Itinerant Antiferromagnetism in RuO2

T. Berlijn,1, 2 P. C. Snijders,3, 4 O. Delaire,3, 5 H.-D. Zhou,4 T. A. Maier,1, 2 H.-B. Cao,6 S.-X.

Chi,6 M. Matsuda,6 Y. Wang,3 M. R. Koehler,7 P. R. C. Kent,1, 2 and H. H. Weitering4, 3
1
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
2
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
3
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
4
Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996
5
Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA
6
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
7
Department of Materials Science and Engineering,
The University of Tennessee, Knoxville, Tennessee 37996
arXiv:1612.09589v1 [cond-mat.str-el] 30 Dec 2016

(Dated: January 2, 2017)

Bulk rutile RuO2 has long been considered a Pauli paramagnet. Here we report that RuO2 exhibits
a hitherto undetected lattice distortion below approximately 900 K. The distortion is accompanied
by antiferromagnetic order up to at least 300 K with a small room temperature magnetic moment
of approximately 0.05 µB as evidenced by polarized neutron diffraction. Density functional theory
plus U (DFT+U ) calculations indicate that antiferromagnetism is favored even for small values of
the Hubbard U of the order of 1 eV. The antiferromagnetism may be traced to a Fermi surface
instability, lifting the band degeneracy imposed by the rutile crystal field. The combination of high
Néel temperature and small itinerant moments make RuO2 unique among ruthenate compounds
and among oxide materials in general.

PACS numbers: 74.70.Pq,75.50.Ee,75.30.Fv

Theories of magnetism in 3d transition metal oxides romagnetic metallic ground states, respectively [4]. Their
(TMOs) are usually framed in the context of strong magnetic ordering temperatures are generally low, al-
Coulomb repulsions and Hund’s rule coupling in the 3d though recently SrRu2 O6 has been reported to host high-
orbitals of the transition metal cation, and their covalent temperature antiferromagnetism with a Néel tempera-
bonding with the oxygen 2p orbitals. Strong on-site elec- ture TN = 563 K [5]. Ruthenium dioxide (RuO2 ), on
tron interactions tend to inhibit double occupancy of the the other hand, has been thought to fall in line with
3d orbital and the overall Coulomb energy of the crystal other binary 4d transition metal oxides [6]; it is a good
is lowered by localizing the valence charge of the cation. metal [7] and believed to be Pauli paramagnetic [8]. From
Covalent bonding delocalizes the d-electron charge and the point of view of correlated electron physics and mag-
thus lowers the kinetic energy. The former mechanism netism, RuO2 seems to be one of the least interesting 4d
favors the formation of local magnetic moments while the TMOs. From a technology perspective, however, RuO2
latter decreases the moment but increases the exchange is by far one of the most important oxides. It has numer-
coupling between the moments through virtual hopping ous applications in electro- and heterogeneous catalysis,
processes. In particular, the anion-mediated Kramers- as electrode material in electrolytic cells, supercapacitors,
Anderson “superexchange” between half-filled 3d orbitals batteries and fuel cells, and as diffusion barriers in mi-
often gives rise to strong antiferromagnetism. Many 3d croelectronic devices [9]. It owes its usefulness in part to
transition metal oxides can be classified as antiferromag- its relatively high electrical conductivity combined with
netic Mott insulators where the on-site Coulomb repul- its excellent thermal and chemical stability [10]. For the
sion U exceeds the electronic band width W . technological applications little attention has been paid
4d TMOs generally have significantly greater band to the potential role of magnetism (with the exception of
widths and smaller U , due to the larger spatial ex- Ref. [11]), presumably because magnetism is generally
tent of the 4d orbitals. With U and W being more-or- believed to be non-existent in bulk RuO2 .
less comparable in magnitude [1–3], they are represen- In this letter we report on the finding that RuO2 is
tative of the less-well understood intermediate coupling distorted from the rutile symmetry (P 42 /mnm) and ex-
regime. Without clear evidence of local moment forma- hibits antiferromagnetic order up to at least 300 K. Our
tion and/or magnetic ordering, many of them are con- DFT+U calculations show that for a reasonable range of
sidered to be metallic Pauli paramagnets. The ruthen- local interactions, the moments within the Ru2 O4 rutile
ate family is a notable exception and features a vari- unit cell strongly prefer to align antiferromagnetically.
ety of magnetically ordered phases. The best-known The predicted magnetic order is confirmed with polarized
examples are the Ca-based Ca2 RuO4 and Ca3 Ru2 O7 , neutron scattering experiments that show structurally
and Sr-based Sr3 Ru2 O7 , Sr4 Ru3 O10 , and SrRuO3 per- forbidden peaks with a significantly decreased non-spin-
ovskites, featuring antiferromagnetic insulating and fer- flip/spin-flip intensity ratio. We conjecture that the rel-
 2

atively high Néel temperature can be attributed to the the AFM configuration decreases relative to that of the
existence of half-filled t2g orbitals, in conjunction with non-magnetic structure. For U = 4 eV, the system is no
a fairly large band width, similar to the recently re- longer metallic and exhibits a band gap gap of about 0.5
ported case of SrTcO3 . Both materials can be described eV, contradicting the experimental fact that RuO2 is a
as strongly covalent intermediate coupling systems. An metal [7]. For U = 2 eV we find that the AFM configura-
important distinction, however, is that RuO2 is metallic tion is more stable than the non-magnetic one by a signif-
and that its magnetism may be traced to a Fermi sur- icant 72 meV per Ru atom, while still retaining a sizable
face instability, whereas SrTcO3 has been predicted to density of states at the Fermi level of about 1 eV−1 per
be a narrow gap insulator. These findings not only pro- Ru atom. The AFM configuration was reproduced using
vide new insights into the origins of antiferromagnetism hybrid functionals. [12, 14] Apparently it requires an in-
in the intermediate coupling regime, but may also have termediate range of interaction strengths in RuO2 to be
important ramifications for the understanding of the re- simultaneously AFM and metallic. We also considered
markable properties that make RuO2 attractive for tech- the influence of the spin-orbit coupling, which we found
nological applications. to have a small effect on the electronic band structure
We begin our investigation of magnetism in RuO2 with and to only make small quantitative changes in the rel-
a DFT analysis (see [12] for technical details). The ma- ative total energies of the magnetic configurations. [12]
jority of theoretical investigations considered bulk RuO2 Ferromagnetic configurations turned out to be unstable
to be non-magnetic. However, almost none of these stud- or high in energy.
ies considered the effects of on-site Coulomb interactions To validate the DFT results, we synthesized RuO2 sin-
among the Ru 4d orbitals. Although these interactions gle crystals via vapor transport in flowing oxygen, and
are expected to be weaker than those in 3d TMOs, they subjected those crystals to extensive x-ray diffraction
should not be ignored. Indeed, Ru L23 X-ray-absorption (XRD), neutron scattering, and magnetic susceptibility
spectra in combination with crystal-field-multiplet calcu- investigations [12]. For the perfect rutile structure, non-
lations indicated the importance of Coulomb interactions magnetic Ru contributions to the (hkl) Bragg reflection
in RuO2 [13]. Ru 3d core-level X-ray photoemission vanish when h + k + l = odd. The non-magnetic oxy-
spectroscopy on RuO2 was found to be most consistent gen contributions vanish when h + l = odd and k = 0,
with a dynamical mean field theory treatment of the sin- or when k + l = odd and h = 0. Indeed, the XRD
gle band Hubbard model, when U is taken to be 1.8 eV data in Fig. 2(a), acquired at room temperature, show
compared to a bandwidth W of 3.6 eV [1]. Since in- that the (100) Bragg peak is absent while the (200) and
teractions always play a critical role in magnetism it is (111) peaks are clearly visible. On the other hand, room
imperative that we include their effects in our theoret- temperature unpolarized neutron diffraction data on a
ical investigation. To this end we employ the PBE+U sample from the same crystal batch (Fig. 2(b)) clearly
functional. reveal significant scattering intensity at reciprocal lattice
points with odd indices such as (100) and (300), but not
at the (001)and (003) locations. This would be consis-
tent with the AFM configuration found from DFT, but
it could also imply the existence of a lattice distortion
that would be invisible when using a conventional x-ray
source. In particular, the x-ray scattering cross section
for light elements such as oxygen is very small. Interest-
ingly, a polarized neutron scattering analysis of the (100)
peak at 300 K [12], indicates that RuO2 is both distorted
and antiferromagnetic at room temperature. While the
majority contribution to the (100) peak intensity seems
FIG. 1: (color online) (a) Total energy per Ru in meV, Ru
to be structural in origin, it does contain a magnetic scat-
magnetic moments within the rutile unit cell in µB , band gap
in eV and density of states at the Fermi energy per eV per tering contribution: the non spin flip/ spin flip intensity
Ru, of the anti-ferromagnetic (AFM), the non-magnetic (NM) ratio R for the (200) peak is 12.8(2), whereas R for the
and the ferromagnetic (FM) configurations calculated with (100) peak is 8.0(2) [12]. While this magnetic scattering
PBE+U (b) Atomic and antiferromagnetic structure of bulk contribution equates to only a small moment of about
rutile RuO2 as predicted by DFT and confirmed by neutron 0.05 µB at room temperature [12], the presence of this
diffraction. magnetic moment is unambiguously demonstrated by the
60% larger R of the (200) peak as compared to the (100)
The DFT results are summarized in Fig. 1(a). First, peak. Given that at 300K the (100) peak intensity is close
we find that even for a weak U of 1.2 eV, the Ru moments to being saturated (c.f. Fig. 2(c)), a significant increase
within the rutile unit cell prefer to align antiferromagnet- of the moment towards lower temperatures is unlikely.
ically (see Fig.1(b)). With increasing U , the energy of The existence of room temperature antiferromag-
 3

netism is thus clearly established. However, the nature previous reports [8, 17, 18]. The 30% rise of the magnetic
of the small lattice distortion is not understood. A sym- susceptibility with increased temperature from 300 K to
metry analysis shows [12] that there are only two possi- 1000 K is also in excellent agreement with Fletcher et
ble tetragonal subgroups of the rutile space group that al. [18], the only study that measured up to 1000 K. Our
could produce finite intensities for the forbidden reflec- measurements, however, either produce a clear, broad
tions like (100) and (300). Yet, a full refinement of maximum in the susceptibility or a significant leveling
the unpolarized neutron diffraction data involving over at the highest temperature, consistent with the presence
one hundred reflections, clearly converges to the rutile of short-range ordering. Due to the extreme difficulty in
symmetry and, consequently, overestimates the magnetic measuring small magnetic signals at such high tempera-
moments. [12] Attempts within DFT+U to find another ture, which is near the limit of our instrument capability,
total-energy minimum by breaking the rutile symmetry as well as the possible loss of oxygen, different crystals
were unsuccessful [12]. At this point we are therefore not produce slightly different behavior above 850 K. It is pos-
able to capture the nature of the distortion with a model sible that this changing magnetic behavior above 900 K
that is consistent with the unpolarized neutron and X- is related to the vanishing of the (100) peak and its un-
ray scattering experiments, or the DFT+U simulations, derlying magnetic and/or structural order.
and leave this question for future investigations. The ab-
sence of the (001) reflection in neutron scattering implies
the lack of a structural deformation along the c-direction
and alignment of the magnetic moments along the ru-
tile c-axis (the unpolarized neutron cross section vanishes
when the scattering vector is parallel to the magnetic mo-
ment). We note that the experimental magnetic moment
of ∼0.05 µB from polarized neutron scattering is much
smaller than the one predicted by DFT. Such discrep-
ancies between DFT and experiment are quite common
in metallic antiferromagnets, such as for example the Fe
based superconductors [15, 16], and probably reflect the
inability of the static mean field DFT to capture charge
and spin fluctuations in time and space.
Fig. 2(c) shows the full temperature dependence of FIG. 2: (a) X-ray and (b) unpolarized neutron diffraction
the (100) and (200) diffraction intensities. The (100) data taken at 295 K at HB-3A. (c) Temperature evolution of
peak vanishes near 1000 K while the (200) peak per- the integrated intensity of the nuclear (200) (left) and mag-
sists to higher temperature and diminishes in intensity netic (100) (right) peak measured at HB-3A and HB-3. (d)
according to the Debye-Waller factor. This rules out Magnetic susceptibility of different multigrain RuO2 samples
as a function of temperature.
multiple scattering as the origin of the (100) reflection,
because the temperature dependences of the (100) and
Given the itinerant nature of the conduction electrons
(200) peaks are clearly different. The concave tempera-
in RuO2 , antiferromagnetism possibly originates from a
ture dependence of the (100) peak intensity furthermore
spin density wave instability of the Fermi surface. To ex-
suggests that the magnetic and/or structural ordering is
plore this possibility, we calculate the Lindhard response
fairly short-range. This is consistent with the Lorentzian
function. To this end, we first map the first-principles
lineshape of the (100) peak, as opposed to the Gaussian
electronic structure from a non-magnetic DFT calcula-
lineshape of the (200) reflection [12].
tion (with U = 0) onto a low energy effective model
The presence of room temperature antiferromagnetism
that only contains the Ru 4d orbital degrees of freedom.
goes against the current lore that RuO2 is a Pauli para-
Specifically we perform a Wannier transformation of the
magnet. This general belief probably stems from the
2×5 Ru 4d bands within the [−2, 6.5] eV energy window.
early work by Ryden et al [8] that concluded Pauli
The resulting tight binding Hamiltonian then allows us
paramagnetism from the quadratic temperature depen-
to efficiently compute the Lindhard response χ0 (q) as a
dence of the magnetic susceptibility within 4-300 K. How-
function of the momentum q according to:
ever, older measurements of the magnetic susceptibil-
ity [17, 18] were performed for much larger temperature X n X hs|µkihµk|tiht|νk + qihνk + q|si
ranges up to 1000 K and demonstrated instead a lin- χ0 (q) =
stµν
Eν (k + q) − Eµ (k)
k
ear increase as a function of temperature. We repeated
o
those measurements while ramping the temperature con- f (Eν (k + q)) − f (Eµ (k)) (1)
tinuously from 4 K to 300 K and from 300 K to 1000
K. The results are presented in Fig. 2(d). The value of where s, t/µ, ν denote the orbital/band indices, respec-
1.7 × 10−4 emu/mol (300 K) is in good agreement with tively, k, k + q the momenta, and f (E) the Fermi dis-
 4

cell equals the structural unit cell. This hot spot mecha-
nism differs from the classical example of chromium [20]
in which nesting takes place between large parallel sheets
of Fermi surface, but is analogous to that being proposed
for charge density waves in 2H-NbSe2 [21]. Whether the
Fermi surface hot spots in RuO2 are capable of driving
the antiferromagnetism or rather play an assisting role,
remains an open question - one that is an integral part
of the longstanding debate on itinerant versus localized
magnetism in metallic systems [22, 23].
The discovery of AFM in RuO2 and particularly its
relatively high Néel temperature (≥300K) is significant
because metallic AFM oxides are rare [24, 25], and their
ordering temperatures are generally low. For example,
within the 3d series, CaCrO3 and Sr0.99 Ce0.01 MnO3 have
a TN of 90 K [26] and 220 K [27], respectively. In the 4d
series, the ruthenates Ca3 Ru2 O7 and Na-doped CaRuO3
(TN = 70 K) display antiferromagnetism at TN = 56
and 70 K, respectively, significantly lower than that of
FIG. 3: (color online) (a) Lindhard response function χ0 (q) RuO2 [28, 29], and they are borderline insulating. In-
in eV−1 . (b) Fermi surface. (c) Momentum resolved contri- deed, the recent discoveries of AFM with high TN in
bution to χ0 (q = (2π, 0, 0)). (d) Band structure at ky = 0 4d transition metal oxides were made in semiconduct-
and kz = 0.5π along the kx direction. Black arrow in various
panels indicates one of eight nested “hot spots”.
ing SrRu2 O6 (TN = 563 K) [5] and SrTcO3 (TN = 1023
K) [30]; the latter is theoretically predicted to be insulat-
ing. [2, 30] While the debate on itinerant versus localized
magnetism in metallic systems [22, 23] shows that it is
tribution function at energy E. As shown in Fig. 3(a), difficult to determine the role of itineracy in AFM order,
the response function χ0 is peaked at q = (2π, 0, 0) and the robust metallicity, small moment, and high magnetic
q = (0, 2π, 0) with a value of roughly 1.4 eV−1 .Therefore ordering temperature of RuO2 places it in a regime that
within the random phase approximation [19] the inter- was hitherto not accessible in transition metal oxides.
acting response function, given by χ = χ0 /(1 − U χ0), The relatively high TN in RuO2 appears to be consis-
will diverge for interactions larger than U ≈(1/1.4)eV) tent with recent explanations of high temperature AFM
driving the system towards a spin density wave instabil- in SrTcO3 [2] and SrRu2 O6 [3]. Here it was argued that
ity. TN maximizes in a regime in which the ratio of the in-
Such a spin density wave (or AFM) modulation would teraction U and the band width W is large enough to
produce magnetic reflections at the (100) and (010) lo- form robust magnetic moments, but small enough to al-
cations in reciprocal space, exactly as predicted by DFT low for significant exchange interactions between those
and observed experimentally. Fig. 3(c) shows the crystal moments. Both high-TN compounds share another im-
momentum resolved contribution to the magnetic suscep- portant feature, namely the existence of a 4d3 electron
tibility, obtained from evaluating the term in the curly configuration. Since at TN SrTcO3 has the ideal per-
bracket in equation (1) as a function of crystal momen- ovskite symmetry (space group Pm3̄m) [30], the three t2g
tum k for the fixed momentum q = (2π, 0, 0). From orbitals are degenerate and thus half-filled. In SrRu2 O6
the heat map in Fig. 3(c) we see that the dominating the RuO6 octahedra are stretched along the c-axis, but
contributions originate from states near (π, 0, π/2) and the C3v symmetry still protects the t2g orbital degener-
symmetry related points. These “hot spots” are located acy [31]. Hence, the 3 t2g orbitals are also half-filled. A
at the neck-shaped regions of the Fermi surface as indi- simple chemical bonding picture by Moriya [32] explains
cated by the black arrows in Fig. 3(b). The energy bands why antiferromagnetism (localized or itinerant) is partic-
near these k-points have very low Fermi velocities (Fig. ularly stable at half filling: the majority spin states on
3(d)) and thus contribute a large density of states to the one magnetic sublattice hybridize with the minority spin
magnetic instability. The hot spots are folded on top of states on the other sublattice, and the resulting “chemical
one another via translation by a reciprocal lattice vector. bond” is most stable at half filling while the stabilization
This can be seen in Fig. 3(d). Note that the bands at the energy is determined by the band width.
hot spot location are doubly degenerate, which is a conse- At first sight, RuO2 does not seem to match this pic-
quence of the rutile symmetry. This degeneracy is lifted ture as it has a 4d4 electron count. However, our orbital
by antiferromagnetism and as a consequence of the hot resolved density of states shows [12] that the 4dx2 −y2 or-
spots being folded on top of each other, the magnetic unit bital is filled with two electrons and resides below EF ,
 5

leaving the remaining two t2g orbitals half filled. Hence Trivedi and J.-Q. Yan, Phys. Rev. B, 92, 100404(R)
the specific crystal field splitting of the edge-shared oc- (2015).
tahedra in the rutile structure ensures that the 4dxz and [4] G. Cao, C. S. Alexander, S. McCall, J. E. Crow and R.
4dyz t2g orbitals that are relevant for the AFM order are P. Guertin, Materials Science and Engineering B63, 76
(1999).
formally half filled, similar to SrTcO3 and SrRu2 O6 . An [5] C. I. Hiley, M. R. Lees, J. M. Fisher, D. Thompsett, S.
important distinction, however, is that RuO2 is a good Agrestini, R. I. Smith, and R. I. Walton, Angew. Chem.
metal whereas SrTcO3 is theoretically predicted to be in- Int. Ed. 53, 4423 (2014).
sulating [2, 30] and SrRu2 O6 has been determined to be [6] L. F. Mattheiss, Phys. Rev. B 13, 2433 (1976).
semiconducting from resistivity measurements. [5] The [7] H. Schäfer, G. Schneidereit and W. Gerhardt, Z. Anorg.
unique combination of good metallicity and high tem- Allgem. Chem, 319, 327 (1963).
perature AFM in RuO2 will allow for a more complete [8] W. D. Ryden and A. W. Lawson, J. Chem. Phys. 52,
6058 (1970).
benchmarking of theoretical models describing the inter- [9] H. Over, Chem. Rev. 112, 3356 (2012).
play between magnetism and metallicity in oxide mate- [10] S. Trasatti, Electrodes and Conductive Metallic Oxides,
rials. Elsevier, New York (1980).
Our discovery of antiferromagnetism in a strongly [11] E. Torun, C. M. Fang, G. A. de Wijs, and R. A. de Groot,
metallic binary oxide material also calls for the reevalua- J. Phys. Chem. C 117, 6353 (2013).
tion of the magnetic properties of other 4d and 5d metal- [12] See supplemental material.
[13] Z. Hu, H. von Lips, M. S. Golden, J. Fink, G. Kaindl,
lic oxide systems. Many of these materials already are F. M. F. de Groot, S. Ebbinghaus and A. Reller, Phys.
of technological importance, often as catalyst or other Rev. B 61, 5262 (2000).
chemical applications, but the existence of itinerant an- [14] Y. Ping, G. Galli and W. A. Goddard, J. Phys. Chem. C
tiferromagnetism in this class of materials would open up 119, 11570 (2015).
a new realm of possibilities, specifically in light of recent [15] J. Phys. Con. Mat. 26, 473202 (2014).
developments in antiferromagnetic-metal spintronics. [33] [16] Y.-T. Tam, D.-X. Yao and W. Ku, Phys. Rev. Let. 115,
Here it may be needed to enhance the magnetic proper- 117001 (2015).
[17] A. N. Guthrie and L. T. Bourland, Phys. Rev. 37, 303
ties, such as the moment, via e.g. alloy substitution or (1931).
dimensional confinement. [18] J. M. Fletcher, W. E. Gardner, B. F. Greenfield, M. J.
TB and PCS contributed equally to this work. We Holdoway and M. H. Rand, J. Chem. Soc. A 653 (1968).
thank Veerle Keppens for the use of her laboratory equip- [19] S. Graser, T. A. Maier, P. J. Hirschfeld and D. J.
ment. The research was supported by the U.S. Depart- Scalapino, New J. of Phys. 11, 025016 (2009).
ment of Energy, Office of Science, Basic Energy Sci- [20] E. Fawcett, Rev. Mod. 60 209 (1988).
[21] S.V. Borisenko, A. A. Kordyuk,V. B. Zabolotnyy, D. S.
ences, Materials Sciences and Engineering Division (TB,
Inosov, D. Evtushinsky, B. Büchner, A. N. Yaresko, A.
PCS, OD, YW, PRCK, HHW). Work by TAM (response Varykhalov, R. Follath, W. Eberhardt, L. Patthey, and
function calculation) was performed at the Center for H. Berger, Phys. Rev. Lett. 102, 166402 (2009).
Nanophase Materials Sciences, a DOE Office of Science [22] E. P. Wohlfarth, J. Mag. Mag. Mat. 7, 113 (1978).
user facility. HDZ (crystal growth, XRD and low tem- [23] T. Moriya and Y. Takahashi, Ann. Rev. Mater. Sci. 14,
perature susceptibility measurements) acknowledges sup- 1 (1984).
port from NSF-DMR-1350002. MRK (high temperature [24] P. A. Bhobe, A. Chainani, M. Taguchi, R. Eguchi, M.
Matsunami, T. Ohtsuki, K. Ishizaka, M. Okawa, M.
susceptibility measurements) acknowledges support from
Oura, Y. Senba, H. Ohashi, M. Isobe, Y. Ueda and S.
the Gordon and Betty Moore Foundations EPiQS Ini- Shin, Phys. Rev. B 83 , 165132 (2011).
tiative through Grant GBMF4416. Research at ORNL’s [25] G. Zhang, Y. Wang, Z. Cheng, Y. Yan, C. Peng, C.
High Flux Isotope Reactor (HBC, MM, SXC) was spon- Wanga and S. Dong, Phys.Chem.Chem.Phys. 17, 12717
sored by the Scientific User Facilities Division, Office of (2015).
Basic Energy Sciences, US Department of Energy. This [26] A. C. Komarek, S.V. Streltsov, M. Isobe, T. Möller, M.
research used resources of the National Energy Research Hoelzel, A. Senyshyn, D. Trots, M. T. Fernández-Dı́az,
T. Hansen, H. Gotou, T. Yagi, Y. Ueda, V. I. Anisimov,
Scientific Computing Center, a DOE Office of Science
M. Grüninger, D. I. Khomskii, and M. Braden, Phys.
User Facility supported by the Office of Science of the Rev. Lett. 101, 167204 (2008).
U.S. DOE under Contract No. DE-AC02-05CH11231. [27] H. Sakai, S. Ishiwata, D. Okuyama, A. Nakao, H. Nakao,
Y. Murakami, Y. Taguchi, and Y. Tokura, Phys. Rev. B
82, 180409(R) (2010).
[28] G. Cao, S. McCall, J. E. Crow and R. P. Guertin, Phys.
Rev. Lett. 78, 1751 (1997).
[1] H.-D. Kim, H.-J. Noh, K. H. Kim and S.-J. Oh, Phys. [29] M. Shepard, G. Cao, S. McCall, F. Freibert, and J. E.
Rev. Lett. 93, 126404 (2004). Crow, J. App. Phys. 79, 4821 (1996).
[2] J. Mravlje, M. Aichhorn and Antoine Georges, Phys. [30] E. E. Rodriguez, F. Poineau, A. Llobet, B. J. Kennedy,
Rev. Lett. 108, 197202 (2012). M. Avdeev, G. J. Thorogood, M. L. Carter, R. Seshadri,
[3] W. Tian, C. Svoboda, M. Ochi, M. Matsuda, H. B. Cao, D. J. Singh and A. K. Cheetham, Phys. Rev. Lett. 106,
J.-G. Cheng, B. C. Sales, D. G. Mandrus, R. Arita, N. 067201 (2011).
 6

[31] D. Wang, W.-S. Wang, and Q.-H. Wang, Phys. Rev. B, Kurosaki, M. Yamada, H. Yamamoto, A. Nishide, J.
92, 075112 (2015). Hayakawa, H. Takahashi, A. B. Shick and T. Jungwirth,
[32] T. Moriya, Sol. St. Comm. 2,239 (1964). Nat. Mat. 10, 347 (2011).
[33] B. G. Park, J. Wunderlich, X. Martı́, V. Holý, Y.

Supplemental Material

CRYSTAL GROWTH

Crystals of RuO2 with the rutile structure were obtained using vapor transport growth in a quartz tube. [1] The
starting material (> 99.95% anhydrous RuO2 powder) was heated to 1350 ◦ C in flowing oxygen, producing RuO3 gas
that was allowed to condense into RuO2 at the cold end of the tube at 1150 ◦ C. Crystals with sizes ranging from 0.5
to 1.5 mm were obtained.

X-RAY DIFFRACTION

FIG. 4: Measured (Y obs) and calculated (Y calc) X-Ray diffraction data, their difference, as well as the expected Bragg
reflection positions for a perfect rutile lattice.

Room temperature X-ray diffraction (XRD) patterns on crushed single crystals were measured with a HUBER
X-ray powder diffractometer. Structural refinements were performed using the software package FullProf Suite [3].
All peaks can be indexed with a rutile structure with space group P 42 /mnm. The mismatch in the Bragg peak
intensities is due to the preferred orientation of grains, which is common for crushed crystal samples measured in the
flat-plate geometry.

NEUTRON DIFFRACTION

Single crystal unpolarized neutron diffraction was performed at the HB-3A Four-Circle Diffractometer (FCD)
equipped with a 2D detector at the High Flux Isotope Reactor (HFIR) at ORNL. A neutron wavelength of 1.005
Å from a bent perfect Si-331 monochromator [2] was used for data collection. The refinement was performed by the
FullProf Suite [3]. A total of 118 reflections was measured by rocking curve scans at 295 K; 89 symmetry allowed
reflections were used for structure refinement within the space group P 42 /mnm. The structural parameters of RuO2 ,
measured at 295 K, are listed in Table I. Given that the unit cell is tetragonal, a symmetry analysis shows that there
 7

are only two possible subgroups (space group 77 and 81) that could show forbidden peaks at the (100) and (300)
locations in reciprocal space [4]. Yet, when using these lower subgroup symmetries, the forbidden reflections still
could not be fitted to the experimental data. Neutrons diffraction intensities were checked at ( 21 12 12 ) and ( 12 10), but
no peak signals were observed here. A G-type antiferromagnetic structure on the Ru sublattice with the magnetic
moments pointing along the c-axis can fit the observed (100) and (300) reflections, and is consistent with the (001)
reflection being absent. The overestimated magnetic moment extracted from this analysis is 0.23 µB per Ru atom.

TABLE I: The structure parameters of RuO2 measured at 295 K by single crystal neutron diffraction at HB-3A. The space group
is P 42 /mnm, a=b= 4.492(2) Å, c=3.1061(15) Å, α=β=γ=90o . Rf =0.0231. χ2 =4.31. The atomic displacment parameters U
have units of Å2 .
atom type site x y z U
Ru1 Ru 2a 0 0 0 0.0021(4)
O1 O 4f 0.30558(17) 0.30558(17) 0 0.0040(4)

FIG. 5: The (100) (left) and (200) (right) Bragg reflections measured using polarized neutron diffraction with the flipper on
and off measured at HB-1. The peaks measured with the flipper off are fit by both a Gaussian (solid line) and a Lorentzian
(dotted line for the (100) peak, and by a Gaussian for the (200) peak. The inset in the left panel shows a magnification of the
tail of the (100) peak measured with the flipper off, revealing that a Lorentzian function fits the (100) lineshape better than a
Gaussian function.

Polarized neutron diffraction experiments were carried out on a thermal neutron triple-axis spectrometer HB-
1 installed at the High Flux Isotope Reactor (HFIR) at ORNL. Heusler alloy (111) crystals were used as
monochromator and analyzer. Neutrons with an incident energy of 13.5 meV were used, together with a horizontal
collimator sequence of 480 –800 –S–800 –2400 . The energy resolution is ∼1.5 meV. Contamination from higher-order
beams was effectively eliminated using PG filters. While the polarized and unpolarized neutron data were recorded
from different samples on different beam lines, the ratio of the (100) and (200) peak intensities were consistent.
To extract the integrated intensities of the measured (200) and (100) Bragg reflections, we attempted to fit the
peaks with a Gaussian and a Lorentzian lineshape. In the left panel of Fig. 5, the results are shown for the (100) peak.
For the flipper off peak (open symbols), the dotted line is a Gaussian fit and the solid line is a Lorentzian fit. It is
clear that the measured peak shape contains tails that are not captured by the Gaussian fit. This is also highlighted in
the inset of the left panel, where the error bars of the measured intensity are also included; they are smaller than the
size of the symbols used for the datapoints. Based on these findings we used a Lorentzian fit to extract the integrated
intensity of the measured (100) peak. The (200) peak instead is fit well with a Gaussian curve, as shown in the right
 8

panel of Fig. 5. Note that the Lorentzian shape of the (100) peak suggests a glassy or short-range ordered state.
Using these fits, a reference flipping ratio of 12.8(2) was measured at the nuclear (200) reflection, corresponding
to an 86% polarization of the beam. In order to determine the magnetic contribution to the (100) reflection, we
determined the intensities in the flipper on and off channels with the neutron spin parallel to the Q direction. The
ratio of the flipper on and off intensities (Ioff /Ion ) is 8.0(2). Considering the instrumental flipping ratio of 12.8(2),
the magnetic fraction at the (100) reflection is estimated to be about 5%. If instead the poorly fitting Gaussian
function is used to fit the (100) peak, the ratio Ioff /Ion becomes 9.3(2).

If we apply the polarized neutron result, that estimates a 5% magnetic scattering fraction at the (100) reflec-
tion, to the unpolarized neutron diffraction data refinement that resulted √in a perfect rutile structure, then the
ordered magnetic moment will be reduced to from 0.23 µB to 0.05 µB (0.23x 0.05µB ). The remaining 95% intensity
of the (100) reflection is likely caused by additional structural distortions and a larger unit cell is likely required to
fit those reflections. The small number of forbidden reflections did not allow us to solve the distorted structure using
a larger unit cell.

MAGNETIC SUSCEPTIBILITY

The dc magnetic susceptibility measurements were performed on as-grown samples using a magnetic field of 2 T while
heating at a rate of 2 K/minute. The low temperature data were measured using a Quantum Design superconducting
interference device (SQUID) magnetometer on multiple grains that were randomly oriented, having a total mass of
89.8 mg.
The high temperature data were measured using a Quantum Design VersaLab with the high temperature VSM
option on multiple crystals that were randomly oriented with a total mass varying from 11 mg to 42 mg, and on a
collection of rod-like RuO2 crystals that were aligned with the rutile c-axis parallel to the magnetic field.

FIRST PRINCIPLES CALCULATIONS

FIG. 6: Orbital resolved density of states from the non-magnetic PBE+U simulation (U =2eV).

The Density Functional Theory (DFT) calculations are performed using the plane wave projector augmented
wave (PAW) method [5] as implemented in the VASP code [6–8]. For the PBE+U calculations the double counting
correction scheme by Dudarev et al.[9] and a 8 × 8 × 12 k-mesh is employed. Atomic positions and lattice constants
are converged down to 1 meV/Å. The density of states calculations of are performed on a 16 × 16 × 24 k-mesh. The
VASP calculations have been performed without the use of symmetry by using the flag ISYM=0 in the input file. This
is done because there appears to be a bug in the symmetry routine of VASP. Only when the symmetry is switched
off do we obtain equal moments on the symmetry equivalent Ru atoms. The number of plane waves is determined
by an energy cut-off of 500 eV. The Wannier calculations where performed with Wannier90 [10]. The number
of maximal localization steps was set to zero in order to obtain symmetry respecting projected Wannier functions [11].
 9

The orbital resolved density of states is shown in Fig. 6. It shows that the 4dx2 −y2 orbital is fully filled.
This result is quite insensitive to the value of U being used. The filling of the 4dx2 −y2 band varies from 97% to 98%
when using U=0 and U=2eV respectively. The 4dx2 −y2 being filled leaves the remaining two t2g orbitals, 4dxz and
4dyz , half filled. Here we follow the convention of Ref. [12]: the 4dx2 −y2 orbital has its lobes pointing in the (001),
(110) and (110) directions.

TABLE II: Total energy per Ru in meV, value of the Ru magnetic moments within the rutile unit cell in µB , band gap in
eV, density of states at the Fermi energy per eV per Ru, lattice constants in Å and internal lattice parameter x that fixes the
oxygen positions for the non-magnetic (NM), the anti-ferromagnetic (AFM) and ferromagnetic (FM) configurations with the
moments along the a and c crystallographic axes calculated with B3LYP and PBE+U without and with spin orbit coupling
(SOC).

Hybrid functional calculations have been performed using B3LYP with an energy cut-off of 400 eV and a 4 × 4 × 6
k-mesh due to the computationally demanding evaluation of the Fock matrix (especially without the use of symmetry
as mentioned above). For the same reason the lattice structure in the B3LYP calculations was taken from x-ray
diffraction reported in Ref. [13] without relaxing the atomic positions or the lattice constants. Table II(a) shows
that the AFM configuration is reproduced using the B3LYP hybrid functional. However, this functional produces a
gap of 1.1 eV, even larger than that of the PBE+U calculation with U =4 eV. The same AFM configuration has also
been reported in Ref. [14] from a calculation with the B3PW91 hybrid functional but no mention was made whether
this configuration was still properly metallic. The B3LYP hybrid functional used here has the same parametriza-
tion as the B3PW91 functional used in Ref. [14]. In particular both functionals [15, 16] contain 20% of Fock exchange.

To investigative the possibility of a structural distortion we removed the symmetry by moving the atoms 0.4
Å in random directions both in the Ru2 O4 unit cell and in a 2 × 2 × 2 supercell. Upon minimizing the forces we
found the atoms to relax back to the positions dictated by rutile symmetry, both in the non-magnetic U =0 and the
anti-ferromagnetic U =2 eV simulation. We also note that a phonon calculation reported in Ref. [17] did not show
phonons with imaginary frequencies.
 10

From comparing Table II(c) and Table II(f) we see that for the U =2 eV simulations the spin-orbit coupling
induces no qualitative changes. The same is the case for the AFM and NM U =4 eV simulations, but for the FM
U =4 eV simulations the SOC does seem to have a strong effect (c.f. Table II(d)(g)). The minimum U for which the
AFM state can be stabilized slightly increases from U =1.2eV to U =1.3eV upon including SOC in the simulations
(c.f. Table II(b)(e)). The effects of the spin-orbit coupling on the non-magnetic (U =0) band structure, used as input
for the Lindhard response calculations in the manuscript, is shown in the band structure close to the Fermi-level
presented in Fig. 7. We note that upon including SOC in our DFT calculation, the band degeneracy at the “hotspot”
location is lifted. This could explain why in the DFT simulations the AFM ground state energy (relative to the NM
ground state energy) increases upon including SOC as the SOC term will partly lift the degeneracy of the “hotspots”
that contribute to the AFM instability. It will be interesting in future investigations to quantify the effects of the
SOC on the Lindhard response function.

FIG. 7: Comparison of the non-magnetic (U =0) band structure without and with spin-orbit coupling (SOC). The arrow
indicates the “hot spot” region of the band structure, discussed in the manuscript.

FIG. 8: Orbital resolved density of states from the non-magnetic PBE+U simulation (U =2 eV).

The orbital resolved density of states obtained from the non-magnetic PBE+U simulation (U =2 eV) presented in
Fig. 8 shows the strong covalency between the Ru 4d and O 2p orbitals. The density of states and bandstructures in
Fig. 9 and 10 respectively illustrate that magnetism gaps out a large portion of the Ru t2g band complex around the
Fermi-level.
 11

FIG. 9: Comparison of the non-magnetic (NM) and anti-ferromagnetic (AFM) density of states from the PBE+U simulation
(U =2 eV).

FIG. 10: Comparison of the non-magnetic (NM) and anti-ferromagnetic (AFM) band structures from the PBE+U simulation
(U =2 eV).
 12

FIG. 11: AFM unit cell doubled in the out-of-plane (left) and in-plane (right) directions.

Within PBE+U with U=2eV, we explored two other AFM configurations depicted in Fig. 11. For the AFM
configuration corresponding to doubling the unit cell in the out-of-plane/in-plane direction we found the total energy
per Ru to be 11meV/107meV higher compared to that of the single unit cell AFM configuration discussed in the
manuscript.

[1] Y. S. Huang, H. L. Park, and Fred H. Pollak, Mater. Res. Bull. 17, 1305 (1982).
[2] B.C. Chakoumakos, H.B. Cao, F. Ye, A.D. Stoica, M. Popovici, M. Sundaram, W. Zhou, J.S. Hicks, G.W. Lynn and R.A.
Riedel, J. Applied Cryst., 44, 655 (2011).
[3] J. Rodriguez-Carvajal, Physica B 192 55 (1993).
[4] International Tables for Crystallography, http://it.iucr.org/Ab/ch7o1v0001/sgtable7o1o136/.
[5] Blochl, P. E. Phys. Rev. B 1994, 50, 1795317979.
[6] G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).
[7] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
[8] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
[9] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
[10] A. A. Mostofi, J. R. Yates, G. Pizzi, Y. S. Lee, I. Souza, D. Vanderbilt, N. Marzari, Comput. Phys. Commun. 185, 2309
(2014).
[11] W. Ku, H. Rosner, W. E. Pickett, and R. T. Scalettar, Phys. Rev. Lett. 89, 167204 (2002).
[12] V. Eyert, Ann. Phys. (Leipzig) 11, 650 (2002).
[13] C. E. Boman, Acta Chem. Scand. 24, 116 (1970).
[14] Y. Ping, G. Galli and W. A. Goddard, J. Phys. Chem. C 119, 11570 (2015).
[15] P.J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem. 98, 11623 (1994).
[16] A. D. Becke, J. Phys. Chem. 98, 5648 (1993).
[17] K.-P. Bohnen, R. Heid, O. de la Peña Seaman, B. Renker, P. Adelmann, and H. Schober, Phys. Rev. B 75, 092301 (2007).