Supporting Information
Engineering domain structures in nanoscale magnetic thin films via strain
Jia-Mian Hu1, Tian-Nan Yang2, Long-Qing Chen1,2, and Ce-Wen Nan1
1School
of Materials Science and Engineering, and State Key Lab of New Ceramics and
Fine Processing, Tsinghua University, Beijing 100084, China;
2Department
of Materials Science and Engineering, The Pennsylvania State University,
University Park, 16802, USA
S1 Free energy analysis, and comparison between the analytical calculations and the
phase-field simulations on the strain-domain stability map for single-domain magnets
Figure S1| (a) Free energy analysis on the six stable magnetic single-domain states and one
rhombohedral domain. The solid circle indicates the minimum of the free energy profile. (b)
Strain-domain stability map for single-domain magnets based on phase-field simulations (solid lines)
and analytical calculations (dashed lines).
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We performed free energy analysis for the six stable magnetic single-domain states (Fig. 1a in
the main text) as well as one metastable rhombohedral domain, where the same strain conditions, i.e.,
(es1, es2), as those in phase-field simulations are assumed. The results are summarized in Fig. S1(a).
Here  and  are the azimuth angles, for example, =0, =0 for a uniform a1 domain. As shown in
Fig. S1(a), the tetragonal (a1, a2, c) and the O12 domains present high energy barriers (i.e., >1000kBT,
where kB and T are the Boltzmann constant and temperature, respectively) which prevent them from
thermal fluctuations. The orthorhombic O13 and O23 domains exhibit lower energy barriers of about
100kBT, which, however, is larger than the 60kBT required by a thermodynamically stable magnetic
domain state. By contrast, the rhombohedral r domain presents an almost flat energy barrier which
induces multiple minimums in the free energy profile, and hence is unstable.
For comparison, we sketch the strain-domain stability map using analytical calculations based
on a previously developed phenomenological model [1], as indicated by the dark dashed lines in Fig.
S1(b). As seen, the calculated map is in good accordance with that obtained via phase-field
simulations, as demonstrated by their superimposed a1(a2) to c phase boundaries and the almost
identical critical switching strain [i.e., the O12 to c domain transition, marked by the central solid
triangle in Fig. S1(b)] of about 0.29%. Such consistency proves the validity of both theoretical
approaches. However, the phase-field model shows advantages in terms of the direct illustration of
the domain structure in three dimensions, and more importantly, it does not put any a priori
assumptions on the magnetization orientations.
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S2 Strain-stabilized multi-domain states in a thinner polycrystalline Ni film of 19219215 nm3
Figure S2| Typical magnetic domain configurations of the seven structural phases in the
polycrystalline Ni film of 19219215 nm3. The biaxial in-plane strains es1 and es2 are the same as
those of 19219235 nm3 Ni films in Fig. 2(a) of the main text. The arrows denote the
magnetization orientations.
As shown in Fig. S2 are the seven strain-stabilized multi-domain states in the polycrystalline Ni
film of 19219215 nm3. Thus a smaller thickness results in the enhancement of the out-of-plane
demagnetization (magnetostatic) energy. Accordingly, the magnetic vectors would more likely align
in the film plane and exhibit larger parts of in-plane domains. Specifically, the mixed a1+a2+c phase
in the 19219215 nm3 Ni film displays more in-plane a1 and a2 domains with the resultant
vortex-type domain state, as compared to the three phase mixtures in the 19219235 nm3 Ni film
[see Fig. 2(a) in the main text]. The area fractions of the in-plane phases, e.g., the a1, a2, and binary
a1+a2, in the corresponding domain stability map would increase as well.
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S3 Domain wall structure illustrations of the seven strain-stabilized multi-domain states
Figure S3| (top view) vector diagram illustration of the typical domain wall structures in each
multi-domain magnetic phase. The biaxial in-plane strains, i.e., es1 and es2, are the same as those of
of 19219235 nm3 Ni films in Fig. 2(a) of the main text. (Color bar) m3 indicates the normalized
magnetization in the out-of-plane x3, i.e., the [001] direction.
As shown in Fig. S3, the in-plane (IP) a1 and a2 phases mostly exhibit 90 out-of-plane (OP)
domain walls resulting from the strong elastic anisotropy which splits a 180 IP Néel wall into two
90 OP walls. The IP a1+a2 and OP c phases show typical 180 IP Néel-type and 180 OP
Bloch-type walls, respectively. The binary a1+c and a2+c phases present both 90 OP and 180 OP
Bloch walls. The a1+a2+c phase displays a mixed characteristic of all types of domain wall structures
mentioned above.
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S4 Critical strain for the out-of-plane magnetic domain switching in 19219215 nm3 Ni films
Figure S4| (a) Time-dependent evolution of the perpendicular c domain in the 19219215 nm3 Ni
films under the biaxial isotropic in-plane strains of 0.8%, 0.85%, and 0.95%, respectively. (b) Vector
diagram of the stable local magnetization distributions of the Ni film under the strain of (0.85%,
0.85%) (Color bar) m3 indicates the normalized magnetization in the out-of-plane x3, i.e., the [001]
direction.
As shown in Fig. S4, the volume fraction of the perpendicular c domain exceeds 50% when the
applied biaxial in-plane strain is higher than 0.85% [Fig. S4(a)], indicating a critical transition from
an in-plane domain dominated structure to an out-of-plane domain dominated one, which is
confirmed by the corresponding magnetic vector diagrams at 0.85% in Fig. S4(b).
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Reference
[1] Hu J M, Nan C W. Electric-field-induced magnetic easy-axis reorientation in
ferromagnetic/ferroelectric layered heterostructures. Physical Review B, 2009, 80(22): 224416.
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