A-initio investigtion on the stility of H-6 Cron Zhris G. Fthenkis A few yers go H-6 Cron hd een proposed s n ll sp 2 three-dimensionl Cron llotrope, with mehnil properties omprle to grphene. However, results on the stility of H-6 Cron presented in the literture re rther ontrditory nd onfusing, nd it is not yet ler if this hypothetil llotrope is stle or not. Studying systemtilly the stility of H-6 Cron, using -initio density funtionl theory nd phonon nd struture lultions, we show tht H-6 Cron is unstle, onverted spontneously to dimond. Aording to our findings, this instility is due to the strin indued y the 60 o rottion of the interonneted zig-zg hins, nd not due to the reltively short distne etween them, s ws suggested. Bsed on this finding we suggest predition for the strutures of the H-n fmily whih ould e stle. 1 Introdution The sientifi ommunity ws lwys interested in Cron llotropes. This is not surprising, sine it is well known the flexiility of Cron to form single, doule nd triple onds, thus llowing (in priniple) the formtion of severl networks nd strutures with different onding nd properties 1 10. H-6 Cron is suh n hypothetil three-dimensionl ll sp 2 struture, whih ws proposed two dedes go 11 s nother Cron llotrope. It onsists of rrys of prllelly rrnged zig-zg hins, whih re rotted with eh other y 60 o (or 120 o ), s shown in Fig. 1(). H-6 Cron ws first studied y Tmor nd Hss 11, using Tight Binding (TB) pproh. They found tht H-6 Cron is stle, metlli nd signifintly hrder thn dimond. A yer lter, Liu et l 12, using density funtionl theory (DFT) in the lol-density pproximtion (LDA) level, found tht H-6 Cron is unstle, reporting tht they lulted severl intermedite strutures long the trnsition pth from H-6 Cron to dimond nd they found tht the energy derese monotonilly long this pth. They lso noted tht the H-6 Cron instility is proly due to the seond nerest neighour (SNN) intertions etween toms elonging to neighouring zig-zg hins, sine their intertomi distne is omprle to the interlyer distne of rhomohedrl grphite, t whih trnsition of rhomohedrl grphite to dimond is fvoured. On the other hnd, Winkler et l 13, s well s Rignnese nd Chrlier 14, reported tht H-6 Cron is stle struture nd ompred its stility with other Cron llotropes inluding grphene nd dimond. Moreover, reently, Bon Zhng, using -initio DFT lultions in the generlized grdient pproximtion (GGA) level with Perdue-Burke- Ernzerhof (PBE) funtionl found tht H-6 Cron is stle nd he lulted its struturl nd mehnil properties 15. All these results re rther onfusing, nd it is not yet ler if H-6 Cron Institute of Eletroni Struture nd Lser, FORTH, P.O. Box 1527, 71110 Herklio, Crete, Greee; E-mil: fthenk@iesl.forth.gr is stle or not. In the present study we try to shed light on the issue of the stility of H-6 Cron. We systemtilly optimize H-6 Cron performing -initio DFT lultions with two different funtionls nd different numer of k-grid points nd mesh utoff vlues. Optimiztions led either to dimond or H-6 Cron struture, independent of the funtionl, k-grid points nd mesh utoff vlues. However, energy lultions with oth funtionls long trnsition pthwy onverting the optimized H-6 Cron struture to dimond, show tht there is not n energy rrier long tht trnsition pth, nd onsequently, H-6 Cron is unstle. This onlusion is verified with phonon nd struture lultion for the H-6 Cron, whih show tht H-6 Cron hs negtive frequenies. On the other hnd, we found n energy rrier long tht trnsition pth, using the TB method used y Tmor nd Hss 11 in the first lultion on H-6 Cron. As we explin, the height of this rrier is overestimted due to utoff funtion whih ws used, nd it might e eliminted for different seletion of the repulsive pir potentil or the sling of the Slter-Koster prmeters nd the introdution of SNN intertions. Moreover, performing severl optimiztion lultions for the H-6 Cron strutures under 10% tensile strin, we show tht inresing the distne etween the prllelly rrnged zig-zg hins, the H-6 Cron struture is not stilized. Consequently, the reltively short interhin distne is not responsile for the H-6 Cron instility, s suggested y Liu et l 12. Performing energy lultions for grphiti-like struture whih ould e otined if the rrys of zig-zg hins of H-6 Cron were not rotted, we find tht tht the grphiti-like struture ould e stilized, whih leds to the onlusion tht the instility of H-6 Cron is due to the strin indued y the 60 o rottion of the interonneted zig-zg hins. Bsed on this we suggest predition for the strutures of the H-n fmily, whih ould e stle. 1 7 1
1/2 1/2 d () () () (d) Fig. 1 (Color online). H-6 Cron struture nd the unit ells of the homeomorphi strutures H-6, dimond nd rhomohedrl grphite. () Side view of H-6 Cron. Different olors indite different lyers of prllelly rrnged zig-zg Cron hins, rotted y 60 o (or 120 o ). () The unit ell of H-6 Cron. () The unit ell of dimond. (d) The unit ell of rhomohedrl grphite. Red dshed lines indite interlyer distnes. 2 The Method As lredy mentioned, H-6 Cron is periodi struture omposed of rrys of prllelly rrnged zig-zg Cron hins. Those lyers re rotted with eh other y 60 o (or 120 o ). The struture is shown in Fig. 1(), where different olors show different lyers of those zig-zg hins. The Brvis lttie is hexgonl, defined y the unit ell vetors = ( ) 1 3 2, 2,0, = ( ) 1 3 2, 2,0 nd =(0,0,1). (1) For the initil vlues of nd for our optimiztions, we dopt the vlues reported y Zhng 15, i.e. =2.618 Å nd =6.295 Å. Oviously, oth nd re norml to, while the ngle etween nd is 120 o. The unit ell, whih is shown in Fig. 1(), ontins six C toms with frtionl oordintes (1/2, 0, 0), (1/2, 0, δ), (1/2, 1/2, 1/3), (1/2, 1/2, 1/3+δ), (0, 1/2, ) nd (0, 1/2, +δ), where δ = 0.228. These positions re identil with those used y Zhng 15. The H-6 Cron struture is optimized using the DFT method s implemented in the SIESTA ode 16. All the energy lultions used in the present work hve een performed using this method. For the exhnge nd orreltion funtionl we utilize the lol density pproximtion (LDA) Ceperley-Adler (CA) funtionl 17 s prmetrized y Perdue nd Zunger 18, nd the generlized grdient pproximtion (GGA) PBE funtionl 19. For the pseudopotentil of C we utilize the norm-onserving Troullier- Mrtins pseudopotentils 20 in the Kleinmn-Bylnder ftorized form 21. The sis for the wvefuntion expnsion in rel spe is n tomi-like doule-zet sis with polriztion oritls. In order to ontrol the effet of the finite numer of k-points of the reiprol spe, s well s the finite vlue of the mesh utoff energy for the determintion of hrge densities nd potentils, on the optimized geometry nd totl energy, we performed severl lultions omining inresing numer of k-grid points with inresing mesh utoff vlues for oth funtionls. Thus, for the k-point grid we used the Monkhorst - Pk sheme 22 with 16 16 7, 24 24 10 nd 32 32 14 points nd for the mesh utoff we used the vlues 100, 200 nd 300 Ry. For the optimiztions we use the onjugte grdient method. Optimiztions inlude not only relxtion of tomi positions, ut lso relxtion of the lttie vetors. The struture is ssumed to e optimized if the mximum tomi fore nd the mximum stress omponent eome smller thn 0.005 ev/å nd 0.01 GP, respetively. H-6 Cron struture is homeomorphi to dimond nd rhomohedrl grphite, whih mens tht from the topologil point of view they re the sme. Both dimond nd rhomohedrl grphite n e onsidered s forming hexgonl lttie, with the unit vetors hving the sme form s in H-6 Cron (see Eq. 1). For dimond =2 0,di nd =4 0,di, where 0,di is the ond length of dimond. The frtionl oordintes of the six toms ontined in its unit ell re (, 0, 0), (, 0, 3/12), (1/3, 1/3, 1/3), (1/3, 1/3, 7/12), (0,, ) nd (0,, 11/12). For rhomohedrl grphite = 3 0,gr nd =3d, where 0,gr is the ond length of grphene nd d is the interlyer seprtion distne. The frtionl oordintes of the six toms ontined in its unit ell re (, 0, 0), (, 0, 1/3), (1/3, 1/3, 1/3), (1/3, 1/3, ), (0,, ) nd (0,, 1). The unit ells of dimond nd rhomohedrl grphite re shown in Figs. 1() nd (d), respetively. Sine H-6 Cron, rhomohedrl grphite nd dimond re homeomorphi with eh other, there is ontinuous trnsformtion onverting H-6 Cron to dimond or rhomohedrl grphite nd vie vers. The simplest trnsition pth, whih linerly onverts H-6 Cron to dimond is defined y the unit ell vetors of 2 1 7
Eq. 1, with = H6 + ( di H6 ) nd = H6 + ( di H6 ), (2) (where H6 nd H6 re the vlues of nd for H-6 Cron, nd di nd di for dimond), nd the frtionl oordintes (1/2+/6, 0, 0), (1/2+/6, 0, δ+(1/4 δ)), (1/2 /6, 1/2 /6, 1/3), (1/2 /6, 1/2 /6, 1/3+δ+(1/4 δ)), (0, 1/2+ /6, ) nd (0, 1/2+/6, +δ+(1/4 δ)) of the six toms ontined in the unit ell, where 0 1. For =0, the struture is the H-6 Cron, while for =1, it is the dimond struture. We will lulte the energy of H-6 Cron long this trnsformtion pth, to show tht H-6 Cron is unstle. 3 Results nd Disussion 3.1 Is H-6 Cron stle? (E-E di )/N (ev) Frequeny (m -1 ) 1 0.8 0.6 0.4 0.2 1500 1000 () LDA/CA GGA/PBE TBH (Tmor & Hss) 0 500 0-500 () Γ M K H L A Γ Fig. 2 () Totl energy per tom of H-6 Cron with respet to tht of dimond versus, lulted using (i) the LDA/CA funtionl, Monkhorst-Pk k-grid of 16x16x7 points nd mesh utoff vlue of 100 Ry (lk solid line), (ii) the GGA/PBE funtionl, Monkhorst-Pk k-grid of 16x16x7 points nd mesh utoff vlue of 300 Ry (red dshed line) nd (iii) the TB pproh with FNN intertions only, used y Tmor nd Hss 11 for their lultion on H-6 Cron (green dotted line). () Phonon nd struture of H-6 Cron long ΓMKHLAΓ points. The optimiztions of the initil H-6 Cron struture desried ove, for different funtionls, nd inresing numer of k-grid points nd mesh utoff vlues, led either to the H-6 Cron or the dimond struture. The optimized unit ell lengths nd Funtionl, Mesh k-grid points nd (Å) utoff 16x16x7 24x24x10 32x32x14 LDA-CA 100-154.5305-155.4046-154.5299 H6 = 2.624 H-6 dimond H-6 H6 = 6.363 200-155.4043-154.5303-154.5302 di = 2.506 dimond H-6 H-6 di = 6.140 300-155.4042-154.5301-154.4046 dimond H-6 dimond GGA-PBE 100-162.3113-162.3113-162.3112 H6 = 2.656 dimond dimond dimond H6 = 6.416 200-162.3129-162.3129-162.3129 di = 2.532 dimond dimond dimond di = 6.203 300-161.4729-162.3139-162.3138 H-6 dimond dimond Tle 1 Totl energy per tom in ev units of the optimized struture for the speified funtionl, mesh utoff vlue (in Ry units) nd Monkhorst-Pk k-grid points. With "H-6" nd "dimond" elow eh energy vlue, we indite whether H-6 Cron or dimond struture ws found s the result of the optimiztion. The vlues of nd of the optimized strutures for the orresponding funtionl re shown in the first olumn. (of either H-6 Cron or dimond) depend only on the funtionl used, they re independent of the numer of k-grid points nd the mesh utoff vlues used in our lultions, nd of ourse they re different for H-6 Cron nd dimond. Their vlues (= H6 nd = H6 for H-6 Cron, nd = di nd = di for dimond) re presented in the first olumn of T. 1. Optimiztions do not ffet the frtionl oordintes of H-6 Cron or dimond, whih remin the sme with those presented in the previous setion. The totl energy per tom of the optimized strutures otined from those optimiztions re lso presented in T. 1. As we n see, inresing the numer of k-grid points or the mesh utoff vlue, does not ffet the ury of the lultions y more thn 1 mev/tom nd the optimized struture found does not depend on the inresing numer of the k-grid points or the mesh utoff vlues. Therefore, either H-6 Cron is stle, (ut there is smll energy rrier etween H-6 Cron nd dimond, whih ws overme during the optimiztion, whenever dimond struture ws found s the optimum struture), or H-6 Cron is unstle, orresponding to sddle point of the potentil energy surfe (PES). The ltter will e orret, if there is t lest one trnsition pthwy onneting H-6 Cron with dimond, long whih the totl energy per tom monotonilly dereses. Seleting the simple trnsition pthwy, desried in setion 2, we lulte the totl energy per tom of the strutures long this pthwy for inresing vlues from 0 to 1, with 0.05 step. The totl energy per tom s funtion of for oth funtionls (LDA/CA nd GGA/PBE) re shown in Fig. 2(). As we n see, the totl energy monotonilly dereses s funtion of for oth funtionls, nd onsequently, there is not n energy rrier in the PES etween H-6 Cron nd dimond. This mens tht, ording to the DFT lultions, the optimized H-6 Cron struture found, does not orrespond to true energy minimum. We verify the ove result performing phonon nd struture lultion for the optimized H-6 Cron struture, whih ws 1 7 3
() 3 1 4 2 2 4 Intertomi distnes (Å) 2.4 2.2 2.0 1.8 1.6 d 12 d 13 d 14 () 1.4 U oh (ev) -6.4-6.6-6.8-7.0 f (d ij )=1-θ(d ij -d 0 ), d 0 =1.7Å () -7.2 f (d ij )=1-θ(d ij -d 0 ), d 0 =2.2Å f used y Tmor nd Hss -7.4 Fig. 3 (Color online) () Intertomi distnes in H-6 Cron. () Intertomi distnes long the trnsition pthwy. () Cohesive energy long the trnsition pthwy ording to the TB method of Tmor nd Hss 11, using different utoff funtions. found using the LDA/CA funtionl with 16x16x7 Monkhorst- Pk grid of k-points nd 100 Ry mesh utoff vlue. We used the vir utility of siest ode, with 5x5x3 superell ontining 450 toms nd 0.02Å displement of eh tom of the entrl unit ell long ±x, ±y nd ±z diretions. The phonon dispersion reltion ws lulted long the pth ΓMKHLAΓ, where the high symmetry points Γ, M, K, H, L nd A of the reiprol spe re defined in frtionl oordintes s Γ=(0, 0, 0), M=(1/2, 1/2, 0), K=(, 1/3, 0), H=(, 1/3, 1/2), L=(1/2, 1/2, 1/2) nd A=(0, 0, 1/2). The otined phonon nd struture is shown in Fig. 2(), where we n see the existene of negtive frequenies ω, whih prove tht indeed H-6 struture is unstle. Interestingly, on the other hnd, if we lulte the energy long the sme trnsition pth using the TB method whih ws used y Tmor nd Hss 11, then it ppers n energy rrier, s shown in Fig. 2() (green dotted line), inditing tht H-6 Cron might e stle ording to the TB lultion. 3.2 Understnding the TB filure The TB Hmiltonin used y Tmor nd Hss utilises the Slter- Koster 23 prmeters of Tománek nd Louie 24 for first nerest neighour (FNN) intertions. The SK prmeters for the intertomi distne d re sled s V ll m(d) = V ll m(d 0 )(d 0 /d) 2 f (d), where d 0 = 1.42 Å nd f (d) is utoff funtion defined to e 1 if d 1.7 Å, 0 if d 2.4 Å nd (1 sin[π(d 2.05)/0.7])/2 if 1.7 < d < 2.4 Å. The ohesive energy U oh is U oh = U tr /N+ U rep /N + ψ 1 n 2 + ψ 2n, where U tr is the sum of the eigenenergies of the oupied spin sttes, U rep is sum of pir potentils E rep (d i j ) of the form E rep (d)=e 3ε (U 0 +U 1 ε+u 2 ε 2 ) f (d), where ε = d/d 0 1, n is the numer of onds per tom defined s n = i> j f (d i j )/N nd U 0, U 1, U 2, ψ 1 nd ψ 2 re onstnts. Oviously, the use of the utoff funtion f is onvenient wy to desrie smooth dey of FNN intertions for d > 1.7 Å, nd ensures tht ny SNN intertions with intertomi distnes d > 2.4 Å re eliminted. However, suh dey introdues unphysil intertions etween toms with intertomi distnes in the rnge [1.7,2.4] Å nd must e used very refully. If the results of interest depend on suh intertomi intertions, then the energy lultions will e ffeted y the unphysil nture of the utoff funtion f, nd it is most likely tht they will e wrong. For H-6 Cron, the intertomi distnes whih n e identified in the rnge 0<d i j < 2.4 Å, re (i) d 12 nd d 12 etween FNN of the sme zig-zg hin, (ii) d 13 etween toms onneting rotted zig-zg hins nd (iii) d 14 nd d 14 etween toms elonging to neighouring zig-zg hins of the sme rry, s shown in Fig. 3() with lue, green (solid) nd red (dshed) lines, respetively. Along the trnsition pthwy d 12 = d 12, ut d 14 = d 14 only for =0. For >0, d 14 < d 14. In Fig. 3(), we show how the intertomi distnes d 1 j, j = 2,3,4 hnge s funtion of, long the trnsition pthwy. As we n see, d 12 nd d 13 inrese from 1.47 nd 1.45 Å, respetively, to 1.53 Å, nd they re lwys smller thn 1.7 Å. On the other hnd, d 14 dereses from 2.37 to 1.53 Å. Thus, the 1-4 intertion is under the influene of the unphysil effets introdued rtifiilly y the f funtion, nd therefore, the energy lultions long the trnsition pthwy, whih re used for the lultion of the energy rrier, re not relile. In order to hve more relile estimtion of the energy long the trnsition pthwy, we perform two different sets of lultions for the ohesive energy U oh, using the TB method of Tmor nd Hss, ut with f (d)=1 θ(d d 0 ), where θ(d d 0 )=0 if d < d 0 nd 1 if d > d 0. For the one set, d 0 = 1.7 Å nd for the other, d 0 = 2.2 Å. The former ignores the 1-4 intertion for d 14 > 1.7 Å nd the ltter, for d 14 > 2.2 Å. Thus, there re not differenes etween the two sets of U oh vlues for d 14 < 1.7 nd d 14 > 2.2 Å, ut they re differenes for 1.7 < d 14 < 2.2 Å. For d 14 slightly smller thn 2.2 Å, (where the 1-4 intertions re still wek), the U oh vlues of the former set re rther more relile. For d 14 slightly lrger thn 1.7 Å, (where the 1-4 intertions re stronger), more relile re the U oh vlues of the lter. In Fig. 3() we plot these U oh vlues ginst nd we lso plot the U oh vlues otined from the originl TB method used y Tmor nd Hss. As we n see, none of the two sets of U oh vlues provide suh lrge energy rrier long the trnsition pthwy, s the originl TB method does. The rrier seems to e 0.2 ev, rther thn 0.6 ev of the originl TB method of Tmor nd Hss. Oviously, this lrge rrier vlue is owed to the unphysil effet of the utoff funtion f. Still, however, either with the one or the other wy, the energy rrier remins in the TB lultions, in ontrst to the DFT results, ut this is not the only differene etween the TB nd the DFT lultions. For instne, the totl energy differene per 4 1 7
tom etween H-6 Cron nd dimond ording to the TB lultion is only 0.35 ev, while for the DFT lultions with oth funtionls (CA nd PBE), the differene is 0.9 ev. This disrepny n e ttriuted to the form of the repulsive pir potentil E rep, whih is rther ritrrily hosen nd proly does not provide n urte desription of the repulsion, s well s to trnsferility prolems of the SK prmeters nd the sene of SNN intertions, whih might lso e of importne. The first indition tht the instility mehnism of H-6 Cron might e different from the mehnism whih onverts rhomohedrl grphite to dimond, is tht the in-lyer C-C onds of the ompressed rhomohedrl grphite norml to the grphiti lyers re tilted towrds tht diretion muh erlier efore its trnsition to dimond, using ukling of the grphiti lyers, while in H-6 Cron the zig-zg hins do not ukle nd toms 1 nd 4 of Fig. 3() remin oplnr with their FNN. If the instε x = ε y ε z d 12 d 13 d 14 0.0 0.0 2.624 6.363 1.472 1.455 2.368 0.1 0.0 2.886 6.363 1.582 1.459 2.593 0.1 0.1 2.886 6.999 1.539 1.577 2.615 0.1-0.1 2.886 5.727 1.583 1.308 2.559 0.0-0.1 2.624 5.727 1.380 1.310 2.340 Tle 2 Unit ell vetor lengths ( nd ) nd ond lengths (d 12, d 13, d 14 ) in Å units of H-6 Cron under strin (ε x, ε y, ε z ) without optimiztion. 3.3 Whih ftor is responsile for the instility? Liu et l 12 suggested tht the instility of H-6 Cron might e relted to the short d 14 distne (d 14 = 2.37 Å) etween the prllelly rrnged zig-zg hins (see Fig. 3()). They noted tht d 14 is omprle to the interlyer seprtion of rhomohedrl grphite t whih the grphite to dimond trnsformtion is fvoured, whih ording to Fhy et l 25 is etween 2.1 nd 2.3 Å. However, s we show next, it seems tht this is not the reson for the instility of H-6 Cron. Totl energy per tom (ev) Totl energy per tom (ev) -154.0-154.2-154.4-154.6-154.8-155.0-155.2 ε x = ε y = ε z =0.0 ε x = ε y = 0.1, ε z = 0.0 ε x = ε y = 0.1, ε z = 0.1 ε x = ε y = 0.1, ε z =-0.1 ε x = ε y = 0.0, ε z =-0.1 () -155.4-154.6-154.8-155.0-155.2-155.4 Grphiti-like H-6 Cron 0.03 ev () () Fig. 4 (Color online) () Energy per tom long the trnsition pthwy onverting strethed H-6 Cron to dimond for severl strin onditions. () Energy per tom long the trnsition pthwy onverting the grphiti-like struture to hexgonl dimond. The energy per tom of H-6 Cron for the orresponding trnsition pth onverting it to dimond is shown for omprison. Snpshots of the grphiti-like struture for =0, 0.5 nd 1 re shown. () The unit ell of H-18 Cron. ility of H-6 Cron ws used y the short d 14 distne, then H-6 Cron would e stilized if d 14 ws inresed, s it hppens with rhomohedrl grphite. We strin the optimized H-6 Cron struture found using LDA y strething nd/or ompression, with four different wys, whih inrese d 14. The ses we studied re (i) ε x = ε y = 0.1, (ii) ε x = ε y = ε z = 0.1, (iii) ε x = ε y = 0.1 nd ε z = 0.1 nd (iv) ε z = 0.1, where ε x, ε y nd ε z is the strin long x, y nd z diretion, respetively. The lengths nd of the unit ell vetors, nd of Eq. 1, s well s the ond lengths d 12, d 13 nd d 14 for eh se re presented in T. 2. As we n see, exluding se (iv), d 14 > 2.55 Å. For suh n interlyer seprtion distne, rhomohedrl grphite does not turn into dimond 25,26. However, optimizing the strined strutures (i.e. optimizing the tomi positions for onstnt nd vlues), the optimum struture for ses (iii) nd (iv) is strined dimond struture, ut for the ses (i) nd (ii), optimiztion stuks gin in the unstle H-6 Cron struture. However, lulting the energy long the orresponding trnsition pthwy desried in setion 2, we did not find ny rrier, s shown in Fig. 4(), nd onsequently, the strined H-6 Cron is gin unstle. Consequently, tensile strin n not stilize H-6 Cron struture, (s it would hppened in rhomohedrl grphite). Thus, the reltively short d 14 distne is not responsile for the H-6 Cron instility. In order to study the effet of the interhin intertions etween the prllelly rrnged zig-zg hins on the H-6 Cron stility, we onsider n hypothetil grphiti-like struture, whih is formed y the rrys of the prllelly rrnged zig-zg hins of H-6 Cron, without ny rottion. The unit ell of this struture (long the trnsition pthwy) is defined y the unit ell vetors =, = nd =, where, nd re defined in Eq. 1, with nd defined in Eq. 2. It ontins only the first four toms of H-6 Cron, with positions whih re defined y the frtionl oordintes (1/2+/6,0,0), (1/2+/6, 0,(3/2)[δ+(1/4 δ)]), (1/2 /6, 1/2 /6, 1/2) nd (1/2 /6,1/2 /6,(3/2)[1/3 δ+(1/4 δ)]). The only differene etween H-6 Cron nd this grphiti-like struture is the rottion of the zig-zg hins. It is worth notiing tht the struture otined for =1 is not the ommon dimond, ut the hexgonl dimond (lso lled lonsdleite), whih is very lose energetilly to the ommon dimond. In Fig. 4(), we present the energy of this grphiti-like struture long the trnsition pthwy whih onverts it to hexgonl dimond, with snpshots of the struture for =0, 0.5 nd 1. For omprison we lso present in the sme figure the orresponding energy of H-6 Cron long the trnsition pthwy whih onverts it to dimond. As we n see, there is 0.03 ev rrier 1 7 5
per tom, whih would stilize the grphiti-like struture, nd ensures tht this grphiti-like struture does not spontneously onvert to hexgonl dimond. Oviously, the grphiti lyers t the interlyer distne d 14 = 2.368 Å of H-6 Cron re repelled nd this n explin the lrge vlue of d 12 in omprison with the grphiti ond length. 1-2 nd 1-2 onds re elongted in order to ommodte the stress of the prllelly rrnged zigzg hins. Consequently, the interhin intertions, not only re not responsile for the instility of the H-6 Cron struture, ut they lso tend to stilize the struture. As in the se of H-6 Cron, the optimized grphiti-like struture, (without ny optimiztion of the unit ell vetors), does not ukle. This ehviour for oth ses is proly due to the reltive rrngement of these lyers, whih is different from tht of rhomohedrl grphite nd it seems tht it does not fvour ukling, thus strengthening the indition tht different mehnisms govern the instility of H-6 Cron nd the onversion of rhomohedrl grphite to dimond. This grphiti-like struture eomes unstle, turning into hexgonl dimond, only for d 14 < 2.15 Å, (whih orresponds to 0.25). Exluding the short interhin distne s the ftor whih is responsile for the H-6 Cron instility, the only ftor whih remins is the strin indued y the rottion of the zig-zg hins. This is very resonle, ering in mind tht struturlly, the only differene etween the hypothetil grphiti-like nd the H-6 Cron struture is the rottion of the zig-zg hins. This rottion inreses the energy of the struture y more thn the energy rrier of the grphiti-like struture, mking it unstle. Therefore, we onlude tht the ftor mking H-6 Cron unstle is the strin energy due to the rottion of the zig-zg hins. 3.4 Could other strutures of the H-n fmily e stle? Aording to the lol tomi environment model, whih hve een suessfully used to urtely predit the energy of grphene flkes 27, the energy of grphene-sed strutures is sum of energy ontriutions depending on the lol tomi environment of eh tom of the struture. Bsed on this model we my onsider tht the totl energy per tom U of the H-6 Cron struture is U = U gr +U rot, where U gr is the ontriution of grphiti-like intertions (whih tends to stilize the struture) nd U rot, the penlty ontriution due to the strin indued y the rottion of the zig-zg hins (whih tends to destilize the struture). This ide my e expnded to over ll strutures of the H-n fmily, nd thus ould e used to predit the vlue of their totl energy per tom. With the term "H-n fmily" we men ll the strutures whih ontin rotted y 60 o (or 120 o ) zig-zg rions, insted of just zig-zg hins, s in H-6 Cron struture. The differene etween the energy per tom of H-6 Cron nd the grphiti-like struture desried ove is E = 0.25 ev/tom (see Fig. 4()). This differene my e ttriuted to the energy ost for the rottion of the rrys of prllelly rrnged zig-zg hins, with respet to the grphiti-like struture. On the other hnd, the totl energy per tom for the grphitilike struture desried ove (i.e. without ny optimiztion) is U gr = 154.78 ev. Bering in mind tht there re 6 toms nd 3 hin rottions per unit ell, the energy ost per rottion per unit ell is E 6/3=0.50 ev. Thus, U rot = 0.50n rot /n, where n rot nd n re the numer of rotted rions nd the numer of toms, respetively, per unit ell. Consequently, the totl energy per tom for struture of the H-n fmily should e U = 154.78+0.50 n rot n in ev units. (3) Eq. 3 is oviously orret for H-6 Cron struture. We test its vlidity for the H-18 Cron struture, whih ontins 3 rotted zig-zg rions per unit ell (i.e. n rot = 3) nd eh rion onsists of 3 zig-zg hins. Consequently, the unit ell ontins 2 3 3 = 18 toms. We optimize H-18 Cron struture using the DFT method with the LDA/CA funtionl. The unit ell of the optimized struture is shown in Fig. 4(). As in H-6 Cron struture, it hs hexgonl unit ell defined y the unit ell vetors of Eq. 1, with =2.632 Å nd =18.969 Å. It hs 18 toms with frtionl oordintes (1/2, 0, 0), (1/2, 0, δ 1 /3), (1/2, 1/2, 1/9), (1/2, 1/2, 1/9+δ 2 /3), (1/2, 0, 2/9), (1/2, 0, 2/9+δ 2 /3), (1/2, 1/2, 3/9), (1/2, 1/2, 3/9+δ 1 /3), (0, 1/2, 4/9), (0, 1/2, 4/9+δ 2 /3), (1/2, 1/2, 5/9), (1/2, 1/2, 5/9+δ 2 /3), (0, 1/2, 6/9), (0, 1/2, 6/9+δ 1 /3), (1/2, 0, 7/9), (1/2, 0, 7/9+δ 2 /3), (0, 1/2, 8/9) nd (0, 1/2, 8/9+δ 2 /3), where δ 1 = 0.229 nd δ 2 = 0.224. In omprison with H-6 Cron struture is slightly longer nd /3 = 6.323 Å slightly shorter. Aording to the ove preditive formul, (for n rot = 3 nd n=18) the totl energy per tom of H- 18 Cron is U = 154.69 ev. Aording to the DFT lultion it is U = 154.688 ev/tom. As we n see the two vlues re prtilly the sme, whih strongly indites tht Eq. 3 n orretly predit the energy of the strutures of the H-n fmily. Bsed on the ove onsidertions, struture of the H-n fmily n e stilized if its energy is lower thn the energy of the rrier of the grphiti-like struture (see Fig. 4()), whih is 0.03 ev/tom lrger thn U gr. Therefore, the ondition for the stiliztion is 0.5n rot /n < 0.03, whih for n rot = 3 yields n > 50. Thus, struture of the H-n fmily might e stle if n>50. Still, however, even with n>50, the H-n struture will not e very stle, sine the energy rrier whih stilizes the struture is no more thn 0.03 ev orresponding to temperture of T 360 K. Thus kinetilly, the struture is most likely to e unstle. 4 Conlusions The im of the present work is to shed light on the stility of H-6 Cron, whih seems to e onfusing. We performed initio DFT lultions for the optimiztion of H-6 Cron struture, using two different funtionls (LDA/CA nd GGA/PBE), nd inresing numer of k-grid points nd mesh utoff vlue. Aording to our findings, optimiztions leds either to H-6 Cron struture or to dimond, independent of the ury of the lultion (i.e. numer of k-grid points nd mesh utoff vlue). Using trnsition pthwy, whih linerly onverts H-6 Cron to dimond, we showed tht there is not ny energy rrier long this trnsition pthwy for oth funtionls nd onsequently, H- 6 Cron is unstle. This onlusion ws verified y phonon nd struture lultion on the H-6 struture, whih finds negtive phonon frequenies. Consequently, H-6 Cron is n unstle 6 1 7
struture, whih is spontneously onverted to dimond. Contrry to these DFT results, erlier optimiztion lultions sed on TB method with FNN intertions only, found tht H-6 Cron is stle. Using the sme TB method we perform energy lultions long the sme trnsition pthwy nd we found tht there is n energy rrier etween H-6 Cron nd dimond. Performing slightly modified TB lultions long the trnsition pthwy we showed tht this rrier is overestimted, ut it still remins. We disussed this TB filure suggesting s possile resons for it the seletion of the repulsive pir potentil, the sene of SNN intertions, s well s the trnsferility prolems of the SK prmeters. Liu et l 12 suggested tht the reson for the H-6 instility might e relted with the short interhin distne etween the prllelly rrnged zig-zg hins, whih is very lose to the distne t whih rhomohedrl grphite turns into dimond. In order to test this hypothesis, we strined the H-6 Cron struture y 10% with 4 different wys, using n inrese of the interhin distne nd we performed energy lultions long the trnsition pthwy whih onverts the strined H-6 ron struture to dimond. Agin we did not find ny rrier, showing tht the inrese of the interhin distne n not stilize H-6 Cron struture. Consequently the short interhin distne is not relted with the H-6 Cron instility. Moreover, in order to study the effet of the interhin intertions on the H-6 Cron instility we onsidered grphiti-like struture, whih ws otined from H-6 Cron struture without rottion of the zig-zg hins. Performing energy lultions long the trnsition pthwy onverting tht grphiti-like struture to hexgonl dimond, we showed tht there is smll rrier whih stilizes the grphiti-like struture nd ensures tht it is not spontneously onverted to dimond. Thus, the interhin intertions not only re not responsile for the instility of the H-6 Cron struture, ut they lso tend to stilize it. Therefore, exluding ll other ftors, the remining one, (whih is the strin indued y the rottion of zig-zg hins), is the ftor whih is responsile for the instility of H-6 Cron struture. Bsed on this nd the lol tomi environment model 27 we showed tht the totl energy per tom of the strutures of the H-n fmily is given y Eq. 3 nd depends on the numer of toms nd the numer of rotted zig-zg rions per unit ell. We verified the preditions of this eqution for H-18 Cron struture nd we used it to predit tht if there is stle struture of the H-n fmily, it should ontin t lest 50 toms per unit ell. However, even those possily energetilly stle strutures, re expeted to e kinetilly unstle, sine the energy rrier whih protets them ginst their trnsition to dimond is of the order of less thn 0.03 ev orresponding to temperture T 360 K. Referenes 1 Z. G. Fthenkis nd N. N. Lthiotkis, Phys. Chem. Chem. Phys., 2015, 17, 16418 16427. 2 A. N. Enyshin nd A. L. Ivnovskii, phys. stt. sol. (), 2011, 248, 1879 1883. 3 M. Côté, J. C. Grossmn, M. L. Cohen nd S. G. Louie, Phys. Rev. B, 1998, 58, 664 668. 4 A. Y. Liu nd M. L. Cohen, Phys. Rev. B, 1992, 45, 4579 4581. 5 C. He, L. Sun, C. Zhng nd J. Zhong, Phys. Chem. Chem. Phys., 2013, 15, 680 684. 6 J.-T. Wng, C. Chen, E. Wng nd Y. Kwzoe, Si. Rep., 2014, 4, 4339. 7 Z. Zhu, Z. G. Fthenkis, J. Gun nd D. Tománek, Phys. Rev. Lett., 2014, 112, 026803. 8 A. Ku nd G. Seifert, Phys. Rev. B, 2006, 74, 214104. 9 S. Zhng, Q. Wng, X. Chen nd P. Jen, Pro. Nt. Ad. Si., 2013, 110, 18809 18813. 10 J.-T. Wng, C. Chen nd Y. Kwzoe, Si. Rep., 2013, 3, 3077. 11 M. A. Tmor nd K. C. Hss, J. Mter. Res., 1990, 5, 2273 2276. 12 A. Y. Liu, M. L. Cohen, K. C. Hss nd M. A. Tmor, Phys. Rev. B, 1991, 43, 6742 6745. 13 B. Winkler, C. J. Pikrd, V. Milmn nd G. Thimm, Chem. Phys. Lett., 2001, 337, 36 42. 14 G.-M. Rignnese nd J.-C. Chrlier, Phys. Rev. B, 2008, 78, 125415. 15 B. Zhng, Comp. Mter. Si., 2014, 82, 540 543. 16 J. M. Soler, E. Artho, J. D. Gle et l., J. Phys.: Cond. Mtter, 2002, 14, 2745. 17 D. M. Ceperley nd B. J. Alder, Phys. Rev. Lett., 1980, 45, 566 569. 18 J. P. Perdew nd A. Zunger, Phys. Rev. B, 1981, 23, 5048 5079. 19 J. P. Perdew, K. Burke nd M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865 3868. 20 N. Troullier nd J. L. Mrtins, Phys. Rev. B, 1991, 43, 1993 2006. 21 L. Kleinmn nd D. M. Bylnder, Phys. Rev. Lett., 1982, 48, 1425 1428. 22 H. J. Monkhorst nd J. D. Pk, Phys. Rev. B, 1976, 13, 5188 5192. 23 J. C. Slter nd G. F. Koster, Phys. Rev., 1954, 94, 1498 1524. 24 D. Tománek nd S. G. Louie, Phys. Rev. B, 1988, 37, 8327 8336. 25 S. Fhy, S. G. Louie nd M. L. Cohen, Phys. Rev. B, 1986, 34, 1191 1199. 26 Z. G. Fthenkis nd D. Tománek, Strething grphite to dimond, to e pulished. 27 Z. G. Fthenkis, Mol. Phys., 2013, 111, 3289 3296. 1 7 7