mperfections, and lattice vibrations, respectively. In semiconductors, the much longer relaxation time and maller effective mass of the electrons makes it much easier to achieve the high field limit. In this limit the esult analogous to Eq (52.15)is[Blatt, 1968, P. 290] ne-n Note that the individual band conductivities do not enter in Eq.(52. 16). Eq.(52.16) is valid provided the cyclotron orbits of the electrons are closed for the particular direction of B used. It is not necessary that the bands be spherical or the t's isotropic. Also, for more than two bands RH depends only on the net difference between the number of electrons and the number of holes. For the case where ne= mh, in general, the lowest order dependence of the Hall electric field on B is B and there is no simple relationship of RH to the number f current carriers. For the special case of the two-band model, however, RH is a constant and is of the same form as Eq (52.15)[Fawcett, 1964 Metals can have geometrically complicated Fermi surfaces wherein the Fermi surface contacts the Brillouin zone boundary as well as encloses the center of the zone. This leads to the possibility of open electron orbits in place of the closed cyclotron orbits for certain orientations of B In these circumstances R can have a variety f dependencies on the magnitude of B and in single crystals will generally be dependent on the exact orientation of B relative to the crystalline axes [Hurd, 1972, P 51; Fawcett, 1964]. R will not, however, have any simpl relationship to the number of current carriers in the material Semiconductors have too few electrons to have open orbits but can manifest complicated behavior of their Hall coefficient as a function of the magnitude of B. This occurs because of the relative ease with which one can pass from the low field limit to the high field limit and even on to the so-called quantum limit with currently attainable magnetic fields. (The latter has not been discussed here. ) In general, these different regimes of B will not occur at the same magnitude of B for all the bands in a given semiconductor, further complicating the dependence of R on B Defining Terms Conducting band: The band in which the electrons primarily responsible for the electric current are found Effective mass: An electron in a lattice responds differently to applied fields than would a free electron or a classical particle. One can, however, often describe a particular response using classical equations by defining an effective mass whose value differs from the actual mass. For the same material the effective mass may be different for different phenomena; e.g., electrical conductivity and cyclotron resonance Electron band: A range or band of energies in which there is a continuum(rather than a discrete set as in, for example, the hydrogen atom) of allowed quantum mechanical states partially or fully occupied by electrons. It is the continuous nature of these states that permits them to respond almost classically to an applied electric field Hole or hole state: When a conducting band, which can hold two electrons/unit cell, is more than half full, unfilled states are called holes. Such a band responds to electric and magnetic fields as if it contained positively charged carriers equal in number to the number of holes in the band. Relaxation time: The time for a distribution of particles, out of equilibrium by a measure d, to return exponentially toward equilibrium to a measure p/e out of equilibrium when the disequilibrating fields are removed(e is the natural logarithm base Related Topic 22 1 Physical Properties c 2000 by CRC Press LLC© 2000 by CRC Press LLC imperfections, and lattice vibrations, respectively. In semiconductors, the much longer relaxation time and smaller effective mass of the electrons makes it much easier to achieve the high field limit. In this limit the result analogous to Eq. (52.15) is [Blatt, 1968, p. 290] (52.16) Note that the individual band conductivities do not enter in Eq. (52.16). Eq. (52.16) is valid provided the cyclotron orbits of the electrons are closed for the particular direction of B used. It is not necessary that the bands be spherical or the t’s isotropic. Also, for more than two bands RH depends only on the net difference between the number of electrons and the number of holes. For the case where ne = nh , in general, the lowest order dependence of the Hall electric field on B is B2 and there is no simple relationship of RH to the number of current carriers. For the special case of the two-band model, however, RH is a constant and is of the same form as Eq. (52.15) [Fawcett, 1964]. Metals can have geometrically complicated Fermi surfaces wherein the Fermi surface contacts the Brillouin zone boundary as well as encloses the center of the zone. This leads to the possibility of open electron orbits in place of the closed cyclotron orbits for certain orientations of B. In these circumstances R can have a variety of dependencies on the magnitude of B and in single crystals will generally be dependent on the exact orientation of B relative to the crystalline axes [Hurd, 1972, p. 51; Fawcett, 1964]. R will not, however, have any simple relationship to the number of current carriers in the material. Semiconductors have too few electrons to have open orbits but can manifest complicated behavior of their Hall coefficient as a function of the magnitude of B. This occurs because of the relative ease with which one can pass from the low field limit to the high field limit and even on to the so-called quantum limit with currently attainable magnetic fields. (The latter has not been discussed here.) In general, these different regimes of B will not occur at the same magnitude of B for all the bands in a given semiconductor, further complicating the dependence of R on B. Defining Terms Conducting band: The band in which the electrons primarily responsible for the electric current are found. Effective mass: An electron in a lattice responds differently to applied fields than would a free electron or a classical particle. One can, however, often describe a particular response using classical equations by defining an effective mass whose value differs from the actual mass. For the same material the effective mass may be different for different phenomena; e.g., electrical conductivity and cyclotron resonance. Electron band: A range or band of energies in which there is a continuum (rather than a discrete set as in, for example, the hydrogen atom) of allowed quantum mechanical states partially or fully occupied by electrons. It is the continuous nature of these states that permits them to respond almost classically to an applied electric field. Hole or hole state: When a conducting band, which can hold two electrons/unit cell, is more than half full, the remaining unfilled states are called holes. Such a band responds to electric and magnetic fields as if it contained positively charged carriers equal in number to the number of holes in the band. Relaxation time: The time for a distribution of particles, out of equilibrium by a measure F, to return exponentially toward equilibrium to a measure F/e out of equilibrium when the disequilibrating fields are removed (e is the natural logarithm base). Related Topic 22.1 Physical Properties R en n H e h = - 1 1