Article

M

ε

EE

P

Eσ

χ

σ

2=

∂

∂∂

=

∂

∂∂

=

∂

∂

ijno ij , (46)

no

n

oij

no

σ

ij

(^22)
keeping in mind that












ε
EE
H
σσ
H
E
P
σ
∂
∂

∂
∂
−
∂
∂

∂
∂
−
∂
∂

∂
∂
ij .( 47 )
nnij ij n
n
ij
According to equation ( 46 ), the electrostriction is a measure of the
dependence of dielectric permittivity on external stress^37. This is
termed the converse electrostriction effect. Therefore, the external
stress would modulate the dielectric permittivity via the electrostric-
tion effect, giving rise to a change in the electric polarization induced
by the electric field En.
Similarly, the second term on the left-hand side of equation ( 43 ) indi-
cates that the electric field En can also induce a converse piezoelectric
effect in a centrosymmetric material. To derive the corresponding
converse piezoelectric coefficient, we extend the Einstein notation
in equation ( 43 )
εsij=+ijklPσMkl ijnnEMni^2 +2 jnqnEEq. (48)
Note that the subscript n ≠ q in above equation. For the case of interest
here, the electric field exerted on a centrosymmetric material consists
of a constant part En, which represents the built-in field, and an alterna-
tive component ΔEn due to an externally applied a.c. voltage. Accord-
ingly, equation ( 48 ) can be rewritten as
εsij=+ijklPσMkl ijnnEMni^22 +2 jnnnEEΔ+niMEjnnnΔ. (49)
The first and second terms on the right-hand side of equation ( 49 ) rep-
resent the static strain induced by the external stress and the built-in
electric field in the space-charged region, respectively; the third term
represents the first-order harmonic strain induced by the dynamic
electric field, that is, the converse piezoelectric effect; the last term
is the second-harmonic strain, which refers to the conventional elec-
trostriction strain. Therefore, the converse piezoelectric coefficient
is the third term of equation ( 49 )
dMnij=2ijnnEn. (50)
In the case that the external field Eq is in a different direction with respect
to the built-in field En, the external-field-induced strain is represented
by the third term on the right-hand side of equation ( 48 ), that is,
εij=2MEijnqnqE. Clearly, the corresponding piezoelectric coefficient is
dMqiji=2 jnqnEM=2ijqnEn. (51)
The piezoelectric coefficient expressed in equations ( 45 ), ( 50 ) and ( 51 )
can be transformed into a unified form given as
dMmklk=2 lmnnE. (52)
In summary, both direct and converse piezoelectric effects with the
same coefficients can occur in centrosymmetric materials once they
are subjected to an electric field En.
Data availability
The data that support the findings of this study are available at the
University of Warwick open access research repository (http://wrap.
warwick.ac.uk/136971) or from the corresponding authors upon
request. Source data are provided with this paper.

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Acknowledgements M.A. acknowledges the Theo Murphy Blue-sky Awards of Royal Society.

The work was partly supported by the EPSRC (UK) through grant numbers EP/M022706/1,

EP/P031544/1 and EP/P025803/1. Z.M. and J.Z. acknowledge the National Natural Science

Foundation of China (11772207); Natural Science Foundation of Hebei Province for

Distinguished Young Scholar (A2019210204) and Shenzhen Peacock Team Program

(KQTD20170810160424889). We acknowledge the discussion with H. Zhang, A. N. Iqbal and

F. Zhuge; and the technical support from M. Crosbie.

Author contributions M.-M.Y. and M.A. conceived the idea, designed the experiments,

collected the data and wrote the manuscript. M.-M.Y. developed the theory. Z.-D.L., Z.M. J.Z.

and S.P.E. were involved in sample preparation.

Competing interests The authors declare no competing interests.

Additional information

Correspondence and requests for materials should be addressed to M.-M.Y. or M.A.

Peer review information Nature thanks Long-Qing Chen and the other, anonymous,

reviewer(s) for their contribution to the peer review of this work.

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