Quantum criticality and disorder in the antiferromagnetic critical point
  of NiS$_{2}$ pyrite
Quantum criticality and disorder in the antiferromagnetic critical point of NiS2 pyrite
N. Takeshita1, S. Takashima2, C. Terakura1, H. Nishikubo2, S. Miyasaka3, M. Nohara2, Y. Tokura1,4, and H. Takagi1,2
1Correlated Electron Research Center (CERC), National Institute of Advanced
Industrial Science and Technology (AIST), Tsukuba 305-8562, Japan
2Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan
3Department of Physics, Osaka University, Toyonaka 560-0043, Japan
4Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
(Dated: November 11, 2021)
A quantum critical point (QCP) between the antiferromagnetic and the paramagnetic phases was
realized by applying a hydrostatic pressure of ∼ 7 GPa on single crystals of NiS2 pyrite with a
low residual resistivity, ρ0, of 0.5 µΩcm. We found that the critical behavior of the resistivity, ρ,
in this clean system contrasts sharply with those observed in its disordered analogue, NiS2−xSex
solid-solution, demonstrating the unexpectedly drastic effect of disorder on the quantum criticality.
Over a whole paramagnetic region investigated up to P = 9 GPa, a crossover temperature, defined
as the onset of T2 dependence of ρ, an indication of Fermi liquid, was suppressed to a substantially
low temperature T ∼ 2 K and, instead, a non Fermi liquid behavior of ρ, T 3/2-dependence, robustly
showed up.
PACS numbers:
A hallmark of strongly correlated electron systems is
the presence of a rich variety of phases often competing
with each other. When two phases meet with each other
in the T = 0 limit by tuning a phase controlling parame-
ter such as pressure and chemical doping, quantum fluc-
tuations often give rise to exotic states of electrons, which
has been attracting considerable interest in condensed
matter research [1]. One of the most fascinating cases is a
breakdown of the Fermi liquid at magnetic quantum crit-
ical points (QCP) in itinerant magnets, which has been
believed to be captured by self-consistent renormalization
theory [2] and scaling analysis [3, 4]. The onset temper-
ature of Fermi liquid coherence, probed by a quadratic
temperature dependence of resistivity ρ(T ) ∝ T 2, is pre-
dicted to be suppressed by the presence of low lying spin
fluctuations near QCP and, right at QCP, a non Fermi
liquid ground state is realized which manifests itself as a
non-trivial power law behavior of resistivity ρ(T ) ∝ T n
down to the T = 0 limit, where n = 3/2 for antiferro-
magnetic (AF) QCP and n = 5/3 for ferromagnetic (F)
QCP [5]. A V-shaped recovery of Fermi liquid behavior,
T 2-resistivity, around QCP is anticipated as a function
of phase tuning parameter.
The critical behavior of ρ(T ) near the AF QCP in
NiS2−xSex solid solution is a textbook demonstration of
standard theories for QCP. NiS2 crystallizes in the pyrite
structure. Ni is divalent and therefore accommodates two
electrons in doubly degenerate eg orbitals (t
g). Due to
a strong onsite Coulomb repulsion among Ni eg electrons,
the system is a S = 1 Mott insulator [6, 7]. By substitut-
ing S with Se, the effective bandwidth can be increased
due to the increase of p-d hybridization. With increasing
x in NiS2−xSex, the system experiences a weakly first or-
der transition to an AF metal with a collinear spin struc-
ture [8] at x ∼ 0.4 and then a second order transition to
a paramagnetic metal at x = 1.0.
In the AF metal phase of NiS2−xSex, the AF transi-
tion temperature is TN ∼ 90 K at x = 0.5 and, with
increasing x, continuously goes down to T = 0 at x =
1.0, giving rise to a well defined AF QCP. The T 3/2 de-
pendence of ρ(T ), expected for AF QCP, is observed at
least down to 1.7 K at x = 1.0. On going away from x =
1.0, T 2-behavior of ρ(T ) quickly recovers and a V-shaped
region with T 2 resistivity is identified around x = 1.0. It
is known that the application of pressure is equivalent to
Se substitution in that it increases the band width. By
applying pressure P on an AF metal NiS1.3Se0.7, sup-
pression of TN analogous to Se substitiution was indeed
observed and, eventually, QCP was approached with P =
2 GPa [9]. The phase diagram and the critical behavior
of the resistivity in pressurized NiS1.3Se0.7 were essen-
tially identical with Se content x simply replaced with P
Recently, however, there has been growing evidence
that, the above mentioned textbook picture is violated
in a variety of intermetallic systems. In a helical magnet
MnSi [10] and a weak ferromagnet ZrZn2 [11], a non triv-
ial power law behavior of resistivity, ρ(T ) ∝ T 3/2, dom-
inates the resistivity down to a very low temperature,
not only right at the QCP but also over a wide range
of paramagnetic phase. At the AF QCP in CePd2Si2,
with increasing the purity of sample, the exponent of the
power law resistivity was found to deviate significantly
from the standard value of 3/2 [12]. The common feature
among these systems is that they are clean with a low
residual resistivity of ρ0 < 1 µΩcm [10, 11]. In contrast,
in NiS2−xSex solid solution where textbook example of
critical behavior of ρ(T ) is observed, Se substitution in-
herently gives rise to disorder. Indeed, ρ0 of NiS2−xSex
around QCP is as large as several 10 µΩcm. Questions
immediately arise. Does the non-trvial behavior in the
intermetallics represent a generic property of magnetic
http://arxiv.org/abs/0704.0591v1
QCP in clean systems? Does standard behavior of QCP
shows up only when the system is disordered? To tackle
these questions experimentally, we attempted to realize
a “clean” QCP in NiS2−xSex. The parent compound of
NiS2−xSex, NiS2, is pure and presumably clean. If one
can approach the QCP of pure NiS2 by pressure without
relying on Se substitution, a clean analogue of AF QCP
in NiS2−xSex can be explored and the impact of disor-
der on QCP can be captured. Recent progress of high
pressure technique enabled us to do so.
In this Letter, we address the issue of criticality and
disorder by examining the critical behavior of resistivity
of pure NiS2 under pressures. The AF QCP of NiS2 was
reached at ∼ 7 GPa, where the system was found very
clean with a low residual resistivity ρ0 of ∼ 0.5 µΩcm.
Not only right at the QCP but over an entire range of the
paramagnetic phase investigated, the recovery of Fermi
liquid T 2 of ρ(T ) is suppressed substantially to a very low
temperature below ∼ 2 K and non Fermi liquid behavior
with T 3/2 dependence of ρ(T ) dominated. We demon-
strate the drastic influence of disorder on this AF QCP
by contrasting the result with previous pressure work on
NiS1.3Se0.7 with a residual resistivity of 60 µΩcm [9].
NiS2 sample used in this study was prepared by a va-
por transport technique. The resistivity measurement
was performed by a conventional four probe technique
under hydrostatic pressure up to ∼ 10 GPa in a cubic
anvil type pressure system down to 3 K and also in a
modified Bridgman-type pressure cell down to 180 mK.
The results obtained by the two different pressure setups
agreed reasonably in the temperature range of overlap,
indicating a very good homogeneity of pressure. Pres-
sure was calibrated by measuring the superconducting
transition temperature of Pb [13].
The inset of Fig. 1 demonstrates ρ(T ) of NiS2 at rel-
atively low pressures below 4 GPa. With applying pres-
sure, the insulating behavior of ρ(T ) switches into metal-
lic behavior, indicating the occurrence of metal-insulator
transition. In between 2.6-3.4 GPa, we observe a discon-
tinuous jump of resistivity as a function of temperature,
which corresponds to a first order metal-insulator transi-
tion line on the phase diagram in Fig. 1. The discontinu-
ous jump appears to terminate around 200 K, indicating
the presence of a critical end point. In the phase dia-
gram of NiS2−xSex solid solution, the first order phase
line terminates at much lower temperature and is hard
to identify [14]. This difference appears to suggest the
strong influence of disorder on the Mott criticality.
As seen in the inset of Fig. 1, ρ(T ) of pure NiS2 showed
metallic behavior above P = 2.6 GPa. The residual re-
sistivity at the critical point was as low as ∼ 0.5 µΩcm,
demonstrating that the system is indeed very clean. Mag-
netic ordering in the AF metal phase manifests itself as a
very weak but sharp kink in ρ(T ) at TN as indicated by
the arrows. The antiferromagnetic transition tempera-
ture TN thus determined systematically goes down upon
1086420
P  (GPa)
3002001000
T  (K)
0,  2.6,  3.0,  3.2, 
3.3, 3.36, 3.4 GPa
 TNInsulator
100500
T  (K)
4.0 GPa
5.0 GPa
6.2 GPa
7.5 GPa
FIG. 1: The electronic phase diagram of clean NiS2 pyrite as
a function of pressure. PM and AFM denote paramagnetic
metal and AF metal, respectively. The inset shows the tem-
perature dependent resistivity under pressures, P = 0 - 3.4
GPa (left) and P = 4.0 - 7.5 GPa (right).
pressure and approaches T = 0 somewhere around 7-7.5
GPa. No superconductivity was observed between P =
6 and 9.1 GPa down to 180 mK, in spite of the low resid-
ual resistivity. This appears to suggest that realizing an
AF QCP in clean systems alone is not enough to achieve
exotic superconductivity as observed in heavy Fermion
compounds [15, 16, 17, 18] and that additional ingredi-
ents such as Kondo physics must be invoked.
The pressure dependence of TN , determined by the
kink in ρ(T ), together with the first order metal insulator
transition, is summarized as a phase diagram in Fig. 1.
TN appears to decrease almost linearly in contradiction
to (Pc − P )
2/3 dependence expected from self consistent
renormalization theory [5]. Unusual linear suppression of
the magnetic transition temperature was also observed
analogously for a helical magnet MnSi [10] and a weakly
ferromagnet ZrZn2 [11] when the sample is very clean. It
may be interesting to infer that, in these clean system,
the magnetic transition as a function of pressure is re-
ported to be a first order rather than a second order. In
the clean NiS2, we cannot rule out the possibility of a
first order transition at this stage, because ρ(T ) is not
very sensitive to TN near the critical point.
The signature of AF criticality in this clean system was
explored. The inset of Fig. 2 demonstrates ρ(T ) below 30
K, plotted as ρ vs. T 3/2. In the antiferromagnetic phase
at P = 6.2 GPa, ρ - T 3/2 curve is linear above TN but
shows superlinear behavior below TN . The temperature
dependence below TN is found to be approximately T
indicative of the formation of coherent quasi particles. In
the paramagnetic phase above ∼ 7 GPa, however, the ρ
- T 3/2 curve shows a linear behavior down to very low
T (K)
T 3/2
6.8 GPa
2001000
T 3/2 (K3/2)
6.2 GPa
FIG. 2: Temperature dependent part of resistivity ρ−ρ0 as a
function of temperature under pressure above ∼ 7 GPa, where
the system is paramagnetic, plotted as log(ρ− ρ0) vs. log T .
The inset shows ρ vs. T 3/2 plot.
temperature which is expected for the antiferromagnetic
critical point due to low lying spin fluctuations. It is
remarkable to see T 3/2 behavior characteristic of the an-
tiferromagnetic critical point over such a wide range of
pressure from ∼ 7 GPa to ∼ 9 GPa. ρ(T ) is surprisingly
insensitive to the pressure in the paramagnetic region
above 7 GPa and it is hard to find a signature of crit-
icality. This is analogous to those observed in a helical
magnet MnSi [10] and a weakly ferromagnetic magnet
ZrZn2 [11] when the sample is clean, implying that un-
usual critical behavior in clean systems is ubiquitous.
To investigate the details of unusual temperature de-
pendence in the paramagnetic phase further, we plotted
the temperature dependence of ρ− ρ0 as log(ρ− ρ0) vs.
logT in the main panel of Fig. 2. The residual resistivity
ρ0 was determined by extrapolating ρ - T
2 curve to T =
0 limit. We note here that the temperature dependent
part ρ− ρ0 is comparable to ρ0 at ∼ 3 K and, therefore,
the ambiguity originating from the estimate of ρ0 does
not influence the temperature dependence of ρ − ρ0 at
least above 1 K. It is again clear that the slope is appar-
ently smaller than 2 and instead close to 3/2 above ∼
2 K. At the lowest temperatures below ∼ 2 K, however,
the slope becomes steeper and T 2-resitivity appears to
recover eventually below 2 K. This crossover tempera-
ture to T 2-resistivity is again insensitive to pressure and
always 2-3 K up to 9 GPa. Close inspection of data indi-
cates that the crossover temperature is the lowest around
7.5 GPa but only slightly lower than the other pressures.
987654
P  (GPa)
6543210
P  (GPa)
NiS1.3Se0.7
FIG. 3: Contour plot of the exponent n of power low depen-
dence of resistivity on pressure-temperature plane, demon-
strating the criticality observed in the temperature depen-
dence of resistivity. The main panel is data for clean NiS2
and the inset shows data for dirty NiS2−xSex. The Néel tem-
perature determined by resistivity anomaly was shown by the
white line.
This strong suppression of the crossover to T 2 behav-
ior in ρ(T ), over a remarkably wide range of pressure
from ∼ 7 GPa up to ∼ 9 GPa, contrasts sharply with
the observation in Se-doped samples, where the recovery
of T 2 resistivity was clearly observed not only for the
magnetic side but also for the paramagnetic side. As a
function of Se-doping, a metal-insulator transition and
antiferromagnetic critical point occurs at around Se con-
tent x = 0.4 and 1.0, respectively, while ∼ 2.5 GPa and ∼
7 GPa are required to reach a metal-insulator transition
and QCP, respectively. This yields a conversion ratio
of phase controlling parameters, ∼ 0.15 Se/1 GPa. In
this disordered NiS2−xSex, the T
3/2-dependence of ρ(T )
dominates at the QCP of x = 1.0. With further doping
of Se up to x = 1.33 which is equivalent of additional
pressure of ≃ 2 GPa, however, the T 2 resistivity is fully
recovered and can be observed below ∼ 80 K [9]. Analo-
gously, in a Se doped NiS1.3Se0.7 crystal under pressure,
on going from the QCP at P ∼ 2 GPa to P = 4 GPa,
T 2 resistivity recovers quickly and shows up below 80 K
with increase of 2 GPa. These should be compared with
the low crossover temperature of 2-3 K, approximately 2
GPa above the critical point.
To visually illustrate these points, we plotted the ex-
ponent of power law dependence of ρ(T ), n as a contour
map on the pressure-temperature plane in Fig. 3. The
exponent was determined by taking the derivative of the
log(ρ − ρ0) - logT curve in Fig. 2. It is clear that the
V-shaped recovery of Fermi liquid behavior around QCP
is absent in clean NiS2. The recovery can be seen only
on the antiferromagnetic side below 7 GPa, where the
region with n = 2 (T 2) develops below TN . Above the
critical point of P ∼ 7 GPa, it is clear that the n =
1.5 (T 3/2) region predominantly occupies a majority of
the paramagnetic phase. A thin region with a different
color is lying at the T = 0 limit. This corresponds to the
marginal recovery of Fermi liquid behavior below ∼ 2 K.
In the inset of Fig. 3, we have constructed the contour
map also for the NiS1.3Se0.7 data under pressure from a
previous report [7]. Note again the V-shaped recovery on
the temperature scale of 100 K over ∼ 2 GPa pressure.
The remarkable contrast in the critical behavior be-
tween pure NiS2 and NiS2−xSex, visually demonstrated
in Fig. 3, indicates that the influence of disorder on
quantum criticality is surprisingly drastic, since the only
difference between the two systems is the disorder pro-
duced by Se substitution. In NiS1.3Se0.7 solid solution,
the residual resistivity ρ0 is approximately 60 µΩcm,
which is larger than those of pure NiS2 by two orders
of magnitude. When the samples are disordered, we do
see a canonical behavior of the QCP as predicted by stan-
dard theories [3, 4, 5]. To our surprise, once the system
becomes clean, the textbook behavior is gone and the
Fermi liquid coherence seen in ρ(T ) is dramatically sup-
pressed. We should note that the magnitude of ρ(T )−ρ0
is roughly 50 µΩcm in the temperature range below 100
K at around QCP. In the Se doped crystal, inelastic scat-
tering is always weaker than or at most comparable to
elastic scattering due to disorder below 100 K. In the
pure NiS2, in contrast, the same situation, ρ − ρ0 < ρ0
occurs only below 2-3 K, where we observed crossover
to T 2-resistivity. This suggests that disorder might be
controlling the appearance of T 2-resistivity.
One of the plausible scenarios for the strong influence
of disorder and robust non-Fermi liquid behavior might
be a dichotomy of the Fermi surface [19]. It is natural
that a specific part of Fermi surface, a “hot spot”, is
coupled strongly with a critical antiferromagnetic fluctu-
ation with a characteristic wave vector Q. There may
remains a region with well defined quasiparticles free
from critical fluctuations, a cold spot. The transport
is then determined by an interplay of these two contri-
butions at high temperatures but eventually a cold spot
with T 2-dependence should dominate the conduction at
very low temperatures. This phase separation in k-space
might explain in part the unusual temperature depen-
dence observed in pure NiS2 but it is not clear whether
the robustness of non Fermi liquid behavior can be prop-
erly described. In this scenario, the strong influence of
disorder can be naturally understood. The strong elas-
tic scatting should mix up the hot spot and cold spot
and the inelastic scattering therefore becomes effectively
isotropic, which might be close to the situation implic-
itly assumed in standard theories [5]. Another scenario
might be a phase separation and the resultant domain or
bubble formation in real space as discussed in clean MnSi
where the helical spin order disappears discontinuously as
a first order transition [10]. These bubbles and domains
have been proposed to be responsible for the robust non
Fermi liquid behavior in the paramagnetic phase. It is
worth checking the possible first order transition carefully
checking the magnetism at ∼ 7 GPa.
In conclusion, we have demonstrated the sharp con-
trast in the quantum critical behavior of ρ(T ) between
the clean and the disordered systems by examining a sin-
gle crystal of NiS2 with a low residual resistivity of ∼ 0.5
µΩcm. Previously, the V-shaped recovery of Fermi liquid
behavior (T 2-behavior of resistivity) around the antifer-
romagnetic critical point was clearly observed as a func-
tion of pressure and Se content in the dirty NiS2−xSex
systems with ρ − ρ0 < ρ0. In sharp contrast, we found
a robust non Fermi liquid behavior over a wide pressure
range in the paramagnetic side of a QCP in the clean sys-
tem with ρ−ρ0 ≫ ρ0. These results clearly demonstrate
that our understanding of the quantum critical point is
still far from complete and some important ingredient
must be missing.
The authors would like to thank M. Imada and H.
Fukuyama for discussion. This work was partly sup-
ported by a Grant-in-Aid for Scientific Research (No.
18043009) from the Ministry of Education, Culture,
Sports, Science and Technology of Japan and CREST,
Japan Science and Technology Agency (JST).
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