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Copyright copy 2010 American Scientific PublishersAll rights reservedPrinted in the United States of America

Nanoscience andNanotechnology Letters

Vol 2 26ndash29 2010

Atomic Formulation of Nano-Piezoelectricity inBarium Titanate

James Chenlowast and James D LeeDepartment of Mechanical and Aerospace Engineering The George Washington University

Washington DC USA 20052

Atomistic Field Theory (AFT) is used to determine nano-piezoelectricity of barium titanate Thecomputational method Generalized Finite Element Method (GFEM) is utilized to formulate a nano-cube made of cubic phase barium titanate (BaTiO3) under a simple tensile loading The results showhow the mechanical displacement induces the polarization and the electric potential Our simulationpredicts how mechanical loading generates the voltage distribution at the nanoscale This studyalso shows at nanoscale a 6 nmtimes6 nmtimes6 nm cube can generate sim272 volts which can be asignificant energy source The effective d33 is sim3times10minus10 mV This theory and numerical schemecan serve as a possible design tool for Nanogenerator nano-piezoelectronics and nano-energyharvesting

Keywords Nano-Piezoelectrics Nanogenerator Polarization Orientation Nano-EnergyHarvesting Atomistic Field Theory Generalized Finite Element Method

In 1984 an atomic definition of polarization for onemolecule was defined as

P =Ngsum=1

q R minus R0int 1

0rminus R0minus R minus R0d (1)

where Ng is the numbers of all the atoms R0 is the centerof molecule R is the position vector of -th atom in themolecule and q is the charge of -th atom in the molecule1

However this atomically iterative calculation is compu-tationally very expensive This concept is not practicalfor the application of nano-electrodynamics but providesa fundamental understanding of nano-electrodynamicsRecently experimentalists have started to explore the elec-tric properties and the application at nanoscale such asNanogenerator and nano-semiconductorNanogenerator has the potential of harvesting energy

from the environment for self-powered nanotechnologyThe concept of the Nanogenerator (NG) was first intro-duced by examining the piezoelectric properties of ZnOnanowires (NWs) with an atomic force microscope2 ZnOhas a wurtzite structure in which the Zn cations andO anions form a tetrahedral coordination Its characteristicis the lack of center symmetry which results in a piezo-electric effect by which a mechanical stressstrain can be

lowastAuthor to whom correspondence should be addressed

converted into electric voltage and vice versa owing tothe relative displacement of the cations and anions in thecrystalPerovskite-type (BaTiO3) is another example for nano-

piezoelectricity So far efforts to make Nanogeneratorshave focused on zinc-oxide nanowires But barium titanatecould lead to better generators because it shows a strongerpiezoelectric effect Experiments demonstrate that peri-odic tensile loading on a BaTiO3 nanowire can produceperiodic voltage generation3 Laboratory experiments alsoshow that a barium-titanate nanowire can generate 16-foldas much electricity as a zinc-oxide nanowire from the sameamount of mechanical vibrations3

Various approaches have been developed for harvest-ing energy from the environments based on thermoelec-tricity and piezoelectricity2ndash4 Innovative nanotechnologieshave been developed for converting mechanical energyinto electric energy experimentally It is noticed that the-oretically at nanoscale the physical phenomena cannotbe explained by classical continuum physics instead oneshould resort to atomistic descriptions Numerous theo-ries for nano-piezoelectricity and finite-size effect havebeen proposed including first principles5ndash7 and moleculardynamics (MD) simulations8 Nevertheless first princi-ple and MD simulation are very difficult to be utilizedin the nano-piezotronic systems (the typical size ofwhich is around several hundreds even thousands lattices)

26 Nanosci Nanotechnol Lett 2010 Vol 2 No 1 1941-490020102026004 doi101166nnl20101048

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IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

because of the massive number of atoms An atom-embedded continuum theory is proposed to be imple-mented to a nano-size system and to bridge the gapbetween microscale (classical continuum) and nanoscale(discrete atoms)This atomistic formulation for nano-piezoelectricity is

derived from Atomistic Field Theory (AFT)9 Crystallinesolids are distinguished from other states of matter bya periodic arrangement of the atoms such a structureis called a Bravais lattice Essentially the regularity dis-played by a crystal lattice is that of a three-dimensionalmesh which divides space into identical parallelepipeds Anumber of atoms referred to as an identical unit cell areplaced at the node of a finite element mesh which can beconsidered as a grain or a single crystal All single crys-tals (grains) are modeled by this field theory a continuumtheory but not just a classical continuum theory in whicha point only has three degrees of freedomIn this field theory the motion could be expressed as

uk= uk+k (2)

where k and indicates the -th atom in the k-th unitcell uk is the displacement of the centroid of the k-thunit cell and k is the relative displacement of the-th atom to the centroid All the physical quantities couldbe expressed in physical and phase spaces and these twospaces are connected through the Dirac delta function and the Kronecker delta function

Ax y t=Nasumk=1

Nasum=1

artptRkminusxrk minus y

(3)

Based on this formulation the balance law of linearmomentum can be obtained as

mux t=minuskBT x t+ fx t+x t (4)

where refers to -th atom within the unit cell f is theinteratomic force derived from specific interatomic poten-tial is body force T is the temperature andminuskBTmay be referred to as the temperature force In this studyonly the interatomic force is consideredFrom Eq (3) it is straightforward to find polarization

density of a lattice at position x and time t as

Px t =Nlsumk=1

Nasum=1

qRk+rkRkminusx

=Nlsumk=1

Nasum=1

qrkRkminusx

(∵

Nlsumk=1

( Nasum=1

q

)RkRkminusx= 0

)(5)

On the grounds of polarization density the induced electricpotential density at position z caused by a unit cell at xis

V zx t=Nlsumk=1

Nasum=1

qrk middot zminusxzminusx3 Rkminusx (6)

Summing over all the unit cells the induced electricpotential at position z and time t can be further derivedas

V z t=int Nlsum

k=1

Nasum=1

qrk middot zminusxzminusx3 Rkminusxdx (7)

From Eq (4) if the solution namely the positionof the k atom Rk +rk is due to interatomic forceon the boundary the resultant effect is called nano-piezoelectricityInteratomic force is usually derived from a specific

interatomic potential We consider Coulomb and Bucking-ham potential10 in this work

Fi = f iT =sumj

Fij =minussumj

13U ij

13ri

U ij = ZiZj

r ij+Aijeminusrij Bij minusCijr ij minus6

Fij =ZiZj

r ij 3+ Aij

Bijr ijeminusrij Bij minus6Cijr ij minus8

riminus rj

(8)

where T is the volume of a unit cellGeneralized Finite Element Method (GFEM) has been

also developed and adopted in this study11 The balancelaw of linear momentum can be re-written as

vk =nsum

l=1

sum=1

f k l

k = 123 n = 123 (9)

where f k l is the interatomic force acting on the-th atom of the k-th unit cell due to -th atom of thel-th unit cell The inner product of Eq (9) with virtualdisplacement uk leads to

vk middotuk

=nsum

l=1

sum=1

f k l middotuk

k = 123 n = 123 (10)

Suppose we have Ne finite elements each with 8 Gausspoints and we have a weak form of GFEM as

NesumIe=1

8sumg=1

sum=1

J Ie gvIe g middotuIe g

minus 12

nsuml=1

sum=1

f Ie g l

middot uIe gminusul= 0 (11)

Nanosci Nanotechnol Lett 2 26ndash29 2010 27

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IP 9816935158Fri 30 Jul 2010 032602

Atomic Formulation of Nano-Piezoelectricity in Barium Titanate Chen and Lee

Fig 1 The configuration of barium titanate nanocube It is subjected toa uniform tensile displacement-type boundary condition in +z directionon the top surface and fixed on the bottom surface The open box isthe initial shape The initial length is 6 nm and the total displacement is2 nm

where J Ie g is the jacobian of the g-th Gauss point ofthe Ie-th element f Ie g l is the force density act-ing on the -th atom in the unit cell located at the g-thGauss point of the Ie-th element due to the interaction withthe -th atom of the l-th unit cell After obtaining atomicpositions polarization of each unit cell is computed anddistributed to nodes (representative unit cell) through shapefunction In a similar way the induced electric potential atnodes is calculated from the polarizations of every otherunit cell using Eqs (6) and (7)In our simulation result a 15-lattice times 15-lattice times

15-lattice (around 6 nm times 6 nm times 6 nm) bariumtitanate cube which initialized as cubic phase is sub-jected to a displacement-type tensile loading (5 lattice

x

y

(a)

(b)

z

Fig 2 Atomic Structure of Barium titanate (a) is a primitive unit cell5

and (b) is a conventional unit cell

Fig 3 Polarization density of a barium titanate nanocube in x minus z

plane (in atomic unit eBohr2) The contour shows the magnitude ofz-direction polarization density The arrow shows the direction of polar-ization density

constants asymp 2 nm) in z-direction as shown in Figure 1Each primitive unit cell of barium titanate has 5 atomsone barium one titanium and three oxygen as shown inFigure 2 The optimized lattice constant of barium titanateis c= 754567634Bohr (1Bohr= 5291772108times10minus11 m)In Figure 2 we show the atomic arrangement of bariumtitanate in a primitive and conventional unit cell It isnoticed that in a primitive unit cell which includes twooxides BaO and TiO2 its atomic arrangement is asym-metric Before applying displacement-type loading thefirst 4000-step relaxation process ensures that the overall

Fig 4 Induced electric potential of a polarized barium titanatenanocube due to axial loading in z direction (in atomic unit eBohr)

28 Nanosci Nanotechnol Lett 2 26ndash29 2010

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IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

energy of the finite specimen is at minimum In Figure 3we show the polarization density of barium titanate nano-cube in xminus z plane with the original shape The contourin Figure 3 shows the magnitude of polarization density inz-direction In Figure 3 the direction of polarization den-sity is pointing to +z-direction ie the arrow is pointingto +z direction This indicates that the direction of polar-ization aligns to the direction of loading12 however AFTcan further decide the direction of polarization which cannot be predicted in classical continuum theory The direc-tion is determined by the polarizability of two oxides (BaOand TiO2 In Figure 4 we show that the voltage distribu-tion in xminus z plane with the original shape Our simulationshows that a 6-nm cube with 2-nm axial loading generatessim1 eBohr (asymp272 volts) The effective d33 is estimated assim 3times10minus10 mV Such an energy source at the nanoscalemay be developed to provide power for micro-robotics andmicro-unmanned vehicle applications Experimental inves-tigations to validate these findings are warrantedThis atom-embedded continuum theory from atom-

istic perspective gives a possible continuum solution tonano-piezoelectricity nano-energy harvesting and nano-electrodynamics It can also be used to study the nanoscalesemiconducting properties (piezoelectric field effect

transducer) theoretically which has been experimentallyobserved in the literature13

References and Notes

1 D P Craig and T Thirunamachandran Molecular Quantum Electro-dynamics An Introduction to Radiation-Molecule Interaction Aca-demic Press Inc Ltd London (1984) p 255

2 Z L Wang and J H Song Science 312 242 (2006)3 A Wang J Hu A P Suryavanshi K Yum and M F Yu Nano

Lett 7 2966 (2007)4 Y Gao and Z L Wang Nano Lett 7 2499 (2007)5 H J Xiang J Yang J G Hou and Q S Zhu Appl Phys Lett

89 223111 (2006)6 Z C Tu and X Hu Phys Rev B 74 035434 (2006)7 R E Cohen Piezoelectricity Springer Berlin Heidelberg (2008)

4718 Y Zhang J Hong B Liu and D Fang Nanotechnology 20 405703

(2009)9 Y Chen and J D Lee Philo Mag 85 4095 (2005)

10 J Chen X Wang H Wang and J D Lee Eng Fract Mech77 736 (2010)

11 J D Lee and Y Chen Theor Appl Frac Mech 50 243(2008)

12 L M Eng H-J Guntherodt G A Schneider U Kopke and J MSaldana Appl Phys Lett 74 233 (1999)

13 Z L Wang Adv Mater 19 889 (2007)

Received 13 February 2010 Accepted 30 March 2010

Nanosci Nanotechnol Lett 2 26ndash29 2010 29

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IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

because of the massive number of atoms An atom-embedded continuum theory is proposed to be imple-mented to a nano-size system and to bridge the gapbetween microscale (classical continuum) and nanoscale(discrete atoms)This atomistic formulation for nano-piezoelectricity is

derived from Atomistic Field Theory (AFT)9 Crystallinesolids are distinguished from other states of matter bya periodic arrangement of the atoms such a structureis called a Bravais lattice Essentially the regularity dis-played by a crystal lattice is that of a three-dimensionalmesh which divides space into identical parallelepipeds Anumber of atoms referred to as an identical unit cell areplaced at the node of a finite element mesh which can beconsidered as a grain or a single crystal All single crys-tals (grains) are modeled by this field theory a continuumtheory but not just a classical continuum theory in whicha point only has three degrees of freedomIn this field theory the motion could be expressed as

uk= uk+k (2)

where k and indicates the -th atom in the k-th unitcell uk is the displacement of the centroid of the k-thunit cell and k is the relative displacement of the-th atom to the centroid All the physical quantities couldbe expressed in physical and phase spaces and these twospaces are connected through the Dirac delta function and the Kronecker delta function

Ax y t=Nasumk=1

Nasum=1

artptRkminusxrk minus y

(3)

Based on this formulation the balance law of linearmomentum can be obtained as

mux t=minuskBT x t+ fx t+x t (4)

where refers to -th atom within the unit cell f is theinteratomic force derived from specific interatomic poten-tial is body force T is the temperature andminuskBTmay be referred to as the temperature force In this studyonly the interatomic force is consideredFrom Eq (3) it is straightforward to find polarization

density of a lattice at position x and time t as

Px t =Nlsumk=1

Nasum=1

qRk+rkRkminusx

=Nlsumk=1

Nasum=1

qrkRkminusx

(∵

Nlsumk=1

( Nasum=1

q

)RkRkminusx= 0

)(5)

On the grounds of polarization density the induced electricpotential density at position z caused by a unit cell at xis

V zx t=Nlsumk=1

Nasum=1

qrk middot zminusxzminusx3 Rkminusx (6)

Summing over all the unit cells the induced electricpotential at position z and time t can be further derivedas

V z t=int Nlsum

k=1

Nasum=1

qrk middot zminusxzminusx3 Rkminusxdx (7)

From Eq (4) if the solution namely the positionof the k atom Rk +rk is due to interatomic forceon the boundary the resultant effect is called nano-piezoelectricityInteratomic force is usually derived from a specific

interatomic potential We consider Coulomb and Bucking-ham potential10 in this work

Fi = f iT =sumj

Fij =minussumj

13U ij

13ri

U ij = ZiZj

r ij+Aijeminusrij Bij minusCijr ij minus6

Fij =ZiZj

r ij 3+ Aij

Bijr ijeminusrij Bij minus6Cijr ij minus8

riminus rj

(8)

where T is the volume of a unit cellGeneralized Finite Element Method (GFEM) has been

also developed and adopted in this study11 The balancelaw of linear momentum can be re-written as

vk =nsum

l=1

sum=1

f k l

k = 123 n = 123 (9)

where f k l is the interatomic force acting on the-th atom of the k-th unit cell due to -th atom of thel-th unit cell The inner product of Eq (9) with virtualdisplacement uk leads to

vk middotuk

=nsum

l=1

sum=1

f k l middotuk

k = 123 n = 123 (10)

Suppose we have Ne finite elements each with 8 Gausspoints and we have a weak form of GFEM as

NesumIe=1

8sumg=1

sum=1

J Ie gvIe g middotuIe g

minus 12

nsuml=1

sum=1

f Ie g l

middot uIe gminusul= 0 (11)

Nanosci Nanotechnol Lett 2 26ndash29 2010 27

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IP 9816935158Fri 30 Jul 2010 032602

Atomic Formulation of Nano-Piezoelectricity in Barium Titanate Chen and Lee

Fig 1 The configuration of barium titanate nanocube It is subjected toa uniform tensile displacement-type boundary condition in +z directionon the top surface and fixed on the bottom surface The open box isthe initial shape The initial length is 6 nm and the total displacement is2 nm

where J Ie g is the jacobian of the g-th Gauss point ofthe Ie-th element f Ie g l is the force density act-ing on the -th atom in the unit cell located at the g-thGauss point of the Ie-th element due to the interaction withthe -th atom of the l-th unit cell After obtaining atomicpositions polarization of each unit cell is computed anddistributed to nodes (representative unit cell) through shapefunction In a similar way the induced electric potential atnodes is calculated from the polarizations of every otherunit cell using Eqs (6) and (7)In our simulation result a 15-lattice times 15-lattice times

15-lattice (around 6 nm times 6 nm times 6 nm) bariumtitanate cube which initialized as cubic phase is sub-jected to a displacement-type tensile loading (5 lattice

x

y

(a)

(b)

z

Fig 2 Atomic Structure of Barium titanate (a) is a primitive unit cell5

and (b) is a conventional unit cell

Fig 3 Polarization density of a barium titanate nanocube in x minus z

plane (in atomic unit eBohr2) The contour shows the magnitude ofz-direction polarization density The arrow shows the direction of polar-ization density

constants asymp 2 nm) in z-direction as shown in Figure 1Each primitive unit cell of barium titanate has 5 atomsone barium one titanium and three oxygen as shown inFigure 2 The optimized lattice constant of barium titanateis c= 754567634Bohr (1Bohr= 5291772108times10minus11 m)In Figure 2 we show the atomic arrangement of bariumtitanate in a primitive and conventional unit cell It isnoticed that in a primitive unit cell which includes twooxides BaO and TiO2 its atomic arrangement is asym-metric Before applying displacement-type loading thefirst 4000-step relaxation process ensures that the overall

Fig 4 Induced electric potential of a polarized barium titanatenanocube due to axial loading in z direction (in atomic unit eBohr)

28 Nanosci Nanotechnol Lett 2 26ndash29 2010

Delivered by Ingenta toGuest User

IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

energy of the finite specimen is at minimum In Figure 3we show the polarization density of barium titanate nano-cube in xminus z plane with the original shape The contourin Figure 3 shows the magnitude of polarization density inz-direction In Figure 3 the direction of polarization den-sity is pointing to +z-direction ie the arrow is pointingto +z direction This indicates that the direction of polar-ization aligns to the direction of loading12 however AFTcan further decide the direction of polarization which cannot be predicted in classical continuum theory The direc-tion is determined by the polarizability of two oxides (BaOand TiO2 In Figure 4 we show that the voltage distribu-tion in xminus z plane with the original shape Our simulationshows that a 6-nm cube with 2-nm axial loading generatessim1 eBohr (asymp272 volts) The effective d33 is estimated assim 3times10minus10 mV Such an energy source at the nanoscalemay be developed to provide power for micro-robotics andmicro-unmanned vehicle applications Experimental inves-tigations to validate these findings are warrantedThis atom-embedded continuum theory from atom-

istic perspective gives a possible continuum solution tonano-piezoelectricity nano-energy harvesting and nano-electrodynamics It can also be used to study the nanoscalesemiconducting properties (piezoelectric field effect

transducer) theoretically which has been experimentallyobserved in the literature13

References and Notes

1 D P Craig and T Thirunamachandran Molecular Quantum Electro-dynamics An Introduction to Radiation-Molecule Interaction Aca-demic Press Inc Ltd London (1984) p 255

2 Z L Wang and J H Song Science 312 242 (2006)3 A Wang J Hu A P Suryavanshi K Yum and M F Yu Nano

Lett 7 2966 (2007)4 Y Gao and Z L Wang Nano Lett 7 2499 (2007)5 H J Xiang J Yang J G Hou and Q S Zhu Appl Phys Lett

89 223111 (2006)6 Z C Tu and X Hu Phys Rev B 74 035434 (2006)7 R E Cohen Piezoelectricity Springer Berlin Heidelberg (2008)

4718 Y Zhang J Hong B Liu and D Fang Nanotechnology 20 405703

(2009)9 Y Chen and J D Lee Philo Mag 85 4095 (2005)

10 J Chen X Wang H Wang and J D Lee Eng Fract Mech77 736 (2010)

11 J D Lee and Y Chen Theor Appl Frac Mech 50 243(2008)

12 L M Eng H-J Guntherodt G A Schneider U Kopke and J MSaldana Appl Phys Lett 74 233 (1999)

13 Z L Wang Adv Mater 19 889 (2007)

Received 13 February 2010 Accepted 30 March 2010

Nanosci Nanotechnol Lett 2 26ndash29 2010 29

Delivered by Ingenta toGuest User

IP 9816935158Fri 30 Jul 2010 032602

Atomic Formulation of Nano-Piezoelectricity in Barium Titanate Chen and Lee

Fig 1 The configuration of barium titanate nanocube It is subjected toa uniform tensile displacement-type boundary condition in +z directionon the top surface and fixed on the bottom surface The open box isthe initial shape The initial length is 6 nm and the total displacement is2 nm

where J Ie g is the jacobian of the g-th Gauss point ofthe Ie-th element f Ie g l is the force density act-ing on the -th atom in the unit cell located at the g-thGauss point of the Ie-th element due to the interaction withthe -th atom of the l-th unit cell After obtaining atomicpositions polarization of each unit cell is computed anddistributed to nodes (representative unit cell) through shapefunction In a similar way the induced electric potential atnodes is calculated from the polarizations of every otherunit cell using Eqs (6) and (7)In our simulation result a 15-lattice times 15-lattice times

15-lattice (around 6 nm times 6 nm times 6 nm) bariumtitanate cube which initialized as cubic phase is sub-jected to a displacement-type tensile loading (5 lattice

x

y

(a)

(b)

z

Fig 2 Atomic Structure of Barium titanate (a) is a primitive unit cell5

and (b) is a conventional unit cell

Fig 3 Polarization density of a barium titanate nanocube in x minus z

plane (in atomic unit eBohr2) The contour shows the magnitude ofz-direction polarization density The arrow shows the direction of polar-ization density

constants asymp 2 nm) in z-direction as shown in Figure 1Each primitive unit cell of barium titanate has 5 atomsone barium one titanium and three oxygen as shown inFigure 2 The optimized lattice constant of barium titanateis c= 754567634Bohr (1Bohr= 5291772108times10minus11 m)In Figure 2 we show the atomic arrangement of bariumtitanate in a primitive and conventional unit cell It isnoticed that in a primitive unit cell which includes twooxides BaO and TiO2 its atomic arrangement is asym-metric Before applying displacement-type loading thefirst 4000-step relaxation process ensures that the overall

Fig 4 Induced electric potential of a polarized barium titanatenanocube due to axial loading in z direction (in atomic unit eBohr)

28 Nanosci Nanotechnol Lett 2 26ndash29 2010

Delivered by Ingenta toGuest User

IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

energy of the finite specimen is at minimum In Figure 3we show the polarization density of barium titanate nano-cube in xminus z plane with the original shape The contourin Figure 3 shows the magnitude of polarization density inz-direction In Figure 3 the direction of polarization den-sity is pointing to +z-direction ie the arrow is pointingto +z direction This indicates that the direction of polar-ization aligns to the direction of loading12 however AFTcan further decide the direction of polarization which cannot be predicted in classical continuum theory The direc-tion is determined by the polarizability of two oxides (BaOand TiO2 In Figure 4 we show that the voltage distribu-tion in xminus z plane with the original shape Our simulationshows that a 6-nm cube with 2-nm axial loading generatessim1 eBohr (asymp272 volts) The effective d33 is estimated assim 3times10minus10 mV Such an energy source at the nanoscalemay be developed to provide power for micro-robotics andmicro-unmanned vehicle applications Experimental inves-tigations to validate these findings are warrantedThis atom-embedded continuum theory from atom-

istic perspective gives a possible continuum solution tonano-piezoelectricity nano-energy harvesting and nano-electrodynamics It can also be used to study the nanoscalesemiconducting properties (piezoelectric field effect

transducer) theoretically which has been experimentallyobserved in the literature13

References and Notes

1 D P Craig and T Thirunamachandran Molecular Quantum Electro-dynamics An Introduction to Radiation-Molecule Interaction Aca-demic Press Inc Ltd London (1984) p 255

2 Z L Wang and J H Song Science 312 242 (2006)3 A Wang J Hu A P Suryavanshi K Yum and M F Yu Nano

Lett 7 2966 (2007)4 Y Gao and Z L Wang Nano Lett 7 2499 (2007)5 H J Xiang J Yang J G Hou and Q S Zhu Appl Phys Lett

89 223111 (2006)6 Z C Tu and X Hu Phys Rev B 74 035434 (2006)7 R E Cohen Piezoelectricity Springer Berlin Heidelberg (2008)

4718 Y Zhang J Hong B Liu and D Fang Nanotechnology 20 405703

(2009)9 Y Chen and J D Lee Philo Mag 85 4095 (2005)

10 J Chen X Wang H Wang and J D Lee Eng Fract Mech77 736 (2010)

11 J D Lee and Y Chen Theor Appl Frac Mech 50 243(2008)

12 L M Eng H-J Guntherodt G A Schneider U Kopke and J MSaldana Appl Phys Lett 74 233 (1999)

13 Z L Wang Adv Mater 19 889 (2007)

Received 13 February 2010 Accepted 30 March 2010

Nanosci Nanotechnol Lett 2 26ndash29 2010 29

Delivered by Ingenta toGuest User

IP 9816935158Fri 30 Jul 2010 032602

Chen and Lee Atomic Formulation of Nano-Piezoelectricity in Barium Titanate

energy of the finite specimen is at minimum In Figure 3we show the polarization density of barium titanate nano-cube in xminus z plane with the original shape The contourin Figure 3 shows the magnitude of polarization density inz-direction In Figure 3 the direction of polarization den-sity is pointing to +z-direction ie the arrow is pointingto +z direction This indicates that the direction of polar-ization aligns to the direction of loading12 however AFTcan further decide the direction of polarization which cannot be predicted in classical continuum theory The direc-tion is determined by the polarizability of two oxides (BaOand TiO2 In Figure 4 we show that the voltage distribu-tion in xminus z plane with the original shape Our simulationshows that a 6-nm cube with 2-nm axial loading generatessim1 eBohr (asymp272 volts) The effective d33 is estimated assim 3times10minus10 mV Such an energy source at the nanoscalemay be developed to provide power for micro-robotics andmicro-unmanned vehicle applications Experimental inves-tigations to validate these findings are warrantedThis atom-embedded continuum theory from atom-

istic perspective gives a possible continuum solution tonano-piezoelectricity nano-energy harvesting and nano-electrodynamics It can also be used to study the nanoscalesemiconducting properties (piezoelectric field effect

transducer) theoretically which has been experimentallyobserved in the literature13

References and Notes

1 D P Craig and T Thirunamachandran Molecular Quantum Electro-dynamics An Introduction to Radiation-Molecule Interaction Aca-demic Press Inc Ltd London (1984) p 255

2 Z L Wang and J H Song Science 312 242 (2006)3 A Wang J Hu A P Suryavanshi K Yum and M F Yu Nano

Lett 7 2966 (2007)4 Y Gao and Z L Wang Nano Lett 7 2499 (2007)5 H J Xiang J Yang J G Hou and Q S Zhu Appl Phys Lett

89 223111 (2006)6 Z C Tu and X Hu Phys Rev B 74 035434 (2006)7 R E Cohen Piezoelectricity Springer Berlin Heidelberg (2008)

4718 Y Zhang J Hong B Liu and D Fang Nanotechnology 20 405703

(2009)9 Y Chen and J D Lee Philo Mag 85 4095 (2005)

10 J Chen X Wang H Wang and J D Lee Eng Fract Mech77 736 (2010)

11 J D Lee and Y Chen Theor Appl Frac Mech 50 243(2008)

12 L M Eng H-J Guntherodt G A Schneider U Kopke and J MSaldana Appl Phys Lett 74 233 (1999)

13 Z L Wang Adv Mater 19 889 (2007)

Received 13 February 2010 Accepted 30 March 2010

Nanosci Nanotechnol Lett 2 26ndash29 2010 29