Electron dynamics in crystalline semiconductors

Electron dynamics in crystalline semiconductors is described by distinguishing between an instantaneous velocity related to electron's momentum and an average velocity related to its quasi-momentum in a periodic potential. It is shown that the electron velocity used in the theory of electron transport and free-carrier optics is the average electron velocity, not the instantaneous velocity. An effective mass of charge carriers in solids is considered and it is demonstrated that, in contrast to the "acceleration" mass introduced in textbooks, it is a "velocity" mass relating carrier velocity to its quasi-momentum that is a much more useful physical quantity. Among other advantages, the velocity mass is a scalar for spherical but nonparabolic energy bands $\epsilon(k)$, whereas the acceleration mass is not a scalar. Important applications of the velocity mass are indicated. A two-band {\bm k}\cdot {\bm \hp} model is introduced as the simplest example of a band structure that still keeps track of the periodic lattice potential. It is remarked that the two-band model, adequately describing narrow-gap semiconductors (including zero-gap graphene), strongly resembles the special theory of relativity. Instructive examples of the "semi-relativistic" analogy are given. The presentation has both scientific and pedagogical aspects.Comment: 7 pages; 1 figur

Compound real Wishart and q-Wishart matrices

We introduce a family of matrices with non-commutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a non-asymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in our previous work for independent complex Wishart matrices. We then analyze the fluctuations about the Marchenko-Pastur law. We show that after centering by the mean, traces of real symmetric polynomials in q-Wishart matrices converge in distribution, and we identify the asymptotic law as the normal law when q=1, and as the semicircle law when q=0

From compositional to systematic semantics

We prove a theorem stating that any semantics can be encoded as a compositional semantics, which means that, essentially, the standard definition of compositionality is formally vacuous. We then show that when compositional semantics is required to be "systematic" (that is, the meaning function cannot be arbitrary, but must belong to some class), it is possible to distinguish between compositional and non-compositional semantics. As a result, we believe that the paper clarifies the concept of compositionality and opens a possibility of making systematic formal comparisons of different systems of grammars.Comment: 11 pp. Latex.

Spin-flip reflection of electrons from a potential barrier in heterostructures

Spin-conserving and spin-flip opaque reflections of electrons from a potential barrier in heterostructures are described. An electric field of the barrier is considered to be the only source of energy spin splitting in the presence of spin-orbit interaction and its form is calculated in the three-level {\kp} model for a nontrival case of unbound electrons. Reflection angles and amplitudes are calculated for oblique incoming directions. It is shown that the system can serve as a source or filter of spin-polarized electron beams. Two unexpected possibilities are pointed out: a) non attenuated electron propagation in the barrier whose height exceeds the energies of incoming electrons, b) total reflection of electrons whose energies exceed barrier's height.Comment: 10 pages, 3 figure