Effective Electron and Hole Masses

We showed previously that the effective electron mass depends on the curvature of the band. In the conduction band in the vicinity of k = 0, the band is curved upward, so the effective mass of a conduction electron is positive. However, near the top of the valence band, the bands curve downward, giving the particles near the top of the valence band negative mass. 

---

InN is at the present time always grown n-type, and this has allowed experimental determinations of the electron effective mass from plasma reflectivity [4,8,24], Hole masses are generally obtained from band structure calculations. TABLE 3 lists some determinations of electron and hole masses of InN in units of mo. Most calculations agree with the experimental electron mass of 0.1 lmo, but the uncertainty regarding hole masses is still large at the present stage. 

The g-factors of electrons and holes reflect the nature of the conduction and valence bands in much the same way as the effective masses. Thus in AgF, AgCl, and AgBr, free electrons and shallowly trapped electrons whose wavefunctions are made up largely of conduction band functions are expected to be isotropic in nature. A free electron at the bottom of the band will have a single effective mass and g-factor. In contrast, free holes near the L-point and shallowly trapped holes whose wavefunctions are largely valence band functions are expected to show anisotropic behavior. A free hole will have parallel and perpendicular g-factors. The available data on electron and hole masses were given in Table 1 and the data on g-factors are given in Table 9. Thermalized electrons and holes in both modifications of Agl will be at the zone center. The anisotropic nature of the wurtzite crystal structure will be reflected in the effective masses and g-factors. 

HOMO and LUMO levels of the NCs are calculated based on reported bulk values, effective electron and hole mobilities of the bulk material, and the relative permittivities following the formula for the effective mass approximation from Brus [34]. 

As mentioned above, the results discussed below are obtained using Ab initio methods. Other methods used to study QDs are effective mass theory (EMT) and the pseudopotential techniques. EMT uses a particle-in-a-box model where the electron and hole masses are given by their bulk values. EMT is an intuitive description that explains general trends seen in experiments. The atomistic pseudopotential technique can be applied to large systems, but requires careful parameterization for each material. Ab initio approaches use minimal parameterization and are applicable to most materials. This makes them particularly useful for studying dopants, defects, ligands, core/shell systems and QD synthesis. The Hartree-Fock (HE) method and density functional theory (DFT) have been around for many decades, while time domain (TD) DFT and non-adiabatic molecular dynamics (NAMD) are more recent areas of research. Currently, ab initio TDDFT/NAMD is the only... 

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. 

Remarkably, although band stmcture is a quantum mechanical property, once electrons and holes are introduced, theit behavior generally can be described classically even for deep submicrometer geometries. Some allowance for band stmcture may have to be made by choosing different values of effective mass for different appHcations. For example, different effective masses are used in the density of states and conductivity (26). 

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. 

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...

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). 

Near k = 0 the electron and hole effective masses must be isotropic and constant. For a nanoparticle, the uncertainty in the exciton position depends upon its size. Ax 2R. If the relation between energy and momentum is independent of particle size the exciton... 

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

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. 

It has been shown in Section 1.3.7 that in semiconductors or insulators the lattice defects and electronic defects (electrons and holes), derived from non-stoichiometry, can be regarded as chemical species, and that the creation of non-stoichiometry can be treated as a chemical reaction to which the law of mass action can be applied. This method was demonstrated for Nii O, Zr Cai Oiand Cuz- O in Sections 1.4.5, 1.4.6, and 1.4.9, as typical examples. We shall now introduce a general method based on the above-mentioned principle after Kroger, and then discuss the impurity effect on the electrical properties of PbS as an example. This method is very useful in investigating the relation between non-stoichiometry and electrical properties of semiconductive compounds. 

---