Magnetic response of magnetic molecules with non-collinear local d-tensors
Investigations of molecular magnets are driven both by prospective applications in future storage technology or quantum computing as well as by fundamental questions. Nowadays numerical simulation techniques and computer capabilities make it possible to investigate spin Hamiltonians with realistic...
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irk-123456789-1199982017-06-11T03:04:39Z Magnetic response of magnetic molecules with non-collinear local d-tensors Schnack, J. Investigations of molecular magnets are driven both by prospective applications in future storage technology or quantum computing as well as by fundamental questions. Nowadays numerical simulation techniques and computer capabilities make it possible to investigate spin Hamiltonians with realistic arrangements of local anisotropy tensors. In this contribution I will discuss the magnetic response of a small spin system with special emphasis on non-collinear alignments of the local anisotropy axes. Актуальнiсть молекулярних магнетикiв обумовлена як перспективами використання у майбутнiх технологiях зберiгання чи квантовому комп’ютингу, так i з фундаментальної точки зору. Сучаснi методи чисельного моделювання та можливостi комп’ютерної технiки дають змогу дослiдити спiновi гамiльтонiани з реалiстичним розташуванням тензорiв локальної анiзотропiї. У цiй статтi обговорюється магнiтний вiдгук малої спiнової системи з спецiальним наголосом на неколiнеарне вилаштування осей локальної анiзотропiї. 2009 Article Magnetic response of magnetic molecules with non-collinear local d-tensors / J. Schnack // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 323-330. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 75.50.Xx, 75.10.Jm, 75.40.Cx, 75.40.Mg DOI:10.5488/CMP.12.3.323 http://dspace.nbuv.gov.ua/handle/123456789/119998 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Investigations of molecular magnets are driven both by prospective applications in future storage technology
or quantum computing as well as by fundamental questions. Nowadays numerical simulation techniques and
computer capabilities make it possible to investigate spin Hamiltonians with realistic arrangements of local
anisotropy tensors. In this contribution I will discuss the magnetic response of a small spin system with special
emphasis on non-collinear alignments of the local anisotropy axes. |
| format |
Article |
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Schnack, J. |
| spellingShingle |
Schnack, J. Magnetic response of magnetic molecules with non-collinear local d-tensors Condensed Matter Physics |
| author_facet |
Schnack, J. |
| author_sort |
Schnack, J. |
| title |
Magnetic response of magnetic molecules with non-collinear local d-tensors |
| title_short |
Magnetic response of magnetic molecules with non-collinear local d-tensors |
| title_full |
Magnetic response of magnetic molecules with non-collinear local d-tensors |
| title_fullStr |
Magnetic response of magnetic molecules with non-collinear local d-tensors |
| title_full_unstemmed |
Magnetic response of magnetic molecules with non-collinear local d-tensors |
| title_sort |
magnetic response of magnetic molecules with non-collinear local d-tensors |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/119998 |
| citation_txt |
Magnetic response of magnetic molecules with non-collinear local d-tensors / J. Schnack // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 323-330. — Бібліогр.: 32 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT schnackj magneticresponseofmagneticmoleculeswithnoncollinearlocaldtensors |
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2025-07-08T17:03:08Z |
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2025-07-08T17:03:08Z |
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1837099059248103424 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 3, pp. 323–330
Magnetic response of magnetic molecules with
non-collinear local d-tensors
J. Schnack1∗
1 Universität Bielefeld, Fakultät für Physik, Postfach 100131, D-33501 Bielefeld, Germany
Received May 26, 2009, in final form June 15, 2009
Investigations of molecular magnets are driven both by prospective applications in future storage technology
or quantum computing as well as by fundamental questions. Nowadays numerical simulation techniques and
computer capabilities make it possible to investigate spin Hamiltonians with realistic arrangements of local
anisotropy tensors. In this contribution I will discuss the magnetic response of a small spin system with special
emphasis on non-collinear alignments of the local anisotropy axes.
Key words: magnetic molecules, single-ion anisotropy, exact diagonalization
PACS: 75.50.Xx, 75.10.Jm, 75.40.Cx, 75.40.Mg
1. Introduction
Since the early investigations of Mn12-acetate [1], single molecule magnets (SMM) have been
at the heart of the investigations of magnetic molecules worldwide. This is due to their properties
which are governed by the anisotropy barrier, as there is observed slow relaxation of the magne-
tization as well as spin tunneling through the barrier [2–5]. From the chemical point of view, it
is striking that until recently no compound could be synthesized with a higher anisotropy bar-
rier which moved hopes of easy application into the more distant future. Theoretical estimates
pointed out that this might be due to very general reasons [6,7]. A recently synthesized manganese
compound put an end to more than 20 years of search [8].
On the theory side, numerically exact evaluations of spin Hamiltonians including anisotropic
terms turned out to be limited to rather small systems such as for instance an antiferromagnetically
coupled Ni4 compound [9–12], which was investigated in great detail or another but ferromagneti-
cally coupled Ni4 [13]. With great numerical effort the exchange constants of Mn12-acetate could
be determined [14], but usually one resorts to models where the interacting spin system is replaced
by one large spin in its effective ligand field (giant spin approximation). But thanks to the technical
progress detailed numerical studies of larger molecules are affordable nowadays. First examples are
given by the simulation of the new Mn6 compound with record anisotropy barrier [15] and by the
investigation of a Mn6Fe compound [16]. Whereas [15] restricts its parameter space to collinear
local anisotropy axes, [16] considers non-collinear ones.
In this contribution the recently developed procedures will be used to discuss the effect of non-
collinear anisotropy axes on the magnetic response of a fictitious molecular compound. In order not
to make the situation too complicated I will restrict the discussion to a triangular arrangement of
equal spins of s = 1 with a molecular C3 symmetry. Then the magnetization will be discussed for
three typical scenarios given by the possible ratios of exchange and anisotropy: |J | > |d|, |J | = |d|,
and |J | < |d|.
∗jschnack@physik.uni-bielefeld.de
c© J. Schnack 323
J. Schnack
2. Hamiltonian and evaluation
2.1. Hamiltonian
It turns out that many magnetic molecules can be well described by a Heisenberg model which
takes the super exchange between the moments of the (typically 3d) ions into account. In this
strong coupling limit anisotropic terms appear as perturbations and are thus often taken care of by
perturbation theory. But for single molecule magnets, that might e. g. contain Mn(III) or V(III)
ions, the situation might be reversed or anisotropy and exchange are at least of the same order.
Therefore, I would like to treat both on the same footing right from the beginning. In the following
Hamiltonian,
H
∼
(~B) = −
∑
i,j
Jij~s
∼
i ·~s
∼
j +
∑
i
di
(
~ei ·~s
∼
i
)2
+ µB
~B ·
∑
i
gi ·~s
∼
i , (1)
the first term models the isotropic Heisenberg exchange interaction. A negative Jij corresponds to
an antiferromagnetic coupling of spins at sites i and j. The second term models the local anisotropy
tensors by their major principal axis which points along ~ei. Depending on the sign of di this is an
easy (di < 0) or hard (di > 0) axis. The neglected two other principal axes of the local d–tensor (e-
terms) are usually much smaller. The last term provides the interaction with the applied magnetic
field. Here gi is the local g-tensor, which might also be anisotropic.
C3
ie
J
Figure 1. Configuration of the fictitious C3-symmetric molecule discussed in this article. The
remaining parameters of the model are the exchange coupling J , the strength of the local
anisotropy d, and the angle ϑ of the local anisotropy axes with respect to the C3 axis.
Since in this contribution the effect of the non-collinearity of the local anisotropy axes shall
be discussed, the Hamiltonian will be restricted to a simplified spin system, that is depicted in
figure 1. Three spins s = 1 of a fictitious C3-symmetric molecule will be considered that interact
with one and the same exchange J . The local anisotropy axes are also related by the C3 symmetry,
so that the anisotropy can be modeled by one strength d and one azimuthal angle ϑ. The g-tensors
are assumed to be isotropic and of value 2.
2.2. Evaluation
Since the various parts of Hamiltonian (1) do not commute in general all eigenvalues and
eigenvectors have to be calculated for each magnetic field ~B, i. e. for each strength and direction of
the field. The magnetization is then evaluated using all eigenvectors. I would like to mention that it
is also possible to obtain the magnetization without using eigenvectors by numerical differentiation
of the energy eigenvalues Eν( ~B), which are functions of ~B.
Very often the investigated substance is only available as a powder. Then an orientational
average has to be performed in order to be able to compare to experiments. Since this average
cannot be performed analytically, one sums over a finite set of special directions. In our evaluations
we use for each absolute value B = | ~B| special sets of directions given by points on the unit
sphere. These sets are called Lebedev-Laikov grids [17]. The directions, which contribute with
various weights, and their total number are chosen such that the angular integration of polynomials
xk ×yl × zm, where k + l +m 6 131, can be performed with a relative accuracy of 2−14 [17]. In our
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Magnetic response of magnetic molecules with non-collinear local d-tensors
evaluations we normally use a Lebedev-Laikov grid with 50 points. These points can be generated
with publicly available software [18]. Equivalently one may cover the unit sphere with the vertices
of regular bodies such as an octahedron, a cube, or a dodecahedron. For practically all examples
we evaluated in the past we can say that averaging over the directions given by the vertices of
a dodecahedron is as accurate as averaging over a Lebedev-Laikov grid with 50 points. On the
contrary, averaging only over a smaller number of directions like ±x, ±y, and ±z is insufficient.
3. Anisotropy versus coupling
In the following main part the magnetic response of the spin system shown in figure 1 will
be discussed for three scenarios: |J | > |d|, |J | = |d|, and |J | < |d|. In most cases the figures
show powder averaged magnetizations for ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 degrees. Solid curves
correspond to ϑ = 0, 10, 20, 30 degrees, dashed ones to ϑ = 40, 50, 60, 70 degrees, and dashed-dotted
curves depict ϑ = 80, 90 degrees. In order to pronounce the details, all magnetization curves are
shown at a low temperature of T = 0.1|J |.
3.1. Strong coupling limit
Figure 2. Magnetization for antiferromagnetic coupling J = −1 and easy axis anisotropy d =
−0.1|J | (l.h.s.) as well as hard axis anisotropy d = +0.1|J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60,
70, 80, 90 degrees.
Figure 3. Magnetization for ferromagnetic coupling J = +1 and easy axis anisotropy d = −0.1|J |
(l.h.s.) as well as hard axis anisotropy d = +0.1|J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90
degrees.
In the strong coupling limit the exchange energy is much bigger than the anisotropy energy,
i. e. |J | > |d|. This is the case for many molecules, e. g. antiferromagnetically coupled iron or
325
J. Schnack
chromium rings [19–28]. What these systems have in common is that a description in terms of
a plain Heisenberg Hamiltonian often provides a very good approximation [29–32]. If needed the
anisotropic terms can be included via perturbation theory.
In figures 2 and 3 the magnetization curves are shown for combinations of anti-/ferromagnetic
coupling J = ∓1 and easy/hard axis anisotropy d = ∓0.1|J |. In the antiferromagnetic case (fig-
ure 2) the low-temperature magnetization curve is a staircase due to the successive ground-state
level crossings in the growing magnetic field whereas in the ferromagnetic case the magnetization
curve is practically given by the Brillouin function of the ground state spin S = 3. Since the
anisotropy is small in magnitude its angular variation is not noticeable.
3.2. Intermediate coupling
Figure 4. Magnetization for antiferromagnetic coupling J = −1 and easy axis anisotropy d =
−|J | (l.h.s.) as well as hard axis anisotropy d = |J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80,
90 degrees.
Figure 5. Magnetization for ferromagnetic coupling J = +1 and easy axis anisotropy d = −|J |
(l.h.s.) as well as hard axis anisotropy d = |J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90
degrees.
In the intermediate coupling regime exchange and anisotropy are of the same order. In the
examples presented in figure 4 and 5 they have the same amplitude |J | = |d|. Looking at the figures
one easily notices that the now larger anisotropy smears out the angle averaged magnetization
curves. Nevertheless, the underlying structure – steps for the antiferromagnetic case and Brillouin
function like for the ferromagnetic case – is still clearly visible. Interestingly, the structure of the
magnetization curve is most similar to the strong coupling limit for intermediate angles ϑ around
50◦ . . . 60◦.
In detail the curves follow the following trend with varying ϑ. At small ϑ (solid curves), i. e.
close to the collinear configuration, the effect of the anisotropy is strong since it acts cooperatively.
326
Magnetic response of magnetic molecules with non-collinear local d-tensors
Then for intermediate ϑ (dashed curves) the effect of the anisotropy is the weakest compared to
the strong coupling limit. This is understandable if one recalls that ϑ = arccos(1/
√
3) = 54, 73◦
is the angle of a perfect octahedral alignment of the anisotropy axes, i. e. the three axis have a
pairwise angle of 90◦. In this configuration the effect of the anisotropic terms on multiplets of the
Heisenberg Hamiltonian is either canceled or rather small. This explains why both at 50◦ and 60◦
anisotropy effects are hardly visible. For ϑ > 60◦ (dashed-dotted curves) the effect of the anisotropy
rises again.
3.3. Weak coupling limit
Figure 6. Magnetization for antiferromagnetic coupling J = −1 and easy axis anisotropy d =
−5|J | (l.h.s.) as well as hard axis anisotropy d = +5|J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70,
80, 90 degrees.
Figure 7. Magnetization for ferromagnetic coupling J = +1 and easy axis anisotropy d = −5|J |
(l.h.s.) as well as hard axis anisotropy d = +5|J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90
degrees.
The weak coupling limit, for which the exchange energy is (much) smaller than the anisotropy
energy, i. e. |J | < |d|, is interesting since this case seems to be relevant for an improvement of single
molecule magnets (SMM). The curves in figures 6 and 7 show that the effect of the anisotropy is –
as expected – strong, and in the case of antiferromagnetic coupling (figure 6) very strong. Here one
can say that a hard axis anisotropy together with antiferromagnetic coupling (r.h.s. of figure 6)
completely destroys any structure of the magnetization curve. A (meta-) stable magnetic ground
state that is separated by an anisotropy barrier from its counterpart of opposite magnetization is
either not created or the low-lying spectrum is so dense that the mechanism of a barrier breaks
down. Zero-field split multiplets do not exist any more and are thus an inadequate picture for such
a situation. In the antiferromagnetic case with easy axis (l.h.s. of figure 6) one can at least for
327
J. Schnack
almost all collinear alignments of the anisotropy axes obtain a non-vanishing ground state moment
that is stabilized by anisotropy.
Figure 8. Magnetization as a function of temperature for gµB/|J | = 0.1 for ferromagnetic
coupling J = +1 and easy axis anisotropy d = −5|J | (l.h.s.) as well as hard axis anisotropy
d = +5|J | (r.h.s.). ϑ = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 degrees.
In the ferromagnetic case a hard-axis anisotropy (r.h.s. of figure 7) also has the effect to weaken
the magnetization compared to a Brillouin function. The interesting case is here again the case of
easy-axis anisotropy (l.h.s. of figure 7). If the magnetization for a small field, here gµB/|J | = 0.1,
is plotted against temperature (figure 8) one again sees that a hard-axis anisotropy tends to smear
out and weaken the magnetization (r.h.s. of figure 8) whereas the curves appear sharper for easy-
axis anisotropy (l.h.s. of figure 8). It is worthwhile to investigate the energy spectrum of the latter
configuration in some detail and compare it to the respective cases of intermediate and strong
coupling.
3.4. Possible SMM behavior
Figure 9. Energy eigenvalues at B = 0 for J = +1 and d = −0.1|J | (l.h.s.), d = −|J | (middle),
and d = −5|J | (r.h.s.). ϑ = 0.
Regarding the question which of the cases is preferential for a good SMM, the energy eigenvalues
have to be studied. This is done for the zero-field case and J = +1 and d = −0.1|J | (l.h.s. of
figure 9), d = −|J | (middle of figure 9), and d = −5|J | (r.h.s. of figure 9). The anisotropy axes
are aligned in a collinear fashion, i. e. ϑ = 0, in order to act constructively. For a good SMM it
would be necessary to obtain a low-lying zero-field split multiplet with both high total spin as
well as high anisotropy barrier. In addition, these levels should be largely separated from higher
lying levels. Looking at figure 9 one notices that in the strong coupling limit the multiplets can
be well separated, but the ground-state multiplet is practically not split. In the opposite case of
large easy-axis anisotropy the ground state multiplet is nicely split but energetically overlaps with
other levels (but not too strong). Such a scenario was for instance discussed for the aforementioned
Mn6 cluster [15]. It turns out that the intermediate case, where exchange and easy-axis anisotropy
328
Magnetic response of magnetic molecules with non-collinear local d-tensors
are of similar magnitude provides the best chances for obtaining a good SMM: the ground-state
multiplet is correctly zero-field split and the higher lying states are well separated.
4. Summary
In this article the effect of non-collinear anisotropy axes on the magnetic response of a small
spin cluster was discussed. The main result is that for good single molecule magnets, ferromag-
netic coupling together with an anisotropy of collinear easy axes that is of similar magnitude is
preferential. Various other scenarios have been discussed.
Acknowledgement
This work was supported by the German Science Foundation (DFG) through the research
group 945.
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Магнiтний вiдгук магнiтних молекул з неколiнеарними
локальними d-тензорами
Ю. Шнак
Унiверситет Бiлєфельда, факультет фiзики, поштова скринька 100131, D–33501 Бiлєфельд,
Нiмеччина
Отримано 26 травня 2009 р., в остаточному виглядi – 15 червня 2009 р.
Актуальнiсть молекулярних магнетикiв обумовлена як перспективами використання у майбутнiх тех-
нологiях зберiгання чи квантовому комп’ютингу, так i з фундаментальної точки зору. Сучаснi методи
чисельного моделювання та можливостi комп’ютерної технiки дають змогу дослiдити спiновi гамiль-
тонiани з реалiстичним розташуванням тензорiв локальної анiзотропiї. У цiй статтi обговорюється
магнiтний вiдгук малої спiнової системи з спецiальним наголосом на неколiнеарне вилаштування
осей локальної анiзотропiї.
Ключовi слова: магнiтнi молекули, однойонна анiзотропiя, точна дiагоналiзацiя
PACS: 75.50.Xx, 75.10.Jm, 75.40.Cx, 75.40.Mg
330
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