SINGLE MOLECULE MAGNET Mn5-CYANIDECONTROL OF THE MAGNETIC ANISOTROPY
A. V. Palii a, S. M. Ostrovsky a, S. V. Kunitsky a , S. I. Klokishner a,
B. S. Tsukerblat b, J. R. Galán-Mascarós c, K. R. Dunbar d
a
Institute of Applied Physics, Academy of Sciences of Moldova,
Academy str. 5, 2028 Kishinev, Moldova
b
Chemistry Department, Ben-Gurion University of the Negev,
Beer-Sheva84105,Israel
c
Instituto de Ciencia Molecular, Universidad de Valencia, Spain
d
Department of Chemistry, Texas A&M University, PO Box 30012, College Station,
Texas 77842-3012 (USA)
In the framework of our study of the magnetic anisotropy in metal clusters here
we report a new model aimed at the explanation of the magnetic properties and single
molecule magnet (SMM) behavior of the trigonal bipyramidal cyanide cluster
{[MnII(tmphen)2]3[MnIII(CN)6]2} (tmphen = 4,5,7,8-tetramethyl-1,10-phenantroline)
exhibiting. The model explicitly takes into account strong single-ion anisotropy
associated with the unquenched orbital angular momenta of two Mn(III) ions and
satisfactory explains the observed temperature dependence of dc magnetic
susceptibility. At the same time the analysis of the model implies the conditions
under which the barrier for reversal of magnetization can exist in such kind of
compounds and provides a theoretical background for the design of cyanide-based
SMMs with controllable barrier and higher blocking temperatures.
1. INTRODUCTION
The phenomenon of single-molecule magnetism was first discovered in the
family of clusters of general formula [Mn12O12(O2CR)16(H2O)4] known as Mn12ac
(see review article (1) and refs therein). These systems show an extremely slow
relaxation of magnetization and quantum tunneling effects at low temperatures. One
of the most important characteristics of SMMs is the blocking temperature, which is
closely related to the magnitude of the spin reorientation barrier. The blocking
temperature for existing magnetic clusters with SMM properties is still not high
enough to be used in applications. The relaxation time in Mn12ac is 108 s at 1.5K
meanwhile the relaxation time acceptable for applications should be at least 15 years
at room temperatures. In search of the new SMMs with the high blocking
temperature chemists are trying to increase the size of the systems and to vary the
metal ions in order to gain the global anisotropy of the system. Nevertheless till
present time Mn12ac remains the system with highest barrier that is estimated to be
of about 60K. The situation looks like some natural limit in this route is already
achieved and a new concept in the field is required.
The general idea of the present consideration is to go beyond spin-clusters and
to essentially enhance the magnetic anisotropy by exploiting strongly anisotropic first
order unquenched orbital magnetism that is inherently related to the orbitally
degenerate metal ions. In striving to exploit this idea we have turned to the cyanide
clusters that have at least two attractive features: 1) the structural motifs of these
clusters contain metal ions in the nearly perfect octahedral sites providing a necessary
condition for the orbital degeneracy and 2) presence of the cyanide groups and
possibility to vary the metal ions and their positions respectively CN group in
3 - 118
heterometallic network make feasible the problem of the design of the series of the
systems with controllable parameters.
2. THE MODEL
We study the magnetic properties of [MnIII(CN)6]2[MnII(tmphen)2]3 (tmphen =
4,5,7,8-tetramethyl-1,10-phenantroline) abbreviated as Mn5-cyanide , ref.(2). Mn(III)
ions are in a nearly perfect octahedral sites (Fig.1) so that strong cubic crystal field
produced by the six carbon atoms of the cyanide ligand leads to the ground orbital
triplet 3T1(t24) that carries first order orbital moment. The ground state of the Mn(II)
in the low field octahedral nitrogen environment is the orbital singlet 6 A1 t 23e 2 .
 
Mn(III)
Mn(II)
N
C
Fig.1. Fragment of the molecular structure of Mn5-cyanide
The model includes the following interactions:
(i) The spin-orbit (SO) coupling operating within the ground cubic 3T1 t24 term of
each Mn(III) ion. Fictitious angular momentum l=1 (see ref. (3)) can be attributed
the orbital triplet T1 , so that SO coupling splits the 3T1 term into three levels: j  0
(A1), j  1 (T1) and j  2 ( E, T2 ) , with j being the quantum number of the
fictitious total angular momentum of the Mn(III) ion in the ground state.
(ii) Trigonal crystal field. The trigonal component of the crystal field (site symmetry
C3 ) splits the ground 3T1 l  1 term into the orbital singlet 3 A2 ( ml  0 ) and the
 
orbital doublet 3 E ( ml  1 ) separated by the gap  . Providing   0 the orbital
part of the magnetic moment is reduced by the trigonal field, meanwhile when   0
the axially anisotropic orbital contribution remains. The trigonal field will be taken
into account along with the SO coupling that can be described by the following
single-ion Hamiltonian acting within the 3T1 manifold of each Mn(III) ion (i=1,2):


2
H i  κλ si li  Δ l iZ
 2/ 3
(1)
,
where  is the orbital reduction factor. The energy levels E j  m j
3 - 119
 of
a trigonally
distorted Mn(III) complex are enumerated by the absolute value of the total angular
momentum projection m j , reflecting thus the axial magnetic symmetry, i =1, 2.
(iii) Magnetic exchange. We assume that only the superexchange interaction
between Mn(II) and Mn(III) ions through the cyanide bridges affects the magnetic
properties. The typical values of the exchange coupling between cyanide bridged
metal ions are of the order of several wavenumbers, ref (4), i.e. they are two order of
magnitude smaller than SO coupling and trigonal field.
The Hamiltonian of the system including Zeeman terms is the following:
H   s l  s l   2J s  s
1 1
s  s  s    l  l  4 / 3 
  H   l  l   g s  s   g s  s  s  ,
ex
2 2
1
1
2
3
2
4
e
5
1
2
2
1Z
1Z
2
3
4
(2)
5
g-factors of all Mn ions are assumed to be isotropic, symbols 3,4,5 enumerate the
Mn(II) ions in the plane.
The wave functions are constructed using the following scheme of the angular
momenta addition:
(3)
s1  l1  j1 , s2  l2  j2 , j1  j2  J12 , s3  s4  S34 , S34  s5  S345 , J12  S345  J
where J is the total angular momentum. This scheme corresponds to the following
labeling of the wave functions that will be used as the basis set for the matrix
representation of the Hamiltonian, Eq. (3):
s1l1  j1  s2l2  j2 J12  s3 s4 S34  s5 S345 J M J   j1  j2 J12 S34 S345  J M J
(4)
The matrix elements of the Hamiltonian are calculated with the aid of the irreducible
tensor operator technique. The problem was approximately treated with the aid of the
irreducible tensor operators technique, reviewed in refs. (5)-(7).
3. RESULTS AND DISCUSSION
Let us discuss how the trigonal crystal field and the SO interaction in the
Mn(III) ion affect the energy pattern of Mn(III) and of the cluster entire. The sign of
the local magnetic anisotropy is defined by the sign of the trigonal component of the
crystal field and we will analyze the two cases ,   0 and   0 , separately. The
energy levels E j m j of a trigonally distorted Mn(III) complex are enumerated by


the absolute value of the total angular momentum projection m j , reflecting thus the
axial magnetic symmetry of the system. They are given by:
E0  2  0  

1
   3 
6

 
3       8  2 2
2
,
(5)
1
E1 0      3    ,
3
1
E1  2  1       3  2  4  2 2  ,
6
1
E2 2      3    .
3

The symbol j = 0, 1, 2 indicates the origin of the level E j m j
3 - 120
 with a given
mj .
Two qualitatively different cases, namely, positive and negative trigonal
field, are to be considered separately on the basis of Eqs.(3). Provided that positive
trigonal field is strong enough, each Mn(III) behaves as a spin S  1 ion with
quenched (to a second order) orbital angular momentum. The Mn5–cyanide cluster
can be considered as a spin-system and the anisotropy of the cluster can be described
by the zero-field splitting Hamiltonian D s12Z  s22Z  with the constant D that proves to
be positive. This parameter obtained accurate within 3 2 is the following:
D      2   .
2
2
3
3
(6)
2
In the limit of strong positive trigonal field the the orbital doublet 3 E ( ml  1 )
separated by the gap  from the singlet 3 A2 is much higher in energy, the SO
splitting (and local anisotropy) is fully suppressed, so the ground state is the orbital
singlet 3 A2 which comprises m j  0 ml  0, ms  0 and m j  1 ml  0, mS  1 .
Since the Mn(II) ions are isotropic, the effective D S value for the isolated
ground state multiplet S  11 2 (under the condition of antiferromagnetic Mn(II)Mn(III) exchange) is positive, in this case the ground level corresponds to
M S  1 2 . In Fig. 2a we show the the low-lying levels calculated for the moderate
positive trigonal field (   250 cm -1) and spin-orbit   180cm1 coupling ( value
for a free ion). This energy pattern demonstrates that the ground state of the system
possesses the minimum projection of the total angular momentum M J  1 2 and that
for the low lying levels the quantum number, M J increases with an increase of the
energy. This energy pattern corresponds to a positive global anisotropy of the system
and thus proves to be incompatible with the experimental observation of the SMM
behavior i.e. the existence of the barrier for the reversal of magnetization.
In the case of a negative trigonal field, one gets a qualitatively different physical
picture due to the fact that the orbital doublet 3 E ml  1 carries a first order orbital
contribution. For a strong trigonal field, the two low lying singlets of Mn(III),
E0 0 and E1 0 , becomes closely spaced, the gap  between these levels in the
case of a strong field becomes (accurate within  3 2 ):
  2    2   .
2
2
3
3
(7)
2
In view of the discussion of the SMM properties it should be underlined that these
levels shows first order magnetic moment ( although m j  0 ) due to the fact that
orbital and spin contributions are not cancelled . In fact, the non-vanishing matrix
elements of the Zeeman interaction, which is operative within this pair of levels, are
found as:
m  0 m  1, m  1    l  g s  H m  0 m  1, m  1
l
    g  H
e
s
e
l
S
.
(8)
Z
One can see that the perpendicular component of the Zeeman interaction vanishes
and the system can be referred to as fully anisotropic. From Fig. 2b, one can see
that the magnetic contributions associated with the next levels E1 1 and E2 2 are
3 - 121
also fully anisotropic. The Hamiltonian of the full Mn5-cyanide system
the restricted 2x2- subspace
acting in
M J M L  1, M S  1 ) and M J  0 ( M L  1, M S  1
can be represented as:
H  4 J s  s
 S  S  S 
  g  s  s  H  g   S  S  S H
1Z
||
1Z
2Z
2Z
3Z
Z
e
4Z
5Z
3
4
.
(9)
5
The explicit form of the Mn(III)-Mn(II) interaction is now transformed into
the Ising type Hamiltonian acting within the ground manifold. The energy pattern
shows that, at low temperatures, the global magnetic anisotropy is negative that is
a condition for the formation of a barrier for magnetization. This is illustrated in
Fig.2b. The main feature of this pattern is that the ground level possesses the
projection M J  15 2 , and that M J decreases with the increase of the energy ,
this result is compatible with the observed SMM behavior of Mn5-cyanide. It should
be stressed that the negative global anisotropy is a consequence of the unquenched
orbital contribution in the case of a negative trigonal field.
Fig. 3 displays the temperature dependence of χT measured for crushed
single crystal at 1000 G over the temperature range 1.8-300 K, ref. (2), and the
theoretical curve calculated for the set of the best fit parameters. The observed room
temperature T value is ~13.7 emuKmol-1 and decreases to 10.4 emuKmol-1 as the
temperature is lowered to 45 K, after which temperature T increases to a maximum
of 15.7 emuKmol-1 at 4.0 K. One can see that the theoretical curve is in a
reasonable agreement with the experimental data in that it reproduces the minimum at
|MJ|
|MJ|
Energy, cm
20
15
10
25
-1
-1
25
7/2
5
20
15
11/2
13/2
10
13/2
5
5/2
3/2
1/2
0
1/2
3/2
7/2
9/2
5/2
9/2
7/2
11/2
9/2
11/2
30
Energy, cm
1/2
7/2
13/2
5/2
9/2
3/2
1/2
7/2
11/2
5/2
3/2
9/2
1/2
30
15/2
0
a
b
Fig.2. Energy schemes of low-lying levels in the cases of positive   250 cm 1 (a),
and negative (b),   251 cm 1 trigonal field, J ex  3.8 cm 1 , κ  0.8
45 K and the slope of the curve in the high temperature region. A relatively small
value of the trigonal field parameter is in accord with the structural data that indicate
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nearly perfect octahedral environments for the Mn(III) ions. The best fit value of the
exchange parameter falls into the range of typical values for the superexchange
parameters mediated by the cyanide bridges, ref. (4).
Elucidation of the role of the key parameters shows promise to the control the
barrier by variation of the magnetic parameters of the system using chemical means,
let say, an appropriate ligand substitution. First, only negative trigonal field leads to
the formation of the barrier. Second, negative crystal field should be strong enough
to provide a favorable condition for a significant orbital magnetic contribution. Fig. 4
demonstrates the low lying energy levels of the Mn5-cyanide cluster calculated with
three different values of the negative parameter  . The level with | M J |  15 2 is
always the ground one, M J decreases (in general, non-monotonically) with the
increase of the energy and reaches its minimum value  | M J |  1 2  for the upper
18
T, emu K mol
-1
16
14
12
10
8
0
50
100
150
200
250
300
Temperature, K
Fig.3. Temperature dependences of T : circles-experimental data, solid line –
theoretical curve calculated with the best fit parameters J ex  3.8 cm 1 ,
  251 cm1 , g  1.95 , κ  0.8 .
level. In all cases we get the barrier for the reversal of magnetization, the height of
the barrier can be associated with the gap b  ε M J  1 2  ε M J  15 2. One
can see that the increase of the negative trigonal field from   250 cm-1 to
  10000 cm-1 almost trebles the barrier (from  30 cm-1 up to  87 cm-1). The
cubic field for the carbon ligands is very strong , 10 Dq  30000 cm-1 in cyanide
complexes , so that the value  10000 cm-1 for the trigonal crystal field falls into the
region of reasonable parameters. The possibility to strongly increase the barrier by
varying  seems to be a specific feature of cyano-bridged SMMs containing metal
ions with unquenched orbital angular momenta distinguishing thus such systems from
the conventional oxo-bridged spin SMMs. In the former systems the control of the
height of the barrier can be attained by the chemical modification of the ligand
surrounding in the apical positions.
4. CONCLUDING REMARKS
The models developed in ref. (8) and in this study represent a first attempt
to reveal the underlying mechanism of the SMM behavior of cyanide clusters
containing highly anisotropic transition metal ions with unquenched orbital angular
3 - 123
momenta. The model includes a trigonal crystal field and a SO interaction operating
within the ground state of the Mn(III) ions, and the isotropic exchange interaction
between Mn(II) and Mn(III) ions. The model was shown to account for the observed
dc magnetic susceptibility of the Mn5-cyanide cluster. The interplay between strong
single ion anisotropy arising from the trigonal crystal field, combined with the SO
interaction and antiferromagnetic Heisenberg-type exchange, was shown to produce
an appreciable barrier for the reversal of magnetization. The proposed model provides
a satisfactory agreement between the observed and calculated dc magnetic
susceptibilities and also confirms the ac susceptibility evidence for SMM behavior of
the Mn5-cyanide cluster. These findings lend insight into the conditions under which
the barrier for the reversal of magnetization appears in clusters containing metal ions
with unquenched orbital angular momenta and provide a theoretical background for
the enhance of the barrier in a controllable way . In contrast to traditional SMMs that
can be treated as spin-clusters, the first-order single ion anisotropy of the Mn(III) ion
was found to be responsible for the formation of the barrier.
|MJ|
|MJ|
|MJ|
-250
-1000
-10000
Trigonal field, cm-1
Fig.4. Low-lying energy levels of Mn5-cluster as a function of the trigonal field
Finally, it should be noted that the model contains several simplifying
assumptions. First, we neglected the orbitally dependent terms in the exchange
Hamiltonian, see refs. (9)-(11) . These terms are expected to contribute to the global
magnetic anisotropy of the system, thus affecting the barrier for the reversal of
magnetization. Second, a truncated basis was used in the magnetic susceptibility
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calculations. Also, the effects of Jahn-Teller vibronic interactions ( see refs. (12),
(13)) for the orbitally degenerate 3T1 term of the Mn(III) ion may be also important
for the formation of the barrier and also for the rate and temperature dependence of
the relaxation processes. These issues remain open and will be considered in future
development of the model.
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