Effective mass of conduction

The first heavy fermion superconductor was discovered in CeCu2Si2 [19]. Common properties in a heavy fermion superconductor are large electronic specific heat coefficient, suggesting that the effective masses of the conduction electrons are 100 times higher than the static mass of an electron, and the unconventional superconductivity is difficult to explain by the BCS theory. It has been believed that the enhancement of the effective masses of conduction electrons in... 

The electronic specific heat coefficient, y, is proportional to the DOSs at the Fermi level. In general, in pure metals, it is of the order of a few mj/ mol K. Simply thinking, the electronic DOSs at the Fermi level is proportional to the effective mass of conduction electrons. The most conspicuous and noticeable systems with respect to their electronic specific heat coefficients are heavy electron systems that include some actinide and cerium compounds to be mentioned later. For example, in the case of a typical heavy electron system CeCug, the electronic specific heat coefficient reaches 1.5 J/mol K, which is more than 1000 times larger than that of a normal metal (Satoh et al., 1989). 

TWg, nif are the effective masses of conduction electrons and electron holes (defined below). Or even ... 

Where b is Planck s constant and m and are the effective masses of the electron and hole which may be larger or smaller than the rest mass of the electron. The effective mass reflects the strength of the interaction between the electron or hole and the periodic lattice and potentials within the crystal stmcture. In an ideal covalent semiconductor, electrons in the conduction band and holes in the valence band may be considered as quasi-free particles. The carriers have high drift mobilities in the range of 10 to 10 cm /(V-s) at room temperature. As shown in Table 4, this is the case for both metallic oxides and covalent semiconductors at room temperature. 

In most metals the electron behaves as a particle having approximately the same mass as the electron in free space. In the Group IV semiconductors, dris is usually not the case, and the effective mass of electrons can be substantially different from that of the electron in free space. The electronic sUmcture of Si and Ge utilizes hybrid orbitals for all of the valence elecU ons and all electron spins are paired within this structure. Electrons may be drermally separated from the elecU on population in dris bond structure, which is given the name the valence band, and become conduction elecU ons, creating at dre same time... 

The effective masses of holes and electrons in semiconductors are considerably less than that of the free electron, and die conduction equation must be modified accordingly using the effective masses to replace tire free electron mass. The conductivity of an intrinsic semiconductor is then given by... 

Fig. 36. Energy levels of excitonic states in CdS particles of various radii. Zero position of the lower edge of the conduction band in macrocrystalline CdS. Exc Energy of an exciton in macrocrystalline CdS. Effective masses of electrons and holes 0.19 m and 0.8 m respectively. The letters with a prime designate the quantum state of the hole...

The Schottky-Mott theory predicts a current / = (4 7t e m kB2/h3) T2 exp (—e A/kB 7) exp (e n V/kB T)— 1], where e is the electronic charge, m is the effective mass of the carrier, kB is Boltzmann s constant, T is the absolute temperature, n is a filling factor, A is the Schottky barrier height (see Fig. 1), and V is the applied voltage [31]. In Schottky-Mott theory, A should be the difference between the Fermi level of the metal and the conduction band minimum (for an n-type semiconductor-to-metal interface) or the valence band maximum (for a p-type semiconductor-metal interface) [32, 33]. Certain experimentally observed variations of A were for decades ascribed to pinning of states, but can now be attributed to local inhomogeneities of the interface, so the Schottky-Mott theory is secure. The opposite of a Schottky barrier is an ohmic contact, where there is only an added electrical resistance at the junction, typically between two metals. 

A and B being constants which need not interest us further. (We may assume that A B, which denotes approximate equality of the effective masses of free electrons and holes.) Thus, the electrical conductivity is diffeient in different cross sections parallel to the adsorbing surface (i.e., at different x). Chemisorption, by changing the bending of the bands, may lead to a noticeable change in the electrical conductivity of the subsurface layer of the crystal, which in the case of a sufficiently small crystal may effect the total electrical conductivity of the sample. Even more, so the very type of conductivity in the subsurface layer may change under the influence of chemisorption n conductivity (e < +) may go over into p conductivity (t > +), or vice versa (the so-called inversion of conductivity). 

Ncv = 2Mc v (27rm v KT/ h2 )3/2 where Mc v — the number of equivalent minima or maxima in the conduction and valence bands, respectively, and m cv = the density of states effective masses of electrons and holes. 

The energy E will necessarily have this minimum, but its value at this point can be positive or negative only in the latter case will a stable self-trapped particle (i.e. a small polaron) form. This is most likely to occur for large effective mass, and thus for holes in a narrow valence band or for carriers in d-bands. If the polaron is unstable then there is practically no change in the effective mass of an electron or hole in equilibrium in the conduction or valence band. 

It is of course possible that a carrier in the conduction band or a hole in the valence band will form a spin polaron, giving considerable mass enhancement. The arguments of Chapter 3, Section 4 suggest that the effective mass of a spin polaron will depend little on whether the spins are ordered or disordered (as they are above the Neel temperature TN). This may perhaps be a clue to why the gap is little affected when T passes through TN. If the gap is U —%Bt -f B2 and Bt and B2 are small because of polaron formation and little affected by spin disorder, the insensitivity of the gap to spin disorder is to be expected. 

Since this calculation of the quantum size effect is demonstrated for direct band gap, this equation must be carefully applied to the indirect band gap semiconductor such as Ti02.9) The effective mass of the conduction band electron reported for rutileTi02 ranges from 3 to 30 with a usual value of 20. For anatase Ti02, on the other hand, the effective electron mass is reported to be l.,0) The smaller effective mass of anatase can be attributed to the higher mobility of the electron, which is 40 times larger than that of rutile.l0)... 

In equation 3, ran is the effective mass of the electron, h is the Planck constant divided by 2/rr, and Eg is the band gap. Unlike the free electron mass, the effective mass takes into account the interaction of electrons with the periodic potential of the crystal lattice thus, the effective mass reflects the curvature of the conduction band (5). This curvature of the conduction band with momentum is apparent in Figure 7. Values of effective masses for selected semiconductors are listed in Table I. The different values for the longitudinal and transverse effective masses for the electrons reflect the variation in the curvature of the conduction band minimum with crystal direction. Similarly, the light- and heavy-hole mobilities are due to the different curvatures of the valence band maximum (5, 7). 

Fig. 21. Real part of the conductivity of YbFe4St>i2- The symbols on die left axis represent dc values at different temperatures. Below T (fv 50 K), a narrow peak at zero frequency and a gap-like feature at 18 meV gradually develop. Inset Renormalized band structure calculated from die Anderson lattice Hamiltonian. % and f denote bands of free carriers and localized electrons, respectively. At low temperatures a direct gap A opens. The Fermi level, Ep is near die top of die lower band,, resulting in hole-like character and enhanced effective mass of die quasiparticles (Dordevic et al., 2001).

Here /ie and are effective masses of electron and hole, respectively. Near to bottom of conductivity band and near to top of valent band where dependence E from k is close to parabolic, electron and hole move under action of a field as particles with effective masses fie — h2l(d2Ec(k)ldk1) and jUh = —h2l( E (k)ldk ) [6]. In particular, in above-considered onedimensional polymer semiconductor /ie — /ih — h2AEQj2PiP2d2 [6]. As a first approximation, it is possible to present nanocrystal as a sphere with radius R, which can be considered as a potential well with infinite walls [6], The value of AE in such nanocrystal is determined by the transition energy between quantum levels of electron and hole, with the account Coulomb interaction between these nanoparticles. 

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