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The planar average of |ψ|2 for the Δ2 and Δ4 states along the direction of quantization for a thin Si(001) slab under a bias of 106 V/cm. The slab is 96 Å thick and terminated with silicon dioxide at the surface. Note the Δ4 density punching through the Δ2 densities at the surface.
The planar average of |ψ|2 for the Δ2 and Δ4 states along the direction of quantization a thin Si(001) slab with no external bias. The slab is 42 Å thick and H-terminated at both surfaces. The bulging effect in the Δ4 density at the surface is more pronounced here compared to Fig. 1 because the well is symmetric, resulting in a narrower spectrum in reciprocal space for the ground state.
(a) Orbital decomposed envelopes of the Δ4 ground state wavefunction in a 4 nm hydrogen passivated silicon quantum well computed with DFT. In-plane orbital Bloch sums that are odd in the quantization dimension result in odd parity envelopes causing a bulging of the wavefunction at the silicon surface. The wavefunctions are plotted on a log scale to reveal the functional form of the odd envelopes. (b) Same plot for the Δ2 ground state, which also exhibits opposite parity for even and odd orbital components. The rapid oscillatory behavior of these envelopes suppresses any bulging effect.
Ground state of a four atom quantum well. For even basis functions such as shown in (a), the envelope must be even in order to maintain even parity of the state. For odd basis functions such as the one shown in (b), the envelope must be odd in order to maintain an even parity. Since the envelopes correspond to , any contribution from even orbitals is suppressed by the modulation at the surface while any contribution from odd orbitals is maximal at the surface.
For a periodic potential with unit cells of length a0 , an even wavefunction is constructed from two Bloch states of the infinite periodic potential with and a Bloch function unk . For such a wavefunction there is a phase shift between the (a) even and odd components of the Bloch function. The spatial extent of this phase shift is inversely proportional to k as can be seen in the contributions to both the wavefunction (b) and density (c),where with the corresponding real space phase shift of 4a0 . (d) Finally, for a wavefunction constructed from Bloch states with large momentum, , the real space phase separation is small as is the redistribution of density—especially compared to a wavefunction constructed from Bloch states with small momentum, .
Trace over the orbital coefficients for spherical harmonic decomposed DFT (solid lines) and ETBM (dashed lines) wavefunctions for the Δ2 (blue lines) and Δ4 (red lines) ground state wavefunctions.
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