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^{1}and Ryan Hatcher

^{1}

### Abstract

A seemingly anomalous enhancement of electron mobility in strained silicon inversion layers has exposed a gap between device physics theory and experiment in recent years. At the root of this discrepancy is a surface bulging effect in the electron Δ_{4}wavefunction, which increases surface roughnessscattering for these states. Complementary metal oxide semiconductor strain engineering reduces Δ_{4} state occupancy, thereby reducing surface roughnessscattering in the channel. The origin of this effect can be explained by moving beyond the effective mass approximation and contrasting the properties of the Δ_{2} and Δ_{4}wavefunctions in a representation that comprehends full crystal and Bloch state symmetry.

The authors acknowledge helpful conversations with Timothy Boykin and Borna Obradovic.

### Key Topics

- Wave functions
- 22.0
- Silicon
- 16.0
- Surface states
- 12.0
- Surface scattering
- 11.0
- Surface measurements
- 10.0

## Figures

The planar average of |ψ|^{2} for the Δ_{2} and Δ_{4} states along the direction of quantization for a thin Si(001) slab under a bias of 10^{6} V/cm. The slab is 96 Å thick and terminated with silicon dioxide at the surface. Note the Δ_{4} density punching through the Δ_{2} densities at the surface.

The planar average of |ψ|^{2} for the Δ_{2} and Δ_{4} states along the direction of quantization for a thin Si(001) slab under a bias of 10^{6} V/cm. The slab is 96 Å thick and terminated with silicon dioxide at the surface. Note the Δ_{4} density punching through the Δ_{2} densities at the surface.

The planar average of |ψ|^{2} for the Δ_{2} and Δ_{4} 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 Δ_{4} 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.

The planar average of |ψ|^{2} for the Δ_{2} and Δ_{4} 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 Δ_{4} 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 Δ_{4} 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 Δ_{2} ground state, which also exhibits opposite parity for even and odd orbital components. The rapid oscillatory behavior of these envelopes suppresses any bulging effect.

(a) Orbital decomposed envelopes of the Δ_{4} 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 Δ_{2} 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.

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 *a _{0} *, an even wavefunction is constructed from two Bloch states of the infinite periodic potential with and a Bloch function

*u*. 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

_{nk}*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 4

*a*. (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, .

_{0}For a periodic potential with unit cells of length *a _{0} *, an even wavefunction is constructed from two Bloch states of the infinite periodic potential with and a Bloch function

*u*. 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

_{nk}*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 4

*a*. (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, .

_{0}Trace over the orbital coefficients for spherical harmonic decomposed DFT (solid lines) and ETBM (dashed lines) wavefunctions for the Δ_{2} (blue lines) and Δ_{4} (red lines) ground state wavefunctions.

Trace over the orbital coefficients for spherical harmonic decomposed DFT (solid lines) and ETBM (dashed lines) wavefunctions for the Δ_{2} (blue lines) and Δ_{4} (red lines) ground state wavefunctions.

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