# Resolved frustrated tunneling ionization in asymmetrical fast oscillation of
above-threshold ionization spectrum

Lifeng Wang1+, Hao Teng1, 3*, Fei Li1, Bingbing Wang1, 2, 3*, Xiaoxin Zhou4,
Peng He3,and Zhiyi Wei1, 2, 3*
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics,
Chinese Academy of Sciences, Beijing 100190, China.
2University of Chinese Academy of Sciences, Beijing 100049, China.
3Songshan Lake Materials Laboratory, Dongguan 523808, China.
4College of Physics and Electronic Engineering, Northwest Normal University,
Lanzhou, 730070, China.
+Currently with Institute for the Frontier of Attosecond Science and
Technology (iFAST),
CREOL and Department of Physics, University of Central Florida,
4111 Libra Drive, Orlando, FL 32816, USA.
*Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tunneling ionization is one of the fundamental electron dynamics, which has
wide applications in ultrafast physics [HMS+18, YMH+18, ENRS09]. When
frustrated tunneling ionization (FTI) is considered [EE14], the tunneling rate
is not equivalent to ionization rate. However, it is hard to resolve the
effects of FTI and direct tunneling ionization (DTI) in ionization spectrum
experimentally. Here we report the first observation of the asymmetrical fast
oscillation in above-threshold ionization (ATI) spectrum of Argon as function
of carrier-envelope phase (CEP), to the best of our knowledge. Simulation
results identify that in the experimental ATI spectrum, the $\pi/5$
oscillation originates from the quantum interference of electrons in FTI,
while DTI is responsible for the asymmetry. Our results provide clear evidence
to resolve the effects of direct tunneling and FTI in a new physical regime.

##  1 Introduction

When atoms or molecules are exposed to intense ultrafast lasers, the tunneling
ionization of electron is a fundamental process for several physical
phenomenon, such as ATI, filamentation and high-order harmonic generation
(HHG) [RWC+19, WG21, BKI+20]. Commonly, the tunneling rates is equal to
ionization rate. However, this is not true if the Coulomb potential is
considered. Attracted by the Coulomb potential, the electron has possibility
to have high negative energy at the tunnel exit and relaxes to Rydberg state
(RS), this process is dubbed as frustrated tunneling ionization[NGS+08,
LXH+21]. The RS atoms or molecules has been proved to have wide applications
[ESE+13, DMR+09, CWC+14]. There are two mechanisms of the formation of RS
atoms or molecules, which are still under debate. One is that RS are populated
by multiphoton resonance[VPT11]. Another one considers that the electron is
recaptured into RS after tunneling ionization[LTM+14]. From the experimental
point of view, the formation of RS is usually investigated in terms of laser
intensity with multicycle femtosecond laser pulses[LZZ+16, XLH+20]. However,
FTI happens with direct tunneling ionization (DTI) at the same time. How to
simultaneously demonstrate the contributions of both ionization processes to
the final ionization probability is still an open question.

For ultrashort laser pulses, especially in few-cycle regime, there is an
important parameter CEP, which has improved to be critical in many physical
phenomena[MPBB06, WLT+16, AOIS21]. However, the CEP effect on the formation of
RS atoms or molecules has not been extensively investigated so far. T.
Nakajima _et al._[NW06] has theoretically investigate the CEP dependent RS
atoms in the multiphoton ionization regime, but without experimental evidence.
H. Yun _et al._[YMH+18] has observed HHG from Hydrogen by 5 fs laser pulses.
The authors contributed the coherent extreme-ultraviolet emission to the
excited atoms through the frustrated tunneling ionization and the population
of excited states is calculated in terms of CEP.J. Liu _et al._[LZH+20]
theoretically optimize the CEP for the creation of RS.A. Gürtler _et
al._[GRVDZN04] experimentally investigate the direction of electron ejected
from RS rubidium with few-cycle radio-frequency pulses in multiphoton
ionization regime. The authors have found that the electron ejection direction
strongly depends on CEP.

Recently, M Kübel _et al._[KWZ+21] have experimentally found that the
modulation periodicity of ATI spectrum from Cs in gas-phase driven by a 3.1
$\mu$m laser is $\pi/3$. The fast oscillation happens in the energy range of
[5Up, 7Up] in the experiment, where Up is the ponderomotive energy. The
authors explain that the unusual behavior originates from the interference of
few recollision quantum orbits of the ionized electron, which is independent
with RS excitation process.

Here we report the first experimental evidence to distinct the effects of FTI
and DTI simultaneously in a new physical regime. The ATI spectrum for Argon is
measured driven by a few-cycle laser pulse. It is observed a modulation of
whole ATI photoelectron spectrum with a period of $\pi$/5 for relative CEP=[0,
$\pi$], which is the fastest CEP-induced modulation period, to the best of our
knowledge. Interestingly, this modulation decreases dramatically in the range
of [$\pi$, 2$\pi$]. Our simulations results, which are well agreed with
experimental data quantitatively, identify the importance of two kinds of
ionization processes. The fast oscillation in ATI spectrum originated from the
interferences of electrons in RS during FTI. While the asymmetry comes from
the electrons DTI in the few-cycle laser fields. Our results are direct
evidence to resolve the different effects of FTI and DTI in ATI spectrum in a
new physical regime. Our findings help to understand the mechanism of
formation of RS atoms in strong laser field and shed light on a new way to
control the electron dynamics in atoms, which can be further developed in
molecules and solids.

##  2 Experiment

Our experiment is performed on a system of isolated attosecond pulse
measurement in Institute of Physics, Chinese Academy of Sciences [ZHY+13]. A
commercial Ti: sapphire chirped-pulse amplifier (Femtolaser Compact PRO)
provides laser pulses with central wavelength of 760 nm, pulse duration of 25
fs, pulse energy of 0.8 mJ laser pulses at 1 kHz repetition rate. The laser
spectrum is broadened in a differentially pumped neon-filled hollow-core fiber
and then temporally compressed by chirped mirrors[RHT+06]. The whole spectrum
range covers from 550 nm to 950 nm. The CEP of the laser pulse is actively
stabilized with a fluctuation of 80 mrad in more than 7 hours by locking both
the oscillator and the amplifier loops. Finally, laser pulses with energy of
0.25 mJ and duration around 7 fs FWHM are obtained, which is around two
optical cycles[ZTY+14]. In our setup, the CEP of the 7 fs laser pulse is tuned
by a pair of fused silica wedges in the optical beam.

The linearly polarized laser pulses are focused into a vacuum chamber with
backing vacuum pressure of $10^{-7}$ mbar. A motorized aperture is set to
control the laser power, which is important in our experiment. The IR
intensity at focus is controlled by the motorized aperture from
$10^{11}W/cm^{2}$ to $10^{14}W/cm^{2}$. In the range of focus, Argon atoms are
injected by a gas jet to interact with IR laser to trigger ATI process. The
photoelectron spectrum for ATI is recorded with a time-of-flight (TOF)
spectrometer in terms of the CEP of input few-cycle IR laser (see Supplemental
material for details of experimental setup and procedures).

Figure 1: The measured ATI spectrogram of Argon driven by few-cycle laser
pulses for (a) the original data and (b) the fitting data, which shows a clear
oscillation of $\pi/5$. (c) The ATI spectra for three different energies in
terms of CEP. (d) The ATI spectra for two different CEPs.

Measured photoelectron energy spectra for ATI in terms of CEP are shown in
Fig. 1 (a), in which CEP is tuned with a step of $\pi/17$, which is calibrated
by HHG spectrum[YHT+14]. The measurement range of the TOF is 5 to 200 eV.
There are several interesting observations in the experimental results.
Firstly, clear oscillation in ATI spectrum is found with a period of $\pi/5$
in the range of [0, $\pi$] after data fitting, as shown in Fig. 1 (b). The
Nyquist frequency of the CEP scan is $8.5/\pi$, which is higher than the
frequency we measured. Secondly, another striking feature of our experimental
results is that the oscillation decreases dramatically in the range of [$\pi$,
$2\pi$]. Compared to experimental results in Ref.[KWZ+21], the CEP-dependent
oscillation repeats in a range of [0, $4\pi$]. The differences between the
observations from Ref.[KWZ+21] and our experiment may indicate that there are
different underlying mechanisms.

To see our experimental finding more clearly, the measured photon electron
counts at three different energy ranges (±0.8 eV) is shown in Fig. 1 (c). The
oscillation for 7.1 eV shows a larger modulation depth than the one for 12.4
eV. The modulation of ATI counts for 23.2 eV is most significant in range of
[0, $\pi$]. Another interesting observation is that the suppression of ATI
spectrum in full spectral range, as shown in Fig. 1 (d). For relative
CEP=5$\pi$/17, the ATI spectral (solid red line) is suppressed in the whole
energy range of [5, 40] eV, compared with the spectrum when CEP=9$\pi$/17. The
origin for this observation will be discussed below.

##  3 Simulation results and discussion

In order to understand the experimental results, the one-dimensional time-
dependent Schrödinger equation (TDSE) is used to describe the interaction
between the atom and the linearly polarized laser pulses. The electric field
of the laser pulses which is expressed as

$E(t)=\sum_{i}E_{i}f_{i}(t)sin(\omega_{i}t+\phi_{i})$ (1)

where $E_{i}$ is the electric field amplitude, $f_{i}(t)=sin^{2}(\pi
t+\phi_{i})$ is the electric field envelope, $\omega_{i}$ is the frequency of
the laser pulse, and $\phi_{i}$ is the CEP of the laser pulse. The initial
state is obtained by diagonalizing the field-free Hamiltonian $H_{0}$ of the
atom and the TDSE is solved by the split-operator method. Hence, the
population of the bound states after the end of laser pulse is obtained by
projecting the final wave function to the eigenstate of $H_{0}$ (see
Supplemental material for details of theory).

Figure 2: Calculated ATI spectra vs CEP along the positive (a) and negative
(b) directions of laser’s polarization. Summation of ionization probability vs
CEP along the positive (c) and negative (d) directions of laser’s
polarization.

Our simulations clearly reproduce the main characterizations in experimental
results. The simulation results for Argon atom are shown in Fig. 2 (a) with
calculation parameters of 660 nm, 760 nm and 860 nm with same CEP (three-color
laser field as represented in Eq. (1)), 7.8 fs and peak intensity $1.9\times
10^{14}W/cm^{2}$. Because there is one TOF spectrometer in our experiment
along laser’s polarization direction, only electrons along one direction are
summed in the calculation. In the range of [0, $\pi$], there is oscillation of
ATI spectrum with a period of $\pi/5$ and the oscillation decrease obviously
in the range of [$\pi$, $2\pi$]. Furthermore, if the electrons along another
direction are summed as shown in Fig. 2 (c), which is not measured in
experiment, the oscillation appears in the range of [$\pi$, $2\pi$]. To see
this oscillation more clearly, the total ionization probability along two
directions in terms of CEP are shown in Fig. 2 (c) and (d) with the energy of
the electron in region of [5, 40] eV. The remarkable agreement between the
simulation results and experimental data give rise to a question: what is the
underlying mechanism of these experimental observations?

Figure 3: (a) The schematic of simplified experimental setup. (b) Two
trajectories of electrons ionization along +x direction. (c) Two trajectories
of electrons ionization along -x direction. GS: ground state, CS: continuum
state and RS: Rydberg state. The red, green and yellow lines are the Coulomb
potential, the external laser field and the total electric field, respectively
in (b) and (c).

Here, we propose a semi-classical picture of electron ionization, as
illustrated in Fig. 3. We simplify the experimental setup, as shown in Fig. 3
(a). The ionization of electron is discussed under two conditions. When the
electric field oscillates along +x direction, as shown in Fig. 3 (b). The
electron is firstly tunneling ionized from ground state directly, which is
labeled as trajectory 1. Then the electron is accelerated by the electric
field and has possibility to be rescattered from ion. On the other hand, the
electron has possibility to be captured by Coulomb potential into RS during
tunneling. Later, the electron may be further ionized from RS to continue
state by laser field, which is labeled as the trajectory 2. The ionized
electrons from both trajectories 1 and 2 are measured in experiment. When the
electric field oscillates along -x direction, as shown in Fig. 3 (c). There
are another two trajectories, similar to Fig. 3 (b). For trajectories 1 and 3,
the electrons move along +x and -x directions, which depend on the electric
field. However, since the Coulomb potential is spatially symmetry, both
trajectories 2 and 4 may contribute to a certain RS, and the ionization from
RS is independent with the direction of laser’s polarization. These
differences have important role in our experimental results, as discussed in
below.

Figure 4: Electric filed with one-color (a) and three-color (b) driving laser
pulse. Calculated ATI spectrum and corresponding population of RS with three-
color driving laser field at intensity of $4\times 10^{13}W/cm^{2}$ (c and d),
and intensity of $1.9\times 10^{14}W/cm^{2}$ (e and f). Calculated ATI
spectrum and corresponding population of RS with one-color driving laser field
at intensity of $1.4\times 10^{14}W/cm^{2}$ (g and h).

Comparing with experimental data, two questions need to be answered: 1. why
the oscillation has a period of $\pi/5$ in range of relative CEP=[0, $\pi$]?
2. Why this oscillation dramatically decreases in the range of relative
CEP=[$\pi$, $2\pi$]? Firstly, we investigate the laser intensity effect on the
ATI spectrum driven by three-color (660 nm, 760 nm and 860 nm with relative
amplitude of electric field 0.6:1:0.6) laser field, as shown in the Fig. 4
(a). To understand the oscillation period, the population of atoms in RS are
also calculated. When the laser intensity is $4\times 10^{13}W/cm^{2}$, the
electrons are mostly ionized by multiphoton ionization. In multiphoton
ionization regime, the ionization probability is independent to the population
of the RS. Therefore, although the population for the principle number of odd
and even RS oscillate with CEP, as shown in Fig. 4 (d), we cannot observe any
oscillation in the ATI spectrum, as shown in Fig. 4 (c). However, when the
laser intensity increases to $1.9\times 10^{14}W/cm^{2}$, the DTI dominate the
electron ionization process. Electrons from trajectories 1 and 2 start to play
an important role. The populations of RS increase four orders of magnitude
($\sim 10^{-4}$), as shown in Fig. 4 (f). More importantly, the CEP dependent
odd and neighboring even RS has a phase shift around $\pi/4$. By using
different absorbing boundaries in calculation, one may find that the
contribution mainly comes from RS with principle number larger than 7 (see
Supplemental material). Consequently, the total RS has an oscillation of
$\pi/5$ by the interference coming from the electrons of trajectory 2. The
higher population of RS, the lower population of ionized electron. Therefore,
the calculated ATI spectrum has an oscillation of $\pi/5$, as shown in Fig. 4
(e) and the whole spectrum is suppressed for different CEPs, as shown in Fig 1
(d).

The population of RS is symmetry for CEP=[0, $\pi$] and [$\pi$, $2\pi$], as
shown in Fig. 4 (f). However, strong asymmetrical ATI spectrum appears in Fig.
4 (e). In order to explain this, we perform simulation with one-color (760 nm)
laser field, as shown in the Fig. 4 (b). The oscillation of ATI spectrum
dependent on CEP is clearly observed with one-color laser field at intensity
of $1.9\times 10^{14}W/cm^{2}$, as shown in Fig. 4 (g), where there is no
different between CEP=[0, $\pi$] and [$\pi$, $2\pi$]. The calculation results
can be understood by the symmetry of electric field. For one-color laser
field, there are more than one peaks to contribute the electron of trajectory
1, like A and B in positive direction or C and D in negative direction, as
shown in Fig. 4 (b). The interference between electrons of trajectory 1 in
different optical cycles washes away the asymmetrical dependence of ATI
spectrum on CEP. However, for the three-color laser field, there is only one
dominate peak A in positive direction and two peaks B and C in negative
direction, as shwon in Fig. 4 (a). The amplitude of A is two time of
amplitudes of D and E in the same direction; therefore, the three-color laser
field is asymmetrical in terms of CEP and also for the electrons of trajectory
1. The comparison of ATI spectrum driven by one-color and three-color lasers
give a clear picture on the contributions of electrons from two different
trajectories. The interference of electrons from trajectory 2 causes the CEP
dependent oscillation in ATI spectrum, furthermore the electrons from
trajectory 1 break the symmetry for relative CEP= [0, $\pi$] and [$\pi$,
$2\pi$] because of the asymmetrical electric field.

Figure 5: Calculated ATI spectrum driven by three-color lasers with 3 optical
cycles at intensity of $3.8\times 10^{14}W/cm^{2}$ (a); by three-color laser
with 2 optical cycles at intensity of $4\times 10^{13}W/cm^{2}$ (b); by three-
color laser with 2 optical cycles at intensity of $1.9\times 10^{14}W/cm^{2}$
(c).

In order to verify our understanding, the laser intensity is experimentally
increased to $3.8\times 10^{14}W/cm^{2}$ and it is found that the fast CEP-
dependent oscillation disappears (see Supplemental material). The
corresponding calculated ATI spectrum is shown in the Fig. 5 (a). It is found
that there is no oscillation in ATI spectrum, where the DTI dominates and the
FTI cannot significantly affect the ionization probability at this intensity.
Furthermore, we theoretically investigate the effect of pulse duration to the
ATI spectrum. When the pulse duration decreases from 7.8 fs (three optical
cycles) to 5.2 fs (two optical cycles), the ATI spectrum changes quite a lot.
At intensity of $4\times 10^{13}W/cm^{2}$, the ATI spectrum is independent of
CEP, as shown in Fig. 5 (b), which is similar to that of the Fig. 4 (c). As
the intensity increase to $1.9\times 10^{14}W/cm^{2}$, there is no CEP-
dependent modulation but a totally different structure in ATI spectrum, as
shown in Fig. 5 (c). Our simulation results indicate that our experimental
results are quite sensitive to the laser parameters, and fruitful physical
phenomena may happen under different experimental conditions.

Last but not least, our experimental results show that our method provides a
potential way to measure the absolute CEP with only one TOF. Calibrated by the
numerical results, the absolute CEP of few-cycle laser pulse can be determined
because of the asymmetry in the measured ATI spectrum. Compared with the
traditional “stereo-ATI” experiment [PLW+03], our results show a potential way
to measure the absolute CEP of input laser with only one TOF, which is
compact, easy to align and lower cost. In Ref. [KWS+17], C. Khurmi _et al._
also propose a method to measure the absolute CEP with one TOF. Their method
relays on that the Schrödinger equation of hydrogen atom in the strong laser
field can be numerically solved with high precision. By comparing the
experimental ATI spectrum from hydrogen atom with the simulation results, the
absolute CEP is determined. However, with other noble gases, significant
differences are observed between experimental data and simulation results.
Compared with Ref. [KWS+17], our experimental data are obtained in Argon
atoms, which may be provide a more versatile way to measure the absolute CEP.
Currently our results only give a potential way to measure the absolute CEP,
more experimental data and sophisticated simulation method are needed to
improve our method into be more reliable and compact, where we can only apply
the one-dimensional simulation because of the limitation of the computer at
present.

##  4 Summary

In summary, ATI spectrum from Argon driven by few-cycle IR laser pulses is
experimentally investigated when the Keldysh parameter is close to 1. For the
first time, we observe a $\pi/5$ oscillation of whole ATI spectrum for
relative CEP=[0, $\pi$]. Interestingly, this oscillation dramatically
decreases for [$\pi$, $2\pi$] under our experimental conditions. The
simulation results are well agreed with our experimental data quantitatively.
We propose a semi-classical physical picture and contribute the experimental
results to two kinds of electrons with different trajectories. The
interference of electrons in FTI causes the fast oscillation while the
electrons in DTI are responsible for the asymmetry in ATI spectrum. We propose
a new experimental approach to modify the electric field to control the
electron dynamics. Our results help to understand the physical mechanism of
CEP modulated FTI process in strong laser field. Furthermore, the experimental
results provide a potential way to determine the absolute CEP of few-cycle
laser pulse with only one TOF spectrometer.

##  5 Acknowledgement

The authors thank Zenghu Chang for useful suggestions. B. B. Wang thanks Jing
Chen for fruitful discussion. The work is finally supported by the National
Natural Science Foundation of China (Grant No. 12034020, 12074418, 11774411,
11864037 and 91850209), and the Synergic Extreme Condition User Facility.

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11institutetext: Cognitive and Systems Neuroscience Group, University of
Amsterdam, Science Park 904, 1098XH Amsterdam, Netherlands 22institutetext:
Machine Learning Group, CWI Amsterdam, Science Park 123, 1098 XG Amsterdam,
Netherlands
22email<EMAIL_ADDRESS>
# Masked Image Modeling as a Framework for Self-Supervised Learning across Eye
Movements

Robin Weiler Equal contribution11 Matthias Brucklacher ✉* 11 Cyriel M. A.
Pennartz 11 Sander M. Bohté 11 2 2

###### Abstract

To make sense of their surroundings, intelligent systems must transform
complex sensory inputs to structured codes that are reduced to task-relevant
information such as object category. Biological agents achieve this in a
largely autonomous manner, presumably via self-super-vised learning. Whereas
previous attempts to model the underlying mechanisms were largely
discriminative in nature, there is ample evidence that the brain employs a
generative model of the world. Here, we propose that eye movements, in
combination with the focused nature of primate vision, constitute a
generative, self-supervised task of predicting and revealing visual
information. We construct a proof-of-principle model starting from the
framework of masked image modeling (MIM), a common approach in deep
representation learning. To do so, we analyze how core components of MIM such
as masking technique and data augmentation influence the formation of
category-specific representations. This allows us not only to better
understand the principles behind MIM, but to then reassemble a MIM more in
line with the focused nature of biological perception. From a theoretical
angle, we find that MIM disentangles neurons in latent space, a property that
has been suggested to structure visual representations in primates, without
explicit regulation. Together with previous findings of invariance learning,
this highlights an interesting connection of MIM to latent regularization
approaches for self-supervised learning. The source code is available under
https://github.com/RobinWeiler/FocusMIM

###### Keywords:

Self-supervised learning Representation learning Generative model

##  1 Introduction

Both biological and artificial intelligent systems must construct useful
object representations (in the general case equivalent to classifiable) from
large amounts of unlabeled data [1]. In self-supervised learning (SSL), two
main approaches have crystalized over the last years: multi-view based
learning (via contrasting [2], latent regularization [3] or distillation [4]),
and masked image-modeling (MIM) [5, 6] which predicts occluded image content
usually at the pixel-level, but to which also the more recently developed
latent predictions [7] can be counted. While constrastive methods have been
suggested as a model of SSL in the brain, structuring representational
geometry via temporal [8] and spatial cues [9], we postulate that the brain
may be engaged in MIM via eye movements and attention shifts (Fig. 1):
Perception, at a singular point in time, is selective, both through foveal
vision and selective attention [10]. But perception is also dynamic: saccadic
eye movements and attention shifts flood the sensory stream with new
information - in a manner that is to some degree predictable from knowledge
about the direction and magnitude of the gaze shift (via a corollary discharge
[11, 12]) and image structure.

Figure 1: Representation learning through eye-movements. (a) Random patch
masks used in artificial MIM approaches, gray patches are hidden from the
network. (b) In our approach, selective masking is achieved through the
inhomogeneous nature of foveal vision. Eye movements depicted as vectors
reveal previously inaccessible or distorted information that is compared to
the prediction $x^{t+\delta}$. (c) After pretraining on the prediction task,
the quality of latent representations in the latent representation is assessed
through classification accuracy in linear probing.

Studying the principles behind MIM is attractive as it requires less
assumptions about temporal sequences and data augmentations than the mentioned
multi-view approaches to biological SSL. Furthermore, such an approach fits
well with predictive theories of cortical processing [13, 14, 15]: both
predictive processing [16] and MIM [17] conceptualize perception as the
process of identifying a hierarchical latent variable model that could have
caused the sensory inputs (this generative aspect is lacking in discriminative
SSL models) and are predictive of future inputs. Experimentally, the frontal
eye fields have been proposed to stabilize perception through predictions
across eye movements [11]. We postulate a larger role of this mechanism: the
learning of visual features useful for downstream tasks such as
classification.
Here, we investigate the influence of multiple variations to MIM in
convolutional neural networks (CNNs) that align well with biological
constraints. First, the influence of different masking strategies is tested,
resulting in a peripheral masking strategy that learns strong representations
and is more in line with the focused nature of biological perception. In doing
so, we discover that representations are markedly improved, and that neurons
become more decorrelated, when peripheral information is completely suppressed
during learning. Second, the influence of data augmentation is tested, showing
that these are especially relevant for the peripheral masking strategy. Third,
we show that the relevant loss signal originates from the main object:
although networks reconstruct the context too, this is not necessary for
representation learning. In fact, we find that under some conditions networks
learn faster when being only rewarded for predicting the foreground, which is
interesting as segmentation may naturally be available from geometric cues or
motion [18, 19]. Indeed, availability of segmentation has been shown to aid
constrastive SSL if leveraged during pretraining [20], while its effect on MIM
has previously not been studied to our knowledge.

##  2 Related Work

The breakthrough for MIM in representation learning arguably was achieved
using random patch masks (Fig. 1a) in a vision transformer [6], an
architecture that differs from information processing in biological networks.
Subsequently, Kong et al. [17] and He et al. [6] found that an optimal
learning ratio falls roughly between 40% and 80%, being sufficiently large but
not overly aggressive.

A more biologically compatible CNN for MIM was developed by [5]. The authors
found that a central mask (predicting from peripheral context to a small
central region) performed worse than randomly distributed patch or region
masks, although quantitative reports were not provided. A more recent CNN-
based MIM is ConvNext-V2 [21], which utilizes random patch masks like the
transformer-based methods in combination with a shallow decoder. Xie et al.
[22] further demonstrated the potential for MIM in CNNs, achieving high
accuracy in a ResNet-based model. Neither Woo et al. [21] nor Xie et al. [22]
investigated a classical self-supervised paradigm with limited amounts of
labeled data and instead finetuned the pretrained model on the full ImageNet
dataset used during pretraining.

Biological approaches to SSL derive mostly from multi-view approaches. They
avoid representation collapse, a catastrophic network state in which different
inputs are projected onto the same latent activity pattern and thus cannot be
distinguished anymore, by introducing an expanding force that pushes
representations apart. Illing et al. [9] used a single negative sample at a
time, interpreted as being obtained by larger eye movements. Their approach
converts contrastive predictive coding [2] into a fully local learning rule.
Interestingly, the model also has a generative component (predicting the next
representation in a sequence of views) as a means to representation learning
that is not investigated further. Local predictive learning (LPL) [8], on the
other hand relies on pushing representations apart based on their temporal
distance, converting variance regularization [3] into a local learning rule.
In contrast, MIM as we employ it, prevents representation collapse in a more
minimalist way, i.e., without the necessity of negative examples [9] or
accumulatie temporal statistics [8], and instead solving a generative problem
in which loss minimization is incompatible with a collapsed solution. As the
approach presented here is an initial step, we use the spatially non-local
backpropagation, however, see [23, 24] and subsequent work for biologically
plausible solutions yielding comparable weight updates.

Focused vision has been combined with eye movements for functional benefits in
a variety of ways, from information gain in line with active inference [25],
to achieving representation invariance and reducing the need for detail when
learning a generative model [26]. The model of Thompson et al. [27] combines
spatially structured eye-movements with visual information processing in an
enactive approach to counting. The abovementioned model of Illing et al. [9]
uses on- vs. off-object eye movements as a heuristic to shape representation
distances in latent space. In comparison, our approach is more reduced in
dynamics and focuses instead on the consequences for neural representation
learning when casting eye movements as a self-supervised predictive task.

##  3 Method

###  3.1 Model

We reduce the task of predicting across eye-movements to its essence and avoid
the burden of learning a spatial transformation, which we argue is orthogonal
to the process of information restoration that drives representation learning
[17]. As a consequence, inputs and predictions take place in the same
reference frame (Fig. 1c), as in other MIM approaches [6, 21, 17]. The network
architecture shown in Fig. 1c resembles an autoencoder where an encoder
network maps input images to a compressed latent code from which a decoder
network reconstructs the image. The encoder reduces the spatial dimension from
96x96 (with three color channels) to a spatial bottleneck of 12x12 (with 128
channels), forming the so called latent representation. Due to the large
receptive fields of the latent space neurons at the output of the encoder,
they map roughly onto neurons in visual area V4 or higher in primates [28].
From here, the inversely structured decoder (with deconvolutions instead of
convolutions) maps the latent representations back onto a 96x96x3 output that
constitutes the model’s expectation of image content. Both encoder and decoder
consist of nine ResNet18-style blocks [29] in three stages without batch
normalization layers, as convolutional networks are closer to biological
networks than transformers.

###  3.2 Self-Supervised Pretraining

As other work in MIM, we define pretraining as the self-supervised training
phase in which the model is presented with a partial view of an image and
attempts to predict the image pixel by pixel in full resolution. The loss
function is given as the mean squared reconstruction error in parts of visual
space with incomplete inputs (masked or blurred, depending on the paradigm),
normalized by the area over which the loss is calculated. This reconstruction
loss is reminiscent of generative model learning in predictive coding [14].
Pretraining was conducted on the unlabeled split of the STL-10 dataset [30],
which was specifically constructed for SSL. We investigated different masking
strategies illustrated in Fig. 2 as inputs during pretraining:

Random patches: The most common masking strategy used, e.g., in [29, 21, 17]
(see also Fig. 1a). In our implementation, we split the image into 144 patches
of 8x8 pixels and replaced the masked image content with the average color of
the image.

Masked periphery: A central circle in full resolution with the periphery
filled in the sample’s average color (see also Fig. 1b). Biologically, this
strict discarding of out-of-focus information is imaginable through
inattentiveness [10].

Figure 2: The masking strategies define the pretraining task. Top row:
Exemplary inputs for different pretraining paradigms with masked areas covered
in the image’s gray average color. Masked periphery and random patches are
shown with 60% masking ratio. Middle row: Predictions from the given inputs
after pretraining. Bottom image: Ground truth.

Blurry periphery and Blurry random patches: Variations of the above paradigms
with Gaussian-blurred information instead of average colors.

Fovea filter: The circular center (”fovea”) of the image is in full resolution
with a very small radius. The further out from the center a pixel is, the
higher the level of blur, mimicking the physiological constraints of human
foveal vision, i.e., the distribution of cone photoreceptors across the retina
[31, p. 244]. We follow the implementation of [32, 33] by means of a
resolution map determining the blur filter’s spatial frequency at a given
distance from the center.

For both random patches and masked periphery (using a square mask instead of a
circle), we experimented with sparse convolutions as in [21] to avoid
processing of borders between masks and image content. However, we found no
significant difference in linear probing accuracy (as described below), so we
continued with standard convolutions. During pretraining, we employed random-
resized cropping (scale $\in[0.08,1.0]$, aspect ratio $\in[0.75,1.33]$). We
used the ADAM optimizer [34] (weight decay=1e-8, batch size=512) and selected
the learning rate from [5e-5, 1e-4, 5e-4] that led to the best classification
performance (described below) for each of the major pretraining tasks (masked
periphery, random patches, foveal filter). This value was then kept when
blurry information was introduced. After 500 epochs of pretraining, we
continued with the model achieving the lowest reconstruction loss for further
analysis.

###  3.3 Linear Probing

To analyze the quality of the learned representations after pretraining,
linear probing was conducted. Here, we followed the standard procedure common
in SSL of discarding the decoder and appending a linear layer to the frozen
encoder. The 128x12x12x10 (latent dimension * classes of the STL-10 labeled
splits) parameters of the linear layer were then fit without further
augmentations on the training split of the STL-10 dataset (5,000 images), of
which we used 1,000 for training validation and early stopping. We used images
without mask or blurring, as these consistently led to higher readout
accuracy. As in pretraining, we employed the ADAM optimizer [34] (learning
rate=1e-4, weight decay=1e-8, batch size=512). Classification accuracy was
then evaluated on the 8,000 test images, averaged across five randomly seeded
runs. We also experimented with multilayer perceptron readout and finetuning
(retraining the whole network), but neither of these approaches led to large
differences in accuracy.

###  3.4 Correlation of Neurons in Latent Space

In addition to linear discriminability, representations can also be analyzed
based on their statistical properties. One desirable property here is
decorrelation [35, 3, 36]. To quantify this, the covariance matrix is required
[3, 36]:

$C(R)=\frac{1}{n-1}\sum_{i=1}^{n}\mathbf{r}_{i}\mathbf{r}_{i}^{T}$ (1)

with $R$ defined as a tensor containing the $n$ latent representation vectors
$\mathbf{r}_{i}$ (that are z-scored across the batch dimension), with $n$
equal to the number of samples, in our case $n=100.000$. As in [3], the sum of
the squared off-diagonal entries in $C$ then quantifies the degree to which
neurons are correlated:

$c(R)=\sum_{i\neq j}[C(R)]^{2}_{i,j}$ (2)

###  3.5 Disentangling Predictions of Object vs. Background

Unlike in the pretraining paradigms outlined so far, eye movements do not
reveal all previously unavailable information at once. Instead, they usually
fall on salient points, often objects [37, 38]. Objects may also be segmented
already before recognition, either from motion signals [19], occlusion
geometry, or both [18], making them the prioritized prediction targets. We
thus evaluated the influence of reduced learning signal availability from the
background. First, we obtained a figure-background segmentation mask using the
rembg-library (https://github.com/danielgatis/rembg). We discarded images in
which less than 100 pixels (out of 96x96) received a confidence score of 0.8
(out of 1) or higher. In these $\sim$8,000 images, no clear segmentation mask
was obtained. Otherwise, as shown in Fig. 3, the masks were generally high-
quality and only small object parts were missed. During pretraining, the
reconstruction loss was then multiplied with the mask value (between 0 and 1)
at each point in space and normalized to account for the reduced prediction
area.

Figure 3: Representative presegmentation masks used to investigate how
discarding loss signals from the background affects network performance,
obtained with rembg. Displayed are the masks obtained with the rembg-library,
with brightness indicating confidence, overlayed onto the respective STL-10
images.

###  3.6 Determining the Optimal Masking Ratio

For each of the major masking paradigms, we varied the masking ratio to choose
a model with optimal classification accuracy. For the random patches, we found
a masking ratio of 60% to lead to the best representations. For the masked
periphery, a higher ratio of 80% performed slightly better. We thus continued
with this fraction for the remaining results and refer to [17, 29] for more
detailed studies on the influence of the masking ratio.

##  4 Results

We applied various combinations of masking strategy and data augmentation, the
results of which are shown in Fig. 4.

Figure 4: Linear readout accuracy quantifies representation quality after
pretraining. a) Influence of masking strategy in comparison to the multi-view
approaches CLAPP [9] and LPL [8], that both come in two variants trained with
a local learning rule or backpropagation. The remaining baselines are pure
autoencoding (AE) from full image to full image in the same network as the
masked methods, and directly conducting linear probing on the input images. b)
Influence of data augmentation. In the left- and rightmost bar, the filled and
outlined bars overlap, i.e., classification accuracy is unaffected by
augmentation. c) Incomplete masking by Gaussian blurring, instead of uniform
average coloring, drastically decreases representation quality. Error bars
indicate the standard deviation across five randomly seeded runs.

###  4.1 Influence of Masking Strategy

Out of the used masking strategies, we identified the masked periphery
condition as the most promising candidate for a biological model. This masking
technique inspired by foveal visual perception led to the second highest
classification accuracy in linear readout (mean value $67.9\pm 0.4\%$) when
crop-and-resize transformations were used during pretraining, outperformed
only by the random patch-wise masks ($70.2\pm 0.4\%$). Strong performance in
these settings supports the assumption that solving the pretraining tasks of
predicting hidden image areas leads to discovery of object representations
[17]. In contrast, deblurring the peripheral region from the foveal filter
resulted in comparably poor representations with readout accuracy not much
higher than pure autoencoding. Presumably, this condition allows the network
to find a local solution that does not require a globally integrated
representation. To more closely investigate the effect of input information
content on classification accuracy, we introduced Gaussian blurred information
in the masked areas (Fig. 2, third column) of the masked periphery and random
patch masks. Despite the low information content, the introduced information
was sufficient to degrade performance substantially (Fig. 4c).

###  4.2 Decorrelation of Latent Space Neurons

To investigate the degree to which neurons in latent space develop independent
tuning properties, we computed the correlation term $c$ using Eq. 2.
Interestingly, both masked periphery and random patches decorrelate neural
representations (Fig. 5), and more strongly than the foveal filter. This
decorrelation, reminiscent of sparse coding [39], emerged without being
explicitly enforced as in other approaches [39, 3, 36].

Figure 5: Masked image modeling implicitly decorrelates latent space neurons.
Shaded regions indicate the standard deviation across five randomly seeded
runs.

###  4.3 Influence of Data Augmentation

Applying crop-and-resize augmentation during pretraining proved necessary for
representation learning in the masked periphery condition. Without this
augmentation technique, readout accuracy dropped from $67.9\pm 0.4\%$ to
$56.7\pm 0.5\%$ (Fig. 4b). Biologically, this augmentation can be approximated
through different viewing angles, especially since no assumption about
temporal co-occurrence is made: differently augmented versions of an image are
presented in different epochs. Interestingly, neither the foveal filter nor
the random patches were affected by ablating the crop-and-resize
transformation. Maintaining the position of the patch masks across epochs
alone did not reduce readout accuracy ($70.3\pm 0.9\%$), but in combination
with ablation of the crop-and-resize augmentation, accuracy reduced to
$62.5\pm 0.8\%$. We conclude from this that the model requires exposure to a
variety of prediction tasks from each object, whether through data
augmentation or through changes in mask position.

###  4.4 Predicting Object vs. Background

Restricting the pretraining loss to the main object instead of the background,
as described in section 3.5, did not affect linear probing accuracy
negatively, suggesting that on-object predictions are sufficient for
classification. While there was no substantial difference in readout accuracy
when disregarding, as opposed to when using, loss signal from the background
in the masked periphery condition (Fig. 6b), the network pretrained using
random patch masks profited from disregarding the background, especially in
early epochs (Fig. 6a).

Figure 6: Influence of presegmentation and masking strategy on training speed.
(a) When using random patches, using presegmentation to weight the
reconstruction loss accelerates training. (b) This is not the case when
masking the periphery. Shaded regions indicate the standard deviation across
five randomly seeded runs.

###  4.5 Reconstruction Quality

Beyond the network’s disciminative function, its generative capacity bears
functional relevance, e.g., for counterfactual reasoning, predictions of
occluded image content, and imagination [16]. An interesting effect from
generative model learning can be observed in the networks pretrained with
uniform (average colored) and blurry masks: These networks also reconstruct in
locations in which no loss was calculated. While is to some degree expected
when the masked area changes shape across image presentations (random
patches), this is somewhat surprising in the masked periphery condition (Fig.
2, second column). Here, also the central region was reasonably reconstructed
although the pretraining loss is indifferent to the network’s predictions.
This holistic percept fits well with the proposed role of visual predictions
in creating a stable visual representation [11, 16].

Across masking paradigms, reconstruction accuracy was uncorrelated with
readout accuracy: while the foveal filter achieved the best reconstructions
(Fig. 2), it performed poorly in linear probing (Fig. 4). Given a pretraining
paradigm, however, longer training on the reconstruction task also improved
readout (Fig. 6).

A third angle to analyze the generative capacity of the network stems from the
fact that eye-movements are often preceded by covert attention shifts, which
increase information sampling from these areas [40]. The sampled information
can then be used to improve predictions about the expected image content. To
test whether the model could exploit such a mechanism for improved
reconstructions, we provided the model with a second circular input sampling
area (covert attention) on top of the peripheral masking while constraining
the error to the foreground (section 3.5). The additionally provided
information indeed led to more truthful predictions (Fig. 7), i.e., reduced
prediction errors (from 2.6e-2 to 2.3e-2), which ties the approach back to
earlier theories of predictions as a stabilizing mechanism [11]. However,
adding this sampling during pretraining, did not alter readout accuracy
significantly, neither when setting the attention circle randomly nor when
placing it on random positions of the main object.

Figure 7: Attention-guided predictions for stable perception. (a) Top row:
ground truth. Middle row: input in the masked periphery condition with a
masking ratio of 80%. Bottom row: reconstructions after pretraining. (b) Same
as (a) with additionally provided information via a second, non-central input
circle (covert attention). The top row depicts the input mask consisting of
these two circular areas (fovea and attention). Adding covert attention
improves reconstructions.

##  5 Conclusion

In this paper, we investigated key components of masked image modeling (MIM)
through the lens of biological vision. As a similarly performant alternative
to the random patch masks common in artificial representation learning [6,
21], we identified the more biologically plausible peripheral masking (Fig.
4a). Provided with differently augmented views of the input images (Fig. 4b),
this approach does not pose restrictions on the temporal sequence in which
inputs are encountered [8] or on knowledge about the spatial extent of an
object to obtain negative samples [9]. An additional advantage of the employed
approach in contrast to purely discriminative methods is learning of a
generative model (Fig. 2, Fig. 7). A general requirement for high
classification accuracy across all investigated masking strategies is strict
intransparency of input masks (Fig. 4c), just as we sometimes do not perceive
objects in our field of view while focusing on another point in space [10].
From a theoretical perspective, implicit decorrelation of latent space neurons
(Fig. 5), together with previous observations of implicit invariance learning
[41], connect the MIM framework to latent regularization SSL [3], where these
objectives are explicitly enforced. This raises the question whether and how
synergies could be efficiently exploited. We conclude that peripheral masking
and prediction is a candidate mechanism for self-supervised learning in the
brain, potentially in addition to previously proposed non-generative methods
[8, 9]. These and other multi-view approaches [42, 43] have successfully
conducted self-supervised learning with more local learning rules than
backpropagation, raising the question whether the same can be done with MIM.
Lastly, as primates employ eye-movements in a strategic and enactive way [27],
an interesting extension would be to learn from multiple peripherally masked
glimpses in a sequence through spatiotemporally masked videos [44].

#### Acknowledgements

This project has received funding from the European Union’s Horizon 2020
Framework Programme for Research and Innovation under the Specific Grant
Agreement No. 945539 (Human Brain Project SGA3 to C.P. and S.B.). We thank
SURF (www.surf.nl) for the support in using the National Supercomputer
Snellius and especially Thomas von Osch for the helpful discussions.

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2021

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Ariel A. Valladares These authors contributed equally to this work.

[1]Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma
de México, Apartado Postal 70-360, Ciudad Universitaria, 04510, CDMX, México

2]Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado
Postal 70-542, Ciudad Universitaria, 04510, CDMX, México

# Could Negative Pressures Turn Bismuth into a Metal? The Case of the
Expanded.

Flor B. Quiroga  David Hinojosa-Romero  Alexander Valladares  Renela M.
Valladares  Isaías Rodríguez<EMAIL_ADDRESS>*  [

###### Abstract

Materials may behave in non-expected ways when subject to unexpected
conditions. For example, when Bi was turned into an amorphous phase (a-Bi)
unexpectedly it became a superconductor at temperatures below $10$ K. We
provided an explanation as to why a-Bi superconducts and the crystalline
(c-Bi) had not been found to do so: we computer calculated their electronic
properties and found that a-Bi has a larger electron density of states, eDoS,
at the Fermi surface than c-Bi and this explained the phenomenon. We even
predicted an upper limit for the superconducting $T_{c}$ of the crystalline
phase, which was experimentally corroborated within the following year. We now
decided to investigate what happens to crystalline (Wyckoff structure) and
amorphous Bi when pressures below the atmospheric are applied (expansion).
Here we show that when expanded, c-Bi becomes more metallic, since the eDoS
increases when the volume increases for the Wyckoff structure, while the
amorphous eDoS decreases. If the crystalline structure is maintained its
$T_{c}$ would rise under expansion, whereas it would diminish for the a-Bi.
Expansion can be obtained in the laboratory by chemically etching Bi-based
alloys, a process also known as dealloying, for example.

###### keywords:

Bismuth, amorphous, pressure, first-principles

##  1 Introduction

The quest for a room temperature superconductor has permeated the field for
decades. Patents and claims have come and gone and with the advent of
laboratory equipment that produces very high pressures for very short periods
of time the quest has increased. Recently, a (controversial) claim, portrayed
in a paper on carbonaceous sulfur hydride Snider_2020  that supposedly is a
room temperature superconductor under very high pressures, provoked an
optimism that gave hope to accomplish this quest for more accessible
pressures. However, the high pressures required made the original optimism
thermalize the hopes and realize that since the difficulties involved in the
process are considerable it does not seem possible to develop a room
temperature superconductor at reasonable pressures. These experimental results
were questioned by some researchers who did not give credit to the claim
Hirsch_2021  (see reference Service_2021  also), disbelieve that has not been
fully clarified. Faced with such controversy, we decided “to look the other
way” and started asking what would happen if we consider pressures below
atmospheric, “negative” pressures, and their effects on the electronic
structure and vibrational properties of condensed matter.

Bismuth is an interesting material since it is one of the few elements that
maintains some properties, like superconductivity. under varied circumstances.
Bismuth was first found to be a superconductor when in the amorphous phase at
ambient pressure Buckel_1954 ; Buckel_1956 . Then, when the Wyckoff
crystalline phase (the structure at room temperature and atmospheric pressure)
was subjected to positive pressures it changed crystalline structures but
maintained the superconducting properties for most of the new topologies (See
Rodriguez_2019  and Valladares_2018 , and references contained therein). Then
we came into the play and predicted, using a simple BCS approach, that
although the crystalline Wyckoff structure had not been found to superconduct,
it should, for temperatures below $1.3$ mK Mata_Pinzon_2016 . A year after our
prediction, experimentalists proved us correct and they found that the Wyckoff
structure superconducts below $0.53$ mK at ambient pressure Prakash_2017 . We
had ventured to estimate superconducting temperatures for bismuth under
various conditions; for the crystalline phase, that had not been found to
superconduct, we succeeded; our other predictions await verification
Valladares_2018 ; Hinojosa_Romero_2018 . So, with this succinct background we
decided that bismuth was the ideal “guinea pig” to test the effect of negative
pressures and we undertook the search to see if it could become metallic or a
semiconductor under expansion, and what the possibilities were to find it in a
superconducting state. This is the origin of this work.

We shall report the changes that take place in the electronic structure of
Wyckoff bismuth when expanded $5\%$, $10\%$, $15\%$ its ambient volume. We
shall then consider the amorphous phases of these expanded structures to
investigate the electronic changes that occur for both the crystal and the
amorphous. To search for superconductivity, we have to calculate the
vibrational densities of states and then apply the approach stated in Ref.
Mata_Pinzon_2016 , which is not the subject of the present paper. For now, we
shall be content to see what happens to the semi-metallicity of bismuth under
expansion; whether it is maintained or whether it evolves to become a metal or
a non-metal.

Studies on the modifications that occur under expansion of other solids have
been reported in the literature like the work of Moruzzi and Marcus whom in
1989 investigated the electronic changes that occur is solid palladium under
negative pressures Moruzzi_1989 . They found that Pd may become magnetic when
expanded. Thirty years later, in 2019, we investigated the electronic
properties of amorphous Rodriguez_2019PRB  and amorphous-porous palladium (to
be published). In both cases we found that when the stable crystalline
structure of Pd is altered, magnetism appears. The field of expanding solids
is not as well researched due to the difficulties of experimentally creating
an expanded sample. However, developments, like the chemical etching of
alloys, seem promising to foster an incursion in this field. Also, from the
computational point of view this does not represent an insurmountable problem
as we shall present in what follows.

In the following section we deal with the methodology that we used. Next, we
present the results of our investigation, to go sequentially to analyze such
results and infer possible consequences of our predictions. Finally, we
elaborate on the conclusions as to what the implications are and to further
our speculations for future work.

##  2 Methodology

When dealing with materials that have not been experimentally realized one has
to very careful not to create science fiction. In this work we assume that the
crystalline Wyckoff structure, stable at room temperature and at atmospheric
pressure, remains the same when expanded up to 15%, which might not be the
case. However, when amorphized, this handicap disappears and no memory of the
original structure remains which then serves for comparison purposes.

We constructed superlattices for the crystalline and for the amorphous
samples. For the crystalline structures the superlattices contained 250 atoms
as a result of multiplying the Wyckoff rhombohedral unit cell $5\times 5\times
5$ times. For the amorphous we constructed a supercell with 216 atoms by
multiplying the diamond-like structure $3\times 3\times 3$ times the cubic
unit cell, an initially unstable structure to propitiate the evolution into a
disordered topology, maintaining the correct density for the non-expanded
structure ($9.81$ g/cm3). After that we expanded both superlattices 5, 10 and
15%, properly scaling the interatomic distances, to investigate the evolution
of the electron densities of states. In Table 1 we specify the lattice
parameters and densities for all our samples. For the amorphous we ran our
undermelt-quench (Valladares_2008, ) procedure to generate the corresponding
samples. Figures 1 and 2 represent the Pair Distribution Functions (PDFs) for
the crystalline and for the amorphous, respectively. In Figure 2 we have
included the experimental results by Fujime (Fujime_1966, ), and the agreement
with our non-expanded amorphous sample is good. Also, in Figure 2 we present
the PDF for the non-expanded Wyckoff crystalline structure, for comparison.

Table 1: Parameters of our crystalline (c-) and amorphous (a-) samples. System |  Lattice parameter [Å] |  Density [g/cm3]
---|---|---
c-100% | 23.73 | 9.81
a-100% | 19.70
c-105% | 24.12 | 9.34
a-105% | 20.02
c-110% | 24.50 | 8.91
a-110% | 20.33
c-115% | 24.86 | 8.53
a-115% | 20.64
|  |

For the sake of completeness and self-containment we briefly reproduce the
description of the undermelt-quench procedure (Valladares_2008, ). We start
from an unstable crystalline structure, with the correct density, to help the
amorphizing procedure; usually for the metallic systems we have studied, we
chose a diamond-like structure to initiate the process. Then we do Molecular
Dynamics (MD) starting from 300 K linearly heating the sample to just below
the melting temperature (undermelt). Next we linearly cool the sample down to
close to absolute zero (quench). At the end of this MD process the structure
is essentially disordered, but evidently unstable. To favor the final locally-
stable structure we perform a Geometry Optimization (GO) procedure which
generates amorphous structures that agree well with experimental results when
available; that is why we consider the process adequate to predict structures
that are presently not known, with their corresponding PDFs
(Rodriguez_2019PRB, ). Once the GO process is completed we proceed to
calculate the electron density of states (eDoS) for all the structures
generated. It should be clear at this point that the amorphous supercell
generated has periodic boundary conditions but we claim that it represents the
material for distances comparable to the dimensions of the supercell. This
periodicity is spurious but helps to calculate the physics of properties that
do not extend beyond the size of the superlattice.

For all calculations, we employed a Density Functional approach as implemented
in the DMol3 code (Delley_1995, ), part of the Materials Studio suite
(biovia_materials_2016, ). The electronic treatment consisted in a double
numerical basis set with d-functions polarization (dnd) and a $6.0$ Å real-
space cutoff. We used the VWN functional (VWN_1980, ) within the Local Density
Approximation for the computation of the exchange-correlation energy and DFT
Semi-core PseudoPotentials (dspp) (Delley_2002, ) for the core treatment. All
calculations were spin-unrestricted with SCF density convergence of $1\times
10^{-6}$ using a thermal smearing of $0.136$ eV.

For the construction of the amorphous samples, the MD process uses a Nosé-
Hoover NVT thermostat with a Nosé Q-ratio of $0.5$ and a time step of $19.4$
fs. The undermelt-quench heating ramp started at 300 K and reached 534 K in
100 steps, followed by a cooling ramp of $2.34$ K per step down to $5.16$ K.
For the GO procedure we sought the convergence by using the following
thresholds: $2.7\times 10^{-4}$ eV, $2\times 10^{-3}$ eV Å-1 and $5\times
10^{-3}$ Å for energy, maximum force and maximum displacement, respectively.

The single-point energy and eDoS calculations for each geometrical structure
were conducted at the $\Gamma$-point of the (periodic) amorphous and
crystalline supercells, and also for a $2\times 2\times 2$ set of k-points in
the first Brillouin zone. In order to represent a bulk material, we applied a
$0.2$ eV gaussian broadening to the eigenvalues obtained from each eDoS
calculation.

Figure 1: (a) PDFs for the crystalline supercell structures. The positions of
the peaks undergo a displacement consistent with the expansion forced on our
structures. (b) 250-atom supercell of the Wyckoff structure where the bilayers
can be seen. Linked atoms are nearest neighbors  Figure 2: PDFs for the
amorphous supercells generated from the non-expanded and the expanded
structures, after undergoing the MD undermelt-quench procedure and the GO
process. The experimental results reported by Fujime (Fujime_1966, ) and the
crystalline non-expanded PDFs, are also included for comparison. The 216-atom
a-Bi 100% supercell is shown in the inset

A word on nomenclature, the non-expanded samples are referred to as the 100%
structures, whereas the samples expanded $x$% are referred to as (10$x$)%, or
to (1$x$)%, when $x$ is a two-digit number.

##  3 Results and Analysis

The calculated eDoS are represented in Figures 3 and 4 for the crystalline
structures, and in Figures 5 and 6 for the amorphous samples after the GO
processes. We also obtained the eDoS for the structures resulting from the MD
processes, Figure 7; the reason to present these results for the non-optimized
superlattices is to generate eDoS that may evolve, in the laboratory, towards
the optimized ones in search for stable topologies and also to observe the
changes that take place due to the optimization procedure. Since the runs for
the crystalline are done on supercells with 250 atoms, whereas for the
amorphous we considered supercells with 216 atoms, we report the eDoS per atom
to normalize our results and to compare them readily.

Figure 3: eDoS per atom at the $\Gamma$-point for the crystalline non-expanded
Wyckoff structure and for the expanded supercells. The inset shows the
behavior of the curves in the vicinity of the Fermi energy Figure 4: eDoS per
atom for $2\times 2\times 2$ k-points in the first Brillouin zone for the
crystalline non-expanded Wyckoff structure and for the expanded supercells.
The inset shows the behavior of the curves in the vicinity of the Fermi energy
Figure 5: eDoS per atom at the $\Gamma$-point for the amorphous expanded and
non-expanded structures after the GO process. The inset shows the behavior of
the curves in the vicinity of the Fermi energy Figure 6: eDoS per atom for
$2\times 2\times 2$ k-points in the first Brillouin zone for the amorphous
expanded and non-expanded supercells after the GO process. The inset shows the
behavior of the curves in the vicinity of the Fermi energy

Figure 7 displays the results for the values of the eDoS at the Fermi level
where trends that appear to be systematic are shown. No sudden changes occur
anywhere for the expansions studied, contrary to what was observed in Ref.
(Hinojosa_Romero_2017, ), which may indicate a tendency to possible
modifications of the structure. However, a geometry optimization of the
expanded crystalline samples is in order to elucidate this point.

Since our purpose is to investigate the behavior of the eDoS curves in the
vicinity of the Fermi energy, in Fig. 3 to 6 we show this behavior in the
insets for the structures considered. Since the parameters used for these
calculations are the same for all samples, the results are directly
contrasted. In this manner we can concentrate on visualizing, comparing and
analyzing all our results, for both non-expanded and expanded, crystalline and
amorphous, structures. An expected result is that the amorphous samples have a
larger eDoS than the crystalline ones in all cases studied here. This was
expected because that was the origin of our proposal to predict that the
Wyckoff structure should superconduct below $1.3$ mK (Mata_Pinzon_2016, ).
However, an unexpected result is to see that the eDoS for the amorphous
diminishes with the expansion, while the opposite occurs for the crystalline
ones. This could be due to the fact that as the amorphous expands, the
tendency is to become more crystalline-like since the appearance of pores due
to the increase in volume fosters the formation of small regions where the
structure is not condition by the amorphicity and tends to crystallize. The
opposite happens with the expanded crystalline cells as they rearrange to cope
with this expansion. It may be possible that eventually, the tendencies will
approach one another and become indistinguishable. Further calculations are
needed to discern this supposition.

What would be the effect of this behavior in the superconductivity of the
cells studied? If we think in terms of the BCS theory, the superconducting
transition temperatures $T_{c}$ would depend not only on the eDoS but on the
vibrational densities of states (vDoS) and on the strength of the Cooper
pairing. According to the theory,

$T_{c}=1.13\ \Theta_{D}\exp\left(-\dfrac{1}{N(E_{F})V}\right),$

where $\Theta_{D}$ is the Debye temperature of the material, $N(E_{F})$ is the
eDoS at the Fermi level that we have obtained, and $V$ is Cooper’s electron-
electron attractive potential. So it would be necessary to get information
concerning $\Theta_{D}$ (through the vDoS) and $V$ to reach reasonable
conclusions.

Figure 7: Comparison of the eDoS at the Fermi level for all the samples
studied after the MD and GO procedures. The results correspond to the
$\Gamma$-point and to the $2\times 2\times 2$ set of k-points in the first
Brillouin zone of the crystalline and amorphous supercell. The curves are
adjusted cubics to our results

##  4 Conclusions

From looking at Figure 7 we conclude that the crystalline structures upon
expansion become more conducting whereas the amorphous ones become less
conducting. Clearly, no changes in the structure of the crystalline cells upon
expansion have been considered and since the Wyckoff structure is one that
would tend to become cubic (look at the closeness of the two nearest-neighbor
peaks in Fig. 1) it would be likely that any perturbation on the topology may
induce rearrangements. However, the idea of this work was to investigate any
changes that would occur upon expansion maintaining the geometry of the
crystalline supercells unchanged to see what the effect of the pressure
“looking the other way” (negative pressures) would have on the electronic
properties, and we have accomplished that. There are many directions to go
from here and hopefully some will be followed as an alternative to studying
the effects that positive pressures have on the physical properties of solids.
The developments due to chemically etching alloys is an active field at the
experimental level. It is likely that as it progresses new alternatives will
be found to consider the effect of negative pressures on solids, in general,
and in particular on bismuth.

Superconductivity is an appealing phenomenon to be studied, but it requires to
calculate the vibrational density of states and the pairing strength proposed
by Cooper to reach conclusions. These calculations will show whether the
expansion becomes a promising process to find new superconductors, with higher
transition temperatures, for the betterment of mankind.

Acknowledgments

F.B.Q. and D.H.-R. acknowledge Consejo Nacional de Ciencia y Tecnología
(CONACyT) for supporting their graduate studies. I.R. thanks PAPIIT, DGAPA-
UNAM, for his postdoctoral fellowship. A.A.V., R.M.V. and A.V. thank DGAPA-
UNAM (PAPIIT) for continued financial support to carry out research projects
under Grants No. IN104617 and IN116520. María Teresa Vázquez and Oralia
Jiménez provided the information requested. Alberto Lopez and Alejandro Pompa
assisted with the technical support and maintenance of the computing unit at
IIM-UNAM. Simulations were partially carried at the Computing Center of DGTIC-
UNAM through project LANCAD-UNAM-DGTIC-131.

## Declarations

  * •

Funding: Financial support was received from DGAPA-UNAM (PAPIIT) under
projects No. IN104617 and IN116520.

  * •

Conflict of interest/Competing interests: The authors have no competing
interests to declare that are relevant to the content of this article.

  * •

Ethics approval: Not applicable.

  * •

Consent to participate: Not applicable.

  * •

Consent for publication: The authors gave their consent for the publication of
this manuscript.

  * •

Availability of data and materials: The datasets analysed during the current
study are available from the corresponding author on reasonable request.

  * •

Code availability: Materials Studio is a licensed software acquired from
Dassault Systèmes BIOVIA.

  * •

Authors’ contributions: Ariel A. Valladares, Alexander Valladares and Renela
M. Valladares conceived this research and designed it with the participation
of Flor B. Quiroga, David Hinojosa-Romero and Isaías Rodríguez. All the
simulations were done by Flor B. Quiroga and David Hinojosa-Romero. All
authors discussed and analyzed the results. Ariel A. Valladares wrote the
first draft and the other authors enriched the manuscript.

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