Structural stability of polymeric nitrogen: A first-principles investigation
Xiaoli Wang, Fubo Tian, Liancheng Wang, Tian Cui,a?Bingbing Liu, and Guangtian Zou
State Key Laboratory of Superhard Materials, College of Physics, Jilin University,
Changchun 130012, People’s Republic of China
?Received 14 November 2009; accepted 17 December 2009; published online 11 January 2010?
The search for the stable single-bonded ?polymeric? solid nitrogen attracted much attention in view
of its potential application as a high energy density material. In this study, the stability of different
candidate polymeric structures of nitrogen has been studied using ab initio calculations based on
density-functional theory for the first time, from the angles of thermodynamic stabilities, mechanical
stabilities, and dynamical stabilities in the pressure range from 0 to 360 GPa, respectively.
According to our results, only Cmcm,A7, rcg, cg, BP, P212121, and Pba2 are competitive structures
and more favorable than sc, ch, LB, and cw strcutrues; their stable pressure range were also
presented. Among the competitive structures, BP, Pba2, and P212121are the novel ones for their
enthalpies are lower than the cg structure above 170 GPa. We further identify that the P212121phase
can transform to cg structure at pressure below 60 GPa. Also a new phase transition sequence with
increasing pressure has been presented, which is from the molecular phase ?-N2to cg at 47 GPa, to
Pba2 at 170 GPa, and then to P212121at 307 GPa. © 2010 American Institute of Physics.
Nitrogen usually consists of molecules in which two at-
oms are strongly triple bonded under ambient conditions, but
high external pressure destabilizes the triple molecular bonds
and leads to various covalent polymeric nitrogen forms, in
which each nitrogen atom is bonded to three nearest neigh-
bors by single covalent bonds.1–6McMahan and LeSar first
predicted this phenomenon.2This unusual transition is also
interesting from the point of view of the chemistry of nitro-
gen and its applications. Due to the uniquely large energy
difference between single and triple bonds, single-bonded
polymeric nitrogen is a high energy density material
?HEDM?. The calculated results show that the energy density
of single-bonded polymeric nitrogen is at least 0.4 eV/cm3,7
which is about three times larger than the most powerful
energetic materials known today. Because nitrogen has five
valence electrons in its outmost electronic shells, there are
many ways for atoms to bond each other and form a variety
of polymeric phases, such as chainlike structures,8layered
structures,1,9and the network structures.1,10,11Owing to the
intriguing properties of the polymeric nitrogen in HEDM,
plenty of structures have been suggested. Besides cubic
gauche ?cg? ?Ref. 12? which was synthesized by Eremets et
al. at high pressure ?110 GPa? and high temperature ?2000
K?, none of the theoretically proposed structures have been
Up to now, almost all of the candidate polymeric nitro-
gen structures are proposed from the aspect of thermodynam-
ics; their mechanical and the dynamical stabilities are not yet
studied. Since unique energy calculations cannot establish
the stability of a crystal, a thorough analysis of the elastic
and dynamical stability is required.
In this study, the stabilities of polymeric nitrogen for the
previously proposed structures have been extensively inves-
tigated using ab initio calculations based on density func-
tional theory. We compared the recently found phases with
other proposed nitrogen phases. The results show that many
of these phases are not stable in a wide range of pressures
and can be excluded. Furthermore, the stable pressure range
of the favorable structures was also present. Finally, the
phase transition sequence with increasing pressure has been
II. THEORETICAL METHOD AND COMPUTATIONAL
All calculations performed in this paper are based on the
functional theory. An ultrasoft pseudopotential is used.
The exchange and correlation effects are described by
the generalizedgradient approximation
?Perdew–Burke–Ernzerhof?13instead of the local-density ap-
proximation ?LDA?, since a number of studies suggested that
GGA functionals are more accurate than LDA ones in co-
valently bonded systems.14–16The convergence tests give ki-
netic energy cutoffs of 500 eV, with 8?8?8, 10?10?12,
10?10?7, 13?13?5, 10?10?7, 12?6?12, 8?8?8,
9?9?9, 9?9?6, 8?8?8, and 8?9?12 Monkhorst–
Pack grids for the electronic Brillouin zone ?BZ? integrations
cis-transchain ?ch?, ?-arsenic ?A7?, layered boat ?LB?, black
phosphorus ?BP?, chaired web ?cw?, cubic gauche ?cg?, rela-
a?Author to whom correspondence should be addressed. Electronic mail;
THE JOURNAL OF CHEMICAL PHYSICS 132, 024502 ?2010?
0021-9606/2010/132?2?/024502/6/$30.00© 2010 American Institute of Physics
tive cg ?rcg?, Pba2 and P212121phases, respectively. For
each selected structure we fully optimized the bond lengths,
cell parameters, and the atomic positions at each pressure.
Elastic constants are obtained from evaluations of the stress
tensor generated by small strains using the density-functional
plane wave technique as implemented in the CASTEP code.17
Hellmann–Feynamn forces are calculated by ab initio pro-
gram CASTEP code and the phonon frequencies are calculated
using PHONON ?Ref. 18? code.
III. RESULTS AND DISCUSSIONS
A. Thermodynamic stability
The structures of most important ones of polymeric ni-
trogen considered in this work are presented in Fig. 1. The
Bravais of lattices of all the selected structures include four
crystal system—cubic ?sc, space group: Pm-3m, 1 atom/cell;
cg, I213, 8 atoms/cell?, orthorhombic ?ch, Cmcm, 4 atoms/
cell; BP, Cmca, 4 atoms/cell; Cmcm, Cmcm, 4 atoms/cell;
Pba2, Pba2, 16 atoms/cell; P212121, 8 atoms/cell?, hexago-
nal ?A7, R-3m, 6 atoms/cell; cw, R-3m, 6 atoms/cell?, and
monoclinic ?rcg, C2/c, 16 atoms/cell; LB, P21/m, 4 atoms/
cell?. From Fig. 1 we can get that the atoms arrangement was
various and forms chainlike, layered, and network structures.
Cmcm and ch phases are chainlike structures. The layered
phase BP is similar to LB andA7, they all have zigzag chains
but the method of connection between them is different. The
rcg phase is very similar to cg and they are network struc-
tures in company with cw, Pba2, and P212121phases.
First, after the fully structural optimized, we explored
the energetics of all the candidate structures of the polymeric
nitrogen through total energy calculations. The enthalpy dif-
ferences of candidate structures relative to cg structure at 0 K
are presented in Fig. 2. The molecular phase ?-N2is found to
be the most stable at pressures lower than 47 GPa, in excel-
lent agreement with previous calculations.19,20The enthalpy
of sc structure is the highest among all these structures,
which indicates that it is an energetically unfavorable one. ch
has the fastest increase rate of enthalpy with increasing pres-
sure, and quickly becomes less energy favorable as the pres-
sure is above 80 GPa. It is clear that the sc and ch phases
cannot exit at pressures lower than 360 GPa.
The enthalpy difference of the other structures ?from 0 to
360 GPa? is within 0.50 eV. A7, LB, Cmcm, and cw have the
same change tendency with pressure. The difference to cg
varies from 0.25 to 0.45 eV. The enthalpies among Pba2,
P212121, BP, and rcg phases remain very close to cg within
?0.25 eV in the whole pressure range studied. Interestingly,
the rcg and cg are all close to each other at all pressures
?within ?0.065 eV?, and the rcg phase has lower enthalpy
than the cg phase above 300 GPa. The relative enthalpy of
these phases in excellent agreement with previous work.9
Among all the polymeric phases, the cg structure is the
most stable one below 170 GPa, in excellent agreement with
the other groups work.19,21At higher pressures, Pba2 is
the lowest-enthalpy structure at least up to 307 GPa. Above
307 GPa the P212121structure is energy favorite. Consider-
ing the enthalpies of all the candidate structures, we can
suggest the new phase transition sequence from the molecu-
lar phase to cg at 47 GPa, to Pba2 at 170 GPa, and then to
P212121at 307 GPa with increasing pressure. The results of
the transition pressures and the structural sequence for the
lower-enthalpy phases agree mostly with the previous
B. Mechanical stability
The mechanical stability of crystal requires the strain
energy to be positive, which implies that the whole set of
elastic constant Cijsatisfies Born–Huang criterion.23The for-
mulas of elastic moduli for cubic, orthorhombic, hexagonal,
and monoclinic crystals were shown in Refs. 24–27,
FIG. 1. Extended structures of the most polymeric phases of nitrogen:
?a? BP, ?b? ch, ?c? cg, ?d? Cmcm, ?e? A7, ?f? rcg, ?g? P212121, ?h? cw, and
FIG. 2. Enthalpy per nitrogen atom difference between the various nitrogen
phases and cg as functions of pressure.
024502-2Wang et al.J. Chem. Phys. 132, 024502 ?2010?
For a cubic crystal, the independent elastic stiffness ten-
sor reduces to three components of C11, C12, and C44. The
mechanical stability criteria are given by C11?0, C44?0,
C11??C12?, and ?C11+2C12??0. Both sc and cg are cubic
phases; the sc structure is the most energy unfavorable one,
so we only calculated the cg elastic coefficients shown in
Fig. 3?a?. It is clear that cg phase is mechanical stability in
the range of 0–360 GPa. We can also get C11, C12exhibits
linearly increasing trends with pressure, however, after
200 GPa C44shows a linear softening trend.
For an orthorhombic crystal, the independent elastic
stiffness tensor consists of nine components of C11, C22, C33,
C44, C55, C66, C12, C13, and C23. The mechanical stability
criteria are given by C11?0, C22?0, C33?0, C44?0, C55
?0, C66?0, ?C11+C22−2C12??0, ?C11+C33−2C13??0,
FIG. 3. The pressure dependence of the elastic constants of ?a? cg, ?b? ch, ?c? Pba2, ?d? cw, and ?e? rcg phases, respectively.
024502-3Structural studies of polymeric nitrogen J. Chem. Phys. 132, 024502 ?2010?
+C23????0. BP, ch, Cmcm, Pba2, and P212121are all ortho-
rhombic phases. From Fig. 3?b?, it can be observed that C66
remains negative under pressures from 0 to 360 GPa, show-
ing that the ch phase is mechanically unstable. From Fig.
3?c? we can get that Pba2 phase is mechanical stability from
0 to 360 GPa. For the same reason, we can get that the
Cmcm phase is mechanical stability from 0 to 360 GPa, the
BP phase is mechanical stability from 20 to 360 GPa, and the
P212121phase is mechanical stability from 60 to 360 GPa.
For a hexagonal crystal, the independent elastic stiffness
tensor reduces to five components of C11, C33, C44, C12, and
C13. The mechanical stability criteria are given by C44?0,
C11??C12?, and ?C11+2C12?C33?2C12
tic constants for cw with pressure are shown in Fig. 3?d?.
From 0 to 360 GPa, all the elastic constants fulfill the me-
chanical stability criteria of the hexagonal crystal. Similarly,
we can get that the hexagonal phase of A7 is mechanically
stable from 60 to 360 GPa and unstable at low pressure from
0 to 60 GPa.
For a monoclinic crystal, the elastic stiffness tensor re-
duces to 13 components of C11, C22, C33, C44, C55, C66, C12,
C13, C23, C15, C25, C35, and C46. The criteria for mechanical
stability are given by C11?0, C22?0, C33?0, C44?0,
?2?C15C25?C33C12−C13C23? + C15C35?C22C13−C12C23? +C25
C35?C11C23− C12C13?? − ?C15
the monoclinic phase of rcg is mechanical stable from 0 to
360 GPa. For the LB phase, ?C22C33−C23
pressure above 120 GPa, which indicates that this phase is
not mechanically stable above 120 GPa.
According to above calculations of the elastic constants,
we can conclude that in all the candidate structures, the cg,
Cmcm, cw, rcg, and Pba2 structures are mechanically stable
from 0 to 360 GPa. The BP phase is stable from 20 to
360 GPa, the A7 and P212121are mechanically stable from
60 to 360 GPa, the LB phase is stable under the pressures
below 120 GPa, and sc and ch are mechanically unstable
from 0 to 360 GPa.
2. The calculated elas-
2??+C55??0. Figure 3?e? shows that
2? + C25
2??0 when the
C. Dynamical stability
The dynamical instability of one structure is associated
with a soft phonon with imaginary frequency.28Eremets et
al. successfully synthesized a crystalline single-bonded cg
structure at temperatures above 2000 K and pressures above
110 GPa. In order to offer theoretical foundations for the
further prospective experimental researches, we calculated
the phonon dispersion curves in the whole BZ of all previ-
ously proposed candidate structures in the pressure range
from 0 to 360 GPa. From the phonon dispersion curves
shown in Fig. 4?a?, we can get that the cg structure is dy-
namical stable from 0 to 360 GPa, but the sc structure is
dynamically unstable at the pressure range considered here
?the phonon dispersion curves are not presented here?.
For the orthorhombic crystals, the ch phase is also both
mechanically dynamically unstable. From the phonon disper-
sion curve of Cmcm in Fig. 4?b?, imaginary phonon frequen-
cies are clearly found at the zone boundary points ?Y and X?
above 200 GPa, suggesting a dynamical instability at higher
pressures, so the dynamically stable range of this structure is
below 200 GPa. Among the candidate structures, BP, Pba2,
and P212121are the interesting ones for their enthalpies are
lower than that of cg structure above 170 GPa. We fully
optimized the cell parameters and atomic positions at each
pressure and found BP to break into zigzag chains below
20 GPa, this finding is in perfect agreement with the results
simulations.29Similarly, we found that the P212121structure
transforms to cg structure at pressure below 60 GPa.
The calculated phonon dispersion curves of BP and
P212121are shown in Figs. 4?c? and 4?d?. The results indi-
cated that the BP structure is dynamically stable in a pressure
range of 20–360 GPa, Pba2 in 80–360 GPa, and P212121in
60–360 GPa, respectively. From the enthalpy and stability of
these three structures, we found that BP, Pba2, and P212121
are thermodynamics stable than cg above 170 GPa and also
dynamical stability at least up to 360 GPa.
For the hexagonal crystals, the calculated phonon disper-
sion curves of A7 structure at different pressures are shown
in Fig. 4?e?. One observes that with increasing pressure, the
frequency of transverse acoustic ?TA? phonon branch de-
creases at Q point and becomes zero at 110 GPa, showing its
structural instability above 110 GPa. Combining the results
of mechanical stability, we can conclude that theA7 structure
is stable from 60 to 110 GPa. Even though cw is mechani-
cally stable from 0 to 360 GPa, the imaginary phonon fre-
quencies are clearly found at the zone points ?T? at 60 and
110 GPa, suggesting its dynamical stability at low pressures.
For the monoclinic crystals LB and rcg, the calculated
phonon dispersion curves of LB structure are shown in
Fig. 4?f?. Imaginary phonon frequencies of the TA phonon
branch are clearly found, suggesting its dynamical instability
in the pressure range we considered here. The LB structure is
not a favorable candidate structure due to its dynamical
unstability, although it is mechanically stable from 0 to
120 GPa. From the phonon dispersion curves of the rcg
structure at different pressures in Fig. 4?g?, in combination
with mechanical properties, we can observe that this struc-
ture is both mechanically and dynamically stable from 0 to
240 GPa, so this structure is another more favorable candi-
date structure in this pressure range.
Finally, we present the calculated results including ther-
modynamic stabilities, mechanical stabilities, and dynamical
stabilities conclusions of all the candidate structures in Table
I. The stable pressure regions of all the structures were given.
We reach a conclusion that cg, Pba2, Cmcm, P212121, BP,
and LB are the more favorable structures.
024502-4Wang et al.J. Chem. Phys. 132, 024502 ?2010?
FIG. 4. The calculated phonon frequencies of ?a? cg phase at pressures of 0, 240, and 360 GPa. ?b? Cmcm phase at pressures of 60, 200, and 240 GPa.
?c? BP phase at pressures of 20, 220, 300, and 360 GPa. ?d? P212121phase at pressures of 60, 240, and 360 GPa. ?e? A7 phase at pressures of 60, 100, and
110 GPa. ?f? LB phase at pressures of 20 and 120 GPa. ?g? rcg phase at pressures of 0, 110, and 240 GPa.
024502-5 Structural studies of polymeric nitrogenJ. Chem. Phys. 132, 024502 ?2010?
In conclusion, we extensively explored the stability of
the different proposed polymeric nitrogen by ab initio calcu-
lations based on DFT. Our results indicate that at pressure-
lower than 47 GPa, the molecular phase ?-N2remains as the
lowest-enthalpy structure in solid nitrogen. In consideration
of the stable criteria and enthalpies, our studies show that
Cmcm, rcg, cg, BP, Pba2, and P212121are competitive
structures. Finally, we suggest the new phase transition se-
quence with increasing pressure from the molecular phase
?-N2to cg at 47 GPa, to Pba2 at 170 GPa, and then to
P212121at 307 GPa.
This work was supported by the National Natural Sci-
ence Foundation of China ?Grant Nos. 10574053 and
10674053?, the National Basic Research Program of China
?Grant No. 2005CB724400?, 2007 Cheung Kong Scholars
Programme of China, Changjiang Scholar and Innovative
Research Team in University ?Grant No. IRT0625?, and
National Fund for Fostering Talents of Basic Science
?Grant No. J0730311?.
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TABLE I. The stabilities of all candidate structures.
Bravais lattice Structure
?the enthalpy difference with cg?
Mechanical stabilities Dynamical stabilitiesConclusion
Unstable Unstable Exclusion
0–360 GPa stable 0–360 GPa stable0–360 GPa favorable
0–360 GPa stable
0–360 GPa stable
60–360 GPa stable
20–360 GPa stable
80–360 GPa stable
60 GPa–200 GPa stable
60–360 GPa stable
20–360 GPa stable
80–360 GPa favorable
60 GPa–200 GPa favorable
60–360 GPa favorable
20–360 GPa favorable
60–360 GPa stable
0–360 GPa stable
60–110 GPa stable
0 GPa stable 60, 120 unstable
60–110 GPa favorable
0–120 GPa stable
0–360 GPa stable
20–120 GPa unstable
0–240 GPa stable
0–240 GPa favorable
024502-6Wang et al.J. Chem. Phys. 132, 024502 ?2010?