# Superconducting and topological properties of compound Lu4H7N

Zheng-Wei Liao  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Xin-Wei Yi  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Jing-Yang
You<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, 2 Science Drive 3, Singapore 117551  Bo Gu<EMAIL_ADDRESS>Kavli
Institute for Theoretical Sciences, and CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China  Gang Su<EMAIL_ADDRESS>School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Kavli
Institute for Theoretical Sciences, and CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China

###### Abstract

A recent experiment has reported a nitrogen-doped lutetium hydride acheving a
remarkable Tc of 294 K at just 1 GPa, significantly reducing the required
pressure for obtaining room temperature superconductivity. However, subsequent
experimental and theoretical investigations have encountered difficulties in
replicating these results, leaving the structure of this Lu-H-N compound
shrouded in uncertainty. Here, we propose a stable structure for Lu4H7N
employing first-principles calculations. Our calculations reveal that Lu4H7N
has a Tc of 1.044 K, which can be substantially enhanced to 11.721 K at 150
GPa, due to the increasing electron-phonon coupling (EPC). Notably, we delve
into the nontrivial Z2 band topology of Lu4H7N, featuring discernible surface
states near the Fermi level, and we explore its spin Hall conductivity
characteristics. Furthermore, we find that the electron doping can enhance the
EPC strength and Tc of Lu4H7N, such as the Lu4H7O structure we predict
simulating electron doping for Lu4H7N with an impressive Tc of 3.837 K. This
work demonstrates the coexistence of superconducting and topological
properties in a Lu-H-N system compound, which holds the promise of guiding the
search for novel topological superconducting materials.

##  I INTRODUCTION

Superconducting materials hold immense promise and practical applications
across diverse fields, including energy, transportation, and healthcare,
thanks to their excellent properties of zero electrical resistance and perfect
diamagnetism. Nonetheless, the vast majority of superconductors exhibit
disappointingly low superconducting temperatures (Tc), limiting their utility
in real-world scenarios. Hence, the search for superconducting materials with
substantially higher Tc, ideally at or near room-temperature, has been a
longstanding aspiration [1]. During the final two decades of the preceding
century, a pivotal breakthrough occurred with the discovery of copper-based
superconductors capable of achieving Tc in the range of liquid nitrogen
temperatures under ambient pressure with a record-high Tc of 130 K reported
for the HgBa2Ca2Cu3O8+x structure [2]. According to the tenets of BCS theory,
attaining high Tc necessitates a high Debye temperature, which, in turn,
demands high vibrational frequency and low atomic mass [3]. Naturally,
hydrogen, characterized by its minuscule atomic mass, emerged as the focus of
attention. However, a significant challenge arose from hydrogen’s gaseous
nature in standard environmental conditions, requiring extreme external
pressures to induce metallic behavior. In 2004, Ashcroft proposed that the
introduction of other elements surrounding hydrogen atoms can create a
chemical pre-compressing environment that could reduce the external pressure
required for hydrogen’s metallization, which may facilitate the development of
high Tc superconductors [4]. Since then, many hydrogen-rich materials with
high Tc have been predicted, including sulfur hydrides [5, 6], rare earth
hydrides [7, 8, 9], alkali metal hydrides [10], alkali earth metal hydrides
[11], and transition metal hydrides [12]. Among them, the LaH10 structure has
been synthesized with the highest Tc, reaching 250-260 K at 170-190 GPa,
remarkably close to room-temperature [8, 13]. However, it is evident that
despite the impressive progress in enhancing the Tc of hydride
superconductors, the critical challenge remains the exceptionally high
pressure conditions required, imposing severe limitations on the practical
applications of these materials. Therefore, the pursuit of methods to reduce
the necessary pressure has emerged as a vital research direction in the realm
of hydride superconductors.

Recently, Dasenbrock-Gammon $et~{}al.$ claimed that they discovered a Lu-H-N
system achieving a Tc of 294 K at a low pressure of 1 GPa, much lower than the
pressures required for H3S and LaH10 [14]. This groundbreaking finding has
sparked substantial interest in the Lu-H-N system as a potential avenue for
high-temperature superconductivity. Their hypothesis revolves around the
possibility of this compound adopting a Lu4H11N structure, wherein a nitrogen
atom replaces the hydrogen atom in either octahedral or tetrahedral
interstitial sites of the $Fm\overline{3}m$ LuH3 lattice. Unfortunately, many
attempts to synthesize Lu-H-N compounds in experiments have not yielded Tc
above 1.5 K within the pressure range from ambient to 50.5 GPa [17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27]. In addition, numerous theoretical works have
sought to explore various Lu-H-N structures and elucidate the reasons behind
the observed color changes in Lu-H-N compounds during experiments [28, 29, 30,
31, 32, 33, 34, 35, 36, 37], but the precise structure remains elusive.

In this article, we found that the Lu4H11N structures by replacing the
octahedral or tetrahedral interstitial H atom with a N atom respectively and
even $Fm\overline{3}m$ LuH3 are dynamically unstable. Therefore, we used
CALYPSO [38] to search Lu-H-N compounds with radios of Lu:H:N from 4:7:1 to
4:4:4, and obtained a new stable structure Lu4H7N with a space group of $C2/m$
with a Tc of 1.044 K and EPC $\lambda$ of 0.413. By electron doping or
applying pressure, the Tc of Lu4H7N can be enhanced. The Tc of Lu4H7O, which
is electron doped relative to Lu4H7N, reaches 3.837 K, and at a pressure of
150 GPa, Lu4H7N exhibits a higher Tc of 11.721 K. Notably, Lu4H7N features
robust topological properties, including a strong Z2 index and clear surface
states near the Fermi level. The spin Hall conductivity (SHC) of Lu4H7N is
evaluated to be about 55$\sim$145 $\hbar\cdot(e\cdot\Omega\cdot cm)^{-1}$ at
the Fermi level, with the potential to soar to 292
$\hbar\cdot(e\cdot\Omega\cdot cm)^{-1}$ at 0.47 eV.

##  II CALCULATION METHOD

The first-principles calculations were based on the density-functional theory
(DFT) as implemented in the Vienna $ab$ $initio$ (VASP) [39], $Wannier$90
[40], $WannierTools$ [41], and QUANTUM-ESPRESSO (QE) packages [42], with the
projector augmented wave method [43]. VASP was used to optimize the structure
until the forces on atoms are less than 1meV/Å, and to calculate the
electronic band structure, within the Perdew-Burke-Ernzerhof parameterization
of the generalized gradient approximation [44]. The plane-wave cutoff energy
was taken as 500 eV. The Monkhorst-Pack scheme was used to sample the
Brillouin zone with a k-mesh of 9$\times$9$\times$9.

$Wannier$90 and $Wanniertools$ packages were used to obtain the effective
tight-binding Hamiltonian, SHC, surface spectra and topological properties.
SHC tensor is calculated by employing the Kubo formula [45, 46]:

$\sigma_{\alpha\beta}^{\gamma}=e\hbar\int\frac{dk}{(2\pi)^{3}}\Omega_{\alpha\beta}^{\gamma}(k),$
(1)

where $\Omega_{\alpha\beta}^{\gamma}(k)$ is the k-resolved term obtained by
integrating the spin Berry curvature $\Omega_{n,\alpha\beta}^{\gamma}(k)$ of
all occupied bands. The k-point mesh for the spin Berry curvature integral
adopts 100$\times$100$\times$100, and an extra 5$\times$5$\times$5 fine mesh
around those points with $\Omega_{n,\alpha\beta}^{\gamma}(k)$ exceeding 100 Å2
is added. Spin-orbit coupling (SOC) is considered in SHC calculation.

The calculations of phonon spectra and superconducting properties were
performed using QE within density functional perturbation theory (DFPT) [47].
The plane-waves kinetic-energy cutoff was set as 90 Ry and the q-point mesh in
the first BZ was taken as 4$\times$4$\times$4\. An unshifted k-point mesh of
16$\times$16$\times$16 was utilized. In the DFPT calculations, Methfessel-
paxton smearing method [48] with a smearing value 0.010 Ry was taken. To
estimate Tc, we use the McMillan-Allen-Dynes equation [49, 50]:

${\rm Tc}=\frac{\omega_{log}}{1.2}{\rm
exp}[{-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}}],$ (2)

where $\omega_{log}$ is the logarithmically averaged phonon frequency, and
$\lambda$ is a dimensionless parameter describing the EPC strength. The
effective screened Coulomb repulsion constant $\mu^{\star}$ was taken as a
typical value of 0.1.

Figure 1: (a) The crystal structure, (b) Brillouin zone (BZ), and (c)
formation enthalpy of Lu4H7N.

##  III Results

###  III.1 Crystal structure and stability of Lu4H7N

Given the prevailing ambiguity surrounding the structural configuration of the
Lu-H-N system, we adopted an approach by replacing the hydrogen atom in
octahedral and tetrahedral interstitial sites of $Fm\overline{3}m$ LuH3 with a
nitrogen atom as suggested by Dasenbrock-Gammon $et~{}al.$, respectively [14].
However, upon conducting thorough calculations, we observed that the phonon
spectra for these two structures have imaginary frequencies, indicative of
dynamic instability. Notably, even LuH3 also displays imaginary phonons
(detailed analysis is given in Sec. S1 of the Supplemental Materials [51]).
Therefore, we focus our exploration on the well-established and stable
$Fm\overline{3}m$ LuH2 structure. Employing CALYPSO, we conducted an extensive
search for potential structures, varying the Lu:H:N ratio from 4:7:1 to 4:4:4,
and obtained a dynamically stable structure Lu4H7N with a $C2/m$ space group
as depicted in Fig.1(a). The optimized lattice constants are $a=6.224$ Å and
$c=5.555$ Å with lattice angles of $\alpha=91.739^{\circ}$ and
$\gamma=141.526^{\circ}$. In Fig.1(b), we present the Brillouin zone,
highlighting the high-symmetry paths
$\Gamma$-$Y$-$F$-$H$-$Z$-$I$-$F_{1}$|$H_{1}$-$Y$-$X$-$\Gamma$-$N$|$M$-$\Gamma$.
In Fig.1(c), we determined the formation enthalpies of Lu4H7N at different
pressures to compare with those of multiple synthesis pathways using some
synthesized structures, such as Lu ($P6_{3}/mmc$), H2 ($P6_{3}/mmc$), N2
($Pa\overline{3}$), LuH3 ($P\overline{3}c1$), LuH2 ($Fm\overline{3}m$), LuN
($Fm\overline{3}m$), and H3N($P2\uline{~{}}1\overline{3}$). It can be seen
that Lu4H7N boasts a lower energy. Furthermore, it’s noteworthy that the
formation energy of Lu2N3, a compound synthesized experimentally with a space
group of $Ia\overline{3}$, stands at 0.86 eV/atom above the hull [52, 53].
This value is notably higher than the formation energy of Lu4H7N at ambient
pressure, which registers at 0.43 eV/atom. Thus, it is possible to obtain this
Lu4H7N structure in experiment.

Figure 2: (a) The electronic band structure, projected density of states (DOS)
without and with spin-orbit coupling (SOC), (b) phonon spectrum, projected
phonon density of states (PhDOS), Eliashberg spectral function
$\alpha^{2}F(\omega)$, and cumulative frequency-dependent EPC
$\lambda(\omega)$ of Lu4H7N.

###  III.2 Superconductivity of Lu4H7N

Through the first-principles calculations, we find that the Tc of Lu4H7N is
1.044 K with the EPC $\lambda$ 0.413. Fig.2(a) gives the electronic band
structures and projected density of states (DOS) for Lu4H7N without and with
SOC, revealing its metallic nature. In particular, Lu atoms contribute to the
majority of DOS near the Fermi level. In Fig.2(b), the phonon spectra,
projected phonon density of states (PhDOS), Eliashberg spectral function
$\alpha^{2}F(\omega)$, and cumulative frequency-dependent EPC
$\lambda(\omega)$ of Lu4H7N are comprehensively analyzed. It’s evident that
vibration modes below 5 THz predominantly originate from Lu atoms, while a
distinct flat band around 8THz results from the vibration of N atoms. On the
other hand, H atoms play a substantial role in contributing vibration modes
higher than 10 THz. Remarkably, the low-frequency vibration region below 10
THz accounts for about 73% of the total EPC, underscoring the pivotal
significance of Lu and N atoms as the principal contributors to the EPC in
Lu4H7N.

Figure 3: (a) The isoenergy slice at 0.2 eV of the surface state spectrum
$A(\bm{k},\omega)$ projected along [010] direction, (b) four larger
independent components of spin Hall conductivity (SHC) tensor as a function of
chemical potential, (c) band structure weighted by spin Berry curvature
$\Omega_{n,zy}^{x}(\bm{k})$ and k-resolved $\Omega_{yz}^{x}(\bm{k})$ by
integrating the spin Berry curvature of all occupied bands along the high-
symmetry paths at the Fermi level for Lu4H7N.

###  III.3 Topological properties of Lu4H7N

Topological superconductivity is a promising method for quantum computing, so
the quest for materials exhibiting both superconducting and topological
properties has become a focal point of intense interest [54, 55, 56]. As shown
in Fig. 2(a), several band gapped points emerge along the $H$-$Z$-$I$ paths at
around 0.2$\sim$0.5 eV close to the Fermi level because of SOC. To delve
deeper into the topological properties, we calculated the Z2 index, yielding a
result of (1,001), a clear indication that Lu4H7N qualifies as a topological
semimetal [57]. Considering the presence of a mirror symmetry within the
$C2/m$ space group, we examined the mirror plane (010) of Lu4H7N. Fig. 3(c)
illustrates an isoenergy slice at 0.2 eV for the surface state spectrum
$A(\bm{k},\omega)$ projected along the [010] direction, which clearly outlines
the surface states.

We further investigate the SHC of Lu4H7N from the intrinsic contribution, an
intrinsic property contingent upon the electronic band structure and
topological characteristics. The crystal symmetry of Lu4H7N imposes constrains
on the SHC tensor, encompassing thirteen independent components [58]. We have
calculated these thirteen independent components and displayed the four most
substantial components $\sigma_{yz}^{x}$, $\sigma_{zy}^{x}$,
$\sigma_{xz}^{y}$, and $\sigma_{zx}^{y}$ in Fig. 3(b). The components exhibit
magnitudes in the range of 55$\sim$145 $\hbar\cdot(e\cdot\Omega\cdot cm)^{-1}$
at the Fermi level, with $\sigma_{zy}^{x}$ notably reaching an impressive 292
$\hbar\cdot(e\cdot\Omega\cdot cm)^{-1}$ at 0.47 eV. To reveal the nature of
SHC, we plot the band structure of Lu4H7N colored by the magnitude of spin
Berry curvature $\Omega_{n,yz}^{x}(\bm{k})$, as well as the k-resolved
$\Omega_{yz}^{x}$ at the Fermi level in Fig. 3(c). It is obvious that
$\Omega_{n,yz}^{x}(\bm{k})$ shows conspicous peaks aligning with positions of
small gapped points, signifying their primary contributions to SHC [59].

Table 1: The DOS at the Fermi level $N(E_{F})$ (in unit of states/spin/eV/cell), volume of primitive cell $V$ (Å3), $\omega_{log}$ (K), $\lambda$, and Tc (K) for Lu4H7N at ambient pressure, 10, 100, 150, and 200 GPa. |  $P$(GPa) |  $N(E_{F}$) |  $V$(Å3) |  $\omega_{log}$(K) |  $\lambda$ |  Tc(K)
---|---|---|---|---|---|---
|  0 |  18.53 |  134.13 |  207.3 |  0.413 |  1.044
|  10 |  17.88 |  120.12 |  213.0 |  0.464 |  1.892
|  100 |  13.10 |  77.28 |  214.5 |  0.724 |  8.097
|  150 |  12.18 |  69.31 |  219.4 |  0.860 |  11.721
|  200 |  11.66 |  64.09 |  237.4 |  0.773 |  10.355
Table 2: The DOS at Fermi level $N(E_{F})$ (in unit of states/spin/eV/cell), volume of primitive cell $V$ (Å3), $\omega_{log}$ (K), $\lambda$, and Tc (K) for Lu4H7N, LuH2, Lu4H7C and Lu4H7O at ambient pressure. |  $N(E_{F}$) |  $V$(Å3) |  $\omega_{log}$(K) |  $\lambda$ |  Tc(K)
---|---|---|---|---|---
Lu4H7N |  18.53 |  134.13 |  207.3 |  0.413 |  1.044
LuH2 |  3.95 |  31.67 |  297.4 |  0.258 |  0.024
Lu4H7C |  8.32 |  136.88 |  232.6 |  0.235 |  0.059
Lu4H7O |  20.89 |  134.81 |  149.7 |  0.624 |  3.837

Figure 4: The phonon spectra, projected PhDOS, $\alpha^{2}F(\omega)$, and
$\lambda(\omega)$ of Lu4H7N at (a) 10, (b) 100, and (c) 150 GPa.
Figure 5: (a) The electronic band structure, projected DOS, (b) phonon
spectrum, projected PhDOS, $\alpha^{2}F(\omega)$, and $\lambda(\omega)$ of
Lu4H7O.

###  III.4 Discussion

To gain a deeper insight into the superconducting properties of Lu4H7N, we
conducted calculations in two aspects. First, we applied pressure to Lu4H7N
and found that upon reaching a pressure of 10 GPa, the structure remained
dynamically stable, and the Tc reaches 1.892 K. However, with continued
pressure increasing, the volume of Lu4H7N decreases, causing the N atoms to
draw nearer to the surrounding Lu atoms which intensifies the repulsive
forces, ultimately leading to dynamic instability. At an elevated pressure of
100 GPa, the positions of N atoms changed, resulting in the structure
regaining its dynamical stability (for a detailed analysis, please refer to
Sec. S3 of the Supplemental Materials [51]). Therefore, we postulate the
existence of another structural form of Lu4H7N in the pressure range between
10 and 100 GPa. For the stable Lu4H7N structure above 100 GPa, we calculated
its superconducting properties, as meticulously detailed in Table 1. As the
pressure increases, the electron DOS at the Fermi level gradually decreases,
while $\omega_{log}$ steadily increases. As observed, the Tc of Lu4H7N reaches
11.721 K, with an enhanced EPC $\lambda$ of 0.86 at 150 GPa. Fig. 4 presents
the superconducting properties of Lu4H7N at 10, 100, and 150 GPa. Compared to
Lu4H7N at ambient pressure and near-ambient pressure (10 GPa), it becomes
evident that at pressures exceeding 100 GPa, the vibrations originating from
Lu atoms, which constitute the primary contribution to $\lambda$, extend from
the 0$\sim$5 THz range to 0$\sim$10 THz,, significantly increasing $\lambda$
and driving an increase of Tc.

In addition, we replaced the N atom in Lu4H7N with a C or O atom to simulate
hole or electron doping, respectively. Subsequently, we conducted thorough
assessments of their superconducting properties, and proceeded to compare them
with the well-known $Fm\overline{3}m$ LuH2, as outlined in Table 2. The
electron DOS of LuH2 and Lu4H7C at the Fermi level are much lower than that of
Lu4H7N, weakening the EPC strength with a Tc less than 0.1 K. In stark
contrast, as shown in Fig. 5, we observe a higher electron DOS at the Fermi
level and PhDOS of Lu atoms in Lu4H7O, which serve as the primary source of
$\lambda$, consequently enhancing the EPC strength and yielding an impressive
Tc of 3.837 K.

##  IV SUMMARY

To summarize, we predicted a new stable Lu-H-N compound Lu4H7N, which has a Tc
of 1.044 K. Additionally, Lu4H7N exhibits a strong Z2 index and clear surface
states near the Fermi level. The SOC opens several band gaps near the Fermi
level, where large spin Berry curvatures exist and contribute significantly to
the SHC of Lu4H7N. Furthermore, we found that the Tc of Lu4H7N can be enhanced
to 11.721 K at 150 GPa. We also performed element substitution based on the
Lu4H7N structure to simulate hole and electron doping and obtained two stable
materials Lu4H7C and Lu4H7O with calculated Tc of 0.059 and 3.837 K,
respectively. The change in Tc is primarily attributed to the modification of
electronic DOS following doping. Our study reveals the coexistence of
superconducting and topological properties within the Lu4H7N structure, making
it valuable for advancing research on novel topological superconducting
materials.

###### Acknowledgements.

This work is supported in part by the Innovation Program for Quantum Science
and Technology (No. 2021ZD0301800), the National Key R&D Program of China
(Grant No. 2018YFA0305800), the Strategic Priority Research Program of the
Chinese Academy of Sciences (Grant No. XDB28000000), the National Natural
Science Foundation of China (Grant No.11834014), and Beijing Municipal Science
and Technology Commission (Grant No. Z118100004218001). B.G. is supported in
part by the National Natural Science Foundation of China (Grant No. 12074378),
the Chinese Academy of Sciences (Grant No. YSBR-030), the Strategic Priority
Research Program of Chinese Academy of Sciences (Grant No. XDB33000000), and
the Beijing Natural Science Foundation（(Grant No. Z190011).

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