# CaMn2Sb${}_{2}:$ a fully frustrated classical magnetic system

I. I. Mazin  Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
Code 6393, Naval Research Laboratory, Washington, DC 20375, USA

(8/9/13)

###### Abstract

We show by means of density functional calculations that CaMn2Sb2 is very
close to a mean-field critical point known for the classical Heisenberg model
on the honeycomb lattice. Three entirely different long range ordered magnetic
phases become degenerate at this point: the Neel phase and two different
spiral phases. We speculate that the unusual physical properties of this
compound, observed in recent experiments, in particular the enigmatic
intermediate temperature phase, are due to this proximity.

Introduction: CaMn2Sb2 is an interesting implementation of a nearly classical
spin system on a honeycomb lattice. Such systems have been attracting the
attention of researchers since the seminal paper of Rastelli $et$ $al$ from
1979Rastelli . The basic model involves the nearest neighbor (nn) interaction
$J_{1}$ and the second neighbor interaction $J_{2}.$ In the classical limit,
the ground state of this $J_{1}-J_{2}$ model, for antiferromagnetic $J_{1,2}$,
is the Neel phase where all nearest neighbor bonds are fully
antiferromagnetic, if $J_{2}/J_{1}<1/6.$ Interestingly, for
$1/6<J_{2}/J_{1}<1/2,$ there are two degenerate solutions, corresponding to
two different spiral phases (in one of them the two spins in one unit cell are
always antiparallel, and the spiral propagation vector is perpendicular to a
nn bond, in the other the two sublattices are rotated by a particular angle
with respect to each other, and the spiral vector is parallel to a bond). At
$J_{2}/J_{1}>1/2$ the ground state is the stripe phase with alternating FM
pairs (Fig. 1). Both critical points, $J_{2}/J_{1}=1/2$ and $J_{2}/J_{1}=1/6$
are triple points: in case there is an additional parameter, for instance the
third neighbor exchange $J_{3},$ three phases meet at these points: the Neel
phase and the two spiral phases at $J_{2}/J_{1}=1/6,$ and the stripe phase and
the two spiral phases at $J_{2}/J_{1}=1/2.$ Whenever a third parameter is
added, be it the third neighbor exchangeRastelli  $J_{3}$, uniaxial
anisotropyRastelli  or biquadratic couplingI  $K,$ complex phase diagrams
arise, with the four phases described above, and additional phases such as
zigzag antiferromagnetism.

Figure 1:  Four antiferromagnetic patterns discussed in the text. Note that
the spiral A coincides with the phase N at $\theta=0$ and spiral B with the
phase N N at $\phi=\pi$ and $\theta=0$, and with S at $\phi=0$ and
$\theta=\pi$. At the same time the two spirals cannot be continuously
transformed into each other except in this limiting cases.

Figure 2: (color online) Crystal structure of CaMn2Sb2, showing examples of
the nearest neighbors ($J_{1}$, red), 2nd neighbors ($J_{2}$, green) and 3rd
neighbors ($J_{3}$), blue, inside a MnSb layer, as well as the nearest
($J_{z1}$, yellow) and 2nd ($J_{z2}$, black) interplanar neighbors. Mn atoms
are shown by gray spheres, Sb by cyan ones and Ca by green ones.

The crystal structure of CaMn2Sb2 is shown in Fig. 2. The Mn sublattice
consists of honeycomb layers in which every other atom is shifted
perpendicular to the plane. This interesting geometry leads to a rather short
Mn-Mn distance, 3.18 Å, with substantial direct overlap between the Mn d
orbitals and as a result sizable direct antiferromagnetic exchange. In
addition, there are two superexchange paths available. One connects the
nearest neghbor Mns, with an Mn-Sb-Mn angle of 70${}^{\circ},$ and the other
connects the second neighbors, with an angle of 108${}^{\circ}.$ Mn in this
compound has valency 2+, and therefore in the high spin state it would have a
magnetic moment of 5 $\mu_{B},$ reduced by hybridization and fluctuations.
Experimentally at low temperature the $J_{1}$-driven Neel phase was found,
with a magnetic moment of 2.8-3.4 $\mu_{B}$. At $T=85$ K this phase gives rise
to another phase of unknown origin, which experimentally resembles weak
ferromagnetism, but no long range antiferromagnetic order was detected.
Finally, at $T\gtrsim 200$ K the material becomes paramagnetic, exhibiting at
$T\gtrsim 300$ K a Curie-Weiss behavior with an effective moment $M_{CW}=1.4$
$\mu_{B}$. The unusual character of the intermediate temperature phase, as
well as the very low Curie-Weiss moment, suggest that frustration
characteristic of honeycomb magnetic models may play a role. The low-
temperature phase shows an activation transport behavior with an activation
gap $\sim 40$ meV, while the intermediate temperature phase exhibits a strong
increase (up to a factor of 100) of the resistivity.Aronson

We have performed first principle calculations of the electronic and magnetic
properties of CaMn2Sb2. We found Mn to be in the high spin state and
$d({5\uparrow})$ configuration. We also found a small gap consistent with the
experiment, and magnetic interactions dominated by the nearest neighbor
exchange. Most interestingly, we found that the second neighbor exchange is
4-6 times smaller than the nearest neighbor one, while the third neighbor
exchange and the biquadratic exchange are very small, and the magnetic
anisotropy is of the easy-plane type. In this regime classical spins on the
honeycomb lattice are highly frustrated, with two spiral and two collinear
phases nearly degenerate. We speculate that this frustration is the cause of
the unusual magnetic phase diagram.

Calculations. The calculations in this paper were performed with the Linear
Augmented Plane Wave code WIEN2kWIEN2K , using the following crystallography:
symmetry group #164, $P\bar{3}m1,$ $a=$4.522 Å, $c=$7.458 Å, $z_{Ca}=0,$
$z_{Mn}=0.3784,$ $z_{Sb}=0.7487,$ and a Generalized Gradient Approximation for
the exchange-correlation potential. In agreement with previous
calculationsAronson  we found that Mn in strongly polarized, and Mn moments
are well localized, as demonstrated by the fact that all magnetic
configurations converge to about the same magnetic moment, $\sim 4$ $\mu_{B},$
and the calculated exchange energies are much smaller than the magnetization
energy. The lowest energy among various collinear states has the Neel state,
in which the nn Mn have opposite spins.

Figure 3: (color online) Calculated band structure of CaMn2Sb2 in the
antiferromagnetic Neel phase. The symbol size is proportional to the Mn
character (spin-up in blue, below the Fermi level, and spin-down in red, above
the Fermi level.  Figure 4: The calculated dielectric function of CaMn2Sb2.
The inset shows the corresponding transmission coefficient (the thickness of
the slab was chosen to roughly match the amplitude of the transmission
coefficient in Ref. Aronson i, and a uniform week absorption was added to
account for the experimentally observed in-gap states).

The band structure and density of states in this configuration are shown in
Fig. 3. One can clearly see that all five $d$ orbitals of Mn are fully
polarized, while the reduction of the moment from 5 to 4 $\mu_{B}$ is due to
hybridization. There is also an indirect excitation gap (of the spin-flip
nature), about 50 meV, consistent with the experimentally measured one. The
calculated optical gap (minimal direct transition energy) is about 0.7 meV,
but the absorption at this energy starts relatively slow (Fig. 4), owing to
the fact that the top of the valence band and the bottom of the conductivity
band are formed in the different spin channels (in a ferromagnetic case, these
transtions would have been optically forbidden; in an antiferromagnetic one,
they are allowed but weak). The main absorption edge is located at 2 eV.
Compared to experimentAronson , the calculated transmission drops at a lower
frequency. However, the optical experiments reported in Ref. Aronson  should
be considered as preliminary. One has to wait for more state-of-the-art
optical data, preferably for the conductivity, for a full comparison.

In order to understand better magnetic interactions in CaMn2Sb${}_{2},$ we
have calculated the total energies of 10 different magnetic configurations,
and mapped the results onto the model Heisenberg that includes 1st, 2nd and
3rd nearest neighbors inside the MnSb layers ($J_{1},$ $J_{2},$ and $J_{3})$,
and 1st and 2nd neighbors between the layers ($J_{z1}$ and $J_{z2}).$ The
corresponding distances and values are listed in the Table. Note that the
definition of $J$ in this paper is such that the pair interaction is
$J\sigma_{i}\sigma_{j},$ where $\sigma=\pm 1.$

Table 1: Calculated exchange parameters and the corresponding Mn-Mn bond lengths. See Fig.2 for graphical examples.  | $J_{1}$ | $J_{2}$ | $J_{3}$ | $J_{z1}$ | $J_{z2}$
---|---|---|---|---|---
$d$ (Å) | 3.179 | 4.522 | 5.528 | 6.219 | 7.458
$J$ (meV) | 91 | 22 | 5 | 11 | 3

Since the number of energy differences (9) is larger than the number of fitted
exchange constants, one can estimate the accuracy of the fitting from the
standard fit error. This is on the order of 7–8 meV. Note that on the mean
field level $J_{z}$s simply renormalize the planar interactions, in other
words, assuming perfect interplanar ordering, one can map the 3D problem onto
2D with $J_{1eff}=J_{1}+J_{z1}=102$ meV.

We have also looked for other possible deviations from the Heisenberg model:
the biquadratic interaction, which plays an extraordinary important role in
Fe-based superconductors, and single-site anisotropy. The former appears to be
zero within computational accuracy, in other words, for all possible angles
between the Mn spins in the same unit cell the energy is perfectly
cosinusoidal, $E=const+0.5J_{1}\cos\alpha.$ The latter (magnetic anisotropy)
is zero in plane (although formally hexagonal symmetry allows for magnetic
anisotropy when the field is rotated by 30${}^{o},$ for all practical purposes
this effect is negligible), and about 2–3 meV/Fe otherwise, with the direction
perpendicular to the plane being the hard axis.

Discussion. The fact that essentially any magnetic pattern can be stabilized
in the calculations, and that the energy cost of suppressing magnetism
entirely (0.8 eV/Fe) is much larger than the exchange constants indicates that
CaMn2Sb2 should be considered a local moment system, and overall one expects
Mn to be subject to considerable Hubbard correlations. The fact that the
nearest-neighbor exchange follows the Heisenberg formula nearly exactly also
speaks in favor of classical superexchange and the Hubbard model. Yet DFT
calculations reproduce the excitation gap well, in fact, better than dynamic
mean field calculationsHaule . The reasons for such an unexpected success of
the DFT are unclear st the moment. Given the sharp increase of the resistivity
in the other, intermediate temperature phase, one may think that the Neel
phase is affected by some cancellation of errors, not operative in the other
phases, which leads to an effective increase of the excitation gap in the
intermediate phase.

In the classical phase diagram of an antiferromagnetic $J_{1},J_{2}$ 2D
honeycomb layer there are five planar phases: The Neel (N) phase, where all
nearest neighbors order antiferromagnetically, the “stripy” (S) phase where
all bond along one direction are ferromagnetic, and forming double stripes
order antiferromagnetically, and two spiral phases (A and B) described in the
introduction. For $J_{2}/J_{1}<1/6$ the ground state is N, for
$1/6<J_{2}/J_{1}<1$ the ground state is degenerate between phases A and B, and
for the larger $J_{2}$ the ground state is S. Adding $J_{3}$ creates a
complicated phase diagram, with a new “zigzag” phase emerging at large
$J_{3},$ which is shown in Fig. 5, where we also for completeness show the
phase diagram $J_{1},J_{2},K$ with a biquadratic interaction, not published
before. In this phase diagram, the calculated interactions correspond to the
point $J_{2}/J_{1eff}\sim 0.21$, $J_{3}/J_{1eff}\sim 0.05,$ but the error in
these numbers is close to $\pm 0.08.$ In other words, according to the
calculations the material is extremely close to a highly frustrated critical
point where three different phases are degenerate, the Neel phase and two
qualitatively different spiral phases, $J_{2}/J_{1}=1/6,$ $J_{3}=0.$

Figure 5: Phase diagram of the classical $J_{1}--J_{2}--J_{3}$ Heisenberg
model. The phase boundaries are given by the lines
$y=(x-\sqrt{x^{2}+2(x-1/2)^{2}})/2$, $y=(1/2+3x-\sqrt{51/12-5x+9x^{2}})/4$,
$y=(6x-1)/4$, $y=(2x+1)/4$, $y=0$, $x=1/2$Rastelli . The square shows the
region in the phase diagram where, within computational accuracy, the actual
system is located. The insert shows the phase diagram for the $J_{1}-J_{2}-K$
model, where K is the biquadratic coupling. The phase boundaries are:
$y=(3x-\sqrt{1-2x+9x^{2}})/2$, $y=\sqrt{x-1/4}-x$, $y=(6x-1)/2$, $y=0$I .

The thermodynamics of the classical Heisenberg model on the honeycomb lattice
has not, to our knowledge, been systematically studied and this subject is
beyond the scope of the current paper. It is however likely that the phase
boundaries between the zero-temperature phase shift as the entropy is
included, so it is tempting to ascribe the transformation at 85 K to a phase
transition between the Neel and one of the two spiral phases (more likely the
spiral A). Very weak net ferromagnetism is then due to relativistic effects,
which can be estimated by the ratio between the magnetic anisotropy energy
(2-3 meV) and magnetization energy (800 meV), which is $\sim 3\times 10^{-3},$
to be compared with the experimental ferromagnetic moment of $7\times 10^{-3}$
$\mu_{B}\approx 3\times 10^{-3}$ $M_{Mn}.$

No long range order has been observed between 85 and 210 K in neutron
scattering; this suggests that the spirals break after relatively short
distance and the emerging spiral chunks are randomly distributed over the
three equivalent crystallographic directions. Such a state would be very
similar to the “magnetic liquid” state in MnSiReznik , also called a
“cholesteric” phase, or a “blue” phase in the original paper. This state is
observed in MnSi near the phase transition into a long-range ordered spiral
phase, on the high-temperature (high-pressure) side of the phase transition.
Neutron scattering in this phase reveals no long range order, but well-defined
spirals of considerable length (several hundred Å), propagating in all
crystallographically equivalent directions with equal weight. It is still not
clear whether the spirals in the “blue phase” of MnSi form domains (possibly
dynamic) or are meandering around, periodically switching directions. Both
options are open as well in the intermediate temperature phase of
CaMn2Sb${}_{2}.$

Another corollary of proximity to the critical point at $J_{2}/J_{1}=1/6$ is
that unusual low energy magnetic excitations should be present in the low-
temperature Neel phase. More detailed experimental spectroscopic studies are
highly desirable.

To summarize, we have shown that CaMn2Sb2 is extremely close (within
computational accuracy) to a critical point where three entirely different
magnetic phases are degenerate on the mean field level. We suggest that
unusual properties of this compound, obtained in recent experiments, are
related to this unique proximity.

The author acknowledges discussions with Girsh Blumberg and Meigan Aronson,
and funding from the Office of Naval Research (ONR) through the Naval Research
Laboratory’s Basic Research Program, and from the Alexander von Humboldt
Foundation.

## References

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