#  Semiconductor to Metal Transition, Dynamical Stability and
Superconductivity of Strained Phosphorene

G. Q. Huang1,2, Z. W. Xing2,3 1Department of Physics, Nanjing Normal
University, Nanjing 210023, China
2National Laboratory of Solid State Microstructures, Nanjing University,
Nanjing 210093, China
3Department of Materials Science and Engineering, Nanjing University, Nanjing
210093, China

###### Abstract

Very recently, field-effect transistors based on few-layer phosphorene
crystals with thickness down to a few nanometres have been successfully
fabricated, triggering interest in this new functional two-dimensional
material. In this work, we apply the first-principles calculations to studying
the evolution of electronic structures and lattice dynamics with vertical
strain for monolayer and bilayer phosphorenes. It is found that, by changing
the thickness of phosphorene or the strain applied on it, its band gap width
can be well tuned, and there will appear a transition from semiconductor to
metal. In particular, the bilayer phosphorene may become a good BCS
superconductor by adjusting the interlayer distance, in which the interlayer
van der Waals coupling is favorable to the dynamical stability against strain.

phosphorene,semiconductor to metal transition,phonon, superconductivity,
first-principles

###### pacs:

63.22.Np, 73.61.Cw, 74.78.-w

##  I Introduction

Black phosphorus (BP) is the most stable allotrope of the element phosphorus
and has many interesting physical properties. BP can server as the electrode
material for Lithium-Ion batteries.1 ; 2  Several structural phase transitions
were found under pressure accompanied by semiconductor-semimetal-metal
transition.3 ; 4  Furthermore, it was reported that the BP single crystal
shows superconductivity with $T_{C}$ higher than 10K under high pressure.5 ; 6
The structure of BP consists of puckered double layers, which are held
together by weak van der Waals (vdW) force. This peculiar layer structure
permits to employ mechanical exfoliation to extract thin black phosphorus from
a bulk crystal. Recently, the few-layer black phosphorus (phosphorene) had
been successfully exfoliated and used them to create field-effect transistors,
which exhibit large current on-off ratios and high mobilities.7 ; 8 ; 9  The
successful fabrication of this novel two-dimensional (2D) semiconducting
material was soon paid amounts of attention and predicted to have a great
potential for practical applications.10 ; 11

Bulk BP is a direct-gap semiconductor with a 0.33 eV band gap.12  However, the
band gap of phosphorene is predicted to be highly sensitive to its thickness
and strain. Previous first-principles calculations show that the band gap
ranges from 1.5 eV for a monolayer to 0.6 eV for a 5-layer.13  A 3$\%$ in-
plane strain can change phosphorene from a direct-gap to an indirect-gap
semiconductor,8  while a vertical compression can induce the semiconductor to
metal transition.14  Strain is an effective method to tailor electronic
properties for 2D semiconductors. For monolayer phosphorene (MLP) and bilayer
phosphorene (BLP), whether there are different strain effects on electronic
properties is one of the motivation of this study. Furthermore, due to the
interlayer coupling bonded by weak van der Waals force, it is also interesting
to see how bilayer phosphorene responses to the change of interlayer distance.

While there have been substantial works on the electronic properties of
phosphorene, investigations on the vibrational properties of phosphorene are
scarce. The Raman spectra was measured for the BP flake with thicknesses
ranging from 1.6 nm to 9 nm.15  The phonon dispersion of MLP was calculated by
first-principles calculations.10 ; 16  The lattice vibrational modes of MLP
with in-plane strain were also reported by Fei et al. through first-principles
simulations.16  However, phosphorene has an anisotropic structure, whether it
has different dynamical behavior for phosphorene with out-of-plane strain?
This problem is our concern. We will pay particular attention to the dynamical
stability of strained phosphorene in order to better use them for practical
application.

In this work, geometric structure, electronic structure and lattice dynamics
of MLP and BLP under vertical strain are studied and compared by first-
principles calculations. The calculated results show that the evolution
behavior of band structure for MLP and BLP under vertical strain is different.
The dynamical stable range of the strained BLP is wider than that of the
strained MLP. Furthermore, we also find that BLP may become a good BCS
superconductor by adjusting the interlayer distance.

##  II  Computational Details

The first-principles calculations have been performed within the density
functional theory through the PWSCF program of the Quantum-ESPRESSO
distribution.17  The ultrasoft pseudo-potential and general gradient
approximation (GGA-PBE) for the exchange and correlation energy functional are
used with a cutoff of 30 Ry for the expansion of the electronic wave function
in plane waves. MLP and BLP are modeled using the slabs, which are separated
by vacuum layer with the thickness of about 35 $\AA$. For the electronic
structure calculations, the Brillouin zone integrations are performed with a
(8,10,1) wavevector $k$-space grid by using the first-order Hermite-Gaussian
smearing technique. Within the framework of the linear response theory, the
dynamical matrixes are calculated for special $\bf{q}$ points in the two
dimensional irreducible Brillouin zone and are Fourier interpolated throughout
the full Brillouin zone. The dense (24,30,1) grid are used in the Brillouin
zone integrations in order to produce the accurate electron-phonon (EP)
interaction matrices. As BP presents a laminar crystal structure, vdW
correction proposed by Grimme (DFT-D2) is included in our calculations.18

##  III  Results and discussion

###  III.1  Geometric structure under vertical strain

BP has a layered orthorhombic structure. Each sheet consists of an atomic
double layers known as a puckered layer. The stacking of the puckered layer
for BP is ABAB $\ldots$ sequence with weak van der Waals bonding between
layers. The edge of the puckered hexagon of A layer is located in the center
of the puckered hexagon of B layer and vice versa. The top and side views of
the atomic structure of a puckered layer are presented in Figs. 1 (a) and (b),
respectively. Each layer may be viewed as consisting of armchair and zigzag
chains along $x$ and $y$, respectively. Each phosphorus atom is covalently
bonded to three neighbors within the puckered layer. Our optimized bond
lengthes, bond angles (as labeled in Fig. 1) and lattice constants for BP are
listed in Table 1. For comparison, the results without vdW correction for bulk
BP are also listed in Table 1\. It can be clearly seen that the inclusion of
vdW interaction changes $a$ and $b$ parameters slightly, but reduces the c
parameter substantially (about 0.6${\AA}$). The overall best agreement with
experimental19  and previous computational results20  is achieved with
inclusion of vdW correction.

Figure 1: The top (a) and side views (b) of the atomic structure of
phosphorene. (c) Vibrational pattern of $B_{3g}$ and $B_{1g}$ modes.

For the free MLP and BLP (see Table 2), lattice constant $b$ and bond lengthes
are very close to those in the bulk BP, while $a$ and bond angles $\theta_{2}$
are larger than those for the bulk BP. This fact shows that the increase of
lattice constants $a$ is not due to the change of bond lengthes, but is due to
the flattening of puckered layer. Next we concentrate mainly on the strained
phosphorene. The vertical strain is modeled by constraint $z$ for the outmost
layer atoms, the unit cell and other atomic positions are then relaxed. The
strain is defined as $\sigma=\frac{h-h_{0}}{h_{0}}$, where $h$ and $h_{0}$ are
the thickness of the strained and the free phosphorene, respectively. The
positive (negative) of $\sigma$ corresponds to the tensile (compressive)
strain. The obtained geometric parameters for MLP and BLP under vertical
strain are listed in Table 2.

For the MLP, when tensile (compressive) strain is applied in the $z$
direction, the lattice constant $a$ decreases (increases) substantially. While
the lattice constant $b$ and the bond length $d_{1}$ show only a weak
variation under strain, reflecting the rigidity of the strong covalent bonding
along the zigzag direction. So the anisotropic geometric structure along
armchair ($x$ ) and zigzag ($y$ ) directions result in their different
responses to the vertical strain. With the decrease of thickness ($h$) of
phosphorene, the bond length $d_{2}$ decreases and the bond angle $\theta_{2}$
increases, reflecting the flattening of the puckered layer. It is need to
mention that the range of strain studied here is not too large, the puckered
character of structure still remains. The previous calculations by Rodin et
al.14  showed that the monolayer approaches a plane square lattice
configuration under severe compression.

For the BLP, when tensile strain is applied, the distance between puckered
layers increases, while the covalent bonds within the puckered layer change
slightly. And we also find that the geometric parameters of BLP are tend to
those in the MLP when tensile strain is increased to $40\%$. This result is
reasonable, since the coupling between puckered layers may be ignored at so
large interlayer distance (about 6 ${\AA}$). With increase of compressive
strain applied, the change of geometric parameters of BLP is qualitatively
same as in the case of MLP but with smaller magnitude. Unlike to that for the
MLP, the lattice constant $a$ of the BLP always increases whatever compressive
or tensile strain is applied along z direction. This peculiar property is same
to that observed in the bulk BP.21  This result implies that BLP has the same
unusual mechanical response as bulk BP.

###  III.2  Electronic structures under vertical strain

The calculated band structures of the strained MLP and BLP are shown in Fig.
2. For the free MLP (Fig. 2c), the band dispersion along the major high
symmetry directions compares very well with the previous calculations by Rodin
et al..14  It is a direct band gap semiconductor with the band gap energy
about 0.83 eV. The top of the valence band and the bottom of the conduction
band have predominantly $p_{z}$ character mixed with $p_{x}$ and $s$
character. The band structure of the free BLP (Fig. 2h) has similar shape to
that of MLP but with smaller gap energy (0.40 eV). The photoluminescence
measurements15  and previous calculations13  indicated a thickness dependent
band structure of the phosphorene.

Figure 2: The band structures of the strained phosphorene, (a)-(e) are for
MLP; (f)-(i)are for BLP. (j) The evolution of the band gap $E_{g}$ with the
strain.

The vertical strain effect on the electronic structure of MLP can be seen from
(a) to (e) of Fig. 2. Both the compressive and tensile strains applied for the
MLP can result in the transition from semiconductor to semimetal or metal. The
band gap $E_{g}$ located at $\Gamma$ point keeps direct and decreases with the
tensile strain, turning into zero when the tensile strain reaches $8\%$ (Fig.
2d), and then converting into semimetal or metal (Fig. 2e) with increasing the
tensile strain continuously. When compressive strain is applied, the band gap
firstly keeps direct and increases slightly, but turns into indirect when the
compressive strain reaches about $4\%$. Then indirect band gap decreases and
turns into zero at about $25\%$ compressive strain (Fig. 2b), and finally
converting into semimetal or metal when the compressive strain continues to
increase(Fig. 2a)

For the BLP, when tensile strain is applied, the band gap $E_{g}$ keeps direct
and increases until tends to the value in the free MLP. The corresponding band
structure (Fig. 2i) is also similar to that of the free MLP (see Fig. 2c).
Under small compressive strain ($1\%$), the band gap soon turns into indirect
and decreases with increasing the compressive strain. It turns into zero when
the compressive strain reaches about $8\%$ (Fig. 2g), and finally converts
into metal (Fig. 2f).

The evolution of the band gap $E_{g}$ with the strain for the MLP and BLP is
summarized in Fig. 2j. The change of band gap is very extensive. So we can
find that the electronic structure of thin BP is very sensitive to its
thickness and the strain applied. This properties help to tailor materials in
electronics. In order to better understand this novel 2D materials and use
them for potential applications, next we turn to discuss the dynamical
stability of the strained phosphorene.

###  III.3  Lattice vibration under vertical strain

The symmetry of phosphorene is described by the $D_{2h}$ point group. For the
MLP, the primitive cell contains four atoms, leading to 12 vibrational modes,
i.e. nine optical and three acoustic phonons branches. The $\Gamma$ point
modes can be decomposed as

$\Gamma_{acoustic}=B_{1u}+B_{2u}+B_{3u}$ (1)

and

$\Gamma_{optical}=B_{1u}+B_{3u}+A_{u}+B_{1g}+B_{3g}+2B_{2g}+2A_{g}$ (2)

Among optical modes, the $A_{u}$ mode is silent, $B_{1u}$ and $B_{3u}$ are
infrared active, while all others are Raman active. The structure of puckered
layer remains the reflection symmetry in the $y$ (zigzag) direction but breaks
the reflection symmetry in the $z$ and $x$ directions. So the vibrations along
the $y$ direction are strict, while those along the $z$ and $x$ directions are
mixed. $B_{3g}$ and $B_{1g}$ Raman modes correspond to the vibrations along
$y$ direction and their vibrational patterns are illustrated in Fig. 1c. For
$B_{3g}$ mode, the vibration of atom in the different layer is out-of-phase,
but is in-phase in the same layer, i.e. two layers beating against each other.
$B_{1g}$ mode corresponds to the bond stretching modes from $d_{1}$ and
$d_{2}$ covalent bonds. The other four Raman active modes have similar
vibrational patterns along $x$ and $z$ directions, but each mode having mixed
$x$ and $z$ characters. The Raman frequencies of free MLP are listed in Table
3. Our calculated frequencies are slightly smaller than those in Ref. [16, ].
For the free BLP, the frequencies of corresponding Raman active modes are also
listed in Table 3. We find that the frequency shift between MLP and BLP is
small except for $A_{g}^{1}$ and $B_{2g}^{2}$ modes, which vibrate mainly
along the $z$ direction. The frequencies of this two modes in BLP is smaller
that those in MLP probably due to the attractive vdW interaction existed in
BLP.

The phonon dispersion of free MLP calculated by the density functional
perturbation theory is presented in Fig. 3 (a), which compares very well with
previous calculations.10 ; 16  Near the $\Gamma$ point, two in-plane acoustic
modes exhibit linear dispersions, while the off-plane acoustic mode exhibits a
parabolic dispersion due to the 2D character of phosphorene. The vibrations of
the bond-breathing modes occupy the high-frequency region, while the
vibrations of layer-breathing modes and the acoustic modes occupy the low-
frequency region. Fig. 3 (b) shows the calculated phonon dispersion of free
BLP which has 24 phonon branches. The optical branches have small splitting
and are nearly double degenerate due to the weak interlayer interaction. Near
the $\Gamma$ point, the splitting has an appreciable magnitude. Three branches
from the opposite vibrations of two puckered layers (i.e. layer-breathing
modes) deviate from three acoustic branches. From Fig. 3a and 3b, we can see
that there is no imaginary frequency in the full phonon spectra, indicating
the dynamical stability for both free-standing MLP and BLP. This conclusion is
in good agreement with the experiment. Raman spectroscopy and transmission
electron microscopy measurements show that the exfoliated flakes of BP are
stable even in free-standing form.15

Figure 3: The phonon dispersion of free MLP (a) and BLP (b).

When vertical strain is increased to a certain value, we find phosphorene
becomes unstable. For MLP, there appears imaginary frequency in the phonon
dispersions when tensile strain reaches 15$\%$ and compressive strain reaches
4$\%$. The appearance of imaginary frequency of phonon modes indicates the
dynamical instability under such strain. The dynamical stable range of MLP for
strain $\sigma$ is about ($-3\%,12\%$). Within this range, the frequencies of
Raman active modes under several strains are also listed in Table 3. The
response of each Raman mode to strain is not the same. Four modes (
$B_{1g},A_{g}^{1},B_{2g}^{1},B_{2g}^{2}$) exhibit monotonic change: their
frequencies are red shifted under tensile strain and blue shifted under
compressive strain. Among them, $A_{g}^{1},B_{2g}^{2}$ modes which are
vibrated mainly along the z direction have prominent shifts. The other two
Raman modes exhibit nonmonotonic change. $B_{3g}$ mode exhibits an abnormal
red shift under small compressive strain. This is possibly due to the fact
that the flattening of the puckered layer under compressive strain is in favor
of the opposite vibrations of two layers along the zigzag chain. $A_{g}^{2}$
mode exhibits a nonmonotonic shift under tensile strain. Its frequency
decreases at small tensile strain and then increases with increasing tensile
strain. This abnormal behavior can be understood from the following:
$A_{g}^{2}$ mode corresponds to the vibration of $d_{1}$ bond-breathing mainly
along $x$ direction. The lattice constant $a$ is contracted when vertical
tensile strain is applied. When the effect of this contraction of $a$
dominates over the effect of increase of thickness of phosphorene, it will
enhance the interatomic interactions, resulting in the increase of frequency
of $A_{g}^{2}$ mode.

For BLP, we find its dynamical stable range for strain $\sigma$ is wider than
that of MLP. When compressive strain reaches 8$\%$, imaginary frequency
appears in the phonon dispersion along $\Gamma$-X direction. Within the stable
dynamical range for strain $\sigma$, the frequencies of Raman active modes are
also listed in Table 3. Under compressive strain, $B_{3g}$ mode exhibits an
abnormal red shift as in the case of MLP, while the frequencies of other Raman
modes are blue shifted as expected as usual. BLP is still stable when tensile
strain reaches 40$\%$. As discussed above, large tensile strain only results
in large interlayer distance with other geometric parameters in close to those
in the MLP. Then BLP may be regarded as two nearly independent MLPs, which are
stable in free-standing form as discussed above. Under the wide range of
tensile strain, the changes of frequencies of four modes (
$B_{3g},B_{1g},A_{g}^{2},B_{2g}^{1}$) are all very small. This shows that the
vertical tensile strain has little effect on the in-plane vibrational modes.
With the increase of tensile strain, $A_{g}^{1},B_{2g}^{2}$ modes vibrated
mainly along the $z$ direction are red shifted at first as expected, but they
are then blue shifted. This abnormal blue shift may be due to the decrease of
attractive vdW interaction at large interlayer distance.

###  III.4 Superconductivity by adjusting the interlayer distance

The above phonon calculations show that BLP is dynamical unstable when it
achieves the transition from semiconductor to metal. Here we propose that
stable metal phase or even a BCS superconductor can be achieved by adjusting
the interlayer distance of BLP. We fix the interlayer distance at different
given values, the unit cell and other atomic positions are then relaxed. Since
the results with increasing the interlayer distance are very similar to those
in the tensile strained BLP, next we only discuss the opposite case.

When the interlayer distance $0.6d_{0}<d<d_{0}$ ($d_{0}$ is the interlayer
distance of free BLP), our calculations show that the lattice constant $a$
increases slightly, $d_{2}$ and $\theta_{2}$ show only a weak variation, while
the lattice constant $b$, $d_{1}$ and $\theta_{1}$ hardly change. This means
that the puckered character of BLP remains and changes little at this range of
interlayer distance. When $d<0.6d_{0}$, the structure of BLP starts to change
substantially. The calculations of band structure show that the band gap
$E_{g}$ goes to zero when $d=0.85d_{0}$. Keeping on decreasing the interlayer
distance will result in the transition from semiconductor to metal. The
calculations of phonon show that BLP is dynamical unstable when $d$ is
$0.65d_{0}$ or less. So the range of $d$ for the stable metal phase of BLP is
about $0.65d_{0}<d<0.85d_{0}$.

The Eliashberg spectral function $\alpha^{2}F(\omega)$ depends directly on the
EP matrix element, $g_{{\bf{k}+\bf{q}}j^{{}^{\prime}}{\bf{k}}j}^{\bf{q}\nu}$,
which can be determined self-consistently using linear response theory. By
plotting $\alpha^{2}F(\omega)$ we can estimate the relative strength of the EP
coupling. The phonon density of states $F(\omega)$ and $a^{2}F(\omega)$ are
plotted in Fig. 4 for three different interlayer distances, which are within
the range of stable metal phase of BLP. When $d=0.8d_{0}$, $d=0.75d_{0}$ and
$d=0.7d_{0}$, $F(\omega)$ is very similar, while $a^{2}F(\omega)$ is
different. With smaller interlayer distance, the Eliashberg spectral function
$a^{2}F(\omega)$ has a significantly enhanced peak at the low-frequency side.
The characteristic phonon modes which dominate the EP coupling are just from
the opposite vibrations of two puckered double layers. So we speculate that
the interlayer vdW attractive interaction may play an important role in
enhancing the EP coupling of BLP with smaller interlayer distance.

Figure 4: The phonon density of states $F(\omega)$ and the Eliashberg spectral
function $\alpha^{2}F(\omega)$ for BLP at three different interlayer
distances.

The superconducting temperature $T_{c}$ can be estimated from the Allen-Dynes
modified McMillan equation22 :

$T_{C}=\frac{\omega_{ln}}{1.2}\exp\left(-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right),$
(3)

where $\omega_{ln}$ is the logarithmically averaged frequency, $\mu^{*}$ is
the Coulomb repulsion parameter. The calculated electronic densities of states
at Fermi level $N(E_{F})$, EP coupling constant $\lambda$, $\omega_{ln}$ and
the estimated $T_{C}$ when taking $\mu^{*}=0.1$ are summarized in Table 4.
With the decrease of interlayer distance, BLP apparently becomes more metallic
and results in the increase of $N(E_{F})$. Most remarkably, we see $\lambda$
increases dramatically. The increase of both $N(E_{F})$ and $\lambda$ is
favorable to enhance $T_{C}$. When $d=0.7d_{0}$, $\lambda$ reaches as high as
1.45 and the estimated $T_{C}$ is about 10K. The superconductivity of BP was
reported early under high pressure.5 ; 6  Our results suggest that BLP may
become a good BCS superconductor by adjusting the interlayer distance.

##  IV Summary

In conclusion, MLP and BLP under vertical strain are studied by density
functional and density-functional perturbation theory. The results show that
the electronic structure of thin BP is very sensitive to its thickness and the
strain applied. The change is extensive. It can increase direct band gap,
decrease indirect band gap or realize the transition from semiconductor to
metal, which help to tailor materials in a variety of settings, from infrared
optoelectronics to high-mobility quantum transport. The shift of zone-center
Raman active modes for strained phosphorenes is analyzed and explained
combining the relaxation of geometric structure. Our study clearly show that
both MLP and BLP in free-standing form are dynamical stable. However, material
becomes dynamical unstable with large vertical strain. We find that the
dynamical stable range for strain $\sigma$ in BLP is wider than that in MLP
due to add the interlayer weak vdW coupling, this additional freedom. This
information is essential for future device fabrication and potential
applications of phosphorene. Furthermore we also find that BLP may become a
good BCS superconductor by adjusting the interlayer distance.

##  V Acknowledgments

The authors acknowledge the support of the Natural Science Foundation of
Jiangsu Province in China under Grant No. BK20141441,the State Key Program for
Basic Researches of China (2014CB921103 and 2010CB923404), the National
”Climbing” Program of China (91021003), and the National Science Foundation of
Jiangsu Province (BK2010012).

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Figure Caption

Figure 1. The top (a) and side views (b) of the atomic structure of
phosphorene. (c) Vibrational pattern of $B_{3g}$ and $B_{1g}$ modes.

Figure 2. The band structures of the strained phosphorene, (a)-(e) are for
MLP; (f)-(i)are for BLP. (j) The evolution of the band gap $E_{g}$ with the
strain.

Figure 3. The phonon dispersion of free MLP (a) and BLP (b).

Figure 4. The phonon density of states $F(\omega)$ and the Eliashberg spectral
function $\alpha^{2}F(\omega)$ for BLP at three different interlayer
distances.

Table 1: Structural parameters of bulk BP. | a(${\AA}$) | b(${\AA}$) | c(${\AA}$) |  $d_{1}$(${\AA}$) |  $d_{2}$(${\AA}$) | $\theta_{1}$ | $\theta_{2}$
---|---|---|---|---|---|---|---
Exp. (Ref.19, ) | 4.376 | 3.314 | 10.478 | 2.224 | 2.244 | 96.34∘ | 102.09∘
Theo.(Ref.20, ) | 4.422 | 3.348 | 10.587 | 2.238 | 2.261 | 96.85∘ | 102.31∘
This work (with vdW) | 4.398 | 3.325 | 10.429 | 2.227 | 2.257 | 96.56∘ | 102.20∘
This work (without vdW) | 4.503 | 3.311 | 11.033 | 2.227 | 2.261 | 96.07∘ | 103.03∘
Table 2:  Structural parameters of strained MLP and BLP.  $\sigma$ | a(${\AA}$) | b(${\AA}$) |  $d_{1}$(${\AA}$) |  $d_{2}$(${\AA}$) | $\theta_{1}$ | $\theta_{2}$
---|---|---|---|---|---|---
|  |  | MLP |  |  |
0.15 | 4.281 | 3.293 | 2.198 | 2.545 | 97.04∘ | 100.27∘
0.10 | 4.379 | 3.294 | 2.210 | 2.451 | 96.38∘ | 101.24∘
0.05 | 4.441 | 3.311 | 2.224 | 2.355 | 96.22∘ | 102.04∘
0.00 | 4.525 | 3.311 | 2.222 | 2.255 | 96.35∘ | 103.35∘
-0.05 | 4.671 | 3.302 | 2.215 | 2.200 | 96.40∘ | 105.10∘
-0.10 | 4.904 | 3.292 | 2.201 | 2.159 | 96.81∘ | 107.74∘
-0.25 | 5.729 | 3.287 | 2.176 | 2.151 | 98.14∘ | 116.00∘
|  |  | BLP |  |  |
0.40 | 4.527 | 3.308 | 2.221 | 2.256 | 96.27∘ | 103.33∘
0.15 | 4.523 | 3.308 | 2.222 | 2.261 | 96.21∘ | 103.31∘
0.10 | 4.513 | 3.310 | 2.223 | 2.265 | 96.20∘ | 103.16∘
0.05 | 4.486 | 3.311 | 2.225 | 2.267 | 96.19∘ | 102.92∘
0.00 | 4.485 | 3.314 | 2.225 | 2.259 | 96.24∘ | 102.98∘
-0.05 | 4.505 | 3.320 | 2.226 | 2.233 | 96.46∘ | 103.33∘
-0.10 | 4.607 | 3.324 | 2.225 | 2.208 | 96.67∘ | 104.41∘
-0.20 | 4.994 | 3.390 | 2.220 | 2.199 | 99.55∘ | 107.91∘

Table 3: Frequencies ($cm^{-1}$)of Raman modes of strained phosphorene.  $\sigma$ | $B_{3g}$ | $B_{1g}$ | $B_{2g}^{1}$ | $B_{2g}^{2}$ | $A_{g}^{1}$ | $A_{g}^{2}$
---|---|---|---|---|---|---
|  |  | MLP111Ref. 16, . |  |  |
0.00 | 196 | 433 | 226 | 427 | 368 | 456
|  |  | MLP222This work |  |  |
0.12 | 170.1 | 417.4 | 152.5 | 276.6 | 147.8 | 455.6
0.10 | 175.2 | 415.8 | 162.8 | 297.1 | 152.0 | 444.0
0.08 | 180.9 | 416.2 | 175.9 | 320.1 | 170.1 | 433.5
0.05 | 187.2 | 418.1 | 196.0 | 357.8 | 266.9 | 431.4
0.00 | 188.2 | 421.9 | 222.9 | 424.5 | 349.7 | 449.9
-0.02 | 187.8 | 424.3 | 226.8 | 437.5 | 354.5 | 459.2
-0.03 | 185.4 | 425.5 | 230.1 | 449.0 | 359.2 | 466.6
|  |  | BLP222This work |  |  |
0.40 | 188.9 | 422.3 | 221.9 | 423.4 | 349.5 | 449.5
0.15 | 189.0 | 421.7 | 221.4 | 420.3 | 345.4 | 448.2
0.10 | 189.1 | 421.2 | 221.1 | 418.0 | 344.9 | 448.5
0.05 | 188.9 | 421.2 | 220.0 | 414.7 | 341.5 | 446.2
0.00 | 188.1 | 422.6 | 221.2 | 419.9 | 346.7 | 450.3
-0.03 | 186.2 | 424.2 | 226.3 | 431.0 | 353.6 | 452.6
-0.05 | 185.8 | 426.0 | 228.1 | 436.7 | 357.8 | 463.0
Table 4: The calculated electronic densities of states at Fermi level $N(E_{F})$, EP coupling constant $\lambda$, the logarithmically averaged frequency $\omega_{ln}$ and the estimated $T_{C}$ for BLP at three different interlayer distances. d/$d_{0}$ | $N(E_{F})(States/eV)$ | $\lambda$ | $\omega_{ln}(K)$ | $T_{C}(K)$
---|---|---|---|---
0.80 | 0.68 | 0.32 | 183.2 | 0.2
0.75 | 0.86 | 0.85 | 107.9 | 5.7
0.70 | 1.08 | 1.45 | 96.1 | 10.6


11institutetext: The University of Queensland, St Lucia, Australia

# Contextuality in distributed systems††thanks: Supported through an
Australian Government Research Training Program Scholarship and Discovery
Grant DP190102142 from the Australian Research Council (ARC).

Nasos Evangelou-Oost  0000-0002-8313-6127 Callum Bannister
0000-0002-8799-054X Ian J. Hayes  0000-0003-3649-392X

###### Abstract

We present a lattice of distributed program specifications, whose ordering
represents implementability/refinement. Specifications are modelled by
families of subsets of _relative_ execution traces, which encode the local
orderings of state transitions, rather than their absolute timing according to
a global clock. This is to overcome fundamental physical difficulties with
synchronisation. The lattice of specifications is assembled and analysed with
several established mathematical tools. Sets of nondegenerate cells of a
simplicial set are used to model relative traces, presheaves model the
parametrisation of these traces by a topological space of variables, and
information algebras reveal novel constraints on program correctness. The
latter aspect brings the enterprise of program specification under the
widening umbrella of _contextual semantics_ introduced by Abramsky et al. In
this model of program specifications, _contextuality_ manifests as a failure
of a consistency criterion comparable to Lamport’s definition of _sequential
consistency_. The theory of information algebras also suggests efficient local
computation algorithms for the verification of this criterion. The novel
constructions in this paper have been verified in the proof assistant
_Isabelle/HOL_.

###### Keywords:

Information algebras Presheaves Refinement lattices.

##  1 Introduction

Lattices of sets of traces have been successful in the field of formal methods
as algebraic models for program specification and verification. For concurrent
programs, two important examples are trace models of Concurrent Kleene Algebra
[DBLP:journals/jlp/HoareSMSZ16] and Concurrent Refinement Algebra
[DBLP:conf/fm/HayesCMWV16]. A core advantage of these models is that they
facilitate _compositional reasoning_ , which mitigates the inherent
difficulties in analysing the exponential proliferation of program behaviours
that occur when programs run in parallel.

These models do not explicitly account for the topological structure inherent
to a distributed system, which can make reasoning about local behaviour, e.g.
local variable blocks, cumbersome. Moreover, trace models for concurrency
often implicitly assume a “global clock” with which all traces progress in
lockstep—and this assumption limits their applicability to systems distributed
over significant distances in space, due to physical (e.g. from relativistic
physics) constraints on synchronisation.

In this article we describe a lattice of program specifications, that encodes
the possible behaviours of a distributed system as subsets of _relative_
traces, as well as its configuration into independent parallel processors.

Inspirations for this work are previous uses of presheaves in concurrency
[DBLP:journals/mscs/Goguen92, DBLP:journals/iandc/JoyalNW96], the
aforementioned refinement algebras [DBLP:journals/jlp/HoareSMSZ16,
DBLP:conf/fm/HayesCMWV16], topological models of concurrency such as
[DBLP:conf/dagstuhl/Porter06, DBLP:books/sp/FajstrupGHMR16], and the diverse
applications of valuation and information algebras [pouly2012generic], their
relationship to sheaves, and their associated notion of _contextuality_
[MR4024458, DBLP:phd/ethos/Caru19].

In Section 2 we introduce the notion of a _relative trace_ , which is a chain
in a proset (preordered set) of states. Such relative traces are assigned to
each variable $v\in V$ in a distributed system, and are modelled as a subset
of the cartesian product $\prod_{v\in V}\mathbf{\Omega}_{v}$ of the prosets
$\mathbf{\Omega}_{v}$ of states for each variable. The variables $v\in V$ are
topologised in a space $(V,\mathcal{D})$ representing the physical
configuration of the variables, and the inclusion ordering of open sets in
this space induces a restriction action on the traces. These traces are called
_relative_ because this restriction action does not preserve the absolute
timing of their states, but only the ordering of state transitions.

In LABEL:sec.specs, we introduce the notion of a _specification_ , which is a
pair $(\mathbf{A},\mathcal{U})$ where $\mathbf{A}$ is a presheaf on
$\mathcal{D}$ whose values are subsets of possible relative traces, and
$\mathcal{U}$ is a maximal cover of $(V,\mathcal{D})$, representing the
distribution of the specification into independent asynchronous components.
Moreover, we explain that such specifications form a lattice
$(\mathbf{\Lambda},\preceq)$ whose ordering represents the refinement relation
between specifications.

In LABEL:sec.info, we define an _information algebra_ and some associated
constructs. We show how the lattice of specifications $\mathbf{\Lambda}$
corresponds to a particular ordered, adjoint information algebra. This
associated information algebra permits the definition of _local_ and _global
consistency_ for specifications, which we introduce in LABEL:sec.consistency.

In LABEL:sec.dining, we show how these consistency criteria can arise in a
classical scenario in distributed systems, namely the _dining philosophers_.

In the following we assume familiarity with the basic definitions of order
theory and category theory, e.g. of a proset, a lattice, a category, a
functor, a natural transformation, etc. Possibly less familiar
structures—topological spaces, presheaves, and (augmented) simplicial sets—are
briefly reviewed.

Several of the constructions and proofs in this paper have been formalised in
the _Isabelle/HOL_ proof
assistant.111https://github.com/onomatic/ramics23-proofs

##  2 Relative traces

A _topological space_ is an abstract model of a geometric space, but without
built-in notions of angle, distance, curvature, etc. It is formalised set-
theoretically as a set $X$ of _points_ , and a set
$\mathcal{T}\subseteq\mathbf{P}X$ of subsets of $X$, called _open sets_ ,
which are closed under unions and finite intersections222Consequently,
$\mathcal{T}$ contains at least $\emptyset$ and $X$, being the union and
intersection respectively of an empty family of open sets.. Roughly speaking,
the open sets measure proximity, where nearby points occupy many open sets in
common.

Let $V$ be a finite333Finiteness is not crucial, but it simplifies our
presentation, and in real-world examples finiteness is a realistic assumption.
set of _variables_ of a distributed computer system and $\mathcal{D}$ a
topology on $V$. We call the open sets of $\mathcal{D}$ _domains_. We consider
that a single computer is also a distributed system (as in
[DBLP:journals/cacm/Lamport78]); then $V$ could be the set of memory locations
over its CPU caches, RAM modules, hard disks, etc., and the topology
$\mathcal{D}$ encodes the connectivity between these parts. Or, $V$ could be
the set of memory locations over a distributed database comprised of many
individual computers, and $\mathcal{D}$ represents the network topology of
this distributed system.

###  2.0.1 The frame functor $\mathbf{\Omega}$.

To each variable $v\in V$ we associate a _nonempty_ proset
$\mathbf{\Omega}_{v}$ of _states_ whose order represents reachability or
causality, and we extend this assignment to open sets $U\in\mathcal{D}$ by
setting

$\mathbf{\Omega}U\vcentcolon=\prod_{v\in U}\mathbf{\Omega}_{v}$ (1)

where the right-hand side is a cartesian product of prosets (for which the
ordering is given componentwise). Moreover, each inclusion of open sets
$U\subseteq U^{\prime}$ induces a function
$\mathbf{\Omega}U^{\prime}\to\mathbf{\Omega}U$ by projection of tuples (i.e.
function restriction). Such restriction maps
$\mathbf{\Omega}U^{\prime}\to\mathbf{\Omega}U$ have the effect of discarding
information involving variables outside $U$.

We evidently have that the function induced by $U\subseteq U$ is the identity,
and if $U^{\prime\prime}\subseteq U^{\prime}\subseteq U$ then the functions
$\mathbf{\Omega}U\to\mathbf{\Omega}U^{\prime\prime}$ and the composite of
$\mathbf{\Omega}U^{\prime}\to\mathbf{\Omega}U^{\prime\prime}$ and
$\mathbf{\Omega}U\to\mathbf{\Omega}U^{\prime}$ are equal—it does not matter
whether we restrict tuples immediately to $U^{\prime\prime}$, or first
restrict to $U^{\prime}$ and then restrict to $U^{\prime\prime}$. These
assignments and properties are summarised in saying that
$\mathbf{\Omega}:{\mathcal{D}}^{\mathbf{op}}\to\mathcal{Pro}$ is a
contravariant functor from the posetal category $\mathcal{D}$ to the category
of prosets, dubbed the _frame functor_.

###  2.0.2 The augmented simplicial nerve functor $\mathbf{N}$.

A _presheaf_ is a contravariant functor valued in sets. An _augmented
simplicial set_ 444An ordinary _simplicial set_ is a presheaf on the full
subcategory $\Delta\subset\Delta_{+}$ consisting of only _nonempty_ linearly
ordered posets. is a presheaf $S:{\Delta_{+}}^{\mathbf{op}}\to\mathcal{Set}$
whose domain is the _augmented simplex category_ $\Delta_{+}$. This is the
category with

  * •

objects as linearly ordered posets $\left\langle
n\right\rangle\vcentcolon=\left\\{i\in\mathbb{N}\mid 0\leq i\leq n\right\\}$
for integers $n\geq 0$, as well as
$\left\langle-1\right\rangle\vcentcolon=\emptyset$,

  * •

morphisms as weakly monotone functions $f:\left\langle
n\right\rangle\to\left\langle m\right\rangle$, i.e. those satisfying $i\leq
j\implies fi\leq fj$.

Augmented simplicial sets are the objects of a category, denoted by
$\mathcal{Set}_{\Delta_{+}}$, with natural transformations as morphisms. It is
conventional to write the application $S\left\langle n\right\rangle$ of an
augmented simplicial set $S$ on $\left\langle n\right\rangle\in\Delta_{+}$ as
$S_{n}$, and the application $Sf$ of $S$ on a morphism $f\in\Delta_{+}$ as
$f^{*}$.

For an augmented simplicial set $S\in\mathcal{Set}_{\Delta_{+}}$, an element
$x\in S_{n}$ is called an _$n$ -cell of $S$_, or a _cell of $S$ (of degree
$n$)_. The cell $x$ is _degenerate_ if there exists a non-injective function
$f\in\Delta_{+}$ and a cell $y$ of $S$ with $x=f^{*}y$. Note that any cell of
degree $-1$ is nondegenerate.

To any proset $\mathcal{P}$ we can produce an augmented simplicial set
$\mathbf{N}\mathcal{P}$, called the _augmented simplicial nerve of
$\mathcal{P}$,_ whose action on objects $\left\langle n\right\rangle$ for
$n\in\mathbb{N}\cup\left\\{-1\right\\}$ is given by

${\left\lparen\mathbf{N}\mathcal{P}\right\rparen}_{n}\vcentcolon=\mathcal{Pro}\left\lparen\left\langle
n\right\rangle,\mathcal{P}\right\rparen$ (2)

where $\mathcal{Pro}\left\lparen\left\langle
n\right\rangle,\mathcal{P}\right\rparen$ is the set of monotone functions from
$\left\langle n\right\rangle$ to $\mathcal{P}$, or equivalently the set of all
_chains of length $n$_ in $\mathcal{P}$, including the empty chain, denoted
$[\,]$, which is said to have length $-1$. Given a morphism
$f:\mathcal{P}\to\mathcal{P}^{\prime}$ in $\mathcal{Pro}$, we define
$\mathbf{N}f:\mathbf{N}\mathcal{P}\to\mathbf{N}\mathcal{P}^{\prime}$ as
postcomposition with $f$, i.e.

$(x:\left\langle n\right\rangle\to\mathcal{P})\mapsto(f\circ x:\left\langle
n\right\rangle\to\mathcal{P}^{\prime})$ (3)

This evidently defines a functor
$\mathbf{N}:\mathcal{Pro}\to\mathcal{Set}_{\Delta_{+}}$. Note that for
$\mathcal{P}\in\mathcal{Pro}$ a proset, a nondegenerate $n$-cell of
$\mathbf{N}\mathcal{P}$ is a chain of length $n$ in $\mathcal{P}$ _that
contains no repeated adjacent elements_. In particular, the empty chain $[\,]$
is nondegenerate.

For each domain $U\in\mathcal{D}$, $(\mathbf{N}\circ\mathbf{\Omega})U$ is an
augmented simplicial set, such that for each
$n\in\mathbb{N}\cup\left\\{-1\right\\}$,
${\left\lparen(\mathbf{N}\circ\mathbf{\Omega})U\right\rparen}_{n}$ is the set
of all possible sequences of states in $\mathbf{\Omega}U$, and the functions
${\left\lparen(\mathbf{N}\circ\mathbf{\Omega})U\right\rparen}_{n}\to{\left\lparen(\mathbf{N}\circ\mathbf{\Omega})U\right\rparen}_{m}$
are generated by _mumbling_ and _stuttering_ maps on traces
[DBLP:journals/iandc/Brookes96] (i.e. maps that omit or repeat elements of a
sequence, respectively).

###  2.0.3 The nondegenerate cells functor $\mathbf{D}$.

A basic result in the theory of simplicial sets is the following:

######  Lemma 1 (Eilenberg-Zilber [MR0210125, II.3.1, pp. 26-27]555The cited
result is stated for ordinary simplicial sets, but it clearly also applies to
augmented simplicial sets as there are no degeneracies of a cell of degree
$-1$.)

For each cell $x$ of an augmented simplicial set $S$ there exists a unique
nondegenerate cell $x^{\prime}$ such that there exists a unique surjection
$f_{x}\in\Delta_{+}$ with $x=f_{x}^{*}x^{\prime}$.

For any augmented simplicial set $S$ we can produce a plain set $\mathbf{D}S$
consisting of only the nondegenerate cells of $S$ (as in Lemma 1). Moreover,
we can extend this assignment to morphisms of $\mathcal{Set}_{\Delta_{+}}$,

$\displaystyle\mathbf{D}$
$\displaystyle:\mathcal{Set}_{\Delta_{+}}\to\mathcal{Set}$ (4) $\displaystyle
S$ $\displaystyle\mapsto\text{set of nondegenerate cells of }S$ (5)
$\displaystyle\alpha$ $\displaystyle\mapsto(x\mapsto(\alpha x)^{\prime})$ (6)

In other words, the action of $\mathbf{D}$ on morphisms of augmented
simplicial sets $\alpha:S\to S^{\prime}$ gives a function
$\mathbf{D}\alpha:\mathbf{D}S\to\mathbf{D}S^{\prime}$ that sends a
nondegenerate cell $x$ of $\mathbf{D}S$ to the nondegenerate cell $(\alpha
x)^{\prime}$ of $S^{\prime}$ that generates $\alpha x$, whose existence and
uniqueness are assured by Lemma 1. These assignments assemble to a functor.

###### Lemma 2

[proof in LABEL:lem.nondegen_functor_proof] $\mathbf{D}$ is a functor.

Postcomposing $\mathbf{N}\circ\mathbf{\Omega}$ by $\mathbf{D}$, we obtain the
presheaf

$\mathbf{\Theta}\vcentcolon=\mathbf{D}\circ\mathbf{N}\circ\mathbf{\Omega}:{\mathcal{D}}^{\mathbf{op}}\to\mathcal{Set}$
(7)

that we call _chaos_.

######  Definition 1 (relative trace)

For $U\in\mathcal{D}$, an element $t\in\mathbf{\Theta}U$ is a _$U$ -relative
trace_, or for short, a _$U$ -trace_ or simply a _trace_.

###### Note 1

We use the standard shorthand for restriction maps of a presheaf
$F:{\mathcal{\mathcal{T}}}^{\mathbf{op}}\to\mathcal{Set}$ (e.g.
$F=\mathbf{\Theta}$) on a topological space $(X,\mathcal{T})$;
$\left.t\right|_{U}\vcentcolon=(Fi)t$ for $t\in FU^{\prime}$ and
$U,U^{\prime}\in\mathcal{D}$ with $U^{\prime}\subseteq U$ and $i$ the unique
morphism $i:U\to U^{\prime}$.