Piezoelectricity in wurtzite polar semiconductor nanowires : A theoretical study

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I. INTRODUCTION
Energy harvesting has been around for centuries in the form of windmills, watermills, and passive solar power systems.Among the different forms of energy harvesting, vibration-based mechanical energy being the most ubiquitous and accessible energy source in the surroundings, harvesting this type of energy exhibits a great potential for remote or wireless sensing, charging batteries, and powering electronic devices.Over the last decade, much research has been focused on energy harvesting using piezoelectric semiconductor nanowires to harvest energy on the microscale and nanoscale. 1,2wing to the small size and high flexibility of the nanowires, the nanogenerators are very sensitive to small level mechanical disturbances and are ideal for powering wireless sensors, microrobots, nano/micro electro-mechanical systems (NEMS/ MEMS) and bioimplantable devices. 3,4Yet, the physics behind the electromechanical phenomena of these semiconductors is poorly studied.In this paper we have presented a detailed study of the electromechanical phenomena in light of the piezoelectric polarization of these materials.
The piezoelectric interaction occurs in all polar crystals lacking an inversion symmetry.On application of an external strain to a piezoelectric crystal, a macroscopic polarization is produced as a result of the displacements of ions.Thus, an acoustic phonon mode will drive a macroscopic polarization in a piezoelectric crystal.The polar crystals are generally of three types: the wurtzite, the zincblende and the rock salt.The wurtzites are the most stable and therefore most commonly considered at ambient conditions.The zinc blende form can be stabilized using substrates with cubic lattice structure and the rarely found rocksalt (NaCl-type) structure is only observed at relatively high pressures.In this paper we have restricted ourselves to the most common type, i.e., the hexagonal wurtzite structure.

II. THEORY
In order to determine the piezoelectric polarization in a wurtzite material, it is necessary to consider the piezoelectric tensor of the material under consideration and to determine the strain components corresponding to the deformation under consideration.The piezoelectric tensor relating the piezoelectric polarization vector and the acoustic strain vector may be expressed in matrix notation for the case of a wurtzite crystal in Cartesian coordinates as e ¼ 0 0 0 0 e x5 0 0 0 0 e x5 0 0 e z1 e z1 e z3 0 0 0 For the case of uniform plane wave propagation at an arbitrary angle g in the XZ plane of a wurtzite crystal, the piezoelectric stress matrix transforms as where the rotation transformation matrix [a] is given by a ½ ¼ cos g 0 À sin g 0 0 1 sin g 0 cos g 0 @ 1 A and the bond stress transformation matrix [M] is derived from [a]. 5 Therefore, the piezoelectric stress tensor e 0 for propagation at an arbitrary angle g in the XZ plane of a wurtzite crystal is given by with e 0 x1 ¼ À e z1 sin gcos 2 g À e z3 sin 3 g À e x5 cos g sin 2g; e 0 x2 ¼ À e z1 sin g; e 0 x3 ¼ À e z1 sin 3 g À e z3 cos 2 g sin g þ e x5 cos g sin 2g ?
½ ; Electronic mail: bananisen@ieee.org.In order to cast e 0 into a more suitable form for cylindrical quantum wires, we made a transformation from the Cartesian coordinate system to the cylindrical coordinate system.In the cylindrical polar coordinate system, the piezoelectric stress tensor e 0 for propagation at an angle g in the XZ plane of a wurtzite crystal transforms as where the coordinate transformation matrix [a 0 ] is given by and the bond stress transformation matrix [M 0 ] for this case is derived from [a 0 ].Therefore, the piezoelectric stress tensor in cylindrical coordinates is given by e 00 ¼ e 00 x1 e 00  The piezoelectrically induced electric polarization vector P is given in terms of the piezoelectric tensor e 00 and the acoustic strain vector S by the matrix equation where P is a three-component vector and S is the six-component strain vector as given below with the strain components with v being the velocity associated with the acoustic phonon displacement, u and x being the harmonic frequency assumed for the phonon field, i.e., v ¼ iwu. 6herefore, the components of the piezoelectrically induced polarization tensor in cylindrical polar coordinate system are given by The static permittivity for a hexagonal wurtzite crystal is given as Now for phonon propagation at an arbitrary angle g in the XZ plane, the permittivity matrix transforms as Again, on coordinate transformation, the static permittivity transforms as In general the electric displacement vector is given by Therefore for the case of no free charge in the nanowire, the piezoelectric potential generated in the nanowire due to the piezoelectric polarization is given by

III. PIEZOELECTRIC STIFFENING
Another important phenomenon in piezoelectric materials is the piezoelectric stiffening.Piezoelectric stiffening is caused by the generation of an electric field from the strain applied and subsequent generation of piezoelectric stress in the piezoelectric materials.According to the Christoffel equation, the change in elastic stiffness (C) is a function of the piezoelectric constants, permittivity and the wave vector direction ðlÞ and in matrix notation, the elastic stiffness tensor at zero electric displacement is given as where often called as the electromechanical coupling constant.The elastic stiffness tensor in matrix notation for the case of a wurtzite crystal in Cartesian coordinates is given as For the case of uniform plane wave propagation at an arbitrary angle g in the XZ plane of a wurtzite crystal, the elastic stiffness matrix transforms as Among the polar semiconductors, the group III nitrides such as AlN and GaN and the group II oxide, ZnO, are gaining more importance for energy related applications.The static dielectric constant ( 0 ) and the piezoelectric constants (e ij ) and the elastic stiffness constants C E ij for these materials are given in Table I.Now for the phonon propagation direction parallel to the z-axis, the electromechanical coupling constant from Eq. ( 14) reduces to 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Substituting the values of the respective coefficients from Table I, the change in elastic stiffness constant due to piezoelectric stiffening for the case of phonon propagation parallel to the z-axis for AlN is by 3%, for ZnO is by 7%, and for GaN 5%.Now for the phonon propagation direction inclined at an angle 90 in the XZ plane, the electromechanical coupling constant from Eq. ( 14) reduces to and substituting the values of the respective coefficients from Table I the electromechanical coupling constant for AlN ZnO, and GaN, respectively, reduces to 0:08 À0:12 À0:12 0 0 0 À0:12 0:01 0:03 0 0 0 À0:12 0:03 0:01 0 0 0 0 0 0 0 0 0 0:09 À0:08 À0:08 0 0 0 À0:08 0:02 0:02 0 0 0 À0:08 0:02 0:02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0:02 À0:02 À0:02 0 0 0 À0:02 0:004 0:01 0 0 0 À0:02 0:01 0:004 0 0 0 0 0 0 0 0 0 Since the change in the elastic stiffness constants due to piezoelectric stiffening is 12% for AlN, ZnO, and GaN, it is clear that useful results may be obtained without including stiffening effects.Accordingly, the results of the next section do not include the small corrections due to stiffening effects; this approximation makes it possible to obtain analytical results.

IV. RESULTS AND DISCUSSION
There has been significant amount of research on energy harvesting from piezoelectric stack coupled in shoe sole during rapid locomotion and cantilever based piezoelectric generators using piezoceramic and polymeric materials. 12,13In all these applications, the strain is applied vertically along the z-axis (c-axis) of the nanowire or bent at an angle with respect to the z-axis.Now considering the first case where the strain is applied along the z-axis (stretched=compressed along the caxis) of the nanowire, @u z @z is the dominant component of the strain.Neglecting the strain components in all other directions, the piezoelectrically induced polarization from Eq. ( 7) reduces to P /zz ¼ À e 0 x3 sin / @u z @z ¼ À Àe z1 sin 3 g À e z3 cos 2 g sin g þ e x5 cos g À sin 2gÞ sin / @u z @z ; (18b) Therefore, the piezoelectric potential generated across the nanowire for @u z @z being the only non-zero strain is The piezoelectric potential distribution as a function of strain applied along z-axis for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire for phonon propagation along z-axis is shown in Fig. 1.On stretching the nanowire such that there is 0.1% strain along z-axis, the piezoelectric voltage generated across the AlN nanowire is À15.1 V, for ZnO nanowire is À11.3 V, and for GaN is À4.9 V.
Figure 2 shows the piezoelectric potential distribution as a function of strain applied along z-axis for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire for the phonon propagation at an angle 45 in the XZ plane.In this case, the magnitude as well as the polarity of the piezoelectric voltage generated exhibits a polar dependence.The maximum potential observed across the AlN nanowire is 2.8 V, across the ZnO nanowire is 3.7 V, and across the GaN nanowire is 2 V.
Figure 3 shows the piezoelectric potential distribution as a function of strain applied along z-axis for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire for phonon propagation at an angle 90 in the XZ plane, i.e., either parallel to x-axis or parallel to y-axis.In this case also, the potential generated varied from positive to negative and back to positive over / and the maximum voltage observed is 3 V for AlN nanowire, 2.6 V for ZnO nanowire, and 1.2 V for GaN nanowire.Now considering the cantilever approach, @u r @z is the dominant component of strain.Neglecting the strain components in all other directions, cylindrical polar components of the piezoelectrically induced polarization from Eq. ( 7) reduces to   þ e x5 sin g cos 2gÞ cos / @u r @z : Therefore, the piezoelectric potential generated across the nanowire for @u r @z being the only non-zero strain is The piezoelectric potential distribution as a function of strain applied radially for 50 nm thick and nm long (a) AlN (b) ZnO, and (c) GaN nanowire for phonon propagation along the z-axis is shown in Fig. 4. On bending the nanowire such that there is a 0.1% strain developed along z-axis, the piezoelectric voltage generated across the AlN nanowire is 0.41 V, for ZnO nanowire is 0.41 V, and for GaN is 0.21 V.
Figure 5 shows the piezoelectric potential distribution as a function of strain applied radially for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire for the phonon propagation at an angle 45 in the XZ plane.In this case, the magnitude as well as the polarity of the piezoelectric voltage generated exhibits a polar dependence.The maximum potential observed across the AIN nanowire is 6 V, across the ZnO nanowire is 4.5 V, and across the GaN nanowire is 2.1 V.
Figure 6 shows the piezoelectric potential distribution as a function of strain applied radially for 50 nm thick and 600  nm long (a) AlN (b) ZnO, and (c) GaN nanowire for phonon propagation at an angle 90 in the XZ plane, i.e., either parallel to x-axis or parallel to y-axis.In this case also, the potential generated varied from positive to negative back to positive over / and the maximum voltage observed is 4.9 V for AlN nanowire, 4.9 V for ZnO nanowire, and 2.5 V for GaN nanowire.In this paper we have estimated the piezoelectrically induced electric potential distribution in AlN ZnO, and GaN nanowires for the zero free-charge case.This (order of magnitude) is in agreement with the experimental results reported by Wang's group for ZnO nanowires. 14With the presence of free charge in the nanowires, there will be electron-phonon as well as hole-phonon interactions and the net output voltage generated will be affected by this phonon scattering as proposed by Alexe's group. 15The importance of the effect produced by this non-zero density of free charge will depend on the specific details of each nanowire piezo generator, such as the contact geometry, the contact materials, the doping of the nanowire, the mobility of the nanowire, and the self-consistent fields in the nanowire; accordingly, these effects are beyond the scope of this present treatment.Indeed, in this treatment, the primary emphasis has been placed on the piezo effects existing in the nanowires of cylindrical geometry as reflected by the use of cylindrical coordinates throughout the formulation.

V. CONCLUSION
The piezoelectrically induced electric polarization vector and the associated potential have been derived in terms of the acoustic phonon mode amplitude displacement for wurtzite nanowires.In this treatment, the semiconductors have been modeled for the case of no free charge.By comparing generated piezoelectric potentials for strains applied in specific directions in these nanowires, it is shown that the maximum piezo energy is generated for the vertical direction (i.e., along z-axis).In terms of energy generation, A1N and ZnO nanowires are found to be superior to GaN nanowires.In order to obtain analytic results, piezoelectric stiffening has been neglected since the change in elastic constants for these semiconductor nanowires is only a few percent.

Àe z1 sin g sin 2g 2 þ e z3 sin g sin 2g 2 þ e x5 cos g cos 2g cos 2 FIG. 1 .
FIG. 1. Piezoelectric potential distribution as a function of strain applied along z-axis for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire.The phonon propagation direction is parallel to z-axis.

FIG. 2 .
FIG. 2. (Color online) Piezoelectric potential distribution as a function of strain applied along z-axis for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire.The phonon propagation direction is inclined at 45 in the XZ plane.

FIG. 4 .
FIG. 4. (Color online) Piezoelectric potential distribution as a function of strain applied radially for 50 nm thick and 600 nm long (a) AlN (b) ZnO, and (c) GaN nanowire.The phonon propagation direction is parallel to the zaxis.

TABLE I .
Static dielectric constants, piezoelectric coefficients, and elastic stiffness constants for AlN ZnO, and GaN (wurtzite structure) from the literature.
b References 8 and 11.c