Superconductivity and Magnetism in
Topological Half-Heusler Semimetals
1
1
1
1
Yasuyuki Nakajima , Rongwei Hu , Kevin Kirshenbaum , Alex Hughes ,
1
1
1
1
Paul Syers , Xiangfeng Wang , Kefeng Wang , Renxiong Wang ,
1
2
2
1
Shanta Saha , Daniel Pratt , Jeffrey W Lynn , Johnpierre Paglione
1
University of Maryland, College Park
2
NIST, Center for Neutron research
Sci. Adv. 1, e1500242
Yasuyuki Nakajima
Outline
• Introduction:
• Topological insulator: a new quantum state of matter
• Half Heusler semimetal RPdBi: promising candidate for a topological material
• Experimental Results for RPdBi:
• Magnetic susceptibility: Localized f electrons
• Neutron diffraction: Antiferromagnetism
• Charge transport and magnetic measurements:
Superconductivity
• Discussion:
• Realization of peculiar superconductivity: Singlet-triplet mixing, Magnetic SC, BCS-BEC crossover
• Summary
Yasuyuki Nakajima
Topological Insulator
Phase transition
Breaking symmetry
RESEARCH ARTICLES
Crystal: Broken
translational symmetry
Quantum Spin Hall Insulator State
Magnet:
Broken
in HgTe
Quantum WellsSuperconductor: Broken
rotational
symmetry
Markus König, Steffen Wiedmann,
gauge symmetry
Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1
1
Laurens W. Molenkamp, * Xiao-Liang Qi,2 Shou-Cheng Zhang2
1
1
New quantum phase of matter
Topological insulator
T
absence of symmetry breaking
e.g. Quantum spin Hall state
2D topological insulator
theoretically that the electronic
he discovery more than 25
Conductance
structure of inverted HgTe quanyears ago of the quantum
channel with
up-spin charge
tum wells exhibits the properties
Hall effect (1), in which the
carriers
that should enable an observation
“Hall,” or “transverse electrical” conof the quantum spin Hall insulaductance of a material is quantized,
tor state. Our experimental obsercame as a total surprise to the physics
vations confirm this.
community. This effect occurs in
These experiments only belayered metals at high magnetic
came possible after the develfields and results from the formaopment of quantum wells of
tion of conducting one-dimensional
sufficiently high carrier mobility,
channels that develop at the edges
combined with the lithographic
of the sample. Each of these edge
techniques needed to pattern the
channels, in which the current moves
sample. The patterning is espeonly in one direction, exhibits a quancially difficult because of the very
tized conductance that is characterConductance
high volatility of Hg. Moreover,
istic of one-dimensional transport. The
channel with
Quantum
down-spin
we have developed a special low–
number of edge channels in the samwell
charge carriers
deposition temperature Si-O-N
ple is directly related to the value of
gate insulator (7), which allows
the quantum Hall conductance. MoreSchematic of the spin-polarized edge channels in a quantum spin Hall
us to control the Fermi level (the
over, the charge carriers in these chaninsulator.
energy level up to which all
nels are very resistant to scattering.
Not only can the quantum Hall effect be observed in macroscopic samples electronics states are filled) in the quantum well from the conduction band,
for this reason, but within the channels, charge carriers can be transported through the insulating gap, and into the valence band. Using both electron
without energy dissipation. Therefore, quantum Hall edge channels may be beam and optical lithography, we have fabricated simple rectangular
useful for applications in integrated circuit technology, where power dis- structures in various sizes from quantum wells of varying width and
chiral boundary state
ded from www.sciencemag.org on November 17, 2009
AUTHORS’ SUMMARY
M. Konig et al. Science (2007)
Yasuyuki Nakajima
Topological Insulator
L. Fu et al. Phys. Rev. Lett. (2007)
3D Topological insulator
Metallic surface state protected against time reversal
NATURE PHYSICS
DOI: 10.1038/NPHYS1274
invariant
perturbations
M. Z. Hasan and C. L. Kane:
news & views
Low
Trivial Topological
a
High
Bi2Se3
Intensity (a.u.)
–
+
–
+
–
Energy
EB (eV)
c
EF in
0.1
0.1
similar in some ways to what is found
a
b
c
K
graphene.
Γ
Γ However, Kif the host is 3D and
M
M
the valence and conduction bands cross
over once (or an odd number of times),
0
0
the 2D surface states are completely
SS
different: indeed, theorists have shown that
SS
Conduction
these even and odd boundary states are
band (host)
–
+
+ ¬0.1 –
+
+
¬0.1
topologically distinct.
En
Boundary
A distinguishing characteristic of the
E0
states
odd FIG.
states12.(which
are known as strong
!Color online" Helical fermions: Spin-momentum
Ep
topological
insulators)
is that
backscattering
locked helical
surface Dirac
fermions
are hallmark signatures
¬0.2
¬0.2
Valence
of topological
!a" ARPES
data can,
for Bi2Se3 reveal
is forbidden:
thisinsulators.
means that
electrons
band (host)
surface electronic states with a single spin-polarized Dirac
in principle,
propagate with little or no
cone. !b" The surface Fermi surface exhibits a chiral leftresistance
the edge
or surface
of structure
the
Momentum
0
0
0
handed along
spin texture.
!c" Surface
electronic
of Bi2Se3
¬0.3
¬0.3
computed
in the
local-density
approximation.
system
— even
if the
host is an
insulator. The shaded regions
describe
bulk
states,
and
the
linestoare
This is a property that could prove
besurface states. !d"
Schematic of the spin-polarized surface-state dispersion in
veryBiuseful
for applications. To see why
2X3 !1;000" topological insulators. Adapted from Xia et al.,
backscattering
is forbidden,
Fig.and
2b:Xia, Qian,
¬0.4
¬0.4
2008, Hsieh, Xia,
Qian, Wray,consider
et al., 2009a,
al.,0.15
2009.
electron
so that its
¬0.15 if anHsieh,
0 Wray,isetbackscattered
¬0.15
0
0.15
momentum
to –k,
0
kx (Ŭ1) (k) is changed from +k¬0.12
ky (Ŭ1)
Allspin
keymust
properties
topological
then its
also beofflipped
fromstates
up tohave kbeen
Manoharan
etband
al.structures.
Nat. Nano
y (Å
Figure 2 H.C.
| Topological
insulators and
a, The(2010)
conduction and valence bands of a
Y.demonstrated
Xia
et al.
Nat.
(2009)
has
the simplest
Dirac
for
Bi2SePhys.
3 which
down,
or vice
versa.
However,
something
is
typical 3D solid (middle section). The shaded regions are the bands in the bulk of the solid, and the
cone
surface
spectrum
and
the
largest
band
gap.
In
needed
to
flip
the
spin,
such
as
a
magnetic
thick black lines are the bands at the surface. (Similar behaviour
is observedNakajima
in a 2D system with a
Bi2Te3 the surface states exhibit large deviations from a
Yasuyuki
impurity
or a magnetic field. If nothing
d
f
EB (eV)
b
–
kind
onic
so-called
f very low
modes
mber
existence of
und result
er index
stence of
in some exotic superconductors including
strontium ruthenate20. The case in which
the normal-state electrons obey Dirac-like
beahviour is predicted to be induced at the
surface of a new class of material called
topological insulators21 or in graphene22,
by exploiting the proximity effect to induce
superconductivity in those materials. Thus,
the race is on to find such exotic realizations
of Majorana’s idea in a variety of systems.
Topological order with symmetry breaking
Unusual collective modes predicted in particle physics
b
Axion
Majorana fermion
Anomalous magnetoelectric effect
Charge
neutral
+
ĉ† = ĉ
M=
c
(e2 /4⇡~c)✓E
✓ : axion field
F. Wilczek et al. Nat. Phys. (2009)
olid state. a, A familiar concept in solid-state physics, holes are
ermi sea of the electronic spectrum, behaving like positively charged
properties of electrons (blue) and holes (grey) are drastically
surrounding sea of Cooper pairs; a hole can attract or bind to a
arge. c, More importantly, Cooper pairs cluster around holes and thin
hat no rigorous distinction between them remains.
Topological order
+
Superconductivity
D43.00015
617
Yasuyuki Nakajima
R. Li et al. Nat. Phys. (2010)
Topological order
+
Magnetism
Half Heulser semimetals
Cubic MgAgAs type
RPdBi
R
Topological
Topological
LuPdBi
LuPdBi
Pd
YPdBi
Trivial
Trivial
Bi
YPdBi
LaPdBi
LaPdBi
Trivial
a
0.08
LETTERS
Topological
L
L
¬0.08
s-like
Lin et al. Nature material (2010)
inversion. a, The crystal structure of half-Heusler compounds MM⇥ X. M, M⇥ and X are denoted by
re formed by M⇥ and X is emphasized by black lines. b–d, Band structures near the point for trivial,
vely. Red dots denote the s-like states near the point. Band inversion occurs in the non-trivial case
ourfold degenerate J = 3/2 states. The degeneracy of the J = 3/2 states is lifted by the lattice distortion
∆
6.66
L
Σ
0
¬0.04
EF
J = 3/2
J = 3/2
LuPdBi
Topological
insulator
Semimetal
Band insulator
YPdBi
Z 2 = ¬1
Γ
Energy (eV)
¬0.4 ¬0.2
0
0.2
d
Z 2 = ¬1
∆
0.4
c
Z 2 = +1
0.04
Energy (eV)
¬0.4 ¬0.2
0
0.2
b
LaPdBi
EΓ6 ¬ E Γ8 (eV)
MAT2771
0.4
Trivial
Al-Sawai et al. PRB (2010)
0.4
Lack of inversion symmetry
LETTERS
Energy (eV)
¬0.4 ¬0.2
0.2
0
Lin et al. Nat. Mat. (2010)
Γ
YPtSb
c
YAuPb
d
TiNiSn
Γ
Σ
Σ
6.67
6.68
6.69
6.70
Lattice constant (Å)
S. Chadov et al. Nature Material (2010)
6.71
Figure 4 | E 6 E 8 difference for YPdBi. The insets show the band structures a
zone. a, E⇥6 E⇥8 changes on scaling the lattice constant from positive (trivial ins
constant (marked by a dashed vertical line) corresponds to the zero gap with a D
coupling at the critical lattice constant, the system can be transformed from trivia
At the border between trivial and topological states
b
∆
Yasuyuki Nakajima
a
b
Rare earth based RPdBi
RPdBi: Promising tunable topological
materials with (multi-)symmetry breaking
Antiferromagnetism
Superconductivity
PHYSICAL REVIEW B 84, 035208 (2011)
YPtBi
BUTCH, SYERS, KIRSHENBAUM, HOPE, AND PAGLIONE
4
0.04
YPtBi
2.5
H
3
1 kHz
0.04 Oe
0.8
0.2
1
χ' (arb. unit)
0.6
0.4
0
0
ng
l.
K. Gofryk et al. PRB (2011)
0
0
0.2 0.4 0.6 0.8
50
100
1
200
250
0.1
0.15
-1
-1
H (T )
xy
60
0.2
∆ρ (mΩ cm)
ρ
xx
20
0.5
0
1
T (K)
1.5
2
0
T=3K
0
5
10
FIG. 3. (Color
online) Superconducting
phase transition in
H
(T)
=
1.0
K.
The
transition
is observed in both
LuPtBi,
occurring
at
T
c
FIG. 1. (Color online) The temperature dependence of the rethe
resistivity
(blue
curve,
right
axis)
and
the
magnetic
susceptibility
sistance of YPtBi is nonmetallic between 0 and 300 K, probably
FIG. 2. (Color online) Magnetotransport properties of
YPtBi
curve,
left axis).
indicative of a semimetallic band structure. The Hall constant RH(red with
magnetic
field along the [001] direction. The low-temperature
T (K)
F. Tafti et al. PRB (2013)
exhibits a similar temperature dependence. Within a one-band model,
(3 K) transverse magnetoresistance ρxx and Hall resistance ρxy
t =a T
/Tc [Fig.
4(b)].
the zero-temperature
the carrier concentration varies from 2 × 1018 cm−3 at 2 K towhere
exhibit
remarkable
linearity
up toUsing
14 T. Above
3.5 T, the Hall
2 × 1019 cm−3 at 300 K. The inset highlights the superconductingrelation
response
magnetoresistance,
both signals, length
quantumξ0 =
Hc2exceeds
= φ0 the
/2π
ξ02 , we extractand
theincoherence
transition with critical temperature Tc = 0.77 K, as seen in resistivity14 nm.
oscillations
are
clearly
visible
at
higher
fields.
Inset:
Shubnikov–de
Comparing the coherence length with the mean free
and ac magnetic susceptibility.
oscillations at 0.1 K, with the background magnetoresistance
path,Haas
we
find our sample satisfying the clean limit condition
subtracted, have a period of approximately 0.02 T−1 (vertical lines)
l ≫ and
ξ0 . an effective mass of 0.15 me . A node due to beating (arrow) at
0.12 T−1 produces a π phase shift in the oscillations (+/− signs).
at least 14 T. Shubnikov–de Haas oscillations are readily
150
apparent, and can be observed up to temperatures of at least
10 K. The inset of Fig. 2 shows five such oscillations measured
below 80(a)
K to 1000 cm2 V−1 s−1 at 300 K. This behavior
at 0.1 K, with background subtracted, which have a frequency
implies
that
most of the reduction in ρ(T ) with increasing
125
RPtBi
single crystal
of approximately 46 T. Assuming a spherical Fermi surface
for simplicity, this corresponds to a carrier density of 1.7 ×
1018 cm−3 , in good agreement with the Hall measurements.
Yasuyuki Nakajima
temperature is nominally due to an unusually large increase in
carrier
100 concentration. Strangely, µ(T ) shows a much less dramatic temperature dependence than that of another low carrier
)
RPdBi
polycrystalline samples
ρ
- -
++
40
0
0
300
N. Butch et al. PRB (2011)
Bi,
ly
ns
ee
bit
ed
nts
inD43.00015
2
80
1
1.2 1.4
150
-0.02
-0.04
0.05
0.5
0.2
0
ρ (µΩ cm)
1
0.4
H
1.5
R (cm /C)
ρ (mΩ cm)
R
3
2
xx
xx,xy
ρ
100
YPtBi
T = 0.1 K
F = 45.8 T
0.02
LuPtBi
χρ(arb.(mΩ
units)
cm)
0.6
PHYSICAL REVIEW B 84, 220504(R) (2011)
SUPERCONDUCTIVITY IN THE NONCENTROSYMMETRIC . . .
0.8
b)
/2
LuPtBi
RAPID COMMUNICATIONS
Sample preparation
Single crystal
Self-flux method
R : Pd : Bi = 1 : 1 : 5 -10
R
Pd
Bi
1 mm
DyPdBi
6
MgAgAs-type
FIG. 1: Evolution of lattice constants as a function of rare earth species R in RPdBi determined by X-ray di↵raction, showing the
lanthanide contraction e↵ect on the half-Heusler cubic (F 4̄3m) crystal structure (inset). The dashed line indicates the critical
lattice constant ac = 6.62 Å demarcation between positive and negative electronic band inversion strength E = E 8 E 6 [12],
and therefore between RPdBi members with predicted topologically non-trivial (squares) and trivial (circles) band structure.
ac = 6.62 Å
S. Chadov et al. Nat. Mat. (2010)
D43.00015
Yasuyuki Nakajima
Magnetic susceptibility
a
d b
7
e
c
2
b
e
FIG. 2: Physical properties of RPdBi single M
crystals, focusing
C on evolutio
Currie-Weiss
behavior
=
Sm, Gd, Tb, Dy, Ho, Er, and Tm. a Magnetic susceptibility
M/H of R
R TN (K) ⇥W (K) µe↵ (µB ) µfree (µB )
Sm
3.4
-258
1.9
0.85
Gd 13.2
-49.6
7.66
7.94
Tb
5.1
-28.9
9.79
9.72
Dy
2.7c
-14.3
10.58
10.65
Ho
1.9
-9.4
10.6
10.6
Er
1.0
-4.8
9.18
9.58
Tm <0.4
-1.7
7.32
7.56
HDyin GdM/H
T denoting
⇥W antiferr
Curie-Weiss behavior and clear, abrupt decreases
Sm
Er
magnetic YPdBi and LuPdBi, exhibiting
b Low-temp
Y diamagnetic
Tm Ho
Tb behavior.
Lu
and c for Gd and Sm, with arrows⇥indicating
Néel temperatures.
(Note: 1 e
temperature
W : Weiss
M at 2 K for magnetic rare earth members R = Sm, Gd, Tb, Dy, Ho, Er
in the temperature range 2 K – 300 K, showing non-monotonic temperature
charge carrier density nH obtained from single-band analysis of Hall e↵ect
sign of all carriers is positive (see supplementary).
f electrons: well-localized
Low temperature
btained from the heat capacity and magnetization, Weiss temperature ⇥ , and the e↵ective
with AFM
rom a fit to the Curie-Weiss expression. The obtained anomaly
e↵ective moments associated
are close to the
W
m.
D43.00015
Yasuyuki Nakajima
Neutron diffraction
Magnetic Brag peak
8
Order parameter
b
a
Pd
R
Bi
RAPID COMMUNICATIONS
type-II order
AFM
= temperature
(1/2,1/2,1/2)
FIG. 3: Characterizationfcc
of antiferromagnetic
with elastic neutron di↵raction.Q
a, Low
magnetic di↵raction
PHYSICAL REVIEW B 90, 041109(R) (2014)
R. A. MÜLLER et al.
cm)
RH(cm3/C)
3.0
1.2
pattern of DyPdBi obtained by subtracting 18 K data from 1.5 K data. Labels indicate the
series of half-integer
antiferro2.66
2.64(0.5,0.5,0.5)
magnetic peaks. b, antiferromagnetic order parameter of single-crystal TbPdBi obtained from the intensity of the
2.5
1.1
2.62
magnetic Bragg peak. Solid curve is a mean-field fit to the data, and the inset presents a schematic of the antiferromagnetic
0
50
2.0
T(K)
spin structure.
1.0
Resistivity (m
Ho, Tb
R. Mong et al. PRB (2010)
R. Muller et al. PRB (2014)
1.0
the same magnetic structure
0.8
0
D43.00015
1.5
0.9
RH(cm3/C)
Topological AF insulator?
FIG. 1. (Color online) The Gd atoms are shown in black (blue),
the Bi as gray (gray), and the Pt as white (yellow). The spins on
the Gd atoms are oriented in ferromagnetic planes which are stacked
Yasuyuki Nakajima
50
100
150
200
250
300
0.5
Temperature (K)
FIG. 2. The solid points show the resistivity ρ(T ) of GdBiPt at
Charge transport
e
Y
Sm
Tm
Er
Ho
Dy
Tb
Gd
Lu
PHYSICAL REVIEW B 82
FENG et al.
Carrier: Hole
Energy(eV)
1.0
(a) LaPtBi - LDA
1.0
Γ8
(b) LaPtBi - MBJLDA
Γ8
1.0
Semi-metallic behavior
cusing on evolution of magnetic order with rare earth species R =
19
-3
bility
M/H
of
RPdBi
members
with magnetic R species, showing
n
~
10
cm
denoting antiferromagnetic transitions. Inset presents data for nonavior. b Low-temperature zoom of M/H for Tb, Dy, Ho, Er, and Tm
Consistent with band calculations
atures. (Note: 1 emu/mol Oe=4⇡⇥10 m /mol.) d Magnetization
0.0
-1.0
D43.00015
1.0
e,
-1.0
Γ6
Γ
X
6
3
(e) YPtBi - LDA
L
Electrical
Γ8
0.0
-2.0
W
L
Γ
X
(f) YPtBi - MBJLDA
1.0
resistivity
of all
-2.0
W
1.0
members
Yasuyuki Nakajima
Γ
(g) LaPdBi - LDA
X
-2.0
W
EF
Γ8
-1.0
Γ7
L
(d) YPdBi - MBJLDA
0.0
Γ6
-1.0
Γ7
1.0
Γ6
Γ6
Γ7
-2.0
W
d, Tb, Dy, Ho, Er, and Tm.
0.0
(c) YPdBi - LDA
Γ7
L
Γ
X
(h) et
LaPdBi
- MBJLDA
Feng
al. PRB
(2010)
1.0
Charge transport
Surface Fermi surface in RPtBi
PHYSICAL REVIEW B 83, 205133 (2011)
CHANG LIU et al.
M
(b)
surface
kz
L
A
M
2
(c)
K
(d)
(e)
DyPtBi
GdPtBi
1
k(Γ-M) [k0]
(a)
H
K
ky
0
-1
kx
LuPtBi
bulk
-2
-2
-1
0
k(Γ-K) [k0]
1
0
1
2
k(Γ-K) [k0]
3
-2
-1
0
k(Γ-K) [k0]
1
2
FIG. 1. (Color online) Surface Fermi maps of half-Heusler compounds RPtBi (R = Lu,Dy,Gd). (a) C1b crystal structure of RPtBi. The
crystallographic axes are rotated so that the (111) direction points along z. The red parallelogram marks the Bi(111) cleaving plane. (b) The
surface and bulk Brillouin zone for the rotated crystal structure in (a). Here kz corresponds to the (111) direction of the fcc Brillouin zone.
(c)–(e) Surface Fermi maps of RPtBi. All data are taken with 48 eV photons at T = 15 K. Yellow lines denote the surface Brillouin zone.
C. Liu et al. PRB (2011)
not inconsistent with TI properties
PHYSICAL REVIEW B 82
FENG et al.
The energy resolution was set at ∼15 meV. All samples were
cleaved in situ, yielding clean (111) surfaces in which atoms
are arranged in a hexagonal lattice. High-symmetry points for
1.0
(b)
(a)
0.5
Energy(eV)
1.0
M
¯
the surface
√ Brillouin zone are defined as !(0,0),
√ K̄(k0 ,0), and
M̄(0,k0 3/2) with unit momentum k0 = 6π/a, where a
LaPtBi
- MBJLDA
(a)
- LDA
is the LaPtBi
lattice constant
for each type of(b)
crystal.
We emphasize
1.0
here that no stress or pulling force is felt by the samples, which
ensures that the measured data reveal the intrinsic electronic
structure of the single crystals.
In the band
for
Γ8 structure and Fermi surface calculation
Γ8
both the bulk and the surface, we have used a full-potential
0.0
linearized augmented plane wave (FPLAPW) method19 with
a local density functional.20 The scalar relativistic method
Γ6
was employed
Γ7 and spin-orbit coupling was included by a
second-variational procedure. The structural data were taken
from a reported experimental-1.0
result.21 For the bulk band
Γ7 in the
calculation Γof6 cubic GdPtBi, we used 1240 k points
irreducible fcc Brillouin zone and set RMT × kmax = 9.0,
where RMT is the smallest muffin-tin radius and kmax is the
plane-wave cutoff. For the surface band calculation, since we
-2.0we generated a hexagonal
are interested in the (111) surface,
W
L
L
Γ of X
Γ
X
cell that has the z axis pointing along the [111] direction
the cubic cell. After that, we constructed supercells with three
(f) these
YPtBi
- MBJLDA
(e)
- LDA
layersYPtBi
and 21.87
a.u. vacuum and used
supercells
to
1.0
calculate the band structures. Although we calculated band
structures of all six possible surface endings (Gd-Bi-Pt-bulk,
k(Γ-M) [k0]
Semi-metallic behavior
19
-3
n ~ 10 cm
Consistent with band calculations
Γ
0.0
-0.5
-1.0
-1.0
0.2
Energy [eV]
0.0
-0.5
(c)
-0.2
0.0
k(Γ-K) [k0]
0.5
1.0 -1.0
-0.5
0.0
k(Γ-K) [k0]
0.5
0.0
K
-1.0
1.0
(d)
-2.0
W
-0.4
-0.6
-0.8
D43.00015
-1.0
1.0
Yasuyuki Nakajima
Carrier: Hole
1.0
(c) YPdBi - LDA
1.0
0.0
Γ6
-1.0
Γ
(g) LaPdBi - LDA
X
-2.0
W
EF
Γ8
-1.0
Γ7
L
(d) YPdBi - MBJLDA
Γ6
Γ8
0.0
-2.0
W
1.0
Γ7
L
Γ
X
(h) et
LaPdBi
- MBJLDA
Feng
al. PRB
(2010)
1.0
Superconductivity
a
c
b
d
Low temperature
Superconductivity except for Gd
Large but non-saturating
screening
Extremely small Hc1
Extremely long
penetration depth due
to low carrier r
density
Hc1 /
FIG. 4: Superconducting
D43.00015
9
2
/
m
n
state properties of RPdBi single
crystals.Nakajima
a, Resistivity of RPdBi at temperatures below 2 K,
Yasuyuki
Superconductivity
T.V. Bay et al. / Solid State Communications 183 (2014) 13–17
9
15
b
a
1 Oe
0.1 Oe
Fig. 2. (Colour online) Ac susceptibility as a function of temperature of YPtBi for
different driving fields Bac as indicated. Lower frame: χ 0 ; upper frame: χ″.
c
Huge field dependence of
magnetic susceptibility
T.V. Bay et al. Solid State Commun. (2014)
Large but non-saturating
screening
Extremely small Hc1
Extremely long
penetration depth due to
the low carrierrdensity
Fig. 4. (Colour online) The lower critical field Bc1 as a function of temperature. The
solid line represents a quadratic dependence Bc1 ðTÞ ¼ Bc1 ð0Þð1 $ ðT=T c Þ2 Þ with
Bc1 ð0Þ ¼ 0:0087 mT and T c ¼ 0:77 K. Inset: Dc-magnetization versus applied field
at T ¼0.17 K. The black arrow indicates where MðBappl Þ deviates from linear
behaviour (black straight line) and flux penetrates the sample.
d
Hc1 /
Fig. 3. (Colour online) χ″ as a function of the internal field Bint at temperatures
as indicated. Bint is obtained by correcting for demagnetization effects: Bint ¼
Bac ð1 $ Nχ 0 Þ. The kink locates the lower critical field Bc1 as indicated by the arrow for
T ¼ 0.17 K.
FIG. 4: Superconducting
D43.00015
2
/
m
n
Fig. 5. (Colour online) DC-susceptibility versus temperature in an applied field of
0.1 mT. After cooling in B¼0 (ZFC), a magnetic field of 0.1 mT is applied. Next the
sample is heated to above Tc and subsequently cooled in 0.1 mT (FC) to demonstrate
flux expulsion.
state properties of RPdBi single
crystals.Nakajima
a, Resistivity of RPdBi at temperatures below 2 K,
Yasuyuki
Lack of heat capacity jump
d
C/T =
n
2
+
T
n
= 0.0 ± 0.5 mJ/mol K
Low carrier density
n ⇠ 1019 cm
⇤
m ⇠ 0.09me
2
3
W. Wang et al. Sci. Rep. (2013)
C/
rystals. a, Resistivity of RPdBi at temperatures below 2 K,
Temperature dependence of upper critical field Hc2 obtained
Bi. The inset shows the normalized upper critical field h⇤ =
t = T /Tc for YPdBi, with dashed line indicating the expectation
samples of RPdBi. The magnitude of the screening below the
perconducting aluminum, confirming bulk diamagnetic
screening
3
A
phase
of
superfluid
He
2
1a fit to the data using C/T = + T , where T
i. Solid line is
2
ecific heat. Inset:D.Vollhardt
enlarged view
the C/T
and of
P.Wolfle,
vs T near Tc ⇠ 1.5
= 1.43
C/T < 0.2 mJ/mol K
Dominant triplet pairing?
The Super fluid Phases of Helium 3 (1990) D43.00015
n Tc
Yasuyuki Nakajima
2
beyond the resolution
interesting aspects to solve are the specific mechanisms
quent heat treatment under high vacuum at 870 C for
of pairing and the symmetry of the superconducting
three weeks. Crystal structure was determined from
condensate [1]. While in conventional superconductors
kappa-CCD single crystal x-ray data and found to be
the binding of electrons into Cooper pairs is normally
tetragonal, space group P4mm (No. 99), isotypic with the
mediated by phonons, the origin of pairing in some heavy
ternary boride CePt3 B [10,11] (see Fig. 1). Crysfermion superconductors is believed to be connected with
tallographic data (standardized) are a % 0:4072#1$ nm
spin fluctuations, giving rise to unconventional superconand c % 0:5442#1$ nm; Ce in site 1(b) at 0:5; 0:5; 0:1468;
ducting phases (see, e.g., Ref. [2]).
Pt(1) in 2(c) at 0:5; 0; 0:6504, Pt(2) in 1(a) at 0; 0; 0 (fixed),
Pronounced electron correlations are generally found
and Si in site 1(a) at 0; 0; 0:4118.
in systems exhibiting the Kondo effect and thus Ce, Yb,
CePt3 Si derives from hypothetical CePt3 with cubic
0
and U based
AuCu3 structure by filling the void with Si, which in
0 compounds are certainly candidates for the
3
occurrence of superconductivity where renormalized quasiparticles form the Cooper pairs. While heavy fermion
superconductors are yet missing in Yb based compounds,
E. Bauer et al. PRL (2004)
such phases were identified for both Ce and U systems.
Ce-based heavy fermion superconductors, however, are
still few in numbers. Prototypic CeCu2 Si2 exhibits superconductivity below Tc % 0:7 K [3]. The application of
0 the 20 to 30 kbar range to members
0 of this
pressure in
structure family such as CeCu2 Ge2 [4], CePd2 Si2 [5], and
PHYSICAL
PRL 94, 197002 (2005)
CeRh2 Si2 [6] is sufficient to trigger superconductivity as
1.0
well. Very recently, a new class of compounds, CeMIn5 ,
q // [100]
(a)
H // [010]
was added where at ambient conditions heavy fermion
0.8
4T
superconductivity occurs for M % Co
10
0 and Ir at Tc % 2:3 0
0.6
Singlet
:
g(
k)
=
g(k),
(
,
)
=
(
,
)
and 0.4 K, respectively [7,8]. Again, pressure initiates
4T
3T
5
max
0.4
2T
superconductivity, e.g., in CeRhIn5 below
0 Tc % 2:1 K 0
1T
0T
Triplet
: g(crystal
k) structure
= g(k),
, ) =can(be, FIG.
) 1 (color online). Crystal structure of CePt3 Si. The bonds
0
[9]. The
of latter(compounds
0.2
s-wave
0
1
2
3
0T
T (K)
κ
/T
considered quasi-two-dimensional variants of CeIn3
ph
indicate the pyramidal coordination (Pt5 )Si around the Si atom.
0.0
0.0
0.2
0.4
0.6
0.8
(AuCu3 -type). Cubic CeIn3 becomes superconducting at
Origin shifted by/T
(0:5; 0:5;
T (K)
⇠0:8532)
N (EforFconvenient
)vF ` comparison
(b)
0.8
&25 kbar [2].
with the parent AuCu3 structure.
Non-centrosymmetric SC
Superconducting paring function
(k) = g(k) ( ,
orbital
)
c.f. heavy fermion SC CePt Si
spin
Nodal SC
Admixture of singlet and triplet
D43.00015
Yasuyuki Nakajima
0.6
1.0
TN
2
027003-1
0.4
κ/T ( W/K m )
No restriction of spin parity
4T
κ/T ( W/K2m )
Lack
of inversion
symmetry© 2004 The American
027003-1
0031-9007=04=92(2)=027003(4)$22.50
/T Physical
/ T Society
ρ ( µΩcm )
κ /T ( W/K2m )
exchange electrons
( k) =
(k)
antisymmetric
0.2
0.0
0.00
0T
0.01
κph/T
Tc
0.5
0T
0.0
0
0.02 0.03
2
2
T (K )
1
2
T (K)
0.04
3
0.05
FIG. 1. (a) Main panel: a-axis thermal conductivity of CePt
K.
Izawa
al.asPRL
for the
H k b axis,et
plotted
!=T vs (2005)
T (", 0 T; ', 0.2 T;
0.5 T; 4, 1 T; $, 1.5 T; %, 2 T; &, 3 T; 5, 4 T). The solid lin
a linear fit in zero field: !=T ! A " BT. The dashed line is
contribution of phonons !ph . The dash-dotted line is the therm
conductivity in s-wave superconductors. Inset: T dependence
a-axis resistivity for the H k b axis. (b) The same data at l
temperatures plotted as !=T vs T 2 . The solid line is a linear fi
the normal state: !=T ! a " bT 2 . The dashed line represe
Nodal SC in half Heusler
Fermi surface from a model Hamiltonian
London penetration depth (Ames) Nodal superconductivity
Nodal SC
(Brydon & Agterberg)
YPtBi
Tuning dominant contribution of
singlet/triplet paring states
Penetration depth
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 86, 220502(R) (2012)
HARADA, ZHOU, YAO, INADA, AND ZHENG
c.f. Li(Pd,Pt)2B3
Line node
(b)
(a)
x=0
(Li Pd B )
2
10
3
(a)
S.Harada et al. PRB (2012)
100
11
B-NMR
(b)
195
Pt-NMR
Superconducting energy gap function
H. Kim et al., unpublished
x = 0.8
s-wave
Asymmetric spin-orbit coupling
Essentially, triplet component
1
Theoretical Fermi surface
1
-1
Fermi surface from a model Hamiltonian
1/T (sec )
Intensity (arb. units)
x = 0.5
&and
Agterberg)
Penetration depth measurements
are consistent
with line nodes unpublished
in the gap, motivating our study to 0.1
P. (Brydon
Brydon
D. Agterberg,
x = 0.9
understand the gap structure and symmetry.
Theoretical model that incorporates spin-orbit coupling suggests nodal superconductivity in YPtBi arising from
a triplet component, with a very unique spin-dependent gapxstructure.
=1
To Do: Verify spin-split band structure to support theoretical (Li
model
Pt B) and conclusion of triplet/gap structure.
2 3
0.01
20
Line node
30
40
50
2θ (deg)
60
70
80
FIG. 1. (Color online) (a) Representative Cu Kα XRD charts
(circles) with Rietveld analysis (solid curves). The error of the
Rietveld fitting (RF ) (Ref. 24) is about 3%. (b) The distorted B(Pd,Pt)6
octahedral units. α is the angle between the two connected octahedra,
and lmax (lmin ) is the longest (shortest) M-M (M = Pd,Pt) bond length.
(c) Tc vs Pt content x.
D43.00015Superconducting energy gap function
Yasuyuki Nakajima
1
10
x=0
x = 0.2
x = 0.5
x = 0.8
x = 0.9
x = 1 #A
T (K)
x = 0.5
*1.5
x = 0.84 *1.2
x = 0.9
x = 1 #A
x = 1 #B /1.4
*2
*4
*8
*35
*70
10
0.7 1
T (K)
4
FIG. 2. (Color online) (a) The T dependence of 11 (1/T1 ). The
straight lines above Tc indicate the 1/T1 ∝ T relation. (b) The T
dependence of 195 (1/T1 ) for x = 0.5, 0.84, 0.9, and 1. For both
(a) and (b), data were offset vertically for clarity by multiplying or
dividing by a number shown in the figure (Ref. 25). Data for x = 1 #A
Finite triplet component
9
b
WHH theory
Hc2 (0) =
N. Werthamer et al. PR (1966)
dHc2
↵Tc
dT
↵ = 0.69 dirty SC
↵ = 0.74 clean SC
T =Tc
exceeding of orbital
depairing field
d
YPdBi: μ0Hc2(0) = 2.7 T, α = 0.82
LuPdBi:
= 2.9 T, α = 0.91
DyPdBi:
= 0.7 T, α = 0.93
Finite triplet component?
D43.00015
Yasuyuki Nakajima
Magnetic superconductivity
Anticorrelation between Tc and TN, well-scaled by de Gennes factor
10
SC is suppressed by magnetic
scattering due to local moments
of f electrons
AFM is induced by RKKY
interaction
SC
AFM
Another canonical material for
magnetic superconductor
c.f. Borocarbide
H. Müller et al. Rep. Prog. Phys. (2001)
Chevrel phases
IG. 5: Phase diagram of RPdBi series, indicating evolution of superconducting (SC) and antiferromagnetic (AFM) ground
tates as a function of de Gennes factor dG = (gJ 1)2 J(J + 1). The superconducting transition Tc (blue) is obtained from
he midpoint of the resistive transition (circles; upper and lower error bars indicate onset and zero resistance) and the onset
f diamagnetism in AC susceptibility (diamonds), and Néel temperature TN (red triangles) is obtained from DC magnetic
D43.00015
Yasuyuki Nakajima
Φ. Fischer et al. Appl. Phys. (1978)
BCS-BEC crossover?
9
b
Hc2 (0) =
0:
⇠
0
2⇡⇠ 2
de
e
⇠ ⌧ de
e
flux quantum
d
Coherence length
⇠ ⇠ 10 nm
YPdBi
Average inter-electron distance
de
⇠n
1/3
⇠ 5 nm
le crystals. a, Resistivity of RPdBi at temperatures below 2 K,
b, Temperature dependence of upper critical field Hc2 obtained
PdBi. The inset shows the normalized upper critical field h⇤ =
ure t = T /Tc for YPdBi, with dashed line indicating the expectation
stal samples of RPdBi. The magnitude of the screening below the
superconducting aluminum, confirming bulk diamagnetic screening
PdBi. Solid line is a fit to the data using C/T = + T 2 , where T
specific heat. Inset: enlarged view of the C/T vs T 2 near Tc ⇠ 1.5
e
n ⇠ 1019 cm
3
⇠ ⇠ de e
Exotic superconducting state?
M. Randeria et al. Nature Physics (2010)
D43.00015
Yasuyuki Nakajima
Summary
We have studied superconductivity and magnetism in the
topological half semimetal RPdBi.
•
fcc type II AFM with Q = (1/2, 1/2, 1/2)
•
Anticorrelation between Tc and TN, well-scaled by de
Gennes factor
•
Anomalous SC: Triplet dominant? BCS-BEC crossover?
Strong candidate for tunable topological materials with
multi-symmetry breaking
D43.00015
Yasuyuki Nakajima
Combination of topological and symmetry-breaking order
D43.00015
SC
AFM
Topo
Y
✓
✘
✘
Lu
✓
✘
✓
Tm
✓
✓(?)
✓
Er
✓
✓
✓
Sm
✓
✓
✘
Ho
✓
✓
✓
Dy
✓
✓
✘
Tb
✓(?)
✓
✘
Gd
✘
✓
✘
Yasuyuki Nakajima