Identifying the order-parameter symmetry of superconducting PrOs Sb by Andreev

Identifying the order-parameter symmetry of superconducting PrOs Sb by Andreev
Identifying the order-parameter symmetry of superconducting PrOs4 Sb12 by Andreev
reflection spectroscopy: Evidence for unconventional pairing involving multiple bands
S. H. Curnoe1 , C. S. Turel2 , S. Bohloul1,3 , W. M. Yuhasz4 , R. Baumbach4 , M. B. Maple4 and J. Y. T. Wei1,5
1
Department of Physics and Physical Oceanography,
Memorial University of Newfoundland, St. John’s, NL, A1B 3X7
2
Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada
3
Centre for the Physics of Materials and Department of Physics, McGill University, Montreal, PQ, H3A 2T8
4
Department of Physics and Center for Advanced Nanoscience,
University of California San Diego, La Jolla, California 92093 and
5
Canadian Institute for Advanced Research, Toronto, ON, M5G1Z8 Canada
Point-contact Andreev reflection spectroscopy was performed on single crystals of the heavyfermion superconductor PrOs4 Sb12 , down to 90 mK in temperature and up to 3 T in magnetic field.
The conductance spectra show multiple structures, including zero-bias peaks and spectral dips, and
are interpreted as evidence for unconventional pairing involving two different bands. Samples with
2% Ru replacing Os were also measured, showing the emergence of a pronounced spectral hump.
The results of a comprehensive set of spectral calculations using the Blonder-Tinkham-Klapwijk
theory of Andreev reflection are presented. All symmetry-allowed gap functions are considered,
including several non-unitary cases. Several candidate gap functions are identified that reproduce
the main features of our experimental data; among these, the superconducting phase with symmetry
C3 × K agrees best with other previous measurements.
PACS numbers: 74.70.Tx, 74.45.+c, 74.25.Dw, 74.20.Rp
I.
INTRODUCTION
Superconductivity in heavy-fermion materials has been
a heavily researched topic, particularly for the role that
strongly-correlated electrons play in the pairing process.
The discovery of superconductivity in the filled skutterudite PrOs4Sb12 has attracted much attention, because it is the first heavy-fermion superconductor containing neither Ce nor U, but rather Pr atoms which
show no ground-state magnetic order1 . Various unconventional properties have been reported in the superconducting state of PrOs4Sb12 2 . Of particular interest is the
appearance of low-energy quasiparticle excitations1,5–10 ,
characteristic of a nodal superconducting gap function.
Also intriguing is the occurence of a second density-ofstates discontinuity just below the superconducting critical temperature (Tc )3,4 , indicative of a second phase
transition. Furthermore, angular magneto-thermal conductivity measurements have observed that the pairing
symmetry of PrOs4 Sb12 undergos a phase transition in a
magnetic field9 . Such a complex order-parameter (OP)
phase diagram suggests the presence of either multiple
superconducting OPs or a multi-dimensional OP, reminiscent of the case of UPt3 11 .
There have been conflicting experimental reports on
the superconducting gap topology of PrOs4 Sb12 , some
indicating the presence of gap nodes while others indicating the Fermi surface to be fully gapped12 . To
reconcile the difference between these reports, a recent proposal has invoked the multi-band dispersion of
PrOs4 Sb12 13 to suggest that there are two superconducting gaps, i.e. a nodal gap in a light-mass band and
a non-nodal gap in a heavy-mass band14,15 . Double
superconducting gaps have been observed in two sepa-
rate thermal-conductivity measurements16,17 . However,
whereas the earlier measurement indicated both gaps to
have s-wave symmetry16 , the more recent measurement
indicated only one s-wave gap with the other gap having
nodal symmetry17 . The latter result suggests that there
are two distinct OPs with different pairing symmetries
and Tc ’s in PrOs4Sb12 . Although still under debate, this
multi-band and multi-symmetry scenario could also explain the two Tc ’s seen in heat-capacity measurements on
Pr(Os1−xRux )4 Sb12 , where the nodal gap becomes dominated by a non-nodal gap for Ru-doping above ≈ 1%7 .
Despite these prior studies, the symmetry of the superconducting OP in PrOs4 Sb12 is still not known. In
this paper, we report on point-contact Andreev reflection spectroscopy (PCARS) measurements of PrOs4 Sb12
single crystals, and identify the OP symmetry by analyzing the data with first-principle spectral simulations. PCARS is an inherently local and phase-sensitive
probe of the superconducting OP20 , and has been used
to study the pairing state of several unconventional
superconductors, including cuprates21 , ruthenates22,23 ,
borocarbides24, MgB2 25 and heavy-fermion metals26,27 .
Our PCARS measurements on PrOs4 Sb12 show conductance spectra with multiple features, including zero-bias
peaks and spectral dips, which are interpreted as evidence of unconventional pairing. The spectral evolution
down to 90 mK in temperature and up to 3 T in field
reveal a phase diagram characteristic of multiband pairing. Samples with 2% Ru replacing Os were also measured, showing the emergence of a pronounced spectral
hump. To relate the data to the superconducting OP
symmetry, we performed spectral simulations based on
all possible OP phases allowed by the crystal symmetry
of PrOs4Sb12 . Our spectral analysis indicates that a sin-
2
gle superconducting phase of the Tg OP, present on two
different bands, can account for the data. We discuss
the most likely of possible OP phases by comparing our
results with other experimental results on PrOs4Sb12 .
12
-1
dI/dV (Ω )
EXPERIMENTAL RESULTS
The single crystals used in our experiment were grown
with a molten metal-flux method28 . For the undoped
PrOs4 Sb12 crystals, electrical resistivity measurements
showed a single Tc at ≈ 1.85 K with ≈ 5 mK transition width, and an upper critical field Hc2 ≈ 2.25 T
at 200 mK. AC susceptibility χ(T ) showed two Tc ’s,
Tc1 ≈ 1.85 K and Tc2 ≈ 1.65 K, similar to previously
reported results3,29 . PCARS measurements were performed in a 3 He-4 He dilution refrigerator, using Pt-Ir tips
on c-axis faces of the crystals. Prior to measurement, the
crystals were etched in a 1:1 HNO3 -HCl mixture to remove any residual Sb flux. Junction impedances were
typically 0.5-1 Ω, resulting in point contacts which were
in the ballistic regime30 . To minimize Joule heating, a
pulsed-signal four-point technique31 was used to acquire
current-vs.-voltage I-V curves, which were numerically
differentiated to yield dI/dV spectra.
Fig. 1(a) shows the temperature evolution of dI/dV
spectra of PrOs4Sb12 in zero magnetic field. At 90 mK,
well below Tc1 ≈ 1.85 K, there is a pronounced zero-bias
peak (ZBP) accompanied by a symmetric dip which begins at ≈ ±0.4 mV, as indicated by δ1 in the figure. An
additional dip is also present at ≈ ±0.2 mV, as indicated
by δ2 . With increasing temperature, the ZBP decreases
in height and the dips move inward. The spectrum flattens out above Tc1 ≈ 1.85 K, consistent with the sample
no longer being superconducting.
Fig. 1(b) shows the detailed temperature evolution of
the ZBP height and of the positions δ1 and δ2 , after normalizing them relative to their maximum values. Also
plotted in Fig.1(b) is the excess spectral area, which
is defined by subtracting each dI/dV spectrum by the
normal-state spectrum and then numerically integrating
between ±1.8 mV. These plots indicate that the evolution of the various spectral features is governed by two
distinct temperature scales, corresponding respectively
to Tc1 ≈ 1.85K and Tc2 ≈ 1.65K.
Figure 2(a) shows the magnetic-field evolution of
dI/dV spectra measured at 90 mK. As the field is increased, the ZBP decreases in height and the dip structures move inward, qualitatively similar to the spectral
evolution with temperature. However, the ZBP height
collapses more quickly with field than with temperature,
relative to the evolution of the excess spectral area. As
shown in Figure 2(b), the ZBP height vanishes above
≈ 1.5 T while the excess area persists up to ≈ 2.3 T. This
field evolution indicates that there are also two different
energy scales governing the field suppression of superconductivity in PrOs4Sb12 . Namely, the low-Z Andreev
states are spectrally robust up to a lower-field bound-
0.09 K
0.56 K
0.91 K
1.12 K
1.34 K
1.51 K
1.86 K
10
8
6
4
2
δ1
δ2
-0.4
-0.2
(a)
0
-0.6
0.0
0.2
0.4
0.6
Voltage (mV)
Excess Area
Peak H eight
1.0
Spectral Features
II.
PrOs4Sb12
δ1
δ2
0.8
0.6
0.4
0.2
0.0
0.0
(b)
0.5
1.0
1.5
Tem perature (K )
2.0
FIG. 1: (color online). (a) Differential conductance as a function of temperature for a Pt-Ir/PrOs4 Sb12 point-contact junction in zero magnetic field. (b) Temperature dependence of
various spectral features, after normalizing relative to their
maximum values. The triangles correspond to zero-bias peak
height, the squares and diamonds to δ1 and δ2 . The circles
correspond to the excess spectral area, as defined in the text.
ary H ′ ≈ 1.5 T at 90 mK, while the high-Z Andreev
states persist up to a higher-field boundary H ′′ ≈ 2.3 T
at 90 mK. It is worth noting that PCARS data taken
on UPt3 38 , which is known to have multi-component superconducting OP11 , also shows spectral features that
vanish at a field boundary lower than Hc2 .
To map out a detailed H-T phase diagram for
PrOs4Sb12 , we carried out a combined analysis of the
temperature and field evolutions of our data. Figure 3
shows a master plot of the full temperature dependencies of H ′ and H ′′ determined from our spectroscopy
data, along with Hc2 (T ) determined from our ρ(T, H)
data. First, it is clear that H ′′ (T ) coincides with Hc2 (T ),
indicating that the higher-field boundary is just the resistive upper-critical field. Second, H ′′ (T ) and H ′ (T )
appear to emerge from different low-temperature asymptotes and gradually approach each other with increasing
temperature. This non-parallel behavior between H ′′ (T )
and H ′ (T ) resembles the H-T phase diagram determined
from angular magneto-thermal conductivity in Ref.9 , in
contrast to parallel phase boundaries determined from
3
-1
dI/dV (Ω )
10
2.5
PrOs4Sb12
0.00 T
0.15 T
0.50 T
0.80 T
1.10 T
1.60 T
2.30 T
8
6
1.5
1.0
4
2
0.5
(a)
0
-0.6
T = 90 mK
-0.4
-0.2
0.0
0.2
0.4
0.6
Voltage (mV)
1.0
Spectral Features
PrOs4Sb12
2.0
µ0H (T)
12
0.0
0.0
ρ
H''
H'
0.5
1.0
1.5
2.0
Temperature (K)
E xcess A rea
P eak H eight
0.8
0.6
0.4
FIG. 3: (color online). Magnetic field vs. temperature phase
diagram determined from our point-contact spectroscopy and
resistivity data on PrOs4 Sb12 . H ′ (triangles) correspond to
the field at which the ZBP vanishes. H ′′ (circles) correspond
to the field at which the excess spectral area vanishes. Hc2
(diamonds) is the upper-critical field measured by resistivity.
Dotted curves are added to guide the eye.
0.2
0.0
0.0
(b)
0.5
1.0
1.5
µ 0 H (T)
2.0
2.5
FIG. 2: (color online). (a) Differential conductance as a function of magnetic field for a Pt-Ir/PrOs4 Sb12 point-contact
junction at 90mK. (b) Field dependence of the zero-bias peak
height and excess spectral area, as defined in the text.
heat capacity in Ref.39 . Finally, our H ′′ (T ) and H ′ (T )
curves appear to approach zero at different temperatures, ≈ 1.85 K and ≈ 1.65 K respectively, agreeing
well with Tc1 and Tc2 measured by χ(T ) on our samples. Here it is important to emphasize that our H-T diagram is determined spectroscopically with a highly local
probe, thus arguing strongly against sample inhomogeneity as the cause of multiple Tc ’s. It should also be noted
that although our H-T diagram is qualitatively similar
with the one reported in Ref.9 , there is a large quantitative difference in the low-temperature asymptote between
our H ′ (T ) and their lower-field boundary H ∗ (T ). Interestingly, surface impedance measurements on PrOs4 Sb12
have also indicated a lower-field phase boundary with a
zero-temperature asymptote of ≈ 1.5 T40 , which is quantitatively consistent with our H ′ (T ).
To probe the effects of Ru-doping, we also measured
Pr(Os0.98 Ru0.02 )4 Sb12 crystals. These 2% Ru-doped
crystals have Tc ≈ 1.8 K and Hc2 ≈ 2.2 T at 200 mK.
Figure 4(a) plots the temperature evolution of a typical
dI/dV spectrum. At 90 mK, a ZBP is also observed,
but its height is lower than in the undoped case. Broad
hump structures clearly emerge at ≈ ±1.4 mV and ≈
±0.6 mV as indicated by δ3 and δ4 , respectively, in the
figure. With increasing temperature, the ZBP decreases
in height while the humps become narrower, and the
spectrum completely flattens out above Tc ≈ 1.8 K. The
inset of Fig. 4(a) shows the spectral evolution with magnetic field at 90 mK. While the ZBP collapses into the
hump structures above ≈ 1.5 T, a spectral hump is still
visible up to ≈ 2.2 T, unlike the undoped case which
shows no pronounced humps (Fig. 2(a)). Figure 4(b)
plots the temperature dependences of the ZBP height,
the excess spectral area, and the positions of δ3 and δ4 ,
using the same data reduction as for Fig. 1(b). Comparing Fig. 4(b) and Fig. 1(b), it is clear that while both
cases show excess spectral area up to the resistive Tc , Fig.
4(b) shows a more rapid ZBP collapse with temperature.
These observations indicate the emergence of a non-nodal
gap function with Ru-doping, in agreement with both
heat-capacity and penetration-depth measurements7,41 .
Further measurements of crystals with higher Ru-doping
levels are under way to elucidate this possibility.
III.
A.
THEORETICAL MODELING OF
CONDUCTANCE SPECTRA
Possible types of superconducting order
parameters for PrOs4 Sb12
PrOs4Sb12 crystallises in the space group Im3̄ with
point group Th . The classification of superconducting
order parameters (OP’s) was described in Refs.42,43 for
the point groups Oh , D6h and D4h and extended to the
point group Th in Ref.44 . Here we summarise these re-
4
5.4
0.00 T
0.50 T
1.00 T
1.50 T
1.75 T
2.00 T
2.25 T
5.2
-1
dI/dV (Ω )
cases. K stands for time reversal; symmetry groups that
do not possess the element K have broken time reversal
symmetry. Some of the symmetry groups (for example,
T (D2 )) have elements which are combinations of rotations and phases and/or time reversal. The details of
these groups may be found in Ref.44 .
The superconducting gap function is a 2 × 2 matrix.
For singlet pairing it takes the form
0
ψ(~k)
∆(~k) =
(1)
−ψ(~k) 0
Pr(Os0.98Ru0.02)4Sb12
5.0
0.09
0.51
1.08
1.75
1.85
1.95
4.8
-3
-2
-1
0
1
2
K
K
K
K
K
K
3
4.6
4.4
(a)
-3
δ3
-2
δ4
-1
0
1
2
3
Spectral Features
Voltage (mV)
1 .0
E x c e s s A re a
P e a k H e ig h t
0 .8
δ3
δ4
0 .6
0 .4
0 .2
(b)
0 .0
0 .0
0 .5
1 .0
1 .5
T e m p e r a tu r e (K )
2 .0
FIG. 4:
(color online).
(a) Differential conductance spectrum as a function of temperature for a PtIr/Pr(Os0.98 Ru0.02 )4 Sb12 point-contact junction in zero magnetic field. The inset shows the spectral evolution versus field
at 90 mK. (b) Temperature dependence of the ZBP height
and excess spectral area, defined in the same way as in Fig. 1.
sults.
The point group Th has one one-dimensional OP
(Ag,u ), one two-dimensional OP (Eg,u ) and one threedimensional OP (Tg,u ) in each of the singlet (subscript
g) and triplet (subscript u) channels. The Ag OP is “conventional”, while the rest are “unconventional”. The appearance of an Ag OP is accompanied by no symmetry breaking apart from gauge symmetry. The other
OP’s always involve a lowering of the point group symmetry. s-wave superconductivity (with gap function
∆(~k) = ∆0 ) belongs to the Ag OP. The Ag OP has no
symmetry-required nodes, although “accidental” nodes
may be present. Because they are one-dimensional, the
Ag,u OP’s have only one phase associated with them.
The 2D and 3D OP’s have potentially more complicated phase diagrams. A catalog of possible phases is
given in Ref.44 ; here we focus phases that are directly accessible from the normal state via a second order phase
transition. The best way to unambiguously label the
various phases is by their point group symmetry. Table
I lists possible SC phases along with representative gap
functions (to lowest order in ~k) for singlet and triplet
where ψ(~k) = ψ(−~k). For triplet pairing, the gap function is
−dx (~k) + idy (~k)
dz (~k)
~
∆(k) =
(2)
dz (~k)
dx (~k) + idy (~k)
~ ~k) = −d(−
~ ~k).
where d(
The magnitudes of the gaps are given by 21 Tr∆∆† (~k) =
~ ~k)|2 ± |~q(~k)|]1/2 for the singlet and triplet
|ψ(~k)| and [|d(
cases respectively. In non-unitary superconductivity,
~ ~k) × d~∗ (~k) 6= 0, and the gap is double-valued.
~q(~k) = d(
The phases T (D2 ), C3 (E) and D2 (E) in the triplet channel are non-unitary.
Table I also lists for each phase possible secondary
OP’s. A secondary OP is one whose symmetry is a supergroup of the primary OP. Secondary OP’s will appear alongside primary OP’s with the same Tc but with
a weaker temperature dependence. It is also worth noting
that among the phases listed in Table I, D2 (E) is accessible from D2 (C2 ) × K via a second order phase transition.
We calculated the conductance spectrum for all of the
trial gap functions listed in Table I, in search of those
that yield ZBP’s as observed in our PCTS results.
B.
BTK theory
We modeled the conductance spectra using the theory invented by Blonder, Tinkham and Klapwijk35 , generalised to include non-unitary superconducting order
parameters. According to our set-up of this model,
an electron is injected into the superconductor from
the −ẑ direction with momentum ~k1 = (k1x , k1y , k1z )
(k1z > 0) and spin s. The normal-superconducting interface is modeled as a δ-function potential of height H
at z = 0. At the interface, the electron may be reflected as a hole (Andreev reflection) with momentum
−~k1 , or it may be reflected as an electron, with momentum −~k1′ = (k1x , k1y , −k1z ). It may also be transmitted into the superconductor as an “electron-like” quasiparticle, with momentum ~k2 = (k2x , k2y , k2z ) (k2z > 0)
or as a “hole-like” quasi-particle with momentum ~k2′ =
(−k2x , −k2y , k2z ). In singlet superconductors, the spin
of reflected electrons is the same as the incident electron,
5
TABLE I: Possible superconducting phases for a crystal with tetrahedral (Th ) point group symmetry. The first column lists
the OP’s, the second column lists the symmetries of the various possible phases, the third and fourth columns give explicit
forms for the gap function in the singlet and triplet cases, and the last column lists secondary order parameters. ε = ei2π/3 ,
and a, b, η1 and η2 are arbitrary real numbers.
gap function
gap function
secondary
~ ~k)
OP symmetry
singlet ψ(~k)
triplet d(
SC OP’s
A T ×K
kx2 + ky2 + kz2
(kx , ky , kz )
none
E T (D2 )
kx2 + εky2 + ε2 kz2
(kx , εky , ε2 kz )
none
E D2 × K
akx2 + bky2 − (a + b)kz2
(akx , bky , −(a + b)kz )
A
T D2 (C2 ) × K
kx ky
(bky , akx , 0)
none
T C3 × K
ky kz + kz kx + kx ky
(akz + bky , akx + bkz , aky + bkx )
A
T C3 (E)
ky kz + εkz kx + ε2 kx ky (εakz + ε2 bky , ε2 akx + bkz , aky + εbkx ) E (T (D2 ))
T D2 (E)
η1 ky kz + iη2 kz kx
(η1 bky + iη2 akz , η1 akx , iη2 bkx )
T (D2 (C2 ))
while the reflected holes have opposite spin to the incident electron; however, in triplet superconductors, if an
electron with a given spin is injected then the reflected
and transmitted particles may have either spin.
Continuity of the wave function at z = 0 requires that
k1x = k2x , k1y = k2y and k1z sin θ1 = k2z sin θ2 , where
θ1,2 are the polar angles on either side of the barrier. We
will assume that EF is constant across the barrier and so
k1z ≈ k2z . The wavefunction is a four component spinor,
which takes the form on the normal side




b↑↑
1
~′  b
~  0 

(3)
+ e−ik ·~r  ↑↓ 
ΨN ↑ (~r) = eik·~r 
0
a↑↑ 
0
a↑↓




0
b↓↑
~  1 
~′  b

ΨN ↓ (~r) = eik·~r 
+ e−ik ·~r  ↓↓ 
(4)
a↓↑ 
0
a↓↓
0
for spin up and down incident electrons, respectively.
For an incident electron beam with spin s and incident
wavevector ~k = k(sin θ cos φ, sin θ sin φ, cos θ), the tunneling conductance is
σSs (E, φ, θ) = 1 + |as↑ |2 + |as↓ |2 − |bs↑ |2 − |bs↓ |2
(5)
where E is the incident electron energy. For an unpolarised incident beam the total conductance is σS =
R 2π
R π/2
dφ 0 sin θdθ[σS↑ + σS↓ ]. The normalised conduc0
tance is σR = σS /σN , where σN is the conductance when
the superconductor is in the normal state. We obtained
analytic expressions for the conductance coefficients as,s′
and bs,s′ , as follows:
aji = kz2 [M −1 ]ij
(6)
bji = −ikz [(Zvu∗−1 + Y uv ∗−1 )M −1 ]ij − δij
(7)
where Z = hH/~2 , Y = Z + ikz and
Mij = [vu∗−1 ]ij Z 2 + [uv ∗−1 ]ij (Z 2 + kz2 )
(8)
u = u(~k) , v = v(−~k ′ )
u∗ = u∗ (~k ′ ) , v ∗ = v ∗ (−~k)
(9)
and
(10)
u and v are 2 × 2 Bogoliubov transformation matrices.
For a unitary gap function, these are given by43
u(~k) =
v(~k) =
[E + ǫ(k)]σ0
~
[E + ǫ(k)]2 + 21 Tr∆∆† (~k)]1/2
−∆(~k)
[E + ǫ(~k)]2 + 21 Tr∆∆† (~k)]1/2
(11)
(12)
where
σ0 is the identity matrix and ǫ(~k) =
q
E 2 − 21 Tr∆∆† (~k). For the non-unitary case, the expressions for u and v are somewhat more complicated
(see Eq. (2.14) of Ref.43 ).
C.
Calculations
We performed numerical calculations of the normalised
conductance σR (E) for the trial gap functions in Table I,
for different domains and domain averaging when necessary, and for a range of the parameters Z, a, b, η1 and η2 ,
resulting in more than 500 plots. We now discuss each of
the gap functions in Table I in detail.
The gap function associated with the Ag order parameter is conventional (s-wave) superconductivity. Fig.
5 shows the result for an s-wave superconductor, first
calculated by Blonder, Tinkham and Klapwick35 . For
small values of Z, it shows a distinctive gap edge at
E = ∆0 = 1. Excess spectral area clearly increases with
increasing Z, a feature that is common to all conductance
curves.
The two phases for the Eg order parameter each have
8 point nodes in the h111i directions. Their conductance
curves (not shown) feature broad peaks at energies associated with the gap maximum, but no ZBP’s.
A ZBP is a feature in the conductance curves of all of
the Tg phases listed in Table I. The phase D2 (C2 ) × K
with gap function ψ(~k) = kx ky has line nodes along
kx = 0 and ky = 0, and there are two other equivalent domains. The conductance curves have ZBP’s for
the domains ψ(~k) = kx kz and ψ(~k) = ky kz , but not for
ψ(~k) = kx ky . The domain average result is shown in
Fig. 6.
6
4
Z
3.5
Norm alized Con ductan ce ΣR
Norm alized Con d u ctan ce ΣR
4.0
0
0.2
0.5
0.8
1.0
1.5
5.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
Z
3
2
1
0
0.0
2.0
0
0.2
0.5
0.8
1.0
1.5
5.0
0.2
0.4
E
FIG. 5: (color online). Normalised conductance for a conventional superconductor (Ag ).
0.6
0.8
E
1.0
1.2
1.4
FIG. 7: (color online). Normalised conductance for the singlet
phase C3 × K, with gap function ψ(~k) = kx ky + ky kz + kx ky +
0.2(kx2 + ky2 + kz2 ).
Z
3.5
4.0
0
0.2
0.5
0.8
1.0
1.5
5.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
Norm alized Con d u ctan ce HΣL
Norm alized Con d u ctan ce ΣR
4.0
Z
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
E
0
0.2
0.5
0.8
1.0
1.5
5.0
0.5
1.0
1.5
2.0
ED0
As for the remaining Tg phases, the phase C3 × K has
6 point nodes in the h001i directions, but these nodes
disappear in the presence of an s-wave secondary OP
(and the ZBP persists), as shown in Fig. 7. Note that for
large Z, the ZBP evolves toward a broad, “hump-like”
structure. The phase C3 (E) (not shown) has 6 point
nodes at h001i and 2 point nodes at h111i. In the presence
of a T (D2 ) secondary OP the h001i nodes disappear, and
the ZBP moves away from E = 0. Finally, the phase
D2 (E) for the domain with gap function ψ(~k) = η1 ky kz +
iη2 kz kx has a line node along kz = 0 and 2 point nodes at
[00 ± 1]. The domain average result yields a ZBP for any
values of η1 and η2 . The results for D2 (E) (not shown)
closely resemble those of D2 (C2 ) × K (shown in Fig. 6).
The triplet phase belonging to Au is a nodeless, unitary superconductor. As shown in Fig. 8, instead of a
gap edge, a sharp minimum in the conductance spectrum appears at the gap energy. This feature has been
seen previously in calculations for Sr2 RuO4 45 .
The triplet phase belonging to the Eu order parameter with symmetry T (D2 ) is non-unitary. It has eight
point nodes in the h111i directions on the lower branch
of the gap function. The spectrum (shown in Fig. 9) has
FIG. 8: (color online). Normalised conductance for the triplet
~ ~k) = (kx , ky , kz ).
phase Au with gap function d(
a strong peak, but not at E = 0. The other phase of the
Eu order parameter, D2 × K, is unitary, and nodeless. Its
spectrum (not shown) has a peak, but not at E = 0.
4
Z
Normalized Conductance HΣL
FIG. 6: (color online). Normalised conductance for the singlet
phase D2 (C2 ) × K with gap function ψ(~k) = kx ky , averaged
over three equivalent domains.
0
0.2
0.5
0.8
1.0
1.5
5.0
3
2
1
0
0.0
0.5
1.0
ED0
1.5
2.0
FIG. 9: (color online). Normalised conductance for the triplet
~ ~k) = (kx , εky , εkz2 )
phase T (D2 ) the gap function d(
The triplet phase belonging to the Tu order parameter
with symmetry D2 (C2 ) × κ has point nodes in the [001]
7
Z
3.5
0
0.2
0.5
0.8
1.0
1.5
5.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
E
FIG. 10: (color online).
Normalised conductance for
~ ~k) =
the triplet phase D2 (C2 ) × K with gap function d(
(bky , akx , 0), averaged over three equivalent domains.
As for the other triplet phases belonging to the Tu order parameter, the phase C3 × K is unitary, and nodeless,
and it displays a broad peak at E 6= 0 (not shown). The
phase C3 (E) is non-unitary and has two point nodes in
the ±[111] directions on the lower branch of the gap function. Its spectrum (not shown) has strong peaks whose
positions vary according to the choice of parameters. Finally, the phase D2 (E) is non-unitary, with nodes that
are not in a direction of high symmetry46 . The domainaveraged spectra display peaks whose positions vary with
the choice of parameters.
Thus, the only phase in the triplet channel that exhibits a ZBP is D2 (C2 ) × κ, which belongs to the three
component OP Tu .
We conclude that there are several different pairing
symmetries for a Th crystal that can yield a conductance
spectrum with a ZBP, namely D2 (C2 ) × K, C3 × K, and
D2 (E) (belonging to the Tg OP) in the singlet channel
and D2 (C2 ) × K (belonging to the Tu OP) in the triplet
channel. In addition, all of these candidate gap functions
can account for both of the key observations of our experiment, namely the ZBP (corresponding to large Z) and
the broad hump-like structure (corresponding to small
Z), which implies that these features may be attributed
to a single OP present on two different bands with two
different values of Z.
IV.
4
Norm alized Con ductan ce HΣL
Norm alized Con d u ctan ce ΣR
4.0
ever, these results can be reconciled with ours if we account for slightly different experimental conditions. First
of all, we repeated the calculations within a narrow polar
angle θ < π/9 in momentum, to model the effect of a finite tunneling cone34,48 which applies to high-impedance
STS junctions in contrast to low-impedance PCS junctions. In addition, we considered different directions for
the incident beam, since Ref.47 did not identify the direction of the tunneling current with respect to the crystallographic axes. Furthermore, they reported that although their observed spectrum was reproduced in different places, it was not found over the whole surface.
We found that for the tunneling current in the [011] and
[111] directions and a narrow tunneling cone, the ZBP
seen in Fig. 6 disappeared, and was replaced by a gap
edge very similar to that in Fig. 5. Furthermore, of all the
calculated spectra exhibiting ZBP’s, this only occurred
for the states belonging to Tg (and not the one belonging
to Tu ). The result of this calculation is shown in Fig. 11.
Therefore, not only does this provide an explanation for
Z
0
0.2
0.5
0.8
1.0
1.5
5.0
3
2
1
0
0.0
0.5
1.0
ED0
1.5
2.0
4
Norm alized Con ductan ce ΣR
directions. This is a unitary case, and it does not exhibit
ZBP’s for d~ = (bky , akx , 0); however ZBP’s are present
for the other two domains. The domain average is shown
in Fig. 10.
Z
0
0.2
0.5
0.8
1.0
1.5
5.0
3
2
1
0
0.0
0.2
0.4
E
0.6
0.8
FIG. 11: (color online). Normalised conductance for the gap
function ψ(~k) = ky kz (top) and ψ(~k) = kx ky + ky kz + kx kz
for a tunneling current in the [011] direction with a narrow
polar angle integration.
DISCUSSION
Our experimental results are apparently in disagreement with a previous measurement of scanning tunneling
spectroscopy (STS) on PrOs4Sb12 by Suderow et al.47 ,
which observed a spectrum which closely resembles that
shown in Fig. 5, i.e., for an s-wave superconductor. How-
the difference between the results of Suderow et al. and
ours, it also strongly suggests that the gap function has
singlet pairing and belongs to the Tg order parameter.
We now discuss our results in the context of other experiments on PrOs4 Sb12 . No experiment has detected
the presence of line nodes in the gap function, which
8
is strong evidence against the phases D2 (C2 ) × K and
D2 (E) in the singlet channel. This leaves C3 × K in the
singlet channel as the best candidate for the superconducting pairing symmetry. Because of the secondary s-wave
OP, the state is nodeless at low temperatures, in agreement with thermal conductivity measurements16 . However, it should be noted that this proposed gap function
disagrees with the µSR measurements of Aoki et al.49 ,
who concluded that superconductivity in PrOs4Sb12 is
non-unitary. It also disagrees with the observation of
broken C3 symmetry, according to the shape of the vortex lattice6,50 .
We now offer a qualitative description of the model
that we propose. Superconductivity appears on a low
Z band with unconventional pairing (with symmetry
C3 × K) accompanied by the simultaneous appearance
of a similar OP on a high Z band, along with an s-wave
component. The growth of the OP on the two different
bands will be governed by the same temperature dependence, but with different magnitudes. Only one Tc is required, but the growth of the s-wave component will be
slower because it is a secondary OP. This may account
for the evolution of the spectral features we observe, but
not for the observation of two distinct Tc ’s. Since the OP
we propose has three components, there are different scenarios that could give rise to two different Tc ’s. Two possibilities are: i) s-wave superconductivity appears first,
followed by the unconventional OP that we described;
or ii) unconventional superconductivity first, followed by
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
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22
23
24
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changes in symmetry below each transition would be of
enormous benefit in discriminating these possibilities.
In summary, we have performed point-contact Andreev
reflection spectroscopy on PrOs4 Sb12 down to 90 mK and
up to 3 T and observed evidence for unconventional superconductivity involving multiple bands. A single superconducting phase of the Tg OP, present on two bands,
can account for our results. Among the possible superconducting phases of this OP, the phase C3 × K with
gap function ψ(~k) = kx ky + ky kz + kx kz + s agrees best
with the results of other experiments. Our observations
corroborate similar observations of multi-band superconductivity from bulk measurements, and highlight the unconventional nature of the pairing state in this heavy
fermion superconductor.
Acknowledgments
Research at the University of Toronto was supported
by grants from NSERC, CFI/OIT, OCE, and the Canadian Institute for Advanced Research under the Quantum
Materials Program. Research at UCSD was supported by
the US Department of Energy under research grant number DE FG02-04ER46105. Research at Memorial University of Newfoundland was supported by NSERC.
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