Crystal static dielectric constant

The parameter is obtained by relating the static dielectric constant to Eg and taking in such crystals to be proportional to a - where a is the lattice constant. Phillips parameters for a few crystals are listed in Table 1.4. Phillips has shown that all crystals with a/ below the critical value of0.785 possess the tetrahedral diamond (or wurtzite) structure when f > 0.785, six-fold coordination (rocksalt structure) is favoured. Pauling s ionicity scale also makes such structural predictions, but Phillips scale is more universal. Accordingly, MgS (f = 0.786) shows a borderline behaviour. Cohesive energies of tetrahedrally coordinated semiconductors have been calculated making use... 

Considering the simple case of an electron bound to a shallow electronic ion in a crystal with static dielectric constant es, when the magnetic field B is oriented along the z axis, the EM Hamiltonian at a non-degenerate band extremum characterized by an effective mass m is ... 

Lux (lx) - The SI unit of illuminance, equal to cd sr m. [1] Lyddane-Sachs-Teller relation - A relation between the phonon frequencies and dielectric constants of an ionic crystal which states that (co., /cOj ) = e(< )/e(0), where co., is the angular frequency of transverse optical phonons, that of longitudinal optical phonons, e(0) is the static dielectric constant, and e(< ) the dielectric constant at optical frequencies. 

In these equations, n is the density of the free charge carriers, ( the density of the charge carriers captured in shallow traps, s the static dielectric constant, the density of states at the transport level Eg, Vext the applied voltage, and G(( ) gives the density of the trapping states per energy interval dE. Ne in disordered fUms or in crystals with narrow conduction bands is equal to the number of molecules per unit volume. 

So far we have been dealing with various forms of the "response" to displacements of atoms. In Section 6 also certain electric fields have been studied, we benefited from the fact that displacing atoms in GaAs generates dipoles and therefore electric fields all the reasonings of Section 6 were, however, limited to polar crystals and e.g. determination of static dielectric constant e as in Section 6.2 would be impossible in Ge or other homopolar substances. From the point of view of studying dielectric properties, the main drawback of Section 6 was our dependence upon the various displacement patterns the electric fields could not be varied at will, as an independent variable. The present Section summarizes the most recent applications of the DF which tend to fill this blank and to open the way to "direct" treatment of dielectric properties of semiconductors, within the framework of the Density Functional. They are the treatment of constant macroscopic electric field imposed from outside (Section 8.1) and "direct" evaluation. of the individual elements of the inverse dielectric matrix s ("q + +"g ) (Section 8.2). 

In this section we wish to consider all the possible contributions to the electric permittivity of liquid crystals, regardless of the time-scale of the observation. Conventionally this permittivity is the static dielectric constant (i.e. it measures the response of a system to a d.c. electric field) in practice experiments are usually conducted with low frequency a.c. fields to avoid conduction and space charge effects. For isotropic dipolar fluids of small molecules, the permittivity is effectively independent of frequency below 100 MHz, but for liquid crystals it may be necessary to go below 1 kHz to measure the static permittivity polymer liquid crystals can have relaxation processes at very low frequencies. 

Since, we know that the liquid crystals are strongly anisotropic and therefore, a static dielectric constant may preclude with the ideal case of dielectric anisotropy. However, the surface stabilization of the molecular director over the free volume boundaries enable us to work with this isotropic approximation because the surface stabilization produces strong network correlation in polymer chains when they template the LC order. In this context our assumption is valid, however, in a more general case we have to relax our approximation considering the anisotropy in dielectric constant. 

Static dielectric measurements [8] show that all crystals in the family exhibit a very large quantum effect of isotope replacement H D on the critical temperature. This effect can be exemphfied by the fact that Tc = 122 K in KDP and Tc = 229 K in KD2PO4 or DKDP. KDP exhibits a weak first-order phase transition, whereas the first-order character of phase transition in DKDP is more pronounced. The effect of isotope replacement is also observed for the saturated (near T = 0 K) spontaneous polarization, Pg, which has the value Ps = 5.0 xC cm in KDP and Ps = 6.2 xC cm in DKDP. As can be expected for a ferroelectric phase transition, a decrease in the temperature toward Tc in the PE phase causes a critical increase in longitudinal dielectric constant (along the c-axis) in KDP and DKDP. This increase follows the Curie-Weiss law. Sc = C/(T - Ti), and an isotope effect is observed not only for the Curie-Weiss temperature, Ti Tc, but also for the Curie constant C (C = 3000 K in KDP and C = 4000 K in DKDP). Isotope effects on the quantities Tc, P, and C were successfully explained within the proton-tunneling model as a consequence of different tunneling frequencies of H and D atoms. However, this model can hardly reproduce the Curie-Weiss law for Sc-... 

Table 1.1. Abundance of the metal in the earths s crust, optical band gap Es (d direct i indirect) [23,24], crystal structure and lattice parameters a and c [23,24], density, thermal conductivity k, thermal expansion coefficient at room temperature a [25-27], piezoelectric stress ea, e3i, eis and strain d33, dn, dig coefficients [28], electromechanical coupling factors IC33, ksi, fcis [29], static e(0) and optical e(oo) dielectric constants [23,30,31] (see also Sect. 3.3, Table 3.3), melting temperature of the compound Tm and of the metal Tm(metal), temperature Tvp at which the metal has a vapor pressure of 10 3 Pa, heat of formation AH per formula unit [32] of zinc oxide in comparison to other TCOs and to silicon...

In compound crystals, the ujn values considered are wlo, the frequency of the longitudinal optical phonons on the high-energy (h-e) side, and wto, the frequency of the transverse optical phonons, on the low-energy side. The dielectric constant at frequencies above c lo is denoted as while that below wto is denoted as s (the index s represents static, despite the fact that s shows a small dispersion between the value just below ujto and the one at radiofrequencies1). It can be seen from expressions (3.14) and (3.15) that above ujo, the ionic contribution decreases such that qo is smaller than s. Typical values are given in Table 3.1. 

In these expressions eQ and are the static and high-frequency dielectric constants of the host, vs is a (mean) velocity of sound therein, Cpe is the piezoelectric coupling constant, CQ a numerical factor that depends on the crystal structure and kD the Debye inverse screening length. 

---