Physica E 42 (2010) 2727–2730

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Theoretical investigation on the structural stability of GaAs nanowires withtwo different types of facets

Tomoki Yamashita �, Toru Akiyama, Kohji Nakamura, Tomonori Ito

Department of Physics Engineering, Mie University, 1577 Kurima-Machiya, Tsu, Mie 514–8507, Japan

a r t i c l e i n f o

Article history:

Received 28 August 2009

Received in revised form

22 December 2009

Accepted 18 January 2010Available online 25 January 2010

Keywords:

Nanowire

GaAs

Structural stability

Side facet

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2010.01.037

esponding author. Tel.: +81 59 232 1211; fax

ail address: yamashita06@nd.phen.mie-u.ac.jp

a b s t r a c t

The relative stability between the wurtzite and zinc blende structures in GaAs nanowires with

f1 1 1g=f1 1 0 0g facets and those of f1 1 0g / f1 1 2 0g facets is systematically investigated using our

empirical interatomic potential calculations and first-principles calculations. Our calculations clarify

that the wurtzite structure is stabilized over entire diameters range for nanowires with f1 1 1g=f1 1 0 0g

facets. In contrast, for nanowires consisting of f1 1 0g=f1 1 2 0g facets, the zinc blende structure is

favorable for diameters larger than 29 nm. This is because the surface energy difference between f1 1 0g

and f1 1 2 0g surfaces is quite small compared to that between {1 1 1} and f1 1 0 0g surfaces. The

calculated results imply that the stability of nanowire side facets is a quite important factor

determining the crystal structure.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Semiconductor nanowires (NWs) are expected to play a keyrole in future nanotechnology owing to their fundamentalimportance in reduced dimensionality and size in optical,electrical, and mechanical properties and their wide range ofpotential application in nanoscale devices. In particular, NWsconsisting of group III–V materials (III–V NWs) have attractedmuch attention because of their specific optical properties andare expected to be applied to light-emitting diodes [1–3],photodetectors [4,5] and lasers [6–8]. Hence, considerable effortshave been devoted to synthesize these NWs by employingvarious methods such as metal-organic vapor-phase epitaxy[9,10] and molecular-beam epitaxy [11–13], and many theoreticalstudies have been carried out [14–19]. It has been known thatthere are some structural characteristics different from bulkcrystals: III–V NWs grown in the [1 1 1] direction often includethe wurtzite (W) segments in the zinc blende (ZB) structure[20–30].

Furthermore, it has been reported that III–V NWs along the[1 1 1] direction have two different types of facets. NWs withf1 1 1g=f1 1 0 0g (in the ZB structure/in the W structure) facets areusually fabricated by the vapor–liquid–solid (VLS) mechanismand tend to have the W structure [20–25]. On the other hand,NWs with f1 1 0g=f1 1 2 0g facets are generally grown by theselective area (SA) growth and tend to have various structures

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(T. Yamashita).

depending on semiconductor materials [26–29]. For example,GaAs NWs grown by the SA growth have f1 1 0g=f1 1 2 0g facetsand exhibit the ZB structure including the W segments as stackingfaults [26]. In spite of these findings, the relationship between theorientation of NW facets and the structural stability is not clearfrom theoretical view point. In our previous study, for III–V NWs,the relative stability between the W and ZB structures consistingof f1 1 0 0g and f1 1 0g facets, respectively, has been successfullyinvestigated based on an empirical interatomic potential calcula-tions [14]. Moreover, we have clarified the structural stability ofGaP NWs with f1 1 1g=f1 1 0 0g facets and those withf1 1 0g=f1 1 2 0g facets [17]. In this study, we evaluate NWcohesive energy and surface energy for f1 1 1g, f1 1 0 0g, f1 1 0gand f1 1 2 0g surfaces using our empirical interatomic potentialcalculations and first-principles calculations, respectively. Basedon the calculated energies, we determine the structural stabilityof GaAs NWs consisting of f1 1 1g=f1 1 0 0g facets and those off1 1 0g=f1 12 0g facets.

2. Computational methods

We employ our simple system energy formula, which is givenby the following equation:

ENW ¼ E0þESurfþDEW�ZB; ð1Þ

E0 ¼1

2

X

i;j

Vij; ð2Þ

ΔENW (ZB3-W)

ΔENW (ZB5-W)

ΔENW (ZB7-W)

ΔEN

W (Z

Bn-

W) (

eV/a

tom

)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0

0.12

dNW (nm)5 10 15 20 25 30 35

T. Yamashita et al. / Physica E 42 (2010) 2727–27302728

ESurf ¼1

2

X

i;j

qiqj

ejRi�Rjj; ð3Þ

where ENW is NW cohesive energy and E0 is cohesive energycalculated by Khor–Das Sarma type empirical interatomic poten-tial Vij within the second nearest neighbors [31–33]. ESurf issurface electrostatic energy between cations and anions [34]. Ri

are the positions of the atoms having charge qi and e is the staticdielectric constant of GaAs, which we take to be 13.2 [35]. Sincethe energy of the Ga dangling-bond states is much higher than theAs lone-pair orbitals, there is a transfer of charge which leaves Aslone-pair bands filled and the Ga-derived bands empty. Formallywe assign a charge of � 3

4 to each of the threefold-coordinated Asatoms and a charge of þ 3

4 to each of the threefold-coordinated Gaatoms. DEW�ZB is energy difference between the W and ZBstructures in the bulk form, which is caused by the electrostaticinteraction between covalent bond charges and ionic charges [36].The value of DEW�ZB used in this study is 8.3 meV/atom obtainedby ab initio calculations [37].

Here, we adopt the NW model proposed by Johansson et al.[20] for the NWs consisting of f1 1 1g=f1 1 0 0g facets (Type 1) andthat to be hexagonal cylinder for the NWs consisting off1 1 0g=f1 1 2 0g facets (Type 2) [38]. Figs. 1(a) and (b) displayrepresentative cross-sectional views of Type 1 and Type 2 models,respectively. In Type 1 model, either surface area of cations oranions terminated by {1 1 1} facets increases as the NW grows inthe ZB stacking sequence, resulting in the emergence of triangularcross-section. Therefore, from the viewpoint of crystal symmetry,periodic (infinite length) NWs are unable to be formed withouttwin planes. In order to treat NWs with infinite length, weconsider four types of NW models including periodic twin plains[17]. They are the W structure (Type 1-W) and the ZB structureswith the W segments periodically. The region of the ZB stackingconsists of 3-bilayer (Type 1-ZB3), 5-bilayer (Type 1-ZB5) and 7-bilayer (Type 1-ZB7). In Type 2 model, the W (Type 2-W) and ZB(Type 2-ZB) structures are considered.

In addition, in order to evaluate the contribution of NW facetsquantitatively, we calculate surface energies for various orienta-tion using ab initio calculations. The calculations are performedwithin pseudopotential approach using the GGA-PBE exchangecorrelation functional [39–42]. The plane-wave basis set with acutoff energy of 16 Ry is used to expand the valence wavefunctions. {1 1 1}, f1 1 0 0g, f1 1 0g and f1 1 2 0g surfaces aresimulated using 1�1 slab models consisting of � 16 atomiclayers and 10 A vacuum region. Both sides of the slab are allowedto relax and four layers at the center of the slab are fixed at idealpositions.

(211)

(121)(112)

(121) (112)

(211)

(101) (110)

(011)(011)

(101)(110)

dNW dNW

Type 1 Type 2

Fig. 1. Cross-sectional views of GaAs NWs with (a) {1 1 1}/f1 1 0 0g (Type 1) and

(b) f1 1 0g/f1 1 2 0g facets (Type 2). Light and dark circles denote Ga and As atoms,

respectively. Type 1 model has {1 1 1} microfacets in the three-dimensional (3D)

geometry although the cross-sectional view becomes a hexagonal shape

terminated by f1 1 2g facets. The arrows represent the NW diameter.

3. Results and discussion

First, we determine the stable crystal structure in Type 1model. Figs. 2(a) and (b) show the calculated ENW and ESurf inEq. (1) of Type 1-ZB3, Type 1-ZB5, and Type 1-ZB7 models withrespect to Type 1-W model as a function of NW diameter (dNW),respectively. Here, we denote the energy differences in ENW (ESurf)between Type 1-ZBn (n = 3, 5, 7) and Type 1-W models asDENWðZBn�WÞ (DESurf ðZBn�WÞ). These energy differences areproportional to the occurrence rate of the twin planes. Ourcalculated results show that Type 1-W model is the most stableover entire diameters range, and DENWðZBn�WÞ increases as theinterval of the W segments becomes large (DENWðZB3�WÞoDENWðZB5�WÞoDENWðZB7�WÞ). Furthermore, DENWðZBn�WÞis found to decrease as dNW becomes large. This structural trendimplies that the ZB type stacking sequence can be incorporatedinto the W type stacking sequence for NWs with large diameters.It should be noted by comparing Figs. 2(a) and (b) that thecontribution of ESurf is quite dominant in GaAs NWs. Thecontributions of E0 and DEW�ZB is negligible compared to that ofESurf. The stabilization of the W structure agrees with the stabilityof NW side facets indicated by surface energy calculation. Thesurface energy for {1 1 1} surface is found to be 1.15 eV/atom

ΔESurf (ZB3-W)

ΔESurf (ZB5-W)

ΔESurf (ZB7-W)

ΔES

urf (

ZBn-

W) (

eV/a

tom

)

0.00

0.02

0.04

0.06

0.08

0.10

0dNW (nm)

5 10 15 20 25 30 35

Fig. 2. Calculated (a) DENWðZBn�WÞ and (b) DESurf ðZBn�WÞ as a function of dNW.

Diamonds, squares, and triangles represent the values of DENWðZBn�WÞ and

DESurf ðZBn�WÞ.

T. Yamashita et al. / Physica E 42 (2010) 2727–2730 2729

corresponding to 80 meV/A2. For f1 1 0 0g surface, the surfaceenergy is 0.431 eV/atom which corresponding to 32 meV/A2. Thesurface energy for f1 1 0 0g surface exhibiting the W structure ismuch lower than that for ZB {1 1 1} surface by � 720 meV=atom.This is because f1 1 0 0g nonpolar surface where cations andanions are alternately aligned in the [0 0 0 1] direction is morestable than {1 1 1} polar surface terminated by only cations oranions. Thus, the stability of NW side facets is a significant factordetermining the structure of NWs with f1 1 1g=f1 1 0 0g facets. As aresult, GaAs NWs of Type 1 model tend to include the W segmentseven for large diameters [24,25].

Next, we discuss the relative stability between Type 2-W andType 2-ZB models. Figs. 3(a) and (b) illustrate the calculated ENW

and ESurf of Type 2-ZB model with respect to those of Type 2-Wmodel as a function of dNW, respectively. Fig. 3(a) indicates thatthe most stable structure in Type 2 model depends on the NWdiameter. the W structure is stabilized for dNWo29 nm and the ZBstructure is favorable for 29 nmodNW. These calculated resultsindicate that GaAs NWs exhibit the ZB structure for largediameters. This is because the energy difference of ESurf is smallcompared to that of Type 1 model as shown in Fig. 3(b), althoughESurf of Type 1-W model is lower than that of Type 1-ZB model.Moreover, the surface energy calculation supports the small

ΔEN

W (Z

B-W

) (eV

/ato

m)

ΔES

urf (

ZB-W

) (eV

/ato

m)

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

dNW (nm)10 20 30 40 50 60

0dNW (nm)

10 20 30 40 50 60

Fig. 3. Calculated ENW and ESurf of Type 2-ZB model with respect to Type 2-W

model as a function of dNW. The vertical dashed line in (a) shows the critical

diameter for structural change (dNW = 29 nm).

energy difference. The surface energy for f1 1 0g surface is foundto be 0.451 eV/atom corresponding to 39 meV/A2, while thesurface energy for f1 1 2 0g surface is 0.399 eV/atom whichcorresponding to 34 meV/A2. The surface energy difference isquite small (� 60 meV=atom) compared to that of Type 1 model.This results in the stabilization of the ZB structure for largediameters in which the contribution of NW facets on the stabilityis negligible. However, the energy difference between the W andZB structures for large diameters still smaller than that in bulkGaAs. Therefore, the W segments can be incorporated in the ZBstructure with large interval along the [1 1 1] direction,qualitatively consistent with the observation of rotational twins[24,26].

4. Conclusion

The relative stability between the W and ZB structures in GaAsNWs with f1 1 1g=f1 1 0 0g facets and those of f1 1 0g=f1 1 2 0gfacets is systematically investigated using our empirical intera-tomic potential calculations and first-principles calculations. Ourcalculated results clarify that the W structure is stabilized overentire diameters range for NWs with f1 1 1g=f1 1 0 0g facets. Incontrast, the most stable structure in NWs with f1 1 0g=f1 12 0gfacets depends on the NW diameter. The NWs consisting off1 1 2 0g facets (the W structure) are favorable for diameters lessthan 29 nm, whereas those consisting of f1 1 0g facets (the ZBstructure) are stabilized for diameters larger than 29 nm. This isbecause the surface energy difference between f1 1 0g and f1 12 0gsurfaces is quite small compared to that between {1 1 1} andf1 1 0 0g surfaces. Although the kinetics of NW growth could playan important role in the formation of the twin planes [43], thestability of NW side facets is of significance determining thecrystal structure.

Acknowledgements

This work was supported in part by Grant-in-Aid for ScientificResearch from JSPS under Contracts no. 21560032. Computationswere performed at RCCS (National Institute of Natural Sciences).

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