J Comput ElectronDOI 10.1007/s10825-014-0620-x

Computational analysis of the contributions to the piezoelectriccoefficient e33 in ZnO nanowires: first-principles calculations

Seong Min Kim · Tae Yun Kim · Jung-Hoon Lee ·Sang-Woo Kim · JeaWook Ha · JinBeak Kim

Received: 7 April 2014 / Accepted: 12 September 2014© Springer Science+Business Media New York 2014

Abstract We investigated the piezoelectric coefficient, e33,of ZnO nanowires, subdividing it into an ionic term, eion

33 ,and an electronic term, eel

33, and calculated the effects ofdifferent diameters on its value using ab initio densityfunctional theory calculations. The eion

33 term was found tobe dominant, with the innermost (outermost) atoms in thenanowires making the largest (smallest) contribution to theterm. Moreover, the density of states (DOS) and projectedDOS data revealed that the DOS tends to increase at thevalence band maximum in the case of the outermost atoms,where the O 2p and Zn 3d orbital peaks increase in mag-nitude, resulting in hybridization and a decrease in bondlength.

Keywords Piezoelectric coefficient e33 · Densityfunctional theory · ZnO nanowires · Size effects

S. M. Kim (B) · J.-H. LeeComputational Science Group, CAS Center, Samsung AdvancedInstitute of Technology (SAIT), Yongin 449-712, South Koreae-mail: smkim417@hotmail.com

T. Y. KimSKKU Advanced Institute of Nanotechnology (SAINT), Centerfor Human Interface Nanotechnology (HINT), SungkyunkwanUniversity (SKKU), Suwon 440-746, Republic of Korea

S.-W. KimSchool of Advanced Materials Science and Engineering,Sungkyunkwan University (SKKU), Suwon 440-746,Republic of Korea

J. Ha · J. KimDepartment of Chemistry, Korea Advanced Institute of Scienceand Technology (KAIST), Daejeon 305-701, Korea

1 Introduction

ZnO is a II–VI semiconductor that is widely used owingto its many desirable properties, including piezoelectricity,pyroelectricity, optical properties, and biocompatibility [1].Moreover, ZnO has a direct wide band gap of 3.37 eV andan exciton binding energy as high as 60 meV. Furthermore,nanostructured ZnO has potential for use in various applica-tions in areas such as optoelectronics, sensors, transducers,and the biomedical sciences [2–9].

The effect of the structural size of piezoelectric materi-als has been the subject of numerous studies, both theo-retical and experimental [10–14]. In particular, it has beenfound that nanostructured materials have larger piezoelec-tric coefficients than those of corresponding bulk materials.This hints at the underlying principle that can be exploitedto enhance the performance of nanoscale piezogenerators.Recently, using first-principle studies, it has been found thatthe polarization per volume of a material with a reduced sizeis related to the increase in its piezoelectric coefficient, e33

[15]. In addition, increasing the degree of free relaxation ofthe outer atoms of a nanowire (NW) when it is stressed alongthe z-axis as the nanowire diameter decreases also enhancese33 [16]. However, a detailed numerical comparative analysisof the main factors contributing to the value of e33 in NWshas not yet been performed.

In this work, (i) we segregate the contributions to e33

into ionic and electronic parts and compare these terms bycalculating the variation in the polarization per unit vol-ume with respect to the strain using first-principles densityfunctional theory (DFT) calculations. (ii) Then, at a certainnanowire diameter, we determine the distribution of atomsthat actually affects e33 by calculating Z∗

33 (the Born effectivecharge along the z-axis) and �ε3 (the variation in the strainalong the z-axis) of the atoms at each radial position using

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density functional perturbation theory (DFPT) calculations.(iii) Finally, using the density of states (DOS) or the projectedDOS (PDOS) data, we compare and analyse the orbital char-acters of the atoms and their bond lengths while moving fromthe inner to the outer atoms along the radial direction.

2 Computational approach

The VASP software [17] was used for performing the DFTcalculations using the projector augmented-wave (PAW)method [18]. The generalized gradient approximation(GGA), performed using the Perdew–Burke–Ernzerhof(PBE) functional, was used for the exchange-correlationpotential. The ionic relaxation of the structure was performedwithout any symmetry constraints, and a conjugate-gradientalgorithm was used. The 3d-orbital electrons of ZnO (atransition metal compound) were modelled as valence elec-trons to describe the interactions during bond formation. Thecalculations related to the supercells were performed using1×1×4 Monkhorst–Pack k-point sampling of the Brillouinzone where the energy differences converged. The cut offenergy of the plane-waves is 400 eV used. The equilibriumlattice parameters for ZnO (wurtzite) are calculate to be a =3.2 Å, c = 5.1 Å, and u = 0.4. It should be noted that the sizesof the NW model were limited to one lattice constant alongthe c-axis (the polar axis) with periodic boundary conditionsto lower the computational costs.

3 Results and discussion

In order to calculate the piezoelectric coefficient (e33), NWswith different diameters were strained along the c-axis tostrains of up to of 3 %. The energy of the strained NWswas first minimized, and then, two polarizations were cal-culated using the modern theory of polarization or theBerry-phase approach [19] for the following two cases: (i)using a centrosymmetric structure as a reference (Pref) and(ii) a strained structure (Pstrained). The actual polarizationused here was Pstrained − Pref. It should be noted that, ide-ally, Pref should be zero for both the ionic and the elec-tronic cases. However, the value was not exactly zero here(Pref (ion) = −0.2, Pref (el) = 0.4 for the diameter of 3.67Å, Pref (ion) = −0.8, Pref (el) = 1.6 for the diameter of9.66 Å, and Pref (ion) = −1.8, Pref (el) = 3.6 for the diam-eter of 15.62 Å, respectively); hence, the above-mentionedpolarization was used.

The total polarization of a strained NW, PT , can be math-ematically expressed as

PTi = Ps

i +eivεv = Psi +

(eel

iv+eioniv

)εv = Ps

i +Peli +P ion

i

(1)

where Psi is the spontaneous polarization of the unstrained

NW, εv is the strain tensor element, and eeliv and eion

iv definethe piezoelectric tensor of the electronic and ionic parts inthe Voigt notation, resulting from the external strain appliedto the system [20]. The two piezoelectric terms multiplied bythe strain tensors then lead to the electronic and ionic polar-izations, Pel

i and P ioni , respectively. In particular, the piezo-

electric coefficient, e33, can be segregated into two parts: eel33

and eion33 . Figure 1a–c show the polarizations per unit volume

of the electronic part and the ionic part plotted against thestrain for three different diameter values. The slopes of thelinear fits through the blue and green data points correspond,respectively, to eel

33 and eion33 . The first term, eel

33, that is, theelectronic contribution, is the clamped-ion or homogeneous-strain contribution evaluated at the vanishing internal strainafter the introduction of an external strain into the system[21] and can be expressed as

eel33 = ∂ PT

3

∂ε3

∣∣∣∣∣u

= ∂

∂ε3

(Ps

3 + Pel3 + P ion

3

)∣∣∣∣u

(2)

= ∂

∂ε3(Pel

3 )

∣∣∣∣u

+ ∂

∂ε3(P ion

3 )

∣∣∣∣u

(3)

where u is an internal coordinate in the system.The second term, eion

33 , is the ionic contribution or the inter-nal strain term attributable to the relative displacements ofthe differently charged ions, induced by the external strain,and can be expressed as

eion33 =

∑k

∂ PT3

∂uk,3

∣∣∣∣∣ε

.∂uk,3

∂ε3(4)

=∑

k

∂

∂uk,3

{Ps

3 + Pel3 + P ion

3

}∣∣∣∣ε

.∂uk,3

∂ε3(5)

=∑

k

{∂ Pel

3

∂uk,3+ ∂ P ion

3

∂uk,3

}∣∣∣∣∣ε

∂uk,3

∂ε3(6)

=∑

k

ea3

�Z∗

k,33.∂uk,3

∂ε3(7)

=∑

k

ea3

�

{Z core

k + Z∗elk,33

}.∂uk,3

∂ε3(8)

where � is the volume, a3 is the lattice parameter alongthe z-axis, k is an atomic index, and Z∗ is the transverseBorn effective charge. The computational result shows thatthe electronic part, eel

33, has a total polarization that variesto a smaller degree with the strain than does the ionic part.Hence, it can be concluded that the main contributor to e33

is eion33 and that e33 can be mathematically expressed as

e33 = eel33 + eion

33 ≈ eion33 (9)

The calculated values of e33 for different diameters are plot-ted in Fig. 1d. It can be seen that the value of e33 decreases asthe diameter increases (and approaches the bulk value); this

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Fig. 1 Ionic and electronicpolarization per unit volumeversus strain for nanowires withrelaxed diameters of a 3.67 Å, b9.66 Å, and c 15.62 Å. Note thatthe slopes are only noticeable inthe case of the ioniccontribution. d Absolute valuesof the piezoelectric coefficient,e33, of the ZnO nanowires as afunction of their diameters. Thevalues presented in the figurewere calculated using the GGA(PBE functional)

Fig. 2 Cross sections ofhexagonally modelled ZnOnanowires with diameters of a9.66 Å and b 15.62 Å afterrelaxation (energy

minimization). NormalizedZ∗

33�ε3

values were calculated for theconstituent atoms using thecross sections of the wires withdiameters of c 9.66 Å and d15.62 Å. Consequently, theinnermost atoms were found tobe the most active contributorsto eion

33

result is consistent with those of previous studies [15,16] andshows clearly the size effects. Here, the eion

33 term is used tocalculate e33, as the variation in the total polarization per unitvolume with the strain is only sensitive to the ionic contribu-tion to the piezoelectric coefficient (Fig. 1a–c and Eq. (9)) as

explained above. Thus, the electronic contribution, eel33, can

be neglected.Cross sections of the {0001} plane of a relaxed ZnO

nanowire with diameters of 9.66 and 15.62 Å are shown inFig. 2a, b, respectively; a clearly hexagonal structure with

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Fig. 3 Calculated total DOS and PDOS values for a O and b Znatoms of a ZnO NW with a diameter of 15.62 Å. The black, red, andblue lines represent the innermost, intermediate, and outermost atoms,respectively. The PDOS analysis showed that the main orbital of Owas Px and that of Zn was d3Z2−r2 . Note that the Fermi levels arelocated at 0 eV. c Side view of a ZnO NW with a diameter of 15.62

Å. The bond lengths α1, α2, and α3 (vertical direction) and β1, β2,and β3 (lateral direction) are marked. d Length of bonds between Znand O plotted with respect to the radial distances R1, R2, and R3. Thebond length becomes shorter in the case of the outer atoms, indicat-ing stronger hybridization between the Zn and O atoms (Color figureonline)

the space group P6mm can be seen. The hexagonal supercellis used to model the ZnO nanowires, and the larger laterallattice constant (∼10 Å) is used to ensure that there are nointeractions between the nanowires. For a diameter of 9.66Å, the cross section with a central hexagon is surrounded bysix hexagons, with the atoms positioned on the inner circleare defined as inner atoms and the atoms on the outer circleas outer atoms, as shown in Fig. 2a. Similarly, for the crosssection corresponding to a diameter of 15.62 Å, the centralhexagon is surrounded by six hexagons, which are, in turn,surrounded by twelve hexagons. The inner atoms are definedon the innermost circle. The intermediate atoms are definedon the next circle. The outer atoms are defined on the outer-most circle, as shown in Fig. 2b. These two nanowires (crosssections shown in Fig. 2a, b) have supercells consisting of 48and 108 atoms, respectively. Note that the lattice constant c(5.05 Å) is the same in all cases for the relaxation condition.Note that in all cases, the strained lattice constant c is fixedduring relaxation.

To analyse and compute the distribution of the constituentatoms that actually affect the dominant term, eion

33 , the ratio ofthe Born effective charge, Z∗

33, to �ε3 is plotted for differentdiameters in Fig. 2c, d on the basis of the diameters shown inFig. 2a, b, respectively, using DFPT calculations [22]. The

ratioZ∗

33�ε3

is closely related to eion33 , which can be expressed as

eion33 =

∑k

ea3

�Z∗

k,33∂uk,3

∂ε3

∼=∑

kZ∗

k,33�uk,3

� ∈3

∼=∑

kZ∗

k,331

�ε3(10)

We assumed that the values of �uk,3 (the variation in theinternal coordinate along the z-axis of the kth atom) areequal, which means that during perturbation, all atoms havean equally weighted effect on the piezoelectric coefficient;

hence, from the value ofZ∗

k,33�ε3

, one can estimate which of the

atoms can affect eion33 and to what extent. It can be seen from

Fig. 2c, d that, as one moves outward, the ratioZ∗

k,33�ε3

(nor-

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malized) makes a smaller contribution to the ionic part of thepiezoelectric coefficient, eion

33 , in the NW system than in theatoms at the very centre of the NW. Physically speaking, theBorn effective charge, Z∗

k,33, is the ratio of the variation ofthe total polarization to the variation of the internal coordi-nate along the z-axis for the kth atom. �ε3 is the change inthe strain induced by the variation of the internal coordinatealong the z-axis (�u3). Therefore, for example, in the case of

the interior atoms of the NWs, if the value ofZ∗

k,33�ε3

is larger,either Z∗

k,33 is larger or �ε3 is smaller or both the conditionsare true.

Figure 3a, b show the DOS and PDOS data for the O andZn atoms along the radial direction when the diameter of theZnO NW is 15.62 Å. The DOS at the valence band maximum(VBM) becomes higher at the outer O (Fig. 3a) and Zn atoms(Fig. 3b). The reason for this is that the exterior atoms can berelatively free to move laterally; thus their bonding is highlydependent on the wall environment. The Zn and O orbitalswill then have a high probability of undergoing hybridization,leading to chemically inert side-wall bonding surfaces {10–10} in the NWs. In particular, from the results of the PDOSanalysis of the O and Zn atoms, the main orbitals were foundto be Px (Fig. 3a) and d

3Z2−r2 (Fig. 3b), respectively. Fig-ure 3c shows the side view of the ZnO NW; the lengths ofthe bonds between the Zn and O atoms are indicated by α1,α2, and α3 (in the vertical direction) and by β1, β2, and β3 (inthe lateral direction). The bond length between the Zn and Oatoms is plotted with respect to the radial distance in Fig. 3d;it decreases as one moves outward. This is another indica-tion that hybridization occurs between the Zn and O atomsowing to the decrease in their bond length, or, alternatively,the increase in their bond strength.

4 Conclusions

In summary, a quantum mechanical computational analy-sis of the contributions to e33 (the ionic part, eion

33 , and theelectronic part, eel

33) in ZnO nanowires was performed. Ourinvestigation of the effects of different diameters, performedusing the Berry-phase approach, revealed that eion

33 is the main

contributor to e33. In particular, the ratioZ∗

33�ε3

was used as aquantitative metric that indicates how many atoms that havea significant effect on eion

33 can be distributed through the NWsystem. Finally, the DOS and PDOS data revealed strongerhybridization between the O 2p and Zn 3d orbitals in the caseof the outer atoms, leading to the formation of short bonds.

Acknowledgments This work was supported by the ComputationalEnergy Harvesting (CEH) project at the Samsung Advanced Instituteof Technology.

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