Solid State Communications 149 (2009) 1212–1215

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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc

Parity violation in a single domain of spin-triplet Sr2 RuO4 superconductors
H. Nobukane a,∗ , K. Inagaki a,b , K. Ichimura a,b , K. Yamaya a,b , S. Takayanagi a,b , I. Kawasaki c , K. Tenya d ,H. Amitsuka b,c , K. Konno a,b , Y. Asano a,b , S. Tanda a,ba

Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japanb

The 21st Century COE Program ‘‘Topological Science and Technology’’ from the MEXT of Japan, Hokkaido University, Sapporo 060-8628, Japanc

Department of Physics, Hokkaido University, Sapporo 060-0810, Japand

Department of Education, Shinshu University, Nagano 380-8544, Japan

articleinfo

Article history:
Received 26 December 2008
Received in revised form
8 April 2009
Accepted 24 April 2009 by S. Miyashita
Available online 3 May 2009
PACS:
73.23.-b74.70.Pq
74.78.Na

abstract
We observed an unconventional parity-violating vortex in single domain Sr2 RuO4 single crystals using atransport measurement. The current–voltage characteristics of submicron Sr2 RuO4 show that the induced
voltage has anomalous components which are even functions of the bias current. The results may suggestthat the vortex itself has a helical internal structure characterized by a Hopf invariant (a topologicalinvariant). We also discuss that the hydrodynamics of such a helical vortex causes the parity violation
to retain the topological invariant.
© 2009 Elsevier Ltd. All rights reserved.

Keywords:
A. Spin-triplet superconductor
D. Parity violation
E. Electric transport

A quantized vortex is a topological matter in superconductors
and superfluids. In metallic superconductors, the Abrikosov vortex
is characterized only by an integer winding number of phase.
In unconventional superconductors, the superfluid of 3 He andspinor cold atoms, internal degrees of freedom of the orderparameter enrich the variety of vortices [1–3]. Although a numberof theoretical studies have predicted the existence of such
unconventional vortices, experimental confirmation of them isstill limited to several studies such as NMR spectroscopy of 3 He
and imaging of spinor cold atoms [4,5]. Here a Cooper pair in
spin-triplet superconductors has electric charge 2e. Thus one canresolve the dynamics of unconventional vortices through electrictransport. We will address this issue in the present Letter.
Sr2 RuO4 [6] is a promising spin-triplet chiral p-wave superconductor candidate (i.e., spin S = 1 and orbital angular momentum L = 1). Since two states with different chirality degenerate in the ground state, bulk Sr2 RuO4 is considered to have chiral domain structures. The transport properties have been studied in relation to Josephson interferometry using bulk Sr2 RuO4

∗ Corresponding author. Tel.: +81 11 706 7293; fax: +81 11 706 6154.
E-mail address: nobukane@eng.hokudai.ac.jp (H. Nobukane).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2009.04.022

crystals to determine the symmetry of Cooper pairs and measure the dynamics of the chiral domains [7,8]. These experimental data on bulk Sr2 RuO4 should be considered as a result of ensemble averaging over possible chiral domain configurations. Thus we need a small enough sample of Sr2 RuO4
rather than the domain size to study phenomena that are peculiar to a single chiral domain such as dynamics of a single
chiral domain, spin supercurrent, and unconventional vortices[9–11]. Transport measurements, however, have never been carried out yet in a single domain because it is also difficult to attach
electrical contacts to submicron Sr2 RuO4 crystals.
In this Letter, we will report an anomalous property of the current–voltage (I–V ) characteristics in a single domain of Sr2 RuO4 .
The creation of vortices gives a finite resistivity even when thetemperature is well below the superconducting transition temperature. In four-terminal measurements, the induced voltage V isusually an odd function of the bias current I. Namely, V changes its
sign when we flip the direction of current to the opposite direction,
which implies parity conservation [12]. However, we find in submicron Sr2 RuO4 samples that V has anomalous components which
are even functions of I. The existence of the anomalous components
means that positive voltage is detected regardless of the current direction and suggests the violation of parity [12]. To understand thenature of the anomalous I–V characteristics, we consider a simple

H. Nobukane et al. / Solid State Communications 149 (2009) 1212–1215

model of a vortex which has a helical internal structure characterized by a Hopf invariant. We also show that hydrodynamics ofsuch a helical vortex violates the parity to retain the topologicalinvariant.
To obtain submicron Sr2 RuO4 single crystals, we synthesized
Sr2 RuO4 crystals with a solid phase reaction and then determinedthe crystal structure of Sr2 RuO4 and the concentration of
impurities. We prepared SrCO3 and RuO2 (both 99.9%, Kojundo
Chem.) powders. The mixed powder was then heated at 990 ◦ C
for 60 h. The mixture was cooled gradually from 990 ◦ C to450 ◦ C over 6 h. The samples were kept at 450 ◦ C for 12 h to
introduce oxide into the crystals and then cooled down slowly
at room temperature. The structure of the Sr2 RuO4 crystals wasanalyzed by using X-ray power diffraction (Rigaku Diffractometer
RINT 2200HK) with Cu Kα radiation. The observed peaks fitted abody-centered tetragonal unit cell of the K2 NiF4 type with lattice
constants a = b = 3.867 (±0.004) Å and c = 12.73 (±0.01) Å[13]. The result of secondary ion-microprobe mass spectrometry(SIMS) shows that the concentration of Al in the sample is less than
100 ppm, while the superconductivity of Sr2 RuO4 is destroyed bynonmagnetic impurities [14].
We selected submicron Sr2 RuO4 single crystals from theobservations of chemical composition and crystallinity. The
samples were dispersed in dichloroethane by sonication anddeposited on an oxidized Si substrate. We found typical samples indiameter of about 50 nm–500 µm. Energy dispersion spectroscopy(EDS; EX-64175 JMU, JEOL) was used to determine the componentsof the submicron samples on the substrate. The molar fractionof the Sr and Ru elements was 2:1. We also confirmed thatthe dispersed crystals had neither boundaries nor ruthenium
inclusions on the sample surface by observing the crystal
orientation using the electron backscatter diffraction pattern(EBSP; OIM TSL [15]).
On the analyzed Sr2 RuO4 , we fabricated gold electrodes using
overlay electron beam lithography. Inset (a) in Fig. 1 showsa micrograph of our samples. The sample size is 2.50 µm ×1.88 µm × 0.10 µm. The sample electrode spacing is 0.63 µm.
Since the fabricated sample surface may have the insulator surfaceof the layer crystals and the residual resist between the sample
and the gold electrodes, it is difficult to form an electricalcontact. Therefore we performed a welding using electron beamirradiation [16]. We heated each electrode on the sample for 15 swith a beam current irradiation of 2 × 10−7 A. As the result, we
succeeded in greatly reducing the contact resistance below 10 at room temperature.
The measurements were carried out in a dilution refrigerator(Kelvinox, Oxford) with a base temperature of 60 mK. All
measurement leads were shielded. The lead lines were equippedwith low pass RC filters (R = 1 k, C = 22 nF). In the DC
measurements, a bias current was supplied by a precise current
source (6220, Keithley) and the voltage was measured with ananovoltmeter (182, Keithley) using four-terminal measurements.
We measured the temperature dependence of the resistivity inthe submicron Sr2 RuO4 . Inset (b) of Fig. 1 shows the temperaturedependence of the resistivity in the ab plane from room
temperature down to 4.2 K. Fig. 1 shows that the resistivityρab (4 K) = 6.0 µ cm. This value is larger than the bulk resistivity
by about three times [14]. We estimated the resistivity ρab from thesample size. Since Sr2 RuO4 has anisotropic resistivity ρab ≈ ρc ×
10−3 µ cm, the resistivity ρab may actually be smaller than theestimation. Here the ratio ρab (300 K)/ρab (4 K) ∼ 40 is comparable
to that of the bulk used in Ref. [6]. Hence we consider there isno degradation of the sample by the solvent. Fig. 1 also shows atransition temperature of Tc = 1.69 K and a broader transition
temperature width of 1T ≈ 200 mK. There was no decreaseof the resistivity when a magnetic field of 3000 G was applied

1213

Fig. 1. Temperature dependence of resistivity of submicron Sr2 RuO4 in zeromagnetic field (0 G) and in a magnetic field (H = 3000 G) applied parallel to the caxis. Flat tail resistivity can be seen at low temperatures below Tc = 1.69 K. Inset(a) shows a micrograph of a submicron Sr2 RuO4 single crystal connected to gold
electrodes. Inset (b) displays the temperature dependence of the resistivity in theab plane from room temperature down to 4.2 K.

parallel to the c axis. Our sample shows neither suppression of
Tc nor enhancement to 3 K [14,17]. Here the resistivity retained
its flat tail below Tc . The result shows that the flow of vortices
can occur by quantum fluctuations of the superconducting phaseθ [18]. The results show transport properties of the submicron
Sr2 RuO4 single crystals because a broader transition temperature
width and quantum fluctuations of the phase are characteristic ofmesoscopic superconductors [19].
We observed anomalous I–V characteristics in zero magnetic
field. Fig. 2(a) shows I–V curves at several temperatures withtypical flat tail resistances of R∗ ≈ 0.16 . In general, the voltage
in I–V curves for metals, quantum Hall systems and Josephson
junction is always an odd function of the bias current, which is a
result of parity conservation. Surprisingly, V is not an odd functionof I at all. In what follows, we define anomalous nonlinear voltage(ANV) as the component of measured voltage given by an evenfunction of I. An ANV implies the violation of parity. The amplitudeof the ANV increases with decreasing temperature in zero magnetic
field and shows a maximum below 200 mK. In order to eliminate
completely the possibility of instrument malfunction in the DC
measurements, I–V curves were measured with a microvoltmeter(AM 1001, Ohkura Electric Co.) with a battery-powered current
source. Furthermore, in the AC measurements, we also measured
the differential resistance dV /dI as a function of the bias current
using lock-in techniques. Fig. 2(b) clearly shows that dV /dI has an
odd component of I. The parity violation in the I–V characteristics
is confirmed in both the DC and AC measurements. Moreover, we
confirmed that the anomalous effect was reproduced in several
samples.
To analyze the ANV in more detail, we subtract the linear part(ohmic contribution to voltage) from the I–V curves in Fig. 2(a).
The results are shown in Fig. 3. We clearly find that the ANV issymmetric with respect to the zero bias current, (i.e., V1 (+I ) =V1 (−I )). Here the voltage V1 represents the ANV of the induced
voltage V = R∗ I + V1 . These curves are described well by a
Lorentzian curve, as shown by the lines.
We discuss the physical difference between parity-violating
I–V characteristics and negative resistance. Negative resistanceitself is an unusual phenomenon. The phenomenon, however, ispossible. It is instructive to compare our result to the negative
resistance of mesoscopic charge density waves (CDWs) reported

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H. Nobukane et al. / Solid State Communications 149 (2009) 1212–1215

Fig. 2. (a): Results of DC measurements. Voltage V is plotted as a function of bias current I in the absence of magnetic field for several choices of temperatures. The amplitudeof the ANV increases with decreasing temperature and shows a maximum below 200 mK. (b): Results of AC measurements. The differential resistance dV /dI versus bias
current I is shown at T = 200 mK and 650 mK. The upper and lower regions of the transverse dotted line represent positive and negative differential resistance, respectively.
As shown in (a) and (b), parity violation is confirmed in two different measurements.

Fig. 3. The anomalous nonlinear voltage (ANV) which is described by the voltage V1
is extracted from I–V characteristics in Fig. 2, where the lines are Lorentzian fitting
curves. The half-widths 1I of the fitting curves are represented by two-headed
arrows. We eliminated the offset voltage of 0.13 µV in order to discuss the ANV.

in Ref. [20]. In their report, the negative resistance was attributed
to the backflow of quasiparticles of the CDW. This does not relate
to parity violation of the I–V curves. On the other hand, ourobservation of V1 (+I ) = V1 (−I ) in the submicron Sr2 RuO4 isa parity violation, which is a qualitatively different phenomenon
from the CDW case. The parity violation of I–V curves must
naturally include both negative resistance and negative differential
resistance. In addition, since the measurement was carried out inthe four-terminal configuration, the observation does not violate
any basic laws, such as energy conservation. From the reasons, we
focus on discovery of the parity violation.
What is the origin of the ANV in the I–V characteristics? The flat
tail of resistivity (which often appears in superconductors owing
to quantum fluctuations of the phase [18]) shown in Fig. 1 maysuggest that the flow of vortices causes the ANV. However, thedynamics of the usual Abrikosov vortex in type-II superconductors
cannot explain the parity violation in the I–V curves. Therefore
we need to consider unconventional vortices characterized by El
E textures as in superfluid 3 He-A. Here the El vector represents
and d
the direction of the pair angular momentum parallel to the c axis
E vector describes the spin configuration of a pair. Theseand the dinternal degrees of freedom are characters of spin-triplet p-wave
superconductivity. Kerr effect measurements of Sr2 RuO4 [21] andJosephson tunneling measurements [8] respectively suggested thechiral domain size to be 50 ∼ 100 µm and 1 µm. Recently, an
experiment on the 3 K phase of Sr2 RuO4 also revealed that thedomain size is ∼10 µm [22]. Since our sample electrode spacing is
0.63 µm, our sample is considered to have a single chiral domain.
E allows the formation
The spin degree of freedom represented by dE textures in the single domain. In bulk Sr2 RuO4 , the spin–orbitof d
E and El in zero magnetic
interaction favors the alignment of dfield. Knight shift measurements have recently suggested that amagnetic field H > 200 G may neutralize the interaction [23].
However, in submicron Sr2 RuO4 , quantum fluctuations of the
E in a particular direction. Thus
phase disturb the alignment of dEd-textures
would be possible in a single domain of Sr2 RuO4 .
In what follows, we consider a spin-triplet Cooper pair asa spin-1 boson with charge 2e. Babaev theoretically predictedthat the ground state of such a boson system can have magnetic
spin textures Es characterized by a topological invariant known as
helicity in zero magnetic field [10]. The equivalence between gaugetransformation and spin rotation causes a term
Bk 2 =

2cM
h̄c− 2 [∇i Jj − ∇j Ji ] + (Es · ∇iEs × ∇jEs)
4e n

4e

(1)

in the Ginzburg–Landau energy functional, where ∇i = dxd , J isithe electric current, M is the mass of a boson, and n is the boson
density. In Ref. [24], the authors described a nontrivial topologicalstructure in the simplest toroidal knot soliton. The main distinction
between knotted solitons in spin-triplet superconductors and the
topological defects of 3 He in Ref. [25] is the appearance of terms ∝(Es · ∇iEs × ∇jEs)2 . This results in the knotted solitons being protectedagainst shrinkage by an energy barrier. According to Ref. [10],the size of the knotted soliton is comparable to the magnetic
penetration length. Our sample satisfies this requirement about
the size. We note that the sample size ≈1 µm and λab ≈ 152 nm.
In the Sr2 RuO4 single domain, a helical vortex could be createdby the spin degree of freedom. If we accept the existence of such
helical vortices, the anomalous I–V characteristics could be understood as a result of the conservation of a topological invariant in
the helical vortex. Now let us consider the hydrodynamics of a helical vortex. In the initial state of the bias current I = 0, the helicalvortex does not move. When we switch on a bias current I > 0,a clockwise helical vortex (ω > 0) exerts the Magnus force in a

H. Nobukane et al. / Solid State Communications 149 (2009) 1212–1215

1215

V

ω >0

I> 0

V

y
ω<0

x

(a) Clockwise.

I< 0(b) Counterclockwise.

Fig. 4. A model of a vortex which violates parity. A topological invariant features a helical structure of the vortex. The blue spiral line on the torus represents a magnetic
helical structure of spin texture described by (Es · ∇i Es × ∇j Es). Under the bias current, the spin texture around the torus moves periodically from the inside to the outside. Openarrows represent the current flow I. Solid arrows show the direction of the Magnus force. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

direction perpendicular to the current, as shown in Fig. 4(a). Here+y indicates the direction of the Magnus force. On the other hand,for a bias current I < 0, the helical vortex changes its rotation
frequency from clockwise (ω > 0) to counterclockwise (ω < 0)in order to retain the topological invariant. As a consequence, thecounterclockwise helical vortex also exerts a Magnus force in the+y direction as shown in Fig. 4(b). Thus the direction of the Magnus
force is independent of the direction of the bias current. According
to Josephson’s equation, the motion of a vortex in the +y direction
induces a voltage across the sample of Sr2 RuO4 in the x direction.
In this way, the presence of helical vortices explains the ANV.
Let us discuss the amplitude voltage of the ANV using thesimple energy conversion formula eV ∼ µB B, where µB is theBohr magneton, in the I–V curves. The energy of the amplitudevoltage V1 = 0.97 µV of the ANV at 63 mK is comparable to theenergy of the magnetic field H ≈ 200 G which neutralizes thespin–orbit interaction in bulk Sr2 RuO4 [23]. Thus we consider thatthe amplitude voltage of the ANV may exhibit the contribution of
the helical vortices.
Finally we briefly discuss the meaning of this experiment.
Although Sr2 RuO4 is a spin-triplet superconductor candidate, thisconclusion is still under debate. We show in this Letter that a
Cooper pair in Sr2 RuO4 has a spin degree of freedom. Thus our
results exhibit a possibility of spin-triplet symmetry [26]. When
helical vortices exist in a sample, such topological defects may
affect the Hall conductivity. This inference stems from an analogy
between the Chern–Simons term in the quantum Hall effect andthe helical spin term in Babaev’s argument. Thus we believe thattransport experiments which are sensitive to probe geometry
would display more interesting phenomena reflecting the internal
degree of freedom of a Cooper pair.
In summary, we have observed an unconventional vortex which
violates the parity in a single domain of Sr2 RuO4 using a transportmeasurement. The I–V characteristics of submicron Sr2 RuO4 showthat the voltage has anomalous components which are evenfunctions of the bias current. We consider a vortex with a helicalinternal structure characterized by a Hopf invariant. The invariantof the vortex is protected while the vortex is moving under the bias
current. By a simple argument, we show that the hydrodynamics
of the helical vortex causes the anomalous I–V characteristics toretain the topological invariant.
Acknowledgements
The authors thank A. Ishii, S. Tsuchiya, T. Toshima, T. Matsuura,
T. Kuroishi, and D. Meacock for experimental help and useful

discussions. We also are grateful to Y. Maeno, and T. Matsuyama.
H.N. acknowledges support from the Japan Society for the
Promotion of Science.
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