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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.

?doi:10.1063/1.3290954?

I. INTRODUCTION

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

obtained experimentally.

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

determined.

II. THEORETICAL METHOD AND COMPUTATIONAL

DETAILS

All calculations performed in this paper are based on the

first-principles plane-wave

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

forthesimple cubic

?sc?,

cis-transchain ?ch?, ?-arsenic ?A7?, layered boat ?LB?, black

phosphorus ?BP?, chaired web ?cw?, cubic gauche ?cg?, rela-

pseudopotentialdensity

?GGA?

PBE

Cmcmchain

?Cmcm?,

a?Author to whom correspondence should be addressed. Electronic mail;

cuitian@jlu.edu.cn.

THE JOURNAL OF CHEMICAL PHYSICS 132, 024502 ?2010?

0021-9606/2010/132?2?/024502/6/$30.00© 2010 American Institute of Physics

132, 024502-1

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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

calculations.22

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

?i? Pba2.

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?

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respectively.

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?

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?C22+C33−2C23??0,

+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,

C55?0,

C66?0,

?C33C55−C35

?C22C33−C23

?C11+C22C33+2?C12+C13+C23???0,

?C22?C33C55−C35

?2?C15C25?C33C12−C13C23? + C15C35?C22C13−C12C23? +C25

C35?C11C23− C12C13?? − ?C15

−C13

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.

and

??C11+C22+C33+2?C12+C13

2. The calculated elas-

2??0,

?C44C66−C46

2??0,

2??0,

2?+2?C23C25C35−C25

2C55−C25

2C33???0, and

2?C22C33− C23

2??+C55??0. Figure 3?e? shows that

2? + C25

2?C11C33

2?+C35

2?C11C22−C12

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

byAlemanyandMartins

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.

withmoleculardynamics

024502-4Wang et al.J. Chem. Phys. 132, 024502 ?2010?

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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?

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IV. CONCLUSIONS

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.

ACKNOWLEDGMENTS

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

Thermodynamic stabilities

?the enthalpy difference with cg?

?eV?

Mechanical stabilities Dynamical stabilitiesConclusion

Cubic

sc

cg

0.6–2.0

0

Unstable Unstable Exclusion

0–360 GPa stable 0–360 GPa stable0–360 GPa favorable

Orthorhombic

ch 0–0.72

0–0.22

0.15–0.45

0–0.25

0–0.15

UnstableUnstableExclusion

Pba2

Cmcm

P212121

BP

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

Hexagonal

A7

cw

0–0.45

0–0.45

60–360 GPa stable

0–360 GPa stable

60–110 GPa stable

0 GPa stable 60, 120 unstable

60–110 GPa favorable

Exclusion

Monoclinic

LB

rcg

0–0.45

0–0.15

0–120 GPa stable

0–360 GPa stable

20–120 GPa unstable

0–240 GPa stable

Exclusion

0–240 GPa favorable

024502-6Wang et al.J. Chem. Phys. 132, 024502 ?2010?