Real-space multiband envelope-function approach without spurious solutions

The envelope-function approach for mesoscopic electronic structure calculations is plagued by spurious solutions that originate in the ill representation of first-order derivatives in real-space bases. We present a symmetry-adapted finite element approach for multiband $\vec{k}·\vec{p}$ envelope-function Hamiltonians that is manifestly free from spurious solutions, numerically efficient, and easy to implement. In addition, a gauge-invariant extension to this method is developed for problems involving magnetic fields. We predict the electrical exciton $g$-factor tuning in single InAs/GaAs embedded quantum dots to be significantly larger than previously found.

Figure 1

Electronic band structure of the White Hamiltonian with the parameters $E_0=1$ eV, $P=1\text{eV}\text{nm}$, and $A=0$. The black line is the calculated continuum dispersion in $k$ space. The squares (blue) and circles (red) are real-space solutions on a periodic $6\text{nm}$ grid of $0.25\text{nm}$ spacing. The squares result from the finite difference method, whereas the circles represent the presently proposed symmetry-adapted finite element (SAFE) method.

Figure 2

Calculated ground-state expectation value of the momentum commutator for a free two-dimensional electron gas in a vertical magnetic field of 25 T. The full line indicates the exact continuum result. The circles (red) are results obtained in terms of the present SAFE method, whereas the squares (blue) have been obtained by a previous gauge-invariant finite difference method (Ref. 12).

Figure 3

Calculated dependence of the electric tuning of the electron and exciton Landé factors $g_e$ and $g_{\mathrm{ex}}$, respectively, of an InAs/GaAs embedded quantum dot as a function of the Indium concentration at the dot apex. The dotted lines indicate numerical results from a gauge-invariant finite difference method (Ref. 16). The full lines indicate the present SAFE results.

Figure 4

Contour graph of the electron density in the $\left(010\right)$ plane in an InAs/GaAs embedded quantum dot through the dot center. There is an applied vertical electric field of $60\mathrm{kV}/\mathrm{cm}$. (a) Finite difference calculations of Ref. 16. (b) Calculated density obtained by the present SAFE method.