††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Role of effective mass anisotropy in realizing a hybrid nodal-line fermion
state

Bikash Patra  Department of Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research, Mumbai 400005, India  Rahul Verma
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India  Shin-Ming Huang  Department of
Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan  Center for
Theoretical and Computational Physics, National Sun Yat-sen University,
Kaohsiung 80424, Taiwan  Physics Division, National Center for Theoretical
Sciences, Taipei 10617, Taiwan  Bahadur Singh<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India

###### Abstract

Understanding the role of lattice geometry in shaping topological states and
their properties is of fundamental importance to condensed matter and device
physics. Here we demonstrate how an anisotropic crystal lattice drives a
topological hybrid nodal line in transition metal tetraphosphides $M$P4 ($M$ =
Transition metal). $M$P4 constitutes a unique class of black phosphorus
materials formed by intercalating transition metal ions between the phosphorus
layers without destroying the characteristic anisotropic band structure of the
black phosphorous. Based on the first-principles calculations and $k\cdot p$
theory, we show that $M$P4 harbor a single hybrid nodal line formed between
anisotropic $M~{}d$ and P states with oppositely-oriented effective masses
unhinged from the high-symmetry planes. The nodal line consists of both type-I
and type-II nodal band crossings whose nature and location are determined by
the effective mass anisotropies of the intersecting bands. We further discuss
a possible topological phase transition to exemplify the formation of the
hybrid nodal line state in $M$P4. Our results offer a comprehensive study for
understanding the interplay between structural motifs-driven mass anisotropies
and topology in anisotropic lattice materials to realize hybrid semimetal
states.

##  I Introduction

Since the discovery of topological insulators, the topology of electronic
states and finding robust topological materials have attracted broad interest
[1, 2, 3, 4]. The initial topological state characterization based on free-
fermion symmetries such as time-reversal was generalized later to include
topological states protected by crystalline symmetries [5, 6, 7, 8]. In this
way, many topological crystalline states such as mirror, inversion, or
rotational Chern insulators, topological semimetals with Dirac, Weyl, nodal-
line, nodal-chain, and higher-fold chiral fermions, among other possibilities,
were proposed with their unique set of nontrivial states and electromagnetic
properties [2, 3, 4, 5, 6, 7, 8, 9]. The topological states in materials are
described uniquely by topological invariants that depend only on the
symmetries of their underlying atomic positions [10, 11, 12, 13]. Such
topological state characterization has allowed unique symmetry-to-topology
mapping that facilitated the identification of topological states in high-
throughput materials searches, revealing many topological materials with
unique nontrivial states [14, 15, 16]. These topological searches often
neglect the effects of the spatial arrangement of atoms and their wavefunction
properties even though they are essential for describing the numbers, energy-
momentum relations, and geometries of the nontrivial states [17, 18, 19].

Among various topological states, topological nodal-line semimetals are
considered parent phases for realizing many exotic topological gapped or
ungapped states through systematic lowering of crystalline symmetries [20, 21,
22, 23, 24]. They form one-dimensional (1D) valence and conduction band
crossings in the momentum space, which realize new quasiparticles beyond the
well-known Dirac, Weyl, and Majorana particles in high-energy physics.
Analogous to Dirac and Weyl point semimetals, topological nodal-line
semimetals can be classified as type-I, type-II, or hybrid nodal line
semimetals depending on the fermiology of the crossings bands [25, 26, 27, 28,
29]. In the type-I case, a nodal line is formed between electronlike and
holelike bands and respects Lorentz symmetry. In type-II semimetals, the nodal
line is formed between either two electronlike or holelike bands, strongly
breaking the Lorentz symmetry. In contrast, the hybrid nodal line consists of
both the type-I and type-II band degeneracies that are naturally expected to
occur if one of the crossing bands has a saddlelike energy dispersion. Such
band crossings constitute a Fermi surface with electron and hole pockets that
touch along specific directions in $k$ space. The hybrid nodal line can
realize amplified electron-correlation effects owing to the saddlelike energy
dispersion and directional-dependent magnetotransport properties due to the
Klein tunneling between the electron and hole pockets, among other phenomena
[20, 29, 28, 30].

Because the hybrid nodal lines constitute both the type-I and type-II band
crossings located along different momentum space directions, their
experimental realization remains a daunting challenge. Here we propose that
effective masses of the intersecting bands in anisotropic lattice materials
can determine the dispersion and location of the nodal band crossings. Of
importance is orthorhombic black phosphorus in which individual two-
dimensional (2D) phosphorene layers are stacked together by weak Van der Waals
interactions [31, 32, 33, 34, 32, 35]. Inside each phosphorene layer, the
phosphorus atoms form puckered honeycomb arrangement along the armchair and a
bilayer configuration in the zigzag directions [see Fig. 1(a)-(d)]. This
unique structural motif drives the anisotropic electronic structure and
physical properties that facilitate the use of black phosphorus in devices
with directional selectivity [31, 32, 33, 34, 32]. The electronic structure
and physical properties of black phosphorus can be engineered by pressure,
doping, adsorption, and electric-field controls to realize insulator-to-metal
transition, topological semimetals with anisotropic Dirac bands, and
superconductivity without destroying their anisotropic character [36, 37, 38,
39, 40]. Importantly, the intercalation of transition metal atoms in black
phosphorus generates varied low-symmetry transition metal tetraphosphides
$M$P4 ($M$ = Transition metal) that preserve the anisotropic electronic bands
of black phosphorus but with a semimetallic ground state [41, 42, 43, 44, 45,
46, 47]. Here, we show that the $M$P4 realizes a single hybrid nodal line
driven by mass-anisotropies of $M$$~{}d$ and P bands. By carefully exploring
the structure-property-topology mapping in $M$P4, our results demonstrate how
an anisotropic crystal structure shapes the low-energy topological states in
materials.

##  II Methods

Electronic structure calculations were performed within the density functional
theory (DFT) framework with the projector augmented wave [48, 49] potentials
as implemented in the Vienna ab-initio simulation package (VASP) [50, 51]. We
used Perdew-Burke-Ernzerhof parameterized generalized gradient approximation
(GGA) [52] and hybrid Heyd-Scuseria-Ernzerof (HSE) functionals for the
exchange-correlation effects. The spin-orbit coupling (SOC) was included self-
consistently to incorporate relativistic effects in the calculations. An
energy cut-off of 400 eV for the plane wave basis set and a $8\times 8\times
10$ $\Gamma$-centered $k$ mesh for the Brillouin zone sampling were used. The
experimental lattice parameters with fully relaxed atomic positions were
adopted to calculate the electronic structure and topological properties of
$M$P4. We constructed material-specific tight-binding model Hamiltonian from
the atoms-centered Wannier functions [53]. $M$ $d$ and P $s$ and $p$ orbitals
were used to construct the Wannier functions. The topological properties and
surface states calculations were performed using the WannierTools package
[54].

Figure 1: Crystal lattice and electronic structure of phosphorene and MoP4.
(a) Puckered honeycomb structure of a single phosphorene layer with distinct
P-P bonding arrangements along the armchair ($x$ axis) and zigzag ($y$ axis)
directions. The in-plane bonding anisotropy is the origin of the directional
selective properties of phosphorene. (b)-(c) Band structure of a single
phosphorene layer along the high-symmetry directions (b) and in the full two-
dimensional Brillouin zone (c). (d) Calculated electron (top) and hole
(bottom) carrier effective masses (in the units of mass of electron $m_{e}$)
of a single phosphorene layer in polar coordinates. $\theta=0^{\circ}$ and
$\theta=90^{\circ}$ mark the armchair and zigzag directions as shown in (a).
(e) Perspective view of the bulk crystal lattice of MoP4 with $C2/c$ (No. 15)
symmetry. The dashed horizontal line marks the glide mirror plane. (f) First-
principles semimetallic band structure of MoP4 without spin-orbit coupling.
Irreducible representations at $\Gamma$ point are marked. The symmetry-
protected band crossings near the Fermi level are highlighted in dashed red
circles. (g) The primitive cell Brillouin zone of MoP4 with high-symmetry
points. The mirror plane is in highlighted grey, and the projected (001)
surface Brillouin zone in blue is shown.

##  III Results

###  III.1 Crystal structure and anisotropic electronic properties

The anisotropic properties of black phosphorus are incepted in the anisotropic
lattice of phosphorene, which is a two-dimensional (2D) building block of
black phosphorus [31, 32, 33, 34]. The P atoms in the phosphorene are
covalently bonded with their three neighboring P atoms to form two in-plane
and one out-of-plane bond, generating a hexagonal puckered lattice (Fig.
1(a)). This atomic arrangement drives a highly anisotropic energy dispersion
such that the conduction and valence bands are nearly flat along the zigzag
direction ($\Gamma-Y$) and significantly dispersive along the armchair
direction ($\Gamma-X$) near the band extremum point (see Figs. 1(b)-(c)).
Based on the energy-dispersion, $E=\frac{\hbar^{2}k^{2}}{2m^{*}}$, where
$m^{*}$ is the effective mass, we calculate the direction-dependent $m^{*}$
and show results in Fig 1(d). The effective masses are higher along the zigzag
direction than the armchair direction for both the valence (hole) and
conduction (electron) bands. The hole-effective masses are $8.62m_{e}$ and
$0.14m_{e}$ ($m_{e}$ is the mass of an electron in vacuum) along the zigzag
and armchair directions that yield a mass anisotropy of $\sim 60$. Band
crossings in such anisotropic bands can realize hybrid nodal dispersion under
appropriate material parameters and symmetries as discussed below.

Using phosphorene anisotropic band structure as a guideline, we searched among
the black phosphorus materials and identified the most robust and ideal hybrid
nodal-line candidate in transition metal tetraphosphides $M$P4 [45, 44, 41].
We discuss the structural and electronic properties of $M$P4 by taking MoP4 as
an exemplary system. The single crystals of MoP4 with a black-phosphorus-
derived structure have been grown, and the transport experiments have reported
it as a semimetal with large positive magnetoresistance [44, 45]. MoP4
crystallizes in the monoclinic Bravais lattice with space group $C2/c$ (No.
15) that has lower symmetry than orthorhombic black-phosphorus $Cmce$ (No.
64). The experimental structural parameters are $a=5.3131$ ÅA, $b=11.1588$ ÅA,
$c=5.8343$ ÅA, and $\beta=110.638^{\circ}$ [45]. The crystal structure is
derived by intercalation of the Mo atoms in between the phosphorene layers,
which reorders the atomic-stacking of phosphorene layers from AB to AC in a
way similar to the alkali metals intercalation in black phosphorus [42, 43].
Figure 1(e) depicts the unit cell of MoP4. It consists of two phosphorene
layers with Mo atoms sandwiched between them. The sandwiched Mo atoms form
zigzag chains extending along [001] with a uniform interatomic distance of
3.20 Å. There are 4 Mo and 16 P atoms in the unit cell. All the P atoms can be
divided into two types. The first type of P atoms is covalently bonded with
three adjacent P atoms and forms a dipolar bond with one Mo atom. The other
type of P atoms is covalently bonded with two neighboring P atoms and forms
dipolar bonds with two Mo atoms. This bonding arrangement distorts the P atoms
from their original in-plane positions of black phosphorus and imposes three-
dimensionality in MoP4. The crystal lattice only respects inversion
$\mathcal{I}$, a two-fold rotation
$\mathcal{\tilde{C}}_{2y}:(x,y,z)\Longrightarrow(-x,y,-z+\frac{1}{2})$, and a
single glide mirror $\widetilde{\mathcal{M}}_{y}$
$:(x,y,z)\Longrightarrow(x,-y,z+\frac{1}{2})$ symmetries. Figure 1(g) shows
the primitive cell Brillouin zone (BZ) and projected (001) surface BZ with
marked $\widetilde{\mathcal{M}}_{y}$ mirror plane and the high-symmetry
points.

Figure 2: Effective-mass anisotropy in MoP4. (a) Mo and (b) P atoms resolved
band structure of MoP4. The atomic weights are encoded in the color scale. An
electron-type Mo band ($\Gamma_{2}^{-}$) intersects the P resolved valence
band ($\Gamma_{1}^{+}$) near the $\Gamma$ point. Calculated effective mass
ellipsoid of (c) Mo and (d) P bands around the $\Gamma$ point. A highly
anisotropic peanut-shaped variation of the effective mass revealed the
anisotropic character of P bands. (e)-(g) Calculated effective masses at
constant planes for the Mo band (blue color) and P band (red color). Angular
direction is given from the horizontal axis in each panel. (h)-(j) Two-
dimensional view of the MoP4 crystal structure in the $b$-$c$, $a$-$c$, and
$a$-$b$ planes. The zigzag and armchair directions are along the $c$ and $a$
directions, respectively. The P atomic layers are stacked along the $b$
direction. The effective mass of the P band is much larger in the zigzag
direction than in the armchair direction and follows the characteristic
anisotropic structure of phosphorene. Table 1: Calculated effective masses of
P ($\Gamma_{1}^{+}$) and Mo ($\Gamma_{2}^{-}$) bands in the $y-z$, $x-z$, and
$x-y$ planes. Anisotropic ratio
$\delta_{\\{\Gamma_{1}^{+},\Gamma_{2}^{-}\\}}=\frac{m^{*}_{max}}{m^{*}_{min}}$,
where $m^{*}_{max}$ and $m^{*}_{min}$ are maximum and minimum effective masses
in each plane for P and Mo bands, is also given. The angular location of
$m^{*}_{max}$ and $m^{*}_{min}$ added in parathesis is calculated w.r.t to the
horizontal axis in each plane (see Fig. 2). Effective masses are given in
units of $m_{e}$.

Plane | P band ($\Gamma_{1}^{+}$) | Mo band ($\Gamma_{2}^{-}$)
---|---|---
| $m^{*}_{max}$ | $m^{*}_{min}$ | $\delta_{\Gamma_{1}^{+}}$ | $m^{*}_{max}$ | $m^{*}_{min}$ | $\delta_{\Gamma_{2}^{-}}$
$y-z$ | $1.82~{}(90^{\circ})$ | $0.13~{}(0^{\circ})$ | $14$ | $0.59~{}(0^{\circ})$ | $0.14~{}(90^{\circ})$ | $4.21$
$x-z$ | $2.34~{}(105^{\circ})$ | $0.40~{}(15^{\circ})$ | $5.85$ | $0.47~{}(165^{\circ})$ | $0.13~{}(75^{\circ})$ | $3.61$
$x-y$ | $0.42~{}(0^{\circ})$ | $0.14~{}(90^{\circ})$ | $3$ | $0.59~{}(90^{\circ})$ | $0.43~{}(0^{\circ})$ | $1.37$

Figure 1(f) shows the calculated band structure of MoP4 without SOC. It is
semimetallic where the $\Gamma_{1}^{+}$ valence band intersects the
$\Gamma_{2}^{-}$ conduction band with an inverted band ordering at the
$\Gamma$ point. Particularly the valence and conduction band crossings stay
ungapped along the $\Gamma-Y$ direction. There are additional two-fold band
crossings along the $Z-M$ direction in the valence and conduction region of
the band structure. Upon the inclusion of SOC, the band structure is locally
gapped at each $k$ point, separating valence and conduction bands in the
entire BZ (see the Supplemental Material [55]). Nevertheless, the SOC-induced
gap at the nodal points is less than 15 meV, which is slightly increased to a
value of 25 meV when SOC artificially scaled to 500%. This bandgap opening
preserves the fermionology of the crossing bands so that SOC effects can be
ignored. Moreover, the band crossings at $Z$ and $M$ points in the valence and
conduction region stay robust, realizing nonsymmorphic symmetry-protected
Dirac states in MoP4 [45].

To uncover lattice-driven anisotropic electronic properties and their
connection to black phosphorus, we present the orbital-resolved band structure
of MoP4 in Figs. 2(a)-(b). The conduction band is derived from the Mo atoms,
whereas the valence band is composed of P atoms. These two bands of distinct
atomic character cross in the vicinity of the Fermi level to generate a
semimetallic state. The nature of the valence and conduction bands is further
probed by calculating the orientation-dependent effective masses around the
$\Gamma$ point (see Table 1). In Figs. 2(c) and 2(d), we present these results
for the Mo and P bands, respectively, in the $x-y-z$ space. The effective mass
associated with the Mo band is less anisotropic and forms a spheroid shape in
space. Whereas the P band effective mass is highly anisotropic, forming a
peanut-like shape and mimicking the anisotropic P lattice structure in MoP4.
The structure-to-mass-anisotropy relation is revealed in results shown on
various plane cuts in Figs. 2(e)-(g) and their associated crystal directions
in Figs. 2(h)-(j). Particularly, the P band effective mass (red markers) along
the zigzag direction is higher than the armchair direction, similar to
phosphorene bands. In contrast, the Mo bands (blue markers) exhibit maximum
and minimum effective masses oriented perpendicular to the P bands. The Mo and
P bands with oppositely oriented effective masses cross in the vicinity of the
Fermi level to generate a semimetallic state.

Figure 3: Hybrid nodal line in MoP4. (a) The location of the nodal line in the
reciprocal space. The color represents the nodal line energy. The arrow shows
the transition from type-I to type-II points. The upper and lower portions of
the nodal line across $\Gamma$ are related by inversion symmetry. (b) Energy-
dependent nodal line configuration on $k_{x}-k_{y}$ plane. An extended energy
region below and above the Fermi level (shown with the gray plane) is
revealed. The color represents the $k_{z}$ location of the nodal line. (c),(e)
Band structure in $k_{x}-k_{y}$ plane at (c) $k_{z}=0$ Å-1 and (e) $k_{z}=0.1$
Å-1 with nodal line (red line). (d),(f) Energy dispersion along the
tangential, radial, and perpendicular directions around the shaded points in
(c),(e). At any point on the nodal line, $k_{R}$ defines radial direction,
$k_{T}$ lies perpendicular to $k_{R}$ in the nodal line plane, and $k_{\perp}$
lies normal to the nodal line plane. From these energy dispersion cuts a type
I character at $k_{z}=0$ Å-1 and a type II character (that is, along
perpendicular directions both the crossings bands have the same sign of
velocity) at $k_{z}=0.1$ Å-1 are revealed.

###  III.2 Hybrid nodal line structure

We now characterize the band crossings between Mo and P bands in Fig. 3. The
gapless valence and conduction bands crossing points in $k_{x}-k_{y}-k_{z}$
space are presented in Fig. 3(a). These crossing points trace a line node that
encloses the $\Gamma$ point. The line node is not hooked to the $k_{y}$ mirror
plane but forms a $\mathcal{I}\mathcal{T}$ symmetric structure primarily
located on the $k_{x}-k_{z}$ plane due to the presence of inversion symmetry.
The symmetry protection of the nodal line is determined by calculating the
Berry phase $\gamma=\oint dk\cdot A(k)$, where $A(k)=i\sum_{n}\langle
u_{n,k}|\nabla u_{n,k}\rangle$ is the Berry connection of the occupied Bloch
bands $|u_{n,k}\rangle$. For a generic closed $k$ loop encircling the line
node, we obtain a Berry phase $\gamma=+\pi$. This results in a nontrivial
winding number $\frac{\gamma}{\pi}=+1$, dictating the topological protection
of the line node. The energy and momentum spread of the line node is shown in
Figs. 3(a)-(b). The line node spans an energy range from $-0.17$ eV to $0.08$
eV that passes through the Fermi level. Such an extended energy range can
enable spectroscopic verifications of the nodal line without fine-tuning the
Fermi level in MoP4.

Figures 3(c) and 3(e) show the $E-k_{x}-k_{y}$ band dispersion at $k_{z}=0$
and $k_{z}=0.1$ Å-1 planes, respectively, with the line node shown in red
color. The isolated nodal points (shaded white circle) lie on the $k_{y}$ axis
at the $k_{z}=0$ plane, and on the $k_{x}$ axis at the $k_{z}=0.1$ plane. We
further plot the energy dispersion away from these nodal points in Figs. 3(d)
and 3(f). Dispersing away from the nodal points, the two bands form nearly
quadratic band crossing along the tangential direction and conical band
crossings along the other two momentum directions on the $k_{z}=0$ plane. In
contrast, the two bands have the same sign of velocity along one momentum
direction (in this case, perpendicular direction to the nodal line plane) on
the $k_{z}=0.1$ Å-1. These results clearly show that the line node is hybrid
in character, having both type-I and type-II band crossings. Exploring the
full momentum space location of type-I and type-II band crossings, we find
that the type of band crossings is hooked to their lattice directions. In
particular, the type-I nodal points lie along the armchair directions whereas
the type-II band touchings lie along the zigzag direction of the P atoms. The
nodal line dispersion continuously evolves from type I to type II in
intermediate directions as shown with an arrow in Fig. 3(a) (see the
Supplemental Material [55] for details). Such a hybrid nodal structure arises
due to crossings of anisotropic Mo and P bands with oppositely oriented
effective masses, which generates a dispersive nodal line by displacing the
crossing bands in opposite momentum directions. The large opposite
displacement between the crossing bands drives type II dispersion along
certain momentum directions (see Sec. III.4).

Figure 4: Topological surface states in MoP4. Surface energy spectrum for the
(001) surface of MoP4 at constant energies (a) $E=-0.169$ eV and (b) $E=0.08$
eV obtained with a surface potential energy of -50 meV. The green dots in (a)
and (b) mark the type-I and type-II projected band crossings along the high-
symmetry axes, respectively. A fermi-arc-type connectively is revealed in the
drumhead surface states at these iso-energy contours. Surface band structure
of the (001) surface along the (c) $\bar{Y}$-$\bar{\Gamma}$-$\bar{Y}$ and (d)
$\bar{X}$-$\bar{\Gamma}$-$\bar{X}$ directions. The left panel shows the
closeup band structure highlighted in dashed boxes. The drumhead surface
states lie inside the projected bulk band crossings.

###  III.3 Topological surface states

The existence of drumhead electron states inside or outside nodal-line
projections on the crystal surface is the hallmark of topological nodal-line
semimetals. To showcase these states, we present the calculated (001) surface
band structure of MoP4 in Fig. 4. In Figs. 4(a) and 4(b), we illustrate the
constant energy contours at $E=-0.169$ eV and $E=0.08$ eV as the
representative case of type-I and type-II bulk nodal energies, respectively.
These plots show a rich electronic structure with surface states located
inside and outside the projected nodal line. The surface states connect two
nodal points at a particular energy cut, forming a double Fermi-arc-type
connectivity. These Fermi arc-type states are part of the highly dispersive
drumhead surface states as seen in the surface band structure along
$\overline{Y}-\overline{\Gamma}-\overline{Y}$ and
$\overline{X}-\overline{\Gamma}-\overline{X}$ directions in Figs. 4(c) and
4(d). Such drumhead electronic states and their Fermi-arc-type constant energy
connectivity are unique to the dispersive hybrid line node and may serve as
spectroscopic fingerprints for their experimental verifications.

###  III.4 Effective Hamiltonian

To better understand the underlying mechanism of mass-anisotropy driven hybrid
nodal line, we drive the low-energy effective Hamiltonian using the theory of
invariants [19, 56]. The first-principles results indicate that anisotropic
$\Gamma^{-}_{2}$ and $\Gamma^{+}_{1}$ bands cross to generate the hybrid nodal
line. A $k\cdot p$ Hamiltonian around $\Gamma$ is constrained by time-reversal
symmetry $\Theta:H^{*}(\mathbf{k})=H(-\mathbf{k})$, glide mirror
$\widetilde{\mathcal{M}}_{y}:H(\mathbf{k})=H(M_{y}\mathbf{k})$ and inversion
symmetry $\mathcal{I}:\sigma_{z}H(\mathbf{k})\sigma_{z}=H(-\mathbf{k}$), where
$\sigma_{z}$ is a Pauli matrix. Based on these symmetry constraints, the
spinless two-band $k\cdot p$ Hamiltonian takes the form

$H(\mathbf{k})=\matrixquantity(\varepsilon^{(\text{u})}(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{*}(\mathbf{k})&\varepsilon^{(\text{l})}(\mathbf{k})),$ (1)

where $\varepsilon^{(\text{u})}$ and $\varepsilon^{(\text{l})}$ are associated
with $\Gamma^{-}_{2}$ and $\Gamma^{+}_{1}$ bands with

$\displaystyle\varepsilon^{(\text{u,l})}(\mathbf{k})$
$\displaystyle=\pm\left\\{\sum_{i=x,y,z}\left(\frac{1}{2}\alpha^{(\text{u,l})}_{i}k_{i}^{2}\right)+\alpha^{(\text{u,l})}_{xz}k_{x}k_{z}+\frac{1}{2}\epsilon_{0}\right\\},$
(2) $\displaystyle\Delta(\mathbf{k})$
$\displaystyle=-iV\left(v_{x}k_{x}-v_{z}k_{z}\right).$ (3)

Here all the parameters are considered real. To have concave upward and
downward curves for $\varepsilon^{(\text{u})}$ and $\varepsilon^{(\text{l})}$,
the inverse of absolute effective masses
$\alpha^{(\text{u})}_{i},\alpha^{(\text{l})}_{i}>0$ for $i=x,y,z$ (here
$\hbar=1$). The $\alpha^{(\text{u,l})}_{xz}$ term is introduced to break the
mirror symmetry on the $k_{x}$ and $k_{z}$ planes, giving tilted effective
mass eclipses as found in our calculations. $\epsilon_{0}<0$ gives the band
inversion between the two bands at $\Gamma$. $\Delta(\mathbf{k})$ describes
the interband coupling with strength $\absolutevalue{V\mathbf{k}}$. The
dimensionless parameters $v_{x}$ and $v_{z}$ with a constraint
$v_{x}^{2}+v_{z}^{2}=1$ describe the anisotropy due to interband coupling. The
eigen energies associated with the Hamiltonian (Eq. 1) are

$E_{\pm}(\mathbf{k})=\xi^{(+)}(\mathbf{k})\pm\sqrt{\xi^{(-)}(\mathbf{k})^{2}+\absolutevalue{\Delta(\mathbf{k})}^{2}},$
(4)

where
$\xi^{(\pm)}(\mathbf{k})=\frac{1}{2}\left[\varepsilon^{(\text{u})}(\mathbf{k})\pm\varepsilon^{(\text{l})}(\mathbf{k})\right]$.
For later use, we define
$\alpha_{X}^{(\text{u,l})}=v_{z}^{2}\alpha_{x}^{(\text{u,l})}+v_{x}^{2}\alpha_{z}^{(\text{u,l})}+2v_{x}v_{z}\alpha_{xz}^{(\text{u,l})}$,
$\alpha_{Z}^{(\text{u,l})}=v_{x}v_{z}(\alpha_{x}^{(\text{u,l})}-\alpha_{z}^{(\text{u,l})})+(v_{x}^{2}-v_{z}^{2})\alpha_{xz}^{(\text{u,l})}$.

Since the two crossing bands have equal mirror eigenvalues, their nodal
crossings are not protected by $\widetilde{\mathcal{M}}_{y}$. Instead, the
inversion symmetry $\mathcal{I}$ can protect a line nodal which is determined
by the conditions $\xi^{(-)}(\mathbf{k})=0$ and $\Delta(\mathbf{k})=0$. The
nodal ring for $\beta\in(-\pi,\pi]$ [see Fig. 5(b)] is positioned at

$\mathbf{k}_{\textrm{node}}(\beta)=\left(v_{z}K_{X}\sin(\beta),\pm
K_{y}\cos(\beta),v_{x}K_{X}\sin(\beta)\right),$ (5)

where $K_{X}=\sqrt{\frac{\absolutevalue{\varepsilon_{0}}}{\alpha_{X}}}$ and
$K_{y}=\sqrt{\frac{\absolutevalue{\varepsilon_{0}}}{\alpha_{y}}}$.

Figure 5: Nodal ring configuration and energy dispersion. (a) Nodal ring
configuration in the global coordinates system ($k_{x},k_{y},k_{z}$). A new
coordinate system ($k_{1},k_{2},k_{3}$) is defined where $k_{2}$ is along
$k_{y}$, $k_{1}$ is perpendicular to the nodal-ring plane, and $k_{3}$ is
perpendicular to $k_{1}$ & $k_{2}$ and lie in the nodal-ring plane. Nodal ring
is parametrized by $\beta\in[-\pi,\pi]$. (b) Variation of nodal ring energy
dispersion as a function of $\beta$ in the first quadrant. The energy
dispersion evolves from type-I to type-II for $\beta=0$ to $\beta=\pi/2$.

To describe the local energy dispersion associated with nodal crossings for
different $\beta$, we define a new coordinate system $(k_{1},k_{2},k_{3})$ as
shown in Fig. 5(a). The transformation of
$(k_{1},k_{2},k_{3})\rightarrow(k_{x},k_{y},k_{z})$ is
$k_{x}=v_{x}k_{1}+v_{z}k_{3}$, $k_{y}=k_{2}$, and
$k_{z}=-v_{z}k_{1}+v_{x}k_{3}$. Here $k_{1}$ is perpendicular to the nodal
ring plane ($\Delta(\mathbf{k})=0$). Taking $k_{2}=K_{y}\cos\beta$ and
$k_{3}=K_{X}\sin\beta$ on the nodal ring, the dispersions of
$\varepsilon^{(\text{u})}$ and $\varepsilon^{(\text{l})}$ along $k_{1}$ takes
the form

$\varepsilon^{(\text{u,l})}(k_{1})=\pm\left\\{\frac{1}{2}A^{(\text{u,l})}k_{1}^{2}+\alpha_{Z}^{(\text{u,l})}K_{X}\sin(\beta)k_{1}\right\\}+E_{\textrm{node}}(\beta).$
(6)

where
$A^{(\text{u,l})}=\alpha_{x}^{(\text{u,l})}+\alpha_{z}^{(\text{u,l})}-\alpha_{X}^{(\text{u,l})}$.
$E_{\textrm{node}}(\beta)=\frac{1}{2}K_{X}^{2}\delta\alpha_{X}\sin^{2}(\beta)+\frac{1}{2}K_{y}^{2}\delta\alpha_{y}\cos^{2}(\beta)$
is the energy of the nodal ring, which agrees with our first-principles
results (Fig. 3(b)) for $K_{X}^{2}\delta\alpha_{X}>K_{y}^{2}\delta\alpha_{y}$.
The energy dispersions $\varepsilon^{(\text{u,l})}(k_{1})$ shows shifted
parabolic bands that cross at $k_{1}=0$ with shifts proportional to
$\alpha_{z}^{(\text{u,l})}\sin(\beta)$. Notably, the band shifts increase with
$\beta$. When the band shifts are opposite,
$\alpha_{Z}^{(\text{u})}\alpha_{Z}^{(\text{l})}<0$, a type II feature emerges
(Fig. 5(b)). Below we show that this is the necessary condition for realizing
a type-II band crossing.

Including the interband coupling, $\Delta(\mathbf{k})=-iVk_{1}$, the slopes of
$E_{\pm}(\mathbf{k})$ along $k_{1}$ are

$\partialderivative{k_{1}}E_{\pm}(\mathbf{k})=K_{X}\delta\alpha_{Z}\sin(\beta)\pm\sqrt{\left[K_{X}\alpha_{Z}\sin(\beta)\right]^{2}+V^{2}},$
(7)

where
$\alpha_{Z}=\frac{1}{2}\left(\alpha_{z}^{(\text{u})}+\alpha_{z}^{(\text{l})}\right)$
and
$\delta\alpha_{Z}=\frac{1}{2}\left(\alpha_{z}^{(\text{u})}-\alpha_{z}^{(\text{l})}\right)$.
The condition for a type II nodal dispersion is

$\absolutevalue{\delta\alpha_{Z}K_{X}\sin(\beta)}>\sqrt{\left[\alpha_{Z}K_{X}\sin(\beta)\right]^{2}+V^{2}},$
(8)

which reduces to

$\left(-\alpha_{Z}^{(\text{u})}\alpha_{Z}^{(\text{l})}\right)\sin^{2}(\beta)>\left(\frac{V}{K_{X}}\right)^{2}.$
(9)

Notably, $\alpha_{Z}^{(\text{u,l})}$ characterizes the dressed mass anisotropy
in the $k_{x}$-$k_{z}$ plane for the two bands. The necessary condition for
realizing a type II band dispersion is that the mass anisotropies of the
crossing bands should be opposite,
$\alpha_{Z}^{(\text{u})}\alpha_{Z}^{(\text{l})}<0$. For small $\beta$ and
finite $V$, Eq. (9) can not be satisfied. This can be understood as for
$\beta=0$, there is no $k$-shift in Eq. (6), and the band crossing should be
type-I. However, when $\beta$ approaches $\pm\pi/2$, a type-II band crossing
can be realized (Fig. 5(b)) in the $k_{1}$ direction provided bands should
have large mass anisotropy :
$\absolutevalue{\beta}>\arcsin\left(\absolutevalue{V}/K_{X}\sqrt{-\alpha_{Z}^{(\text{u})}\alpha_{Z}^{(\text{l})}}\right)$.
The dispersion along $k_{2}$ always remains type-I as found in our first-
principles results.

We emphasize that Eq. (9) is the main result of our work. Taking
$\sin^{2}(\beta)=1$ describes the condition for a hybrid nodal ring. A large
opposite mass anisotropy of the crossing bands is an ingredient to realizing a
hybrid nodal line in materials. More precisely, the anisotropy includes the
effect of particle-hole asymmetry between the bands. When the conduction and
the valence bands have opposite mass anisotropies as found in $M$P4, the
crossing bands form a hybrid nodal line.

###  III.5 Materials tunability and phase transition

We now consider the robustness of the hybrid node line and demonstrate a
semimetal-to-insulator transition in $M$P4. Owing to the presence of $M$$-d$
bands, the electronic correlations may play an important role in dictating the
nature of the nodal line in $M$P4 since they could shape the energy dispersion
and effective masses of the crossings bands. In Fig. 6(a), we present the band
structure of MoP4 obtained with HSE06 hybrid functional that incorporates a
part of the exact Fock exchange [57]. The calculated energy dispersions of
valence and conduction bands stay preserved although their band overlap is
reduced compared to GGA results. On exploring the mass anisotropies and nodal
line energy dispersion, we find that the hybrid nodal line is quite robust to
the changes to the exchange-correlation functionals (see the Supplemental
Material [55]). However, due to reduced overlap between the valence and
conduction bands, MoP4 lies close to a semimetal-to-insulator transition
point.

In order to discuss the semimetal-to-insulator transition and the tunability
of hybrid nodal line, it is useful to calculate the HSE06 band structures of
CrP4 and WP4 (Figs. 6(b) and 6(c)). Since CrP4 and WP4 are isostructural to
MoP4, they show similar mass anisotropies in the valence and conduction bands
as seen in MoP4 [44, 45, 46]. However, the magnitude of effective masses and
anisotropy are materials dependent owing to distinct interband coupling
effects (see the Supplemental Material [55] for details). This can be further
understood based on the different structural parameters of $M$P4. Due to the
different effective radii of the $M$ atoms, the lattice constant is decreased
in CrP4 ($a=5.19$ ÅA, $b=10.16$ ÅA, and $c=5.77$ ÅA) and increased in WP4
($a=5.34$ ÅA, $b=11.19$ ÅA, and $c=5.87$ ÅA) as compared to the MoP4. The
change in lattice parameters results in different crystal-field effects and
electronic states in CrP4 and WP4. Particularly, CrP4 realizes an insulator
state with a band gap of 0.54 eV at the $\Gamma$ point, whereas WP4 forms a
hybrid nodal line with an increased band inversion strength. These results
demonstrate that CrP4 and WP4 lie electronically on the opposite sides of
MoP4. Alternatively, the Cr doping in MoP4 will be expected to reduce the
valence and conduction bands overlap and push the material toward the trivial
insulator state. In contrast, W doping in MoP4 will increase the band overlap
and realize a hybrid nodal line with increased band inversion strength.

Figure 6: Hybrid nodal-line to insulator transition and phase tunability. Band
structures of (a) MoP4, (b) CrP4, and (c) WP4 obtained using the HSE hybrid
exchange-correlation functional. (d) Calculated band structure of CrxW1-xP4
along $Y-\Gamma-Y$ direction for different $x$ values. (e) Variation of energy
gap at $\Gamma$ for CrxW1-xP4 as a function of $x$. The related topological
state is highlighted. The red line is a guide to the eye. (f) Schematic
representation of calculated hybrid-functional band structure of
representative compounds in the $M$P4 family. Band inversion strength
increases in WP4 whereas CrP4 realizes a fully gapped state.

Figure 6(d) shows energy dispersion of CrxW1-xP4 alloys for various Cr
concentrations $x$ calculated along the $k_{y}$ axis. The virtual crystal
approximation is adopted to model the band structure using the Wannier
Hamiltonian obtained with the HSE06 functional. When $x=0$, a clear overlap
between $M$$-d$ and P$-p$ states is seen so that the system realizes a nodal
line semimetal state. As $x$ is increased, the $M$-$d$ and P-$p$ state move in
opposite energy directions and touch at the $\Gamma$ point for $x_{c}=0.32$.
With a further increase in $x$, the overlap between these states vanishes, and
the system transitions to an insulator state. Notably, we calculate the
orientation-dependent effective masses of valence and conduction bands and
nodal line dispersion of CrxW1-xP4 for $x=0.2$. The mass anisotropy of bands
and the hybrid nature of the nodal line dispersion remain preserved, showing
that these features are robust in $M$P4 materials. The evolution of bandgap of
CrxW1-xP4 as a function of $x$ at $\Gamma$ is shown in Fig 6(e). A hybrid
semimetal-to-insulator transition can thus be achieved by varying the Cr
concentration. A similar tunable system can be achieved by doping MoP4 with Cr
atoms (see Fig. 6(f)). Since Cr, W, and Mo atoms are isovalent and form
isostructural materials with similar mass anisotropies, one can create Crx(Mo,
W)1-xP4 material that can be tuned among various topological states.

##  IV Discussion

Understanding the structure-to-property relationship lies at the heart of
physics research and provides concepts for engineering new device design [35].
It remains largely unexplored in topological materials, even though the
numbers and energy dispersions of nontrivial states depend on the structural
and positional motifs of materials. Based on the first-principles calculations
and $k\cdot p$ theory, we have systematically demonstrated how the lattice
geometry-driven effective mass anisotropies result in unique hybrid nodal line
states in black phosphorus materials $M$P4. Taking MoP4 as an explicit
example, we show that it retains characteristic anisotropic energy dispersion
of the phosphorene and harbors a single hybrid nodal line between Mo-$d$ and
P-$p$ states. The nodal line constitutes both the type-I and type-II band
crossings and spreads across the Fermi level around the $\Gamma$ point. We
reveal that the P bands have a peanut-type effective mass variation such that
the large and small effective masses orient along the zigzag and armchair
directions of the P atoms, respectively. The anisotropic P band crosses with
the Mo-$d$ band to form the hybrid nodal line where the type-I and type-II
nodal points are hooked to the armchair and zigzag lattice directions. Through
in-depth symmetry and Hamiltonian analysis, we show that the strong mass
anisotropy in the crossing bands is a necessary condition to form the hybrid
nodal line in $M$P4. We further propose the realization of the hybrid nodal
line state with increased inversion strength and a metal-to-insulator
transition through isoelectronic chemical substitutions such as Crx(W,
Mo)1-xP4. $M$P4 materials have been synthesized in experiments and explored in
connection with high diffusion anisotropy of sodium or other metal ions in the
phosphorous layers [41, 42, 43, 44, 45, 46, 47]. More recent work discusses
the high-pressure synthesis of MoP4 and reveals its semimetallic nature with
high-mobility electrons [45]. Since the single crystalline samples are grown,
and the hybrid nodal line runs across the Fermi level, the experimental
validation of the nodal line could be done, for example, in photoemission
experiments without any Fermi level tuning. Moreover, orientation-dependent
effective masses can be determined from the transport experiments. Their
anisotropic behavior could serve as experimental fingerprints to determine the
nodal line dispersion in topological semimetals. These facts highlight that
$M$P4 materials constitute an ideal platform to explore mass-anisotropy-driven
hybrid nodal states and advance our understanding of the role of lattice
geometry in shaping the topological state dispersion in materials.

## Acknowledgements

This work is supported by the Department of Atomic Energy of the Government of
India under Project No. 12-R$\&$D-TFR-5.10-0100 and benefited from the
computational resources of TIFR Mumbai. S.M.H. is supported by the NSTC-AFOSR
Taiwan program on Topological and Nanostructured Materials, Grant No.
110-2124-M-110-002-MY3.

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Bansil  _et al._ [2016] A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys. 88, 021004 (2016).
  * Armitage  _et al._ [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * Singh  _et al._ [2022] B. Singh, , H. Lin, and A. Bansil, Topology and symmetry in quantum materials, Adv. Mater. , 2201058 (2022).
  * Fu [2011] L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106, 106802 (2011).
  * Hsieh  _et al._ [2012] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Topological crystalline insulators in the snte material class, Nat. Commun. 3, 982 (2012).
  * Fang and Fu [2019] C. Fang and L. Fu, New classes of topological crystalline insulators having surface rotation anomaly, Sci. Adv. 5, eaat2374 (2019).
  * Bradlyn  _et al._ [2016] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond dirac and weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353, aaf5037 (2016).
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  * Slager  _et al._ [2013] R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, The space group classification of topological band-insulators, Nature Physics 9, 98 (2013).
  * Kruthoff  _et al._ [2017] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).
  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017).
  * Song  _et al._ [2018] Z. Song, T. Zhang, Z. Fang, and C. Fang, Quantitative mappings between symmetry and topology in solids, Nat. Commun. 9, 3530 (2018).
  * Zhang  _et al._ [2019] T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological electronic materials, Nature 566, 475 (2019).
  * Vergniory  _et al._ [2019] M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature 566, 480 (2019).
  * Tang  _et al._ [2019] F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Comprehensive search for topological materials using symmetry indicators, Nature 566, 486 (2019).
  * Lin  _et al._ [2013] H. Lin, T. Das, Y. Okada, M. C. Boyer, W. D. Wise, M. Tomasik, B. Zhen, E. W. Hudson, W. Zhou, V. Madhavan, C.-Y. Ren, H. Ikuta, and A. Bansil, Topological dangling bonds with large spin splitting and enhanced spin polarization on the surfaces of ${{\mathrm{Bi}}}_{2}{{\mathrm{Se}}}_{3}$, Nano Lett. 13, 1915 (2013).
  * Singh  _et al._ [2016] B. Singh, H. Lin, R. Prasad, and A. Bansil, Role of surface termination in realizing well-isolated topological surface states within the bulk band gap in TlBiSe2 and TlBiTe2, Phys. Rev. B 93, 085113 (2016).
  * Singh  _et al._ [2018a] B. Singh, X. Zhou, H. Lin, and A. Bansil, Saddle-like topological surface states on the ${T}{T}^{{}^{\prime}}{X}$ family of compounds (${T},{T}^{{}^{\prime}}$ = transition metal, ${X}=\mathrm{Si}$, ge), Phys. Rev. B 97, 075125 (2018a).
  * Burkov  _et al._ [2011] A. A. Burkov, M. D. Hook, and L. Balents, Topological nodal semimetals, Phys. Rev. B 84, 235126 (2011).
  * Fang  _et al._ [2015] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Topological nodal line semimetals with and without spin-orbital coupling, Phys. Rev. B 92, 081201 (2015).
  * Bian  _et al._ [2016] G. Bian, T.-R. Chang, H. Zheng, S. Velury, S.-Y. Xu, T. Neupert, C.-K. Chiu, S.-M. Huang, D. S. Sanchez, I. Belopolski, N. Alidoust, P.-J. Chen, G. Chang, A. Bansil, H.-T. Jeng, H. Lin, and M. Z. Hasan, Drumhead surface states and topological nodal-line fermions in TlTaSe2, Phys. Rev. B 93, 121113 (2016).
  * Chan  _et al._ [2016] Y.-H. Chan, C.-K. Chiu, M. Y. Chou, and A. P. Schnyder, ${{\mathrm{Ca}}}_{3}{{\mathrm{P}}}_{2}$ and other topological semimetals with line nodes and drumhead surface states, Phys. Rev. B 93, 205132 (2016).
  * Singh  _et al._ [2018b] B. Singh, S. Mardanya, C. Su, H. Lin, A. Agarwal, and A. Bansil, Spin-orbit coupling driven crossover from a starfruitlike nodal semimetal to dirac and weyl semimetal state in CaAuAs, Phys. Rev. B 98, 085122 (2018b).
  * Xu  _et al._ [2017] S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Belopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng, M.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R. Chang, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Strocov, H. Lin, S. Jia, and M. Z. Hasan, Discovery of lorentz-violating type ii weyl fermions in LaAlGe, Sci. Adv. 3, e1603266 (2017).
  * Chang  _et al._ [2019] T.-R. Chang, I. Pletikosic, T. Kong, G. Bian, A. Huang, J. Denlinger, S. K. Kushwaha, B. Sinkovic, H.-T. Jeng, T. Valla, W. Xie, and R. J. Cava, Realization of a type-ii nodal-line semimetal in ${{\mathrm{Mg}}}_{3}{{\mathrm{Bi}}}_{2}$, Adv. Sci. 6, 1800897 (2019).
  * Wang  _et al._ [2019] B. Wang, B. Singh, B. Ghosh, W.-C. Chiu, M. M. Hosen, Q. Zhang, L. Ying, M. Neupane, A. Agarwal, H. Lin, and A. Bansil, Topological crystalline insulator state with type-ii dirac fermions in transition metal dipnictides, Phys. Rev. B 100, 205118 (2019).
  * Li  _et al._ [2016] F.-Y. Li, X. Luo, X. Dai, Y. Yu, F. Zhang, and G. Chen, Hybrid weyl semimetal, Phys. Rev. B 94, 121105 (2016).
  * Zhang  _et al._ [2018] X. Zhang, Z.-M. Yu, Y. Lu, X.-L. Sheng, H. Y. Yang, and S. A. Yang, Hybrid nodal loop metal: Unconventional magnetoresponse and material realization, Phys. Rev. B 97, 125143 (2018).
  * Shao  _et al._ [2020] Y. Shao, A. N. Rudenko, J. Hu, Z. Sun, Y. Zhu, S. Moon, A. J. Millis, S. Yuan, A. I. Lichtenstein, D. Smirnov, Z. Q. Mao, M. I. Katsnelson, and D. N. Basov, Electronic correlations in nodal-line semimetals, Nat. Phys. 16, 636 (2020).
  * Kou  _et al._ [2015] L. Kou, C. Chen, and S. C. Smith, Phosphorene: Fabrication, properties, and applications, J. Phys. Chem. Lett. 6, 2794 (2015).
  * Xia  _et al._ [2019] F. Xia, H. Wang, J. C. M. Hwang, A. H. C. Neto, and L. Yang, Black phosphorus and its isoelectronic materials, Nat. Rev. Phys. 1, 306 (2019).
  * Qiao  _et al._ [2014] J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus, Nat. Commun. 5, 4475 (2014).
  * Xia  _et al._ [2014] F. Xia, H. Wang, and Y. Jia, Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics, Nat. Commun. 5, 4458 (2014).
  * Li  _et al._ [2017] X. Li, L. Tao, Z. Chen, H. Fang, X. Li, X. Wang, J.-B. Xu, and H. Zhu, Graphene and related two-dimensional materials: Structure-property relationships for electronics and optoelectronics, Appl. Phys. Rev. 4, 021306 (2017).
  * Xiang  _et al._ [2015] Z. J. Xiang, G. J. Ye, C. Shang, B. Lei, N. Z. Wang, K. S. Yang, D. Y. Liu, F. B. Meng, X. G. Luo, L. J. Zou, Z. Sun, Y. Zhang, and X. H. Chen, Pressure-induced electronic transition in black phosphorus, Phys. Rev. Lett. 115, 186403 (2015).
  * Kim  _et al._ [2015] J. Kim, S. S. Baik, S. H. Ryu, Y. Sohn, S. Park, B.-G. Park, J. Denlinger, Y. Yi, H. J. Choi, and K. S. Kim, Observation of tunable band gap and anisotropic dirac semimetal state in black phosphorus, Science 349, 723 (2015).
  * Liu  _et al._ [2017] Y. Liu, Z. Qiu, A. Carvalho, Y. Bao, H. Xu, S. J. R. Tan, W. Liu, A. H. Castro Neto, K. P. Loh, and J. Lu, Gate-tunable giant stark effect in few-layer black phosphorus, Nano Lett. 17, 1970 (2017).
  * Ghosh  _et al._ [2016] B. Ghosh, B. Singh, R. Prasad, and A. Agarwal, Electric-field tunable dirac semimetal state in phosphorene thin films, Phys. Rev. B 94, 205426 (2016).
  * Christiansson  _et al._ [2022] V. Christiansson, F. Petocchi, and P. Werner, Superconductivity in black phosphorus and the role of dynamical screening, Phys. Rev. B 105, 174513 (2022).
  * Gong  _et al._ [2018] N. Gong, C. Deng, L. Wu, B. Wan, Z. Wang, Z. Li, H. Gou, and F. Gao, Structural diversity and electronic properties of 3d transition metal tetraphosphides, $\mathrm{{TMP}}_{4}$ ($\mathrm{{TM}}=\mathrm{{V}},\mathrm{{Cr}},\mathrm{{Mn}}$, and $\mathrm{{Fe}}$), Inorg. Chem. 57, 9385 (2018).
  * Sun  _et al._ [2015] J. Sun, H.-W. Lee, M. Pasta, H. Yuan, G. Zheng, Y. Sun, Y. Li, and Y. Cui, A phosphorene–graphene hybrid material as a high-capacity anode for sodium-ion batteries, Nat. Nanotechnol. 10, 980 (2015).
  * Cheng  _et al._ [2017] Y. Cheng, Y. Zhu, Y. Han, Z. Liu, B. Yang, A. Nie, W. Huang, R. Shahbazian-Yassar, and F. Mashayek, Sodium-induced reordering of atomic stacks in black phosphorus, Chem. Mater. 29, 1350 (2017).
  * Jeitschko and Donohue [1972] W. Jeitschko and P. C. Donohue, The high pressure synthesis, crystal structure, and properties of CrP4 and MoP4, Acta Crystallogr., Sect. B 28, 1893 (1972).
  * Mayo  _et al._ [2021] A. H. Mayo, J. A. Richards, H. Takahashi, and S. Ishiwata, High-pressure synthesis of a massive and non-symmorphic dirac semimetal candidate $\mathrm{{Mo}P}_{4}$, J. Phys. Soc. Jpn. 90, 123704 (2021).
  * Kinomura  _et al._ [1983] N. Kinomura, K. Terao, S. Kikkawa, and M. Koizumi, Preparation and properties of $\mathrm{{WP}}_{4}$, J. Solid State Chem. 48, 306 (1983).
  * Khan  _et al._ [2020] M. R. Khan, K. Bu, J.-S. Chai, and J.-T. Wang, Novel electronic properties of monoclinic $\mathrm{M}\mathrm{P}_{4}$ ($\mathrm{M}=\mathrm{Cr},\mathrm{Mo},\mathrm{W}$) compounds with or without topological nodal line, Sci. Rep. 10, 11502 (2020).
  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Mostofi  _et al._ [2014] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised wannier functions, Comput. Phys. Commun. 185, 2309 (2014).
  * Wu  _et al._ [2018] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Wanniertools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018).
  * [55] See Supplemental Material.
  * Liu  _et al._ [2010] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Model hamiltonian for topological insulators, Phys. Rev. B 82, 045122 (2010).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003).


(First published in Astrophysics & Space Science, 275, 90 (2001). Some
references to more recent works have been added as footnotes to the original
text)

###### Abstract

A number of late [WC] stars have unique infrared properties, not found among
the non-[WC] planetary nebulae, and together define a class of IR-[WC] stars.
They have unusual IRAS colours, resembling stars in the earliest post-AGB
evolution and possibly related to PAH formation. Most or all show a double
chemistry, with both a neutral (molecular) oxygen-rich and an inner carbon-
rich region. Their dense nebulae indicate recent evolution from the AGB,
suggesting a fatal-thermal-pulse (FTP) scenario. Although both the colours and
the stellar characteristics predict fast evolution, it is shown that this
phase must last for $10^{4}\,\rm yr$. The morphologies of the nebulae are
discussed. For one object in Sgr, the progenitor mass ($1.3\,\rm M_{\odot}$)
is known. The stellar temperatures of the IR-[WC] stars appear much higher in
low metallicity systems (LMC, Sgr). This may be indicative of an extended
’pseudo’ photosphere. It is proposed that re-accretion of ejected gas may slow
down the post-AGB evolution and so extend the life time of the IR-[WC] stars.

##  1 Introduction

The nebulae surrounding the hydrogen-poor emission-line [WC] stars are thought
not to differ from those surrounding non-[WC] stars, except for a higher
expansion velocity (Gorny & Stasinska 1995) and higher internal turbulence
(Gesicki et al. 1998), both of which can be understood as the effect of the
hydrogen-poor wind on the nebula (Mellema, these proceedings). The lack of
chemical peculiarities in the nebulae shows that the ejected occured before
the star became hydrogen poor. The normal range of ionized masses indicates
that this occured on the AGB, in a normal AGB superwind phase.

The hydrogen-poor nature of the [WC] star is probably related to a thermal
pulse. This can have occured during three possible phases: (1) on the white
dwarf cooling track (a very late thermal pulse or VLTP); (2) during the
hydrogen-burning post-AGB evolution (a late thermal pulse or LTP); (3) at the
very end of the AGB, with the pulse terminating the AGB evolution (a fatal
thermal pulse or FTP). The VLTP model is preferred from stellar modelling
(e.g. Herwig et al. 1999). The nebulae surrounding VLTP [WC] stars should be
highly evolved, since the pulse occurs $\sim 10^{4}\,\rm yr$ after the AGB.
The envelope mass of the star prior to VLTP is very small
($10^{-4}{M_{\odot}}$), and so the PN cannot be resupplied with significant
amounts of hydrogen-rich gas. Some of the [WC] PNe are so compact that they
must have been ejected on the AGB very recently. For these objects the FTP
scenario may be preferred (Zijlstra et al. 1991).

The compact [WC] PNe are extremely bright infrared emitters. I will show that
their nebulae have unexpected characteristics, and do not have a counterpart
among non-[WC] PN. 111For a general recent review of planetary nebulae,
published after these proceedings, see Kwitter & Henry 2022. A list of
Galactic [WC] stars is in Muthumariappan & Parthasarathy 2020

##  2 Infrared flux and colours

Figure 1. : Infrared IRAS colours. The axes are defined in terms of 12$\mu$m,
25$\mu$m and 60$\mu$m IRAS flux densities, without colour correction. Panel a:
the open circles show all PNe detected in all three bands; the filled circles
are all the [WC] stars. Panel b: the [WC] stars; the size of the circles is
proportional to the subclass (class 11 are the largest circles). The boxes
show the expected location of HII regions (I), PNe (II), PostAGB stars (III),
Miras (IV) and bipolar outflow sources (V) (from Zijlstra et al. 2001). Panel
c: the open circles are the [WC] stars; the size of the symbols is now
proportional to the integrated IRAS flux. The dots represent all IRAS sources
with integrated flux comparable to the IR-brightest [WC] stars. Panel d: All
IRAS sources with good detections in all three bands.

Table 1 lists all known Galactic [WC] stars in the catalog of Gorny (these
proceedings), with IRAS detections (flux quality 2 or 3) at 12, 25 and 60
micron. IRAS21282+5050 is included: although this object does not satisfy the
[WC] criteria due to the lack of higher ionization levels (Crowther et al.
1998), it shows characteristics very similar to belong to the late [WC] stars.
PC14 is not included: it has very peculiar IRAS colours but the IRAS position
is more than 1 arcmin from the PN and confusion is likely. The total sample
contains 40 objects. There is no excess of objects towards the Bulge
suggesting all are Galactic disk objects.

TABLE 1. : IRAS data for Galactic [WC] stars, in order of Galactic longitude

Name [WC] subtype $F_{12}$ $F_{25}$ $F_{60}$ [Jy] [Jy] [Jy] SwSt1 9 17 86 24
IR-[WC]? Cn1-5 4 1.7 7.5 9.4 NGC6369 4 9.0 65.5 109 Hb4 3.5 1.3 10.3 20.9
M1-25 6 0.8 6.4 8.5 M3-15 5 0.53 5.66 8.02 M1-32 4.5 4.2 15.7 17.9 PM1-188 11
3.8 15.3 14.8 M1-61 4 2.19 26.5 17.0 M1-60 4 0.5 5.6 8.5 M1-51 5 3.6 23.3 31.4
M2-43 8 15.7 53.8 21.3 IR-[WC]? NGC6751 4 4.0 18.4 23.6 He2-429 4.5 0.7 4.4
7.2 PM1-310 11 2.1 8.4 6.0 NGC6905 3 0.4 6.2 8.6 BD+30o3639 9 89.4 234.5 161.7
IR-[WC] He2-459 9 4.5 37.8 39.1 NGC7026 3 2.4 18.4 42.7 IRAS21282+5050 10 51.0
74.4 33.4 IR-[WC] NGC40 8 14.5 71.9 64.8 IC1747 4 0.7 3.7 8.2 NGC1501 4 1.2
5.6 15.2 M4-18 11 4.2 9.5 4.0 IR-[WC]? IC2003 3 0.5 4.1 3.4 NGC2452 3 0.5 5.0
6.6 NGC2867 3 2.2 14.9 17.0 PB6 3 0.92 6.0 5.1 Pe1-1 4.5 1.8 18.4 11.9
IRAS07027$-$7934 11 22.7 82.0 42.0 IR-[WC] NGC5189 2 1.3 13.5 33.7 He2-99 9
1.2 9.2 11.8 NGC5315 4 7.5 72.0 84.0 He2-113 11 92.4 310.5 176.6 IR-[WC]
PM1-89 4 2.3 12.9 14.2 He2-142 9 13.8 31.0 15.6 IR-[WC]? He3-1333 11 144. 257.
199. IR-[WC] Pe1-7 9 6.5 50.0 47.6 K2-16 11 9.2 36.9 29.4 IC1297 3 0.4 2.9 5.7

Figure 1 shows the IRAS colour–colour diagram for the objects of Table 1. In
Fig. 1a the 40 [WC] stars (filled circles) are shown together with all other
PNe (open circles). The full PNe sample was selected as all PNe in the ESO-
Strasbourg catalogue of PNe, with flux quality of 2 or 3 in all three bands,
and where the IRAS position is within 20 arcsec of the optical position in the
catalogue. The [WC] and other object cover the same region of the diagram, but
the [WC] stars are more predominant among the bluer objects. The [WC] stars
have average IRAS colours of (0.75,0.01) in Fig. 1a, while the non-[WC] PNe
have on average (0.85,0.02). Thus, the [WC] stars tend to have a relatively
stronger $F_{12}$ flux ratio but the $F_{60}/F_{25}$ is no different. Sczcerba
(these proceedings) shows that ISO spectra of some [WC] objects show faint PAH
features: these could raise the 12-micron flux and cause a blueward shift in
the diagram. It thus appears that the dust characteristics differ between [WC]
and non-[WC] stars.

Figure 1b shows the IRAS colours as function of [WC] subclass, where the size
of the circles is proportional to the [WC] subclass; i.e. the largest circles
correspond to the coolest central stars. The coolest stars have the hottest
dust, implying their dust is nearest to the star. This agrees with a general
evolution towards hotter stars while the nebula expands: the [WC] stars evolve
towards higher temperatures on time scale comparable to the nebular expansion.
The difference in distribution in Fig. 1a appears to be caused by the cooler
[WC] stars only.

The boxes in Fig 1b indicate the typical location of different classes of
objects, as indicated in the Figure caption. The [WC] stars are found on the
blue side of the PNe box (as discussed before), with several having colours
typical for young Post-AGB stars, with nebular time scales of only a few
hundred years. A few [WC] stars have a 60-$\mu$m excess which puts them in a
region where objects with bipolar outflows are found (Zijlstra et al. 2001).

In Fig. 1c the size of the open circles is proportional to the log of the
integrated IRAS flux. The integrated flux is obtained by summing the in-band
flux of the three bands and is thus a lower limit to the integrated dust
emission, especially for hot objects. The objects with the hottest dust are
also very bright in IRAS. This trend is affected by the unknown distances: the
flux will scale with $d^{-2}$ where $d$ ranges from perhaps 1 kpc for the
nearest objects to 8 kpc for Bulge nebulae. However, the dust emission
declines so strongly as the nebula expands that the distance spread is a
secondary parameter.

The dots in Fig. 1c are all entries in the IRAS point source catalogue with
flux quality 3 at 12, 25 and 60$\mu$m, and with total in-band flux
$F_{ir}>10^{-11}\,\rm W\,m^{-2}$ which is similar to the brightest [WC] stars.
There are very few such IRAS sources especially in the PNe region, and those
show colours clearly distinct from the [WC] stars. The bright [WC] stars form
a well defined group among IRAS sources with little confusion with other types
of objects. For comparsion, Fig. 1d shows all IRAS sources with flux quality 3
at 12, 25 and 60$\mu$m (for all values of total flux). The AGB and its
termination point are clearly visible, with the bright [WC] stars located near
this point.

There is a gap in the distribution of bright IRAS sources in Fig 1c, at
($0.5,-0.3$). This gap corresponds to the AGB–Post-AGB transition where
evolution is extremely fast ($\sim 100\,\rm yr$), when the envelope first
becomes detached. The bright [WC] stars are seen to straddle this gap.

##  3 Defining the IR-[WC] stars

Figure 2. : Histogram of the distribution of total in-band IRAS flux, for [WC]
stars (drawn line) and other PNe (dashed line), for all objects with IRAS
detections at 12, 25 and 60$\mu$m

Figure 2 shows the distribution of total in-band IRAS fluxes, with the drawn
line representing the [WC] stars and the dashed line all non-[WC] PNe of Fig.
1a. The two distributions peak at a similar flux ($10^{-12}\rm\,W\,m^{-2}$),
but the [WC] stars dominate at the highest integrated fluxes. The brightest
IRAS PNe are almost all [WC] stars. These [WC] stars are the objects which are
located near the post-AGB gap of Fig. 1c; all have late subtypes, or cool
central stars. These are the IR-[WC] stars, defined as [WC] stars with IRAS
in-band flux $F_{ir}>0.8\times 10^{-11}\,\rm W\,m^{-2}$ and IRAS colours near
or in the post-AGB gap.

In Table 1, the IR-[WC] stars are indicated either as confirmed or possible.
Confirmed stars (BD+30o3639, IRAS07027$-$7934, He 2-113 and He 3-1333 (also
known as CPD$-$56o8032) ) have colours near or in the gap and IRAS in-band
flux $F_{ir}>0.8\times 10^{-11}\,\rm W\,m^{-2}$. The suspected candidates have
similar colours but lower flux (possibly because of a larger distance). The
[WC] subclasses range from 9 to 11, with one possible candidate (M2-43) having
8. This very limited range of subclass is not part of the selection criterium
but indicates the likely common origin of the IR-[WC] stars.

Figure 3. : Total in-band IRAS flux and 60/12 micron colour versus [WC]
subclass for the IR-[WC] stars (filled circles), possible IR-[WC] stars (half-
filled circles) and other [WC] stars (open circles)

Fig. 3 shows the IR-[WC] stars (filled circles), the possible ones (partly
filled circles) and the other [WC] stars of Table 1 (open circles). The
IR-[WC] stars very clearly separated from the other [WC] stars. The gap at
intermediate subclasses (5–7) is clear (Gorny & Stasinska 1995). The IR-[WC]
stars all belong to the late [WC] stars. However, not all late [WC] stars
belong to this class. Leaving out the IR-[WC] stars, the gap at subclasses 5–7
is much less significant.

The right panel effectively measures the distance in the colour–colour diagram
from the AGB, showing the IR-[WC] stars should be in an extremely early post-
AGB phase. These colours are however very different from those of the well-
known bipolar post-AGB stars (such as, the Egg nebula and similar objects;
Zijlstra et al. 2001) which are found near region V of Fig. 1b.

These IR-[WC] stars with colours similar to young post-AGB stars have few or
no counterparts among the non-[WC] PNe. They define a separate class of
objects, and point at a genuine difference in nebular evolution between some
late [WC] stars and other PNe.

##  4 The silicate problem

Zijlstra et al. (1991) found an OH maser originating from the [WC11] star
IRAS07027$-$7934\. The IRAS LRS spectrum shows strong PAH features. This
indicated a chemical dichotomy with both carbon-rich and oxygen-rich regions
being present. The OH maser showed strong 1612MHz emission, a transition which
requires long path lengths at moderate temperatures (Field 1985). The PAH
emission require a strong UV field. This implies that the oxygen-rich material
is situated in the outer regions, while the carbon-rich dust is found close to
the star. This was seen as evidence for a very recent (less than 500 yr ago)
change of the star from oxygen to carbon rich, probably related to the thermal
pulse. Thus, IRAS07027$-$7934 appears a candidate for a [WC] star which can
only have formed through a fatal thermal pulse scenario.

Figure 4. : ISO spectra of two IR-[WC] stars, showing the PAH features at
$\lambda<13\mu$m, and the crystalline silicate bands at $\lambda>20\mu$m. Thin
marks indicate olivine bands and thick marks indicate pyroxenes (from Waters
et al. 1998).

Although previously a unique case, ISO spectroscopy found a similar chemical
dichotomy in other [WC] stars, with strong PAHs bands at short wavelengths and
bands of crystalline silicates (olivines and pyroxenes) at longer wavelength.
This has been seen in He 2-113 and BD+30o3639 (Waters et al. 1998) and in He
3-1333 (Cohen et al. 1998). The temperature of the crystalline dust is lower
than the PAH particles, indicating there is a carbon-rich region close to the
star but an oxygen-rich region at larger distances. In BD+30o3639 and He
3-1333 the ionized region is known to be strongly carbon rich and the likely
seat of the PAH emission (Siebenmorgen et al. 1994). The chemical change in
the stellar wind was dated to about 1000 yr ago for both He 2-113 and
BD+30o3639, and 250 yr for He 3-1333, consistent with the estimates for
IRS07027$-$7934\. In all cases, the silicates appear to be located in the
molecular part of the envelope.222For some relevant results after these
proceedings were published (2001), see Toala et al. 2019, Guzman-Ramirez et
al. 2015, Clayton et al. 2011, Garcia-Hernandez et al. 2008

The observations show that probably all confirmed IR-[WC] stars show the
double chemistry. IRAS 21282+5050 has not yet been observed at long
wavelengths, but it also shows very strong PAH emission (Geballe et al. 1994).

Chemical dichotomy (showing both a carbon-rich and an oxygen-rich region) is
very rare in post-main-sequence objects is. Novae show this behaviour (Evans
et al. 1997). The Red Rectangle (Waters et al. 1996) shows PAHs band and
strong crystalline silicates. IRAS08005$-$2356 (Bakker et al. 1997, Zijlstra
et al. 2001) shows carbon molecules and an OH maser. Both the last two stars
are too cool to ionize their nebulae. They are binary stars with the
crystalline silicates trapped in an ancient disk: they are probably not
related in any way to the [WC] stars. Roberts 22 also has PAHs, an OH maser
and crystalline silicates. It has an inner ionized region (Zijlstra et al.
2001) but is probably a supergiant (Molster, priv comm.). Among PNe, NGC 6302
has very weak PAH emission and crystalline features.

It appears likely that the presence in PNe of strong PAH features in
combination with OH masers and/or crystalline silicates is uniquely related to
the IR-[WC] stars. 333Work done since these proceedings have shown that weak
PAH emission is common in planetary nebulae, both oxygen-rich and carbon-rich,
but strong PAHs are rare and indicate carbon-rich nebulae: Cerrigone et al.
2009, Guzman-Ramirez et al. 2014. Although the IR-[WC] stars are strong PAH
emitters, they do not appear to show fullerene emission: Otsuka et al. 2014

##  5 Time scales

The IR colours, the IR fluxes and the chemical dichotomy all point at very
short time scales of the order of $10^{2}$–$10^{3}\,$yr. The apparent
conclusion is that the IR-[WC] stars were on the AGB before that time. The
time when the star became hydrogen-poor may be even more recent: BD+30o3639
shows evidence for hydrogenated PAHs (Waters et al. 1998) implying that the
PAHs formed after the star became carbon rich but before it became hydrogen
poor.

Cohen et al. (1998) note that such short time scales are implausible from
evolutionary considerations, and suggest that the silicate grains do not
participate in the nebular expansion and date back to a much earlier epoch.
Support for their argument comes from a separate reasoning: if the timescale
are so short, the stars should evolve rapidly in the HR diagram. One would
expect to see hotter stars with a similar chemical dichotomy. Such stars are
not known. On the other hand, hydrogen-poor stars must have very low stellar
envelope mass, and the cool [WC] stars have wind with very high mass-loss
rates. (Note that the extreme C/O ratio of He 3-1333 and He 2-113 (de Marco et
al. 1998) suggest a very low envelope mass (e.g. Frost et al. 1998)). Low
envelope mass would lead to very fast evolution towards higher temperatures.
The lack of hot IR-[WC] stars is therefore puzzling.

The number of IR-[WC] stars can be used to obtain limits on the evolutionary
age of the IR-[WC] stars. The IRAS in-band flux is about
$10^{-11}\rm\,W\,m^{-2}$ which corresponds to a luminosity of approximately
$10^{29}d^{2}\rm\,W$ where $d$ is the distance in kpc. The in-band flux will
underestimate the total IR flux typically by a factor of 2 for objects with
hot dust. This gives $L_{\rm IR}\approx 500\,d^{2}\,\rm L_{\odot}$. The
distance should be less than about 3 kpc for post-AGB luminosities. In fact
distance estimates for IR-[WC] stars are mostly 1–2 kpc.

The local column density of PNe is about 25 kpc-2 (Zijlstra & Pottasch 1991).
The IR-[WC] PNe account for 1–2% of all PNe. For a life expectancy of $3\times
10^{4}\,\rm yr$ for a PN (Zijlstra & Pottasch 1991), we can expect one object
within 2 kpc for every 100-yr phase of PN evolution. The number of IR-[WC]
stars than implies a minimum life time of 500–1000 yr for a IR-[WC] PN if all
PNe pass through this phase. The oberved fraction of [WC] stars (8%) can be
used as an upper limit to the fraction of PNe which may have been IR-[WC]
stars. We find a minimum life time for an IR-[WC] star of $5\times 10^{3}\,\rm
yr$, with a likely life time in excess of $10^{4}\,\rm yr$.

This is in contradiction with the implied short nebular time scales, and
agrees with the suggestion of Cohen et al. The discrepancy can be reduced by
assuming a higher distance and luminosity for the IR-[WC] stars. However, PNe
with stellar luminosity $>10^{4}\,\rm L_{\odot}$ are very rare and would
evolve rapidly, again decreasing the expected numbers. The observed number of
objects shows that IR-[WC] stars are comparatively long-lived. Both nebulae
and stars must evolve much slower during the IR-[WC] phase than their compact
nebulae suggest.

##  6 Morphology

A way to slow down evolutionary time scales of the nebulae is by assuming the
existence of a slowly expanding torus or disk (e.g. Waters et al 1998). This
would lead to bipolar morphologies, with possibly high velocities
perpendicular to the torus.

BD+30o3639 is among the most spherically symmetric PNe known. However, Bryce &
Mellema (1999) have shown that in fact it has a high-velocity bipolar outflow
and thus is likely bipolar. The nebula is seen almost pole-on hiding its true
structure.

CPD$-$56o8032 and He2-113 are very compact but HST images have been published
by de Marco et al. (1997). Both show irregular but ring-like structures. The
structures are consistent with bipolar geometries but do not proof it.
Curiously, both objects show that the ’central’ star is in fact located off-
centre of the nebulosity. The ionized core of IRAS 07027$-$7934 has not been
imaged, being subarcsecond. There are a few indications of asphericity (the
split OH profile, the fact that the CO expansion velocity is half that of the
OH, and the fact that the ionized core is not quite at the centre of the
H$\alpha$ scattering halo, Zijlstra et al. 1991), but no conclusive evidence.
IRAS 21282+5050 has a strong bipolar morphology (Bregman et al. 1992). 444More
recent morphological work is in Garcia-Hernandez et al 2006

The evidence is at present inconclusive, but bipolar, torus-like morphologies,
are possible for the IR-[WC] nebulae; these would allow for slower nebular
expansion. Note that similar bipolar morphologies are seen in 10–15% of PN
(Manchado et al. 1996). Off-centre central stars are however very rare.

##  7 Extra galactic [WC] stars: the effect of metallicity

There are few or no Bulge [WC] stars in Table 1, as argued above. However,
there are [WC] stars known in nearby galaxies of which three have IRAS
detections: He2-436 in the Sagittarius dwarf spheroidal at 25 kpc (Sgr:
Zijlstra & Walsh 1996), and SMP 58 and SNP61 in the LMC (Zijlstra et al.
1994). Their IRAS data is listed in Table 2 where the flux densities for He
2-436 were re-determined using IRAS Software Telescope data (SRON, Groningen).

TABLE 2. : IRAS data for extra-galactic [WC] stars

Name [WC] subtype $F_{12}$ $F_{25}$ $F_{60}$ He 2-436 (in Sgr) 3/4 0.22 Jy
0.49 Jy 0.11 Jy SMP58 (in LMC) 4/5 0.15 Jy 0.22 Jy – SMP61 (in LMC) 4/5 0.08
Jy 0.13 Jy $<$0.16 Jy

Sgr has a metallicity of [Fe/H]$=-0.55$ and a main-sequence turn-off mass of
1.3 M⊙ (Dudziak et al. 2000). Its two [WC] stars are the only such stars where
these parameters are known.

The IR colours and IR flux (when scaled to a distance of 2 kpc) are similar to
those of the IR-[WC] stars. However, Fig. 5 shows their [WC] subclass is much
earlier than for Galactic stars with similar IR characteristics. Possibly the
stars evolve faster to higher temperature at lower metallicity. Another
possibility is that the subclass is affected by metal blanketing in an
extended wind, in which case lower metallicity causes the $\tau=1$ surface to
be closer to the real stellar surface. (This would however imply that the late
IR-[WC] stars have much hotter stars than the wind emission lines suggest.)

Figure 5. :  $F_{12}/F_{25}$ as function of [WC] subclass, for Galactic [WC]
stars (open circles), and the extra-galactic [WC] stars (filled circles)

The data suggests that the group of the IR-[WC] stars is not confined to the
Galactic disk. However, the detailed properties show a dependence on
metallicity. Given the AGB populations of Sgr and the LMC, it is likely that
their progenitor stars were already carbon-rich before the ejection of the PN
on the AGB. A chemical dichotomy similar to the Galactic IR-[WC] stars is
therefore not expected.

##  8 The problem of evolution

There are three main problems regarding the IR-[WC] stars: (1) A unique
nebular double chemistry seen only around IR-[WC] stars; (2) The conflict
between the very short time scales implied by the nebulae and by the low
envelope mass expected for a hydrogen-poor Post-AGB star, and the much longer
timescales implied by other considerations; (3) The fact that the dense
nebulae favour a fatal-thermal pulse (FTP) scenario whereas stellar model
calcuations favour a VLTP to obtain hydrogen-poor stars.

Long time scales require slow post-AGB evolution of the star. Slow evolution
is expected for stars with very low core mass. However, their luminosity would
also be very low, which is not observed.

The only other way to ensure slower post-AGB evolution is by replenishing the
envelope of the star, to compensate for the effects of nuclear burning and
wind mass loss. This would require accretion of mass from the ejecta back onto
the star.

An accretion scenario could explain the link between dense, compact nebulae
and the peculiar evolution. It requies that the hydrogen is efficiently mixed
down in the star to maintain the [WC] characteristics. In such a model, the
PAHs molecules could in fact form in the accreting gas, where CO is
dissociated (as in novae, Evans et al. 1997). Furthermore, slow accretion
could trigger a VLTP on the cooling track where otherwise this might not
occur.

Such a model is highly speculative. It is however clear that the FTP and VLTP
scenarios, while adequate for the other [WC] stars, do not as yet provide a
good explanation for the IR-[WC] stars.

## References

  * [1]
  * [2] Bakker E.J., van Dishoeck E.F., Waters L.B.F.M., Schoenmaker T., 1997, A&A, 323, 469
  * [3]
  * [4] Bregman J.D., Booth J., Gilmore D.K., Kay L., Rank D., 1992, ApJ, 396, 120
  * [5]
  * [6] Bryce M., Mellema G., 1999, MNRAS, 309, 731
  * [7]
  * [8] Cerrigone L., Hora J. L., Umana G., Trigilio C., 2009, ApJ, 703, 585
  * [9]
  * [10] Clayton G. C., De Marco O., Whitney B. A., Babler B., Gallagher J. S., Nordhaus J., Speck A. K., et al., 2011, AJ, 142, 54.
  * [11]
  * [12] Cohen M., Barlow M.J., Sylvester R.J., Liu X.-W., Cox P., Lim T., Schmitt B., Speck A.K., 1998, ApJ, 513, L135
  * [13]
  * [14] Crowther P.A., De Marco O., Barlow M.J., 1998, MNRAS, 296, 367
  * [15]
  * [16] De Marco O., Crowther P.A., 1998, MNRAS, 296, 419
  * [17]
  * [18] De Marco O., Barlow M.J., Storey P.J., 1997, MNRAS, 292, 86
  * [19]
  * [20] Dudziak G., Pequignot P., Zijlstra A.A., Walsh J.R., 2000, A&A, 363, 717
  * [21]
  * [22] Evans A., Geballe T.R., Rawlings J.M.C., Eyres S.P.S., Davies J.K., 1997, MNRAS, 292, 192
  * [23]
  * [24] Field D., 1985, MNRAS, 217, 1
  * [25]
  * [26] Frost C.A., Cannon R.C., Lattanzio J.C., Wood P.R., Forestini M., 1998, 332, L17
  * [27]
  * [28] García-Hernández D. A., Manchado A., García-Lario P., Cañete A. B., Acosta-Pulido J. A., García A. M., 2006, ApJ, 640, 829
  * [29]
  * [30] Geballe T.R., Joblin C., D’Hendecourt L.B., Jourdain de Muizon M., Tielens A.G.G.M., Leger A., 1994, ApJ, 434, L15
  * [31]
  * [32] Gesicki K. Zijlstra A.A., Acker A,, Szczerba R., 1998, A&A, 329, 265
  * [33]
  * [34] Górny S. K., 2001, Ap&SS, 275, 67
  * [35]
  * [36] Gorny S.K., Stasinska G., 1995, A&A, 303, 893
  * [37]
  * [38] Guzman-Ramirez L., Lagadec E., Wesson R., Zijlstra A. A., Muller A., Jones D., Boffin H. M. J., et al., 2015, MNRAS, 451, L1
  * [39]
  * [40] Guzman-Ramirez L., Lagadec E., Jones D., Zijlstra A. A., Gesicki K., 2014, MNRAS, 441, 364
  * [41]
  * [42] Herwig F., Blöcker T., Langer N., Driebe T., 1999, A&A, 349, l5
  * [43]
  * [44] Kwitter K. B., Henry R. B. C., 2022, PASP, 134, 022001
  * [45]
  * [46] Manchado A., Guerrero M.A., Stanghellini L., Serra-Ricart M., 1996, The IAC Morphological Catalog of Nortern Galactic Planetary Nebulae
  * [47]
  * [48] Mellema G., 2001, Ap&SS, 275, 147
  * [49]
  * [50] Muthumariappan C., Parthasarathy M., 2020, MNRAS, 493, 730
  * [51]
  * [52] Otsuka M., Kemper F., Cami J., Peeters E., Bernard-Salas J., 2014, MNRAS, 437, 2577
  * [53]
  * [54] Siebenmorgen R., Zijlstra A.A., Krügel E., 1994, MNRAS, 271, 449
  * [55]
  * [56] Szczerba R., Górny S. K., Stasińska G., Siódmiak N., Tylenda R., 2001, Ap&SS, 275, 113
  * [57]
  * [58] Toalá J. A., Ramos-Larios G., Guerrero M. A., Todt H., 2019, MNRAS, 485, 3360
  * [59]
  * [60] Waters L.B.F.M., Beintema D.A., Zijlstra A.A., de Koter A., Molster F.J., Bouwman J., de Jong T. Pottasch S.R., de Graauw Th., , 1998, A&A, 331, L61
  * [61]
  * [62] Zijlstra A. A., Chapman J. M., te Lintel Hekkert P., Likkel L., Comeron F., Norris R. P., Molster F. J., et al., 2001, MNRAS, 322, 280
  * [63]
  * [64] Zijlstra A. A., Walsh J. R., 1996, A&A, 312, L21
  * [65]
  * [66] Zijlstra A.A., Potasch S.R., 1991, A&A, 243, 478
  * [67]
  * [68] Zijlstra A.A., Gaylard M.J., te Lintel Hekkert P., Menzies J.W., Nyman L.-A., Schwarz, H. E. 1991, A&A, 243, L9
  * [69]
  * [70] Zijlstra A.A., van Hoof P.A.M., Chapman J.M., Loup C., 1994, A&A, 290, 228
  * [71]