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n if the magnetic field is along the 2 direction, B — Bêz, the Hamiltonian can be expressed as

-B2r2 sin2 6

Here, Lz is the z component of the angular momentum operator, and 6 is the usual polar angle in these coordinates. The magnetic field preserves the azimuthal symmetry (i.e., m is still a "good" quantum number). As only the ground state is studied, the value m = 0 is required; thus the Hamiltonian [Eq. (184)] will retain the quadratic dependence on B only.

Because of confinement, the solution of the Schrodinger equation associated with Eq. (184) must satisfy t(r — r0, 6,ç) — 0

m = 0 is chosen, thus the Hamiltonian [Eq. (188)] will not have a linear dependence on magnetic field B, just like the symmetric case studied previously.

As we shall see, the trial wavefunction [Eq. (189)] is flexible enough to describe the case when the radius of the confining sphere becomes infinite and the hydrogenic impurity is close to its boundary. The latter would correspond to the case of a hydrogenic impurity close to a plane, which allows us to investigate the ground state edge level of the spherical quantum dot.

The binding energy (eb) for the hydrogenic impurity is defined as the ground state energy of the system without Coulombic interaction (ew) minus the ground state energy including the presence of the electron-nucleus interaction (e0); that is, eb — ew eo

In order to use the variational method, the trial wave-function for the calculation of the ground state energy is chosen as

where N is the normalization constant and a is a varia-tional parameter to be determined. The last term has been obtained considering that in absence of A the system holds spherical symmetry. A similar choice of the trial wavefunc-tion to calculate the ground state energy has widely been used in the case of hydrogenic impurities within spherical quantum dots [65] and cylindrical quantum wires [77] without magnetic field.

In the case of a hydrogenic impurity displaced a constant distance a from the center of symmetry of a quantum dot, the Hamiltonian of Eq. (182) is modified as

where the confining potential, Vb(r), is still given by Eq. (183), and ez is a unit vector on the z-axis. Hence, the corresponding Hamiltonian and trial wavefunction, Eqs. (184) and (186), are given by

B2r2 sin2 6

$(r) = N exp(-a\r — aêz\)exp^ — g Br sin 0j(ro — r)

respectively.

Notice that the wavefunction [Eq. (189)] satisfies the boundary condition imposed by Eq. (185). Again, since

The binding energy defined previously is a positive quantity. The ew term of Eq. (190) is the ground state energy of an electron confined inside a spherical quantum dot of radius r0; that is,

The values of the physical parameters pertaining to a GaAs quantum dot used in these calculations, for the sake of comparison, are m* = 0.067m0 and k = 12.5; thus a*B = 98.7 A and R*y = 5.83 meV Moreover, the linear dependence term on B, in Eq. (184), can be scaled down to ym —

l0Bm

where /x0 = 5.78838263 x 10—2 meV T—1 is the Bohr magneton; then the value B = 6.75 T corresponds to an effective Rydberg (7 = 1).

The results for the symmetric case (on-center impurity) are displayed in Figure 32, in which the binding energy of the hydrogenic impurity is plotted as a function of the quantum dot radius for different magnetic field strengths. The qualitative and quantitative behaviors of the results are in good agreement with those of Xiao et al. [101] and Branis et al. [102], who studied the magnetic field dependence over the binding energies of hydrogenic impurities in quantum dots and quantum well wires, respectively. In the strong spatial confinement case, (r0 < a*B), the binding energy is relatively insensitive to the magnetic field and diverges as the dot radius r0 ^ 0, as in the case of zero magnetic field. The latter reflects that the main contribution to the binding energy is due to electron spatial confinement, which prevails over magnetic field confinement. The latter can be estimated through the magnetic length defined as lB — J — — 484.82 x B—1/2 eB

where lB is expressed in effective Bohrs, and B is given in Teslas. For dot radius r0 = a*B, the different magnetic field m

0 0

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