Effective mass of electron

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... 

Lead(II) sulfide occurs widely as the black opaque mineral galena, which is the principal ore of lead. The bulk material has a band gap of 0.41 eV, and it is used as a Pb " ion-selective sensor and IR detector. PbS may become suitable for optoelectronic applications upon tailoring its band gap by alloying with II-VI compounds like ZnS or CdS. Importantly, PbS allows strong size-quantization effects due to a high dielectric constant and small effective mass of electrons and holes. It is considered that its band gap energy should be easily modulated from the bulk value to a few electron volts, solely by changing the material s dimensionality. 

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...

Although the De Broglie wavelength of free electrons is 0.1 nm, the value of an electron in a small crystallite can be much larger because the effective mass of electrons in a small particle is considerably smaller. Energy levels evolve from HOMO and LUMO to those of clusters, Q-sized particles, and finally bulk semiconductor. Figure 7.7 shows the energy levels in bulk- and Q-sized particulate semiconductors. 

One further effect of the formation of bands of electron energy in solids is that the effective mass of electrons is dependent on the shape of the E-k curve. If this is the parabolic shape of the classical free electron theory, the effective mass is the same as the mass of the free electron in space, but as this departs from the parabolic shape the effective mass varies, depending on the curvature of the E-k curve. From the definition of E in terms of k, it follows that the mass is related to the second derivative of E with respect to k thus... 

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 effective masses of electrons (md) and holes (m ) represent the masses that these charges appear to have when moving in the solid rather than in free space, and these vary from material to material. (In the size quantized regime, they can also vary with crystal size, particularly for small quantum dots, hence the limitations of the effective mass model). 

Within the approximation of the effective mass, consideration of the field created by the condensed media is confined to substitution of the real electron mass by the effective mass. Precise calculation of the effective mass is equivalent to solution of the Schrodinger equation with the consideration of the field created by the medium, and, consequently, as noted before, is hardly possible. Thus, as far as the problem of electron tunneling is concerned, the effective mass must be considered as a phenomenological parameter. In the case of tunneling with the energy I of the order of 1-5 eV, the field created by the medium apparently increases considerably the probability of electron tunneling, and the effective mass of electron can be noticeably lower than the real mass. 

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. 

Ion Scattering Spectroscopy mass, molar mass effective mass of electron concentration of ions or charge carriers concentration of acceptors concentration of donors coordination number of shell j complex refraction index photo ionization cross-section electric charge gas constant... 

The proportionality constant between the applied electric field and the resulting drift velocity is called the charge carrier mobility, jx. For electrons, = q r /ml ), for holes, ftp = 7(Trn/mj ). It should be noted that, owing to differences in the effective masses of electrons and holes, their mobilities within a semiconductor may be markedly different. The electrical conductivity, a, of a semiconductor is related to the free carrier concentrations by ... 

Figure 7. Schematic view of polysilane band structure. The abbreviations and symbols are defined as follows me, effective mass of electrons mh effective mass of holes lumi., luminescence abs., absorption T, k = 0 point and k,...

The effective masses of electrons and holes are estimated by parabolic approximation a large curvature corresponds to a small effective mass and a small curvature corresponds to a large mass. With this band concept, light absorption and luminescence are interpreted as follows Light is absorbed by the transition from valence band to conduction band. Therefore, the broadening of the absorption spectrum originates basically from the one dimensionality of the joint density of states, which is described by (E - g) . Excited electrons and holes relax to the bottom of the bands and then recombine radiatively. Therefore, the photoluminescence of the spectrum is very sharp. The energy difference between two peaks is called the Stokes shift. 

Current-voltage characteristics of the CoFe/MgO/Si nanostructure were calculated for the work ftinction of CoFe of 4.8 eV, the MgO effective mass of electrons of 0.35 We> MgO electron affinity of 1.5 eV, Si electron affinity of 4.15 eV. The current density tends to saturation at increasing of the applied voltage from 0.2 to 0.5 eV (Fig. 3). In case of changing of temperature from 50 to 360 K the tunnel current through the nanostructure varies insignificantly. Qualitatively the same I-V characteristic behavior was observed for similar structures in [4]. 

The n-type conductivity is determined by the product of the reciprocal effective mass of electrons and the concentration of carriers in a semi-classical viewpoint. The effective mass is often calculated by fitting the dispersion curve near the bottom of the conduction band. The bottom is mainly composed of M-4s/5s orbitals in the oxides of the present interest, and the curvature is mainly determined by the M-M interactions. The situation is schematically shown in Fig. 4. The strong interaction among M-4s/5s orbitals should bring about wider M-4s/5s band-width and smaller effective mass. On the other hand, weaker M-4s/5s interactions result in narrower band-width and larger effective mass. With the increase of the atomic number in the same row of the periodic table, the M-4s/5s orbitals tend to be contracted. Assuming the M-M distance is the same, the M-4s/5s interactions should be weakened with the atomic number, in general. 

The overlap population between M4s orbitals at the lowest unoccupied molecular orbital (LUMO) obtained by the cluster calculation is shown in Fig. 5. Differently from 2 Jdy y, the value is smaller in CU2O than in ZnO. This can be explained by the remarkably smaller M4s population in the LUMO only in CU2O, as can be seen in Fig. 5. LUMO of CU2O cannot be treated similarly to the later oxides. Except for CU2O, however, M4s is dominant at the bottom of the conduction band. Therefore, the M4s interactions should play a determining role for the effective mass of electrons. With the increase of the atomic number, the effective mass is expected to increase, thereby reducing... 

Here, h is the Plank constant, p = e, h stands for the conduction and valence bands, respectively, are the effective masses of electron (p = e) and holes (p = h), and ( )n,i is the -th zero of the spherical Bessel functions of the order /, gi (( )n i) = 0, and R is the nanocrystal radius. Two lowest energy levels in the conduction band and two highest energy levels in the valence band are shown in Figure 1, where letters S and P stand for the levels with the angular momenta 1 = 0 and 1=1, respectively. 

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