# First-principles Investigation into the Heterostructures and Moiré Bilayers
of Transition Metal Dichalcogenides (TMDCs)

Yu-Hsiu Lin  Department of Chemical Engineering & Materials Science, Michigan
State University, East Lansing, Michigan 48824, United States.  Jose L.
Mendoza-Cortes<EMAIL_ADDRESS>Department of Chemical Engineering &
Materials Science, Michigan State University, East Lansing, Michigan 48824,
United States.  Department of Physics, Michigan State University, East
Lansing, Michigan 48824, United States.

###### Abstract

Layered two-dimensional (2D) materials unveil exceptional characteristics
unseen in their individual counterparts, opening avenues for exploring a vast
array of materials. In this study, we focus on MX2 transition metal
dichalcogenides (TMDCs), where M is either Mo or W and X is S, Se, or Te,
utilized as foundational elements for creating both hetero- and homo-bilayers
with diverse stacking configurations. Recent scientific interest has surged in
the study of twisted homogenous bilayers, particularly due to the discovery
that specific “magic angles” between layers can drastically transform material
properties. We emphasize MX2 TMDC bilayers with such twisted angles in this
work, as they readily form Moiré patterns, which are pivotal in understanding
and harnessing these unique material behaviors. In the exploration of
heterostructures, we meticulously computed 30 distinct combinations derived
from six types of MX2 materials, incorporating two varieties of stacking
configurations. These computations reveal that the choice of building blocks
and stacking arrangements not only influences the stability of the
heterostructures but also significantly impacts their band gap energy (Eg). A
notable finding within this array of combinations is the MoTe2/WSe2
heterostructure, which, when shifted by 60°, uniquely exhibits a direct Eg,
highlighting its potential for novel applications in material science. In the
realm of homobilayers, our investigation encompassed both fully relaxed and
low-strain scenarios, each examined under a variety of stacking styles and
twisting angles. We observed that in fully optimized bilayers, the band gap
energy (Eg) is notably influenced by the stacking alignment. Intriguingly,
under low-strain conditions with twisted layers, materials such as 1T-MoS2,
WS2, and WSe2 demonstrated the remarkable ability to exhibit either direct or
indirect Eg at specific angles of twist. Additionally, 1T-MoS2 exhibited a
fascinating versatility, showing the potential to transition between
semiconductor and conductor states, underscoring the rich diversity of
electronic properties these homobilayers can exhibit. The identified “magic
angles”, specifically 17.9° and 42.1° (which correspond to 77.9° and 102.1° ,
respectively), in twisted bilayers of WS2 and WSe2, have been found to create
highly symmetrical Moiré patterns. These patterns are pivotal in leading to a
direct band gap energy (Eg). Intriguingly, the magnitude of Eg in these
structures can be finely tuned through varying the twist angles. Additionally,
the degree of twist significantly influences the flatness of electronic bands.
In comparison to monolayers, most TMDC bilayers exhibit enhanced stability, a
trait that can also be tailored through the strategic selection of specific
building blocks. The use of heterostructures and carefully controlled twisted
angles emerges as a remarkably potent approach in material engineering,
offering the ability to manipulate a multitude of electronic behaviors in
these advanced materials.

##  I Introduction

Layered two-dimensional (2D) materials exhibit unique characteristics distinct
from their bulk counterparts.[1] The presence of weak van der Waals (vdW)
interactions between these layers permits stacking in non-equilibrium
configurations, unveiling a plethora of fascinating phenomena,[2, 3]. This
versatility is not confined to heterostructures alone; homogeneous 2D
materials also stand at the forefront of scientific inquiry for their ability
to manipulate a wide range of properties. The field, abundant with a diverse
array of building blocks and customization strategies, still holds many
secrets yet to be uncovered. In navigating this vast landscape, first-
principles calculations emerge as a crucial tool, providing efficient and
insightful guidance in the ongoing research within this domain.

Figure 1: Core schematic concepts of our study, which shows the construction
of bilayer structures, followed by the identification of potential target
candidates. The final stages entail comprehensive calculations of various
geometrical configurations and the associated properties of these bilayers.

Within the vast landscape of van der Waals (vdW) heterostructures, a variety
of layered materials such as graphene, boron nitride (BN)[4], transition metal
dichalcogenides (TMDCs)[5, 6], and metal carbides and nitrides (MXenes) have
been extensively utilized as fundamental building blocks. In this context,
even simple bilayer configurations of TMDCs emerge as a robust platform for
engineering applications, given their shared properties and symmetries.
Notably, in certain bilayer TMDC heterostructures, the generation of direct
band gaps (Eg) has been observed, paving the way for groundbreaking
advancements in optoelectronics[7, 8]. The role of symmetry is pivotal in
determining the electronic characteristics of these materials. Transitioning
TMDCs from their bulk form to 2D structures often involves breaking inversion
symmetry, which can lead to fascinating changes in band structure, including
the lifting of spin degeneracy[9, 10, 11]. Moreover, the hybridization of the
density of states from different layers fosters interlayer charge transfer.
This charge transfer plays a crucial role in creating an electric dipole
moment, contributing significantly to the unique electronic properties of
these heterostructures[12].

Beyond heterostructures, homogeneous layered transition metal dichalcogenides
(TMDCs) have been shown to display a wide range of electronic behaviors,
heavily influenced by their geometric arrangements[13, 14]. For example,
bilayer WS2 has been observed to alternate between direct and indirect band
gaps (Eg) depending on its stacking configuration. However, comprehensive
studies exploring the full gamut of TMDC material families across various
stacking geometries remain limited. Additionally, the interlayer coupling in
2D layered materials is not only sensitive to stacking types but also to
twisted angles[15, 16]. The discovery of exceptional superconductivity in
twisted bilayer graphene (TBG) at a specific “magic angle” of 1.1° has sparked
significant interest in the study of twisted bilayer structures[17, 18]. This
research has since expanded, exploring the topological and electronic
properties of TBG and similar systems[19, 20, 21]. At these magic angles,
symmetric superlattices can form, often characterized by Moiré patterns which
create spatial confinement and enhance interlayer coupling. This leads to a
flattened band structure and modulates the electronic properties, a phenomenon
also observed in double bilayer graphene[22, 23]. Given their hexagonal
symmetry and van der Waals (vdW) interlayer interactions, TMDCs are postulated
to offer similar opportunities for property manipulation through twisting of
homobilayers[24, 25]. The optoelectronic behavior of MoS2, for instance, has
been closely linked to its twist angles[26]. Unlike TBG, where flat bands
occur within a narrow range of angles, TMDCs are expected to exhibit flat
bands across a broader spectrum of twisted angles[27, 28, 29]. This makes
TMDCs particularly versatile for engineering band structures. Therefore,
twisting the layered configurations of homogeneous TMDC bilayers presents a
promising approach to effectively modify their intrinsic properties.

In this study, MX2 transition metal dichalcogenides (TMDCs), where M is Mo or
W and X is S, Se, or Te, were selected as the foundational elements for
constructing both hetero- and homo-bilayers, aiming to unearth novel
phenomena. A significant observation is that most bilayers demonstrate
enhanced stability compared to their monolayer counterparts. This work delves
into how the choice of building blocks influences this stability. Initially,
we analyzed combinations of these TMDCs in different stacking configurations,
focusing on their band gap energy (Eg) and band structures. This was followed
by a comparison between two types of stacking in homobilayers and their
corresponding monolayers, setting the stage for a discussion on twisted
homobilayers. In the context of heterobilayers, it was found that the
selection of building blocks and stacking types critically influences the
electronic properties. A noteworthy discovery was the direct Eg in the
MoTe2/WSe2 heterobilayer. For homobilayers, while stacking types present a
method to adjust properties, twisting emerges as a more dynamic variable for
engineering interlayer couplings. Within specific ranges of twisted angles,
homobilayers exhibiting symmetric Moiré patterns can induce regular shifts in
Eg, toggle between types of Eg, and affect the flatness of electronic bands.
This research paves the way for further exploration of innovative materials,
with potential applications in energy conversion, electronics, and
optoelectronics, marking a significant advancement in the field of material
science.

Figure 2: Comparative analysis of the band gap energy (Eg) shown for 15
bilayer TMDC heterostructures, each evaluated at either a 0 or 60-degree
shift. Hollow symbols denotes structures with an indirect Eg, while filled
symbols indicate those with a direct Eg. In this plot, the 1H phase of MoS2 is
utilized, whereas Figure S2 incorporates the 1T phase of MoS2, highlighting
the variations in Eg due to different structural phases of the same material.
Figure 3: Top and side views of transition metal dichalcogenides (TMDCs)
bilayers are shown. Panels (a-b) display the A-A and A-B stacking arrangements
in the 1T phase of TMDCs, while panels (c-d) depict the A-A and A-B stacking
in the 1H phase of TMDCs. Panel (e) shows the comparison of the band gap
energy (Eg) for six fully optimized TMDCs, showcasing their properties in
monolayer form as well as in A-A and A-B stacked bilayer configurations.
Hollow symbols are used to indicate structures with an indirect Eg, and filled
symbols represent those with a direct Eg. It’s noteworthy that this plot
utilizes 1H-MoS2, in contrast to Figure S3, which presents the corresponding
data for the 1T phase of MoS2, thus providing a comparative understanding of
the influence of structural phases on Eg.  Figure 4: Geometries and band
structures of bilayer WSe2 at various twisted angles are shown for (a) 0° ,
(b) 17.9° , (c) 27.8° , (d) 32.2° , (e) 42.1° , and (f) 60° . Each panel
provides both a top view and a side view of the bilayer geometries (shown in
the bottom left of each section). For a broader perspective, analogous
analyses for other materials including 1H-MoS2, 1T-MoS2, MoSe2, MoTe2, WS2,
and WTe2 are comprehensively presented in Figures S7-S12. This extensive
representation allows for a detailed comparison across a range of materials
and twisted angles, offering a deeper understanding of the relationship
between geometric configurations and electronic properties in these layered
structures.

##  II Methodology

The methodology of our study is bifurcated into two segments: the
establishment of prototypes and density functional theory (DFT) computations.
In the initial phase, we employed a Python script, adapted from existing
literature, to assemble monolayers into both heterostructures and homogeneous
bilayers[30]. This script was enhanced to control not only the rotational
angle and the size of the unit cell but also the vertical distance between
layers. It also calculated the corresponding strain as the structures were
constructed. For the prototypes of the fully relaxed heterostructures and
homogeneous bilayers, we confined the rotational angles to either 0° or 60° ,
in alignment with the hexagonal symmetry characteristic of the TMDCs studied.
We also considered multiple unit cell sizes for each angle, ensuring the total
number of atoms remained below 30 to accommodate computational limitations.
For the initial models of low-strain, yet symmetrized homogeneous bilayers, we
explored rotational angles ranging from 0° to 120° , in increments of 0.1° .
Structures exhibiting a strain greater than 0.001 were screened out. The
remaining structures were then symmetrized using FINDSYM, adhering to a
tolerance level of 0.001 for the lattice and 0.01 for atomic positions[31].
The size of these symmetrized structures was restricted to a maximum of 40
atoms per cell, balancing the need for detailed structural representation with
computational feasibility.

In the second phase of our study, the previously established prototypes
underwent geometric optimization and property analysis using unrestricted and
periodic density functional theory (DFT). We employed the HSE06 hybrid
functional[32, 33], as implemented in the CRYSTAL17 software[34], and
integrated it with the D3 dispersion correction[35]. For the Gaussian basis
sets, the triple-zeta valence with polarization (TZVP)[36] was selected.
During geometric optimization, we conducted full optimization for both
heterostructures and homogeneous bilayers. For the low-strain but symmetrized
homogeneous bilayers, a partial optimization was performed, keeping the unit
cell size constant. The optimization process adhered to stringent convergence
criteria, with an RMS force threshold of 1.54 × 10–2 eV/Å, a maximum force of
2.31 × 10–2 eV/Å, an RMS displacement of 6.35 × 10–4 Å, and a maximum
displacement of 9.53 × 10–4 Å. To extract detailed electronic information in
the property calculations, we projected the first Brillouin zone ($\approx$
$2\pi$ × $1/60$ Å–1) onto a dense Pack-Monkhorst k-mesh grid ($k_{a}$ ×
$k_{b}$ × $k_{c}$). The mesh grid employed much higher shrinking factors (with
$ak_{a}$ $\geqq$ 60, $bk_{b}$ $\geqq$ 60, $ck_{c}$ $\geqq$ 60) compared to
those used in the geometric optimization phase (where $ak_{a}$ $\geqq$ 40,
$bk_{b}$ $\geqq$ 40, $ck_{c}$ $\geqq$ 40).[37] This approach ensured the
acquisition of high-resolution data essential for a comprehensive
understanding of the electronic properties of these complex systems.

The calculation of the binding energy per atom ($E_{BE}$) for the bilayers in
our study is performed using the following equation:

$E_{BE}=2\frac{E_{BL}}{N_{BL}}–\frac{E_{ML1}}{N_{ML1}}–\frac{E_{ML2}}{N_{ML2}}$
(1)

In this equation, $E_{BL}$ denotes the total energy per unit cell of the
bilayer, while $E_{ML1}$ and $E_{ML2}$ represent the total energies per unit
cell of the first and second monolayer building blocks, respectively. The
terms $N_{BL}$, $N_{ML1}$, and $N_{ML2}$ correspond to the number of atoms in
the unit cell of the bilayer and the first and second monolayers,
respectively.

##  III Discussion

The foundational elements of this research are succinctly encapsulated in
Figure 1, which sketches how the qualified materials are built, screened, and
simulated. The ensuing discussion is divided into different categories of
layered materials. These include conventionally stacked hetero- and homo-
bilayers, as well as twisted homobilayers.

In this work, we systematically explored TMDC heterostructures due to their
remarkable material properties and structural versatility. Utilizing six MX2
TMDCs (M = Mo, W; X = S, Se, Te) as building blocks, we formed 15 bilayer
heterostructures. These were further categorized into two stacking types: 0°
and 60° shifts, resulting in 30 unique structures. Each structure underwent
comprehensive geometric optimization and property analysis. Stability, a key
factor, was assessed using equation (1), which calculates the binding energy
by comparing the energy of bilayers with their constituent monolayers.
According to Figure S1, most heterobilayers exhibit negative binding energy,
indicating greater energetic favorability than separate monolayers. Notably,
bilayers with WTe2 tend to have positive binding energy, suggesting
thermodynamic instability. Conversely, bilayers with MoTe2 show stronger
binding, highlighting how the choice of building blocks significantly
influences stability. The stacking method slightly alters the energy, as seen
in the Eg analysis of these heterostructures in Figure 2. Although Eg varies
with stacking in all heterostructures, those with 1T-MoS2 consistently exhibit
conductive properties, as Figure S2 illustrates, underscoring the impact of
bonding types on electronic characteristics. Remarkably, MoTe2/WSe2 can toggle
between indirect and direct Eg ( 1.04 eV), dependent on the stacking
arrangement. These findings offer valuable insights into constructing TMDC
heterostructures and tailoring their properties for electronic applications.

Figure 5: Variation in binding energy as a function of twisted angles for six
twisted bilayer cases: (a) MoS2, (b) MoSe2, (c) MoTe2, (d) WS2, (e) WSe2, and
(f) WTe2. The red and gray dashed lines denote the binding energies for A-A
and A-B stacked homobilayers, respectively, providing a reference for
comparison against the fluctuating binding energies observed in the twisted
bilayers.  Figure 6: Variation in band gap energy (Eg) across a range of
twisted angles for six twisted bilayer materials: (a) MoS2, (b) MoSe2, (c)
MoTe2, (d) WS2, (e) WSe2, and (f) WTe2. The graph captures the changes in Eg
as a function of the twisting angle in these bilayer structures.

TMDC heterostructures and homogeneous bilayers both offer versatile avenues
for engineering material properties. This discussion first addresses fully
optimized TMDC bilayers, followed by those with slight strain. We considered
six MoX2 TMDCs, each with A-A and A-B stacking types. In A-A stacked bilayers,
viewed from above, the cations of each monolayer overlap, whereas in A-B
stacking, cations of one layer overlap with anions from the other, as depicted
in Figure 3(a)-(d). Compared to their monolayer counterparts, A-A and A-B
stacked bilayers of TMDCs exhibit unique electronic properties, as summarized
in Figure 3(e). Key observations include: First, the Eg of these MX2 TMDCs
spans from 0.8 to 2.7 eV. Bonding types significantly influence these
properties. For instance, both monolayer and bilayer forms of 1T-MoS2
consistently exhibit conductive behavior, as shown in Figure S3, a
characteristic attributable to its distinctive structure and interlayer
coupling, differing from 1H-TMDCs. Second, the Eg typically follows the trend:
A-B stacking bilayer $<$ A-A stacking bilayer $<$ monolayer. Third, monolayers
of 1H-MoS2, MoSe2, and WS2 present direct Eg, unlike others. This evolution
from monolayer to bilayer demonstrates significant electronic changes, hinting
at the vast potential inherent in multilayered TMDCs for material engineering.

In this study, we shifted our focus from conventional A-A and A-B stacking to
the more intricate relationships between structure and property in homogeneous
TMDC layers, facilitated by varying twisted angles. To refine the scope of our
investigation, we initially screened twisted bilayers based on strain
magnitude, setting a threshold of less than 0.001 post-formation. The
comprehensive trends of strain versus twisted angles for all six TMDCs are
depicted in Figure S4. These trends are further elaborated in Figure S5, which
zooms into regions showcasing geometries that meet the strain criteria. This
screening approach is predicated on the hypothesis of an inverse relationship
between structural stability and strain. Subsequently, the low-strain
structures underwent symmetrization, and those with more than 40 atoms per
unit cell were excluded to optimize computational efficiency. The selected
final candidates, adhering to these criteria, are highlighted in Figure S5,
marking a strategic narrowing down of potential structures for detailed
analysis. This process not only aids in identifying structurally viable
configurations but also sets the stage for a deeper exploration of the nuanced
interplay between twisted angles and material properties in TMDC bilayers.

Simultaneously, Moiré patterns emerged prominently among these selected
candidates. Subsequently, these structures underwent symmetrical optimization
while maintaining fixed unit cell sizes, preserving both low strain and the
captivating Moiré patterns capable of eliciting novel material properties. For
instance, consider bilayer WSe2: Figure 4 vividly illustrates the evolution of
Moiré patterns across twisted angles from 0° to 60°. Correspondingly, band
structures exhibit notable variations, with pronounced flattening due to Moiré
patterns’ influence on interlayer coupling interactions. This enhanced
flatness facilitates energy conversion by mitigating the impact of crystal
momentum. Furthermore, the thermodynamic stability of these twisted bilayers
was meticulously examined, with Figure 5 showcasing the binding energy as a
function of twisted angles.

In essence, the binding energy of twisted bilayers exhibits a strong
correlation with their geometry, oscillating between the values observed for
A-A and A-B stacked bilayers. Notably, except for the WTe2 bilayer with a 0°
twist, the majority of twisted homobilayers display a negative binding energy,
indicating their spontaneous formation from monolayer counterparts.
Intriguingly, 1T-MoS2 is predicted to experience a substantial reduction in
binding energy at specific twist angles. These findings underscore the role of
van der Waals interactions in enabling non-equilibrium stacking
configurations, providing insights into the construction of layered
twistronics. Additionally, Figure 6 visually depicts how the bandgap (Eg) of
various TMDC homobilayers varies with twisted angles, revealing distinct
behaviors among different bilayers.

For example, the behavior of 1T-MoS2 is particularly intriguing, as it
exhibits a transition between a conductor and a semiconductor depending on the
twisted angle, as depicted in Figure 6(a). Remarkably, at a twist angle of
92.2°, 1T-MoS2 even achieves a direct bandgap (Eg) with a value of 0.232 eV.
In contrast, MoSe2 appears to be less responsive to changes in the Eg-twist
relationship, maintaining an indirect bandgap within a limited range of
variation (1.79 - 1.81 eV) across different twist angles, as shown in Figure
6(b). Conversely, the other four TMDCs (MoTe2, WS2, WSe2, and WTe2) exhibit
symmetric Eg-twist relationships, as illustrated in Figure 6(c)-(f), with
respect to 60° twist angles. These TMDCs exhibit maximum Eg values at 0° twist
and minimum Eg values at 60° twist. Notably, both WS2 and WSe2 undergo a
transition from indirect to direct bandgaps at specific twist angles, namely
17.9° (equivalent to 102.1° ) and 42.1° (equivalent to 77.9° ), respectively.
These findings highlight that the Eg of these four twisted TMDCs fluctuates
between the Eg values observed in their A-A and A-B stacked bilayers, as
demonstrated in the fully optimized bilayers section.

The tunability of Eg is evident across a range of twist angles, as: 1H-MoS2:
1.87 $\sim$ 2.16 eV; 1T-MoS2: 0.00 $\sim$ 0.80 eV; MoSe2: 1.79 $\sim$ 1.81 eV;
MoTe2: 0.81 $\sim$ 1.49 eV; WS2: 1.62 $\sim$ 2.28 eV; WSe2: 1.55 $\sim$ 2.10
eV; WTe2: 0.92 $\sim$ 1.63 eV; In short, twisted bilayers offer a
sophisticated approach for finely tuning electronic behavior. The presence of
highly symmetrical Moiré patterns is believed to induce diverse electronic
interactions in low-strain, homogeneous bilayer materials.

##  IV Conclusion

In summary, the manipulation of stacking and twist in bilayer MX2 TMDCs (M =
Mo, W; X = S, Se, Te) takes advantage of weak vdW interactions, enabling
highly adjustable material properties.

Beyond the manipulation of Eg magnitude, both hetero- and homo-bilayers offer
detailed property modifications. Among the 30 heterobilayer combinations,
MoTe2/WSe2 exhibits the ability to transition between indirect and direct Eg,
suggesting potential in energy conversion. In the case of homobilayers,
precise twist, rather than conventional stacking, enables fine-tuning of
structure-property relationships. Diverse electronic behaviors emerge due to
varying interlayer coupling interactions, especially in bilayers with highly
symmetric Moiré patterns and minimal strain.

In essence, Moiré bilayers exhibit flattened band structures, blurring the
boundary between direct and indirect Eg and enhancing energy conversion.
Notably, 1T-MoS2 can transition between conductor and semiconductor, with the
ability to possess either direct or indirect Eg by varying twisted angles.
Magic angles of 17.9° and 42.1° induce direct Eg in WS2 and WSe2 Moiré
bilayers. Additionally, the choice of building blocks impacts binding energy,
with MoTe2-based bilayers being more stable and WTe2-based bilayers less
stable. In conclusion, sophisticated twisting and heterostructures serve as
valuable engineering tools for designing innovative materials with diverse
scientific applications. This concise work provides crucial insights into
bilayers, laying the foundation for future research on multilayer systems.

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##  V Supporting Information

Figure S 1: Binding energy are shown for 15 bilayer TMDCs heterostructures
with two types of stacking, either 0 or 60 degrees of shifting. Figure S 2:
The Eg of 15 bilayer TMDCs heterostructures compared with two types of
stacking, either 0 or 60 degrees of shifting. Hollow signs represent indirect
Eg while filled sign stands for direct Eg. Compared to this plot where 1T-MoS2
was applied, 1H-MoS2 was applied in Figure 2. Figure S 3: The Eg are shown
of 6 fully optimized TMDCs in forms of monolayer, A-A stacking bilayer, and
A-B stacking bilayer. The MoS2 in this plot belongs to 1T instead of 1H.
Hollow signs represent indirect Eg while filled signs stand for direct Eg.
Figure S 4: The overall fluctuation of strain according to the twisted
angles of 1H bilayers are shown for the cases of (a) MoS2, (b) MoSe2, (c)
MoTe2, (d) WS2, (e) WSe2, and (f) WTe2. Figure S 5: The zoom-in fluctuation
of strain below 0.001 according to the twisted angles of 1H bilayers in the
cases of (a) MoS2, (b) MoSe2, (c) MoTe2, (d) WS2, (e) WSe2, and (f) WTe2. The
low-strain twisted TMDCs bilayers being symmetrized and further computed in
this work are dotted in the plots. Figure S 6: (a) The overall fluctuation
of strain according to the twisted angles of bilayers in the cases of 1T-MoS2.
(b) The zoom-in fluctuation of strain below 0.001 according to the twisted
angles of bilayers in the cases of 1T-MoS2. The low-strain twisted TMDCs
bilayers being symmetrized and further computed in this work are dotted in the
plots. Figure S 7: The schematic geometries and band structures of bilayer
1H-MoS2 with the twisted angles of (a) 0° , (b) 27.8° , (c) 38.2° , (d) 60° ,
and (e) 98.2° . These include the top view and the side view of geometries
(bottom left). Figure S 8: The schematic geometries and band structures of
bilayer 1T-MoS2 with the twisted angles of (a) 0° , (b) 27.8° , (c) 38.2° ,
(d) 60° , and (e) 92.2° . These include the top view and the side view of
geometries (bottom left). Figure S 9: The schematic geometries and band
structures of bilayer MoSe2 with the twisted angles of (a) 0° , (b) 27.8° ,
(c) 32.2° , and (d) 60° . These include the top view and the side view of
geometries (bottom left). Figure S 10: The schematic geometries and band
structures of bilayer MoTe2 with the twisted angles of (a) 0° , (b) 17.9° ,
(c) 27.8° , (d) 32.2° , and (e) 60° . These include the top view and the side
view of geometries (bottom left). Figure S 11: The schematic geometries and
band structures of bilayer WS2 with the twisted angles of (a) 0° , (b) 17.9° ,
(c) 27.8° , (d) 32.2° , (e) 42.1° , and (f) 60° . These include the top view
and the side view of geometries (bottom left). Figure S 12: The schematic
geometries and band structures of bilayer WTe2 with the twisted angles of (a)
0° , (b) 17.9° , (c) 27.8° , (d) 32.2° , (e) 42.1° , and (f) 60° . These
include the top view and the side view of geometries (bottom left).


# Grey solitons in the ultracold fermions at the full spin polarization

Pavel A. Andreev<EMAIL_ADDRESS>Faculty of physics, Lomonosov
Moscow State University, Moscow, Russian Federation, 119991.  Peoples
Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street,
Moscow, 117198, Russian Federation

###### Abstract

A minimal coupling quantum hydrodynamic model of spin-1/2 fermions at the full
spin polarization corresponding to a nonlinear Schrodinger equation is
considered. The nonlinearity is primarily caused by the Fermi pressure. It
provides an effective repulsion between fermions. However, there is the
additional contribution of the short-range interaction appearing in the third
order by the interaction radius. It leads to the modification of the pressure
contribution. Solitons are considered for the infinite medium with no
restriction on the amplitude of the wave. The Fermi pressure leads to the
soliton in form of the area of decreased concentration. However, the center of
solution corresponding to the area of minimal concentration has nonzero value
of concentration. Therefore, the grey soliton is found. Soliton exist if the
speed of its propagation is below the Fermi velocity.

degenerate fermions, hydrodynamics, non-linear Schrodinger equation, dark
soliton, Fermi pressure.

##  I Introduction

Solitons are fundamental nonlinear structures existing in various physical
systems including the quantum gases Alotaibi PRA 17 , Andreev MPL B 12 , Wang
NJP 14 , Andreev 2009 , Katsimiga NJP 17 , Shamailov PRA 19 , Syrwid PRA 17 ,
Syrwid PRA 18 , Wang Dai PRA 18 , Zhao PRA 20 . Vortexes, shock waves, and
skyrmions are among main nonlinear objects considered in atomic quantum gases.
Solitons and other nonlinear phenomena are well studied in the bosonic atoms
experimentally and theoretically. The theoretical approach is mainly based on
the mean-field nonlinear Schrodinger equation called the Gross-Pitaevskii
equation which describes the bright and dark solitons in the Bose-Einstein
condensates (BECs), where the nonlinearity is caused by the interparticle
interaction Dalfovo RMP 99 . The form of solitons is related to the sign of
interaction, so the bright and dark solitons appears in the attractive and
repulsive BECs, correspondingly. For instance, the experimental study of the
dark solitons in BECs is presented in Ref. Becker Nat Ph 08 . Majority of
study of fermionic gases are focused on the superfluid phase or BCS state
(Bardeen-Cooper-Schrieffer state), where pairs of fermionic atoms with
opposite spins and momentum form the Cooper pairs and demonstrate the boson-
like behavior Giorgini RMP 08 . The dark solitons in superfluid Fermi gases
are considered in Refs. Antezza PRA 07 , Scott PRL 11 , Liao PRA 11 ,
particularly, the Bogoliubov-de Gennes equations are used in Antezza PRA 07 .
A heavy soliton in a fermionic superfluid is experimentally observed in Ref.
Yefsah Nat 13 . Solitons in the superfluid Fermi gases are considered in terms
of the nonlocal generalization of the Ginzburg-Landau model Barkman PRR 20 ,
following Ref. Buzdin PLA 97 . Here, the basic and fundamental solitons are
considered in degenerate fermionic atoms with the full spin polarization. The
spin-1/2 atoms are chosen, but the same analysis is correct for the fermions
with higher spins. The presented theoretical work is based on the quantum
hydrodynamics which is straightforwardly derived from the microscopic many-
particle Schrodinger equation (from the full quantum theory). The minimal
coupling model of fermions is composed of two hydrodynamic equations: the
continuity and Euler equation. These hydrodynamic equations allow to obtain
the corresponding nonlinear Schrodinger equation, where the nonlinearity is
mainly caused by the Fermi pressure. It is nonlinearity of fractional degree
$7/3$. However, the interaction between fermions gives the additional
nonlinearity Andreev 2001 .

Unpolarized fermions are mostly discussed in literature regime for fermions
Kulkarni PRA 12 , Babadi PRA 12 , Andreev 1912 . If we have system of spin-1/2
Fermi atoms with equal population of the spin-up state and the spin-down state
we can observe interesting phases of matter. They are the Bose-Einstein
condensate of molecules, crossover superfluid, and the BCS state. The spin
orbit coupling can also be engineered in the unpolarized fermions.

If we consider bosons being in the Bose-Einstein condensate state it can be
described by the Gross-Pitaevskii equation. The interaction appears in the
first order by the interaction radius in term of the hydrodynamic derivation
of the Gross-Pitaevskii equation. Or it can be interpreted as the s-wave
scattering in terms of the scattering theory. For the polarized fermions,
there is no contribution of the interaction in the first order by the
interaction radius (FOIR), due to the antisymmetry of the wave function.
Hence, main selfaction of the fermion fluid comes from the Fermi pressure.
However, the additional contribution of the interaction appearing in the third
order by the interaction radius (TOIR) can be derived Andreev 2001 . This
contribution can be interpreted via the p-wave scattering in terms of the
scattering theory Parker PRA 12 , Roth PRA 01 , Roth PRA 02 .

The spin polarized fermions is the system where all fermions occupy the single
spin state. This systems shows rather avaricious phase. Nevertheless, it also
demonstrates some interesting fundamental nonlinear phenomena. To some extend
the system of polarized degenerate fermions can be describe by the effective
macroscopic single particle wave function $\Phi(\textbf{r},t)$. This
possibility follows from the quantum hydrodynamic equations restricted by the
particle density and the momentum density evolution. However, complete
description of polarized fermions requires the momentum current evolution,
which is the kinetic pressure of polarized fermions. The pressure evolution
equation gives more accurate value of the speed of sound Andreev 2001 ,
Andreev 1912 , Tokatly PRB 99 . Nevertheless, the minimal coupling model based
on the particle density and the momentum density shows good qualitative
description of fermions.

This paper is organized as follows. In Sec. II the quantum hydrodynamics is
presented in two regimes: the mean-field approximation and up to the TOIR
approximation. In Sec. III solution of the hydrodynamic equations in the one-
dimensional regime in the form of the grey soliton is obtained by the Sagdeev
potential method. In Sec. IV a brief summary of obtained results is presented.

##  II Quantum hydrodynamic equations

Here, we present two quantum hydrodynamic models for the degenerate fermions
being in the same spin state (the regime of the full spin polarization). The
first model is obtained in the FOIR approximation, where the interaction gives
the zero contribution. The second model contains the contribution of the
interaction in the TOIR approximation.

###  II.1 First order by the interaction radius: A minimal coupling
hydrodynamic model for the full spin polarization

Nonzero contribution of the interaction in the first order by the interaction
radius exists for nonpolarized or the partially polarized systems of fermions.
However, the fully polarized fermions have zero contribution of interaction in
this case.

In all regimes we have same form of the continuity equation:

$\partial_{t}n+\nabla\cdot(n\textbf{v})=0,$ (1)

In the FOIR approximation we also have the Euler (momentum balance) equation

$mn(\partial_{t}+\textbf{v}\cdot\nabla)\textbf{v}-\frac{\hbar^{2}}{2m}n\nabla\frac{\triangle\sqrt{n}}{\sqrt{n}}$
$+\nabla p=-n\nabla V_{ext},$ (2)

where $p$ is the Fermi pressure

$p=\frac{(6\pi^{2})^{\frac{2}{3}}\hbar^{2}n^{\frac{5}{3}}}{5m^{2}}.$ (3)

Minimal coupling assumes the application of the continuity and Euler equation
with no account of the pressure evolution, but application of the equation of
state for the reduction of the pressure evolution to the concentration
evolution.

Equations (1)-(3) correspond to the nonlinear Schrodinger equation for
fermions at the potential velocity field Andreev 2001 :

$\imath\hbar\partial_{t}\Phi=\biggl{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+\frac{(6\pi^{2}n)^{\frac{2}{3}}\hbar^{2}}{2m}+V_{ext}\biggr{)}\Phi,$
(4)

where $n=\mid\Phi\mid^{2}$. The effective macroscopic wave function $\Phi$ is
defined via the hydrodynamic wave functions $n(\textbf{r},t)$ and
$\textbf{v}(\textbf{r},t)$:

$\Phi(\textbf{r},t)=\sqrt{n}e^{\imath m\phi/\hbar},$ (5)

where $\textbf{v}=\nabla\phi$. The contribution of the Fermi pressure (3) is
presented by the second term on the right-hand side of equation (4).

Equation similar to NLSE (4) are used in literature Adhikari PRA 04 ,Adhikari
PRA05 , Adhikari JPB05 , Adhikari NJP06 , Bludov PRA06 , Rizzi PRA08 ,
Maruyama PRA08 , Karpiuk PRA06 . Different forms have different
justifications. Equation (4) is justified via the quantum hydrodynamics.
Moreover, the partial or full spin polarization is not included there.

Equations (1) and (2) are applied below to consider the possibility of
solitons in the systems of neutral atomic degenerate fermions. To complete the
description of model we present the hydrodynamics containing the contribution
of the interaction between fermions with the same spin polarization. Absence
of the interaction in equations (2) and (4) shows that the equilibrium
condition cannot be reached in such systems. However, we have interaction
between fermions which is presented below. It provides the additional transfer
of the momentum and a mechanism of reaching of the equilibrium state.

###  II.2 Hydrodynamic equations and nonlinear Schrodinger equation for
fermions with the interaction included up to the TOIR approximation

Derivation of the macroscopic equations by the many-particle quantum
hydrodynamics method Maksimov QHM 99 , Andreev 2005 , MaksimovTMP 2001 ,
Andreev 2007 , Andreev LP 19  shows that the hydrodynamic equations appear
first. Next, in some simplified regimes the nonlinear Schrodinger equation can
be found Andreev 2001 , Maksimov QHM 99 .

The nonlinear Schrodinger equation can be derived in the chosen approximation
Andreev 2001

$\imath\hbar\partial_{t}\Phi=\biggl{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+\frac{(6\pi^{2}n)^{\frac{2}{3}}\hbar^{2}}{2m}+V_{ext}-\frac{4}{5}g_{2}(6\pi^{2})^{\frac{2}{3}}n^{\frac{5}{3}}\biggr{)}\Phi,$
(6)

the additional term caused by the interaction is the last term in equation
(6). The additional term is obtained in the TOIR approximation. It contains
the interaction constant $g_{2}$ which is defined via the potential of
interatomic interaction $U$:

$g_{2}=\int r^{2}U(r)d\textbf{r}.$ (7)

The positive interaction constant decreases the pressure. However, the model
is obtained in the weak interaction limit. So, the contribution of interaction
should be small in compare with the Fermi pressure.

Let us present the corresponding hydrodynamic equations. The continuity
equation has same form (1). The Euler equation contains the additional term

$mn(\partial_{t}+\textbf{v}\cdot\nabla)\textbf{v}-\frac{\hbar^{2}}{2m}n\nabla\frac{\triangle\sqrt{n}}{\sqrt{n}}$
$+\nabla p=-n\nabla V_{ext}+\frac{5m^{2}}{2\hbar^{2}}g_{2}\nabla(np),$ (8)

which is the last term in equation (8). The interaction term appear via the
kinetic pressure $p$ Andreev 2001 . The gradient of the kinetic pressure
itself also presented by the last term on the left-hand side of equation (8).
In this paper we use the equation of state in form of the Fermi pressure (3).
Hence, the Euler equation (8) is truncated and its final form corresponds to
the nonlinear Schrodinger equation (6).

The model presented above is obtained for the fermions with the full spin
polarization. Hydrodynamic model of degenerate spin-1/2 fermions with the
partial spin polarization in the mean-field approximation for the interaction
between fermions with different spin projections Andreev LPL 18 .

##  III Large amplitude grey solitons

Let us present the analysis of equations for the solitons obtained up to the
TOIR approximation. Let us consider solitons in uniform infinite medium.
Hence, the symmetry of the system allows to consider the nonlinear waves with
the plane wave front. The wave appears as the one dimensional solution. We
consider the wave propagation in the arbitrary direction and choose the
cartesian coordinates with axis $Ox$ in the direction of the wave propagation.
We seek the stationary solutions of the nonlinear equations. We consider the
steady state in the comoving frame. Hence, all hydrodynamic functions depend
on $\eta=x-ut$ and $u$, where the parameter $u$ is the constant velocity of
the nonlinear solution. The perturbations vanish at
$\eta\rightarrow\pm\infty$.

###  III.1 One dimensional limit of hydrodynamic equations

For the nonlinear plane waves we have the following simplified continuity
equation

$-u\partial_{\eta}n+\partial_{\eta}(nv^{x})=0,$ (9)

where the time derivative $\partial_{t}$ is replaced by $-u\partial_{\eta}$ in
accordance with the variable $\eta$ introduced for the stationary solution.
Similar simplification is made for the Euler equation

$-umn\partial_{\eta}v^{x}+mnv^{x}\partial_{\eta}v^{x}-\frac{\hbar^{2}}{2m}n\partial_{\eta}\frac{\partial_{\eta}^{2}\sqrt{n}}{\sqrt{n}}$
$=-\frac{(6\pi^{2})^{2/3}\hbar^{2}}{2m^{2}}n\partial_{\eta}n^{2/3}+g_{2}\frac{4(6\pi^{2})^{2/3}}{5}n\partial_{\eta}n^{5/3},$
(10)

where the terms placed on the right-hand side are represented via construction
$n\partial_{\eta}n^{a}$ useful for the further transformations, with $a$ is
the arbitrary degree.

The one dimensional continuity equation (9) can be integrated

$n(v^{x}-u)=-un_{0},$ (11)

where the boundary conditions $n(\eta\rightarrow\pm\infty)=n_{0}$, and
$v^{x}(\eta\rightarrow\pm\infty)=0$ are used. Equation (11) allows to express
the velocity field via the concentration

$v^{x}=\frac{u(n-n_{0})}{n}.$ (12)

All terms in the Euler equation (10) are proportional to the concentration, so
we can drop it. Next, the Euler equation can be integrated. As the result we
find

$m\biggl{(}\frac{1}{2}v_{x}^{2}-uv^{x}\biggr{)}-\frac{\hbar^{2}}{2m}\frac{\partial_{\eta}^{2}\sqrt{n}}{\sqrt{n}}$
$+\frac{(6\pi^{2})^{2/3}\hbar^{2}}{2m^{2}}n^{2/3}-g_{2}\frac{4(6\pi^{2})^{2/3}}{5}n^{5/3}$
$=\frac{(6\pi^{2})^{2/3}\hbar^{2}}{2m^{2}}n_{0}^{2/3}-g_{2}\frac{4(6\pi^{2})^{2/3}}{5}n_{0}^{5/3},$
(13)

where all terms existing in equation (10) are placed on the left-hand side
while the right-hand side contains the result of application of the boundary
conditions.

Figure 1:  The Sagdeev potential $V=V(\Delta)$ (16) is demonstrated for the
noninteracting limit for three different values of the dimensionless speed of
the soliton $\alpha=0.2$ (the upper red dotted line), $\alpha=0.1$ (the middle
black continuous line), $\alpha=0.05$ (the lower green dashed line).

We substitute the velocity field (12) in the integrated Euler equation (13).
Moreover, we see that equation (13) contains the second derivative on
$\sqrt{n}$. It shows that we should find solution relatively $\sqrt{n}$.
Equation (13) can be integrated to obtain the ”energy integral” in the
following manner

$\frac{1}{2}(\partial_{\eta}\sqrt{n})^{2}+\tilde{V}_{eff}(\sqrt{n})=0,$ (14)

where the first term can be considered as the effective kinetic energy of
soliton, while $\tilde{V}_{eff}(\sqrt{n})$ is the effective potential energy
called the Sagdeev potential Schamel PF 77 , Witt PF 83 , Mamun PRE 97 , Shah
PoP 10 , Marklund PRE 07 , Akbari-Moghanjoughi PP 17 . The Sagdeev potential
$\tilde{V}_{eff}(\sqrt{n})$ appears in the following form

$\tilde{V}_{eff}(\sqrt{n})=\frac{1}{2}(6\pi^{2})^{2/3}\biggl{(}1+\frac{8mg_{2}}{5\hbar^{2}}n_{0}\biggr{)}(n-n_{0})$
$+\frac{m^{2}u^{2}}{2\hbar^{2}}\biggl{(}n+\frac{n_{0}^{2}}{n}-2n_{0}\biggr{)}-\frac{3}{10}(6\pi^{2})^{\frac{2}{3}}\biggl{(}n^{\frac{5}{3}}+\frac{mg_{2}}{\hbar^{2}}n^{\frac{8}{3}}\biggr{)}$
$-\frac{3}{10}(6\pi^{2})^{2/3}\biggl{(}n_{0}^{5/3}+\frac{mg_{2}}{\hbar^{2}}n_{0}^{8/3}\biggr{)}.$
(15)

Equations can be solved for parameter $\sqrt{n}$, but the traditional form of
the presentation of the results including the zero value of the effective
potential and its first derivative on parameter $\sqrt{n}$ requires to
consider dependence on $\sqrt{n}-\sqrt{n}_{0}$. Let us to choose the
dimensionless form of the chosen parameter
$\Delta\equiv(\sqrt{n}-\sqrt{n}_{0})/\sqrt{n}_{0}$ for the further analysis.
Moreover, the coordinate in the comoving frame $\eta$ can be presented in the
dimensionless form as well $\xi=\eta n_{0}^{1/3}$.

Figure 2: The Sagdeev potential $V=V(\Delta)$ (17) is demonstrated at the
account of the interaction up to TOIR approximation. The Sagdeev potential
$V=V(\Delta)$ is demonstrated for the fixed speed of perturbation
$\alpha=0.03$ for different values of the dimensionless interaction constant
$\Lambda=0$ (the upper red dotted line), $\Lambda=0.01$ (the second from above
black continuous line), $\Lambda=0.1$ (the third from above green dashed
line), $\Lambda=0.3$ (the lower blue dashed line).

###  III.2 Grey soliton in the mean-field approximation

The mean-field approximation corresponds to the FOIR limit of the hydrodynamic
equations. Formally, it can be obtained from equations (14) and (15) at
$g_{2}=0$. we discuss the Sagdeev potential in the dimensionless form.

The dimensionless form of the Sagdeev potential
$V(\Delta)\equiv\tilde{V}_{eff}(\sqrt{n})n_{0}^{-5/3}$ presented in the mean-
field approximation has the following form

$V(\Delta)=\frac{1}{2}\alpha^{2}\biggl{[}(\Delta+1)^{2}+\frac{1}{(\Delta+1)^{2}}\biggr{]}$
$+\frac{1}{2}(\Delta+1)^{2}-\frac{3}{10}(\Delta+1)^{10/3}-\biggl{[}\alpha^{2}+\frac{1}{5}\biggr{]},$
(16)

where $\alpha\equiv u/v_{Fe}$. The dimensionless Sagdeev potential (16) is a
part of the following dimensionless equation
$(1/2)(\partial_{\xi}\Delta)^{2}+V(\Delta)=0$.

Figure 3:  A part of the Sagdeev potential $V=V(\Delta)$ (17) is demonstrated.
It is a part of Fig. (2) which corresponds to the point of crossing of the
Sagdeev potential of the line of zero potential.  Figure 4:  This figure is
similar to Fig. (2), but this figure is obtained for the larger speed of the
soliton. The Sagdeev potential $V=V(\Delta)$ (17) is demonstrated at the
account of the interaction up to TOIR approximation. The Sagdeev potential
$V=V(\Delta)$ is demonstrated for the fixed speed of perturbation $\alpha=0.1$
for different values of the dimensionless interaction constant $\Lambda=0$
(the upper red dotted line), $\Lambda=0.01$ (the second from above black
continuous line), $\Lambda=0.1$ (the third from above green dashed line),
$\Lambda=0.3$ (the lower blue dashed line).

Fig. (1) shows the single illustration of the Sagdeev potential in the mean-
field regime (16). Value $\Delta_{0}\neq 0$ corresponding to $V(\Delta_{0})=0$
shows the amplitude of the soliton . First, we see that $\Delta_{0}$ is
negative. Hence, there is the decrease of concentration in the soliton
$n<n_{0}$. However, $\Delta_{0}$ does not reach value $-1$. Consequently, the
concentration of particles in the soliton is always nonzero $n>0$. The soliton
appears as the area of decreased concentration, which is above the zero value
at the center of soliton. Such solitons are called the gray soliton. While the
dark soliton is the limiting case of the grey soliton with the zero
concentration in its center.

Fig. (1) shows that the increase of the speed of the soliton propagation up to
the Fermi velocity decreases the amplitude of the soliton down to the zero
value at $\alpha\approx 0.6$. Moreover, no solution exists at $\alpha>0.6$.
Obtained behavior shows similarity to the dark soliuton in the BECs, where the
speed of soliton propagation is limited by the Landau critical velocity Busch
PRL 00 .

Dimensionless velocity $\alpha$ is the single parameter in the mean-field
approximation. This dependence is discussed. Further analysis of the
properties of soliton can be made in the TOIR approximation.

Presented here soliton solution for the spin polarized fermions. The spin-0
BECs demonstrate two fundamental solitons in the mean-field regime. Moreover,
the spin-0 BEC show the beyond mean-field bright soliton in the repulsive BEC
regime. The boson-boson and boson-fermion mixtures show some additional
soliton related effects. Particularly, the boson-fermion mixture of the spin-0
BEC and spin-polarized spin-1/2 fermions is considered in Ref. Andreev LP 21 ,
where focus is made on the modification of properties of the beyond mean-field
bright soliton existing in the repulsive BECs under influence of the fermions.
Hence, there is no direct relation between the grey soliton given here and the
fermion part of the relation demonstrated in Ref. Andreev LP 21 .

###  III.3 Generalization of the grey soliton solution up to the TOIR

Complete expression of the dimensionless form of the Sagdeev potential
$V(\Delta)\equiv\tilde{V}_{eff}(\sqrt{n})n_{0}^{-5/3}$ obtained from the
expression (15) derived up to the TOIR approximation can be written in the
following form

$V(\Delta)=\frac{1}{2}\alpha^{2}\biggl{[}(\Delta+1)^{2}+\frac{1}{(\Delta+1)^{2}}\biggr{]}$
$+\frac{1}{2}(\Delta+1)^{2}(1+\Lambda)-\frac{3}{10}(\Delta+1)^{10/3}$
$-\frac{3}{16}\Lambda(\Delta+1)^{16/3}-\biggl{[}\alpha^{2}+\frac{1}{2}(1+\Lambda)-\frac{3}{10}-\frac{3}{16}\Lambda\biggr{]}.$
(17)

Additional terms in compare with the FOIR approximation (16) are proportional
to $\Lambda$.

Contribution of the short-range interaction obtained in the TOIR approximation
in the Sagdeev potential is demonstrated in Figs. (2), (3), and (4). Figs. (2)
and (3) are obtained for the relatively small speed of soliton $\alpha=0.03$.
In this case, there is small modification of the amplitude of soliton under
the change of the interaction constant. So, this modification is demonstrated
in Fig. (3). Fig. (4) presents the Sagdeev potential for the same values of
the interaction constant, but it is obtained for the larger velocity
$\alpha=0.1$. The contribution of interaction is larger in this velocity
regime. Further increase of the velocity $\alpha\rightarrow 1$ gives large
modification of the amplitude under influence of the interaction. This limit
is not presented in figures since it is beyond the area of applicability of
the model, which corresponds to the weak interaction regime.

##  IV Conclusion

Grey soliton has been found in the system of weakly interacting fermions being
in quantum states with the same spin projection. It has been found from the
quantum hydrodynamic equations corresponding to the nonlinear Schrodinger
equation. The soliton has been found and studied within the Sagdeev potential
method. The form of the Sagdeev potential allows to find the amplitude and
width of the soliton. Particularly, it has been found that the concentration
is decreased, but it does not reach the zero value. Thus, the soliton is
classified as the grey soliton. The change of the grey soliton parameters at
the modification of the speed of the soliton and the interaction constant has
been analyzed. We have concluded that the polarized fermions demonstrate the
existence of one kind of soliton, i.e. the soliton of the partial rarification
called the grey soliton. It is in contrast with the Bose-Einstein condensate,
where two kinds of solitons are possible for the uniform medium. The forms of
solitons for the bosons depend on the sign of the interaction between the Bose
atoms. The dark (bright) soliton corresponds to the repulsive (attractive)
interaction. The degenerate fermions are mainly affected by the Fermi pressure
which provides the effective repulsion. So the dark/grey solitons is the
possible structure.

##  V Acknowledgements

Work is supported by the Russian Foundation for Basic Research (grant no.
20-02-00476). This paper has been supported by the RUDN University Strategic
Academic Leadership Program.

##  VI DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study, which is a purely theoretical one.

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Consciência e mecânica quântica: uma abordagem filosófica
Raoni Wohnrath Arroyo

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Harvard University Press

titleQuantum mechanics and experience

titleInterpreting the Many Worlds Interpretation

journaltitleFoundations of science
titleOntological frameworks for scientific theories

PPGFIL UFRN

booktitleProceedings of the 3rd Filomena Workshop
titleBridging the Gap Between Science and Metaphysics, with a Little Help from Quantum Mechanics

titleBack to the question of ontology (and metaphysics)

titleOn physics, metaphysics, and metametaphysics

journaltitleScientiae studia
titleIndistinguibilidade, Não Reflexividade, Ontologia e Física Quântica

Peeters Online Journals

journaltitleLogique et Analyse
titleContradiction, Quantum Mechanics, and the Square of Opposition

noteTradução, introdução e notas por Lucas Angioni
seriesClássicos da Filosofia: Cadernos de Tradução
titleMetafísica, Livros I, II e III

São Paulo

noteTradução, textos adicionais e notas por Edson Bini

Universidade Estadual de Maringá (UEM)

titleO Problema Ontológico da Consciência na Mecânica Quântica
typeDissertação apresentada como requisito parcial para a obtenção do título de Mestre em Filosofia

booktitleQuanta and Mind: Essays on the Connection between Quantum Mechanics and the Consciousness
seriesSynthese Library
titleBetween physics and metaphysics: A discussion of the status of mind in quantum mechanics

titleAgainst `Interpretation': Quantum Mechanics Beyond Syntax and Semantics

journaltitlePrincipia: An international journal of epistemology
titleOn Quantum Mechanics, Phenomenology, and Metaphysical Underdetermination

journaltitleGriot: Revista de Filosofia
titleNotas sobre o misticismo racional de Erwin Schrödinger

journaltitlePhysical Review Letters
titleExperimental Test of Bell's Inequalities Using Time-Varying Analyzers

journaltitleReviews of Modern Physics
titleThe Statistical Interpretation Of Quantum Mechanics

Metaphysics Research Lab, Stanford University

booktitleThe Stanford Encyclopedia of Philosophy
editionWinter 2018
titleEverett's Relative-State Formulation of Quantum Mechanics

Oxford University Press

titleThe Quantum Mechanics of Minds and Worlds

titleThe mind of Wigner's friend

journaltitleThe British Journal for the Philosophy of Science
titleThat von Neumann Did Not Believe in a Physical Collapse

Pergamon Press

titleA Survey of Hidden-Variable Theories

titleOn the Einstein-Podolsky-Rosen Paradox

Cambridge university press

titleSpeakable and unspeakable in quantum mechanics: Collected papers on quantum philosophy

journaltitleFoundations of Physics
titleThe conceptual and the anecdotal history of quantum mechanics

journaltitleStudies in History and Philosophy of Science
titleThe later work of E. Schrödinger

journaltitleAnnals of mathematics
titleThe logic of quantum mechanics

journaltitlePhenomenology and the Cognitive Science
titleThe problems of other minds: a debate between Schrödinger and Carnap

Englewood Cliffs

titleQuantum Theory

titleWholeness and Implicate Order

journaltitlePhysical Review
titleA suggested interpretation of the quantum theory in terms of `hidden' variables, I

titleThe Undivided Universe: An Ontological Interpretation of Quantum Theory

titleThe Quantum Postulate and the Recent Development of Atomic Theory

titleQuantum Physics and Human Cultures

New York

Dover Publications

titleAtomic physics and human knowledge

La Nuova

booktitlePhilosophy in Mid-Century: A Survey
titleQuantum Physics and Philosophy: Causality and Complementarity

American Association of Physics Teachers

journaltitleAmerican Journal of Physics
titleAtomic theory and the description of Nature

Ox Bow Press

titleThe Philosophical Writings of Niels Bohr, Volume II: Essays 1958–1962 on Atomic Physics and Human Knowledge

Princeton University Press

booktitleQuantum Theory and Measurement
titleCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?

titleDie philosophischen Voraussetzungen der exakten Naturwissenschaften

Duncker und Humblot

titleGeisteswissenschaften und Naturwissenschaften

Kluwer Academic Press

booktitleJohn von Neumann and the Foundations of Quantum Physics
titlevon Neumann, Gödel and Quantum Incompleteness

booktitleQuanta and Mind: Essays on the Connection between Quantum Mechanics and the Consciousness
seriesSynthese Library
titleIs There a Place for Consciousness in Quantum Mechanics?

journaltitleStudies in History and Philosophy of Science Part A
titleStructural realism, mathematics, and ontology

New York

titleQuantum Theory and Reality

journaltitleReviews of Modern Physics
titleThe Measuring Process in Quantum Theory

M.I.T. Press

titleExperience and Conceptual Activity: A Philosophical Essay Based Upon the Writings of A. N. Whitehead

titleThe quantum theory of measurement

journaltitleStudies in History and Philosophy of Modern Physics
titleBohr, Heisenberg, and the Divergent Views of Complementarity

titleAn Account of the Principles of Measurement and Calculation

New York

titleUncertainty: the Life and Science of Werner Heisenberg

journaltitleAmerican Journal of Physics
title“Answer To The Question: When Did The Indeterminacy Principle Become The Uncertainty Principle?

booktitleHandbook Of Quantum Logic And Quantum Structures
titleQuantum Logic and Nonclassical Logics

journaltitleStudies in Hisory and Philosophy of Science
titlePhysics, metaphysics, dispositions, and symmetries – À la French

Santa Cruz

University of California

titleThe Conscious Mind: In Search Of A Theory Of Conscious Experience

Unicamp: Coleção CLE

titleAspectos da Descrição Física da Realidade

journaltitleRevista Brasileira de Ensino de Física
titleCertezas e Incertezas Sobre as Relações de Heisenberg

Princeton University Press

titleIntroduction to mathematical logic

Editions Frontieres

booktitleSchrödinger: Philosophy and the Birth of Quantum Mechanics
titleSome Notes on Schrödinger and Mysticism

titleQuantum Mechanics, Volume I: Basic Concepts, Tools, and Applications

Princeton University Press

booktitlePhilosophy – East and West
titleEastern and Western Metaphysics

São Paulo

Livraria da Física

booktitleTeoria quântica: estudos históricos e implicações culturais
titleMecânica Quântica e a cultura em dois momentos

University of Chicago Press

titleQuantum Mechanics: Historical Contingency and the Copenhaguen Hegemony

Kluwer Academic Publishers

booktitleBohmian Mechanics and Quantum Theory: An Appraisal
titleThe Causal Quantum Theory Program

New York

titleIn Search of Reality

Perseus Books

titleConceptual Foundations of Quantum Mechanics

Princeton University Press

titleOn physics and philosophy

São Paulo

titleEnsaio Sobre os Fundamentos da Lógica

São Paulo

titleLógica Indutiva e Probabilidade

journaltitlePrincipia: An international journal of epistemology
titleLogic and Ontology

booktitleThe Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today
titleThe Logic of Complementarity

Oxford University Press

booktitleEssays on Actions and Events
titleThe Logical Form of Action Sentences

journaltitleFoundations of Physics
titleCan We Falsify the Consciousness-Causes-Collapse Hypothesis in Quantum Mechanics?

New York

Back Bay Books

titleConsciousness Explained

journaltitlePhysics today
titleQuantum mechanics and reality

Addison Wesley Publishing Company

titleIntroduction to Quantum Mechanics

New York

State University of New York Press

booktitlePhysics and Whitehead: Quantum, Process, and Experience
titlePhysics and Whitehead: Quantum, Process, and Experience

New York

booktitleAlbert Einstein: Philosopher-Scientist
titleAutobiographical Notes

New York

booktitleAlbert Einstein: Philosopher-Scientist
titleRemarks to the Essays Appearing in this Collective Volume

New York

Philosophical Library

titleOut of My Later Years

booktitleThe Bohr-Einstein Letters
titleQuantum Mechanics and Reality

Princeton University Press

booktitleQuantum Theory and Measurement
shorthandEPR, 1983
titleCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?

New York

Fordham University Press

titleQuantum Mechanics and the Philosophy of Alfred North Whitehead

journaltitleReviews of modern physics
title`Relative state' formulation of quantum mechanics

Kluwer Academic Publishers

booktitleNiels Bohr and Contemporary Philosophy
seriesBoston Studies in the Philosophy of Science
titleNiels Bohr and Realism

Kluwer Academic Publishers

titleNiels Bohr: His Heritage and Legacy, an Anti-Realist View of Quantum Mechanics

Kluwer Academic Publishers

booktitleNiels Bohr and Contemporary Philosophy
seriesBoston Studies in the Philosophy of Science
titleNon-Locality or Non-Separability? A Defense of Bohr's Anti-Realist Approach to Quantum Mechanics

Springer Science & Business Media

titleNiels Bohr: His heritage and legacy: An anti-realist view of quantum mechanics

Metaphysics Research Lab, Stanford University

booktitleThe Stanford Encyclopedia of Philosophy
editionSpring 2019
titleCopenhagen Interpretation of Quantum Mechanics

University of Chicago Press

titleThe Shaky Game: Einstein, Realism and the Quantum Theory

journaltitleFoundations of Physics
titleEinstein and Ensembles: Response

North Holland

titleThe Philosophy of Niels Bohr: The Framework of Complementarity

Kluwer Academic Publishers

booktitleNiels Bohr and Contemporary Philosophy
seriesBoston Studies in the Philosophy of Science
titleBohr's Framework of Complementarity and the Realism Debate

journaltitleHistorical Studies in the Physical and Biological Sciences
titleScience and Exile:David Bohm, the Cold War, and a New Interpretation of Quantum Mechanics

New York

titleThe Quantum Dissidents: Rebuilding the Foundations of Quantum Mechanics (1950-1990)

São Paulo

Editora Livraria da Física

booktitleFundamentos da Física 1
titleSobre A Recepção Do Programa Causal De David Bohm

journaltitleStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
titleA phenomenological solution to the measurement problem? Husserl and the foundations of quantum mechanics

journaltitleStudies in History and Philosophy of Science Part A
titleDefending eliminative structuralism and a whole lot more (or less)

Palgrave Macmillan

seriesNew Directions of the Philosophy of Science
titleInterpreting Quantum Theory: A Therapeutic Approach

journaltitlePhysical Review D
shorthandGRW, 1986
titleUnified dynamics for microscopic and macroscopic systems

Cambridge University Press

titleParticles and Paradoxes: The limits of quantum logic

São Paulo

Martins Fontes

titleVocabulário Grego de Filosofia

Harvard University Press

booktitleFrom Frege To Gödel: A Source Book In Mathematical Logic, 1979-1931
titleOn Formally Undecidable Propositions Of Principia Mathematica and Related Systems 1

journaltitlePhysics Essays
titleThe Idealistic Interpretation of Quantum Mechanics

New York

Penguin Putnam

titleThe Self-Aware Universe: How Consciousness Creates the Material World

New York

titleThe Physicists' View of Nature – Part 2: The Quantum Revolution

Long Grove

Waveland Press

titleQuantum Mechanics

Kegan Paul

titleThe Philosophy of the Upanishads and Ancient Indian Metaphysics

Cornell University Press

titleReenchantment without Supernaturalism: A Process Philosophy of Religion

State University of New York Press

booktitleProcess Approaches to Consciousness in Psychology, Neuroscience, and Philosophy of Mind
titleConsciousness as Subjective Form: Whitehead's Nonreductionist Naturalism

Cambridge University Press

titleIntroduction to quantum mechanics

journaltitlePhyics Essays
titleThe Einstein-Podolsky-Rosen Paradox in the Brain: The Transferred Potential

University of Chicago Press

titleThe Physical Principles of Quantum Theory

New York

titlePhysics and Philosophy: The Revolution in Modern Science

American Institute of Physics

Niels Bohr Library & Archives

titleInterview with Thomas S. Kuhn at the Max Planck Institute, Munich, Germany, February 25, 1963

booktitleNiels Bohr: His Life and Work As Seen By His Friend and Colleagues
titleQuantum Theory and its Interpretation

Princeton University Press

booktitleQuantum Theory and Measurement
titleOn The Physical Content Of Quantum Theoretical Kinematics And Mechanics

Rio de Janeiro

titleA Parte e O Todo

New York

booktitleWolfgang Pauli: Scientific Correspondence With Bohr, Einstein, Heisenberg
titleIs a Deterministic Completion of Quantum Mechanics Possible?

Metaphysics Research Lab, Stanford University

booktitleThe Stanford Encyclopedia of Philosophy
editionWinter 2016
titleThe Uncertainty Principle

New York

Basic Books

titleGödel, Escher, Bach: An Eternal Golden Braid

Oxford University Press

booktitleOntology after Carnap
titleCarnap's Big Idea

Metaphysics Research Lab, Stanford University

booktitleThe Stanford Encyclopedia of Philosophy
editionSummer 2018
titleLogic and Ontology

journaltitlePhysics Reports
titleEnsemble interpretations of quantum mechanics. A modern perspective

journaltitlePhysics Reports
titleEnsemble Interpretations Of Quantum Mechanics: A Modern Perspective

Kluwer Academic Publishers

booktitleNiels Bohr and Contemporary Philosophy
seriesBoston Studies in the Philosophy of Science
titleWhat Makes a Classical Concept Classical? Towards a Reconstruction of Niels Bohr's Philosophy of Physics

The University of Chicago Press

journaltitlePhilosophy of Science
titleWho invented the `Copenhagen Interpretation'? A study in mythology

Metaphysics Research Lab, Stanford University

booktitleThe Stanford Encyclopedia of Philosophy
editionFall 2017
titleEinstein's Philosophy of Science

Harvard university press

titleThe structure and interpretation of quantum mechanics

journaltitleFoundations of Physics
titleOn Math, Matter and Mind

Oxford University Press

titleThe Perennial Philosophy

New York

titleThe Philosophy Of Quantum Mechanics: The Interpretations Of Quantum Mechanics In Historical Perspective

University of Helsinki

titleQuantum Metaphysics: The Role of Human Beings within the Paradigms of Classical and Quantum Physics
typeDissertation (Mesters in Philosophy)

State University of New York Press

booktitleProcess Approaches to Consciousness in Psychology, Neuroscience, and Philosophy of Mind
titleThe Interpretation and Integration of the Literature on Consciousness from a Process Perspective

São Paulo

titleCausalidade e Teoria Quântica

booktitleJohn von Neumann and the foundations of quantum physics
titleWhy von Neumann Rejected Carnap's Dualism of Information Concepts

Editorial Universidad del Valle

booktitleEinstein: científico y filósofo
title¿Pueden Ser Separables Las Entidades Indiscernibles? Ensayo Sobre Posibles Consecuencias De Una Posición De Einstein Frente A La Indiscernibilidad

seriesRumos da Epistemologia
titleÁlgebra Linear com um Pouco de Mecânica Quântica

titleThe Logical Foundations of Scientific Theories: Languages, Structures, and Models

booktitlePhilosophy of Logics
seriesHandbook of the Philosophy of Science
titleParaconsistent Logics and Paraconsistency

São Paulo

titleA Estrutura das Revoluções Científicas

booktitleConcepts for Neural Networks: a Survey
titlePenrose's Philosophical Error

titlePsychologische Untersuchungen

New Jersey

Basil Blackwell

titleMind, brain and the quantum: The compound `I'

New York

Dover Publications

titleSuperfluids, Macroscopic Theory of Superconductivity

Princeton University Press

booktitleQuantum Theory and Measurement
origtitleLa thé́orie de l'observation en mé́canique quantique
titleThe theory of observation in quantum mechanics

New York

Open Court Publishing Company

titleThe revolt against dualism

Rio de Janeiro

Sociedade Brasileira de História da Ciência

booktitleAnais do 15 Seminário Nacional de História da Ciência e da Tecnologia: Volume III
titleA Teoria Quântica e a Apropriação do Conhecimento Científico: O uso da História e Filosofia da Ciência pelos Misticismos

journaltitleFoundations of Physics
titleA Whiteheadian Approach to Bell's Correlations

journaltitleFoundations of physics
titleThe collapse of quantum states: A new interpretation

New York

Oxford University Press

titleNature loves to hide: Quantum physics and reality, a western perspective

New York

State University of New York Press

booktitlePhysics and Whitehead: Quantum, Process, and Experience
titleWhitehead's Philosophy and the Collapse of Quantum States

journaltitleFoundations of Physics
titleFounding Quantum Theory on the Basis of Consciousness

journaltitlePhilosophy of Science
titlePhilosophical Problems Concerning The Meaning Of Measurement In Physics

journaltitleAnnals of Physics
titleMeasurements in Quantum Mechanics

titleThree measurement problems

Oxford Universty Press

booktitleThe Oxford Handbook of Metaphysics
titleDistilling Metaphysics from Quantum Physics

noteTranslated by G. M. Temmer
titleQuantum Mechanics, Volume I

journaltitleFoundations of Physics
titleThe World According to Quantum Mechanics (Or the 18 Errors of Henry P. Stapp)

Kluwer Academic Publishers

booktitleNiels Bohr and Contemporary Philosophy
seriesBoston Studies in the Philosophy of Science
titleThe Bohr-Einstein Dispute

Universidade Federal de Santa Catarina (UFSC)

noteTese (Doutorado em Filosofia)
titleA Realidade Através Do Espelho: Schrödinger E Russell No País Da Objetivação

New York

State University of New York Press

booktitlePhysics and Whitehead: Quantum, Process, and Experience
titleWhitehead and the Quantum Experience

journaltitleReview of Metaphysics
titleOn Heisenberg's Key Statements Concerning Ontology

journaltitleInternational Journal of Theoretical Physics
titleThe Self-contradictory Foundations of Formalistic Quantum Measurement Theories

São Paulo

titleMatéria Roubada: A apropriação crítica do objeto da física contemporânea

Oxford University Press

titleThe Emperor's New Mind: Concerning Computers, Minds, And The Laws Of Phyisics

Oxford University Press

titleShadows of the Mind: A Search for the Missing Science of Consciousness

Fundação Calouste Gulbenkian

titlePlatão, A República. Introdução, tradução do grego e notas

journaltitleCadernos de História e Filosofia da Ciência
titleO Problema Da Medição Em Mecânica Quântica: Um Exame Atualizado

São Paulo

Livraria da Física

titleConceitos De Física Quântica, Volume I

São Paulo

Livraria da Física

booktitleTeoria quântica: estudos históricos e implicações culturais
titleO fenômeno cultural do misticismo quântico

titleEinführung in die Psychologie

Fundação Calouste Gulbenkian

noteIntrodução, tradução do grego e notas por Maria H. Pereira
titleA República

New York

booktitleQuantum Mechanics and Reality
titleQuantum Mechanics Without `The Observer'

La Salle

Library of Living Philosophers

booktitleThe Philosophy of Karl Popper
titleReplies To My Critics

Kluwer Academic Publishers

booktitleErwin Schrödinger's World View: The Dynamics Of Knowledge And Reality
titleThe Notion Of Consciousness In Schrödinger's Philosophy Of Nature

titleKuhn's The Structure of Scientific Revolutions: a Reader's Guide

Oxford University Press

journaltitleBritish Society for the Philosophy of Science
titleA Philosopher Looks at Quantum Mechanics (Again)

New York

Random House

titleThe Ways Of Paradox And Other Essays

journaltitleInternational Journal of Ethics
titleThe Vedanta Philosophy and the Doctrine of Maya

Clarendon Press

titleIncompleteness, Nonlocality, And Realism: A Prolegomenon To The Philosophy Of Quantum Mechanics

University of California Press

titlePhilosophic Foundations of Quantum Mechanics

Lawrence Berkeley Laboratory

journaltitleAmerican Journal of Physics
titleDoes God Play Dice? A Discussion Of Some Interpretations Of Quantum Mechanics

titleOn Denoting

Oxford University Press

booktitleMany Worlds? Everett, Quantum Theory, and Reality
titleMany Worlds? An Introduction

New York

titleQuantum Mechanics

journaltitleStudies in History and Philosophy of Modern Physics
titleA snapshot of foundational attitudes toward quantum mechanics

Cambridge University Press

titleMy View of the World

Cambridge University Press

booktitleWhat is Life & Mind and Matter
titleWhat is life? The Physical Aspect of the Living Cell

Princeton University Press

booktitleQuantum Theory and Measurement
titleThe Present Situation in Quantum Mechanics

journaltitleAmerican Journal of Physics
titleRole of the observer in quantum theory

Cornell University Press

booktitlePhilosophy in America
titleQuantum physics and the philosophy of Whitehead

journaltitleThe British Journal for the Philosophy of Science
titleContextual hidden variables theories and Bell's inequalities

journaltitleQuantum Information Processing
titleDialogue Abner Shimony–Shimon Malin

New York

booktitleQuantum Reality, Relativistic Causality, and Closing the Epistemic Circle Essays in Honour of Abner Shimony
titleBistro Banter: A Dialogue with Abner Shimony and Lee Smolin

New York

Oxford University Press

booktitleConsciousness: New Philosophical Perspectives
titleWhy Cognitive Scientists Cannot Ignore Quantum Mechanics

titleOne mind or many: A Note on the Everett Interpretation of Quantum Theory

titleQuantum Theory and the Relation between the Conscious Mind and the Physical World

journaltitleNuovo Cimento
titleAre Superluminal Connections Necessary?

journaltitleFoundations of Physics
titleWhiteheadian Approach to Quantum Theory and the Generalized Bell's Theorem

journaltitleFoundations of Physics
titleMind, matter, and quantum mechanics

journaltitleFoundations of Physics
titleThe 18-Fold Way

John Wiley & Sons

journaltitleZygon: Journal of Religion and Science
titleQuantum Interactive Dualism: An alternative to materialism

journaltitleMind & Matter
titleWhitehead, James, and the Ontology of Quantum Theory

booktitleJohn von Neumann and the foundations of quantum physics
titleOpportunistic axiomatics – von Neumann on the methodology of mathematical physics

Basic Books

titleQuantum mechanics: The theoretical minimum

Princeton University Press

booktitleQuantum Theory and Measurement
titleOn The Decrease Of Entropy In A Thermodynamic System By The Intervention Of Intelligent Beings

Cambridge University Press

titleAn introduction to metametaphysics

journaltitleLogic, semantics, metamathematics
titleThe concept of truth in formalized languages

journaltitleFoundations of Physics
titleThe Mathematical Universe

journaltitleChaos, Solitons & Fractals
titleConsciousness as a state of matter

Oxford University Press

booktitleIn Metaphysics in the Philosophy of Science: New Essays
titleAgainst Bracketing and Complacency: Metaphysics and the Methodology of the Sciences

Princeton University Press

origtitleMathematische Grundlagen der Quantenmechanike
titleMathematical Foundations of Quantum Mechanics

State University of New York Press

booktitleProcess Approaches to Consciousness in Psychology, Neuroscience, and Philosophy of Mind
titleWhitehead as a Neglected Figure of Twentieth-Century Philosophy

booktitleThe Essential David Bohm
titleDialogues With Scientists And Sages: The Search For Unity

State University of New York Press

booktitleProcess Approaches to Consciousness in Psychology, Neuroscience, and Philosophy of Mind
titleConsciousness and Causation in Whitehead's Phenomenology of Becoming

journaltitleJournal of Consciousness Studies
titleThe Mind-Body Problem and Whitehead's Nonreductive Monism

Cambridge University Press

titleAn Enquiry Concerning the Principles of Natural Knowledge

New York

Free Press

titleProcess And Reality: An Essay In Cosmology

Princeton University Press

booktitleQuantum Theory and Measurement
titleRemarks On The Mind-Body Question

Quest Books

titleThe Spectrum of Consciousness

journaltitleFoundations of Physics
titleThe Problem Of Conscious Observation In Quantum Mechanical Description

[title=Referências bibliográficas,heading=bibintoc]