Hall effect equations

Examples of even processes include heat conduction, electrical conduction, diflfiision and chemical reactions [4], Examples of odd processes include the Hall effect [12] and rotating frames of reference [4], Examples of the general setting that lacks even or odd synnnetry include hydrodynamics [14] and the Boltzmaim equation [15]. 

This same equation is, of course, also used to rationalise the general electronic behaviour of metals, semiconductors and insulators. The quantitative application of Eqn (2.1) is handicapped for ionic conductors by the great difficulty in obtaining independent estimates of c,- and u,-. Hall effect measurements can be used with electronic conductors to provide a means of separating c, and u,- but the Hall voltages associated with ionic conduction are at the nanovolt level and are generally too small to measure with any confidence. Furthermore, the validity of Hall measurements on hopping conductors is in doubt. 

If the sample conductivity is dominated by only one type of carrier, then a simple Hall-effect analysis is sufficient. The appropriate equations for a Hall-... 

Examples of the observational equations are given in Table 2. In that table, r H and vb are transition frequencies in hydrogen and deuterium such as those given in Table 3 below, Kj is the Josephson constant, which is characteristic of the Josephson effect, and Rk is the von Klitzing constant, which is characteristic of the quantum Hall effect. Note that Ex(riLj)/h is proportional to cRoo and independent of h, hence h is not an adjusted constant in these equations. 

The literature abounds with reports of thermal activation energies for shallow donors in GaN, obtained from Hall effect measurements over a range of temperatures, above and below room temperature, though their interpretation is rendered problematic by a number of complicating factors. At low temperatures there is clear evidence for impurity band conduction (see, for example, [31]) which severely limits the temperature range over which data may usefully be fitted to the standard equation for free carrier density n in terms of the donor density ND and compensating acceptor density NA ... 

Experimentally, the quantities that can be determined are [e ] and [h+] as obtained from suitable electrical measurements as the Hall effect these measurements are made on crystals which have been equilibrated at some high temperature and quenched. Details as to how such measurements can lead to estimates of the quantities C and thus to the concentration of defects obtained by solving graphically the set of equations are given by de Nobel (26). The stoichiometry of the solid at a given T, F, and /xCd is given by ... 

As indicated in an earlier review," the Hall effect has been the subject of numerous investigations. The Hall constant Rjj is defined by the classical electric field equation... 

Figure 5.22 shows p, i H, and S ( T)/T for disordered (Au, Ag, Cu)-Sn systems versus x. TK and the MDOS have already been presented above (Fig. 5.13 and Fig. 5.17). A close overall similarity exists among the different alloys as well as to the corresponding liquid state. For alloys with 27-30 at. % Sn, where Kpe = 2fcF is fulfilled, p is maximal while RH and S ( T)/T deviate from the corresponding free-electron value. These deviations are obviously related to FsBz-effects showing that f or fcF cannot be deduced from Hall-effect data in alloys with a high peak at Kpc as mentioned above. Equations (5.7) and (5.8) are always found to be a better approach.

This generalized Ohm s law differs from the conventional one because it takes into account the electron pressure gradient and the [J B] term related to the Hall effect. Solution of equation (3-263) with respect to electric current is comphcated because the current is present in two terms. The generalized Ohm s law can be simplified if plasma condnctivity is high (a oo) ... 

The role of time will be discussed in two steps. This chapter deals with reversible behaviors of systems, that is, conversions without dissipation, in studying oscillators and the role of space-time in various systems (propagating waves. Maxwell equations in electrodynamics. Hall effect). Chapters 10 and 11, subsequently, will be devoted to irreversible behaviors, when conduction and dissipation play a non-negligible role in conversion processes. 

The two last case studies belong to electrodynamics (as for case study F6 Light Propagation ) and are concerned with Maxwell equations in free space and to the Hall effect in a conducting material. The latter is a process working through the synchronization of two space-time velocities, which is an example of internal coupling. (This subject will be tackled in Chapter 13.)... 

It is important to recall the generality expressed by this nonlinear relationship to help remember that the Euler equation (see Equation K6.1) is valid only in the peculiar case of linear media. From now, the relation in Equation K6.5 will be utilized in this restricted case with a scalar and constant volumic mass pfg. The surface density of percussion F,/ is linked to the previous variable according to the specificity of the particular derivative, analogous to the expression of the Lorentz force in electrodynamics (see case study F8 Hall Effect in chapter 9), and its expression can also be given in the linear case in extracting the volumic mass from the scope of the operators ... 

We shall now go into greater detail with these equations in order to render them appropriate for a more in-depth study of Hall-effect thrusters. 

A number of authors have established balance equations for these flows, using simplifying hypotheses. For examples, see Barral et al. [BAR 03] and Gascon, Dudeck and Barral [GAS 03] in the steady-state case, or Boeuf and Garrigues [BOE 98] for an initial approach to the low-frequency oscillations in plasma thrusters and, more recently, Barral and Miedzik [BAR 11] for a more elaborate model of these phenomena in the context of a closed-loop study of Hall-effect accelerators. Also see Dabiri et al. [DAB 13]. 

The system of vmsteady equations shown in Table 8.1 has been applied to the study of looped control of Hall-effect thrusters with the aim of controlling the osdllations of the plasma [BAR 11],... 

The Appendix gives supplementary information about the balance laws with an electromagnetic field, before going on to describe the methodology used to establish onedimensional equations for a ffow with active walls, as is the case with certain Hall effect thrusters. 

We want to derive the quantization of energy and flux of the Landau levels in the quantum Hall effect. Assume that the magnetic field is generated by a vector potential A = (-yH, 0,0), which is known as the Landau gauge. The Schrodinger equation for electrons confined in the xy plane is... 

We shall see in later chapters that it is possible in semiconductors (and in some metals) for holes in the electrons band structure to act as positively charged carriers. Had the carriers been holes, the signs in the equation of motion would have been reversed resulting in a positive Hall coefficient. (Note This simplified derivation of the Hall effect is only... 

One can easily see that if n p, Eh reduces to -1/ne. Hall measurements are easiest to interpret in doped materials when either n 3> p orn < p. Otherwise one is faced with four unknowns, which require other measurements to resolve. For example, except for the difference between electron and hole mobilities, the Hall effect would be zero for intrinsic materials. One can also see that doing Hall measurements as a fimction of temperature offers a means of determining the occupancy number and energy levels of the various impurity states in the freeze-out region through Equation 20.22. 

This simplified discussion has neglected the effects of axial current flow, ie. Hall current, induced by the axial field. At a local region in the channel, equation 1 can be written in more general form as... 

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