Signatures of quantum criticality in pure Cr at high pressure.
ABSTRACT The elemental antiferromagnet Cr at high pressure presents a new type of naked quantum critical point that is free of disorder and symmetry-breaking fields. Here we measure magnetotransport in fine detail around the critical pressure, Pc approximately 10 GPa, in a diamond anvil cell and reveal the role of quantum critical fluctuations at the phase transition. As the magnetism disappears and T-->0, the magnetotransport scaling converges to a non-mean-field form that illustrates the reconstruction of the magnetic Fermi surface, and is distinct from the critical scaling measured in chemically disordered CrV under pressure. The breakdown of itinerant antiferromagnetism only comes clearly into view in the clean limit, establishing disorder as a relevant variable at a quantum phase transition.
- SourceAvailable from: 220.127.116.11[Show abstract] [Hide abstract]
ABSTRACT: Quantum criticality emerges when a many-body system is in the proximity of a continuous phase transition that is driven by quantum fluctuations. In the quantum critical regime, exotic, yet universal properties are anticipated; ultracold atoms provide a clean system to test these predictions. We report the observation of quantum criticality with two-dimensional Bose gases in optical lattices. On the basis of in situ density measurements, we observe scaling behavior of the equation of state at low temperatures, locate the quantum critical point, and constrain the critical exponents. We observe a finite critical entropy per particle that carries a weak dependence on the atomic interaction strength. Our experiment provides a prototypical method to study quantum criticality with ultracold atoms.Science 03/2012; 335(6072):1070-2. · 31.20 Impact Factor
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ABSTRACT: One of the common features of unconventional, magnetically mediated superconductivity as found in the heavy-fermions, high-transition-temperature (high-Tc) cuprates, and iron pnictides superconductors is that the superconductivity emerges in the vicinity of long-range antiferromagnetically ordered state. In addition to doping charge carriers, the application of external physical pressure has been taken as an effective and clean approach to induce the unconventional superconductivity near a magnetic quantum critical point (QCP).[2,3] Superconductivity has been observed in a majority of 3d transition-metal compounds,[4-9] except for the Cr- and Mn-based compounds in the sense that the low-lying states near Fermi level are dominated by their 3d electrons. Herein, we report on the discovery of superconductivity on the verge of antiferromagnetic order in CrAs via the application of external high pressure. Bulk superconductivity with Tc ~ 2 K emerges at the critical pressure Pc ~ 8 kbar, where the first-order antiferromagnetic transition at TN = 265 K under ambient pressure is completely suppressed. Abnormal normal-state properties associated with a magnetic QCP have been observed nearby Pc. The close proximity of superconductivity to an antiferromagnetic order suggests an unconventional pairing mechanism for the superconducting state of CrAs. The present finding opens a new avenue for searching novel superconductors in the Cr and other transitional-metal based systems.06/2014;
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ABSTRACT: We describe a technique for making electrical transport measurements in a diamond anvil cell using an alcohol pressure medium, permitting acute sensitivity while preserving sample fidelity. The sample is suspended in the liquid medium by four gold leads that are electrically isolated by a composite gasket made of stainless steel and an alumina-loaded epoxy. We demonstrate the technique with four-probe resistivity measurements of chromium single crystals at temperatures down to 4 K and pressures above 10 GPa. Our assembly is optimized for making high precision measurements of the magnetic phase diagram and quantum critical regime of chromium, which require repeated temperature sweeps and fine pressure steps while maintaining high sample quality. The high sample quality enabled by the quasi-hydrostatic pressure medium is evidenced by the residual resistivity below 0.1 μΩ cm and the relative resistivity ratio ρ(120 K)∕ρ(5 K) = 15.9 at 11.4 GPa. By studying the quality of Cr's antiferromagnetic transition over a range of pressures, we show that the pressure inhomogeneity experienced by the sample is always below 5%. Finally, we solve for the Debye temperature of Cr up to 11.4 GPa using the Bloch-Gruneisen formula and find it to be independent of pressure.The Review of scientific instruments 10/2012; 83(10):103902. · 1.52 Impact Factor
Signatures of quantum criticality
in pure Cr at high pressure
R. Jaramilloa, Yejun Fengb,c, J. Wangc, and T. F. Rosenbaumc,1
aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138;
Argonne, IL 60439; and
bThe Advanced Photon Source, Argonne National Laboratory,
cThe James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637
Edited by Laura H. Greene, University of Illinois at Urbana–Champaign, Urbana, IL, and approved June 28, 2010 (received for review April 13, 2010)
The elemental antiferromagnet Cr at high pressure presents a new
type of naked quantum critical point that is free of disorder and
symmetry-breaking fields. Here we measure magnetotransport
in fine detail around the critical pressure, Pc∼ 10 GPa, in a diamond
anvil cell and reveal the role of quantum critical fluctuations at the
phase transition. As the magnetism disappears and T → 0, the
magntotransport scaling converges to a non-mean-field form that
illustrates the reconstruction of the magnetic Fermi surface, and is
distinct from the critical scaling measured in chemically disordered
Cr∶V under pressure. The breakdown of itinerant antiferromagnet-
ism only comes clearly into view in the clean limit, establishing
disorder as a relevant variable at a quantum phase transition.
antiferromagnetism ∣ spin density waves ∣ electric transport
exotic ground states such as non-Fermi liquid metals and uncon-
ventional superconductors (1). This observation has motivated
several decades of work to understand the physics of magnetic
quantum phase transitions (QPT) (2–7). A substantial part of
the effort has been directed at the materials science challenges
that are inherent to realizing nearly-magnetic states of matter
and to the fine tuning of materials so that the phase transitions
can be probed systematically. The fundamental limitations that
remain are uncertainty over the role of disorder (2, 4, 8), as well
as a predilection for first-order transitions that shroud the quan-
tum critical behavior (3, 5). Recent X-ray measurements identi-
fied a continuous disappearance of magnetic order in the
elemental antiferromagnet Cr near the critical pressure Pc∼
10 GPa, and concurrent measurements of the crystal lattice
across the transition failed to detect any discontinuous change
in symmetry or volume (9). These results identify Cr as a stoichio-
metric itinerant magnet with a continuous QPT—where the
effects of the critical point should be manifest—and present a
rare opportunity to study quantum criticality in a theoretically
tractable system that is free from the effects of disorder. More-
over, the use of hydrostatic pressure as a tuning parameter avoids
the introduction of any confounding symmetry-breaking fields.
For the experimentalist, studying elemental Cr shifts the sig-
nificant technical difficulties from solid state chemistry to high
pressure experimentation. Here we report on high-resolution
measurements of the electrical resistivity and Hall coefficient
of Cr as the system is tuned with pressure in a diamond anvil cell
across Pc. Magnetotransport is a sensitive probe of quantum cri-
ticality and is widely used to identify and characterize quantum
matter (4, 5, 8, 10). At ambient pressure Cr orders antiferromag-
netically at the Néel temperature, TNðP ¼ 0Þ ¼ 311 K. Below
TN, electrons and holes form magnetic pairs and condense into
a spin density wave (SDW), in a process with strong analogies to
the Bardeen-Cooper-Schrieffer (BCS) formulation of electron
pairing in superconductors (11). The quantum critical point
where TN→ 0 can be reached either through applied pressure or
chemical doping. Previous transport measurements of Cr1-xVx,
x ¼ 3.2%, under pressure revealed a wide regime of quantum cri-
tical scaling in this substitutionally disordered system (8). Doping
ompetition between magnetic and nonmagnetic states of mat-
ter in the zero-temperature limit underlies the emergence of
with electron-poor V to near-critical levels lowered the critical
pressure, making Pcaccessible with a conventional clamp cell.
Accessing the QPT in the pure system, on the other hand, re-
quires high sensitivity measurements on submillimeter single
crystals in a diamond anvil cell at low temperature (12, 13).
We present here the results from experimental runs with seven
different samples, including two that were studied in fine detail
in the critical regime. An overview of the resistivity for 0 < P <
10 GPa is shown in Fig. 1A. The Néel transition is marked by a
sharp upturn in the resistivity, ρðTÞ, as the reduction in metallic
carrier density closely tracks the growth of the energy gap, gðTÞ,
just below TN. This data is analyzed by first subtracting the para-
magnetic background resistivity ρPMðTÞ, yielding the normalized
magnetic excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ. This quantity is
then fit to a model function which accounts for the formation
of a BCS-like energy gap below TNand the resulting freezing-
out of carriers. This model function was successfully applied in
an important early study of Cr under pressure by McWhan
and Rice (13). By analyzing Δρ∕ρ (see Methods) we obtain the
phase diagram of Fig. 1B. TNðPÞ evolves exponentially with pres-
sure for P < 7 GPa with the form TNðPÞ ¼ TN;0expð−cPÞ,
TN;0¼ 310.9 ? 0.9 K, c ¼ −0.163 ? 0.001 GPa−1. Above 7 GPa
this BCS-like exponential ground state breaks down as the system
approaches the QPT.
The data analysis in the immediate vicinity of the QPT is pre-
sented in logical progression in Fig. 2. We plot in Fig. 2Athe elec-
trical resistivity measured in fine detail in the quantum critical
regime. For T < 50 K the paramagnetic resistivity displays a
dominant T3dependence. This is demonstrated in Fig. 2B where
we plot ρðT3Þ, and for each pressure we limit the temperature
range to T > TNin order to emphasize ρPMðTÞ. The T3coeffi-
cient b varies by less than 6% between samples and is well
described by metallic transport due to phonon scattering in the
presence of a weakly inelastic nonphonon scattering channel (14).
Theory gives b∕d ¼ ð4.8∕Θ2Þ ¼ 1.74 × 10−5K−2, where Θ is the
Debye temperature (Θ ¼ 525 K (15)) and d is the linear tempera-
ture coefficient of resistivity at high temperature. The coefficient
d is determined from data for T > 315 K at P ¼ 0, and b is de-
termined from data for T < 25 K in the paramagnetic phase at
high pressure. For the sample presented in Fig. 2 we find
b∕d ¼ 1.95 ? 0.15ð10−5K−2Þ, in reasonable agreement with the
theoretical expectation. The T3resistivity in this temperature
range (vs. a T5form) is typical for metallic samples with residual
resistivities ρ0≥ 1 nΩ · cm (14, 16); our single-crystal Cr is
99.996 þ % pure and has ρ0≈ 0.1 μΩ · cm (compared to ρ0≈
1.4 μΩ · cm in critically doped Cr∶V 3.2% (8)). The electron
mean-free path in our samples is estimated to be λ > 400 Å at
base-T for all pressures P < Pc, where λ is calculated from the
Author contributions: R.J., Y.F., and T.F.R. designed research; R.J., Y.F., and J.W. performed
research; R.J. and T.F.R. analyzed data; and R.J., Y.F., J.W., and T.F.R. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: email@example.com.
www.pnas.org/cgi/doi/10.1073/pnas.1005036107PNAS ∣ August 3, 2010 ∣ vol. 107 ∣ no. 31 ∣ 13631–13635
measured Hall mobility. We note that the presence of finite
quenched disorder in our samples is a necessary precondition
for measuring a pressure-dependent residual resistivity. However,
the extremely low level of disorder suggests that pure Cr is a
benchmark for how closely a QPT in a real solid state system
can approach the clean limit.
We plot in Fig. 2C the excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ
calculated from the data in Fig. 2A and the ρPMbackground
(see Fig. 2B and Methods). As P approaches Pc, it is preferable
to analyze data for experimental cuts which are close to perpen-
dicular to the increasingly steep phase diagram TNðPÞ. We extract
such cuts from the data by considering isotherms of Δρ∕ρ. These
isotherms Δρ∕ρjTare then fit to a power law Δρ∕ρjTðPÞ ¼
aðPc;T− PÞβ, convolved with a Gaussian that accounts for the
finite pressure variation across the sample (see Methods: Data
Analysis in the Critical Regime). The 5 K isotherm and fit are
plotted on the projected (P, Δρ∕ρ) plane in Fig. 2C, and a scaling
plot of the data approaching the low-temperature limit is shown
in Fig. 3A. The phase diagram is given by the fit parameters Pc;T,
and the exponent β relates to the breakdown of the SDW energy
gap and the reemergence of nested Fermi surface area; in the
T → 0 limit β directly reflects the critical reconstruction of the
We present in Fig. 3 the resistivity scaling results for the quan-
tum critical regime. The exponent β converges to 0.24 ? 0.01 for
temperatures T ≤ 8 K (Fig. 3B). This exponent speaks to a rapid
reconstitution of the Fermi surface that takes place in a narrow
quantum critical regime, and stands in contrast to the value β ¼
2∕3 which is seen at all temperatures in the pressure-driven quan-
tum critical regime in Cr1-xVx, x ¼ 3.2% (8). β increases with tem-
perature above 8 K, approaching the mean-field value, β ¼ 1∕2,
or perhaps even β ¼ 2∕3, as the quantum critical point recedes
from sight. However, due to limited data density and the difficulty
of modeling ρPMat higher temperatures, we are not able to follow
β to the point at which it settles at a high temperature limit. The
crossover in temperature demonstrated in Fig. 3B is strongly re-
miniscent of the crossover from quantum to classical critical scal-
ing that is expected at finite temperatures in a system of itinerant
fermions (6), although the applicability of the usual Landau-
Ginzburg-Wilson (LGW) critical analysis to the case of nested
Fermi surfaces remains in question (7). The critical phase dia-
gram TNðPÞ ∝ ðPC− PÞγis shown in Fig. 3C. The exponent γ de-
termined from the two samples is 0.55 ? 0.03 and 0.48 ? 0.05,
respectively, giving a best estimate γ ¼ 0.53 ? 0.03, consistent
with the mean-field expectation, γ ¼ 1∕2, also observed for
Cr1-xVx, x ¼ 3.2%.
The critical reconstruction of the nested Fermi surface is
further demonstrated by the Hall coefficient, RHðPÞ. The Hall
effect is acutely sensitive to the quantum critical point, changing
ICDW(P, T < 8 K)
pressure. Data and results shown for all seven samples measured. (A)
Resistivity ρðTÞ. (B) Antiferromagnetic phase diagram TNðPÞ. Black ¼
determined directly fromρðTÞcurves;
X-ray measurements of the CDW diffraction intensity ICDWat low tempera-
ture (9), from which the phase diagram can be calculated using the harmonic
relationship TN∝ ICDW
order, and is anticipated by thermal fluctuations (TF) for T > TN. At low tem-
perature and for pressures P < 7 GPa the SDW is well described by the mean-
field BCS-like theory and the phase diagram evolves exponentially with P (19,
20). For pressures above 9 GPa this mean-field ground state is continuously
suppressed by quantum critical (QC) fluctuations. Red shaded region indi-
cates the quantum critical regime which is the focus of this paper.
Data overview and phase diagram for antiferromagnetic Cr under
Blue ¼ determined indirectly from
1∕4. At low pressure the Néel transition is weakly first
T = 5 K
T3 (104 K3)
both samples, but the different pressure conditions and residual resistivities make it difficult to clearly present raw data for both samples on the same plot. The
pressure colorbar applies to A–C. (A) Resistivity ρðTÞ. (B) ρðTÞ plotted against T3for T < 50 K, with each curve truncated just above TN. Over this temperature
range the paramagnetic background ρPMðTÞ is dominated by the shown T3dependence. (C) The magnetic resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ, calculated from ρðTÞ
and the modeled ρPMðTÞ. Also shown (dashed red line) is the McWhan-Rice fit to the lowest pressure curve at 9.13 GPa, for which TN¼ 37.9 ? 0.03 K and
g0∕kBTN¼ 1.36 ? 0.01 (error bars represent 1-σ variations from the nonlinear fit routine). (C, offset) Data and power law fit to the Δρ∕ρ isotherm at 5 K. The
exponent β ¼ 0.23 ? 0.03 and the Gaussian pressure inhomogeneity is 0.24 GPa (FWHM).
Data for 9 < P < 10 GPa for one of the two samples which were measured in detail in the quantum critical regime. The scaling results are the same for
www.pnas.org/cgi/doi/10.1073/pnas.1005036107Jaramillo et al.
by 300% across the narrow critical regime at low temperature
(Fig. 4). For P < Pc, the data can be described by the scaling form
ing a SDW (17). According to mean-field theory (17, 18), small
deviations from ideal nesting will cause both RHand Δρ∕ρ to
scale linearly with the SDWenergy gap g0in the T → 0 limit. Un-
der the conservative assumption that g0∝ TNthe data indicate
that the gap scales with the mean-field exponent of 1∕2, while
the Hall coefficient and the excess resistivity behave differently.
The non-mean-field scaling which we observe for both RHand
Δρ∕ρ implies that the observed critical behavior is driven not
by the SDW energy gap, but by fluctuations that restore flat sec-
tions of Fermi surface. Moreover, the lengthening of the SDW
ordering wavevector Q through the critical regime, in contrast to
the monotonically decreasing dependence of Q on P for
P < 7 GPa, also has been interpreted as evidence for quantum
critical fluctuations (9).
H∝ ð1 − Δρ∕ρÞ2, as predicted for a flat Fermi surface support-
The complete magnetic phase diagram of Cr can be considered in
three parts (illustrated in Fig. 1B). At low pressures the Néel tran-
sition is weakly discontinuous, and the temperature regime T >
TNis marked by incipient antiferromagnetic fluctuations that go
beyond the mean-field theory of the SDW (19). As a function of
pressure, however, this BCS-like theory accurately describes the
observed exponential dependence of both the phase boundary
TNðPÞ and the saturated order parameter at low temperature
(19, 20). The exponential evolution results from a competition
between exchange energy and bandwidth that is tuned by applied
pressure while preserving the Fermi surface nesting condition
(21). Above 7 GPa the exponential ground state breaks down and
the phase diagram approaches the QPT. Finally, at higher pres-
sure the coherent SDW breaks down, quantum critical fluctua-
tions dominate, and the nested Fermi surface reappears at a
We have shown that the critical scaling of this breakdown is
different in pure Cr than in Cr1-xVx, x ¼ 3.2%, which establishes
that substitutional disorder is a relevant variable at the pressure-
driven QPT. Although the number and identity of relevant vari-
ables are well known across the many categories of classical phase
transitions, the same is not true for their quantum counterparts.
In both the clean and disordered limits we find that the exponent
β is distinct from the exponent γ ¼ 1∕2 that describes the scaling
of the phase boundary, thus suggesting that the QPTs in both pure
and V-doped Cr are driven by fluctuations that couple to the
resistivity (18). However the value of β differs between the two
cases, as does the scaling RHðPÞ ∝ ðPc− PÞα, which is mean-field
in Cr∶V and distinctly non-mean-field with an exponent α close to
1∕4 in pure Cr. V-doping is more efficient at driving the QPT, in
that the phase diagram departs from the exponential curve at a
larger TN(or, equivalently, at a larger SDW coupling constant)
for Cr1-xVxthan for pure Cr under pressure (9). Furthermore,
with V-doping the body centered cubic lattice expands and the
SDW wavevector decreases monotonically, in contrast to the
behavior under pressure. This decrease in Q with V-doping re-
sults from the fact that the band filling varies with electron-poor
doping. However, barring the unrealistic scenario in which pure
Cr remains perfectly nested at all pressures, this change in band
filling is not expected to alter the critical scaling at the QPT. Our
scaling results therefore demonstrate that the distinct micro-
scopic effects of chemical doping (or “chemical pressure”) and
hydrostatic pressure lead to distinct phase transitions, and
γ = 1/2
(∆ρ/ρ)4 (× 10−3)
β = 1/2
β = 1/4
phase boundary TN. (A) Scaling plot of Δρ∕ρ shows that the low-temperature
isotherms are well described by β ¼ 1∕4 and can be differentiated clearly
from the mean-field result, β ¼ 1∕2. (B) The Δρ∕ρ scaling exponent β con-
verges to 0.24 ? 0.01 for T ≤ 8 K. Results are shown for both samples (black
and gray, respectively) measured in detail in the critical regime. There is a
crossover at higher T to a larger exponent, verging towards approaching
either mean-field behavior or the β ¼ 2∕3 found for Cr∶V. (C) The critical
scaling of TN, which is consistent with the mean-field exponent γ ¼ 1∕2
for both samples (blue and red, respectively). A nearly constant offset of
0.12 GPa was found between the phase diagrams measured for the two sam-
ples. This offset characterizes the systematic uncertainty in our experiment,
and although it does not affect error bars in the relative quantity (P-Pc), it
does limit the accuracy to which we can determine Pcitself, which we report
as 9.71 ? 0.08 GPa.
Quantum critical scaling of the magnetic resistivity Δρ∕ρ and the
(RH)−1 (104 C/cm3)
(RH)−1 (104 C/cm3)
measured at T ¼ 4.9 ? 0.4 K. Inset shows R−1
predicted for a flat Fermi surface supporting a SDW; solid line is a linear
fit to the data. Resistivity ρPM∕ρAF¼ 1 − Δρ∕ρ (see Results) was recorded at
5.0 ? 0.1 K, and the plotted range is limited to P < Pc. The scaling relationship
is consistent with α ¼ β ≈ 1∕4, where the exponent α describes the T → 0
quantum critical scaling of the Hall coefficient.
Overview and critical scaling of the inverse Hall coefficient R−1
H∝ ðρPM∕ρAFÞ2scaling which is
Jaramillo et al.PNAS
August 3, 2010
indicate that substitutional disorder must be considered a rele-
vant variable for antiferromagnetic QPTs.
For superconducting copper oxides, the relevance of substitu-
tional disorder at the postulated QPT remains an outstanding
question. Recent transport measurements on La2-xSrxCuO4at
high magnetic fields showed no clear signature of quantum
criticality near optimal hole doping, raising the prospect that
the tuning parameter of the postulated QPT is substitutional dis-
order (4). This situation bears similarities to the subject at hand:
the scaling of Δρ∕ρ with pressure in disordered Cr1-xVxis broad
and extends throughout the entire pressure-temperature plane,
while pure Cr has a narrowly defined quantum critical regime.
The role of substitutional disorder is somewhat better understood
in heavy fermion systems, and well characterized quantum critical
points have been found in a number of stoichiometric materials
(7, 22). However, the critical spin density wave model, which un-
doubtedly applies to Cr, does not capture the physics of heavy
fermion quantum criticality. Crucially, the lack of local magnetic
moments and the absence of effective mass divergences through-
out the Brillouin zone separate the QPT in Cr from the heavy
fermion examples (23).
Our results also stand in interesting contrast to a body of work
on weak itinerant ferromagnets. For these systems, a line of con-
tinuous thermal phase transitions terminates at a first-order QPT.
The quantum critical regime is inaccessible, but both the mag-
netic and nonmagnetic ground states are often characterized
by strong quantum fluctuations that destabilize the Fermi liquid
(3, 5, 24). By contrast, the magnetic ground state of Cr is well
described by mean-field theory, with signatures of quantum fluc-
tuations only developing within the narrow quantum critical re-
gime. The outstanding feature common to both Cr and itinerant
ferromagnets appears to be a tricritical point in the pressure-tem-
perature plane, where the quartic stiffness of the order parameter
passes through zero.
The nature of the quantum fluctuations at the QPTremains an
open question. Assuming the applicability of the traditional LGW
formalism to nested fermions in three-dimensions, dimensional
arguments allow effects beyond mean-field in the quantum re-
gime. The relation γ ¼ z∕ðd þ z − 2Þ implies a dynamical expo-
nent z ¼ 1 and a scaling dimension d þ z ¼ 4, which is not over
the upper critical dimension (6, 22). Furthermore, the quasi-one-
dimensional dispersion relation at the nested Fermi surface
(which is the origin of the R−1
result in a reduced effective dimension for the critical fluctua-
tions, as has been observed for quantum critical heavy fermion
systems (25). The critical reconstruction of the nested Fermi sur-
face in Cr is accompanied by the reemergence of nested fermions
with greatly enhanced exchange interactions and as the quantum
critical point is uncovered, it drives a weak coupling BCS-like
state towards strong-coupling physics (19, 26). The persistence of
strongly interacting fermions above Pcalso opens the possi-
bility for the ground state that replaces the SDW to be charac-
terized by short-coherence length pairing, akin to the BCS-BEC
(Bose Einstein condensate) crossover observed in ultracold
gasses, or to a pseudogap-like state of dynamical pair fluctuations.
H∝ ð1 − Δρ∕ρÞ2scaling (17)) may
Magnetotransport in a Diamond Anvil Cell at Cryogenic Temperatures. All
measurements were performed in a low-temperature diamond anvil cell
equipped with a He gas membrane for fine pressure control. The pressure
medium was a methanol∶ethanol 4∶1 mixture. Pressure was measured in situ
using the ruby fluorescence method. The pressure P is calculated from the
wavelength λ of the ruby R1fluorescence by P ¼ A · lnðλ∕λ0Þ, where λ0is
the (temperature-dependent) R1wavelength at ambient pressure. A has
been calibrated directly (12) at 5 K (A5 K¼ 1;762 ? 13 GPa) and room-T
(A295 K¼ 1;868 GPa). To interpolate between these two temperatures we
assume that A is constant up to 100 K, above which it varies linearly with
temperature, in qualitative accordance with the temperature dependence
of the bulk modulus of Al2O3. Resistivity ρ and Hall coefficient RHwere mea-
sured in the four probe van der Pauw geometry on single-crystal Cr plates
using an ac resistance bridge. RHwas derived from data taken in the range
−3;500 < H < 3;500 Oe, which is in the low-field limit for all pressures. The
microscopic samples, size ð200 × 200 × 40Þ μm3with (111)-oriented faces,
were derived from large Cr single-crystal discs (Alfa Aesar, 99.996 þ %) by
a procedure described elsewhere (27). The gold leads were spot-welded to
the sample and insulated from the metallic gasket using a mixture of alumina
powder and epoxy.
McWhan-Rice Model. Forpressuresupto∼0.3 GPabelowPcthephasediagram
was determined by fitting Δρ∕ρ to the McWhan-Rice model (13). This model
hasthreefreeparameters:theNéeltemperatureTN,theT → 0energygapg0,
and the magnetic Fermi surface fraction q (note the typo in Eq. 6 of ref. 13,
where E3∕2is written instead of E3; for the correct expression see Eq. 6.16 of
ref. 28). We implemented this model with an additional free parameter dTN
which describes the width of a Gaussian distribution in TN. This convolution
allows for pressure inhomogeneity and is valid as long as the variation in TN
implemented numerically, holding q constant and scaling g0linearly with TN.
Modeling ρPMðTÞ is easy at low temperatures (approximately TN< 50 K or
P > 8.8 GPa), where it obeys the expected form ρPMðTÞ ¼ ρ0þ bT3þ cT5
and the T3dependence dominates (14, 16). In this regime the McWhan-Rice
fit parameters are robust. At higher T modeling ρPMðTÞ is difficult, and the
McWhan-Rice fit results for g0and q are strongly correlated with the form
assumed for ρPMðTÞ. The result for TN, however, remains robust. As a check
we also estimated TNfrom ρðTÞ by simply finding the temperature at which
ρðTÞ has the most negative slope. This point is assumed to correspond to that
temperature at which the energy gap gðTÞ grows the fastest, which is identi-
fied with TN. For all P < 9 GPa the discrepancy between these results and the
McWhan-Rice approach is less than the size of the data points in Fig. 1B; for
P > 9 GPa this simpler technique fails due to the increasing influence of the
pressure inhomogeneity as the phase diagram steepens near Pc.
Data Analysis in the Critical Regime. For pressures within ∼0.3 GPa of PCthe
McWhan-Rice fits fail for two reasons. First, our lowest measurement tem-
perature of 4.5 K is too high for the Δρ∕ρ form to fully develop (Fig. 2C),
and as a result the fit parameters are poorly determined. Second, the finite
pressure inhomogeneity produces a smearing of TNthat diverges at Pc. It is
preferable to consider the isotherms Δρ∕ρjT, which are fit to a power law
Δρ∕ρjTðPÞ ¼ aðPc;T− PÞβconvolved with a Gaussian pressure distribution.
The 5 K isotherm and best-fit curve are shown in Fig. 2C, and a scaling plot
of the data approaching the low-temperature limit is shown in Fig. 3A. The
critical exponents βðTÞ are plotted in Fig. 3B, and the fit parameters Pc;T
define the phase diagram which is plotted in Fig. 3C.
For the two samples studied in fine detail in the critical regime, the best-fit
FWHM of the Gaussian pressure distribution was 0.24 and 0.33 GPa, respec-
tively, for the T ¼ 5 K data. For a given sample this fit parameter is then held
constant for fits to all T > 5 K isotherms to reduce systematic correlations be-
tween fit parameters. The Gaussian pressure distributions correspond to a 2-σ
width of 0.43 and 0.58 GPa, respectively, somewhat smaller than the 0.72 GPa
base width that was found for the pressure inhomogeneity over a ð200×
200Þ μm2area in a recent study of the pressure conditions in the same
methanol∶ethanol medium at 10 GPa and 5 K (12). The pressure condition
is characteristic of a given cell assembly and depends mainly on choice of
pressure medium and the sample-to-chamber volume ratio. A nearly con-
stant offset of 0.12 GPa of the critical phase boundaries measured for two
different samples could result from several systematic issues, most notably
the position of the ruby chips (the ruby is positioned to the side of the
sample, close to the gasket wall where the pressure gradients are highest).
ACKNOWLEDGMENTS. We acknowledge Arnab Banerjee and Peter Littlewood
for enlightening discussions. The work at the University of Chicago was
supported by National Science Foundation (NSF) Grant DMR-0907025.
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