# Crystal structures and high-temperature superconductivity in molybdenum-
hydrogen binary system under high pressure

Aiqin Yang  MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Advanced Functional
Materials and Mesoscopic Physics, School of Physics, Xi’an Jiaotong
University, 710049, Xi’an, Shaanxi, P.R.China  Xiangru Tao  MOE Key Laboratory
for Non-equilibrium Synthesis and Modulation of Condensed Matter, Shaanxi
Province Key Laboratory of Advanced Functional Materials and Mesoscopic
Physics, School of Physics, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi,
P.R.China  Yundi Quan  MOE Key Laboratory for Non-equilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Advanced
Functional Materials and Mesoscopic Physics, School of Physics, Xi’an Jiaotong
University, 710049, Xi’an, Shaanxi, P.R.China  Peng Zhang
<EMAIL_ADDRESS>MOE Key Laboratory for Non-equilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Advanced
Functional Materials and Mesoscopic Physics, School of Physics, Xi’an Jiaotong
University, 710049, Xi’an, Shaanxi, P.R.China

###### Abstract

Motivated by advances in hydrogen-rich superconductors in the past decades, we
conducted variable-composition structural searches in Mo-H binary system at
high pressure. A new composition-pressure phase diagram of thermodynamically
stable structures has been derived. Besides all previously discovered
superconducting molybdenum hydrides, we also identified series of
thermodynamically metastable superconducting structures, including I4/mmm-
Mo3H14, I4cm-MoH9, P4/nmm-MoH10 and P4212-MoH10, with the superconducting
transition temperatures from 55 to 126 K at 300 GPa. In these superconducting
molybdenum hydrides, vibrations of the Mo-atoms contributes significantly to
the electron-phonon coupling and the superconducting transition temperature,
in complementary to the contributions by the vibrations of the H-atoms. Our
works highlight the importance of compounds with non-integer composition ratio
and metastable states in material searches, for example the potential high
temperature superconductors.

##  I Introduction

In 1911, Onnes discovered superconductivity in mercury, which started an new
era in science. [1] However, the progress in searching for high-temperature
superconductors is limited. Even after more than a century of exploration, the
highest superconducting transition temperature at ambient pressure is 133 K in
cuprates [2], and that of conventional superconductors is only 39 K in MgB2
[3]. Early in 1968, Ashcroft suggested that metallic hydrogen could be a
candidate for high temperature superconductors due to its high-frequency
phonons and strong electron-phonon coupling. [4] Unfortunately, the
synthesizing of metallic hydrogen by physical compression requires extremely
high pressure, which leads to ambiguities and discrepancies in experiments.
[5, 6, 7, 8, 9] In 2004, Ashcroft further proposed an alternative that pre-
compression of hydrogen atoms by chemical pressure from surrounding atoms in
hydrides could reduce the required physical pressure for superconductivity.
[10] Following the strategy, series of high-temperature superconducting
hydrides have been found in the last decade. The superconducting temperature
of H3S is 203 K at 155 GPa [11, 12], and that of LaH10 is 260 K at 180-200
GPa. [13, 14, 15, 16, 17, 18] Very recently, Dasenbrock-Gammon et al. [19]
even claimed that lutetium-nitrogen-hydrogen has $T_{\text{c}}$ of 294 K at 1
GPa, although most of the subsequent works cannot reproduce their results.
[20, 21, 22, 23, 24, 25, 26] The efforts in searching for new superconducting
hydrides are still on-going.

Molybdenum, crystallized in a body centered cubic structure under ambient
conditions, is a typical transition metal with high melting point and
toughness. [27] As early as 2003, the pressure-temperature phase diagram of
the Mo-H binary system was reported by Fukai et al. [28]. Several follow-up
works have studied the crystal structure phase transitions in this system as
well. [29, 30, 31] The superconducting transition temperatures in MoH and MoD
of P63/mmc structure are 0.92 K and 1.11 K at ambient pressure. [32] The
hydrogen content in molybdenum hydrides will increase under compression. [28,
31] Fukai et al. [28] showed that the H/Mo ratio in the body-centered cubic
$\alpha$-phase of Mo-H system is small at the ambient pressure, which
increases significantly at high pressures to a stoichiometeric composition of
around 1 in the hexagonal close packed $\epsilon$-phase or face-centered cubic
$\gamma$-phase.

The density functional theory (DFT) based crystal structure prediction
algorithms [33, 34, 35] are proved successful in searching for superconducting
hydrides, including the Mo-H binary superconductors. In 2016, Feng et al. [36]
investigated the MoHn (n = 1, 2 and 3) compounds below 100 GPa using the
crystal structure prediction method. MoH2 (P63mc) of the hexagonal structure,
with the same arrangement of metal atoms as MoH (P63/mmc), was predicted
stable at the pressure of 9-24 GPa, then transformed into orthorhombic
structure (Pnma) under higher pressure. In 2020, Liao et al. [37] conducted
structural searches of MoHn (n = 1-6) up to 300 GPa, and found four new
superconducting phases including I4/m-MoH4 (Tc of 41-49 K at 250 GPa), I4/mmm-
MoH4 (Tc of 64-73 K at 300 GPa), C2/m-MoH6 (Tc of 3.1-5.9 K at 150 GPa), and
Immm-MoH6 (Tc of 74-82 K at 300 GPa). In 2021, Du et al. [38] investigated the
phases of MoHn (n=1-12) compounds at pressures up to 300 GPa, where three new
superconducting phase Cmcm-MoH5 (Tc of 67-72 K at 150 GPa), Pnma-MoH6 (Tc of
22-27 K at 100 GPa) and Cmmm-MoH11 (Tc of 107-117 K at 250 GPa) were
predicted.

The structural searches above are limited to compounds of MoHn-type
stoichiometry, while the potentially stable or superconducting Mo-H compounds
of MomHn-type stoichiometry are missing. Actually, binary compounds of m:n
composition ratio could be very important. Recent work by Tikhonov et al. [39]
reported thermodynamically stable vanadium hydrides of non-integer composition
ratio, including I4/mmm-V3H2, C222-V6H5, R$\bar{3}$m-V3H7, Ibam-V2H5 and
I$\bar{4}$2m-V4H11, using the variable-composition crystal structure
prediction method. Another example is the lutetium hydrides. Li et al. [40]
discovered superconductivity of $T_{\text{c}}$ $\approx$ 71 K at 218 GPa in
lutetium polyhydrides, with candidate of Pm$\bar{3}$n-Lu4H23 stoichiometry.

In this work, we carried out comprehensive variable-composition structural
searches of the Mo-H binary system up to 300 GPa. An updated pressure-
composition phase diagram of Mo-H systems is derived, including five
unprecedented molybdenum hydrides of non-integer H/Mo ratios. Besides, we also
discovered series of thermodynamically metastable high-temperature
superconducting structures, including I4/mmm-Mo3H14, I4cm-MoH9, P4/nmm-MoH10
and P4212-MoH10, with estimated Tc from 55 to 126 K at high pressure.
Therefore, our works established an example to stress the importance of
structures of non-integer stoichiometry or thermodynamical metastability in
searching for novel compounds such as high temperature superconductors.

Figure 1: (a) Formation enthalpies of the Mo-H system at 100, 200 and 300 GPa.
Red dots indicate thermodynamically stable phases and blue ’x’ indicate
metastable phases above the convex hull of black solid line. (b) The pressure-
composition phase diagram of the Mo-H system in the pressure range of 0-300
GPa.

##  II Method

The evolutionary algorithm USPEX [35] is adopted in our structure prediction
calculations. Totally more than 3000 structures are generated in our variable-
composition searches in 50 generations, and more than 400 structures are
generated in each fixed-compositions search in 8-12 generations. In both
searches up to 40 atoms are included in the unit cell. The Vienna ab initio
simulation program (VASP) [41] was used in structure prediction calculations
for structural relaxation and derivations of the total energy. The convergence
criteria for force and energy are 3 meV/Å and 1$\times$10-5 eV, repectively.
We used projector augmented planewave (PAW) [42] potentials with an energy
cutoff of 500 eV and the $\Gamma$-centered k-point meshes of 2$\pi\times$ 0.06
Å-1 resolutions for Brillouin zone sampling to ensure the convergence of total
energy. We also conducted higher precision calculations to pin down the convex
hull and the composition-pressure phase diagram. More accurate convergence
criteria for force and energy at 0.01 meV/Å and 1$\times$10-9 eV were token.
The PAW energy cutoff is increased to 800 eV and the k-point meshes of the
Brillouin zone is 2$\pi\times$ 0.03 Å-1.

The electronic structures and the phonon related calculations are performed
using the Quantum Espresso package [43, 44]. The optimized norm-conversing
pseudopotentials [45, 46] were used with the valence electrons 4s24p64d55s1
for Mo and 1s1 for H. The kinetic energy cutoff and the charge density cutoff
for wavefunctions were set to 70 and 280 Ry, respectively. The Methfessel-
Paxton [47] smearing with a spreading of 0.02 Ry was used for the Brillouin
zone integration. A uniform $\Gamma$-centered k-point of
12$\times$12$\times$12 were employed in the electronic self-consistent
calculations. The structures were optimized until the maximum energy and force
were less than 1.0$\times$10-7 Ry and 1.0$\times$10-5 Ry/Bohr, respectively.
Phonon dispersion and electron-phonon coupling (EPC) were calculated within
density functional perturbation theory [48]. Self-consistent electron density
and electron-phonon coefficients were calculated with 24$\times$24$\times$24
k-point meshes and 4$\times$4$\times$4 q-point meshes.

The superconductive transition temperatures were estimated using the McMillan
equation,

$\displaystyle
T_{c}=\frac{\omega_{log}}{1.2}\exp\left(-\frac{1.04(1+\lambda)}{\lambda-\mu^{\ast}\left(1+0.62\lambda\right)}\right)$
(1)

The electron-phonon coupling constant $\lambda$ and the logarithmic average
frequency $\omega_{log}$ can be obtained from Eliashberg function
$\alpha^{2}F(\omega)$ by

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
2\int\frac{\alpha^{2}F(\omega)}{\omega}d\omega$ (2)

and

$\displaystyle\omega_{log}$ $\displaystyle=$
$\displaystyle\exp\left(\frac{2}{\lambda}\int\frac{d\omega}{\omega}\alpha^{2}F(\omega)\ln(\omega)\right)$
(3)

##  III Results and discussion

###  III.1 Convex hull and phase stabilities

We have performed comprehensive variable-composition and fixed-compositions
structure searches in the Mo-H system at the pressures of 100, 200 and 300
GPa. Comparing with previous studies, thermodynamically stable phases of non-
integer H/Mo composition ratio are discovered in our study. All
thermodynamically stable structures at the convex hull are marked by red dots,
as shown in Fig. 1(a). At 100 GPa, there are seven thermodynamically stable
phases, the new P31m-Mo6H7, C2/m-Mo4H5, C2/m-Mo2H3 and P$\bar{1}$-Mo4H7
structures and the previously known P63/mmc-MoH, Pnma-MoH2 and Pnma-MoH6
structures. [36, 37, 38] Based upon the output structures of the searches, we
further derived the pressure-composition phase diagram of Mo-H system in the
range of 0-300 GPa as summarized in Fig. 1(b). The Pnma-MoH2 structure remains
stable up to 300 GPa, consistent with previous work [37, 38]. In contrast, MoH
undergoes a structural phase transition from P63/mmc structure to R$\bar{3}$m
structure at around 195 GPa. The Pnma-MoH6 structure remains stable in the
pressure range of 100-139 GPa, which is narrower than the range of 75-175 GPa
predicted by Du et al. [38]. Moreover, we found that P4/nmm-MoH4 is stable in
the range of 18-67 GPa. The I4/mmm-MoH4 and the Immm-MoH6 structures
discovered by Liao et al. [37] are thermodynamically metastable in our study.
P31m-Mo6H7 structure is thermodynamically stable from 100 to 110 GPa. P4/mbm-
Mo3H14 structure is thermodynamically stable above 140 GPa. The Cmmm-MoH11
structure predicted by Du et al. [38] becomes thermodynamically metastable due
to the appearance of P4/mbm-Mo3H14 phase in our work. Mo2H3 of C2/m structure
transforms into more energetically favorable P21/m structure at about 115 GPa
and remains thermodynamically stable until 300 GPa. Mo4H5 of C2/m structure
transforms into P$\bar{1}$ structure above 137 GPa, which remains stable until
200 GPa.

In addition to the thermodynamically stable structures, we also examined the
superconductivity of energetic competitive metastable phases within 50
meV/atom above the convex hull. Totally nine metastable molybdenum-hydrides
are found superconducting at 300 GPa, including P42/mnm-MoH3 (5.7 meV/atom
above hull), I4/m-MoH4 (11.8 meV/atom above hull), I4/mmm-Mo3H14 (2.5 meV/atom
above hull), P4/mmm-Mo3H14 (2.3 meV/atom above hull), R3-Mo3H14 (37.3 meV/atom
above hull), P31m-Mo3H14 (39.6 meV/atom above hull), I4cm-MoH9 (12.5 meV/atom
above hull), P4/nmm-MoH10 (20.6 meV/atom above hull) and P4212-MoH10 (27.0
meV/atom above hull). As a matter of fact, metastable materials, from
polymorphs of metals [49, 50, 51, 52] to fullerene C60 [53], have long been
synthesized and implemented. [54] P63/mmc-NdH9 is predicted 35 meV/atom above
the convex hull at 150 GPa, and has been successfully synthesized at high
pressure with Tc $\approx$ 4.5 K. [55] Previous structural searches also
predicted the metastable ternary Li2MgH16 of $\sim$20 meV/atom above the
convex hull, which has a very high Tc $\approx$ 473 K at 250 GPa [56, 57].
Therefore, the metastable nature of these discovered molybdenum-hydrides in
our study doesn’t necessarily exclude the possibility for the synthesis of
them in experiments. [58]

###  III.2 Superconductivity, phonon spectra and electron-phonon coupling

Table 1: The calculated electronic DOS of H at the Fermi level NH(EF), the total DOS at the Fermi level N(EF), the EPC parameter $\lambda$, the logarithmic average frequency $\omega_{log}$, and the superconducting transition temperature Tc of molybdenum-hydrides at 300 GPa, where the thermodynamically stable structures are shown in bold. Mo-H | Space group | NH(EF) | N(EF) | $\lambda$ | $\omega_{log}$ | Tc |
---|---|---|---|---|---|---|---
|  | (states/meV/Å3) |  | (cm-1) | (K) |
MoH | R$\bar{3}$m | 2.6 | 57.9 | 0.56 | 411 | 7-11 |
MoH3 | P42/mnm | 3.0 | 35.3 | 0.47 | 521 | 4-7 |
MoH4 | I4/m | 5.8 | 47.2 | 0.65 | 545 | 17-23 |
Mo3H14 | P4/mmm | 4.1 | 43.6 | 0.65 | 404 | 13-17 |
Mo3H14 | P4/mbm | 4.2 | 48.0 | 0.60 | 518 | 12-17 |
Mo3H14 | P31m | 5.8 | 28.1 | 0.62 | 678 | 18-25 |
Mo3H14 | R3 | 6.0 | 28.2 | 0.58 | 670 | 14-20 |
Mo3H14 | I4/mmm | 16.4 | 60.9 | 1.09 | 597 | 60-68 |
MoH9 | I4cm | 9.3 | 25.9 | 0.88 | 811 | 55-66 |
MoH10 | P4/nmm | 17.1 | 32.0 | 2.24 | 432 | 90-96 |
MoH10 | P4212 | 15.5 | 29.7 | 1.77 | 665 | 116-126 |
Figure 2: The phonon dispersion relation, the total and the projected PHDOS,
the Eliashberg function $\alpha^{2}F(\omega)$ and the EPC parameter $\lambda$
of P4/nmm-MoH10 (a) P4212-MoH10 (b) I4/mmm-Mo3H14 (c) and I4cm-MoH9 (d) at 300
GPa.

The superconducting transition temperature Tc at 300 GPa of all eleven new
structures discovered in this study are listed in Table 1, together with their
total electronic density of states (DOS) at the Fermi level N(EF), the
hydrogen-DOS at the Fermi level NH(EF), the EPC parameter $\lambda$ and the
logarithmic average frequency $\omega_{log}$. Among all thermodynamically
stable structures, only R$\bar{3}$m-MoH and P4/mbm-Mo3H14 are found
superconducting with transition temperature at 7-11 K and 12-17 K,
respectively. The hydrogen-DOS at the Fermi level NH(EF), the EPC parameter
$\lambda$, and the superconducting temperatures Tc of I4/mmm-Mo3H14, I4cm-
MoH9, P4/nmm-MoH10 and P4212-MoH10 are significantly larger than these of
other seven structures. Obviously, in our study the hydrogen-DOS at the Fermi
level NH(EF) play critical roles in control the EPC parameter $\lambda$ and
consequently the superconducting transition temperature Tc of molybdenum
hydrides. In order to explain the origin of high-temperature superconductivity
in molybdenum hydrides, we mainly focus on these four superconducting
structures, I4/mmm-Mo3H14, I4cm-MoH9, P4/nmm-MoH10 and P4212-MoH10, in later
contents.

The phonon spectra and the phonon density of states (PHDOS) of P4/nmm-MoH10,
P4212-MoH10, I4/mmm-Mo3H14 and I4cm-MoH9 at 300 GPa are shown in Fig. 2.
Absence of imaginary frequency modes in the whole Brillouin zone suggests the
dynamical stability of these phases. As shown in the projected PHDOS, the
phonon modes of these phases can be classified into two parts, the high
frequency modes originating from vibrations of H-atoms above 500 cm-1 and the
low frequency modes being dominated by vibrations of Mo-atoms below 500 cm-1.

Figure 3: Crystal structures of P4/nmm-MoH10 (a, b) P4212-MoH10 (d, e) I4/mmm-
Mo3H14 (h) and I4cm-MoH9 (k) at 300 GPa. (c) and (f) depict the Mo-centered
H17 cage of P4/nmm-MoH10 and P4212-MoH10, respectively. Each H17 cage contains
5 quadrilateral and 20 triangles. (i) and (j) depicts the Mo-centered H14 cage
in I4/mmm-Mo3H14. The Mo1-H14 cage contains 24 isosceles triangle faces, and
the Mo2-H14 cage contains 4 irregular quadrilaterals and 16 isosceles
triangles. The small and large spheres represent H- and Mo-atoms,
respectively.

The Eliashberg function $\alpha^{2}F(\omega)$ and the accumulated EPC
parameter $\lambda$ are presented in the rightmost panel of each subfigure of
Fig. 2. The vibrations of Mo-atoms at the low-frequency region contribute
about 53%, 23%, 45% and 24% of the total $\lambda$ for P4/nmm-MoH10,
P4212-MoH10, I4/mmm-Mo3H14 and I4cm-MoH9, respectively. This indicates that
the low-frequency vibrations of Mo-atoms plays a non-negligible role in
enhancing the superconducting temperature, especially for P4/nmm-MoH10 and
I4/mmm-Mo3H14 structures. The resulting EPC parameters $\lambda$ of P4/nmm-
and P4212-MoH10 are quite large at values of 2.24 and 1.77 at 300 GPa. With
typical Coulomb pseudopotential parameters $\mu^{*}$ from 0.1 to 0.13, Tc of
P4/nmm- and P4212-MoH10 reach 90-96 K and 116-126 K at 300 GPa, respectively.
For P4212-MoH10, despite the relatively small $\lambda$ compared to P4/nmm-
MoH10, it has even higher Tc owing to its larger $\omega_{log}$ at 665 cm-1.

For I4/mmm-Mo3H14, a noticeable feature of the phonon spectrum is the flat
dispersion related to the vibrations of the H2 molecule at very high
frequencies (above 3500 cm-1). Although the H-DOS of I4/mmm-Mo3H14 is
comparable to that of P4212-MoH10 as presented in Table 1, its Tc is estimated
to be only 60-68 K due to the relatively weak electron-phonon coupling.

The distribution of phonon frequencies in I4cm-MoH9 is relatively narrower.
However, it has a logarithmic average frequency $\omega_{log}$ at 811 cm-1,
which indicate that high-frequency phonons contribute significantly to the EPC
strength. Although the H-DOS at the Fermi level NH(EF) and the EPC strength of
I4cm-MoH9 are the smallest among the four high-temperature superconducting
phases, at 9.3 state/meV/Å3 and 0.88, the estimated upper limit of the
superconducting temperature Tc of I4cm-MoH9 is 66 K owing to the large
$\omega_{log}$.

Figure 4: The electronic band structure and density of states (DOS) of P4/nmm-
MoH10 (a) P4212-MoH10 (b) I4/mmm-Mo3H14 (c) and I4cm-MoH9 (d) at 300 GPa.

###  III.3 Crystal structures and electronic properties

The crystal structures of P4/nmm-MoH10, P4212-MoH10, I4/mmm-Mo3H14 and I4cm-
MoH9 at 300 GPa are shown in Fig. 3. The detailed structure parameters are
listed in Table.S1 of the Supplementary Information. For P4/nmm-MoH10 and
P4212-MoH10, the two crystal structures are very close. In both structure,
each Mo-atom has 17 nearest neighbour H-atoms that form a cage surrounding the
Mo-atom at the center. The H17 cage consists of 25 faces, including 5
irregular quadrilaterals and 20 triangles (see Fig. 3(c) and (f)). Two
adjacent hydrogen cages share a quadrilateral face, and four neighbour cages
share a vertex. In the Mo-H17 cages of P4/nmm-MoH10 and P4212-MoH10, the
nearest H-H distances are 1.15 and 1.08 Å, and the average Mo-H distances are
1.72 and 1.73 Å, respectively.

In I4/mmm-Mo3H14, each Mo-atom is surrounded by a hydrogen cage consists of 14
nearest neighbour H-atoms. There are two types of H14 cages in I4/mmm-Mo3H14,
the one enclosing the Mo1-atom has 24 isosceles triangles (see Fig. 3(i)), and
the other enclosing the Mo2-atom has 4 irregular quadrilaterals and 16
isosceles triangles (see Fig. 3(j)). The adjacent Mo2-H14 cages share an
irregular quadrilateral face. In particular, I4/mmm-Mo3H14 structure contains
two H2 molecular like units with a bond length of 0.765 Åat 300 GPa. Different
from the calthrate structures in P4/nmm-MoH10, P4212-MoH10 and I4/mmm-Mo3H14,
I4cm-MoH9 crystal is composed of alternating Mo- and H-layers, as shown in
Fig. 3(k).

The electronic band structures and density of states of P4/nmm-MoH10,
P4212-MoH10, I4/mmm-Mo3H14 and I4cm-MoH9 at 300 GPa are shown in Fig. 4. In
all four high-Tc molybdenum-hydrides, both Mo-4d and H-s have significant
contributions to the total DOS near the Fermi level. In particular, the H-s
DOS in P4/nmm-MoH10 and P4212-MoH10 are comparable to the Mo-4d DOS. Previous
experience in the rare-earth hydrides is that the transfer of rare-earth-4f
electrons to the H-s orbitals would enlarge the effective number of H-s
electrons participating the EPC and then enhance the superconducting
transition temperature Tc. In electronic band structures of P4/nmm-MoH10 and
P4212-MoH10, the Mo-4d electrons and the H-s electrons are strongly hybridized
around the Fermi level (see Fig. 4(a, b)), which suggests the transfer of
Mo-4d electrons to the H-s orbitals. It explains the large H-s DOS and the
high superconducting transition temperature of P4/nmm-MoH10 and P4212-MoH10.
In contrast, in I4/mmm-Mo3H14 and I4cm-MoH9 the hybridization between Mo-4d
and H-s orbitals around the Fermi level is less pronounced as shown in Fig.
4(c) and (d), which is consistent with their relatively smaller H-s DOS at the
Fermi level and their relatively lower superconducting transition temperature.
There is no van Hove singularity in the electronic DOS of these
superconducting hydrides in our study.

###  III.4 Role of H2umbbells

There are H2-like units in I4/mmm-Mo3H14, P4/mmm-Mo3H14 and P4/mbm-Mo3H14
phases. The presence of H2-like units were also observed in previous
structrual searches in MoH11, NbH11, TaH11 and WH11 of Cmmm structure. [38]
Established experience about superconducting hydrides with hydrogen clathrate
structures tells us that H2-like unit doesn’t help in enhancing the
superconducting transition temperature, since it will not contribute to either
the H-DOS at the Fermi level NH(EF) or the EPC parameterized by $\lambda$.
[13] In our study of I4/mmm-Mo3H14, P4/mmm-Mo3H14 and P4/mbm-Mo3H14, NH(EF) is
16.4, 4.1, 4.2 states/meV/Å3 and $\lambda$ is 1.09, 0.65, 0.60, as shown in
Table. 1. As a result, the superconducting transition temperature Tc of
I4/mmm-Mo3H14 is 60-68 K, much higher than the Tc of P4/mmm-Mo3H14 and P4/mbm-
Mo3H14 at 13-17 K and 12-17 K, respectively.

On the other hand, weakening or even elimination of the H-H bonding will help
promoting the Tc of hydrides with hydrogen clathrate structures. The H-H bond
length in I4/mmm-Mo3H14 equals to 0.765 Å, which is longer than the H-H bond
lengths of P4/mmm-Mo3H14 and P4/mbm-Mo3H14 at 0.736 and 0.744 Å, respectively.
The relatively longer H-H bond length of I4/mmm-Mo3H14 suggests suppressed
bonding of the H2-like unit, as well as enhanced NH(EF), $\lambda$ and Tc. By
introduction of extra charges, for example replacing the Mo-atom by heavier
W-atom, the strength of H-H bond in I4/mmm-Mo3H14, P4/mmm-Mo3H14 and P4/mbm-
Mo3H14 could be suppressed, therefore the H-DOS at the Fermi level, the EPC
parameter, and the superconducting transition temperature could be enhanced.

##  IV Conclusion

In summary, we have performed comprehensive variable-composition structural
searches for Mo-H binary system at pressures up to 300 GPa. We reported an
updated phase diagram of the thermodynamically stable structures in Mo-H,
including structures of five unprecedented non-integer H/Mo composition
ratios, Mo2H3, Mo4H5, Mo6H7, Mo4H7 and Mo3H14. Besides all previously known
superconducting molybdenum hydrides, series of thermodynamically metastable
superconducting structures are identified, in which I4/mmm-Mo3H14, I4cm-MoH9,
P4/nmm-MoH10 and P4212-MoH10 have relatively high superconducting transition
temperatures from 55 to 126 K at 300 GPa. Further analysis indicates that
high-Tc of these superconducting molybdenum hydrides originates from
collaborating contributions by both the H-atoms and the Mo-atoms. Our results
confirm that in superconducting hydrides with hydrogen clathrate structures,
H2-like unit plays negative role for elevating the Tc. This research
underlines the importance of phases of non-integer composition ratio or
metastable states in structural searches, therefore sheds new lights on future
explorations of materials with novel properties, such as high temperature
superconductors.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China
(11604255) and the Natural Science Basic Research Program of Shaanxi
(2021JM-001). The calculations are performed at the HPC platform of Xi’an
Jiaotong University.

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# Degenerate Distributed Feedback Photonic Structure with Double Grating
Exhibiting Degenerate Band Edge

Tarek Mealy, and Filippo Capolino  Department of Electrical Engineering and
Computer Science, University of California, Irvine, CA 92697 USA
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

We propose a degenerate version of the Bragg condition, associated to a
degenerate band edge (DBE). A standard Bragg condition can be implemented
using a periodic grating that operates at the regular band edge. The structure
we propose to realize a DBE is made of two stacked identical gratings that
form two coupled periodic waveguides with broken mirror symmetry. The
occurrence of the DBE is verified by using both an eigenmode solver and the
calculations of scattering parameters, using full-wave simulations. The
proposed structure is a good candidate to conceive a degenerate distributed
feedback lasers operating at the DBE.

###### Index Terms:

Exceptional point of degeneracy; Double grating waveguide; Degenerate band
edge laser; Distributed feedback laser.

We present a degenerate version of the Bragg condition in a double grating,
leading to a 4th order degeneracy, namely a degenerate band edge (DBE), where
four eigenmodes coalesce forming a single degenerate eigenmode. Compared to
previous studies [1, 2, 3, 4, 5, 6, 7] devoted to the fundamental aspects of
the DBE, here we build on those concepts and specifically show a realistic
structure that supports a DBE, based on two coupled dielectric layers with
periodic gratings, shown in Fig. 1. The standard Bragg condition $kd=m\pi$ for
a periodic waveguide, where $k$ is the Bloch wavenumber, $d$ is the waveguide
period, and $m$ is an integer, is associated with the regular band edge (RBE)
[8, 9]. A Bragg condition implies that two modes with power flux in opposite
directions merge at a band edge to form a degenerate mode with vanishing group
velocity. The Bragg condition is used in distributed feedback (DFB) lasers so
that light is reflected in a distributed fashion within a cavity rather than
only at the end mirrors [8, 10, 11, 9]. The Bragg condition is implemented
through a periodic grating that acts as a dielectric waveguide. DFB lasers
require the presence of an active medium in addition to satisfying the Bragg
condition $kd=m\pi$ to form a self sustained resonant mode that keeps
oscillating [9]. The “degenerate DFB” concept presented here could improve
laser properties.

The DBE is a particular kind of exceptional point of degeneracy (EPD) of order
$4$ in a waveguide without loss and gain, where $4$ eigenmodes coalesce in
their wavenumbers and polarization states. The concept of degeneracy of the
eigenvalues and eigenvectors and their perturbation was discussed in [12, 13,
14, 15], and more recently also in [1, 2, 16, 7, 17, 18]. Here, we use those
concepts to show that the double grating in Fig. 1 exhibits a DBE (i.e., an
EPD of order four) by engineering the structure’s geometrical dimensions. At a
DBE the dispersion diagram is locally represented as
$\omega_{d}-\omega=h\left(k-k_{d}\right)^{4}$, where $\omega_{d}$ and $k_{d}$
are the DBE angular frequency and wavenumber, respectively, and the parameter
$h$ describes the flatness of the dispersion curve. Therefore, because of the
very flat dispersion, not only the degenerate mode has a vanishing group
velocity $v_{g}=0$, but also its second and third derivatives vanish. Photonic
devices operating at the DBE have the potential to exhibit better performance
than those operating at the RBE [3, 4, 5, 6, 19, 7, 20, 21].

Figure 1: (a) Degenerate distributed feedback photonic cavity with double
grating, based on a DBE, obtained by braking mirror symmetry. (b) Dispersion
diagram of the eigenmodes in the double grating (a) when the geometry is
optimized to exhibit a DBE (red curve). We also show the dispersion of modes
in a single grating (blue curve). Note the much flatter dispersion of the mode
in the double graing (red curve). (c) Electric field distribution of the DBE
mode calculated at $kd=\pi$.

The waveguide with double grating in Fig. 1 consists of a standard grating
over a dielectric layer, coupled to another grating obtained by a mirror
operation in the $x$ direction, followed by a translation $s$ along $z$. The
shift $s$ breaks mirror symmetry, because mirror symmetry prohibits the
existence of a DBE in two coupled waveguides due to the existence of even and
odd modes that are always decoupled from each other. The double grating in
Fig. 1 is made of two silicon on insulator (SOI) waveguides with silicon
refractive index $n_{\mathrm{Si}}=3.45$, and with a cladding material of
silicon dioxide with refractive index $n_{\mathrm{SiO2}}=1.97$.

The modal dispersion relation in Fig. 2 is obtained using the full-wave
eigenmode solver implemented in CST Studio Suite. We used periodic boundaries
at two virtual cross sections at a distance of a period $d.$ We look for modes
polarized along $y$, hence we use perfect electric conductor (PEC) walls at
planes with constant y to account for field invariance in the $y$ direction.
The computational domain is truncated by PEC also at $x=\pm 12t$, where $t$ is
the dielectric layer thickness as illustrated in Fig. 1, and it is
sufficiently large not to affect the field in the grating area, to ensure that
the studied modes decay away from the waveguide in the positive and negative
$x$ directions.

The dielectric waveguide has thickness $t=67$ nm, the gratings has period
$d=$335 nm, width $w=87$ nm, and height $h=121$ nm. The rest of parameters,
the coupling gap $g$ and the shift between the waveguides $s$, are left to be
optimized to achieve a DBE. An optimization procedure based on having a DBE in
the dispersion relation leads to $g=100$ nm and $s=20$ nm. The dispersion
relation obtained from the eignmode solver for the optimized unit cell in the
Fig. 1 shows a DBE at around 193 THz (red curve). The flatness of the DBE is
compared to that of the the dispersion of modes in the single grating
structure (blue curve) with the same waveguide and grating parameters except
for the grating height that is now 42 nm in order to have a regular band edge
more or less at the DBE frequency of 193 THz. It is clear from the figure that
the dispersion relation for the double grating structure (red curve) is
flatter than that of single grating structure (blue curve). The field
distribution of the DBE mode at $k=\pi/d$ is shown in Fig. 1.

Figure 2: Dispersion diagram of modes in the double grating in Fig. 1 for
three distinct gap values. The change of the dispersion curve from a split
band edge ($g=70$ nm) to regular band edge ($g=130$ nm) implies the existence
of DBE using a value of $g$ between 70 nm and 130 nm. Indeed, the figure shows
that the DBE occurs when $g=100$ nm (red curve).

Figure 2 shows the dispersion relation of three distinct designs of the double
grating structure in Fig. 1 based on three distinct values of the coupling gap
$g$, the rest of the dimensions are the same as the ones mentioned for the
case above. Note that for $g=70$ nm the dispersion shows a split band edge
[22], with four modes around $f=193$ THz (black curve), which coalesce when
using $g=100$ nm. For larger values of $g$, the green curve shows a RBE.
Indeed, when a parameter is swept (e.g., $g$) and one finds both a RBE and a
split band edge, there is a proper intermediate value (here, $g=100$ nm) that
leads to a DBE [22].

Figure 3: (a) Finite-length structure used to estimate the transfer matrix
$\underline{\mathbf{T}}_{\mathrm{U}}$ of one unit cell in the middle. (b)
Structure used to de-embed the effect of the two transition regions connected
to the left and right sides of the unit cell in (a). (c) Dispersion diagram
showing four complex-valued wavenumber versus frequency illustrating the
coalescence of four wavenumbers at the DBE point $k_{d}d=\pi$ , which is the
degenerate Bragg condition. The dispersion is obtained based on the transfer
matrix $\underline{\mathbf{T}}_{\mathrm{U}}$ found by dembedding the structure
in (b) from (a). The black-dashed curve shows the dispersion of the purely
real branch obtained from the eigenmode solver, already plotted in Fig. Fig.
1, for validation.

We also verify the existence of the DBE in the proposed double grating
structure by finding the scattering matrix using frequency-domain full-wave
simulations. Open (radiation) boundaries are used in this simulation and we
use waveports as shown in Fig. 3 where two modes (even and odd) are excitable
on each port. Note that simulating one unit cell would not lead to an accurate
transfer matrix because it would not fully account for the structure
periodicity since a waveport supports a waveguide mode and not a Bloch mode.
Therefore, in the simulation we include two segments, on the left and right
sides of the unit cell, that act as adiabatic transitions between the
dielectric waveguide mode and the grating, as shown in Fig. 3. The transition
is a chirped grating with height steps of 30 nm, until we reach the whole
grating height $h=120$ nm. However our goal is to find the transfer matrix
$\underline{\mathbf{T}}_{\mathrm{U}}$ only. Therefore, we also simulated
another structure with two chirped gratings as shown in Fig. 3, which is the
same structure as in Fig. 3 but without the unit cell at the center. The
structure in Fig. 3 is used to deembed the effect of the two transition
segments from the results relative to Fig. 3, similarly to what was done in
[23, 24].

Using even and odd modes in the ports shown in Fig. 3 and 3, and defining
their associated incident and reflected waves

$\begin{array}[]{c}\mathbf{b}=\left[\begin{array}[]{cccc}b_{\mathrm{e}1},&b_{\mathrm{o}1},&b_{\mathrm{e}2},&b_{\mathrm{o}2}\end{array}\right]^{T},\\\
\mathbf{a}=\left[\begin{array}[]{cccc}a_{\mathrm{e}1},&a_{\mathrm{o}1},&a_{\mathrm{e}2},&a_{\mathrm{o}2}\end{array}\right]^{T}.\end{array}$
(1)

such that $\mathbf{b=\underline{\mathbf{S}}a}$. Using full-wave simulations,
we find the $4\times 4$ scattering matrix expressed as

$\underline{\mathbf{S}}=\left[\begin{array}[]{cc}\mathbf{S}_{11}&\mathbf{S}_{12}\\\
\mathbf{S}_{21}&\mathbf{S}_{22}\end{array}\right].$ (2)

It is convenient to define the state vectors
$\mathbf{\mathbf{\Psi}}_{1}=\left[\begin{array}[]{cccc}a_{\mathrm{e}1},&a_{\mathrm{o}1},&b_{\mathrm{e}1},&b_{\mathrm{o}1}\end{array}\right]^{T}$
and
$\mathbf{\mathbf{\Psi}}_{2}=\left[\begin{array}[]{cccc}b_{\mathrm{e}2},&b_{\mathrm{o}2},&a_{\mathrm{e}1},&a_{\mathrm{o}1}\end{array}\right]^{T}$,
relative to port 1 and port 2, respectively. They are defined in way that the
first two elements represent the amplitudes of waves propagating in the
positive $z$ direction, whereas the second two elements represent the
amplitudes of waves propagating in the negative $z$ direction. As a
consequence, the transfer matrix $\underline{\mathbf{T}}$ relating them,
$\mathbf{\mathbf{\Psi}}_{2}=\underline{\mathbf{T}}\mathbf{\mathbf{\Psi}}_{1}$,
is found as [25]

$\underline{\mathbf{T}}=\left[\begin{array}[]{cc}\mathbf{S}_{21}-\mathbf{S}_{22}\mathbf{S}_{12}^{-1}\mathbf{S}_{11}&\mathbf{S}_{22}\mathbf{S}_{12}^{-1}\\\
-\mathbf{S}_{12}^{-1}\mathbf{S}_{11}&\mathbf{S}_{12}^{-1}\end{array}\right].$
(3)

The T-matrices of the two structures shown in Figs. 3 and 3 are
$\underline{\mathbf{T}}_{\mathrm{15}}=\underline{\mathbf{T}}_{\mathrm{R}}\underline{\mathbf{T}}_{\mathrm{U}}\underline{\mathbf{T}}_{\mathrm{L}}$
and
$\underline{\mathbf{T}}_{\mathrm{14}}=\underline{\mathbf{T}}_{\mathrm{R}}\underline{\mathbf{T}}_{\mathrm{L}}$,
respectively, where $\underline{\mathbf{T}}_{\mathrm{R}}$ and
$\underline{\mathbf{T}}_{\mathrm{L}}$ are the T-matrices of the two chirped
grating structures. The chirped structure represented by
$\underline{\mathbf{T}}_{\mathrm{R}}$ is a flipped version, in the $z$
direction, of the other one represented by
$\underline{\mathbf{T}}_{\mathrm{L}}$, and therefore they satisfy
$\underline{\mathbf{T}}_{\mathrm{R}}=\mathbf{F}\underline{\mathbf{T}}_{\mathrm{L}}^{-1}\mathbf{F}^{-1}$,
if $\underline{\mathbf{T}}_{L}$ is not singular, where $\mathbf{F}$ is a
transformation matrix that flips the direction of wave propagation, made by
two $\mathbf{0}$ in the diagonal blocks and two identity matrices in the
codiagonal blocks.

Hence, we calculate the new T-matrix
$\underline{\mathbf{T}}_{\mathrm{n}}=\underline{\mathbf{T}}_{\mathrm{15}}\underline{\mathbf{T}}_{\mathrm{14}}^{-1}=\underline{\mathbf{T}}_{\mathrm{R}}\underline{\mathbf{T}}_{\mathrm{U}}\underline{\mathbf{T}}_{\mathrm{R}}^{-1}$
whose eigenvalues are the same of those of the unit-cell T-matrix
$\underline{\mathbf{T}}_{\mathrm{U}}$, similarly to what is done in [23, 24].
The T-matrices $\underline{\mathbf{T}}_{\mathrm{15}}$ and
$\underline{\mathbf{T}}_{\mathrm{14}}$ are obtained by transformation of the
scattering matrices associated to the 4-port structures in Figs. 3 and Fig. 3
based on3.

According to Floquet-Bloch theory, we look for periodic solutions of the state
vector as $e^{-jkd}$ where $k$ is the Floquet-Bloch complex wavenumber, that
satisfy
$\mathbf{\boldsymbol{\Psi}}^{\prime}=\lambda\mathbf{\boldsymbol{\Psi}}$, with
$\lambda\equiv e^{-jkd}$, where $\mathbf{\boldsymbol{\Psi}}$ and
$\mathbf{\boldsymbol{\Psi}}^{\prime}=\mathbf{\underline{T}}_{\mathrm{U}}\mathbf{\boldsymbol{\Psi}}$
are the state vectors at the input and output of a unit cell. The eigenvalue
problem is then formulated as

$\mathbf{\underline{T}}_{\mathrm{U}}\mathbf{\boldsymbol{\Psi}}=\lambda\mathbf{\boldsymbol{\Psi}},$
(4)

where the eigenvalues $\lambda_{n}\equiv e^{-jk_{n}d}$, with $n=1,2,3,4$, are
obtained by solving the dispersion characteristic equation
$D(k,\omega)\equiv\mathrm{det}(\mathbf{\underline{T}}_{\mathrm{U}}-\lambda\mathbf{\underline{1}})$,
with $\mathbf{\underline{1}}$ being the $4\times 4$ identity matrix.

We show in Fig. 3 the Floquet-Bloch complex-valued wavenumber dispersion
diagram of the four eigenmodes of the structure in Fig. 1. The dispersion
diagram confirms the existence of the DBE at a frequency $f\approx 193$ THz,
where four branches coalesce. The black-dashed curve is the dispersion diagram
with a purely real wavenumber obtained via the eigenmode solver using phase-
shift boundary conditions, shown here for comparison purposes. Indeed, a good
matching is found for the curve with purely real wavenumber, which validates
the method described above based in the dembedding procedure, that confirms
the existence of the DBE.

In conclusion, we have proposed a photonic structure with double grating that
exhibits a DBE in its dispersion diagram where a degenerate Bragg condition
occurs at $kd=\pi$. We have demonstrated the occurrence of the DBE using two
methods: (i) by finding the dispersion relation of modes with purely real
wavenumbers using an eigenmode solver and observing the very flat dispersion,
and (ii) by finding the four complex-valued wavenumbers from the scattering
parameters by using full-wave simulations. The degenerate Bragg condition we
have found in the double grating structure can be used to conceive a
“degenerate distributed feedback laser”, that may lead to interesting
properties, even augmented when compared to a DFB laser. The first DBE laser
analysis was carried out in [19] using a transmission line method for an ideal
structure. Interesting properties were found in that ideal DBE structure like
stability of the lasing oscillation frequency when varying the load, and lower
threshold than the one in the RBE laser, even in the case when the RBE cavity
has the same quality factor of the DBE cavity. Future studies shall be devoted
to study the cavity effect and the lasing properties of the double grating
with degenerate Bragg condition presented in this letter.

## Acknowledgment

This material is based on work supported by the Air Force Office of Scientific
Research award number FA9550-18-1-0355, and by the National Science Foundation
under award NSF ECCS-1711975. The authors are thankful to DS SIMULIA for
providing CST Studio Suite that was instrumental in this study.

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