# Electronic structure of boron and aluminum $\delta$-doped layers in silicon

Quinn T. Campbell  Center for Computing Research, Sandia National
Laboratories, Albuquerque NM, USA  Shashank Misra  Sandia National
Laboratories, Albuquerque NM, USA  Andrew D. Baczewski  Center for Computing
Research, Sandia National Laboratories, Albuquerque NM, USA

###### Abstract

Recent work on atomic-precision dopant incorporation technologies has led to
the creation of both boron and aluminum $\delta$-doped layers in silicon with
densities above the solid solubility limit. We use density functional theory
to predict the band structure and effective mass values of such $\delta$
layers, first modeling them as ordered supercells. Structural relaxation is
found to have a significant impact on the impurity band energies and effective
masses of the boron layers, but not the aluminum layers. However, disorder in
the $\delta$ layers is found to lead to significant flattening of the bands in
both cases. We calculate the local density of states and doping potential for
these $\delta$-doped layers, demonstrating that their influence is highly
localized with spatial extents at most 4 nm. We conclude that acceptor
$\delta$-doped layers exhibit different electronic structure features
dependent on both the dopant atom and spatial ordering. This suggests
prospects for controlling the electronic properties of these layers if the
local details of the incorporation chemistry can be fine tuned.

##  I Introduction

Heavily $\delta$ doping a layer of silicon can be used to create a number of
technologically interesting quantum devices for nanoelectronic applications
Oberbeck  _et al._ (2002); Ruess  _et al._ (2004); Qian  _et al._ (2006); Rueß
_et al._ (2007). $\delta$ doping creates a narrow, highly doped layer of
silicon that generates an approximate $\delta$ potential within the
semiconductor device, causing the trapped dopant carriers to form a two-
dimensional electron or hole gas. Atomic-precision advanced-manufacturing
(APAM) techniques combining selective lithography techniques such as scanning
tunneling microscopy or pulsed UV laser Katzenmeyer  _et al._ (2021) with
surface chemistry and subsequent silicon growth have been used to create
highly confined $\delta$-doped layers, typically using phosphorus as the
dopant atom Ruess  _et al._ (2004); Ward  _et al._ (2020); Bussmann  _et al._
(2021). Recent work, however, has demonstrated the ability to create
$\delta$-doped layers using acceptor precursors, which incorporate aluminum or
boron Škereň  _et al._ (2020); Radue  _et al._ (2021); Dwyer  _et al._ (2021),
raising the intriguing possibility of precision patterning of superconducting
regions in silicon Bustarret  _et al._ (2006); Bourgeois and Blase (2007);
Shim and Tahan (2014); Duvauchelle  _et al._ (2015); Bonnet  _et al._ (2022).
The electronic structure of these layers will play a role in determining the
performance of any quantum devices utilizing these structures.

Previous work on the electronic structure of $\delta$-doped layers has largely
focused on donor-doped layers using a phosphine precursor within the APAM
process. A variety of theoretical methods, from tight binding Cartoixa and
Chang (2005); Lee  _et al._ (2011); Smith  _et al._ (2014), to planar Wannier
orbitals Qian  _et al._ (2005), to density functional theory (DFT) Carter  _et
al._ (2009, 2011); Drumm  _et al._ (2013a, b), to effective mass theory Drumm
_et al._ (2012), have been used to predict the electronic structure of
phosphorus $\delta$-doped layers in silicon. These theoretical predictions
have largely been confirmed by subsequent experimental characterization Miwa
_et al._ (2013); Mazzola  _et al._ (2014a, b); Miwa  _et al._ (2014); Mazzola
_et al._ (2018); Holt  _et al._ (2020). The electronic structure of these
$\delta$-doped layers has been shown to be crucial to the behavior of quantum
devices, with Hagmann et al. demonstrating that the electronic structure of
Si:P $\delta$-doped layers can be used to measure the confinement of carriers
in these layers based on non-destructive electrical transport measurements
Hagmann  _et al._ (2018). The success of DFT in describing the electronic
structure of phosphorus $\delta$-doped layers suggests that extending this
methodology to acceptor $\delta$-doped layers will lead to useful predictions
of electronic structure behavior.

In this manuscript, we use DFT calculations to predict the electronic
structure of boron and aluminum $\delta$-doped layers, calculating energies
and effective masses of the induced impurity bands. We demonstrate that boron
$\delta$-doped layers induce a significant amount of stress that leads to
broadening of the $\delta$ potential and a slight electron-like curvature
around the $\Gamma$ point in the highest valence band. We also calculate the
band structure of an acceptor $\delta$-doped layer with significant disorder
in the placement of dopant atoms, resulting in significant flattening of the
impurity bands. We compare these results to the band structure of aluminum
$\delta$-doped layers, which induce less stress and come closer to a pure
$\delta$ potential and its resulting band structure. Our work shows that both
the choice of dopant and the spatial ordering of dopants allow for control of
the gaps and effective masses of acceptor $\delta$-doped layers, suggesting
that these features can be controlled if local incorporation chemistry schemes
can be fine tuned.

##  II Methods

All electronic structure calculations are done using the quantum espresso
package Giannozzi  _et al._ (2009). We use norm-conserving pseudopotentials
from the PseudoDojo repository van Setten  _et al._ (2018) and the Perdew-
Burke-Ernzerhof exchange-correlation functional Perdew  _et al._ (1996). We
use kinetic energy cutoffs of 50 Ry and 200 Ry for the plane wave basis sets
that define the Kohn-Sham orbitals and charge density, respectively. We use a
2$\times$2$\times$1 Monkhorst-Pack grid Monkhorst and Pack (1976) to sample
the Brillouin zone in our initial self-consistent calculation and then a
4$\times$4$\times$1 Monkhorst-Pack grid for non self-consistent calculations
before band structures are calculated. Unless otherwise noted, we allow the
atoms to undergo geometric relaxation at a fixed supercell shape until the
interatomic forces fall below a threshold of 50 meV/Å. The cell size is kept
fixed to mimic embedding the $\delta$-doped layer in a larger bulk-like
silicon crystal.

To predict the electronic structure of $\delta$ layers, we create a 2$\times$2
Si(100) slab 110 Å long, with a single layer of 1/4 monolayer concentration of
boron or aluminum atoms at substitutional sites, as seen in Fig. 1a. The
silicon lattice constant is set at the PBE minimum lattice constant of 5.468 Å
based on calculations from the Materials Project Jain  _et al._ (2013).
Because we are using a plane wave DFT code that enforces periodic boundary
conditions in the Kohn-Sham orbitals and potential, the modeled slab repeats
in the (100) direction and the $\delta$ layer interacts with an infinite
number of its images spaced with a period of 110 Å. This distance was found to
be sufficient to mitigate the effect of these spurious images on our results.
To test the impact of the ordering of the acceptor dopants comprising the
$\delta$ layer, two structures are considered: an ordered 2$\times$2 cell
shown in Fig. 1b, and a disordered 2$\times$4 cell shown in Fig. 1c. The
placement of the atoms in the disordered cell is taken from a Kinetic Monte
Carlo (KMC) simulation of dopant placement using a BCl3 precursor within a
typical APAM process Campbell  _et al._ (2022). The same placement of dopant
atoms is used for the later aluminum calculations.

One difficulty of using DFT to predict the $\delta$-doped layer’s band
structure is disentangling the impact of the placement of dopants in a
$\delta$-layer structure from the overall concentration of dopants within the
DFT supercell. We examine this issue in more detail in Appendix A. We
otherwise note that all of our calculations suffer from the well-known
tendency of semilocal DFT to underestimate band gapsPerdew (1985). For ease of
comparison to bulk-like silicon, all of the band diagrams in this manuscript
include the band structure of a comparable silicon slab as gray-shaded
background. This is similar to the convention introduced in Ref. Carter  _et
al._ , 2009 for phosphorus-doped $\delta$ layers. The band gap we predict for
the pure silicon structure is approximately 0.6 eV. The conduction band of the
systems remains entirely unaffected with the introduction of acceptor layers,
and only the valence band is raised. Throughout, we reference the band
energies to the valence band maximum of silicon to avoid confusion regarding
the band gap of the material. We expect that the conduction band will be
unaffected by the acceptor $\delta$ layer and the band gap will increase by
$\approx$0.5 eV in real-world materials.

Figure 1:  (a) Side view of the silicon slab used to calculate the electronic
structure of $\delta$-doped layers. The purple box in the middle highlights
the location of the $\delta$-doped layer within the overall structure. (b) Top
view of the placement of dopant atoms within the ordered 2$\times$2 cell.
Silicon atoms are shown in blue, and dopant atoms in purple. The darker
spheres indicate the atoms present at the $\delta$ layer and the lighter atoms
are silicon at lower layers within the structure. (c) Top view of the
placement of dopant atoms within the disordered 4$\times$2 cell.

##  III Results

###  III.1 Boron $\delta$-doped layers

Figure 2:  The electronic band structure of boron $\delta$-doped layers when
(a) all atoms are frozen from their default position in silicon (i.e. no
geometry optimization), (b) after the system has been allowed to relax atomic
positions in response to stress, and (c) after atomic position relaxation with
spin-orbit coupling included in the calculation. In these figures, the gray
background shading represents our calculated band structure of bulk silicon
and the red line represents the Fermi level.

The band structure of a $\delta$ layer of boron in silicon without ionic
relaxation is shown in Fig. 2a. It is worth noting that we refer all energies
to a zero point set by the bulk valence band maximum (VBM) of any given
structure. Furthermore, these band diagrams appear less distinct compared to a
typical silicon primitive cell band diagram because the larger supercell used
in this work generates Brillioun zone folding. There are three impurity bands
associated with the $\delta$ layer, two of which form the degenerate valance
band maximum (VBM) 248 meV above the bulk-like VBM. The curvature of the third
band (1$\Delta$) becomes slightly positive (electron-like) near the $\Gamma$
point, with the maxima occurring at 0.1$\times$2$\pi$/a in the $X$ direction
and a local minima at the $\Gamma$ point. Each of the impurity bands is
partially occupied, albeit far from the $\Gamma$ point, with the Fermi level
at 77 meV above the pure silicon VBM.

The stress induced in the surrounding silicon by boron $\delta$ layers is
significant (148 MPa per unit cell) and thus we examine the impact of relaxing
the atomic coordinates on the calculated band structure in Fig. 2b. The
lattice constant is held constant, however, to mimic the embedding of the
$\delta$ layer within the larger silicon structure. Most prominently, the
order of the impurity bands away from the VBM is reversed. Here we determine
the character of the bands under the reordering in terms of the two-fold
$\Gamma$-point degeneracy (1$\Gamma$ and 2$\Gamma$) and the non-$\Gamma$ VBM
(1$\Delta$). The 1$\Delta$ band now moves up in energy 49 meV to become the
VBM of the $\delta$-doped layer electronic structure, this time only 219 meV
above the bulk Si VBM, increasing the overall band gap at the $\Gamma$ by 29
meV. The 1$\Gamma$ and 2$\Gamma$ bands move down by 37 meV, moving to below
the 1$\Delta$ band, while still maintaining their double degeneracy. A small
gap of 5 meV is created at the $\Gamma$ point between the 1$\Delta$ and
1$\Gamma$ bands. This switching of ordering now means that the top valence
band has the opposite curvature as typical around the $\Gamma$ point and that
the VBM occurs at 0.14$\times$2$\pi$/a in the $X$ direction. The Fermi level
for the $\delta$-doped layer nonetheless stays essentially constant at 79 meV.

Spin-orbit coupling has a smaller influence on the band structure of the boron
$\delta$-doped layer than the ionic relaxation in response to stress, as shown
in Fig. 2c where spin-orbit coupling is included in the relaxed structure. The
overall ordering and curvature of bands remains largely constant, with the
1$\Delta$ band remaining as the top impurity band. The band gap of the
$\delta$-doped layer at the $\Gamma$ point increases by 28 meV, with the
1$\Delta$ band now only 200 meV above the bulk VBM at the $\Gamma$ point. The
1$\Gamma$ and 2$\Gamma$ bands remain doubly degenerate with the gap between
them and the 1$\Delta$ point only increasing from 5 to 8 meV, indicating a
relatively small contribution of spin-orbit coupling. The Fermi level again
remains constant at 83 meV. The orbital energies of each band at the $\Gamma$
point are summarized in Table 1.

We also calculate the effective masses at different $k$–points using
$m_{ij}^{*}=\hbar^{2}\left(\partial^{2}E(k)/\partial k_{i}\partial
k_{j}\right)^{-1}$, where we approximate the second derivatives using a five-
point stencil and the results are reported in Table 2. For the sake of
comparison, we note that the heavy- and light-hole effective masses for bulk
silicon are respectively 0.49 and 0.16 $m_{e}$. We find that the
$\Gamma$-point effective masses for the heavy- (1$\Gamma$) and light-hole
(2$\Gamma$) impurity bands are respectively lighter and heavier than their
bulk counterparts, but only very slightly. We also note the positive
(electron-like) curvature of the 1$\Delta$ band at the $\Gamma$ point. Most
relevant to the transport properties of these layers are the effective masses
near the Fermi level at points with $\Delta$ and $\Lambda$ symmetries, where
the bands flatten out and have much higher effective masses.

Table 1: Energy of bands at the $\Gamma$ point for B $\delta$-doped layers, corresponding to the diagrams of Fig. 2. All energies are reported in meV relative to the VBM of pure Si. Band | Unrelaxed (2a) | Relaxed (2b) | Relaxed, SOC (2c)
---|---|---|---
1$\Gamma$ | 248 | 214 | 186
2$\Gamma$ | 248 | 214 | 186
1$\Delta$ | 167 | 219 | 194
EF | 77 | 79 | 83
ECB | 624 | 627 | 630
Table 2: Effective mass of the different bands at varying $k$ points in the case of a B $\delta$-doped layer which has been calculated including geometry optimization and spin-orbit coupling. All masses are reported in units of me. The sign convention is taken such that a valence band with negative curvature (i.e. the expected curvature for a valence band) has a positive mass and bands with positive curvatures have negative masses. Band | $\Gamma$ | EF crossing L | EF crossing X
---|---|---|---
1$\Gamma$ | 0.43 | 2.07 | 2.65
2$\Gamma$ | 0.24 | 3.73 | 0.87
1$\Delta$ | –0.56 | 0.95 | 0.88
Figure 3:  Planar average of the local density of states (LDOS) for boron
$\delta$-doped layers (a) before and (b) after atomic position relaxation.
Dark blue regions indicate no occupied states and red and green regions
indicated positions of high occupancy.

Next, we consider the spatial extent of the impurity bands by examining the
local density of states in Fig. 3. Without relaxing the stress created by the
$\delta$ layer, the associated states remain well localized within $\approx$1
nm of the layer itself. In contrast, after relaxation these states delocalize
and separate. While the impurity band edge becomes slightly more diffuse it is
largely still concentrated within $\approx$1 nm of the layer itself. However,
the bands below this edge delocalize over the entire $\pm$ 5 nm extent of the
supercell and there are evident gaps in energy separating these delocalized
states from both the impurity band edge and the bulk valence bands. The
calculated band gap remains a relatively constant 0.6 eV throughout the
structure with both the valence and conduction bands forming a
$\delta$-potential around the $\delta$-doped layer. This band gap corresponds
to the typical DFT underestimation of the silicon band gap and could be
improved with more computationally expensive methods, e.g. the use of hybrid
exchange-correlation functionals Heyd  _et al._ (2003).

Figure 4:  The electronic band structure of a disordered boron $\delta$-doped
layer with substitutional boron sites taken from a KMC calculation Campbell
_et al._ (2022) and positions subsequently relaxed. The gray background
shading is the calculated band structure of bulk silicon and the red line
represents the Fermi level.

While we have seen that structural relaxation tends to encourage
delocalization of the impurity bands, it is worth noting that these results
have assumed an ordered supercell. We next consider the impact of realistic
disorder on our results by studying the KMC-derived layer displayed in Fig.
1c, but allowing the structure to relax. We do not include spin-orbit coupling
in these calculations. The associated band structure shown in Fig. 4. In
contrast to the ordered structures shown before, the bands are flattened. The
overall band gap increases by 31 meV compared to the relaxed ordered case
(equivalently without spin-orbit coupling included). The 1$\Delta$ band is
again the bottom impurity band, meaning the VBM does occur at the $\Gamma$
point in this structure. The disordered structure also lifts the degeneracy of
all the bands, creating a gap of 3 meV between the 1$\Gamma$ and 2$\Gamma$
band and a gap of 17 meV between the 2$\Gamma$ and 1$\Delta$ bands. The Fermi
level is also notably pulled down by 41 meV, placing it only slightly above
the bulk valence band maximum. The bands are generally much flatter and have
higher effective masses, although the 2$\Gamma$ band still exhibits
significant curvature around the $\Gamma$ point, as detailed in Table 3. The
1$\Delta$ band continues to exhibit opposite curvature as to be expected
around the $\Gamma$ point, resulting in a large negative effective mass of
–2.32. Such large effective masses might be useful for realizing high kinetic
inductance superconducting elements Masluk  _et al._ (2012); Niepce  _et al._
(2019).

Table 3: The energy and effective mass of bands in a disordered B $\delta$-doped layer structure. Energies are given in units of meV relative to the Si VBM, and effective masses are given in units of me. The sign convention is such that a valence band with negative curvature (i.e. the expected curvature for a valence band) has a positive mass and bands with positive curvatures have negative masses. Band | Energy at $\Gamma$ | Effective mass at $\Gamma$
---|---|---
1$\Gamma$ | 188 | 1.46
2$\Gamma$ | 185 | 0.36
1$\Delta$ | 168 | –2.32
EF | 42 |
ECB | 583 |

###  III.2 Aluminum $\delta$-doped layers

Figure 5:  The electronic structure after atomic relaxation of an aluminum
$\delta$-doped layer that is (a) ordered and (b) disordered. The gray
background shading represents our calculated band structure of bulk silicon
and the red line represents the Fermi level.

Aluminum $\delta$-doped layers experience significantly less distortion due to
stress than boron $\delta$-doped layers. The calculated band structure for an
aluminum $\delta$-doped layer, which has undergone geometric optimization and
includes spin-orbit coupling, is shown in Fig. 5. This is notably similar to
the band structure of the boron $\delta$-doped layer before any stress induced
relaxation is included, suggesting that an aluminum $\delta$-doped layer will
have a much smaller impact on its neighboring layers (and indeed induces less
stress, at 85 MPa per unit cell). This is consistent with chemical intuition–
an aluminum atom is closer in size to a silicon atom than a boron atom and
therefore generates less strain to the lattice. Even with spin-orbit coupling
included, there is still a degeneracy in the topmost 1$\Gamma$ and 2$\Gamma$
bands. The bands are also pulled closer to the bulk valence bands with the
band gap of the structure increasing by 28 meV and the Fermi level being
pulled down by 33 meV. The exact values of the band and Fermi level energies
are displayed in Table 4 along with the effective masses. The top 1$\Gamma$
and bottom 3$\Gamma$ band are both extremely flat with effective masses
$\gtrapprox$ 2 me. The middle band has an effective mass of 0.32 me, between
the heavy- and light-hole masses in bulk silicon.

Table 4: The energy and effective mass of bands in an ordered Al $\delta$-doped layer structure. Energies are given in units of meV relative to the VBM, and effective masses are given in me. Band | Energy at $\Gamma$ | Effective mass at $\Gamma$
---|---|---
1$\Gamma$ | 199 | 1.96
2$\Gamma$ | 199 | 0.32
3$\Gamma$ | 71 | 2.68
EF | 46 |
ECB | 625 |
Table 5: The energy and effective mass of bands in a disordered Al $\delta$-doped layer structure. Energies are given in units of meV, and effective masses are given in me.  Band | Energy at $\Gamma$ | Effective mass at $\Gamma$
---|---|---
1$\Gamma$ | 307 | 1.75
2$\Gamma$ | 226 | 0.36
3$\Gamma$ | 138 | 5.33
EF | 109 |
ECB | 636 |

Disordered aluminum $\delta$-doped layers behave qualitatively similarly to
disordered boron $\delta$-doped layers, with the induced bands both being
flattened significantly, as demonstrated in Fig. 5b. However, the valence
bands are significantly raised and the overall band gap is lowered by 108 meV
in comparison with the ordered aluminum structure. All degeneracies in the
bands at the $\Gamma$ point are lifted with the 1$\Gamma$ and 2$\Gamma$ bands
separated by 81 meV, and the 2$\Gamma$ and 3$\Gamma$ bands separated by 88
meV. The Fermi level in the case now rests below the 3$\Gamma$ point,
introducing another band above the Fermi level. The 1$\Gamma$ and 3$\Gamma$
bands flatten significantly with effective masses over 1 me for both, with the
exact values given in Table 5. The 2$\Gamma$ band remains highly curved, with
an effective mass near the same magnitude as the This flattening of the
1$\Gamma$ and 3$\Gamma$ bands is also potentially promising for
superconducting applications requiring large kinetic inductances Masluk  _et
al._ (2012); Niepce  _et al._ (2019).

Figure 6:  The doping potential of ordered (a) boron and (b) aluminum
$\delta$-doped layers both before and after relaxation, causing atomic
displacement. The aluminum doping potential much more closely resembles a pure
$\delta$ potential within silicon.

The difference in the extent of the impact of geometry optimization on boron
versus aluminum $\delta$-doped layers can be most clearly seen in the doping
potential for each, shown in Fig. 6. To calculate the doping potential we
subtract the electrostatic potential of the $\delta$-doped layer after
geometry optimization from the electrostatic potential of a pure silicon slab.
The geometry optimization in both the aluminum and boron $\delta$ layers leads
to widespread displacement in the exact position of neighboring silicon atoms
which accounts for the rapid oscillations in doping potential after relaxation
seen in Fig. 6. These displacements are much more pronounced around the boron
$\delta$-doped layer, however; the aluminum doping potential has a clear peak
in the middle corresponding to the $\delta$-doped layer. The boron doping
potential, in contrast, extends $\sim$ 10 nm further out from the $\delta$
layer in either direction and with a much greater magnitude of potential at
these tails of influence. The smaller impact of atomic displacement on
aluminum systems matches previous work on phosphorus $\delta$-doped layers,
where DFT work has shown that relaxation of the underlying position of the
donors leads to negligible changes in the electronic structure.Drumm  _et al._
(2013a) This further emphasizes the unique nature of boron as a $\delta$-doped
layer dopant: it induces significant atomic displacement in the surrounding
system that leads to widespread changes in the charge, position, and potential
of neighboring layers which in turn causes a significant shift in the overall
band structure, a feature which has not been seen in any other $\delta$-doped
layer dopants. Aluminum does not induce anywhere near the same amount of
displacement and thus the corresponding $\delta$-doped layer behaves much more
like a pure $\delta$ potential in silicon.

Overall, while atoomic displacement plays little role in the band structure of
aluminum $\delta$ layers, the exact placement of the dopant atoms within the
$\delta$ layer will have a large impact on the curvature, and thus effective
mass, of the resulting electronic structure. We further explore the band
structure of an Al ‘line’ structure Radue  _et al._ (2021) within the
$\delta$-doped layer in Appendix B.

##  IV Conclusion

We have reported the results of DFT calculations of the band structure of
boron and aluminum $\delta$-doped layers embedded in silicon. They demonstrate
that the band structure of boron $\delta$-doped layers depends strongly on the
detailed geometry of the $\delta$ layer, with both geometry optimization in
response to induced stress and disorder in the location of substituted boron
producing effects. The band structure calculations for aluminum $\delta$-doped
layers show that these layers induce less atomic displacement due to stress,
and consequently behave much more like a pure $\delta$ potential in silicon.

It should be noted that this DFT model will not correspond perfectly to the
behavior of real-world $\delta$-doped systems. Our silicon does not include
any background level of defects/dopants as this would be computationally
prohibitive within our model. Furthermore, Mazzola et al. have recently shown
that non–perfect confinement of donors to a single layer can lead to less
valley splitting and additional bands crossing the Fermi level beyond what DFT
predicts.Mazzola  _et al._ (2020) Additionally, we do not include any chlorine
that may also become trapped inside the silicon during a $\delta$-doping
process which uses aluminum or boron trichloride as dopant precursors.
Significant levels of chlorine may significantly impact the electronic
characteristics of these layers.

Our work indicates that different acceptor chemistries can be used to alter
the electronic structure of $\delta$-doped layers. While boron $\delta$-doped
layers avoid degeneracy at the top valence band (particularly in disordered
states) and move close to a direct band gap, they also induce significant
stress within the system that can lead to inversion of the curvature of the
top valence band. In contrast, ordered aluminum $\delta$-doped layers are
expected to be more similar to a pure $\delta$ potential, and might provide
greater control over factors like $\delta$ potential width. They do, however,
introduce degeneracies at the top levels of the induced valence bands and have
relatively large effective masses. Disordered aluminum $\delta$-doped layers,
in contrast, have no degeneracy, but even larger effective masses. Recent work
has proposed using a guided self-assembly process to create atomically precise
patterning of $\delta$-doped layers,Radue  _et al._ (2022) which would enable
the creation of the ordered band structures similar to those displayed here,
provides another axis for controlling the electronic structure. Given this
choice of both ordered versus disordered and boron versus aluminum, we
conclude that the band structure of an acceptor based $\delta$-doped layer can
be designed for desirable characteristics such as absence of degeneracy at the
$\Gamma$ point or large/small effective mass.

###### Acknowledgements.

We gratefully acknowledge useful conversations with Evan Anderson, Ezra
Bussmann, Jeff Ivie, Tzu-Ming Lu, Jonathan Moussa, Scott Schmucker, and Bob
Butera. This work was supported by the Laboratory Directed Research and
Development program at Sandia National Laboratories under project 213017 (FAIR
DEAL) and project 226347. This article has been authored by an employee of
National Technology & Engineering Solutions of Sandia, LLC under Contract No.
DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all
right, title and interest in and to the article and is solely responsible for
its contents. The United States Government retains and the publisher, by
accepting the article for publication, acknowledges that the United States
Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce the published form of this article or allow others to
do so, for United States Government purposes. The DOE will provide public
access to these results of federally sponsored research in accordance with the
DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan
.

## References

  * Oberbeck  _et al._ (2002) L. Oberbeck, N. Curson, M. Simmons, R. Brenner, A. Hamilton, S. Schofield, and R. Clark, Applied physics letters 81, 3197 (2002).
  * Ruess  _et al._ (2004) F. J. Ruess, L. Oberbeck, M. Y. Simmons, K. E. J. Goh, A. R. Hamilton, T. Hallam, S. R. Schofield, N. J. Curson, and R. G. Clark, Nano Letters 4, 1969 (2004).
  * Qian  _et al._ (2006) M. Qian, C. Fong, K. Liu, W. E. Pickett, J. Pask, and L. Yang, Physical review letters 96, 027211 (2006).
  * Rueß  _et al._ (2007) F. J. Rueß, W. Pok, T. C. Reusch, M. J. Butcher, K. E. J. Goh, L. Oberbeck, G. Scappucci, A. R. Hamilton, and M. Y. Simmons, Small 3, 563 (2007).
  * Katzenmeyer  _et al._ (2021) A. M. Katzenmeyer, S. Dmitrovic, A. D. Baczewski, Q. Campbell, E. Bussmann, T.-M. Lu, E. M. Anderson, S. W. Schmucker, J. A. Ivie, D. M. Campbell, _et al._ , Journal of Micro/Nanopatterning, Materials, and Metrology 20, 014901 (2021).
  * Ward  _et al._ (2020) D. R. Ward, S. W. Schmucker, E. M. Anderson, E. Bussmann, L. Tracy, T.-M. Lu, L. N. Maurer, A. Baczewski, D. M. Campbell, M. T. Marshall, _et al._ , arXiv preprint arXiv:2002.11003 (2020).
  * Bussmann  _et al._ (2021) E. Bussmann, R. E. Butera, J. H. Owen, J. N. Randall, S. M. Rinaldi, A. D. Baczewski, and S. Misra, MRS Bulletin 46, 607 (2021).
  * Škereň  _et al._ (2020) T. Škereň, S. A. Köster, B. Douhard, C. Fleischmann, and A. Fuhrer, Nature Electronics , 1 (2020).
  * Radue  _et al._ (2021) M. S. Radue, S. Baek, A. Farzaneh, K. Dwyer, Q. Campbell, A. D. Baczewski, E. Bussmann, G. T. Wang, Y. Mo, S. Misra, _et al._ , The Journal of Physical Chemistry C 125, 11336 (2021).
  * Dwyer  _et al._ (2021) K. J. Dwyer, S. Baek, A. Farzaneh, M. Dreyer, J. R. Williams, and R. E. Butera, ACS Applied Materials & Interfaces 13, 41275 (2021).
  * Bustarret  _et al._ (2006) E. Bustarret, C. Marcenat, P. Achatz, J. Kačmarčik, F. Lévy, A. Huxley, L. Ortéga, E. Bourgeois, X. Blase, D. Débarre, _et al._ , Nature 444, 465 (2006).
  * Bourgeois and Blase (2007) E. Bourgeois and X. Blase, Applied physics letters 90, 142511 (2007).
  * Shim and Tahan (2014) Y.-P. Shim and C. Tahan, Nature communications 5, 4225 (2014).
  * Duvauchelle  _et al._ (2015) J.-E. Duvauchelle, A. Francheteau, C. Marcenat, F. Chiodi, D. Débarre, K. Hasselbach, J. R. Kirtley, and F. Lefloch, Applied Physics Letters 107, 072601 (2015).
  * Bonnet  _et al._ (2022) P. Bonnet, F. Chiodi, D. Flanigan, R. Delagrange, N. Brochu, D. Débarre, and H. le Sueur, Physical Review Applied 17, 034057 (2022).
  * Cartoixa and Chang (2005) X. Cartoixa and Y.-C. Chang, Physical Review B 72, 125330 (2005).
  * Lee  _et al._ (2011) S. Lee, H. Ryu, H. Campbell, L. C. Hollenberg, M. Y. Simmons, and G. Klimeck, Physical Review B 84, 205309 (2011).
  * Smith  _et al._ (2014) J. Smith, J. Cole, and S. Russo, Physical Review B 89, 035306 (2014).
  * Qian  _et al._ (2005) G. Qian, Y.-C. Chang, and J. Tucker, Physical Review B 71, 045309 (2005).
  * Carter  _et al._ (2009) D. J. Carter, O. Warschkow, N. A. Marks, and D. R. McKenzie, Physical Review B 79, 033204 (2009).
  * Carter  _et al._ (2011) D. J. Carter, N. A. Marks, O. Warschkow, and D. R. McKenzie, Nanotechnology 22, 065701 (2011).
  * Drumm  _et al._ (2013a) D. W. Drumm, A. Budi, M. C. Per, S. P. Russo, and L. C. Hollenberg, Nanoscale research letters 8, 1 (2013a).
  * Drumm  _et al._ (2013b) D. Drumm, J. Smith, M. Per, A. Budi, L. Hollenberg, and S. Russo, Physical review letters 110, 126802 (2013b).
  * Drumm  _et al._ (2012) D. W. Drumm, L. C. Hollenberg, M. Y. Simmons, and M. Friesen, Physical Review B 85, 155419 (2012).
  * Miwa  _et al._ (2013) J. A. Miwa, P. Hofmann, M. Y. Simmons, and J. W. Wells, Physical review letters 110, 136801 (2013).
  * Mazzola  _et al._ (2014a) F. Mazzola, M. T. Edmonds, K. Høydalsvik, D. J. Carter, N. A. Marks, B. C. Cowie, L. Thomsen, J. Miwa, M. Y. Simmons, and J. W. Wells, ACS nano 8, 10223 (2014a).
  * Mazzola  _et al._ (2014b) F. Mazzola, C. M. Polley, J. A. Miwa, M. Y. Simmons, and J. W. Wells, Applied Physics Letters 104, 173108 (2014b).
  * Miwa  _et al._ (2014) J. A. Miwa, O. Warschkow, D. J. Carter, N. A. Marks, F. Mazzola, M. Y. Simmons, and J. W. Wells, Nano letters 14, 1515 (2014).
  * Mazzola  _et al._ (2018) F. Mazzola, J. Wells, A. Pakpour-Tabrizi, R. Jackman, B. Thiagarajan, P. Hofmann, and J. Miwa, Physical review letters 120, 046403 (2018).
  * Holt  _et al._ (2020) A. J. Holt, S. K. Mahatha, R.-M. Stan, F. S. Strand, T. Nyborg, D. Curcio, A. K. Schenk, S. P. Cooil, M. Bianchi, J. W. Wells, _et al._ , Physical Review B 101, 121402 (2020).
  * Hagmann  _et al._ (2018) J. A. Hagmann, X. Wang, P. Namboodiri, J. Wyrick, R. Murray, M. Stewart Jr, R. M. Silver, and C. A. Richter, Applied Physics Letters 112, 043102 (2018).
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, _et al._ , Journal of physics: Condensed matter 21, 395502 (2009).
  * van Setten  _et al._ (2018) M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, Computer Physics Communications 226, 39 (2018).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters 77, 3865 (1996).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Physical review B 13, 5188 (1976).
  * Jain  _et al._ (2013) A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, _et al._ , APL materials 1, 011002 (2013).
  * Campbell  _et al._ (2022) Q. Campbell, K. J. Dwyer, S. Baek, A. D. Baczewski, R. E. Butera, and S. Misra, arXiv preprint arXiv:2201.11682 (2022).
  * Perdew (1985) J. P. Perdew, International Journal of Quantum Chemistry 28, 497 (1985).
  * Heyd  _et al._ (2003) J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of chemical physics 118, 8207 (2003).
  * Masluk  _et al._ (2012) N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, and M. H. Devoret, Physical review letters 109, 137002 (2012).
  * Niepce  _et al._ (2019) D. Niepce, J. Burnett, and J. Bylander, Physical Review Applied 11, 044014 (2019).
  * Mazzola  _et al._ (2020) F. Mazzola, C.-Y. Chen, R. Rahman, X.-G. Zhu, C. M. Polley, T. Balasubramanian, P. D. King, P. Hofmann, J. A. Miwa, and J. W. Wells, npj Quantum Materials 5, 1 (2020).
  * Radue  _et al._ (2022) M. S. Radue, Y. Mo, and R. Butera, Chemical Physics Letters 787, 139258 (2022).
  * Carter  _et al._ (2013) D. Carter, O. Warschkow, N. Marks, and D. McKenzie, Physical Review B 87, 045204 (2013).
  * Yao and Kanai (2017) Y. Yao and Y. Kanai, The Journal of chemical physics 146, 224105 (2017).

##  Appendix A Concentration Effects within DFT

Figure 7:  The band structure of a system with the same supercell as used
throughout the work and the same concentration of dopants, but the dopants are
scattered through the system instead of fixed in a $\delta$-doped layer.

One factor complicating the use of DFT to predict the band structure of
$\delta$-doped layers is disentangling the effect of the arrangement of
dopants in a $\delta$ layer from the concentration effect of a finite number
of dopants in a periodically repeated cell. For the supercell illustrated in
Fig. 1a and b, the concentration of dopants within the $\delta$-doped layer is
1/4 ML. Within the entire cell, however, the concentration is
$7.64\text{\times}{10}^{19}\text{\,}\mathrm{c}\mathrm{m}^{-3}$, a relatively
high concentration for a typical semiconductor. It might be expected that for
this high level of concentration, we should see large induced valence bands,
regardless of whether the boron atoms are forming a $\delta$-doped layer or
not. To test this effect, in Fig. 7, we predict the band structure of the
2$\times$2 supercell used throughout the work with the same concentration of B
dopants (Figs. 1a and b), but with the positions randomly determined, instead
of forming a controlled $\delta$-doped layer.

The shape of the impurity bands is very similar to the predicted shape of a
boron $\delta$-doped layer (see Fig. 2a), with a doubling of the bands similar
to what Carter et al. report for two interacting phosphorus $\delta$-doped
layers Carter  _et al._ (2013) (indeed, given the setup of our structure, this
could alternately be viewed as the band structure of two spatially separated
$\delta$-doped layers with a 1/8 ML concentration of acceptors). The magnitude
of the impurity bands is somewhat reduced with a 1$\Gamma$ energy of 158 meV,
compared to a 1$\Gamma$ energy of 248 meV in the $\delta$-doped structure.
This demonstrates that while a significant amount of the band structure can be
attributed to the concentration effect of using a limited supercell within
DFT, the magnitude and clarity of the structure is still related to the
arrangement of atoms into a $\delta$-doped layer. We expect a similar effect
would be seen for an analogously randomized Al structure. Our current
technique for measuring the $\delta$-doped layer electronic structure matches
that of the literature with phosphorus $\delta$-doped layers Carter  _et al._
(2009, 2011); Drumm  _et al._ (2013a, b). These theoretical techniques have
largely been subsequentally confirmed by experimental characterization Miwa
_et al._ (2013); Mazzola  _et al._ (2014a, b); Miwa  _et al._ (2014); Mazzola
_et al._ (2018); Holt  _et al._ (2020). We thus have some confidence that
while the impact of $\delta$-doped layers and the concentration of dopants in
a DFT supercell are unavoidably somewhat entangled, our work nevertheless
provides a useful first-order approximation of the electronic structure of
boron and aluminum $\delta$-doped layers.

##  Appendix B Line features

Figure 8:  (a) Top view of the placement of dopant atoms making a line feature
within a 2$\times$4 cell. (b) Band structure of boron $\delta$-doped line
feature. (c) Band structure of aluminum $\delta$-doped line feature.

Radue et al. noted that using AlCl3 as an APAM precursor leads to the
formation of large “line” structures, where AlCl fragments migrate to form a
linear superstructure on the silicon surface Radue  _et al._ (2021). We
accordingly examine the band structure of $\delta$-doped layers with a linear
structure of dopant atoms for both boron and aluminum, as shown in Fig. 8.
While these line structures have not been observed in boron $\delta$-doped
layers, they nonetheless provide a useful comparison to the observed aluminum
structures.

The band structure of the boron line structure, shown in Fig. 8a, combines the
features of the relaxed ordered and disordered band structures shown earlier.
Similar to the fully relaxed, ordered structure, the 1$\Delta$ band is pulled
to be the highest energy valence band, creating an inversion of the curvature
around the $\Gamma$ point, and moving the VBM to 0.12 $\times$2$\pi$/a in the
$X$ direction. Similar to the disordered structure, however, the impurity
bands exhibit no degeneracy with a separation of 23 meV between the 1$\Delta$
and 1$\Gamma$ band and 17 meV between the 1$\Gamma$ and 2$\Gamma$ band. While
the 1$\Delta$ band is highly flat with an effective mass of –1.64, the
1$\Gamma$ and 2$\Gamma$ bands are curved with effective masses of 0.51 and
0.61, respectively. The exact values of the bands and effective masses are
displayed in Table 6. Overall, our results highlight that the exact band
structure of boron $\delta$ layers is highly dependent on the exact placement
of boron atoms.

In aluminum line structures, however, the top impurity band is moved
dramatically up, as shown in Fig. 8b. The 1$\Gamma$ band is moved to 468 meV
above the bulk VBM, with a separation of 272 meV between it and the next
2$\Gamma$ band. Both the 1$\Gamma$ and 3$\Gamma$ bands are fairly flat, with
effective masses greater than unity. The 2$\Gamma$ band is curved, however,
with an effective mass of only 0.4. This indicates that the aluminum line
feature, which is favored in experiments, has dramatic results on the
accompanying electronic structure. The exact values for the impurity band
energies and effective masses can be found in Table 7. It should again be
noted that these band structures were calculated at the GGA level and future
work using more accurate approximations such as hybrid functionals would be
useful to accurately determine the exact shift of the impurity bands,
particularly relative to the larger band gap. We should also note that due to
the periodic nature of these calculations, the line feature of four aluminum
atoms is repeated in every supercell, but does not form a continuous line.
Radue et al. found that these line structures were typically only three
aluminum atoms long Radue  _et al._ (2021), so the proposed line structure of
Fig. 1d used throughout this work will likely be somewhat close to the
experimentally realized structure.

While this line leads to an unusual band structure when using aluminum
acceeptors, our finding of an elevated 1$\Gamma$ band remains consistent even
when a variety of methodologies are used. We have examined the same line
feature using rotated supercells as well as supercells extended in the further
in the slab direction, increasing the distance between periodic images and
still found either the same elevated band or a band spanning the entire width
of the band gap. To investigate whether this was due to unusual ordering of
the atomic positions, we additionally ran ab-initio molecular dynamics (AIMD)
at a temperature of 25 C for 0.5 ps and then measured the band structure,
producing the same elevated band. Finally, we also used SCAN pseudopotentials
Yao and Kanai (2017), which have been shown to improve the band gap
estimation, which similarly produced a 1$\Gamma$ band which spanned the entire
band gap. While it seems likely that real world devices with this line feature
will have a band structure more closely resembling those seen throughout the
main text of this work, the multiplicity of methods giving the same results
indicate that some interesting physics is at work in this case and further
work is warranted.

Table 6: The energy and effective mass of bands in a Boron $\delta$-doped layer structure forming a line feature. Energies are given in units of meV, and effective masses are given in me.  Band | Energy at $\Gamma$ | Effective mass at $\Gamma$
---|---|---
1$\Delta$ | 186 | –1.64
1$\Gamma$ | 163 | 0.61
2$\Gamma$ | 146 | 0.51
EF | 33 |
ECB | 581 |
Table 7: The energy and effective mass of bands in an Aluminum $\delta$-doped layer structure forming a line feature. Energies are given in units of meV, and effective masses are given in me.  Band | Energy at $\Gamma$ | Effective mass at $\Gamma$
---|---|---
1$\Gamma$ | 468 | 2.13
2$\Gamma$ | 196 | 0.40
3$\Gamma$ | 133 | 4.85
EF | 77 |
ECB | 620 |


health impacts. - The role of digital social platforms in influencing drinking
habits, especially among younger demographics, and how this relates to
psychological and physical health outcomes. - The impact of personalized
genetics and precision medicine on the efficacy of public health policies
designed to mitigate the harms of alcohol. - Cross-national comparative
studies to explore how different legal and cultural environments shape alcohol
consumption patterns and health outcomes, potentially uncovering new variables
for the model. In summary, the model excels in its integrative and innovative
nature but may face challenges in maintaining conciseness and clarity due to
its complexity. Further development should focus on elucidating the multi-
directional interactions of the domains and exploring new patterns that may
emerge from data, particularly in the rapidly changing social and policy
landscapes.

### Appendix 60: Second Iteration Model for Cardiovascular Health Effects of
Alcohol

Academia-OS-generated model text:

Model Name: Dynamic Socio-bio-psychological Model of Alcohol Impact (DSBPAI)
Model Description: The Dynamic Socio-bio-psychological Model of Alcohol Impact
(DSBPAI) is a theoretical construct designed to synthesize and build upon
existing literature to explore the intricate relationships between alcohol
consumption and diverse health outcomes. This model explicitly acknowledges
the multidimensionality and dynamism of factors at play, categorizing and
integrating biological (bio), psychological (psycho), and sociodemographic
(socio) influences within a complex, adaptive system. The DSBPAI model
conceptualizes the interactions across different levels—individual, social,
and policy—and maps out the reciprocal feedback mechanisms that modulate these
interactions over time. Model Structure: 1. Individual Well-being Sphere: -
The core of the model is individual health and well-being, integrating
physical (bio), mental (psycho), and social (socio) health dimensions as
dynamic and interrelated. 2. Aggregate Dimensions: - Biochemical Reaction
Network * Denotes biological processes including metabolic pathways, genetic
predispositions, and physiological responses to alcohol, providing a causal
mapping with individual health outcomes. - Psychological Complexity Framework
* Captures cognitive, emotional, and behavioral responses to alcohol,
incorporating both individual differences and common psychological patterns,
facilitating correlational associations and feedback loops with health
outcomes. - Socio-Environmental Influence Fabric * Constitutes the spectrum of
social, cultural, economic, and policy-related factors that influence and are
affected by alcohol consumption, highlighting the correlation and potential
for causal relationships on both individual and population health levels.
Relationships and Narratives: 1. Biochemical Reaction Network → Individual
Well-being Sphere (Causal) - Direct causal biochemical and physiological
reactions result from alcohol consumption, affecting health outcomes through
discrete, quantifiable biological changes. 2. Psychological Complexity
Framework $\leftrightarrow$ Individual Well-being Sphere (Correlational and
causal) - Psychological dispositions and experiences influence and are
influenced by individual health outcomes, forming a bidirectional relationship
where feedback loops are common and causality can be context-dependent. 3.
Socio-Environmental Influence Fabric $\leftrightarrow$ Psychological
Complexity Framework (Correlational) - Socio-environmental factors correlate
with psychological approaches to alcohol use and vice versa, reflecting the
influence of society on individual mentality and behavior towards alcohol. 4.
Socio-Environmental Influence Fabric → Biochemical Reaction Network (Causal) -
The social dimension, through policy impacts and environmental components,
impacts biological processes by influencing the patterns and nature of alcohol
consumption, which can cause health outcomes. 5. Socio-Environmental Influence
Fabric $\leftrightarrow$ Individual Well-being Sphere (Correlational and
Causal) - The fabric of socio-environmental influence has a bidirectional
relationship with individual health outcomes, whereby it shapes, and is shaped
by, the collective health and consumption behaviors of individuals. 6.
Psychological Complexity Framework $\leftrightarrow$ Biochemical Reaction
Network (Correlational) - The interplay between psychology and biology
acknowledges the psychosomatic nature of health, where psychological states
can correlate with and sometimes affect biochemical reactions to alcohol. 7.
Biochemical Reaction Network $\leftrightarrow$ Socio-Environmental Influence
Fabric (Causal) - Emerging evidence suggests biological factors can inspire
shifts in the socio-environmental domain, as understanding of health
risks/benefits of alcohol influences social norms and policies. Model
Applications: The DSBPAI model is designed to: - Act as a heuristic tool for
researchers and clinicians to identify and analyze the feedback loops between
individual behavior and broader societal structures in the context of alcohol
consumption and health outcomes. - Inform health policies and public health
strategies by identifying leverage points within the socio-bio-psychological
nexus where interventions could be most effective. - Guide educational
campaigns to tailor messages to diverse audiences, acknowledging the complex
interactions between individual, societal, and biological components related
to alcohol. - Provide a framework for the development of predictive models for
public health outcomes related to alcohol by integrating individual and public
health data across the biological, psychological, and social domains. Model
Innovation: The DSBPAI model introduces the following original components: -
Dynamic Feedback Systems: The model adopts systems thinking to demonstrate how
feedback loops dynamically modulate the relationships between consumption,
psychology, biology, and social context. - Scalable Modularity: Designing the
components as modules within a larger system allows for the model to be
expanded or focused, depending on the level of analysis, from molecular to
societal. - Contextual Health Matrix: This matrix integrates genetic,
lifestyle, and environmental health determinants to offer nuanced insights
into population-specific alcohol health impacts. The DSBPAI model thus
integrates and advances existing knowledge into a comprehensive, dynamic, and
multidimensional framework, acknowledging the complex realities of alcohol’s
impact on health and society. It provides a cohesive narrative that can evolve
and adapt to new data, offering a sophisticated, systemic understanding of the
alcohol-health nexus.

### Appendix 61: Temperature Test 0 Outcome

Academia-OS-generated model text:

Model Name: The Dynamic Synergy Model (DSM) The Dynamic Synergy Model (DSM) is
a theoretical framework designed to elucidate the complex interplay between
various aggregate dimensions that influence and drive organizational success
and sustainability. The DSM posits that the interrelationships between these
dimensions are not merely additive but synergistic, leading to emergent
properties that can significantly enhance an organization’s performance and
resilience. The DSM is composed of eight core aggregate dimensions: 1.
Organizational Growth 2. Professional Advancement 3. Financial Strategy 4.
Sustainability Initiatives 5. Technological Integration 6. Marketing
Effectiveness 7. Operational Excellence 8. Community Relations Each dimension
is further broken down into specific codes that represent the operational
elements within that dimension. The DSM emphasizes the dynamic and reciprocal
relationships between these dimensions, suggesting that changes in one
dimension can have cascading effects on others. **Theoretical Constructs and
Relationships:** 1. **Organizational Growth and Technological Integration:** -
The relationship between these two dimensions is causal. Technological
Integration can lead to Innovation and Product Development, which is a sub-
code of Organizational Growth. Conversely, the pursuit of Business Development
may necessitate advancements in Technology and Digital Transformation. 2.
**Professional Advancement and Community Relations:** - This relationship is
correlational. Networking and Relationships, a sub-code of Professional
Advancement, can enhance Community and Social Engagement. Community Relations
can also provide opportunities for Career and Personal Development. 3.
**Financial Strategy and Sustainability Initiatives:** - The relationship here
is bidirectional and causal. Financial Management and Investment decisions can
impact Sustainability and Environmental Impact initiatives, while a focus on
sustainability can lead to long-term financial stability and new investment
opportunities. 4. **Marketing Effectiveness and Organizational Growth:** - A
causal relationship exists where effective Branding and Marketing, as well as
Product and Service Delivery, drive Market and Customer Engagement, leading to
Organizational Growth. 5. **Operational Excellence and Financial Strategy:** -
This is a causal relationship. Addressing Operational Challenges can lead to
more efficient Financial Management and Investment, as operational
efficiencies often translate into cost savings and improved financial
performance. 6. **Technological Integration and Marketing Effectiveness:** -
The relationship is causal. Technology and Digital Transformation can enhance
Branding and Marketing efforts, making them more effective and efficient. 7.
**Sustainability Initiatives and Community Relations:** - This relationship is
correlational. Sustainability and Environmental Impact initiatives can improve
Community and Social Engagement, as organizations that prioritize
sustainability may be viewed more favorably by the community. 8.
**Professional Advancement and Technological Integration:** - A causal
relationship where Career and Personal Development often requires staying
abreast of new technologies, which in turn can drive Technology and Digital
Transformation within the organization. **Narrative of the DSM:** The Dynamic
Synergy Model suggests that an organization’s ability to thrive in a
competitive landscape is contingent upon its capacity to foster synergies
between these eight dimensions. The model posits that no single dimension
operates in isolation; rather, each dimension is intricately connected to
others, creating a complex web of interactions that can either propel an
organization forward or hold it back, depending on how well these interactions
are managed. For instance, an organization that invests in Technological
Integration may find that it not only enhances its Operational Excellence but
also amplifies its Marketing Effectiveness. Similarly, a focus on Professional
Advancement can lead to improved Organizational Growth through enhanced
Networking and Relationships, which in turn can lead to better Community
Relations. The DSM encourages organizations to adopt a holistic approach to
strategy development, recognizing that the pursuit of excellence in one area
can have far-reaching implications across the organization. By understanding
and leveraging the dynamic synergies between these dimensions, organizations
can create a virtuous cycle of growth, sustainability, and innovation.

### Appendix 63: Temperature Test 0.5 Outcome

Academia-OS-generated model text:

Model Name: Integrated Framework for Sustainable Market Leadership (IFSML) The
Integrated Framework for Sustainable Market Leadership (IFSML) is a
theoretical model designed to synthesize the multidimensional aspects of
business operations, strategy, and brand management with a clear focus on
sustainability, ethical practices, and customer-centric growth. This model is
novel in its comprehensive approach to intertwining these dimensions while
maintaining a focus on achieving market leadership through sustainable means.
**Model Overview:** The IFSML model is structured around four core pillars:
Strategic Business Synthesis (SBS), Brand and Customer Integration (BCI),
Operational and Sustainability Nexus (OSN), and Networking and Developmental
Dynamics (NDD). Each pillar represents an aggregate dimension that encompasses
various codes from the provided list. **Pillar 1: Strategic Business Synthesis
(SBS)** - Entrepreneurial Journey - Product Innovation and Development -
Sustainable Business Practices - Market Disruption and Differentiation -
Business Development **Pillar 2: Brand and Customer Integration (BCI)** -
Brand and Marketing - Product Identity - Marketing Strategies - Brand Elements
- Customer and Market Understanding - Customer Experience and Service -
Customer Engagement **Pillar 3: Operational and Sustainability Nexus (OSN)** -
Technology and Infrastructure - Production and Manufacturing - Finance and
Investment - Environmental Impact and Sustainability - Sustainable Business
Practices (cross-listed with SBS) **Pillar 4: Networking and Developmental
Dynamics (NDD)** - Networking and Relationships - Personal Development and
Career **Model Dynamics:** The IFSML model posits that the pathway to
sustainable market leadership is through the strategic alignment and dynamic
interaction of these four pillars. The relationships are as follows: 1.
**Causal Relationships:** - The SBS influences BCI by determining the
overarching strategy that shapes brand identity and marketing initiatives. -
The SBS also has a direct impact on OSN, as the strategic choices made by a
company will dictate operational priorities and investments in technology and
sustainable practices. - NDD affects all other pillars by fostering
relationships and personal growth that can lead to strategic partnerships,
brand collaborations, and an understanding of evolving market needs. 2.
**Correlational Relationships:** - There is a strong correlation between BCI
and OSN, as the brand’s promise to customers often requires the operational
capacity to deliver on that promise sustainably. - Similarly, the
effectiveness of NDD activities may correlate with the strength of a brand
(BCI) and the company’s strategic positioning (SBS). **Narrative of the
Model:** The IFSML model suggests that a business that effectively integrates
its strategic business synthesis with its brand and customer management will
be better positioned to achieve market differentiation and leadership. This
integration is supported by a robust operational framework that is not only
efficient but also committed to sustainability, ensuring that the business
practices are in line with environmental and ethical standards. Networking and
developmental dynamics serve as a catalyst for growth and innovation,
providing the necessary insights and relationships to keep the business agile
and responsive to market changes. The emphasis on personal development within
NDD ensures that the company’s leadership and workforce are equipped to handle
the challenges of a dynamic business environment. The IFSML model is original
in its comprehensive approach to business leadership, combining strategy,
branding, operations, sustainability, and networking in a cohesive framework.
It provides a roadmap for businesses seeking to achieve and maintain market
leadership by prioritizing sustainability and ethical practices without
compromising on customer engagement and operational excellence.

### Appendix 64: Temperature Test 1 Outcome

Academia-OS-generated model text:

Model Name: The Integrated Framework for Sustained Entrepreneurial Advancement
(IFSEA) The Integrated Framework for Sustained Entrepreneurial Advancement
(IFSEA) is a theoretical model designed to encapsulate key aggregate
dimensions relevant to the lifecycle of entrepreneurial ventures from
inception through various stages of growth. This model seeks to establish
relationships and potential causal pathways between innovation, strategy,
marketing, entrepreneurial experience, financial management, consumer
relations, and sustainability. Each of these dimensions represents a vital
component of a thriving business ecosystem. Theoretical Constructs: 1.
Innovation Engine - Combines "Innovation and Product Development" with
elements of "Strategic Business Insights" to drive the creation of unique
products and the forming of strategies to maintain a competitive edge. 2.
Brand Orbit - Entails "Marketing and Branding" and integrates "Consumer
Relations and Outreach" to resonate with consumer experiences and engagement,
fostering brand loyalty and market identity. 3. Entrepreneurial Odyssey -
Encompasses "Entrepreneurial Experience" and infuses personal growth
narratives within the business development process. 4. Fiscal Architecture -
Includes "Financial Management and Growth" and is concerned with the financial
scaffolding that supports business operations and expansion. 5. Ethical
Compass - Engages with "Sustainability and Ethics" to guide business practices
and product development towards ethical and sustainable frontiers. The IFSEA
model posits that these constructs are intrinsically linked and the impact of
these relationships on a business’s success is both causal and correlational.
For instance, innovation can directly influence the fiscal architecture by
generating new revenue streams (causal), while strong financial management can
correlate with more resources for innovation (correlational). Relationship
Narrative & Causality: 1. Innovation Engine -> Fiscal Architecture:
Breakthroughs in product development can lead to the expansion of market
share, directly affecting financial growth. 2. Brand Orbit -> Consumer
Relations and Outreach: Effective marketing and branding increase consumer
loyalty, which in turn enhances customer experience and engagement strategies.
3. Entrepreneurial Odyssey -> Innovation Engine: The entrepreneur’s
experiences and insights fuel innovation pipelines and imbue product
development with a unique identity and competitive advantage. 4. Fiscal
Architecture -> Brand Orbit: Availability of financial resources can lead to
more substantial marketing campaigns and branding initiatives. 5. Ethical
Compass -> Innovation Engine: A commitment to sustainability and ethics can
inspire product development towards eco-friendly and socially responsible
solutions. 6. Brand Orbit + Consumer Relations and Outreach -> Entrepreneurial
Odyssey: Market recognition, combined with consumer feedback, can shape the
personal and professional growth of the entrepreneur. The IFSEA model also
acknowledges feedback loops, where success in one domain can reinforce and
strengthen another. For instance: - The success of Brand Orbit feeds back into
the Fiscal Architecture as better brand positioning can lead to higher sales
and investment attraction. - Fiscal Architecture’s robustness can enable more
sustainable and ethical practices, reinforcing the Ethical Compass. This
framework aims to provide a holistic blueprint whereby researchers and
practitioners can evaluate the interdependencies of business domains, and
devise strategies that promote synergy and sustainable growth. IFSEA offers a
lens through which the complexity of a business’s ecosystem can be
appreciated, and potential areas for intervention can be identified, promoting
a dynamic approach to entrepreneurism in the ever-evolving market landscape.

### Appendix 65: Temperature Test 1.5 Outcome

Academia-OS generated model:

Name: *Continual Agile Nexus in Dynamic Organizational Strategies (CANDOS)*
Model **I. Overview of the Theoretical Model:** The Continual Agile Nexus in
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development, organizational adaptability, stakeholder engagement, and risk
management through a lens that elucidates the robustness and innovation
capacities of a company within changing environments. This model is not
unidirectional but demonstrates bi-directional relationships, iterating that a
change in one dimension can concurrently inspire adaptations across others.
**II. Description of Aggregate Dimensions:** 1. **Strategic Development:** -
Factors: Environmental Vision and Action, Financial Journey, Company
Foundations, Growth and Changes. 2. **Organizational Adaptability:** -
Factors: Resilience and Commitment, Engaging Innovations. 3. **Stakeholder
Engagement:** - Factors: Communication and Interaction. 4. **Risk
Management:** - Factors: Security and Conflict Minimization. **III.
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remains, directing it toward sustaining innovation while navigating growth and
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to engaging in innovating processes—it feeds back into reassessing and
considering the refinements in their strategic aims and execution
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# Mesh motion in fluid-structure interaction with deep operator networks

Ottar Hellan  Simula Research Laboratory, Oslo, Norway<EMAIL_ADDRESS>
###### Abstract.

A mesh motion model based on deep operator networks is presented. The model is
trained on and evaluated against a biharmonic mesh motion model on a fluid-
structure interaction benchmark problem and further evaluated in a setting
where biharmonic mesh motion fails. The performance of the proposed mesh
motion model is comparable to the biharmonic mesh motion on the test problems.

##  1\. Introduction

Recently, machine learning algorithms have been developed to learn mappings
between function spaces, such as the solution operators of PDEs, in a field
called operator learning, with examples including the deep operator network
[9, 10] and Fourier neural operator [8]. The motivating applications for these
models is to replace entire solvers with data-driven models, but the training
of these is challenging, sometimes resulting in relative errors of several
percent [10].

Alternatively, operator learning can be applied to replace only certain
components of solvers, for instance targeting those steps which are based on
heuristics or where classical approaches yield insufficient performance. One
example of such components is mesh motion, where a computational reference
domain is moved to fit the domain of interest. The outcome of the final solver
depends on a minimal level of quality of the mesh movement.

In this work, we present a deep operator network-based mesh motion operator,
trained on data from an FSI benchmark problem. We evaluate the learned mesh
motion on this FSI benchmark, comparing it with the biharmonic mesh motion it
is trained on, as well as a setting constructed such that the biharmonic mesh
motion fails.

##  2\. Mesh motion

The problem of mesh motion is prominent in a number of applications involving
a change-of-coordinates between reference and target domains. In these cases,
a mesh motion operator is used to move a mesh of the reference domain to
triangulate the target domain. Examples include fluid-structure interaction
(FSI) in the arbitrary Lagrangian-Eulerian (ALE) formulation [17, 6], where
the mesh of the fluid domain is moved to fit a time-changing solid domain, or
in shape optimization via the method of mappings, where a reference mesh is
moved to fit the desired shape [5]. The change-in-coordinates given by the
mesh motion are typically based on PDEs, with Dirichlet boundary conditions
representing the geometry of the target domain [14], but recently data-driven
approaches have been suggested [1, 15, 4].

Let the domain $\Omega$ be a bounded subset of $\mathbb{R}^{d}$,
$d\in\\{2,3\\}$. We define a mesh motion operator as a mapping $g\mapsto u$,
where $g:\partial\Omega\to\mathbb{R}^{d}$ represents the deformation of the
boundary $\partial\Omega$ and the resulting deformation field
$u:\Omega\to\mathbb{R}^{d}$ defines the change-of-coordinates

$\chi:\Omega\to\mathbb{R}^{d},\quad x\mapsto x+u(x),$ (2.1)

with the boundary condition $u|_{\partial\Omega}=g$. For this change-of-
coordinates to be well defined and suitable for our computations, it is
necessary that it is bijective, that $u|_{\partial\Omega}=g$, and that

$J=\mathrm{det}(\nabla\chi)>0\text{ in }\Omega.$ (2.2)

In addition to these constraints the transformation of $\Omega$ should produce
a suitable domain for the computational problem, for instance having a
Lipschitz boundary, and the resulting mesh should be of high quality.
Nevertheless, there is considerable freedom in the choice of $u$, allowing the
definition of many different mesh motion operators.

The simplest mesh motion operator is the harmonic mesh motion [17], which is
defined by the solution operator of the Laplace equation

$-\Delta u=0\text{ in }\Omega,\quad u=g\text{ on }\partial\Omega,$ (2.3)

where $\Delta\cdot$ is the vector-valued Laplacian $(\Delta
u)_{i}=\sum_{j=1}^{d}\partial^{2}u_{i}/\partial x_{j}^{2}$ for $j=1,...,d$.
Another common mesh motion operator, which can deal with larger deformations
than harmonic mesh motion, is the biharmonic mesh motion [17], one variant of
which is stated as

$\Delta^{2}u=0\text{ in }\Omega,\quad u=g\text{ on }\partial\Omega,\quad\nabla
u\cdot\mathbf{n}=0\text{ on }\partial\Omega.$ (2.4)

Other versions of biharmonic mesh motion exist which use different sets of
boundary conditions [17].

##  3\. Deep Operator Networks

The deep operator network (DeepONet) is a neural network model that is used to
learn operators, that is, mappings between function spaces [9]. We wish to
learn mesh motion operators, and thus we map boundary deformations
$g\in\mathcal{G}$ to the deformation map $u$ that defines the change-of-
coordinates (2.1).

For DeepONets with scalar output functions, we evaluate the output function
$\mathscr{D}(g)$ of the learned operator at a point $x$ by the inner product
of two separate networks,

$\mathscr{D}(g)(x)=\mathscr{B}(\mathscr{E}_{\mathscr{B}}(g))\cdot\mathscr{T}(x).$
(3.1)

The neural networks
$\mathscr{B}:\mathbb{R}^{b_{\mathrm{in}}}\to\mathbb{R}^{p}$ and
$\mathscr{T}:\mathbb{R}^{t_{\mathrm{in}}}\to\mathbb{R}^{p}$ are known as the
branch and trunk networks respectively and can be of any given architecture,
for instance multi-layer perceptrons (MLPs), convolutional neural networks, or
graph neural networks [10]. The mapping
$\mathscr{E}_{\mathscr{B}}:\mathcal{G}\to\mathbb{R}^{b_{\mathrm{in}}}$ encodes
the input function $g$ to discrete, finite-dimensional values that neural
networks can process. Typically, $\mathscr{E}_{\mathscr{B}}$ is the evaluation
of $g$ at a fixed set of locations called sensors, but can also be projection
onto a predefined basis [10]. To work with vector-valued output functions, we
use the second of the four approaches presented in [10]. The first $p/d$
components of $\mathscr{B}(\mathscr{E}_{\mathscr{B}}(g))$ and $\mathscr{T}(x)$
are used to construct $u(x)_{1}$, the first component of $\mathscr{D}(g)(x)$,
the next $p/d$ components for $u(x)_{2}$, and so on, producing an output
vector in $\mathbb{R}^{d}$.

In our DeepONet models, we let $\mathscr{E}_{\mathscr{B}}$ be the
concatenation of $g$ evaluated at a number of sensors placed appropriately in
the domain of $g$, and let $\mathscr{B}$ and $\mathscr{T}$ be MLPs. The output
of an MLP is defined by

$\displaystyle\mathcal{N}(z)=W_{L}z_{L-1}+b_{L},$ (3.2a) $\displaystyle
z_{l}=\sigma(W_{l}z_{l-1}+b_{l}),\text{ for }l=2,...,L-1,\quad z_{1}=z,$
(3.2b)

with $\sigma$ a chosen activation function. The matrices
$W_{2}\in\mathbb{R}^{w\times\mathrm{dim}(z)}$, $W_{l}\in\mathbb{R}^{w\times
w}$, $W_{L}\in\mathbb{R}^{\mathrm{dim}(\mathcal{N}(z))\times w}$ and vectors
$b_{l}$ are trainable parameters of the network. The depth of the network is
$L$ and the width of the network is $w$.

For mesh motion, it is important to satisfy the Dirichlet boundary constraints
exactly. If this is not the case, the geometry will be represented
incorrectly, causing numerical artifacts. We therefore utilize a hard
constraint boundary condition technique to ensure the DeepONets produce a
valid mesh motion operator. Following [11, 10], we define the evaluation at
$x$ of our DeepONet mesh motion model $\mathscr{U}$ as

$\mathscr{U}(g)(x)=h(g)(x)+l(x)\,\mathcal{D}(g)(x),$ (3.3)

where $h(g)$ is a function satisfying the Dirichlet boundary conditions
exactly and $l:\overline{\Omega}\to\mathbb{R}$ is a function satisfying $l=0$
on $\partial\Omega$ and $0<l(x)<1$ for $x\in\Omega$.

##  4\. FSI benchmark problem

To train and evaluate our DeepONet mesh motion model, we implement the FSI2
benchmark problem [16], simulating the interaction of an incompressible
Navier-Stokes fluid with an obstacle comprised of a rigid cylinder with an
elastic solid made of compressible St. Venant-Kirchhoff material attached
behind it, see Figure 1. The problem setup results in periodic oscillations of
the elastic solid.

The governing equations of the solid and fluid, in the ALE formulation on a
fixed reference domain $\Omega=\Omega_{s}\cup\Omega_{f}$, are given as

$\displaystyle\rho_{s}\partial^{2}u_{s}/\partial
t^{2}=\mathrm{div}(J\sigma_{s}F^{-T})\text{ in }\Omega_{s},$ (4.1a)
$\displaystyle\rho_{f}\partial v_{f}/\partial t+\rho_{f}(\nabla
v_{f})F^{-1}(v_{f}-\partial u/\partial
t)=J^{-1}\mathrm{div}(J\sigma_{f}F^{-T})\text{ in }\Omega_{f},$ (4.1b)
$\displaystyle\mathrm{div}(Jv_{f}F^{-T})=0\text{ in }\Omega_{f},$ (4.1c)
$\displaystyle\sigma_{s}=J^{-1}F\left(\lambda_{s}(\mathrm{tr}\,E)\mathrm{I}+2\mu_{s}E\right)F^{T},\quad\sigma_{f}=-p_{f}\mathrm{I}+\rho_{f}\nu_{f}(\nabla
v_{f}+\nabla v_{f}^{T}),$ (4.1d)

where, $F=\mathrm{I}+\nabla u$, $J=\mathrm{det}(F)$, and
$E=(F^{T}F-\mathrm{I})/2$ [16, 6]. Here, $\rho$ is the density of the
material, $\nu_{f}$ is the viscosity of the fluid, and $\lambda_{s}$,
$\mu_{s}$ are the Lamé parameters of the solid. To complete the formulation,
there are appropriate initial, boundary, and coupling conditions, in
particular enforcing the stress and mass balance on the fluid-solid interface.
In $\Omega_{s}$, $u$ represents the deformation of the solid, and in
$\Omega_{f}$, $u$ represents the deformation field that defines the ALE
change-of-coordinates (2.1). The ALE formulation is introduced to reconcile
the natural formulation of the solid problem in a Lagrangian frame and the
fluid problem in an Eulerian frame [2, 6].

Figure 1. Partial view of the FSI2 benchmark problem’s geometry, centered on
the submerged solid. The displacement of the particle $A(t)$ is a reported
quantity in the benchmark.

A detailed description of a solver for (4.1) of monolithic type is given in
[6] and the performance of different mesh motion operators in monolithic ALE-
FSI solvers is investigated in [17]. We solve (4.1) using a monolithic solver
from a previous work [4], which solves the resultant nonlinear system in
several steps, solving for $u$ in $\Omega_{f}$ separately and the solid and
fluid equations together, allowing us to easily change the mesh motion model
used in our tests. The solver uses second order Lagrange finite elements for
$u$ and $v$ and linear Lagrange elements for $p$ over a triangular mesh. We
use the FSI2 benchmark problem and this solver to generate our dataset and to
evaluate the trained DeepONet mesh motion.

##  5\. Network details

###  5.1. Training process

We train our DeepONet models in a supervised fashion, with biharmonic mesh
motion as target data. Using our FSI solver with biharmonic mesh motion, we
simulate one period of the oscillatory motion in the FSI2 benchmark, after a
15 second period to allow the oscillations to fully develop, producing 207
different snapshots $u_{\mathrm{bih}}^{k}$ that make up our dataset. We
randomly split this dataset into 70% training set and 30% validation set.

The cost function used is

$\mathcal{J}=\frac{1}{K}\sum_{k}\sum_{i}\frac{\left(h(g^{k})(x_{i})+l(x_{i})\,\mathcal{D}(g^{k})(x_{i})-u_{\mathrm{bih}}^{k}(x_{i})\right)^{2}}{\left(h(g^{k})(x_{i})-u_{\mathrm{bih}}^{k}(x_{i})\right)^{2}+\varepsilon}.$
(5.1)

Here, $K$ is the number of snapshots in the batch, $g^{k}$ is the boundary
deformation of snapshot $k$ and $u_{\mathrm{bih}}^{k}$ is the solution of
(2.4) with boundary condition $g^{k}$. The evaluation points $x_{i}$ are the
mesh vertices of our mesh of the FSI2 geometry. The parameter $\varepsilon>0$
adds stability, since $h(g^{k})(x_{i})-u^{k}_{\mathrm{bih}}(x_{i})$ is close
to zero for some $i$ and $k$.

Our models and training are implemented in the PyTorch [12] deep learning-
framework. We use the built-in Adam optimizer [7], with learning rate
$10^{-5}$ and otherwise default parameters, and the built-in ReduceLROnPlateau
learning rate scheduler, with the reduction factor set to $0.5$ and otherwise
default parameters, for $40000$ epochs.

###  5.2. Hyperparameters

Following [4], we define our DeepONet mesh motion model by (3.3), with $h(g)$
chosen as the harmonic extension (2.3) of the boundary deformation $g$.
Harmonic mesh motion is suitable for small boundary deformations and satisfies
the boundary deformation exactly. It is also a well studied PDE, with fast,
scalable solvers [3].

The function $l:\overline{\Omega}\to\mathbb{R}$ in (3.3) is determined by
solving a Poisson problem

$-\Delta\tilde{l}(x)=f(x)\text{ in }\Omega,\quad\tilde{l}(x)=0\text{ on
}\partial\Omega,$ (5.2)

with $f(x)=2(x+1)(1-x)\mathrm{exp}(-3.5x^{7})+0.1$ specifically chosen to
weight areas of the FSI2 benchmark problem’s domain where harmonic mesh motion
is inadequate [4]. From classical PDE theory, it holds that $0<\tilde{l}$ in
$\Omega$ and $\tilde{l}\in C^{2}(\Omega)\cap C(\overline{\Omega})$. We then
take $l=\tilde{l}/\left\lVert\tilde{l}\right\rVert_{\mathrm{sup}}$, such that
$0<l<1$ in $\Omega$. With $f\equiv 1$ the same properties hold for our and
also for many other relevant geometries [4].

We choose the sensor locations of $\mathscr{E}_{\mathscr{B}}$ encoding $g$ to
be the mesh vertices of the solid domain’s boundary, as this encodes the
geometry of the deformed solid in our simulations. On our mesh, this entails
evaluating $g$ at $206$ vertices, resulting in
$b_{\mathrm{in}}=412$-dimensional input-vectors to the branch network. The
trunk network takes inputs of size $t_{\mathrm{in}}=d=2$.

With our DeepONet mesh motion model and training process defined, we search
for appropriate values of the model’s hyperparameters, to obtain accurate
results on our test problems. We choose the hyperparameters of our DeepONet by
performing a grid search over a selection of hyperparameter values, leaving
the rest fixed at values we found were robust choices for many different
architectures in the exploratory phase of the research. The branch and trunk
network architectures are constrained to be equal, except for the size of the
input layer, and we vary the depth $d$, width $w$, and output size $p$ of the
two MLPs.

To evaluate our mesh motion models, we use the scaled Jacobian mesh quality
measure on the deformed meshes, computed using the visualization toolkit [13].
This mesh quality measure is scale invariant and gives the value of 1 for
equilateral triangles and goes to zero as a cell degenerates. Since the
networks are trained in a supervised fashion against the biharmonic extension,
we assume their performance will be worse than the biharmonic extension and
select the DeepONet with the lowest drop in quality. We compute for each
snapshot $k$ the difference in minimal cell value of mesh quality between the
biharmonic and DeepONet mesh motion, and select the DeepONet with the lowest
such difference.

The hyperparameter values used in the grid search are listed in Table 1. All
hyperparameter combinations are tested twice, using the random seeds 0 and 1,
to account for random initialization. All other network and training
parameters are unchanged between runs. The number of network parameters ranges
from $160\,576$ to $3\,430\,528$.

hyperparameter | depth ($d$) |  | width ($w$) |  | output size ($p$)
---|---|---|---|---|---
values | $\left\\{4,5,6,7\right\\}$ |  | $\left\\{128,256,512\right\\}$ |  | $\left\\{32,64\right\\}$
Table 1. Hyperparameter values used grid search hyperparameter optimization.
Figure 2. Minimum cell value of scaled Jacobian mesh quality measure over the
dataset with biharmonic extension and best trained DeepONet ($d=7$, $w=512$,
$p=32$, seed 1).

Figure 2 shows the minimal cell value of the scaled Jacobian mesh quality for
the biharmonic extension and the best trained DeepONet extension over the
dataset. The resulting extension is of high quality and the worst meshes from
the DeepONet mesh motion are not poorer than the worst meshes from the
biharmonic mesh motion in this dataset. The chosen architecture is one of the
largest, with both depth $d$ and width $w$ being the largest possible values,
but the networks’ output size $p$ is the lower of the two possible values. We
are testing on data included in the training set, but with a test metric
different from the cost function.

To investigate the robustness of the model and training process, we
investigate the dependence of random initialization on the quality of the mesh
motion. We train 20 different random initializations of the DeepONet model
selected in the hyperparameter search. All parameters are kept constant,
except for the random seeds, which are chosen from $\\{0,...,19\\}$.

Figure 3. Quantiles of validation loss history DeepONet training (left) and
resulting scaled Jacobian mesh quality over FSI2 dataset (right) for 20 random
initializations of the best performing hyperparameters found in grid search.

In Figure 3, we report quantiles of the validation loss history and the
minimum cell value of scaled Jacobian mesh quality at dataset index $k$ for
the fully trained DeepONet mesh motion models. The results indicate that the
model and training process result in a high quality mesh motion model, with
$0.1$-$0.9$ quantiles of minimal cell value scaled Jacobian mesh quality
almost on par with biharmonic mesh motion. Of the 20 random initializations,
19 of them give results similar to those shown in Figure 3, while one
initialization stagnated in a local minimum almost immediately, after
approximately 10 epochs.

##  6\. FSI test

To evaluate the DeepONet mesh motion, we use it as the mesh motion operator in
the FSI2 benchmark problem with our monolithic solver. We evaluate the
DeepONet mesh motion on a GPU, while the coupled fluid and solid equations are
solved in serial on the CPU. The mesh motion is performed by computing
$\mathcal{D}(g)(x_{i})$ for every degree of freedom-location $x_{i}$ of the
finite element space of $u$, not only the mesh vertices, thereafter inserting
the values in the basis coefficient array of the finite element function. The
mesh used in the FSI simulations is the same on the solid boundary as in the
training process, up to ordering, so to encode the input function $g$, we
access the finite element basis array using a pre-built vector of indices. If
the mesh vertices and sensors do not align, this could be done similarly with
a pre-built, sparse interpolation matrix. The functions $h(g)$ and $l$ are
computed using finite elements, with the same function space as $u$ is
represented in.

We report some quantities of interest from the FSI2 benchmark to validate the
DeepONet mesh motion. The simulation is run for 15 seconds, allowing the
periodic oscillations of the solid to fully develop, using the DeepONet mesh
motion. Thereafter, we compute for the period
$[15\,\mathrm{s},16\,\mathrm{s}]$ the drag and lift forces applied to the
solid by the fluid flow and the minimal value in $\Omega$ of $J$, see (2.2).
We also report the $y$-displacement $A_{2}$ of a point in the solid, see
Figure 1. The same values are also computed using biharmonic mesh motion,
initialized at $t=15\,\mathrm{s}$ with the same state as the DeepONet mesh
motion.

Figure 4. Drag (top left), lift (top right), $y$-displacement $A_{2}$ of point
on tip of solid (bottom left), and minimum value in $\Omega$ of $J$ (bottom
right) in FSI2 benchmark problem with biharmonic and DeepONet mesh motion.

Figure 4 shows the computed quantities of interest. The drag, lift, and
$y$-dis-placement values are identical for the two mesh motion models and
match the reference values of [16] well. This indicates the DeepONet mesh
motion is suitable for FSI computations in ALE formulation. The value of $\min
J$ is slightly lower at certain times for the DeepONet mesh motion, indicating
a slightly lower quality mesh motion than the biharmonic. This is in line with
the results in Figure 3.

##  7\. Gravity driven deformation test

To further explore the performance of the DeepONet mesh motion model, we apply
it to a test problem [14, sec. 4.1] constructed to push biharmonic mesh motion
to degeneracy. With the same geometry, material model, and parameters as in
the FSI2 benchmark, the solid is subjected to a uniform gravitational load
$(0,f_{g})$ and the solid deformation at its maximum deformation is extended
to the fluid domain with different mesh motion models. The solid deformations
are computed using implicit Euler time discretization and second order
Lagrange finite elements in space.

Figure 5. Histograms of scaled Jacobian mesh quality for biharmonic and
DeepONet mesh motion models with solid deformations caused by uniform
gravitational load $(0,f_{g})$, $f_{g}\in\\{1,2,2.5\\}$.

Figure 5 shows histograms of the scaled Jacobian mesh quality produced by the
biharmonic and DeepONet mesh motions with the maximal solid deformations
caused by the gravitational loads $f_{g}\in\\{1,2,2.5\\}$ as boundary
conditions. The biharmonic mesh motion handles the first two deformations
well, but breaks down for the third due to collapsing cells, with mesh quality
going to zero. Although the DeepONet mesh motion is trained to learn the
biharmonic mesh motion, it is able to handle even the largest deformation.

##  8\. Conclusion and future work

We have presented a DeepONet-based mesh motion model trained on biharmonic
mesh motion data, with performance on the test problems comparable to the
biharmonic mesh motion it is trained on. The training data comes from the same
problem as one of the test problems and future work should therefore
investigate training the model without access to this data. Training the
network in an unsupervised fashion and applying the method to problem settings
where current mesh motion operators struggle would be especially interesting.
Other work on the efficiency and scalability of the method can also be
considered.

## References

  * [1] A. Aygun, R. Maulik, and A. Karakus.  Physics-informed neural networks for mesh deformation with exact boundary enforcement.  Engineering Applications of Artificial Intelligence, 2023.
  * [2] J. Donea, A. Huerta, J. Ponthot, and A. Rodríguez-Ferran.  Arbitrary Lagrangian-Eulerian Methods.  In Encyclopedia of Computational Mechanics. Wiley, 2004.
  * [3] A. Gholami, D. Malhotra, H. Sundar, and G. Biros.  FFT, FMM, or Multigrid? A comparative Study of State-Of-the-Art Poisson Solvers for Uniform and Nonuniform Grids in the Unit Cube.  SIAM Journal on Scientific Computing, 2016.
  * [4] J. Haubner, O. Hellan, M. Zeinhofer, and M. Kuchta.  Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction, 2024.
  * [5] J. Haubner, M. Siebenborn, and M. Ulbrich.  A Continuous Perspective on Shape Optimization via Domain Transformations.  SIAM Journal on Scientific Computing, 2021.
  * [6] J. Hron and S. Turek.  A monolithic FEM solver for an ALE formulation of fluid-structure interaction with configuration for numerical benchmarking.  2006\.
  * [7] D. P. Kingma and J. Ba.  Adam: A Method for Stochastic Optimization, 2017.
  * [8] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar.  Fourier Neural Operator for Parametric Partial Differential Equations.  2021\.
  * [9] L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis.  Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.  Nature Machine Intelligence, 2021.
  * [10] L. Lu, X. Meng, S. Cai, Z. Mao, S. Goswami, Z. Zhang, and G. E. Karniadakis.  A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data.  Computer Methods in Applied Mechanics and Engineering, 2022.
  * [11] K. McFall and J. Mahan.  Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions.  IEEE Transactions on Neural Networks, 2009.
  * [12] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala.  PyTorch: an imperative style, high-performance deep learning library.  In Proceedings of the 33rd International Conference on Neural Information Processing Systems. Curran Associates Inc., 2019.
  * [13] W. J. Schroeder, K. Martin, W. E. Lorensen, L. S. Avila, K. W. Martin, and B. Lorensen.  The visualization toolkit: an object-oriented approach to 3D graphics ; [visualize data in 3D - medical, engineering or scientific ; build your own applications with C++, Tcl, Java or Python ; includes source code for VTK (supports UNIX, Windows and Mac)].  Kitware, Inc, 2006.
  * [14] A. Shamanskiy and B. Simeon.  Mesh moving techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency.  Computational Mechanics, 2021.
  * [15] W. Song, M. Zhang, J. G. Wallwork, J. Gao, Z. Tian, F. Sun, M. Piggott, J. Chen, Z. Shi, X. Chen, and J. Wang.  M2N: Mesh Movement Networks for PDE Solvers.  In Advances in Neural Information Processing Systems. Curran Associates, Inc., 2022.
  * [16] S. Turek and J. Hron.  Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow.  In Fluid-Structure Interaction. Springer Berlin Heidelberg, 2006.
  * [17] T. Wick.  Fluid-structure interactions using different mesh motion techniques.  Computers & Structures, 2011.

††Ottar Hellan acknowledges support from the Research Council of Norway, grant
303362.