Publisher Summary The chapter presents a discussion on the theoretical aspects of the electron-hole liquid in semiconductors. In an intrinsic semiconductor such as germanium or silicon, there are essentially no free carriers at very low temperatures. However, high densities of positive (hole) and negative (electron) charge carriers can be generated in these crystals by photoexcitation with a photon energy greater than the forbidden gap. In recent years, experiments in Ge and Si have revealed that optically pumped electrons and holes at high densities can undergo a phase transition at liquid helium temperatures into a metallic, liquid state. This condensate represents a state of matter particularly unique in nature because the electrons and holes, by reason of their small masses, are simultaneously in the quantum limit. This chapter presents a discussion on the collective behavior of electrons and holes at low temperatures and high densities. The chapter presents an historical introduction, mentions a few of the important early experiments performed, and presents a comparison of theory and experiment. Semiconductors can be divided into two classes. Those in which the conduction band minimum and valence band maximum occur at the same value of the wavevector are known as direct gap semiconductors, while those in which they are separated in k space are known as indirect gap semiconductors. The chapter discusses the latter.

The Electron-Hole Liquid in Semiconductors: Theoretical Aspects

The bulk upper critical field ${H}_{c2}$ of single crystals of Nb and V has been measured for various crystallographic orientations within the temperature range between the respective critical temperature and $T\ensuremath{\approx}0.06$ \ifmmode^\circ\else\textdegree\fi{}K. The normalized critical field averaged over all crystallographic directions $〈h(t)〉=\frac{〈{H}_{c2}(t)〉}{{(\ensuremath{-}\frac{d{H}_{c2}}{\mathrm{dt}})}_{t=1}}$, where $t=\frac{T}{{T}_{c}}$, and the relative anisotropy of ${H}_{c2}$ for the two metals are found to be similar. The measurements support predictions of the Hohenberg-Werthamer calculation that attributes the observed anisotropy of ${H}_{c2}$ to the effects of an anisotropic Fermi surface. Some differences in the behavior of ${H}_{c2}(t)$ for the present samples can be understood qualitatively to arise from the effects of a shorter electron-collision time $\ensuremath{\tau}$ for the less pure V specimen. At very low temperatures, ${H}_{c2}(0)\ensuremath{-}{H}_{c2}(t)$ for V does not follow a ${t}^{2}\mathrm{ln}(t)$ temperature dependence, previously observed for Nb, but can be characterized by an empirical formula ${t}^{2}\mathrm{ln}(t+\ensuremath{\alpha})$ in which the constant $\ensuremath{\alpha}$ is conjectured to be inversely proportional to $\ensuremath{\tau}$.

Bulk Upper Critical Field of Clean Type-II Superconductors: V and Nb

In type-II superconductors with uniaxial symmetry the three critical fields (Hc1, Hc2, Hc3) are calculated near the transition temperature Tc using the anisotropic Ginzburg-Landau equations. It is shown that in general an induced magnetic field δH⊥ perpendicular to the applied magnetic field exists and that the anisotropy removes the degeneracy in the direction of the flux-line lattice. The pattern and the orientataion of the flux-line lattice are in good agreement with the experimental results of technetium.



In Supraleitern II. Art mit einachsiger Symmetrie werden die drei kritischen Feldstarken (Hc1, Hc2, Hc3) in der Nahe der Ubergangstemperatur Tc mit den anisotropen Ginzburg-Landau-Gleichungen berechnet. Es wird gezeigt, das im allgemeinen ein induziertcs Magnetfeld δH⊥ senkrecht zum auseren Magnetfeld existiert und das die Anisotropie die Entartung in Richtung des Flusliniengitters aufhebt. Das Muster und die Richtung des Flusliniengitters sind in guter Ubereinstimmung mit den experimentellen Ergebnissen an Technetium.

Magnetic properties of superconductors with uniaxial symmetry

The upper critical field in two-dimensional superconductors is calculated by taking into account quantum corrections due to the Anderson localization to the theory of dirty superconductors in the order of ( e F τ 0 ) -1 where e F and τ 0 are the Fermi energy and the relaxation time of an electron, respectively. The upper critical field is enhanced at low temperatures. This is because the diffusion constant of electrons is depressed and the Coulomb repulsive interaction modified by the localization becomes small in the magnetic field. Discussion is given on the upward curvature of upper critical field of layered superconductors in the field perpendicular to the layers.

Upper Critical Field in Two-Dimensional Superconductors

We present a theory of the upper critical field for an anisotropic polycrystal. This theory is based on the well-known separable model for the electron-phonon anisotropy which is expanded to describe the Fermi-velocity anisotropy. It is assumed to be sufficient to describe these anisotropies by the mean-square parameters and , respectively. Comparison with existing N-band models shows that our model only allows formation of Cooper pairs from quasiparticles which belong to the same Fermi-surface sheet. Predictions of this theory are discussed using a model system emphasizing the deviation of the upper critical field of the anisotropic system from an equivalent isotropic one. Finally, a procedure is suggested which should enable experimentalists to make a more detailed analysis of experimental data in terms of anisotropy effects.

Upper critical field of superconducting anisotropic polycrystals.

There existed quantitative descrepancies in the upper critical field Hc2 of type-II superconductors between experimental observations and theory which as­ sumes the spherical Fermi surface and which is based on the BCS weak­ coupling model (GLAG theory). Werthamer and McMillan 1 l have investigated the correction due to the strong-coupling effects and concluded that they make up only a negligible portion of the descrepancy between the weak-coupling theory and experimental observations if one considers a normalized field h = -He~ j(dHc2/ dt)t=l· Meanwhile, Reed et al. 2 l have observed that Hc2 has anisotropic nature in several type-II superconductors which have a cubic crystal structure, and that the relative anisotropy of Hc2 decreases as the reduced temperature t increases and approaches zero as t goes to unity. These natures were attributed to the Fermi surface anisotropy produced by the crystal structure and to the non-local correc­ tion to the GL-equation by Hohenberg and W erthamer. 3 l Moreover, they found that their Hc2 curves are in better agreement with the experimental observations on Nb and V. Farrell et al. 4 l have made precise measurements of Hc2 for Nb crystals and analyzed their result assuming a four-term Kubic harmonic expression ( 4-parameter

The Upper Critical Field of Superconducting Alloys

The condensation of excitons in a highly excited state of semiconductors or insulators is investigated without an assumption of excitons being bosons. The discussion is based on the quasi-chemical equilibrium theory and the dynamical correlation between electrons and/or holes is taken into account with an appropriate screening parameter. It turns out that the concentration of excitons in the ground level attains a finite value when n exc exceeds a value n exc c which is identical to the critical concentration for the Bose-Einstein condensation in an ideal boson system. It is also seen that the excitons can be treated as bosons for n c n exc c . It is also mentioned how to determine the concentration of free pairs and of excitons against the total concentration of excited pairs. A favorable situation for the observation of the exciton condensation is also discussed.

Condensation of Excitons in a Highly Excited State of Semiconductors

The investigation is carried out of the upper critical field H c 2 of a superconducting alloy in the case where an appreciable variation of the transition temperature T c is observed on alloying. It is assumed that the density of states is not a smooth function of energy at and near the Fermi level and that the BCS coupling parameter is anisotropic. It turns out that the shift of Fermi level on alloying does not modify the expression for H c 2 usually employed if one interprets T c and Fermi velocity in the expression as those of the alloy. It is also discussed that a recent investigation by Teichler on the effect of anisotropic coupling ignores an important term. For the case of Nb–Ta system, it becomes clear that the shift of Fermi level is small enough so that Fermi velocity remains constant even though there exists a large variation in T c with alloying. Qualitative discussions are given on the effect of anisotropic coupling, and it turns out that more detailed experimental investigations are needed t...

Theory of Upper Critical Field of Superconducting Alloys

In order to understand a general trend of the photoconductivity in a highly excited state of semiconductors, the determination of carrier density against the total excitation has been carried out in a relatively low density region where electrons, holes and excitons coexist. The assumption of chemical equilibrium among electrons, holes and excitons enables one to determine the density of charge carriers. The variation of binding energy of an exciton with the excitation density is taken into account. Carrier density appears to increase abruptly at a certain critical region of total excitation density.

Thermodynamical approach to the highly excited

The upper critical field Hc2 and the K 2-parameter of a pure superconductor with the cubic crystal symmetry are calculated near the transition temperature. The result obtained for I-Ic 2 is in good agreement with the experimental result of Farrell et al. It is found that the direc­ tion of the induced magnetic field is not parallel to that of the external magnetic field applied in an arbitrary direction. Therefore, the t.:: 2-parameter becomes a tensorial quantity. The ratio MxiMz is found to be about 0.015 at the reduced temperature t=0.7, where iY.lx and Mz are respectively magnetizations perpendicular and parallel to the applied external field. § l. Introduc11:im:Ji Reed et al. have experimentally shown that the upper critical field Hc2 has an anisotropic nature in several cubic type II superconductors and that the re­ lative anisotropy of Hc2 decreases as the reduced temperature t increases, and approaches zero as t-> 1. 1 ) They have attributed this nature to the anisotropy of the Fermi surface and to the non-locality of the Ginzburg-Landau equation. Hohenberg and W erthamer derived expressions for Hc2 in the nearly local limit and also in the limit of small anisotropy. 2 ) They have successfully shown that the anisotropy of Hc2 comes from the non-local correction to the original G-L equation as well as from the cubic symmetry of the Fermi surface. Farrell et al. have recently made precise measurements of the upper critical field of niobium crystal and analyzed their result assuming a four-term Kubic harmonic expression. 3 ) The calculation of Hohenberg and W erthamer in the nearly local limit contains only three-term Kubic harmonics. In this paper, we derive a theoretical expression for 1-Ic 2 involving four-term Kubic harmonics, extending the work of Hohcnberg and Werthamer to a higher order term in the non-locality. Further, we investigate a possible coupled effect from the non-locality and the anisotropic nature of the cubic crystal on the tc 2parameter. We restrict our attention to the study of 1-Ic 2 and tc 2 for a pure superconductor only near the transition temperature region (G-L region). In § 2, we will give the basic G-L equations for our system, and an expres­ sion for 1-Ic2 will be derived in § 3. In § 4 is contained the discussion of the effect of anisotropy on the induced magnetic field and also of the tc 2-parameter.

Hc2 and the κ2-Parameter of a Superconductor with Cubic Symmetry

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