Supplementary material
Strain-mediated electric-field control of multiferroic domain structures
in ferromagnetic films
N. A. Pertsev
A. F. Ioffe Physico-Technical Institute of the Russian Academy of Sciences
and St. Petersburg State Polytechnical University, St. Petersburg, Russia
It should be emphasized that the domain patterns with in-plane magnetizations, which
are discussed in this paper, appear under certain strain-thickness conditions only. Indeed, in
the considered case of ferromagnets with cubic magnetocrystalline anisotropy (MCA), the
spontaneous magnetization Ms of a ferromagnetic film deposited on a dissimilar substrate can
have in-plane, perpendicular-to-plane, or intermediate orientation [1]. Whilst the
magnetostatic shape anisotropy always favors in-plane directions of Ms in extended films, the
magnetoelastic coupling [2,3] facilitates such orientation only at substrate-induced in-plane
strains S11 and S22 of a certain type (compressive when the magnetoelastic coefficient B1 is
positive and tensile at B1 < 0). As a result, in the usual case of an isotropic biaxial misfit strain
(S11 = S22 = Sm), in-plane magnetic states form at S m  S m* in CoFe alloys having negative B1
and at S m  S m* in CoFe2O4 and other ferromagnets with B1 > 0 [1]. For example, an in-plane
magnetization was observed in the 400-nm-thick CoFe2O4 film subjected to small tensile
strain [4]. Owing to the surface magnetic anisotropy, the critical misfit strain S m* depends on
the film thickness t [5], but this dependence becomes pronounced only in ultrathin films with t
not exceeding a few nanometers. Importantly, according to our numerical estimates, in thick
extended Co40Fe60 films S m* < 0 so that any tensile strain is sufficient to stabilize an in-plane
magnetization orientation.
In contrast to the perpendicular-to-plane magnetic state, the appearance of in-plane
0
0
magnetization is accompanied by anisotropic strains S110  S 22
or shear strains S120  S21
in the
1
lattice planes parallel to the film surfaces. Indeed, in a ferromagnetic film with K1 > 0 and Ms
parallel to the [100] crystallographic axis, these spontaneous strains are equal to
0
S110  (2 / 3) B1 /( c11  c12 ) , S22
 (1 / 3) B1 /(c11  c12 ) , and S120  0 , whereas for a film with K1 <
0
0 and Ms oriented along the [110] direction they become S110  S 22
 (1 / 6) B1 /(c11  c12 ) and
S120  (1 / 2) B2 / c44 . Since the film is clamped by a thick substrate, in-plane spontaneous
strains must be compensated by elastic strains of opposite sign, which gives rise to an elastic
energy stored in the ferromagnetic film. Importantly, in films deposited on (001)-oriented
cubic and tetragonal substrates, this elastic energy can be minimized by forming patterns with
the magnetization alternating between two orthogonal crystallographic directions ([100] and
[010] axes at K1 > 0 and [110] and 1 1 0 axes at K1 < 0). This feature results from the
0
elimination of mean strain anisotropy S110  S 22
or average shear strains
S120
at equal
widths of the a1 and a2 or the aa1 and aa2 domains, respectively.
Although homogeneous in-plane magnetic states always tend to break into 90
domains, well-defined multiple a1/a2/a1/a2 and aa1/aa2/aa1/aa2 patterns form only in a certain
range tmin < t < tmax of film thicknesses. Here the minimum thickness can be found as tmin =
max{ts, tL1}, where ts(Sm) is the film thickness at which the surface magnetic anisotropy [6]
makes an out-of-plane magnetization direction energetically more favorable than the in-plane
orientation, whereas tL1 is the thickness below which the period D(t) of domain pattern
becomes comparable to the in-plane size L of a ferromagnetic layer. The existence of tL1 is
due to the fact that the equilibrium period D* of domain structure increases rapidly with
decreasing film thickness after reaching a minimum value at t = t* (see Fig. 7 in Ref. [7]). In
turn, the maximum film thickness is defined by the relation tmax = min{td, tL2}, where td is the
thickness at which the relaxation of elastic strains compensating in-plane spontaneous strains
via the generation of misfit dislocations becomes energetically favorable, and tL2 > t* > tL1
represents another film thickness at which the domain period increases up to a value
2
comparable to L.
The characteristic thicknesses tL1 and tL2 can be evaluated using the dependence of the
equilibrium domain period D* on the normalized film thickness t/t0 calculated in Ref. [7]. For
cubic ferromagnets with positive and negative K1, the thickness unit t0 is given by the
relations t0  2 (c11  c12 ) / B12 and t0  4c44 / B22 , respectively. Since in the case of Co40Fe60
layers considered in this work t0 is about 400 nm (see the main text), we find, for example,
that tL1 ~ 400 nm and tL2 ~ 4 m at the in-plane size L = 100 m.
In turn, the thickness td can be estimated from the modified Matthews-Blakeslee
criteria for the initial introduction of misfit dislocations [8]. Taking into account the influence
of elastic anisotropy on the dislocation energy [9], we obtain the following equation for the
threshold thickness of a ferromagnetic film with K1 > 0:
td  
3(c112  c122 )
(c11  c12 )c44
b 2  4td 
ln 
,
4B1 (2c11  c12 ) (c11  c12  2c44 )c11 b100  b 
(S1)
where b is the magnitude of the dislocation Burgers vector b, and b100 represents the
projection of b on the [100] crystallographic axis. For bcc CoFe alloys, b = (1/2) a0 [111] and
b100 = a0/2, where a0  0.284 nm is the lattice parameter. With these data and the numerical
values of B1, c11, c12, and c44 listed in Ref. [5], Eq. (S1) gives td  1.4 m for the Co40Fe60 film,
which is smaller than the thickness tL2 given above.
Finally, the thickness ts(Sm) can be set equal in the first approximation to the critical
thickness, at which the size-induced spin reorientation transition between homogeneous inplane and out-of-plane magnetic states takes place. Hence for ferromagnetic films having
positive MCA and negative surface anisotropy constant Ks we obtain [5]
ts ( S m ) 
2K s
.
0 M ( N11  N 33 )  B /(3c11 )  2 B1 1  2c12 / c11 S m
2
s
2
1
(S2)
With Ks = 1.3 mJ m2 [10] and theoretical demagnetizing factors [11], Eq. (S2) gives ts(Sm =
3
0) < 1 nm for extended Co40Fe60 films and ts(Sm = 0)  1.5 nm for the Co20Fe60B20 nanolayers,
which is in excellent agreement with observations [10]. Although the formation of out-ofplane 180 stripe domains should increase the thickness ts above the value given by Eq. (S2),
ts(Sm  0) cannot exceed a few nanometers, as confirmed by the observation of the a1/a2/a1/a2
domain pattern in the 15-nm-thick Co60Fe40 film [12]. Therefore, the critical thickness ts
should be much smaller than the thickness tL1 discussed above.
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