PHYSICAL REVIEW B 75, 212405 共2007兲
Ferromagnetic quantum critical point in URhGe doped with Ru
N. T. Huy,1 A. Gasparini,1 J. C. P. Klaasse,1 A. de Visser,1,* S. Sakarya,2 and N. H. van Dijk2
der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
of Radiation, Radionuclides & Reactors, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands
共Received 13 March 2007; published 8 June 2007兲
We have investigated the thermal, transport, and magnetic properties of URh1−xRuxGe alloys near the critical
concentration xcr = 0.38 for the suppression of ferromagnetic order. The Curie temperature vanishes linearly
with x and the ordered moment m0 is suppressed in a continuous way. At xcr, the specific heat varies as c
⬃ T ln T, the ␥ value 兩c / T兩0.5 K is maximum, and the temperature exponent of the resistivity ␳ ⬃ Tn attains a
minimum value n = 1.2. These observations provide evidence for a ferromagnetic quantum phase transition.
Interestingly, the coefficient of thermal expansion and the Grüneisen parameter ⌫ remain finite at xcr 共down to
T = 1 K兲, which is at odds with recent scaling results for a metallic quantum critical point.
DOI: 10.1103/PhysRevB.75.212405
PACS number共s兲: 75.30.Mb, 71.10.Hf, 75.40.Cx
In recent years, interest has continued to grow in materials
that exhibit a quantum phase transition 共QPT兲, i.e., a transition at zero temperature driven by quantum fluctuations.1
QPTs are fundamentally different from their classical counterparts at finite T, where the transition is due to thermal
fluctuations of the order parameter. QPTs can be induced in a
wide range of materials, such as correlated metals,2 cuprate
superconductors,3 common metals,4 and the two-dimensional
electron gas.5 This is accomplished by adjusting a control
parameter 共e.g., pressure p, doping x, magnetic field B, or
electron density兲 in order to tune the system to a quantum
critical point 共QCP兲. At this point, the quantum critical fluctuations give rise to unusual temperature laws 关non-Fermiliquid behavior 共nFL兲兴 for the magnetic, thermal, and transport parameters,6,7 and new collective states may emerge,
e.g., unconventional superconducting8 or electronic states.9
This in turn calls for novel concepts and theories.2,10,11 In
order to provide a fruitful testing ground, it is important to
identify new systems and to investigate their critical behavior.
Strongly correlated electron systems, notably heavyfermion compounds based on the f elements Ce, Yb, or U,
are especially suited to study magnetic-to-nonmagnetic
QPTs, because the ordering temperatures are low 共⬃10 K兲
and the exchange interaction can be modified relatively easily by an external control parameter. Currently, there are two
central questions that are being addressed by studying QPTs
in these materials. The first issue is the fate of the quasiparticles when the antiferromagnetic 共AF兲 or ferromagnetic
共FM兲 phase is entered. In the conventional scenario, a spin
density wave is formed6,7 and the quasiparticles preserve
their itinerant character 关as in CeIn3−xSnx 共Ref. 12兲兴. Because
the itinerant model is unable to account for the nFL behavior
in certain materials, an alternative local quantum criticality
model has been put forward.2,10,11 Here, the quasiparticles
共Kondo-screened moments兲 decompose at the critical point
in conduction electrons and local f moments that undergo
magnetic order 关as in CeCu6−xAux 共Ref. 2兲 and
YbRh2共Si1−xGex兲2 共Ref. 13兲兴. The second captivating issue is
the emergence of unconventional superconducting 共SC兲
states near the pressure induced QCPs in CePd2Si2, CeIn3,8
and UGe2.14 Evidence is at hand that in these materials un1098-0121/2007/75共21兲/212405共4兲
conventional pairing is realized 共d-wave pairing for the AF
and p-wave pairing for the FM systems兲. This strongly suggests Cooper pairing mediated by AF or FM spin fluctuations
rather than by phonons. The coexistence of FM order and SC
in UGe2 关and possibly in UIr 共Ref. 15兲兴 under pressure is
uncommon in nature and attracts much attention.
In this Brief Report, we provide evidence for a ferromagnetic QPT in URhGe doped with Ru. Our research is motivated by the unique properties of the parent compound
URhGe at ambient pressure: 共i兲 SC below Ts = 0.25 K coexists with itinerant FM order 共Curie temperature TC = 9.5 K兲16
and 共ii兲 reentrant SC is induced by applying a large magnetic
field 共B ⬃ 12 T兲.17 These observations immediately prompted
the question whether one can tune URhGe to a FM QCP by
mechanical or chemical pressure, with the objective of probing the quantum critical fluctuations and possibly linking
these to the SC pairing mechanism. Resistivity measurements under hydrostatic pressure, however, revealed that TC
increases at a rate of 0.065 K / kbar.18 Also, upon the application of uniaxial pressure TC increases as was extracted
from the Ehrenfest relation.19 As regards to chemical pressure, the best candidate dopants are Ru and Co, since among
the neighboring isostructural UTX compounds 共T
= transition metal and X = Ge or Si兲 only URuGe and UCoGe
have a paramagnetic ground state.20,21 Indeed, FM order in
URhGe can be suppressed by replacing Rh by Ru and vanishes at 38 at. % Ru.22,23 Here, we investigate the thermal,
transport, and magnetic properties of URh1−xRuxGe alloys
near the critical concentration xcr = 0.38. The observed nFL T
dependencies of the specific heat and electrical resistivity,
together with the smooth suppression of the ordered moment,
provide evidence for a continuous FM QPT. This classifies
URh1−xRuxGe as one of the scarce f-electron systems in
which a FM QCP can be reached by doping 共a FM QPT was
also reported for CePd1−xRhx,24 but here the transition is
Polycrystalline URh1−xRuxGe samples with 0.0艋 x
艋 0.60 were prepared by arc-melting the constituents U, Rh,
and Ru 共all 3N兲 and Ge 共5N兲 under a high-purity argon atmosphere in a water-cooled copper crucible. The as-cast
samples were wrapped in Ta foil and annealed under high
vacuum in quartz tubes for 10 days at 875 ° C. Samples were
©2007 The American Physical Society
PHYSICAL REVIEW B 75, 212405 共2007兲
FIG. 1. f-electron specific heat of URh1−xRuxGe plotted as cm / T
vs log T for 0 艋 x 艋 0.50 as indicated. For x 艋 0.10, the data are
fitted to cm共T兲 = ␥T + ␦T3/2e−⌬/kBT with ␥, ␦, and ⌬ / kB values of
0.150, 0.146, and 0.136 J / mol K2, 0.024, 0.041, and
0.094 J / mol K5/2, and 0, 6.5, and 10.6 K for x = 0, 0.05, and 0.10,
respectively 共solid lines for x = 0 and x = 0.10; data for x = 0.05 not
shown兲. The arrow indicates TC for x = 0.35. For xcr = 0.38, cm / T
⬃ ln T over one and a half decade in T 共straight solid line兲.
cut by spark erosion. Electron probe microanalysis showed
the single phase nature of the samples within the resolution
of 2%. X-ray powder diffraction confirmed the orthorhombic
TiNiSi structure 共space group Pnma兲.25,26 Upon substituting
Ru, the unit-cell volume ⍀ = 224.3 Å3 of URhGe decreases
linearly at a rate of 0.067 Å3 / at. % Ru 共i.e., ⌬⍀ = 1.1% at
xcr兲 in an anisotropic way, the main effect being the reduction
of the a lattice parameter.23
The specific heat c共T兲 was measured down to 0.4 K using
a semiadiabatic method in a homebuilt 3He system. Electrical resistivity ␳共T兲 data were collected in a commercial 3He
system 共Heliox, Oxford Instruments, T 艌 0.25 K兲 using a
low-frequency ac-resistance bridge. The thermal expansion
␣共T兲 was measured using a parallel-plate capacitance
dilatometer in the T range 1 – 15 K. The dc magnetization
M共T兲 共T 艌 1.8 K兲 was obtained using a Quantum Design superconducting quantum interference device magnetometer.
Temperature scans in magnetic fields B up to 5 T were made
after field cooling.
The overall effect of Ru doping on ferromagnetism in
URhGe is presented in Fig. 1, where we have plotted the
f-electron specific heat cm, obtained after subtracting the lattice contribution 关clat = ␤T3 for T 艋 20 K with ␤ = 0.60
⫻ 10−3 J / mol K4 共Ref. 25兲兴, as cm / T vs log T for 0 艋 x
艋 0.50. Upon doping, TC initially increases, but for x
艌 0.10 the ordering peak shifts toward lower T and weakens.
Values of TC共x兲, identified by the inflection points in c / T vs
T 共on a linear T scale兲 at the high T side of the peaks, are
traced in Fig. 2共a兲 and are in excellent agreement with the
values determined from M共T兲 and ␳共T兲.22 For x 艌 0.20, TC
decreases linearly with x at a rate of 0.45 K / at. % Ru. For
x = 0, the magnetic specific heat for T 艋 5 K is described by
cm共T兲 = ␥T + ␦T3/2, where ␥ is the linear coefficient of the
electronic specific heat and the second term is the spin-wave
contribution.27 The values for ␥ and ␦ extracted by fitting the
data 共see Fig. 1兲 are in good agreement with the values reported in Ref. 25. Upon doping Ru, an energy gap ⌬ opens in
FIG. 2. 共a兲 Curie temperature of URh1−xRuxGe determined from
c共T兲 共䉱兲, ␳共T兲 共쎲兲, and M共T兲 共夝兲. The critical Ru content is xcr
= 0.38 共vertical dashed line兲. 共b兲 Magnetization M at 2 K in B
= 0.01 共쎲兲 and 1 T 共䊊兲. Inset: Arrott plot for x = 0.38 at 1.8 K 艋 T
艋 6 K. 共c兲 c / T at T = 0.5 K 共䊏兲 and the exponent n 共䊐兲 of ␳ ⬃ Tn.
The horizontal dashed line indicates n = 2.
the magnon spectrum and the specific heat for x = 0.05 and
0.10 now follows the relation 共T 艋 5 K兲 cm共T兲 = ␥T
+ ␦T3/2e−⌬/kBT 共Ref. 27兲 共see fits in Fig. 1兲. The most important result of our specific-heat experiments, however, is the
pronounced cm共T兲 = −bT ln共T / T0兲 dependence for xcr, where
b = 0.062 J / mol K2 and T0 = 41 K. This nFL term is observed
over one and a half decade in T 共0.5– 9 K兲. At xcr
兩c / T兩0.5 K共x兲 has a maximum 关Fig. 2共c兲兴. The total f-electron
entropy obtained by integrating cm / T vs T between 0.5 and
⬃15 K amounts to ⬃0.48R ln 2 for x = 0 and decreases to
0.33R ln 2 at xcr. Its small value confirms the itinerant nature
of the FM transition 关the ordered moment m0 is 0.4 ␮B for
x = 0 共Refs. 16 and 25兲兴.
The electrical resistivity of URh1−xRuxGe 共x 艋 0.60兲 at
high T 共Ref. 23兲 shows the behavior typical for a FM Kondo
lattice. The data for x = 0.38 are shown in the inset in Fig. 3,
where the maximum near 130 K signals the formation of the
Kondo lattice. For the FM compounds at low T, a kink in
␳共T兲 关and maximum in d␳共T兲 / dT兴 marks TC. For all doped
samples, the total resistivity drop in the T interval 0 – 300 K
is ⬃150– 250 ␮⍀ cm, which is usual for uranium
intermetallics.21 However, the residual resistivity values ␳0
are large 共⬃200– 300 ␮⍀ cm兲, which is due to the brittleness of the samples 共cracks兲. Consequently, the residual
resistance ratio values 关R共300 K兲 / R共0 K兲兴 are small
共⬃2兲. In Fig. 3, we show ␳共T兲 at low T for 0.10艋 x 艋 0.60.
For a FM with gapped magnon modes ␳共T兲 = ␳0 + ATn
+ BT⌬e−⌬/kBT共1 + 2kBT / ⌬兲,28 where the second term is the
electron-electron scattering term 共i.e., the FL term when n
= 2兲 and the third term yields the scattering from magnons.
For x = 0.10 and 0.20, fits reveal that the second term is
dominant 共A Ⰷ B兲 and ␳共T兲 ⬃ T2.0±0.1 over a wide T range in
the FM state 共see Fig. 3兲. Therefore, we conclude that scat-
PHYSICAL REVIEW B 75, 212405 共2007兲
FIG. 4. Coefficient of volume thermal expansion ␤共T兲 共solid
line兲 and specific heat c共T兲 共•兲 of URh0.62Ru0.38Ge. Inset: Grüneisen
ratio ⌫ as a function of T.
FIG. 3. Resistivity of URh1−xRuxGe for 0.10艋 x 艋 0.60. The bar
gives the absolute scale. The arrows for x = 0.30 and 0.35 indicate
TC obtained from additional data sets. The solid lines are fits to
␳共T兲 = ␳0 + ATn. For x 艋 0.3, n = 2.0± 0.1. For xcr = 0.38, n = 1.2 is
minimum. Inset: Resistivity for x = 0.38 up to 300 K.
tering from magnons can be neglected in our polycrystalline
samples and we restrict the analysis to fitting ␳共T兲 = ␳0
+ ATn 共see Fig. 3兲. The values of n extracted 共by taking the
best fit over the largest T interval兲 are shown in Fig. 2共c兲.
n共x兲 attains a minimum value n = 1.2 at xcr, followed by a
slow recovery to the FL value n = 2 there above.
The magnetization M共T兲 for all samples was measured in
B = 0.01 and 1 T down to 1.8 K. In addition, M共B兲 was measured at fixed T in order to produce Arrott plots 共M 2 vs
B / M兲. 兩M兩2 K values are traced in Fig. 2共b兲. For pure
URhGe, 兩M兩2 K in 1 T ⯝ 0.2␮B in agreement with the polycrystalline average 21 m0 for a uniaxial FM 共m0 = 0.4␮B
directed along the c axis16兲. In 0.01 T, a reduced
value 兩M兩2 K ⯝ 0.11␮B is observed due to demagnetizing effects. Values of TC 关Fig. 2共a兲兴 were determined from the
inflection points in M共T兲 in 0.01 T and from the Arrott plots.
For x 艌 0.38, the Arrott plots 共T 艌 1.8 K兲 no longer indicate
magnetic order 关see inset in Fig. 2共b兲 for x = 0.38兴. The most
important feature of the data is the gradual decrease
of 兩M兩2 K共x兲. For B = 0.01 T, 兩M兩2 K共x兲 smoothly goes to 0 at
x = 0.35 共TC = 1.3± 0.1 K兲, while for B = 1 T a finite field induced 兩M兩2 K remains. We conclude that the FMparamagnetic transition as a function of x is a continuous
共second order兲 phase transition.
In Fig. 4, we show the coefficient of volume thermal expansion ␤共T兲 for xcr = 0.38 at T 艌 1 K. The data 共solid line兲
are obtained by averaging ␣i共T兲 measured for three orthogonal directions on the polycrystalline sample 共␤ = ⌺i␣i兲 in order to eliminate possible anisotropy effects due to crystallites
with preferred orientations. The T dependence of ␤ at low T
is weaker than that of the specific heat 共see Fig. 4兲. Concurrently, the Grüneisen ratio ⌫ = Vm␤ / ␬c decreases below T
⬃ 7 K 关here the molar volume Vm = 3.36⫻ 10−5 m3 / mol and
isothermal compressibility ␬ ⯝ 10−11 Pa−1 共Ref. 23兲兴. The
quasilinear behavior of ⌫共T兲 for 1 K 艋 T 艋 5 K suggests an
unusual T variation of ␤, i.e., roughly proportional to T2 ln T.
Having documented the critical behavior of the
URh1−xRuxGe alloys, we conclude that our c共T兲, ␳共T兲, and
M共T兲 data provide evidence for a continuous FM QPT with
xcr = 0.38. The most compelling evidence is the specific heat
ccr ⬃ T ln共T / T0兲 observed over one and a half decade in T
共Fig. 1兲7 and the concomitant maximum in 兩c / T兩0.5 K共x兲 关Fig.
2共c兲兴. The temperature T0 = 41 K is large, which indicates
that our c共T兲 experiments down to T = 0.4 K 共T / T0 ⯝ 0.01兲
indeed probe the quantum critical regime. It will be interesting to investigate whether the c / T ⬃ ln T behavior persists
even at lower T. Eventually, however, c / T will saturate because of crystallographic disorder inherent to the
URh1−xRuxGe alloys. Further support for a QCP is provided
by the critical behavior in the resistivity ␳cr ⬃ T1.2 up to 2 K.
The exponent n共x兲 has a pronounced minimum at xcr 关Fig.
2共c兲兴. The value n = 1.2 is smaller than the value n = 5 / 3 predicted for a clean FM QCP.29 This is not unexpected as disorder reduces n.30 The itinerant nature of the FM state and
the smooth suppression of m0 pointing to a continuous phase
transition strongly suggest that the QPT in URh1−xRuxGe is
of the Hertz-Millis type,6,7 albeit with modified exponents
due to the effects of doping 共notably emptying the d band
and alloy disorder兲. For instance, for an itinerant clean FM
QPT, one expects TC ⬃ 共xc − x兲3/4 共dimension d = 3, dynamical
critical exponent z = 3兲, while we obtain TC ⬃ 共xc − x兲 over a
wide range 0.20艋 x 艋 0.35. Deviations from the clean behavior are also observed in f-electron materials with a pressure
induced continuous FM QPT, such as CeSi1.81.31 On the other
hand, for d-electron alloys with a continuous FM QPT 关e.g.,
NixPd1−x 共Ref. 32兲 and Zr1−xNbxZn2 共Ref. 33兲兴, the data are
to a large extent in agreement with the itinerant model. Further theoretical work is required to clarify these issues.
Finally, we discuss our results for the thermal expansion
and the Grüneisen parameter. The finite ⌫ value at low T is at
variance with the recent prediction of a diverging Grüneisen
ratio ⌫ ⬃ T−1/z␯ at the QCP 共␯ is the correlation length
exponent兲.34 For the case of an itinerant FM QCP, the scaling
results are ␤cr ⬃ T1/3 and ccr ⬃ T ln共1 / T兲, whence ⌫cr
PHYSICAL REVIEW B 75, 212405 共2007兲
⬃ ␤cr / ccr ⬃ 关共T2/3 ln共1 / T兲兲兴−1.34 While the specific heat follows the expected behavior, the thermal expansion clearly
does not 共␤ ⬃ T2 ln T for 1 K 艋 T 艋 5 K兲. With the value T0
= 41 K extracted from ccr, we calculate that ⌫cr within the
scenario of Ref. 34 should have a minimum near 8 K and
diverge at lower T. This is obviously not the case experimentally 共Fig. 4兲. The only other system for which the Grüneisen
ratio near a FM QPT has been investigated so far
is CePd1−xRhx.35 In this system, a nondiverging
共T-independent兲 ⌫ was also observed in the critical regime.
In conclusion, we have investigated the thermal, transport,
and magnetic properties of URh1−xRuxGe near the critical
concentration for the suppression of FM order. At xcr = 0.38
c ⬃ T ln T, the ␥ value 兩c / T兩0.5 K has a maximum and the T
exponent in the resistivity attains the nFL value n = 1.2. Together with the gradual suppression of the ordered moment
m0, the data provide evidence for a continuous FM quantum
phase transition. This offers the sole opportunity thus far to
investigate FM spin fluctuations in URhGe under quantum
critical conditions. The identification of the FM QCP at ambient pressure in URhGe doped with Ru paves the road to a
host of experiments on this unique material.
*Electronic address:
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