Giant piezoelectric size effects in zinc oxide and gallium nitride nanowires. A first principles investigation.

Abstract

Nanowires made of materials with noncentrosymmetric crystal structure are under investigation for their piezoelectric properties and suitability as building blocks for next-generation self-powered nanodevices. In this work, we investigate the size dependence of piezoelectric coefficients in nanowires of two such materials - zinc oxide and gallium nitride. Nanowires, oriented along their polar axis, ranging from 0.6 to 2.4 nm in diameter were modeled quantum mechanically. A giant piezoelectric size effect is identified for both GaN and ZnO nanowires. However, GaN exhibits a larger and more extended size dependence than ZnO. The observed size effect is discussed in the context of charge redistribution near the free surfaces leading to changes in local polarization. The study reveals that local changes in polarization and reduction of unit cell volume with respect to bulk values lead to the observed size effect. These results have strong implication in the field of energy harvesting, as piezoelectric voltage output scales with the piezoelectric coefficient.

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Published:January 11, 2011
r2011 American Chemical Society
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dx.doi.org/10.1021/nl104004d|Nano Lett. 2011, 11, 786–790
LETTER
pubs.acs.org/NanoLett
Giant Piezoelectric Size Effects in Zinc Oxide and Gallium Nitride
Nanowires. A First Principles Investigation
Ravi Agrawal and Horacio D. Espinosa*
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3111, United States
ABSTRACT: Nanowires made of materials with noncentro-
symmetric crystal structure are under investigation for their
piezoelectric properties and suitability as building blocks for
next-generation self-powered nanodevices. In this work, we
investigatethesizedependence ofpiezoelectriccoefficientsin
nanowires of two such materials - zinc oxide and gallium
nitride. Nanowires, oriented along their polar axis, ranging
from 0.6 to 2.4 nm in diameter were modeled quantum
mechanically. A giant piezoelectric size effect is identified for both GaN and ZnO nanowires. However, GaN exhibits a larger
andmoreextended size dependence thanZnO.The observed sizeeffectisdiscussed inthecontextofchargeredistributionnearthe
free surfaces leading to changes in local polarization. The study reveals that local changes in polarization and reduction of unit cell
volume with respect to bulk values lead to the observed size effect. These results have strong implication in the field of energy
harvesting, as piezoelectric voltage output scales with the piezoelectric coefficient.
KEYWORDS: Gallium nitride, zinc oxide, nanowires, density functional theory, piezoelectric properties
P
orviceversa.Itiscausedbythenonsymmetriccrystalstructureof
certain materials, which results in an effective change in polariza-
tion in response to an applied mechanical strain. The strained
material behaves like a charged capacitor with an electrostatic
potentialacrossit,whichcanbeutilizedinsensing,actuation,and
energy harvesting applications. Therefore, piezoelectricity pro-
vides a direct means of conversion between mechanical and
electrical energy. However, at the macroscale, the electrical
energy output is relatively low in comparison to mechanical
energy required to strain the material. Thus at large scales,
applications of piezoelectric devices are typically limited to
sensors (e.g., pressure sensors1) and actuators (e.g., in atomic
force microscopy2) where efficiency is not critical. By contrast,
nanoscale offers an advantage, that is, the forces required to
deform nanostructures made of piezoelectric materials are small
enough to be extracted from natural sources of mechanical
energy (e.g., ambient noise, wind energy, body movements,
and flowing water). Therefore, thin films and nanowires are
considered suitable building blocks for next-generation energy-
harvesting devices.3,4Inaddition, as the sizeof these structures
is reduced to the nanoscale (thin films and nanowires (NWs)),
the conversion efficiency can be improved dramatically for the
following reasons: (i) nanomaterials tolerate relatively large
deformations prior to failure, which is critical as the electro-
static potential generated (V) is proportional to the applied
strain (ε) and piezoelectric coefficient (d33), namely V ? d33ε;
(ii) material properties are often enhanced at the nanoscale
relative to the bulk due to surface effects and high surface-to-
volume ratios.
iezoelectricity is a phenomenon in which an electric field is
generated inside a material subjected to a mechanical strain
Recently, Wang et al. showed that ZnO NWs can act as
piezoelectricnanogeneratorsbothbydeflectingindividualNWs5
and by integrating them in hybrid microfiber assemblies.6NW
nanogenerators were also used to convert biomechanical energy
(movement of a human finger and body motion of a hamster) to
electrical energy.7
Variousexperimentalstudieshaveprobedeithertheelectrical8-10
or mechanical11-16behavior of nanowires separately; however,
electromechanicalcouplinganditspotentialsize-effectshavenot
been adequately addressed. The challenges associated with such
experiments include(i)difficultiesinsamplemanipulationatthe
nanoscale, (ii) making appropriate electrical measurements
accounting for contact resistances, (iii) measuring currents and
voltages with sufficiently high resolution, and so forth. The
difficulties in conducting such experiments seem to be the
primary reason for a large scatter observed in the experimentally
reported piezoelectric coefficients for ZnO nanostructures. For
example, piezoresponse force microscopy (PFM) has revealed
piezoelectric coefficient ranging from 4.41 to 7.5 pm/V for ZnO
nanorods with diameters in the 150-500 nm range,17,18as well
asvaluesrangingfrom14.3to26.7pm/VforZnOnanobeltstens
of nm in thickness.19On the contrary, a resonance shift method
has revealed a value as high as 12000 pm/V for a 230 nm ZnO
nanowire,20as compared to a bulk value of 12.4 pm/V.21,22
In this work, we investigate piezoelectric size effects from a
computational standpoint. First principles-based density func-
tional theory (DFT) calculations were performed to model
Received:
Revised:
November 15, 2010
December 19, 2010

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nanowires of zinc oxide and gallium nitride with hexagonal
cross sections and with diameters ranging from 0.6 to 2.4 nm.
Size dependent trends for piezoelectric coefficients were identi-
fied and an analysis of distribution of charges and dipole
moments was performed in order to understand the observed
size dependence.
The SIESTA23software was used for conducting the DFT
calculations. The generalized gradient approximation (GGA)
using the Perdew-Burke-Ernzerhof (PBE) functional and the
revisedPBEfunctionalwithdouble-ζpolarization(DZP)orbital
basis sets was used. Pseudopotentials for all the atomic species
were generated using the Troullier-Martins scheme,24and are
available via the SIESTA homepage.25As d-orbitals play an
important role in bonding in case of transition metals, the effect
of d-orbital cutoff radius was also investigated for the case of
GaN.Asmallercutoffradius(thanthecutoffradiusavailablewith
SIESTA pseudopotentials) was used following the work of
Carter et al.26-28The cutoff radii used for generating two sets
ofpseudopotentialsforGaNareshowninTable1.ForbothZnO
and GaN, 3d-orbital electrons were modeled as the valence
electrons to allow for their interaction in bond formation as
should be the case with transition metal elements.
To keep the computational costs low, the model sizes were
limited to one lattice constant along the polar axis of the
nanowire with periodic boundary conditions applied. The con-
vergence of density of k-space mesh-points was studied and
Monkhorst pack grid of 1 ? 1 ? 5 was used for these
calculations.29To validate the modeling approach and pseudo-
potentials employed, bulk piezoelectric coefficients were first
calculated for both GaN and ZnO. Table 2 shows the bulk
properties as calculated for ZnO and GaN. A comparison
between our calculations and values reported in the literature
is also provided. The piezoelectric coefficient for bulk GaN, as
calculated using PSP1, was found to be smaller than the value
reported for bulk. As we were interested in the size dependent
trends,calculations werepursuedwithbothpseudopotentials for
GaN to investigate how the definition of the d-orbital affects the
results.
Nanowire cross sections as a function of wire diameter are
shown inFigure1together withtherelaxed atomicpositions.To
calculate the piezoelectric coefficients (d33), the nanowires were
strainedalongthepolaraxis atfixedincrementsof 0.5% strainup
to 4% strain. The energy of the strained configurations was first
minimized and then their polarization was calculated using the
Berry-phase approach.33The polarization per unit volume was
plotted as a function of strain, the slope of which yielded the
piezoelectric coefficient for a given nanowire material and
size.34
Figure 2a shows the piezoelectric coefficients for ZnO and
GaN nanowires, as a function of their diameter, calculated using
the PBE functional. In Figure 2b, the values are normalized with
respect to the bulk piezoelectric coefficients to reveal the size
dependent enhancement. These results suggest that, for both
ZnO and GaN nanowires, approximately 2 orders of magnitude
improvement in piezoelectric coefficients can be attained if the
nanowire diameter is reduced to less than 1 nm. The results also
showthatforGaNthestrongpiezoelectricenhancementextends
to diameters well in excess of 2.5 nm. By contrast, the enhance-
ment is much attenuated for ZnO nanowires with diameters
larger than 1.5 nm.
The investigation reveals that the computed piezoelectric
coefficients are highly dependent on the chosen pseudopoten-
tials and functionals. Table 3 compares the predictions of PBE
and RPBE functionals for NWs made of the two materials. The
RPBE functional revealed smaller piezoelectric coefficient as
compared to PBE functional in the case of ZnO; however, the
differencesinPBE/RPBEpredictionsweresmallforGaN.These
differences suggest that further insight from experiments is
necessary to validate the use of one functional over another for
modeling electromechanical properties in nanowires.
In Table 4, the results obtained for GaN nanowires using two
different pseudopotentials for Ga-atoms are compared. PSP1
with higher cutoff radius for the 3d-orbital, predicts higher
piezoelectric coefficients as compared to PSP2. It is noteworthy
that the PSP1 underestimates the bulk value as compared to
PSP2. This means that PSP1 predicts stronger size dependence
as compared to PSP2.
To further understand the origin of this size dependence, the
effect of atomic restructuring was decoupled from the change in
absolute value of polarization. Figure 3a shows the polarization
Table 1. Difference in Cut-off Radii of the Two Sets of
Pseudopotentials (PSP) Used for Ga and N Atoms
Cut-off radii for s, p, d, and f orbitals (Å?)
atom type
PSP1PSP2
Ga
N
2.18, 2.35, 2.18, 2.59
1.48, 1.48, 1.48, 1.48
2.0, 2.0, 1.2, 2.38
1.5, 1.5, 1.5, 1.5
Table 2. Bulk Properties of ZnO and GaNa
literatureour calculations
material property
ZnOGaNZnOGaN (PSP1)GaN (PSP2)
lattice constant a (Å?)
c/a
c33(GPa)
e33(C/m2)
aFor GaN, values obtained using two different PSPs are reported.
3.26,303.28631
1.616,301.59531
3.198,323.2828
1.634,321.61928
35432
0.6632
3.25
1.634
3.283
1.628
344
0.255
3.281
1.617
323
0.5541.1931
1.18
Figure 1. Cross sections of modeled GaN nanowires after energy
minimization.

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LETTER
(per atom) as a function of strain for GaN nanowires of different
diameter using PSP1 and the PBE functional. It is noteworthy
that the absolute value of polarization for nanowires is smaller
thanthatofbulk.Onthecontrary,whenthepolarizationperunit
volume is plotted (shown in Figure 3b), the trend is reversed.
This asserts that the reduction in volume of nanowires due to
restructuring of the surface atoms (surface reconstruction) plays
a significant role in enhancing the piezoelectric properties of the
nanowires. Note that the volume for each nanowire was com-
puted from the diameter (d, as shown in Figure 1) using the
formulaV=(3√3/8)d2L,withLbeingthelengthoftheunitcell.
Furthermore, to understand why the overall polarization per
atom fornanowires issmaller thanthat of bulk, Mulliken charges
wereusedtocalculatethefirst-orderdipolemomentofatomicpairs
ofGaandNatoms.Ateachradiallocation,localdipolemomentof
each pair of atoms was calculated using the following summation
μdipole¼
X
qdij
ð8Þ
where q is the average charge on each atomic pair and dijis the
interatomic distance between them along the axial direction. This
first-order dipole moment is plotted as a function of atomic radial
coordinate in Figure 4b-d for two GaN nanowires of different
diameters. Figure 4a shows the radial coordinate, r, for a set of
atoms lying onthe dotted circlefor a 1.8 nm nanowire. Thedipole
momentperatomicpair, asmeasuredforbulkGaN,isalsoplotted
for reference as a dashed line in Figure 4.
The reduced dipole momentwith respect to bulk, as observed
for nanowires, is in general agreementwith the reducedpolariza-
tion for nanowires as shown inFigure3. However, the volume of
nanowires is smaller as compared to the bulk value (as shown in
Figure 5a). This plays an important role in enhancing the
piezoelectric coefficient, which depends on the polarization per
unit volume. The charge redistribution in GaN nanowires was
alsoanalyzedandisshowninFigure5b,whichrevealsthatcharge
deviates from bulk behavior primarily on the surface of the
nanowires. These findings assert that interatomic rearrangement
also plays an important role in affecting polarization. The overall
charge redistribution and interatomic rearrangement in the axial
direction of nanowires has the net effect to reduce polarization.
However, the contraction in the radial direction, due to surface
relaxation, leads to a reduction in overall nanowire volume with
respect to a bulk crystal with the same number of atoms. This
reduction in volume, in essence, causes the observed enhance-
ment in piezoelectric coefficients.
In summary, quantum mechanical computational estimates of
piezoelectric coefficients in nanowires were reported. The in-
vestigation revealed giant piezoelectric coefficients in both GaN
andZnOnanowiresasaresultofsizeeffects.ForGaN,diameters
well in excess of 2.5 nm are required to converge to bulk values.
These results demonstrate that the full potential of nanowire
systems in actuation, sensing and energy harvesting applications
isfarfrombeingfullyexploited.Particularly,inthefieldofenergy
harvesting to develop self-powered devices,35piezoelectric out-
put voltage, which is proportional to piezoelectric coefficient,36
can be improved by 1-2 orders of magnitude by reducing the
sizeofnanowiresusedinthedesignofsuchsystems.Thefindings
reported here, therefore, suggest new research directions. In
particular, the need for experimental confirmation of the theore-
tical predictions is highly needed to gain further insight into the
phenomenon. Clearly, measurement of piezoelectric coefficients
innanowireswithdiametersbelow20nmisverychallengingand
atopicofhighinterestwithinthenanotechnologycommunity.In
thisregard,thisworkalsoprovidesdirectionforthedevelopment
of new computational tools, for example, based on classical
molecular dynamics,37which can be used to model larger
nanowires that cannot be investigated using DFT.
Onthebasisofthechargedistributionanalysisandcalculation
of first-order dipole moments, overall polarization is found to be
reduced in nanowires as compared to bulk. However, the piezo-
electric coefficients are found to be much higher due to surface
relaxation induced volume reductions in nanowires. The study
Figure 2. Piezoelectric coefficients of GaN and ZnO nanowires as a function of their diameter. (a) Absolute values; (b) normalized with respect to the
respectivebulkvalues.ValuesreportedinthesefiguresusedPBEandRPBEfunctionalsforGaNandZnO,respectively. ForGaN,resultsfromPSP1are
reported.
Table 3. Differences in Piezoelectric Coefficient Computed
Using PBE and RPBE Functional
ZnO (C/m2)GaN (PSP1, C/m2)
diameter (nm)
PBERPBEPBERPBE
0.6
1.2
97.1
35.5
50.4
18.1
51.0
23.3
58.8
19.9
Table4. Effectof3d-OrbitalRadiusChosenforModelingGa
Atoms in GaN Nanowires
GaN (C/m2)
diameter (nm)
PSP1PSP2
0.6
1.2
bulk
51.0
23.3
0.255
25.8
7.6
0.554

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also uncovered the fact that the absolute values of piezoelectric
coefficients appear to be strongly influenced by the functionals
and/or pseudopotentials being employed. More insight from
experimentationishighlydesirabletovalidatetheapplicabilityof
the approximations used in the reported quantum mechanical
computations.
Figure 3. (a) Polarization (per atom) as a function of strain for GaN nanowires of different diameters; (b) polarization per unit volume for different
nanowires.
Figure4. (a)Schematicshowingtheradialcoordinate,r,forasetofatomslyingonthedottedcircle.(b-d)First-orderdipolemomentcalculatedbased
oninteratomicdistancesandMullikenchargesfornanowiresofdifferentdiameter.UnitsofdipolemomentareeÅ,whereeistheelectroniccharge.Note
that no atoms lie on r = 0.
Figure5. (a)ReductioninvolumeofNWperGa-NpairwithrespecttobulkvolumeforGaNandZnONWs;(b)chargedistribution perGa-Npair
asafunctionofradialcoordinatefor2.4nmdiameterGaNnanowireatdifferentstrains.Dottedlinerepresentsthebulkvalueof18foreachGa-Npair.

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’AUTHOR INFORMATION
Corresponding Author
*Phone:(847) 467-5989.Fax:(847) 491-3915.E-mail:espinosa@
northwestern.edu.
’ACKNOWLEDGMENT
H.D.E. acknowledges the support from the National Science
Foundation through Award Numbers DMR-0907196 and
CMMI-0555734. We are thankful to Dr. Jeffrey T. Paci for
insightful discussions. A special thanks is due to George Schatz
andSusanTrolierMcKinstryforreadingandcommentingonthe
manuscript.ThisworkwasalsosupportedinpartbytheNational
Science Foundation through TeraGrid resources provided by
Indiana University. In particular, the authors would like to thank
Ray Sheppard at Indiana University for hardware/software
support.
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