PiezoElectric constant 관련 논문 자료 첨부합니다.

Material Properties에서 보이는 PiezoElectric constant 값의 근거 자료입니다.





2 Strain, Piezoeffect, and Spontaneous Polarization

An LED heterostructure is considered as a stack of epitaxial layers pseudo-morphically

grown on an underlying template/substrate layer. For structures which are not latticematched

to the substrate, such as GaN/sapphire, a buffer layer serves as the template

due to strain relaxation at the buffer/substrate interface. By default, other epilayers are

assumed to have the lateral lattice constant equal to that of the template layer. Optionally,

user can directly specify the lateral lattice constant astrained of some layer or specify a

degree of relaxation, , in a layer (these two options can not be used simultaneously). If

relaxation degree is specified, the strained lattice constant in the layer is calculated as

astrained = afree + (1 􀀀 ) a0

strained (2.1)

where afree is the free-standing lattice constant for the layer of specified composition

and a0

strained is the strained lattice constant at the top of the underlying (previous) layer.

The first layer is assumed to have no strain, i.e. astrained = afree, until opposite is not

specified explicitly by setting the value of the strained lattice constant. Eventually, the

evolution of the strained lattice constant astrained is calculated for the whole structure

starting from the first layer and going up layer by layer. In vast majority of practical

cases, the degree of relaxation is zero for all the layers and so the strained lattice constant

equals to that of the first layer. Use of degree of relaxation or direct specification of the

strained lattice constant are used rarely.

Let us define the coordinate system related to the epilayers (x0; y0; z0) in such a way

that axis z0 is directed normal to the epilayers. Let the coordinate system related to

the crystollagraphic axis (x; y; z) be rotated compared to epilayers around x axis, so that

x = x0. Below, we will denote with 0 symbol the values defined in the coordinate system

related to the epilayers. Note that all material parameters, such as stiffness constants

3

Cij and piezoelectric constants eij , are defined in the Voigt notation [1] in the coordinate

system related to the crystallographic axis. In analysis of the strain effect on the band

structure in Sections (3) and (4) we also use strain components in the coordinates related

to the crystal lattice.

NB Outside this section and Sections (3) and (4), we will denote by z the direction

normal to the epilayers.

NB In literature, there are two definitions for the components of the strain tensor, u

and ", which are same for normal strain and differs by the factor of 2 for shear strain

uxx = "xx uyy = "yy uzz = "zz

uxy = 1

2"xy uyz = 1

2"yz uxz = 1

2"xz :

(2.2)

NB Starting from version 6.3, output for strain components presents strain components

uij in the coordinate system related to the crystallographic axes. In older versions,

strain components " was outputted (the difference is only in the shear strain component).