Semiconductors effective mass

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

Fig. 2. Electron drift velocities as a function of electric field for A, GaAs and B, Si The gradual saturation of curve B is characteristic of all indirect semiconductors. Curve A is characteristic of direct gap semiconductors and at low electric fields this curve has a steeper slope which reflects the larger electron mobiUty. The peak in curve A is the point at which a substantial fraction of the electrons have gained sufficient energy to populate the indirect L minimum which has a much larger electron-effective mass than the F minimum. Above 30 kV/cm (not shown) the drift velocity in Si exceeds that in...

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... 

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. 

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. 

Electron-donor end group, 20 504 Electron donors, in Ziegler-Natta polymerization, 26 518-521 Electron effective mass, in direct gap semiconductors, 22 143—144 Electronegativities, Pauling scale of,... 

Enright B, Fitzmaurice D (1996) Spectroscopic determination of electron and hole effective masses in a nanociystalline semiconductor film J Phys Chem 100 1027-1035... 

Other CD semiconductors have been shown to exhibit size quantization. PbSe shows the effect very clearly, since quantum size effects can be clearly seen in this material, even in crystals up to several tens of nanometers in size (due to the small effective mass of the excited electron-hole pair). Shifts of greater than 1 eV have been demonstrated, from the bulk bandgap of 0.28 eV to 1.5 eV. 

There are many theoretical models to correlate the increase in semiconductor bandgap with crystal size. However, for our purposes we will show only the original model, known as the effective mass model, since this is the easiest to understand, in spite of its limited accuracy. 

The other semiconductor, apart from CdSe, that has been studied with deliberate emphasis on quantum size effects, is PbSe (see Table 10.3). There are several reasons for this. One is the long-known use of CD to deposit PbS and later PbSe. Second, and of particular importance, the electron/hole effective mass in PbSe is very... 

It is interesting that, apart from this study, quantum size effects have not been described in CD PbS films, in contrast to PbSe ones. Although PbS does show weaker quantum effects than does PbSe (because of its larger effective mass), it still should show strong quantum size effects—greater than CdSe, for example. For some reason, PbS seems to grow with larger crystal size than many other semiconductors. However, there is no a priori reason to indicate that size-quantized PbS could not be deposited by CD, and it is likely that an effort to do so would bear fruit. 

Perhaps the most direct evidence of the high effective mass in such materials comes from the work of Shapira et al. (1972) on the conductivity of EuTe and the work of Shapira and Reed (1972) on EuS, an antiferromagnetic material, with sufficient non-stoichiometry to make it a degenerate n-type semiconductor. Figure 3.5 shows the temperature variation of the resistivity for various magnetic... 

Many papers have been published on the theory of die Kondo effect, including some exact solutions. We recommend the 260 page review by Tsvelich and Weigmann (1983). Our aim in giving a simple non-mathematical account is to point out the similarity between the enhancement of the effective mass that occurs in crystalline metallic systems near to the conditions for a Mott transition (Chapter 4), and also to address the possible effects of free spins in doped semiconductors near the transition (Chapter 5). 

In relation to this problem is the fact that as the top of a band is approached the effective mass of a carrier changes and the range of allowed k values is small. Thus, the mobility of a carrier either in a narrow band conductor or at the top of an almost filled band must inevitably be small (9). In these cases it is probably not correct to assume the mass of the carrier and an electron to be the same. Under some circumstances the transfer of charge in a narrow band semiconductor is better described as an activated hopping process. 

---