Nature - USA (2020-08-20)

d=Uσijd+εEij mmd+PTd,S (28)

where ε is the strain, P is the electric polarization, T is the temperature
and S is the volume density of entropy. Note that all the subscripts of
the variables, that is, i, j, m and so on, in this section refer to elements
of {1,2,3} and Einstein summation convention is used here. Here, we
choose (σ, E, S) as the independent variables, with the (ε, P, T) as the
dependent variables. Accordingly, we introduce the enthalpy H per
unit volume, defined by

HU=−σεijij−.EPmm (29)

Hence,

d=Hε−−ijdσij PEmmd+TSd. (30)

As can be seen from above equation, the dependent variables can be
expressed as

ε

H

σ

=−

∂

ij ∂ij

P

H

E

=−

∂

∂

m (31)

m

T

H

S

=

∂

∂

.

Since the entropy is a constant in the adiabatic conditions, the depend-
ent variables of interest (ε, E) are a function of stress σ and polarization P

εεij=(σEkl,)nm;=PP(,σEkl n). (32)

Expanding the above functions to the second order about the position
of zero strain and zero electric polarization, we obtain

















ε

ε

σ σ

ε

EE

ε

σσ

σσ

ε

σE

σE

ε

EE

EE

=

∂

∂ +

∂

∂

+

1

2!

∂

∂∂

+2

∂

∂∂

+

∂

∂∂

(33)

ij

ij

klkl

ij

nn

ij

kl qr

klqr

ij

kl n

kln

ij

no

no

222

P

P

E

E

P

σ

σ

P

σσσσ

P

σEσE

P

EEEE

=

∂

∂

+

∂

∂

+

1

2!

∂

∂∂ +2

∂

∂∂ +

∂

∂∂.

(34)

m

m

n

n

m

kl

kl

m

kl qr klqr

m

kl n kln

m

nono

^222















The first two differentiation terms in equations ( 33 ) and ( 34 ) represent
the elastic compliance and the dielectric susceptibility, respectively.

ε

σ

s

∂

∂

ij= (35)

kl

ijkl

P

P

E

χ

∂

∂

m=. (36)

n mn

σ

The fourth-order tensor sijPkl is the elastic compliance measured
at constant electric polarization and the second-order tensor χmnσ
denotes the dielectric permittivity measured at constant stress. The
second first-order differentiation in equations ( 33 ) and ( 34 ) represents
the converse and direct piezoelectric effects in non-centrosymmetric
materials.

ε

E

P

σ d

∂

∂ =

∂

∂ =. (37)

ij

n

n

ij nij

For simplicity, we assume the piezoelectric constant remains constant

under external stress. Thus,

ε

σE

d

σ

∂

∂∂

=

∂

∂

ij =0 (38)

kl n

nij

kl

2

P

σσ

d

σ

∂

∂∂

=

∂

∂

m =0. (39)

kl qr

mkl

qr

2

It is also reasonable to assume that the elastic constant of the crystals

remains unchanged under external stress, namely, =0

ε

σσ

∂

∂∂

ij

kl qr

2

, and the

dielectric susceptibility remains constant under small external electric

field, that is,∂∂EE∂noPm =0

2

(ref.^6 ). Also, the other second-order partial

derivatives are correlated

ε

EE

H

σEE

P

σE

M

∂

∂∂

=−

∂

∂∂∂

=

∂

∂∂

ij =2. (40)

no ij no

o

ij n

ijno

(^232)
The derived fourth-rank tensor Mijno is the electrostriction coefficient
in the unit of m^2  V−2. Therefore, the strain εij and electric polarization
Pm induced by external stress σkl and electric field En can be written as
εsij=+ijklPσdkl nijEMni+ jnonEEo (41)
Pχm=+σmnEdnmklσMkl+2 klnmEσnkl. (42)
In the case of materials without inversion symmetry, the external
applied electric field induces mechanical strain via both the con-
verse piezoelectric effect and the electrostriction effect. The strain
induced by the electrostriction effect in piezoelectric materials
is normally much smaller than that induced by the piezoelectric
effect and thus is generally ignored. When just applying external
stress to the piezoelectric materials without applying an electric
field, it would generate an electric polarization only by the direct
piezoelectric effect.
In the case of centrosymmetric materials, piezoelectric constants
dnij are all zero. Equations ( 41 ) and ( 42 ) can be rewritten as
εsij=+ijklPσMkl ijnoEEno (43)
Pχm=+mnσ EMnk2.lnmnEσkl (44)
According to equation ( 43 ), the external electric field can induce
mechanical strain through only the electrostriction effect. In contrast,
homogenous mechanical stress cannot induce electric polarization
along in these materials due to the inversion symmetry. However, the
second term on the right-hand side of equation ( 44 ) indicates that exter-
nal stress can modulate electric polarization via the electrostriction
effect if there is an electric field En, which could be either an externally
applied field or a built-in space charge field. The effective piezoelectric
effect is given as
dMmklk=2 lnmnEM=2 klmnEn. (45)
This can be understood as the electric field, that is, a unidirectional
vector, breaking the inversion symmetry in native centrosymmetric
materials, inducing electric polarization and giving rise to a piezo-
electric effect via the electrostriction. We can unveil the underlying
mechanism by rewriting equation ( 40 ) as