Effective Bohr radius

The total energies and wavefunctions of the Hamiltonian (1) have been calculated as the eigenvalues and eigenvectors of a Cl matrix. Full Cl has been used for all calculations of quasi-one-dimensional quantum dots and for quasi-two-dimensional quantum dots with N = 2, while multi-reference Cl has been used for quasi-two-dimensional quantum dots with N = 3 and 4. The results are presented in atomic units. They can be scaled by the effective Bohr radius of 9.79 nm and the effective Hartree energy of 11.9 meV for GaAs semiconductor quantum dots [25,26]. 

As the nuclei become heavier, the strong attraction of the electrons by the very large nuclear charge causes the electrons to move very rapidly and behave relativistically, i.e. their relative mass (m) increases according to equation 1, and the effective Bohr radius (ao) for inner electrons with large average speeds decreases according to equation 230. 

Here itot is the lifetime of the energy donating molecule, y = 2r0/L, where a constant L (called the effective Bohr radius) falls in the 0.1-0.2nm range, and r0 is defined by... 

In the semiconductors of greater polarity, the dielectric constants are smaller and the effective masses larger, and the same evaluation leads to 0.07 eV in zinc selcnidc, for example many of the impurity states can be occupied at room temperature. As the energy of the impurity states becomes deeper, the effective Bohr radius becomes smaller and the use of the effective mass approximation becomes suspect the error leads to an underestimation of the binding energy. Thus, in semiconductors of greatest polarity- and certainly in ionic crystals— impurity states can become very important and arc then best understood in atomic terms. We will return to this topic in Chapter 14, in the discussion of ionic crystals. 

Assuming for the sake of simplicity an isotropic centre, the effective Bohr radius a can be obtained directly from the hyperfme constant A, since this scales as the third power of the radius ... 

Here L is an effective Bohr radius and K is a constant involving the spatial overlap of the electron wave functions. 

A value of the energy corresponding to the Coulomb term e2/ (47r o saod ) for a value of r equal to the effective Bohr radius a0d is also given for comparison in Table 5.3. For H, this energy is 27.2 eV. 

The situation for InSb is somewhat different, the reason being the small electron effective mass and large dielectric constant, which result in a rather large effective Bohr radius pm for n = 2 state) and a small value... 

The sharpest donor lines reported in natGe are Ap (/l (I I, O) and 4/ i (D (H, O), with FWHMs 6.4 peV (0.05 cm 1). The model of Nishikawa and Barrie [113] has been used by Navarro et al. [109] to compare with the calculated line width the residual width of the 2p line of the D (H, O) donor in ultrapure natGe, measured by Zeeman tuning of the energieer frequencies. The broadening calculated by taking into account the interaction of the 2p 1 level with the four nearest levels (2po, 2s, 3po, and 3s) was 2.5 peV, compared to an experimental value of 8.6 peV for the FWHM of the 2p 1 line, but the calculated value was found to be sensitive to the value of the effective Bohr radius used in the calculation. For the acceptors, the sharpest line reported... 

The approximate analytical solutions for wave functions and energy levels had shown the following. P-type contribution is dominant at the defect distance zo from the surface 0 10 ag. Here is effective Bohr radius that depends strongly on effective mass p, and dielectric permittivity 82 S2/p- (see Fig. 4.13). It is seen from the Fig. 4.13... 

Then, scaling lengths by the effective Bohr radius,... 

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