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Long-period systems and superlattices, with additional periodicity, have new effects on the energy spectrum and wave functions. Most approaches adjust theories for infinite systems, which is acceptable for large but not small number of unit cells $n$. In the past 30 years, a theory based entirely on transfer matrices was developed, where the finiteness of $n$ is an essential condition. The theory of finite periodic systems (TFPS) is also valid for any number of propagating modes, and arbitrary potential profiles (or refractive indices). We review this theory, the transfer matrix definition, symmetry properties, group representations, and relations with the scattering amplitudes. We summarize the derivation of multichannel matrix polynomials (which reduce to Chebyshev polynomials in the one-propagating mode limit), the analytical formulas for resonant states, energy eigenvalues, eigenfunctions, parity symmetries, and discrete dispersion relations, for superlattices with different confinement characteristics. After showing the inconsistencies and limitations of hybrid approaches that combine the transfer-matrix method with Floquet's theorem, we review some applications of the TFPS to multichannel negative resistance, ballistic transistors, channel coupling, spintronics, superluminal, and optical antimatter effects. We review two high-resolution experiments using superlattices: tunneling time in photonic band-gap and optical response of blue-emitting diodes, and show extremely accurate theoretical predictions., 57 pages, 33 figures

Using the genuine superlattice eigenvalues and eigenfunctions, and their parity symmetries, new transition selection rules are derived, and a novel approach for the explicit calculation of the optical spectra of periodic structures presented. The optical response and infrared spectra published in the last 40 years for (In, Ga)N and (Al, Ga)As superlattices, with different number of unit cells n (varying from ∼ 10 to ∼ 400) and barrier and valley widths in the range of 2.5 nm to 40 nm, and different properties have been revisited. Among the various examples discussed here, we study the narrow peaks clustered in groups reported in high resolution spectra of blue-emitting devices, that could not be explained until now. These properties are faithfully reproduced and fully understood. We show that these effects are related to and depend on the surface energy levels and their tuneable detachment. We shown also that the optical transitions experimentally observed, but forbidden by the current optical physics, are really allowed transitions. Since the number of energy eigenvalues E μ , ν r , v and eigenfunctions Ψ μ ′ , ν ′ r , v ( z ) is now much greater, the number of allowed matrix-elements can be extremely large (of the order of n 2 n c n v ∕ 2 , with n c and n v the number of subbands in the conduction and valence bands). We show that this number can be reduced by using the eigenfunctions’ symmetries, which are behind the parity symmetry selection rules and the leading order selection rules. These rules reduce the number of matrix-elements evaluations from ∼ n 2 n c n v ∕ 2 to ∼ n n c n v ∕ 2 , i.e., depending on the superlattice, from about 1000 to 100. We discuss also a third rule, that collects the contributions of the surface and edge states, reducing the number of transitions even more to N s ≤ n c n v . With these rules, the main peaks are conserved and their number practically matches that of the real spectrum. Excellent agreements are found with experimental results.

Recently, an approach for metallic superlattices based on the finite periodic systems theory was introduced \cite{Pereyra2020}. Unlike most, if not all, of the published approaches that are valid in the $n \rightarrow \infty $ limit, the finite periodic approach is valid for any natural number $n$ and allows one to determine analytical expressions for scattering amplitudes and dispersion relations. It was shown, for frequencies below $\omega_{p}$ and large metallic-layer thickness, that under the common assumption that fields inside conductors move along the so-called "true" angle that defines the orientation of the constant-phase planes, anomalous results appear with an apparent parity effect. This issue is addressed here and it is shown that those results are due to the lack of unitarity and the underlying phenomena of absorption and loss of energy. Two compatible approaches are presented here to solve the lack of unitarity and to account for the absorption phenomenon. We show that by keeping the complex angles, the principle of flux conservation is fully satisfied, above and below $\omega_p$. This approach, free of assumptions, gives us light to improve the formalism when the real angle assumption is made. We show that by taking into account the induced currents and the requirement of flux conservation, we end up with an improved approach, with new Fresnel and transmission coefficients, fully compatible with those of the complex-angle approach., Comment: 16 pages and 12 figures

We study the time evolution of the survival probability and the spin polarization of a dissipative nondegenerate two-level system in the presence of a magnetic field in the Faraday configuration. We apply the Extended Gaussian Orthogonal Ensemble approach to model the stochastic system-environment interaction and calculate the survival and spin polarization to first and second order of the interaction picture. We present also the time evolution of the thermal average of these quantities as functions of the temperature, the magnetic field, and the energy-levels density, for ρ(ϵ)∝ϵs, in the subohmic, ohmic, and superohmic dissipation forms. We show that the behavior of the spin polarization calculated here agrees rather well with the time evolution of spin polarization observed and calculated, recently, for the electron-nucleus dynamics of Ga centers in dilute (Ga,N)As semiconductors.

The reason why the effective-mass approximation, derived for wave packets constructed from infinite-periodic-systems' wave functions, works so well with nanoscopic structures, has been an enigma and a challenge for theorists. To explain and clarify this issue, we re-derive the effective-mass approximation in the framework of the theory of finite periodic systems, i.e., using energy eigenvalues and fast-varying eigenfunctions, obtained with analytical methods where the finiteness of the number of primitive cells per layer, in the direction of growth, is a prerequisite and an essential condition. This derivation justifies and explains why the effective-mass approximation works so well for nano-structures. We show also with explicit optical-response calculations that the rapidly varying eigenfunctions $\Phi_{\epsilon_0,\eta_0}(z)$ of the one-band wave functions $\Psi^{\epsilon_0,\eta_0}_{\mu,\nu}(z)= \Psi^{\epsilon_0}_{\mu,\nu}(z) \Phi_{\epsilon_0,\eta_0}(z)$, can be safely dropped out for the calculation of inter-band transition matrix elements., Comment: 6 pages

We study the contact resistance of strongly doped ferromagnetic/non-magnetic semiconductors structure ( p + - F/n + -N ), working as spin injectors and spin extractors. Because of the strong effect that the barrier shape has on the tunneling probabilities, we evaluate, as accurately as possible, the quantum-mechanical spin-dependent transmission across the Esaki barrier built up at the p-n interface. To simplify the discussion and calculation of transmission coefficients through the Esaki barrier, we concentrate here on the structure p + - F/n + -N , without a stop layer I . We evaluate the spin injection and spin extraction transmission coefficients T ↑↑ and T ↓↓, and the spin transmission polarization as functions of bias potential, exchange interaction energy and Fermi energy level for specific realizations of the structure p + -Ga 1-x Mn x As/n + -GaAs .

Using the analytical expressions for the genuine eigenfunctions $\varphi_{\mu\nu}(z)$ and eigenvalues $E_{\mu,\nu}$, of open, bounded and quasi-bounded finite periodic systems, we derive the eigenfunctions space-inversion symmetry relations. The superlattice eigenfunctions symmetries, closely related with the symmetries and zeros of the Chebyshev polynomials of the second kind $U_n$, are fully written in terms of the number of unit cells $n$, the subband index $\mu$ and the intra-subband index $\nu$., Comment: 10 pages, 14 figures

Based on numerical evaluations of the transition probabilities, and using true superlattice eigenfunctions derived from the theory of finite periodic systems, new superlattice selection rules are obtained. We show that the evaluation of the large number N of the possible recombination transitions reduces by a factor of two. We also show that taking into account these selection rules, the theoretical calculations for specific blue emitting GaN ⧹ ( In x Ga 1 - x N ⧹ In y Ga 1 - y N ) n ⧹ GaN superlattices agree with those reported by Nakamura et al. [1] .

The transmission probabilities of an accurate and an approximate (triangular) Esaki barrier are evaluated. We show that the approximate results for transmission coefficients differ by orders of magnitude from the correct ones, however the approximate methods provide good results for physical quantities defined as ratios of transmission coefficients.

We explicitly show that the well-known transmission and reflection amplitudes of planar slabs, obtained via an algebraic summation of Fresnel amplitudes, are completely equivalent to those obtained from transfer matrices in the scattering approach. This equivalence makes the finite periodic systems theory a powerful alternative to the cumbersome Fresnel coefficients in the analysis of superlattices and photonic crystals. The versatile transfer matrices, with their compact formulae, can be useful in calculating transmission properties not only of superlattices made of dielectrics but also of left-handed materials and conducting media.

We apply the Theory of Finite Periodic Systems to study the photoluminescent behavior of GaN ∖ ( In x Ga 1 − x N ∖ In y Ga 1 − y N ) n ∖ GaN superlattices. Using this formalism, where intra-subband energy levels are neatly identified, we study the optical response for different barrier and well thicknesses, and show that this quantity is very sensitive to small changes in the superlattice parameters. Varying the barrier thickness (for fixed well-width), or the well-width (for constant barrier thickness), the intensities of the relevant PL-peaks change drastically. We make clear also that this highly accurate formalism can be used to design optimal superlattice-optoelectronic devices.

Based on the Sylvester and Frobenius theorems, we drastically enhance the feasibility of the transfer-matrix approach to deal with problems involving a large number of propagating and interfering modes, which require the solution of coupled differential equations and the evaluation of functions of matrix variables. We report closed formulas for the spectral decomposition of this type of functions. As specific example, besides the calculation of simple and well-known 1D one channel transfer matrices, we derive the multi-channel transfer matrix for an electron gas in the presence of a transverse electric field.

We study the spin dynamics of a 2DEG moving across a magnetic superlattice. We describe the spin transitions induced by sectionally homogeneous tilted magnetic fields and the spatial evolution of spin-1/2 wave functions: amplitude and phase, in terms of the tilting angle, the Fermi energy and the magnetic field strengths. Clear signatures of coherent spin mixing and coherent spin flipping processes are shown. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Using a simple approach that requires neither the Bloch functions nor the reciprocal lattice, new, compact, and rigorous analytical formulas are derived for an accurate evaluation of resonant energies, resonant states, energy eigenvalues and eigenfunctions of open and bounded n -cell periodic systems with arbitrary 1D potential shapes, provided the single cell transfer matrix is given. These formulas are applied to obtain the energy spectra and wave functions of a number of simple but representative open and bounded superlattices. We solve the fine structure in bands and exhibit unambiguously that the true eigenfunctions do no not fulfill the periodicity property | Ψ μ , ν ( z + l c )| 2 = | Ψ μ , ν ( z )| 2 , with l c the single cell length. We show that the well known surface states and surface energy levels come out naturally. We analyze the surface repulsion effect and calculate exactly the surface energy levels for different potential discontinuities an the ends.

We have studied the behavior of the Shubnikov-de Haas oscillations in magnetoresistance under applied magnetic fields with an arbitrary orientation. Different procedures to measure the Lande-g factor have been used in the literature. We propose four unambiguous methods and formulas for the evaluation of this quantity. The Lande-g factor obtained using these formulas compares within the 5% with the reported results of Brosig et al. [1] in InAs – AlSb quantum wells, Bompadre et al. [2] for Bismuth crystals, Kobayashi et al. [3] measuring magnetoresistance as a function of the gate Voltage and F. Fang et al. [4] for InAs – GaSb heterostructure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study the Shubnikov-de Haas oscillations in the magnetoresistance and Landauer conductance of a three strip quasi-2D semiconductor wave guide with thickness δ z and transversal width Wy. We assume that the strip in the middle, of length l H , is subject to a homogeneous magnetic field, tilted by θ H with respect to the normal z of the 2DEG. The structure of the Shubnikov-de Haas oscillations, associated with spin parallel and antiparallel to the z-component of the magnetic field, and the finiteness of l H , lead us to define, related with the limits between the corresponding Landau levels and the continuous spectrum, two characteristic saturation fields B s a t , đ= 2Φ o /l H cosθ H (2n s π - 1/δ 2 z ) 1 / 2 and B s a t , Đ defined by the positive root of B 2 s a t , Đ + 4gΦ o /πl 2 H cos 2 θ H B s a t , Đ - B 2 s a t , đ= 0. Here g is the Lande factor, Φ o the unit magnetic flux and n s the charge concentration. We also obtain the polarization field B p = Φ 0 /g 2n s - π/δ 2 z .

We study the time reversal and space inversion symmetry properties of the transfer matrices mostly used in the calculation of energy spectra and transport-process quantities. We determine the unitary transformation relating transfer matrices. We consider the Kohn-Luttinger model for a quasi-2D system and show that even though the system studied in the (4 × 4) scheme satisfies all the symmetry requirements, the (2 × 2) subspaces do not fulfill such constrains, except in the Γ point of the Brillouin Zone. We find new exchange properties between the (2 × 2) subspace quantities.

The optical response of an AlGaN/GaN/(InxGa1−xN)n/GaN/AlGaN heterostructure is obtained from precise, and comparatively simple, transition probability calculations. A comprehensive approach to evaluate these quantities from rigorous expressions of the heterostructure's energy eigenvalues and eigenfunctions is given. Taking full account of the longitudinal quantum wells and the lateral quantum dot confinement, quite precise intraband energy levels and wave functions have been obtained. Photoluminescence results for sectionally constant band-edge profile differ from the experimental measurements by approximately 45 meV. We show that the band-edge modulation arising from charge polarization, is responsible for this difference.

We study the effects of charge polarization on the extended physical properties of superlattices, such as transmission coefficients and valence band structure. We consider both linear and parabolic modulation of the band edge. Based on the theory of finite periodic systems, analytic expressions and high precision calculations of the relevant physical quantities for n-cell systems are obtained. Well-known features of these systems are identified. Besides the well-known energy band structure, we have also the field band structure, with interesting characteristics. Wider field gaps at stronger internal electric fields and higher density of field bands for larger layer widths are some of these characteristics. Well defined level density asymmetries identify the minibands induced by charge polarization or the so-called quantum confining Stark effect. We present the n-cell transmission amplitudes, transmission coefficients and miniband structures for different values of the relevant parameters.

A phenomenological theory to describe the electromagnetic properties of granular superconductors, based on known bulk superconductors expressions and conventional Josephson's junctions tunneling currents, is presented and succesfully used to fit distinct experimental results for the magnetic susceptibility as a function of the temperature and the applied magnetic field of rather different samples., Comment: 6 pages, 6 figures

In the new approach to study the optical response of periodic structures, successfully applied to study the optical properties of blue-emitting InGaN/GaN superlattices, the spontaneous charge polarization was neglected. To search the effect of this quantum confined Stark phenomenon we study the optical response, assuming parabolic band edge modulations in the conduction and valence bands. We discuss the consequences on the eigenfunction symmetries and the ensuing optical transition selection rules. Using the new approach in the WKB approximation of the finite periodic systems theory, we determine the energy eigenvalues, their corresponding eigenfunctions and the subband structures in the conduction and valence bands. We calculate the photoluminescence as a function of the charge localization strength, and compare with the experimental result. We show that for subbands close to the barrier edge the optical response and the surface states are sensitive to charge polarization strength., Comment: 6 figures

An important drawback of the standard theory, describing light-emitting periodic structures, is the lack of explicit knowledge of the initial and final states. In the new approach, based on the theory of finite periodic systems, the superlattice eigenvalues and eigenfunctions, are rigorously and explicitly determined. This approach provides the possibility of a truly quantum calculation of the optical response of periodic structures. New selection rules, tunable surface-energy-levels repulsion and interesting optical-spectra effects, related to intra-subband and surface states, are reported. Most of the actually observed transitions, “forbidden” for the standard approach, become allowed. The unexplained phenomenon of clustering of narrow peaks, observed in high-resolution blue-laser spectra by Nakamura et al. , are now fully explained and faithfully reproduced. The higher accuracy of this approach and the feasibility to control the detachment of the surface energy levels are important improvements relevant to optical-physics and the technological and biomedical applications.

A different approach to calculate the I−V characteristics of p+(Ga,Mn)As/n+GaAs spin injectors is presented. The vanishing of the spin-extraction transmission coefficients at the spin-split valence-band edges leads us to predict a dip or plateau in the I−V characteristics of this kind of diodes. We show that this minimum, or the inflection point, shifts with the exchange energy. Within this approach and using Kane-like densities of states with a detached impurity band, and exchange energies reported in the literature, excellent agreement is found with observed low-bias features of I−V characteristics.

A simple expression for the transport critical current, which describes rather well the Ambegaokar-Baratoff (A-B) to Ginsburg-Landau (G-L) crossover, is obtained from an elementary analysis of a Josephson junction subject to external magnetic fields within the Ginsburg-Landau theory. We show that the order parameter in the intergrain region depends crucially on the interplay of two characteristic lengths: the superconducting coherence length $\ensuremath{\xi}\ensuremath{\propto}(1\ensuremath{-}{T/T}_{c}{)}^{\ensuremath{-}1/2}$ and the junction-induced superconducting decoherence length ${\ensuremath{\zeta}}_{q}\ensuremath{\propto}(1\ensuremath{-}{T/T}_{c}{)}^{\ensuremath{-}\ensuremath{\alpha}}.$ Data for the transport critical current in granular YBCO samples are presented, and the A-B to G-L crossovers successfully described.

New formulas for an exact and simple evaluation of the phase time associated with the passage of electrons or photons through a finite superlattice are derived. This time, named here the superlattice-tunneling time ${\ensuremath{\tau}}_{n}$, exhibits a resonant-band structure or a superluminal phase time behavior depending on whether the particle's energy lies in a band or a gap. In the band, ${\ensuremath{\tau}}_{n}$ remains larger than the free motion time ${\ensuremath{\tau}}_{f}$, while in the gap it can be less than ${\ensuremath{\tau}}_{f}$ (with strong substrate effects), but it is larger than the single-cell phase time ${\ensuremath{\tau}}_{1}$. Extremely good agreements with optical-pulse and superluminal delay times measured by Spielmann et al. and Steinberg et al. are found, including the superlattice-tunneling-time limit and the substrate effects. Conditions for measurements of earlier electron arrival are also analyzed.

A theoretical analysis and calculation of the excitonic states in asymmetric quantum dots is carried out in the presence of magnetic fields. The lack of rotational symmetry, introduced by strains and structural factors, produces splittings of the excitonic states with corresponding consequences on the optical oscillator strengths and polarization dependence. For example, we find that the asymmetry produces Zeeman splittings that are smaller than those for symmetric dots at small fields, which could be used as an additional diagnostic of the geometry of the structure. We focus our calculations on naturally occurring quantum dots due to layer fluctuations in narrow quantum wells. Moreover, we observe that increasing magnetic fields produce an interesting crossover to pure angular momentum states for all the excitonic eigenstates, regardless of the degree of asymmetry of the dots and their size. Explicit calculations of photoluminescence excitation yields are presented and related to the different degrees of freedom of the system. (c) 2000 The American Physical Society.

Using the transfer matrix method and the multi-channel scattering approach, we solve analytically the Schrodinger equation for spin 12 electrons moving through a homogeneous semiconductor wave guide subject to an external magnetic field, acting on alternating stripes. We study the space-time evolution of Gaussian spin-wave packets inside and outside of the homogeneous magnetic superlattices (MSL) for different tilting angles. We discuss interfering processes of spin-wave packets, defined in the band gap and in the resonant energy regions of homogeneous MSL based on an InSb semiconductor. We show also that the time spent inside the MSL is in good agreement with the phase time predictions.

The transfer matrix method was used to calculate the transmission of electromagnetic waves through alternate layers of thin silver films separated by air. It was found that the plasma frequency is a clear divide between good and poor transmission. Equally the calculations show that for frequencies above the plasma frequency the transmission coefficient shows a Bragg-like behavior with oscillations that manifest coherent reflections on the entrance side of the layer at favored wavelengths.

Using the transfer-matrix technique to describe transport properties in multichannel-multilayer systems, a three-term non-commutative matrix recurrence relation is deduced and solved. The matrix polynomials obtained in this way allow one to write compact expressions for the n-cell transmission amplitudes, from channel to channel i. In the one-dimensional, one-channel limit, the non-commutative polynomials reduce to the well known Chebyshev orthogonal polynomials. To illustrate the role of these polynomials in the resonant tunnelling and channel-mixing behaviour, we discuss one- and two-channel examples.

Compact and closed expressions for the evaluation of $n$-cell transmission amplitudes, in superlattices with $N$ open channels, are written in terms of the single-cell transmission amplitudes and a set of noncommutative polynomials, which, in the one-channel case, reduce to the Chebyshev polynomials. Interesting and nontrivial features, due to channel mixing and multiple interference phenomena occurring along the $n$-cell system, emerge when applying to specific potential profiles.

We show here that a transverse electric field, as in field-effect devices, leads to interesting interfering phenomena in the electronic transport. Using the Sylvester Theorem for matrix-valued functions, we study the transmission properties of a multi-propagating modes (multichannel) quasi-two-dimensional electron gas in the presence of a channel mixing transverse electric field. We observe interesting resonant coupling effects, which explain the fundamentals of the giant superlattice-conductance as well as giant superlattice-resistance effects.

The magnetic-field and temperature behavior of the critical current exponents in poycrystalline YBa2CuxOy (with x = 3,4 and different grain sizes), has been measured in the low magnetic-field region. The critical exponent α of small-grain samples with dominant weak links, increases with increasing the magnetic field and temperature. Except for very low magnetic fields and temperatures far away from Tc, the exponent α becomes larger than 1. For big grain sizes and dominant Josephson-like junctions the critical exponent oscillates between 1/2 and 1.

An appropriate combination of the scattering theory and the transfer matrix formalism, for the solution of a (4×4) Kohn–Luttinger model, allow us to study the multichannel-multiband transmission process of heavy and light holes through a (GaAs/AlAs) n superlattice. Appealing effects and interesting channel coupling phenomena, mediated by quasi-bond states, are clearly foreseen.

From careful four probe measurements at different magnetic fields, we deduce the critical transport current density Jc(B) behaviour for polycrystalline Y and Gd-based superconducting samples which, on top of the characteristic Fraunhofer-like “weak-link” gross structure, exhibit anomalous oscillations, due to quantum interference effects in a finite number of current loops in mesoscopic networks of Josephson junctions.

Useful and new representations of the transfer matrices are explicitly obtained for systems with and without time reversal or spin rotation symmetries. In the symplectic, orthogonal, and unitary cases, the whole group of transfer matrices contains a compact and a noncompact subgroup, and the generalized Bargman’s polar parametrization is obtained in a straightforward way. It is shown that the transfer matrices used by Luttinger, to prove the Saxon–Hutner conjecture on the forbidden energy levels, belong to the noncompact Sp(2,C) subgroup.

Important and well-established superlattice properties are strongly determined by phase coherence, phase shift and phase interference phenomena. We study both the relation between: (a) the Bargmann's phase, the Chebyshev polynomials and the superlattice band structure; and (b) the giant conductance-resistance effects and the scattering amplitudes phase-shifts.

Rotation is a fundamental motion in the real world. Thus an important and an appropriate number of physical variables have been introduced to study this type of motion, both in the classical and the quantum description. The relevant physical quantity is the angular momentum. In the quantum description, the angular momentum is defined exactly as in the classical physics, but with the momentum written as an operator, i.e. as the product