1. No category

Solid State Communications 166 (2013) 60–65
Contents lists available at SciVerse ScienceDirect
Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Critical behavior of the in-plane weak ferromagnet Sr2IrO4
Min Ge a, Lei Zhang b,n, Jiyu Fan c, Changjin Zhang b, Li Pi a,b, Shun Tan a,b, Yuheng Zhang a,b
a
b
c
Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China
High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China
Department of Applied Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 25 January 2013
Received in revised form
4 April 2013
Accepted 13 May 2013
by S. Miyashita
Available online 18 May 2013
Critical behavior of the single-layered Sr2IrO4, which exhibits weak ferromagnetism in the ab plane, is
investigated by the bulk magnetization study. Critical exponents β ¼ 0:284 7 0:008 and γ ¼ 1:726 7 0:004
with the critical temperature TC ¼230.8 70.9 are obtained by the Kouvel–Fisher method, and
δ ¼ 4:443 7 0:008 is generated by the critical isotherm analysis at TC ¼231 K. The obtained critical
exponents of Sr2IrO4 are close to prediction of the theoretical model with fd : ng ¼ f2 : 1g (where d is the
partial dimensionality and n is the spin dimensionality), which indicates one-dimensional ferromagnetic
interaction in the two-dimensional layered material. On the other hand, the critical exponents lie
between the prediction of long-range and short-range models, indicating that the short-range Heisenberg interaction alone cannot totally describe the critical behavior. It is suggested that the Dzialoshinsky–
Moriya interaction caused by rotation of the octahedra and the symmetric anisotropy result into a lower
spatial short-range interaction, which competes with the long-range interaction caused by expansion of
5d orbits.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
A. Magnetically ordered materials
D. Exchange and superexchange
1. Introduction
The iridium-based compounds have triggered great interests
due to the abundant physical phenomena, such as topological
quantum phase transition in Na2IrO3 [1], orbital magnetism in
Ba2IrO3 [2], charge-orbital density wave in IrTe2 [3,4], orbitally
driven Peierls instability in CuIr2S4 [5], the theoretically predicted
topological semimetal in Y2Ir2O7 [6], etc. The layered perovskite
Srn+1IrO3n+1 has been deserved considerable attention due to the
strong spin–orbital coupling and magnetic ordering, especially the
single-layered Sr2IrO4 with n ¼1 [7–11]. The cell of Sr2IrO4 belongs
to K2NiF4-type structure with space group I41/acd [12–15]. A
rotation of about 111 along the c-axis happens to IrO6 octahedra
in each unit cell, which corresponds to a distorted in-plane Ir1-O2Ir1 bond angle [16–18]. The distorted bond angle plays an
important role in the electronic structure. The ground state of
electronic structure in Sr2IrO4 consists of a completely filled band
with total angular momentum Jeff ¼3/2 and a narrow half-filled
Jeff ¼ 1/2 band near the Fermi level, where the Jeff ¼1/2 band is split
into an upper Hubbard band (UHB) and a lower Hubbard band
(LHB) due to on-site Coulomb interactions [17]. In addition, it
almost exhibits antiferromagnetic ordering of effective Jeff ¼1/2
moments below TN ¼240 K [13,19]. However, the Jeff ¼1/2
moments are significantly canted, resulting in weak ferromagnetic
n
Corresponding author. Tel.: +86 551 65595141; fax: +86 551 655951149.
E-mail address: zhanglei@hmfl.ac.cn (L. Zhang).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ssc.2013.05.007
moments of ∼0:1μB =Ir within each IrO2 (ab) plane [13,19]. A Mott
insulating gap between UHB and LHB in the Jeff ¼ 1/2 bands is
induced by the Coulomb repulsion U (about 0.5 eV). Therefore,
magnetic coupling representing the complex spin–orbital states
areanticipated in this spin–orbital coupling dominated compound
[20]. Recent investigations revealed that Sr2IrO4 exhibits a twodimensional Heisenberg antiferromagnetic behavior of Jeff ¼1/2
isospins even in the paramagnetic state, where the estimated
antiferromagnetic coupling constant as large as J∼0:1 eV comparable to the small Mott gap ð o 0:5 eVÞ [7]. It is suggested that
Sr2IrO4 is a unique system in which Slater- and Mott–Hubbardtype behavior coexist [8].
As mentioned above, Sr2IrO4 exhibits a weak ferromagnetic
behavior within the ab plane, which suddenly vanishes at a
pressure of ∼17 GPa with the material retaining insulating behavior
to much higher pressures [21]. In this work, critical behavior of the
weak ferromagnetic Sr2IrO4 is investigated by bulk magnetization.
It is found that the weak ferromagnetic interaction of Sr2IrO4 lies
between the long-range and short-range models with {d: n}¼{2:1}
(where d is the partial dimensionality and n is the spin dimensionality), ambiguously indicating a one-dimensional weak ferromagnetic coupling in the two-dimensional layered material.
2. Experiment
Polycrystalline Sr2IrO4 was prepared by the solid-reaction
method which was described elsewhere [22]. The phase purity
M. Ge et al. / Solid State Communications 166 (2013) 60–65
61
has been checked by the X-ray diffraction (XRD). The magnetic
properties were measured by the Magnetic Properties Measurement System (Quantum Design MPMS). The isothermal magnetization was measured from 0 T to 6.5 T. The magnetization was
performed after the sample was heated well above the critical
temperature TC for long time enough to ensure every curve was an
initial magnetizing curve. The sample was shaped into long
ellipsoid, and the magnetic field was applied along the long axis
to decrease the demagnetization energy.
3. Results and discussion
Fig. 1 shows the temperature dependence of magnetization
[M(T)] under H ¼0.2 T. A magnetic phase transition from paramagnetic (PM) state to weak ferromagnetic (WFM) state was
detected with temperature decreasing. In addition, a divergency
appears between the zero-field-cooling (ZFC) and field-cooling
(FC) curves, which means spin canting behavior [22]. The inset of
Fig. 1 depicts dM/dT vs T of the ZFC curve, where TC∼231 K was
determined at the peak of the curve. Isothermal magnetization
around TC was measured at intervals of 2 K to investigate the
critical behavior, as given in Fig. 2(a). In high field region, the effect
of charge, lattice, and orbital degrees of freedom is suppressed in a
ferromagnet and the order parameter can be identified with the
macroscopic magnetization [23]. Fig. 2(b) gives the Arrott plot of
M2 vs H/M. Generally, M2 vs H/M should be a series of parallel
straight lines in the high field range in the Arrott plot. The
intercept of the M2 as a function of H/M on the H/M-axis is
negative above TC, while it is positive below TC. The line of M2 vs H/
M at TC should pass through the origin. According to the criterion
proposed by Banerjee [24], the order of the magnetic transition
can be determined from the slope of the straight line: the positive
slope corresponds to the second order transition while the
negative to the first order one. Apparently, the positive slope of
the M2 vs H/M implies that the PM-WFM transition is a second
order one, as shown in Fig. 2(b). However, all curves in the Arrott
plot are nonlinear and show curvature downward even in the high
field region, which indicates that the long-range Landau meanfield theory with β ¼ 0:5 and γ ¼ 1:0 are not satisfied for Sr2IrO4
according to Arrott–Noakes equation of state ðH=MÞ1=γ ¼
ðT−T C Þ=T C þ ðM=M 1 Þ1=β [25]. Hence, a modified Arrott plot should
be employed to obtain the critical parameters.
Is is well known that for a second order ferromagnetic phase
transition, its critical behavior can be studied through a series of
critical exponents. In the vicinity of a second order magnetic phase
transition, the divergence of correlation length ξ ¼ ξ0 jðT C −TÞ=T C j−ν
leads to universal scaling laws for the spontaneous magnetization
MS and initial susceptibility χ 0 . The mathematic definitions of the
Fig. 1. (Color online) Temperature dependence of magnetization [M(T)] for Sr2IrO4
(the inset shows the dM/dT vs T of the ZFC curve).
Fig. 2. (Color online) (a) Initial isothermal magnetization around TC measured at
intervals of 2 K. (b) The Arrott plot of M2 vs H/M.
exponents from magnetization can be described as [26,27]
M S ðTÞ ¼ M 0 ð−εÞβ ;
ε o 0; T o T C
χ 0 ðTÞ−1 ¼ ðh0 =M 0 Þεγ ;
M ¼ DH1=δ ;
ε 4 0; T 4T C
ε ¼ 0; T ¼ T C
ð1Þ
ð2Þ
ð3Þ
where ε ¼ ðT−T C Þ=T C is the reduced temperature; M0/h0 and D are
critical amplitudes. Parameters β (associated with MS), γ (associated with χ 0 ) and δ (associated with TC) are critical exponents.
Four kinds of trial models of 3D-Heisenberg model (β ¼ 0:365,
γ ¼ 1:386), tricritical mean-field model (β ¼ 0:25, γ ¼ 1:0), 3D-Ising
model (β ¼ 0:325, γ ¼ 1:24), and 3D-XY model (β ¼ 0:345,
γ ¼ 1:316) are used to make a modified Arrott plot, as given in
Fig. 3. All these models yield quasi-straight lines in high field
region, and the line of M 1=β vs ðH=MÞγ at TC just passes through the
origin. In order to distinguish which model is the best, the
normalized slope (NS), which is defined as NS ¼S(T)/S(TC) (where
S(T) is the slope of M 1=β vs ðH=MÞ1=γ , are plotted in Fig. 4. In an ideal
model, all values of NS should be equal to ‘1’ because the modified
Arrott plot should consist of a series of parallel straight lines. For
Sr2IrO4, NS of the 3D-Heisenberg model is the one close to ‘1’
mostly, and the line of M2 vs H/M at 231 K passes through the
origin. Moreover, Jackeli et al. and Katukuri et al. suggested a
62
M. Ge et al. / Solid State Communications 166 (2013) 60–65
Fig. 3. (Color online) Isotherms of M 1=β vs ðH=MÞ1=γ with (a) 3D-Heisenberg model. (b) Tricritical mean-field model. (c) 3D-Ising model. (d) 3D-XY model.
Fig. 4. (Color online) The normalized slopes (NS) as a function of temperature
defined as NS ¼ S(T)/S(TC ¼ 231 K).
short-range antiferromagnetic Heisenberg interaction theoretically [28,29]. Therefore, the short-range 3D-Heisenberg model is
chosen as the starting model to generate the initial MS and χ −1
0 .
As can bee seen, the initial MS and χ −1
0 are obtained by the
linear extrapolation from high field region to the intercepts. Then,
a set of β and γ was yielded, which is used to reconstruct a new
modified Arrott plot. Then, a new set of β and γ can be obtained.
This procedure was repeated until that β and γ do not change. The
last obtained MS and χ −1
0 are shown in Fig. 5(a). According to Eqs.
(1) and (2), it is obtained from the modified Arrott plot that
β ¼ 0:279 7 0:002 with TC ¼230.4 70.2 and γ ¼ 1:749 7 0:004 with
TC ¼ 230.3 70.1. One can see that TC obtained from the modified
Arrott plot is in good agreement with that obtained from the M(T)
curve. Moreover, β and γ obtained by this iteration method are
independent of the starting model.
More accurately, critical exponents β and γ can be determined
according to the Kouvel–Fisher (KF) method [30]
M S ðTÞ
T−T C
¼
dM S ðTÞ=dT
β
ð4Þ
Fig. 5. (Color online) The MS and χ −1
0 vs T with the fitting curves. (b) KF plot for MS
(left) and χ −1
0 (right) (solid lines are fitted).
χ −1
0 ðTÞ
−1
dχ 0 ðTÞ=dT
¼
T−T C
γ
ð5Þ
M. Ge et al. / Solid State Communications 166 (2013) 60–65
63
−1
According to the KF method, MS(T)/[dMS(T)/dT] and χ −1
0 ðTÞ=½dχ 0
ðTÞ=dT are as linear functions of temperature, and the slopes are
1=β and 1=γ, respectively. Fig. 5(b) plots MS(T)/[dMS(T)/dT] and
−1
χ −1
0 ðTÞ=½dχ 0 ðTÞ=dT vs T, where the exponents are obtained as
β ¼ 0:284 7 0:008 with TC ¼ 230.870.9 and γ ¼ 1:726 70:04 with
TC ¼230.570.2. The exponents obtained by the KF method are
consistent with those generated by the modified Arrott plot, indicating
reliability of these critical exponents.
As confirmation, the critical exponents can be tested according
to the prediction of the scaling hypothesis. In the critical asymptotic region, the magnetic equation can be written as [27]
MðH; εÞ ¼ εβ f 7 ðH=εβþγ Þ
ð6Þ
where f 7 are regular functions with f+ for T 4 T C and f− for T o T C .
Eq. (6) indicates that MðH; εÞε−β vs Hε−ðβþγÞ forms two universal curves
for T 4 T C and T o T C , respectively. The isothermal magnetization
around TC is plotted as this prediction of the scaling hypothesis in
Fig. 6, with the log–log scale in the inset of Fig. 6. All experiment data
collapse onto two independent curves. The obedience of Eq. (6) over
the entire range of the normalized variables further indicates the
reliability of these obtained critical exponents.
The critical exponent δ can be determined by the critical
isotherm analysis from M(H) at TC following Eq. (3). It is determined TC ¼231 K from the obtained critical exponents. Thus, the
isothermal magnetization at T¼ 231 K is given in Fig. 7, where the
inset plots the log–log scale. The lg(M)–lg(H) relation yields
straight line at higher field range ðH 41:5 TÞ with the slope 1=δ,
where δ ¼ 4:443 7 0:008 is obtained. According to the statistical
theory, these three critical exponents should agree with the
Widom scaling relation δ ¼ 1 þ γβ−1 [31]. Thus, there is
δ ¼ 7:269 70:04 from Fig. 5(a) and δ ¼ 7:077 7 0:09 from Fig. 5
(b). However, δ obtained from the Widom scaling law deviates
from that generated by the critical isotherm analysis. It should be
mentioned that the Widom scaling law is well obeyed for a
ferromagnet with fully polarized spins. Nevertheless, Sr2IrO4 is a
canted antiferromagnetism with incomplete polarized spins in the
ab plane [22], which is very different from the conventional
ferromagnet with spin–spin interaction. The incomplete polarization of spins in Sr2IrO4 may be the reason of this discrepancy. In
addition, the strong spin–orbital coupling is suggested to renormalize the critical exponents significantly [33]. As we know, the
strong spin–orbital coupling of Sr2IrO4 is as large as λ∼380 meV,
which may be another fact causing the discrepancy of the critical
exponents from theoretical values [32]. Besides, it is demonstrated
that the weak ferromagnetism is caused by the Dzialoshinsky–
Moriya (DM) interaction due to the lattice distortion, which can
also result into the deviation from the Widom scaling law [15]. In a
word, the weak ferromagnetic mechanism in Sr2IrO4 is very
Fig. 6. (Color online) Scaling plot below and above TC using β and γ determined by
the KF method (only several typical curves are shown).
Fig. 7. (Color online) Isothermal M(H) at TC ¼ 231 K (the inset shows the same plot
in log–log scale and the solid line is the linear fitting).
different from that in a common ferromagnet, which results into
the discrepancy from the Widom scaling law.
The critical exponents of Sr2IrO4, as well as several theoretical
models [34–36], are listed in Table I for comparison. As you can
see, although the starting model is 3D-Heisenberg model, the final
critical exponents by the iteration method are independent of the
starting parameters. Moreover, the obedience of the scaling
equations also indicates the reliability of these obtained critical
exponents. However, the critical exponents of Sr2IrO4 does not
belong to any single universality class.
As we know, for a homogeneous magnet, the universality class
of the magnetic phase transition depends on the exchange interaction J(r). A renormalization group theory analysis suggests that
the long-range exchange interaction decays as JðrÞ∼r −ðdþsÞ (where d
is the spatial dimension and s is a positive constant), and the
short-range exchange interaction decays as JðrÞ∼e−r=b (where b is
the spatial scaling factor) [34,37,38]. The long or short range of
spin interaction depends on the s. According to the theory of
renormalization group, s is determined by [34,40,39]
"
#
2Gð2dÞð7n þ 20Þ
4n þ 2
8ðn þ 2Þðn−4Þ
1þ
Δs þ
1þ
ð7Þ
Δs2 ¼ γ
2
dnþ8
ðn−4Þðn þ 8Þ
d ðn þ 8Þ2
where Δs ¼ ðs− 2dÞ, Gð2dÞ ¼ 3− 14 ð2dÞ2 , d is the spatial dimensionality, n
is the spin dimensionality. The short-range Heisenberg model is
valid when s 4 2, while the long-range mean-field model is
satisfied when s o3=2. In present case, based on the experimental
γ ¼ 1:726, it can be obtained that s ¼ 1:582 following Eq. (7). It can
be seen that s lies between the long-range and short-range
interaction ð3=2 o s o 2Þ. In addition, one can see that although
γ ¼ 1:726 is close to the short-range interaction with {d:n}¼{2:1}
(β ¼ 0:125, γ ¼ 1:75 and δ ¼ 15:0), β ¼ 0:284 approaches the longrange interaction with {d:n}¼{2:1} (β ¼ 0:298, γ ¼ 1:392 and
δ ¼ 5:67) (as listed in Table 1). Here d¼ 2 indicates that the spatial
dimensionality of Sr2IrO4 belongs to two-dimensional materials,
and n¼1 generally implies uniaxial or Ising-like magnetic interaction. The experimental results ambiguously indicate that the critical
behavior of Sr2IrO4 lies between long-range and short-range interaction with {d:n}¼{2:1}, which demonstrates one-dimensional
weak ferromagnetic coupling in the two-dimensional layered
Sr2IrO4. In addition, it is found that the experimental critical
exponents are slight smaller that the theoretical values. In fact, for
spin–spin and density–density correlations, the critical exponents
decrease with the growth of the spin–orbital interactions [33].
The weak ferromagnetism of Sr2IrO4 is very different from the
conventional ferromagnetism. As we know, the weak ferromagnetism of Sr2IrO4 correlates intimately with the antiferromagentism, which makes this system can also be regarded as canted
antiferromagnetism. Previously theoretically investigations have
64
M. Ge et al. / Solid State Communications 166 (2013) 60–65
Table 1
Critical exponents of Sr2IrO4 as well as different theoretical models (MAP ¼ modified Arrott plot; KF¼Kouvel–Fisher method; CI ¼ critical isotherm analysis).
Material
Ref.
Method
TC (K)
β
γ
δ
Sr2IrO4
Sr2IrO4
Sr2IrO4
{d:n} ¼{2:1}
This work
This work
This work
MAP
KF
CI
230.4 70.2
230.8 70.9
231
0.2797 0.002
0.284 7 0.008
–
1.7497 0.004
1.726 70.004
–
–
–
4.4437 0.008
2D short-range ðJðrÞ∼e−r=b Þ
{d:n} ¼{2:1}
[34]
Theory
–
0.125
1.75
15.0
2D long-range ðJðrÞ∼r −ðdþsÞ Þ
{d:n} ¼{3:1}
3D-Ising model
{d:n} ¼{3:2}
3D-XY model
{d:n} ¼{3:3}
3D-Heisenberg model
Mean-field model
Tricritical mean-field model
[34]
Theory
–
0.298
1.393
5.67
[35]
Theory
–
0.325
1.24
4.82
[35]
Theory
–
0.345
1.316
4.80
[35]
[35]
[36]
Theory
Theory
Theory
–
–
–
0.365
0.5
0.25
1.386
1.0
1.0
4.8
3.0
5.0
demonstrated that the antiferromagentism of Sr2IrO4 belongs to
the short-range Heisenberg interaction [7,28,29]. Thus, the weak
ferromagnetic interaction tends to be short-range interaction.
However, as one can see from the experimental results that the
short-range interaction alone cannot describe the weak ferromagnetic behavior. In fact, it is concluded that the weak ferromagnetism of Sr2IrO4 originates from the DM interaction from the
structural aspect [15]. Therefore, the total Hamiltonian is written
! !
! ! !
as: H ¼ J S i S j þ J z Szi Szj þ D ð S i S j Þ (where the first item is
the short-range Heisenberg interaction, the second one is symmetric anisotropy, and the third one describes the DM interaction)
[28]. The DM interaction, which gives rise to short ranged but of
lower spatial symmetry, will explain the short-range and low
spatial interaction with {d:n} ¼{2:1} in Sr2IrO4 [38]. Moreover, the
second item of the Hamiltonian, i.e. symmetric anisotropy, also
indicates a lower spatial interaction. Concerning the long-range
interaction, it is suggested that the 5d orbits are more expansive
than 3d or 4d orbits, which tends to forming long-range interaction [14]. Besides, the transport behavior implies negligible longrange Coulomb repulsions between electrons in this temperature
region [14]. The critical behavior of Sr2IrO4 is determined by the
competition of the short-range and long-range interactions.
4. Conclusion
In summary, critical behavior of Sr2IrO4, which exhibits weak
ferromagnetism in the ab plane, is investigated by the bulk
magnetization study. Critical exponents β ¼ 0:284 70:008 and
γ ¼ 1:726 7 0:004 with TC ¼ 230.870.9 are obtained by the KF
method, and δ ¼ 4:443 7 0:008 is generated by the critical isotherm analysis at TC ¼230 K. The obtained critical exponents of
Sr2IrO4 lie between the predictions of long-range and short-range
theoretical models with {d:n}¼{2:1} (where d is the partial
dimensionality and n is the spin dimensionality), which indicates
one-dimensional ferromagnetic coupling in the two-dimensional
layered material. On the other hand, the results indicate the
ferromagnetic interaction distance lies between the long-range
and short-range interaction.
Acknowledgments
This work was supported by the National Natural Science
Foundation of China through Grant nos. 11204288, 11174262 and
11004196, the Postdoctoral Science Foundation of China through
Grant no. 2012M521226, the Knowledge Innovation Program of
the Chinese Academy of Sciences through Grant no. 106CS31121
(Hefei institutes of Physical Science, CAS), and the State Key Project
of Fundamental Research of China through Grant no.
2010CB923403.
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