Mostovoy Reply: In their Comment [1] Kenzelmann
and Harris argue against the conclusion made in [2] that
spiral magnets are in general ferroelectric. First of all, I
believe, this conclusion was proved experimentally. The
systematic search for ferroelectricity in magnets with spi-
ral ordering recently led to a discovery of new multifer-
roic materials, such as CoCr2O4 [3], MnWO4 [4, 5] and
LiCu2O2 [6].
Furthermore, Kenzelmann and Harris argue that the
continuum theory outlined in [2] leads to misleading pre-
dictions about the magnetically-induced electric polar-
ization. To prove their point, they consider two hy-
pothetical spin configurations shown in Fig. 1 (c) and
(d) of their Comment, and argue that the results of the
continuum theory are incompatible with crystal symme-
tries. While one cannot deny the importance of symme-
try considerations, the arguments Kenzelmann and Har-
ris are themselves very misleading. They incorrectly as-
sert that for the spin configurations shown in Fig. 1 (c)
and (d) ‘the spiral theory’ would predict electric polar-
ization along, respectively, the c and a axes.
The continuum model of multiferroics [2] is based on
assumption that the spin state can be described by a sin-
gle magnetization vector. For TbMnO3 (see Fig. 1b),
where the wave vector of the magnetic spiral is along the
b axis and spins are rotating in the bc plane, it predicts
electric polarizationP along the c axis, in agreement with
experiment. The magnetic structures (c) and (d) are of a
different kind, as they are made of spirals rotating in op-
posite directions. Thus in the configuration (c) there are
two counter-rotating bc spirals in each ab plane, which is
why the net polarization along the c axis is zero. Simi-
larly, in the configuration (d) the ab spirals in neighboring
bc planes rotate in opposite directions, resulting in zero
net Pa.
It is not difficult to modify the continuum model con-
sidered in [2] to describe these more general magnetic
orders. For more than one magnetic ion per unit cell
one can introduce several independent magnetic order
parameters, which increases the number of possible mag-
netoelectric coupling terms. For instance, all three spin
configurations shown in Fig. 1 of the Comment can be
described by three antiferromagnetic order parameters
L1 = S1 + S2 − S3 − S4,
L2 = S1 − S2 + S3 − S4,
L3 = S1 − S2 − S3 + S4
(the labels of the 4 Mn ions in the unit cell of TbMnO3
are the same as in [7]). The spiral configuration (b)
can be described by a single order parameter L1 with
nonzero Lb
and Lc
. As discussed in [2], the magneto-
electric coupling linear in the gradient of the magnetic
order parameter (Lifshitz invariant) allowed by symme-
tries has the form P c
, which gives
rise to magnetically-induced P c. The configuration (c)
is described by two different order parameters, Lb
. The term Lc
does not transform like
any of the components of P, so that the induced polar-
ization is zero. Finally, for the configuration (d) with
nonzero Lb
and La
, the only possible coupling term is
, allowing for nonzero P c.
The point is, however, that the spin configurations (c)
and (d) considered by Kenzelmann and Harris, are very
artificial, as it is difficult to find a system where interac-
tions between spins would favor the simultaneous pres-
ence of counter-rotating spirals. The average interaction
between counter-rotating spirals is zero, while for spirals
with spins rotating in the same direction some interac-
tion energy can always be gained by properly adjusting
their relative phases. This is the reason why the simple
model of Ref. [2] with a single vector order parameter
successfully describes thermodynamics and magnetoelec-
tric properties of many spiral multiferroics.
Maxim Mostovoy
Materials Science Center, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
[1] M. Kenzelmann and A. B. Harris, Comment
arXiv/cond-mat0610471.
[2] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).
[3] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T.
Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204
(2006).
[4] K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, and T.
Arima, Phys. Rev. Lett. 97, 097203 (2006).
[5] O. Heyer et al., J. Phys. Condens. Matter 18, L471
(2006).
[6] S. Park, Y. J. Choi, C. L. Zhang and S.-W. Cheong, to
be published.
[7] A. B. Harris and G. Lawes, arXiv/cond-mat0508617.
http://arxiv.org/abs/0704.0545v1
	References

  In response to the comment of Kenzelmann and Harris I show how the continuum
theory of spiral multiferroics can be modified to describe general magnetic
orders and discuss why the microscopic mechanism of magnetically-induced
ferroelectricity usually makes such modifications unnecessary. This explains
why the simple model with a single vector order parameter successfully
describes thermodynamics and magnetoelectric properties of many spiral
multiferroics.

Mostovoy Reply: In their Comment [1] Kenzelmann
and Harris argue against the conclusion made in [2] that
spiral magnets are in general ferroelectric. First of all, I
believe, this conclusion was proved experimentally. The
systematic search for ferroelectricity in magnets with spi-
ral ordering recently led to a discovery of new multifer-
roic materials, such as CoCr2O4 [3], MnWO4 [4, 5] and
LiCu2O2 [6].
Furthermore, Kenzelmann and Harris argue that the
continuum theory outlined in [2] leads to misleading pre-
dictions about the magnetically-induced electric polar-
ization. To prove their point, they consider two hy-
pothetical spin configurations shown in Fig. 1 (c) and
(d) of their Comment, and argue that the results of the
continuum theory are incompatible with crystal symme-
tries. While one cannot deny the importance of symme-
try considerations, the arguments Kenzelmann and Har-
ris are themselves very misleading. They incorrectly as-
sert that for the spin configurations shown in Fig. 1 (c)
and (d) ‘the spiral theory’ would predict electric polar-
ization along, respectively, the c and a axes.
The continuum model of multiferroics [2] is based on
assumption that the spin state can be described by a sin-
gle magnetization vector. For TbMnO3 (see Fig. 1b),
where the wave vector of the magnetic spiral is along the
b axis and spins are rotating in the bc plane, it predicts
electric polarizationP along the c axis, in agreement with
experiment. The magnetic structures (c) and (d) are of a
different kind, as they are made of spirals rotating in op-
posite directions. Thus in the configuration (c) there are
two counter-rotating bc spirals in each ab plane, which is
why the net polarization along the c axis is zero. Simi-
larly, in the configuration (d) the ab spirals in neighboring
bc planes rotate in opposite directions, resulting in zero
net Pa.
It is not difficult to modify the continuum model con-
sidered in [2] to describe these more general magnetic
orders. For more than one magnetic ion per unit cell
one can introduce several independent magnetic order
parameters, which increases the number of possible mag-
netoelectric coupling terms. For instance, all three spin
configurations shown in Fig. 1 of the Comment can be
described by three antiferromagnetic order parameters
L1 = S1 + S2 − S3 − S4,
L2 = S1 − S2 + S3 − S4,
L3 = S1 − S2 − S3 + S4
(the labels of the 4 Mn ions in the unit cell of TbMnO3
are the same as in [7]). The spiral configuration (b)
can be described by a single order parameter L1 with
nonzero Lb
and Lc
. As discussed in [2], the magneto-
electric coupling linear in the gradient of the magnetic
order parameter (Lifshitz invariant) allowed by symme-
tries has the form P c
, which gives
rise to magnetically-induced P c. The configuration (c)
is described by two different order parameters, Lb
. The term Lc
does not transform like
any of the components of P, so that the induced polar-
ization is zero. Finally, for the configuration (d) with
nonzero Lb
and La
, the only possible coupling term is
, allowing for nonzero P c.
The point is, however, that the spin configurations (c)
and (d) considered by Kenzelmann and Harris, are very
artificial, as it is difficult to find a system where interac-
tions between spins would favor the simultaneous pres-
ence of counter-rotating spirals. The average interaction
between counter-rotating spirals is zero, while for spirals
with spins rotating in the same direction some interac-
tion energy can always be gained by properly adjusting
their relative phases. This is the reason why the simple
model of Ref. [2] with a single vector order parameter
successfully describes thermodynamics and magnetoelec-
tric properties of many spiral multiferroics.
Maxim Mostovoy
Materials Science Center, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
[1] M. Kenzelmann and A. B. Harris, Comment
arXiv/cond-mat0610471.
[2] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).
[3] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T.
Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204
(2006).
[4] K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, and T.
Arima, Phys. Rev. Lett. 97, 097203 (2006).
[5] O. Heyer et al., J. Phys. Condens. Matter 18, L471
(2006).
[6] S. Park, Y. J. Choi, C. L. Zhang and S.-W. Cheong, to
be published.
[7] A. B. Harris and G. Lawes, arXiv/cond-mat0508617.
http://arxiv.org/abs/0704.0545v1
	References
