions in transition-metal dichalcogenides offer unique opportunities to study rich many-body physics. Recent experiments in monolayer WSe$_2$ and WS$_2$ have shown that while the low-temperature photoluminescence from neutral excitons and three-body complexes is suppressed in the presence of elevated electron densities or strong photoexcitation, new dominant peaks emerge in the low-energy side of the spectrum. I present a theory that elucidates the nature of these optical transitions showing the role of the intervalley Coulomb interaction. After deriving a compact dynamical form for the Coulomb potential, I calculate the self-energy of electrons due to their interaction with this potential. For electrons in the upper valleys of the spin-split conduction band, the self energy includes a moderate redshift due to exchange, and most importantly, a correlation-induced virtual state in the band-gap. The latter sheds light on the origin of the luminescence in monolayer WSe$_2$ and WS$_2$ in the presence of pronounced many-body interactions., Comment: Figure 1 includes two animations (activated by clicking on the figures). I will appreciate any feedback provided on the content or missing references

We report the experimental demonstration of a magnetologic gate built on graphene at room temperature. This magnetologic gate consists of three ferromagnetic electrodes contacting a single layer graphene spin channel and relies on spin injection and spin transport in the graphene. We utilize electrical bias tuning of spin injection to balance the inputs and achieve "exclusive or" (XOR) logic operation. Furthermore, simulation of the device performance shows that substantial improvement towards spintronic applications can be achieved by optimizing device parameters such as device dimensions. This advance holds promise as a basic building block for spin-based information processing., Comment: 23 pages, 4 figures

We present a seamless integration of spin-based memory and logic circuits. The building blocks are magnetologic gates based on a hybrid graphene/ferromagnet material system. We use network search engines as a technology demonstration vehicle and simulate a high-speed, small-area, and low-power spin-based circuit.

The polarization analysis of optical transitions in monolayer and bilayer transition-metal dichalcogenides provides invaluable information on the spin and valley (pseudospin) degrees of freedom. To explain optical properties of a given monolayer transition-metal dichalcogenide, one should consider (i) the order of its spin-split conduction bands, (ii) whether intervalley scattering is prone to phonon bottleneck, (iii) and whether valley mixing by electron-hole exchange can take place. Using these principles, we present a consistent physical picture that elucidates a variety of features in the optical spectra of these materials. We explain the differences between optical transitions in monolayer MoSe$_2$ and monolayer WSe$_2$, finding that indirect excitons in the latter correspond to several low-energy optical transitions that so far were attributed to excitons bound to impurities. A possible mechanism that can explain the vanishing polarization in MoSe$_2$ is discussed. Finally, we consider the effect of an out-of-plane electric field, showing that it can reduce the initial polarization of bright excitons due to a Rashba-type coupling with dark excitons., we welcome any feedback (including missing references if prior to the official peer-reviewed publication)

We present a theoretical model for gain and noise saturation in quantum dash (QDash) semiconductor optical amplifiers. The model is based on the density matrix formalism and addresses static saturation spectra. The calculations are confirmed by a series of experiments which highlight the unique properties of these amplifiers. We demonstrate a high gain, a wide bandwidth, and high saturation power. The saturation spectrum is shown to be asymmetric, emphasizing saturation at short wavelength. The asymmetry stems from the high energy tail of the density of state function in those quantum wire (QWire) like gain media as well as from the interactions with the wetting layer.

We present a general theoretical model for the optical differential gain in semiconductor lasers. The model describes self assembly quantum dots (QDs), self assembly quantum wires (QWRs) and single quantum-well lasers. We have introduced the inhomogeneous broadening due to size fluctuations in the assembly cases. At each dimensionality, we have considered the carrier populations in the excited states and in the reservoirs, where conduction and valence bands are treated separately. We show that for room temperature operation the differential gain reduction due to increased size inhomogeneity is more pronounced in QDs than in QWRs. We show this reduction to be smaller than the one-order reduction attributed to state filling in conventional dot and wire assemblies operating at room temperature. The integration prefactor coefficient of the differential gain in zero-dimensional cases exceed one- and two-dimensional coefficients only for low temperatures where the homogenous broadening is considerably smaller than the thermal energy. The differential gain of QDs, QWRs, and compressively strained single quantum-well lasers operating at room temperature and close to equilibrium is nearly the same.

We describe a detailed model for the dynamical and spectral properties of quantum dash (quantum wire assembly) lasers. We use a self-consistent semiclassical theory for a multimode laser field which interacts with an inhomogeneously broadened assembly of quantum wires via the quantum mechanical radiation-matter interaction. Our comprehensive coupled equations are spectrally resolved enabling to study accurately the effect of the gain inhomogeneity. Carrier-carrier and carrier-phonon scattering are also included. We highlight the effective capture rate which is determined by the ratio between the number of states in the reservoir and in the assembly, the energetic region into which carriers are captured and the width of the inhomogeneously broadened gain. Specifically, we demonstrate that a large number of states ratio lowers both the linear optical differential gain and the nonlinear gain coefficient. We show that gain suppression dominates when a realistic energy range into which capture takes place is considered as well as for small number of states ratios. In addition, we show that the width of the inhomogeneous broadening plays a relatively small role. We conclude that the differential gain and nonlinear damping can not be optimized simultaneously. These results point therefore to the clear advantages offered by laser structures which employ non conventional carrier injection schemes such as tunnelling barrier or n-type /spl delta/-doping regions.

We present a theory for resonance-tunneling magnetoresistance (MR) in Ferromagnetic-Insulator-Nonmagnetic junctions. The theory sheds light on many of the recent electrical spin injection experiments, suggesting that this MR effect rather than spin accumulation in the nonmagnetic channel corresponds to the electrically detected signal. We quantify the dependence of the tunnel current on the magnetic field by quantum rate equations derived from the Anderson impurity model, with important addition of impurity spin interactions. Considering the on-site Coulomb correlation, the MR effect is caused by competition between the field, spin interactions and coupling to the magnetic lead. By extending the theory, we present a basis for operation of novel nm-size memories., Comment: 6 pages, 3 figures and 5-page supplemental material, bringing it closest to the published version

We present a theory for spin relaxation of electrons due to scattering off the central-cell potential of impurities in silicon. Taking into account the multivalley nature of the conduction band and the violation of translation symmetry, the spin-flip amplitude is dominated by this short-range impurity scattering after which the electron is transferred to a valley on a different axis in $k$-space (the so called $f$-process). These $f$-processes dominate the spin relaxation at all temperatures, where scattering off the impurity central-cell dominate at low temperatures, and scattering with $\Sigma$-axis phonons at elevated temperatures. To the best of our knowledge, the theory is the first to explain and accurately quantify the empirically-found dependence of spin relaxation on the impurity identity. Accordingly, the new formalism fills a longstanding gap in the spin relaxation theory of $n$-type silicon, and it is valuable for characterization of silicon-based spintronic devices., Comment: 4 pages of main text, 7 pages of appendices, 2 Figures

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 longitudinal magnetic field. 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 ns at low temperature, far beyond any other available experimental results., Comment: to appear in Phys. Rev. Lett

The strain effects on the spin-orbit induced splitting of the valence band maximum and conduction band minimum in monolayer MoS2 and the gap in graphene are calculated using first-principles calculations. The dependence of these splittings on the various symmetry types of strain is described by means of an effective Hamiltonian based on the method of invariants and the parameters in the model are extracted by fitting to the theory. These splittings are related to acoustic phonon deformation potentials, or electron-phonon coupling matrix elements which enter the spin-dependent scattering theory of conduction in these materials., 9 pages, 5 figures

Spin orientation of photoexcited carriers and their energy relaxation are investigated in bulk Ge by studying spin-polarized recombination across the direct band gap. The control over parameters such as doping and lattice temperature is shown to yield a high polarization degree, namely larger than 40%, as well as a fine tuning of the angular momentum of the emitted light with a complete reversal between right- and left-handed circular polarization. By combining the measurement of the optical polarization state of band-edge luminescence and Monte Carlo simulations of carrier dynamics, we show that these very rich and complex phenomena are the result of the electron thermalization and cooling in the multivalley conduction band of Ge. The circular polarization of the direct-gap radiative recombination is indeed affected by energy relaxation of hot electrons via the X valleys and the Coulomb interaction with extrinsic carriers. Finally, thermal activation of unpolarized L valley electrons accounts for the luminescence depolarization in the high temperature regime. © 2013 American Physical Society.

We present a theory that elucidates the major momentum and spin relaxation processes for electrons, holes and hot excitons in monolayer transition-metal dichalcogenides. We expand on spin flips induced by flexural phonons and show that the spin relaxation is ultrafast for electrons in free-standing membranes while being mitigated in supported membranes. This behavior due to interaction with flexural phonons is universal in two-dimensional membranes that respect mirror symmetry and it leads to a counterintuitive inverse relation between mobility and spin relaxation., The published version. 5 pages main text is followed by 5 pages supplemental materials

At low temperatures, electrons in semiconductors are bound to shallow donor impurity ions, neutralizing their charge in equilibrium. Inelastic scattering of other externally-injected conduction electrons accelerated by electric fields can excite transitions within the manifold of these localized states. Promotion of the bound electron into highly spin-orbit-mixed excited states drives a strong spin relaxation of the conduction electrons via exchange interactions, reminiscent of the Bir-Aronov-Pikus process where exchange occurs with valence band hole states. Through low-temperature experiments with silicon spin transport devices and complementary theory, we reveal the consequences of this previously unknown spin depolarization mechanism both below and above the impact ionization threshold.

We performed spin-resolved photoluminescence measurements on Ge/SiGe multiple quantum wells with different well thickness and using different exciting power densities. The polarization of the direct emission strongly depends on the relative weight of electrons photoexcited from the light and the heavy hole subbands. The study of the polarization as a function of the exciting power highlights the role of the carrier-carrier interactions in determining spin depolarization. © 2013 AIP Publishing LLC.

We report spectrally resolved dynamical properties of 1550nm quantum-dash lasers. The dynamics are governed by the inhomogeneous broadening of the gain medium and by wavelength-dependent carrier capture and escape rates.

Several recent measurements of magnetoresistance in ferromagnet/insulator/semiconductor tunnel junctions have led to controversial claims of spin accumulation, where the signal amplitude and derived spin relaxation time often deviates from theoretically calculated and experimentally confirmed values by several orders of magnitude. These discrepancies cast doubt on the physical origin of the measured magnetoresistance, which was initially attributed to spin precession and dephasing in the semiconductor. More recently, models incorporating transport through localized defects have shown that they can account for device behavior, without any spin accumulation. To directly investigate the role of localized states in this signal, we subject CoFe/SiO2/n-Si junctions to varying doses of proton irradiation. Weak radiation doses not only have little effect on the electrical current-voltage relationship of the junction but also modify the magnetoresistance substantially. Our interpretation of this phenomenon involves the hydrogen passivation of defects within the tunnel barrier, and is consistent with the emerging consensus that defects within the barrier play a crucial role in the physical mechanism behind junction magnetoresistance in this class of devices.

Dynamical properties of cross-gain modulation (XGM) within the inhomogeneously broadened gain spectrum of an InP quantum dash optical amplifier operating at 1550 nm are examined. The small-signal XGM modulation bandwidth increases with the carrier escape time, which is achieved at long probe wavelengths. The nature of the XGM dynamics is confirmed by spectrally resolved optical modulation response measurements in quantum dash lasers.

We present a methodology to systematically and analytically treat phonon-induced spin relaxation of conduction electron in silicon. All leading order contribution from all phonon modes and scattering processes are considered and the results for spin-flip matrix elements and spin lifetime are summarized. We show the explicit dependence of matrix elements on the electron wavevectors, spin orientation and phonon polarization. These results are shown to be powerful especially under symmetry-breaking conditions when an averaging rough evaluation of the matrix elements is not sufficient. Corrections due to the special two-band degeneracy in the X point (near the conduction valley minima) are also discussed. Numerical calculation are used to confirm the analytical results.

We investigate the intrinsic spin relaxation of conduction electrons in germanium due to electron-phonon 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 spin-flip matrix element is derived from the eigenstates of a compact spin-dependent $\mathbf{k}$$\cdot$$\mathbf{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., Comment: 13 pages, 2 figures

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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., Comment: 4 pages, 1 figure, 1 table

We show that the electric field-induced thermal asymmetry between the electron and lattice systems in pure silicon substantially impacts the identity of the dominant spin relaxation mechanism. Comparison of empirical results from long-distance spin transport devices with detailed Monte-Carlo simulations confirms a strong spin depolarization beyond what is expected from the standard Elliott-Yafet theory already at low temperatures. The enhanced spin-flip mechanism is attributed to phonon emission processes during which electrons are scattered between conduction band valleys that reside on different crystal axes. This leads to anomalous behavior, where (beyond a critical field) reduction of the transit time between spin-injector and spin-detector is accompanied by a counterintuitive reduction in spin polarization and an apparent negative spin lifetime.