Table Of ContentNew allotropesofphosphorene withremarkable stabilityandintrinsic piezoelectricity
ZhenqingLi,1,2ChaoyuHe,1,2,∗TaoOuyang,1,2ChunxiaoZhang,1,2CaoTang,1,2,†RudolfA.Ro¨mer,3andJianxinZhong1,2,‡
1HunanKeyLaboratoryforMicro-NanoEnergyMaterialsandDevices,XiangtanUniversity,Hunan411105,P.R.China;
2School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105, China.
3Department of physics, the university of warwick coventry, CV4 8UW, UK.
(Dated:January20,2017)
In this letter, we show that a new class of two-dimensional phosphorus allotropes can be constructed via
assemblingthepreviouslyproposedultrathinmetastablephosphorusnanotubeintoplanarstructuresindifferent
7
stacking orientations. Based on first-principles method, thestructures, stabilitiesand fundamental electronic
1
0 properties of these new two-dimensional phosphorus allotropes are systematically investigated. These
2 two-dimensionalphosphorusallotropespossessremarkablestabilitiesduetothestronginter-tubevanderWaals
interactions, which cause an energy release of about 30-70 meV/atom depending on their stacking manners.
n
Ourresultsshow thatmost of thesetwo-dimensional van der Waalsphosphorene allotropesareenergetically
a
morefavorablethantheexperimentallyviableblackα-Pandblueβ-P.Threeofthemshowingrelativelyhigher
J
probability to be synthesized in future are further confirmed to be dynamically stable semiconductors with
9
strain-tunablebandgapsandremarkablepiezoelectricity,whichmayhavepotentialapplicationsinnano-sized
1
sensors,piezotronics,andenergyharvestinginportableelectronicnano-devices.
]
i
c PACSnumbers:73.20.At,61.46.-w,73.22.-f,73.61.Cw
s
-
l
r Black phosphorene (α-P) is considered as a formidable
t
m competitor to graphene and other two-dimensional materi-
als for applications in future nano-electronicsdue to its sig-
.
at nificant band gap1,2 and high carrier mobility3. It was suc-
m cessfullyexfoliatedfromitsthree-dimensionalcounterpartin
20143,4 forthefirsttimeandnowcanbesynthesizedthrough
d- low-cost and high-yieldmethods, such as liquid exfoliation5 3 (a)
n and chemical deposition methods6,7. Blue phosphorene (β- 2.1
o P),anothertwo-dimensionalphosphorusallotrope,wasfirstly V)2 ) 1.8
v2 [c pi2mar0lao1gtpe4roroaiswanelddtshbhrmeyacsZeestnhhtituomlydeu9ts..layaPlntebrtohdagessesrieidgzsneosidnfiincbfiaysrnsyZttn-hiptnharteinensgrciesiespottl.sfeaistlnwtchoaerxl-ocdpuuilmlgoahretiinneosgpniisont8anexiwan-l -Energy (e101 Eg=2.083 eV Egap (eV 011...925
5 two-dimensionalphosphorusmaterials. -2
7 0.6
0 Considered that the possible crystal structures for two- -3 -10 -8 -6 -4 -2 0 2 4 6 8 10
5 dimensional phosphorus synthesized from deposition meth- (b) (c) Stain (%)
0 ods or epitaxial growth methods always depend on the sub-
. strate, pressure, temperature and other experimental condi- FIG.1: Crystallineviewsoftheum-PNTfromdifferentdirections
1
0 tions, many other potential10–17 candidates beyond black α- (a).Thecalculatedelectronbandstructures(b)andstrain-dependent
7 P and blue β-P have beentheoretically proposedand are ex- bandgaps(c)oftheum-PNT.
1 pected to be synthesized in future. For example, the atomic
: thinglayersofγ-P10,δ-P10,θ -P11andredphosphorene12,the
v 0
i diatomic thin layers η-P13, θ-P13 and their transformations14 which indicate that they are non-piezoelectric. In this let-
X (G1, G2, B1, B2 and B3) with pentagons, the multi-atomic ter three dynamically stable two-dimensional van der Waals
r thinlayerofHittorfene15, aswellastheporousphosphorene phosphorus crystals are obtained via planer stacking of the
a
allotropeswerepreviouslyproposed16,17throughthetopolog- previously proposed ultrathin metastable phosphorus nan-
ical modeling method. Recently, a new Helical Coil phos- otube (um-PNT, P∞18) and confirmed energeticallymore fa-
phoruswastheoreticallypredicted18anditscorrespondingal- vorable than black α-P and blue β-P. Our results show that
lotropeswasexperimentallysynthesized19incarbonnanotube these three new phosphorene allotropes are excellent candi-
reactor. dates for potential application in nano-electronics according
Phosphorene allotropes with lone pairs on the surface to their strain-turnable band gaps and remarkable piezoelec-
like SnS and SnSe20,21 are expected to be excellent two- tricity.Weexpectfutureeffortswillbepaidtosynthesizethem
dimensional piezoelectric materials22–25. However, the ex- duetotheirremarkableenergeticstabilities.
perimental viable black β-P and blue α-P, as well as most Ourcalculationswerecarriedoutbyusingthedensityfunc-
of all the previouslyproposedtwo-dimensionalallotropesof tional theory (DFT) within generalized gradient approxima-
phosphorusarecentrosymmetricandconsequentlynon-polar, tions(GGA)26 asimplementedinViennaabinitiosimulation
package (VASP)27,28. The interactions between nucleus and
2
FIG.2: Sketchmapofthetranslationandself-rotationforassemblingone-dimensionalum-PNTintotwo-dimensionalphosphorusallotropes
(a). Thecalculatedaverageenergyforthe21newtwo-dimensional vanderWaalsphosphorene allotropes(b). Thecrystallineviewsofthe
moststablethreeones,Tube-P-0-0.5,Tube-P-30-0.5andTube-P-180-0.25(c).
the 3s23p3 valence electrons of phosphorus atoms were de- stablethanourum-PNT.Theradiusofum-PNTisonly1.6Å.
scribedbytheprojectoraugmentedwave(PAW)method29,30. Thus,um-PNTcanbeconsideredasthesmallestphosphorus
Toensuretheaccuracyofourcalculations,aplane-wavebasis tube so far. We then investigatedthe fundamentalelectronic
withacutoffenergyof500eVwasusedtoexpandthewave property of such an um-PNT. The calculated band structure
functions and the Brillouin Zone (BZ) sample meshes were showninFig.1(b)indicatesthatum-PNTisasemiconductor
setto bedenseenough(lessthan0.21Å−1) foreachsystem. withindirectbandgapofabout2.02eV.Furtherinvestigation
The structuresof all phosphoreneallotropeswere fully opti- showsthatum-PNTisa goodcandidateforone-dimensional
mizedupto theresidualforceoneveryatomless than0.001 semiconductorapplicationsinviewofthatitsbandgapcanbe
eV/Åbeforepropertyinvestigations.Theoptimizedexchange effectivelymodulatedbystrains.
vanderWaalsfunctional(optB86-vdW)31,32 wasalsoapplied As shown in Fig.1 (a), two um-PNTs are arranged in an
totakeintoaccountvanderWaalsinteractionsinthesystems. orthorhombiccell as a starting structure, in which the violet
The vibrational properties of the three new phosphorene al- tubeis fixed andthe yellowone is freefor operation. Based
lotropesmorefavorablethanblackα-Pandblueβ-Pwerein- on the well defined operators T (Translation, i=0, 0.25 and
i
vestigated through the PHONON package33 with the forces 0.5 b) and R (Self-rotation, j=0◦, 30◦, 60◦, 90◦, 120◦, 150◦
j
calculatedfromVASPtoconfirmtheirdynamicalstabilities. and180◦)asindicatedinFig.1(a),21initialstructurescanbe
Thecrystalstructureoftheum-PNTconsistsofphosphorus constructed.WenamethemasTube-P-j-iaccordingtotheop-
pentagonsis shown in Fig.1 (a). It is structurally very simi- eration{R ,T}appliedonthefreeyellowtube,wherejmeans
j i
lartothefundamentalstructure-unitsinvioletphosphorusand the rotation angle and i means the translation. For example,
redphosphorus15,34,35. Ourcalculatedresultsshowthatitisof Tube-P-180-0.25 means the mutation of the initial structure
about44meV/atomhigherthantheblackα-Pinenergy.And Tube-P-0-0 through a self-rotation of 180◦ and a translation
itismorefavorablethanmostofthepreviouslyproposedtwo- of0.25balongthetube(Ydirection)ontheyellowtube.
dimensionalphosphoreneallotropes. Especially, um-PNT is We then optimized the crystalline structures of these 21
alsomorestablethanmostofthepreviouslyproposeda-PNTs new phosphorus allotropes through first-principles calcula-
and z-PNTs36. The smallest phosphorus tube proposed be- tions. The calculated total energiesof black α-P and the 21
foreisthea-PNTwithtuberadiusofabout2.3Å,whichisof new two-dimensional van der Waals phosphorus allotropes
about100meV/atomhigherthanblackphosphoreneandless are shown in Fig.2 (b). According to our results, all these
3
new two-dimensional phosphorus allotropes are more stable
15
than the one-dimensionalum-PNT in energy of about 30-70 14
rmeleeVa/saetoomfredmepaernkdabenleteonnertghyeaafstseermasbsleinmgblminagnninetro. pWlaintahrtahre- -1 m )111123
ray,mostofthesenewtwo-dimensionalphosphorusallotropes y (c109
are energetically more stable than the viable black α-P and enc 78
blueβ-P.Fromthe resultsin Fig.2(b), wecan see thatsome qu 6
e 5
initial structuresare degeneratein energyafter optimization. Fr 4
3
We checkedtheoptimizedstructuresandfoundthatTube-P- 2
0-0.25, Tube-P-30-0.25 and Tube-P-60-0.25 are degenerate 1
0
to Tube-P-0-0.5, Tube-P-30-0.5 and Tube-P-60-0.5, respec-
tively. Both Tube-P-120-0.25 and Tube-P-150-0.25 are de- ol) Tube-P-0-0.5 Tube -P-30-0.5 Tube-P-180-0.25
m
generate to Tube-P-180-0.25. We further confirmed that the J/0.0 Black phosphorene
K
most stable three ones are Tube-P-0-0.5, Tube-P-30-0.5and E (-0.5
Tube-P-180-0.25,whoseenergiesareofabout22meV/atom, F
23meV/atomand21 meV/atomlowerthanthatofthe black e H-1.0 333.92 K Tube-P-30-0.5
v-1.5
phosphorene. The crystal structures of the most stable three ati Tube-P-0-0.5
ones are shown in Fig.2 (c), where we can see that they are el-2.0 Tube-P-180-0.25
R
obviouslydifferenttoeachother. WealsofoundthatTube-P- 0 500 1000 1500 2000 2500 3000
0-0.5 keeps its initial structure well in the optimization pro- Temperature (K)
cess, but Tube-P-30-0.5 and Tube-P-180-0.25 have obvious FIG.3: Thecalculatedphonon bandstructures(top)ofTube-P-0-
changesinorientationangleasindicatedinFig.2(c). 0.5,Tube-P-30-0.5andTube-P-180-0.25aswellastheirHelmholtz
From the view of thermodynamics,a structure with lower freeenergies(bottom)relativetothatofblackα-P.
energy generally means higher probability to be synthesized
in experiments if it is dynamically possible. Tube-P-0-0.5,
Tube-P-30-0.5 and Tube-P-180-0.25with remarkable stabil-
systemafterassembling. We noticethatsuchaphenomenon
ityexceedingblackα-Pareexpectedtobesynthesizedinfu-
isverysimilartothatofstackingsinglelayersofblackα-Por
ture usingvapordepositionmethods. Thedynamicalstabili-
blueβ-Pintomulti-layer8. Accordingtosuchaphenomenon,
tiesofthesethreenewphosphoreneallotropesarethusneeded
wecaneffectivelymodulatetheelectronicpropertiesofthese
tobeconfirmedthepossibilityofsynthesis. Westudiedtheir
newtwo-dimensionalphosphorusallotropesbyadjustingthe
vibrational properties through PHONON package combined
inter-tubedistancebystrain.
with VASP. The results are shown in Fig.3. We can see that
As shown in Fig.5 (a), (b) and (c) are the calculated band
the phonon band structures of Tube-P-0-0.5, Tube-P-30-0.5
gaps of Tube-P-0-0.5, Tube-P-30-0.5 and Tube-P-180-0.25
and Tube-P-180-0.25are free of soft modes associated with
underdifferentstrains. Wecanseethatthecompressive(ten-
structural instability. We also examined the whole Brillouin
sile)strainscrossthetubereduce(increase)theinter-tubedis-
Zone and found no imaginary states in their phonon den-
tance and correspondinglyreduce (increase)the band gapof
sity of states. The results show that allotrope Tube-P-0-0.5,
the systems. Such results are very similar to those in three-
Tube-P-30-0.5 and Tube-P-180-0.25 are dynamically viable
dimensionalbulk black phosphorus, whose band gap can be
too. Based on these vibrational spectra, we calculated the
effectivelymodulatedbyinter-layerdistance.Themodulating
Helmholtzfree energiesand comparedthemwith thatof the
effectsofstrainsalongthetubeonthebandgapsofTube-P-0-
black α-P. As shownin Fig.3, we can see thatTube-P-0-0.5,
0.5,Tube-P-30-0.5andTube-P-180-0.25areexpectedsimilar
Tube-P-30-0.5andTube-P-180-0.25arealwaysenergetically
tothatonthesingleum-PNTwherebothcompressiveandten-
more favorable than black α-P in the temperature range of
silestrainsreducethebandgaps(seeinFig.1(c)). However,
0-1700K. Tube-P-0-0.5and Tube-P-30-0.5are always more
we can see that the results in Fig.5 (a), (b) and (c) are lit-
stablethanTube-P-180-0.25inthewholetemperaturerange.
tledifferenttothatinFig.1(c),especiallyinthecompressive
Interestingly,Tube-P-30-0.5isenergeticallymorestablethan
area. For example, in single um-PNT, the band gap always
Tube-P-0-0.5blow 333.29K and it becomesless stable than
decreasesasthecompressivestrainincreasesbutthebandgap
Tube-P-0-0.5whenthetemperatureishigherthan333.29K.
of Tube-P-0-0.5 slightly increases at first under small com-
The electronic band structures of allotropes Tube-P-0-
pressive strain and then decreases under larger compressive
0.5,Tube-P-30-0.5andTube-P-180-0.25wereinvestigatedby
strain. Sucha phenomenonisdueto thefeatureoftheinter-
first-principles calculations. As shown in Fig.4, we can see
tubevanderwaalsinteractioninTube-P-0-0.5,Tube-P-30-0.5
that Tube-P-0-0.5, Tube-P-30-0.5 and Tube-P-180-0.25 are
andTube-P-180-0.25. Thecompressivestrainalongthetube
indirectbandgapsemiconductorswithbandgapsof1.430eV,
firstreducethebandgapofthetubeitself. Meanwhileitwill
1.522eVand1.574eV,respectively. Asdiscussedabove,the
alsoincreasetheinter-tubedistanceandthuscorrespondingly
um-PNT is an indirect band gap semiconductorwith a band
increase the band gap of the system. Therefore, when small
gapof 2.083eV, which is largerthan those of Tube-P-0-0.5,
compressivestrainalongthetubeisapplied,thebandgapof
Tube-P-30-0.5andTube-P-180-0.25.Thatistosay,theinter-
thesystemslightlyincreases.
tube van der Waals interactions reduce the band gap of the
Phosphorene allotropes with lone pairs on the surface
4
FIG.4:Thecalculatedelectronbandstructures(a)ofTube-P-0-0.5,Tube-P-30-0.5andTube-P-180-0.25,aswellasthecorrespondingmodu-
latingeffectsofstainsontheirbandgaps(b).
like SnS and SnSe20,21 are expected to be excellent two- tion P2D along X directionto strain ε and ε alongX and
1 11 22
dimensional piezoelectric materials22,23. However, black α- Y directions, respectively. Response of polarization P 2D to
1
P, blue β-P and most of all the previously proposed two- sharestrainε isnotconsideredinourcalculation. Polariza-
12
dimensional allotropes of phosphorus are centrosymmetric tion P2D along Y direction is always zero due to the mirror
2
and consequently non-polar, indicating that they are non- symmetry. Using theVoigtnotation37, we simplifythe coef-
piezoelectric.Here,wenoticethatTube-P-0-0.5(Amm2)and ficients as e2D=e2D, e2D=e2D. The piezoelectric coefficients
11 111 12 122
Tube-P-30-0.5 (Pmn21) are allowed to be piezoelectric ac- e2D ande2D forTube-P-0-0.5andTube-P-30-0.5areobtained
cordingtotheirnon-centrosymmetricandpolarspacegroups. 11 12
byleast-squaresfittingofthepolarizationchangeperunitarea
Their puckered C2v symmetries are very flexible along the
toeq. 2andeq. 3,respectively.
direction perpendicular to the tube, which are expected to
further enhance the piezoelectricity. Based on the widely P2D(ε ,ε =0)−P2D(ε =0,ε =0)=e2Dε (2)
1 11 22 1 11 22 11 11
usedmethodology20,21,24,25fortwo-dimensionalmaterials,the
piezoelectriccoefficientse2DofTube-P-0-0.5andTube-P-30-
ijk P2D(ε =0,ε )−P2D(ε =0,ε =0)=e2Dε (3)
0.5arecalculatedasthethird-ranktensorsastherelatepolar- 2 11 22 2 11 22 22 22
izationvectorP2D tostrainε asshownineq. 1:
i jk In our calculations, we chose the ”Relaxed-ion” mode
∂P2D to calculate the change of polarization. Namely, all the
e2D = i (1) atomic positions are fully relaxed under each strain before
ijk ∂ε
jk polarization calculation. We first considered the single
With mirror symmetry along the tube direction (Y direc- layered SnSe as reference system. Accordingto our results,
tion),theindependentpiezoelectriccoefficientsforTube-P-0- its e and e are 30.212 10−10C/m and 4.882 10−10C/m,
11 22
0.5andTube-P-30-0.5are e2D, e2D ande2D = e2D. Indices respectively. These values are in good agrement with those
111 122 212 221
1and2 correspondtotheX andY directionsasindicatedin inpreviousreports20,21. Ourcalculatedresultsofpolarization
Fig.2 (a). Here, we just consider the responses of polariza- changes for Tube-P-0-0.5 and Tube-P-30-0.5 are shown in
5
0.00
0.00
)-0.02
) m
m
-0.02 /
/ C-0.04
C
p
(p e11 GGA P (-0.06 e11 GGA
P -0.04
e12 GGA -0.08 e12 GGA
e11 GGA-optB86 e11 GGA-optB86
-0.10
e12 GGA-optB86 e12 GGA-optB86
-0.06
0.000 0.0010.002 0.003 0.004 0.005 0.000 0.0010.002 0.003 0.004 0.005
(a) (b)
strains ( and ) strains ( and )
22 22
) 36 e11
m 32 Previous works e11 6This work
C/ 28 B8 86
B
p 24 pt t
10 20 e11 o op
- 12 (10 11826 e12 e12 e11 e11 e11 e e12 GGA- GGA GGA- GGA
e 4 11
0
d A
n -4 G e11 e11 e11 e11
a -8 G
11e --1162 e12 e12 e12 e12
-20
e e
12 12
SnSe SnS GeSe GeS MoS2 h-BN SnSe Tube-P-0-0.5 Tube-P-30-0.5
(c)
FIG.5:PolarizationchangesforTube-P-0-0.5(a)andTube-P-30-0.5(b)calculatedfromGGAandGGA-opt86methods.Previouslycalculated
piezoelectriccoefficientsfordifferenttwo-dimensionalmaterials(left)andthecalculatedpiezoelectriccoefficientsforTube-P-0-0.5andTube-
P-30-0.5with/withoutvanderwaalsfunction(right)(c).
Fig.5 (a) and (b), respectively. After a liner fitting for the thesinglelayeredGeSeandGeSandthepiezoelectriccoeffi-
polarization change lines, we got the slope for each line, cientsofTube-P-30-0.5arecomparabletothoseofthesingle
which is considered as the corresponding coefficient. The layered SnSe and SnS. These results show that tube-P-0-0.5
piezoelectriccoefficientsofTube-P-0-0.5and Tube-P-30-0.5 andTube-P-30-0.5possessremarkablepiezoelectricityhigher
calculatedwith/withoutvanderWaals functionareshownin thanthoseofh-BNandMoS . Especially,Tube-P-0-0.5and
2
Fig.5(c). WithoutvanderWaalsfunction,thecalculatede Tube-P-30-0.5possessing remarkable energetic stability and
11
and e for Tube-P-0-0.5 are -0.844 10−10C/m and -10.871 positivedynamicalstabilityarehighlyexpectedtobesynthe-
22
10−10C/m, respectively. And the calculated e and e for sizedinfutureexperimentsforpiezotronicsapplications.
11 22
Tube-P-30-0.5 are -0.652 10−10C/m and -20.311 10−10C/m. In summary, based on the previously proposed ultrathin
The consideration of van der Waals function just slightly metastable phosphorus nanotube, we predicted a new class
adjusts these values for Tube-P-0-0.5 and Tube-P-30-0.5. of two-dimensional van der Waals crystals for phosphorus
These values are larger than that of MoS 22–25, indicating through operations of translation and self-rotation. Such a
2
that Tube-P-0-0.5 and Tube-P-30-0.5 are good piezoelectric newclassoftwo-dimensionalphosphoreneallotropespossess
materials for applications in mechanical-electrical energy remarkable stabilities due to the strong inter-tube van der
conversion. Waals interactions. They are of about 30-70 meV/atom
In Fig.5 (c), the previously reported piezoelectric co- lower than the single um-PNT depending on the assembling
efficients of some well-know 2D materials20–25 are also manner. Our calculated results show that most of these
summarized for comparison. We can see that the piezoelec- two-dimensional van der Waals phosphorene allotropes are
tric coefficientsof Tube-P-10-0.5are comparableto those of energeticallymorestablethantheexperimentallyviableblack
6
α-P and blue β-P. Three of them showing relatively higher Foundation of China (Grant Nos. A040204, 11474244 and
probability to be synthesized in future deposition methods 11204261), the National Basic Research Program of China
werealsoconfirmedtobedynamicallystablesemiconductors (2015CB921103),theYoungScientists Fundof theNational
with strain-tunable band gaps. Especially, two of them NaturalScienceFoundationofChina(GrantNo. 11204260),
belonging to polar space groups were predicted possess theScientificResearchFoundofHuNanProvincialEducation
remarkable piezoelectricity higher than h-BN and MoS , department(No. 14C1095), the Natural Science Foundation
2
whichmayhavepotentialapplicationsfornano-sizedsensors, of Hunan Province, China (GrantNo. 2016JJ3118),and the
piezotronics, and energy harvesting in portable electronic Program for Changjiang Scholars and Innovative Research
nano-devices. TeaminUniversity(IRT13093).
This work is supported by the National Natural Science
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