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Jan/13/’14, Aspen Winter Conference "Beyond quasiparticles: New paradigms for quantum fluids"
Yosuke Matsumoto Institute for Solid State Physics, University of Tokyo, Japan
Collaborators:
S. Nakatsuji, K. Kuga, Y. Karaki, Y. Shimura, T. Sakakibara, T. Tomita,
Y. Uwatoko, A. H. Nevidomskyy, P. Coleman
Contents
Introduction: Kondo lattice vs. mixed valence systems
Kondo lattice behavior in the mixed valence systems: b-, a-YbAlB4 Quantum criticality without tuning in b-YbAlB4 Strange metal phase detached from magnetism in b-YbAlB4
Kondo lattice vs. Mixed valence systems
Mixed valence systems
Large TK Pauli paramagnetism below TK g ( = C/T ) value is not large.
“Kondo lattice” with integer valence
Doniach like phase diagram
Kondo vs. RKKY Quantum criticality based on critical
spin fluctuations associated with
magnetic QCP.
Basically, no QC has been expected.
A. L. Cornelius et al.,
PRL 88, 117201 (2002).
Example: YbAl3 Yb valence: 2.71 Suga et al., JPSJ 74,
2880 (2005).
TK ~ 300 K
In contrast,
a-, b-YbAlB4 provide unique examples of quantum criticality
in the mixed valence systems.
What is the origin ?
Novel metallic state ?
HF Compounds: b-YbAlB4
Space Group : Cmmm,
a = 7.3080 Å, b = 9.3150 Å, c = 3.4980 Å
Thickness: ~10 mm
0.5 mm
c
a
b
c
Yb Al B
Yb form distorted hexagonal lattice.
Layered but Yb-Yb distance is shortest in c axis.
The first heavy fermion SC (Tc = 80 mK) in Yb based system.
Pronounced NFL behavior in the normal state @ B = 0.
S. Nakatsuji et. al., Nature Phys. 4, 603-607 (2008).
0
0.2
0.4
0.6
0.8
0 0.5 1
T 1.5
(K1.5
)
a
b (
m
cm
)
0
0.5
1
1.5
2
0 1 2 3 4T (K)
a
b (
m
cm
)
ab T
@T > 1 K
ab T1.5
@ T < 1 K
SC
SC
SC in b-YbAlB4
0
50
100
150
200
0 20 40 60 80 100
Bc2 (
mT
)
T (mK)
B // ab
B // c
b-YbAlB4
0
0.2
0.4
0.6
0 50 100150
B (mT)
ab (
m
cm
)
50
mK
20
mK
B // ab
-30
-20
-10
0
0 50 100 150
T (mK)
Bext
= 3 mT // c
-0.4
-0.2
0
M/H
ext (
em
u/m
ol-
Yb)
Bext
= 60 mT // ab
b-YbAlB4
40%
7%
15%
5%
ZFC
FC
ZFC
FC
K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji,
PRL101, 137004 (2008).
0
0.5
1
1.5
2
0 0.5 1
b-YbAlB4
a
b (
m
cm
)
T1.5 (K1.5)
sample A
sample B
sample C
0
1
2
0 50 100
T (mK)
sample C
BA
Type-II SC in the clean limit (l = 1.2 mm > x = 0.06 mm) with Tc = 80 mK
RRR dependence of Tc ⇒ unconventional, non-s-wave SC ?
Considerable amounts of volume fractions → Bulk SC
Anisotropic Bc2 (B // ab: orbital critical fields, B // c: paramagnetic effect)
= -2.6 T/K for B // ab → m*/m0 ~ 180 CT
C TB /2
HF Compounds: b-, a-YbAlB4
Space Group : Cmmm,
a = 7.3080 Å, b = 9.3150 Å, c = 3.4980 Å
The first HFSC (Tc = 80 mK) in
Yb based system.
Pronounced NFL behavior in
the normal state @ B = 0. R. T. Macaluso et al., Chemistry of Materials 19 1918 (2007)
Thickness: ~10 mm
0.5 mm
c
FL ground state @ B = 0
Close to QC ?
Metamagnetic behavior @ B ~ 3 T
a
b
c
Yb Al B
Space Group : Pbam
a = 5.9220(2) Å b = 11.468(5) Å c = 3.5060(5) Å
c
~ 1 x 1 x 3 mm3
a
b
c
Yb form distorted hexagonal lattice.
Layered but Yb-Yb distance is shortest in c axis.
b-YbAlB4 a-YbAlB4
0 100 200 3000
50
100
150
T (K)
Strong Valence Fluctuation in a, b-YbAlB4
a-YbAlB4 : 2.73 0.02
b-YbAlB4 : 2.75 0.02
b phase is the first example of QC
in mixed valence system.
M. Okawa et al., PRL 104, 247201 (2010) .
Inte
nsity (
arb
. u
nits)
Coherence peak of a resistivity
@ T ~ 200 ~ 250 K
TTTSTCM /ln// 000
T0 ~ 200 K obtained from C.
Y. Matsumoto et al., PRB 84, 125126 (2011).
(m
c
m)
a-YbAlB4
T (K)
ab
b-YbAlB4 ab
m(ab)
B = 0
m(ab)
a-LuAlB4
b-LuAlB4
m = (YbAlB4) - (LuAlB4)
Large T scale peculiar to mixed valence systems
from HXPES
Y. H. Matsuda et al., JKPS 62, 1778 (2013).
Yb valence measured by X-ray adsorption
spectra revealed T scale of ~ 290 K.
Almost identical c for a and b @ T > 8 K
Ising like susceptibility with moments
aligned along the c-axis
Curie-Weiss behavior in cc far below the
valence fluctuation T scale
Shoulder in cc @ T* ~ 8 K
Energy scale of Kondo lattice which
emerges in low T.
Below T* ~ 8 K,
a phase: FL ⇔ b phase: NFL
C/T ~ 130 mJ/K2 mol @ 0.4 K
Y. Matsumoto et al., Science 331, 316 (2011), Y. Matsumoto et al., PRB 84, 125126 (2011).
Kondo lattice behavior with small T scale of ~ 8 K
How does the small T scale 8 K coexists with
large valence T scale of 200 ~ 300 K ?
101 1020
0.01
0.02
0.03
M/H
(e
mu
/mo
l-Y
b)
T (K)T (K)
cc
cab
T*
c (
em
u/m
ol)
0 100 200 3000
100
200
H/M
(e
mu-
1m
ol-
Yb
)
T (K)
cc -1
(e
mu
-1m
ol)
b-YbAlB4
a-YbAlB4
B = 0.1 T
T (K)
b-YbAlB4
a-YbAlB4
b-YbAlB4
a-YbAlB4
B = 0.1 T
Local moment behavior in ESR spectra of b-YbAlB4
171Yb hyperfine lines.
g value increases from 2.4 to 3.0 below T ~ 100 K.
anisotropy in g value
H (kOe)
Unique dual behavior
L. M. Holanda et al., PRL 107, 026402 (2011).
b-RAlB4 – fine powder (X-Band) b-RAlB4 – oriented crystals (X-Band)
T independent intensity which is typical in ESR signal
for conduction electron.
Two component Hall effect E. C. T. O’Farrell, Y. Matsumoto, S. Nakatsuji, PRL 109, 176405 (2012).
B (T)
Large T dependence of RH
RH has a peak at T ~ 40 K ⇒ Coherence develops below 40 K.
xy becomes non-linear in B below 100 K. This can be explained
by two band model with
1. hole-like component (band 89?) and
2. electron-like component with low carrier density
(band 91?).
A. Nevidomsky and P. Coleman, PRL 102,
077202 (2009). Also by E. C. T. O’Farrell et
al., H. Harima et al, and W. Pickett et al.
The elecrton-like component is responsible for
the Kondo lattice behavior and QC ?
T (K)
m (1
04c
m2/V
s)
RH (
10
-10m
3C
-1K
)
2
0
-2
-4
-6
-8
-10
-12 0 50 100 150 200 250 300 350
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6 B (T)
0 1 2 3 4 0 1 2 3 4
xy (
10
-10m
3C
-1K
)
0
-1.0
-2.0
-3.0
de
via
tion
fro
m initia
l slo
pe
(%
)
0
20
40
20 K
40
60
80 K
100
20 K
40
60
80 K
100
B
T
-dM
/dT
Superconductivity
QCP
Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karaki, N. Horie, Y. Shimura, T. Sakakibara,
A. H. Nevidomskyy, P. Coleman, Science 331, 316 (2011).
Superconductivity
D = (T) – (0) = Ta
Fermi Liquid
Non-FL
S. Nakatsuji et al., Nature Phys. 4, 603-607 (2008).
QCP at B = 0 under P = 0 ?
NFL behavior @ B = 0 in b-YbAlB4
ab T1.5
M/H T-0.5
C/T ln(T*/T )
B = 0:
ab T2
M/H = const.
C/T = const.
Non Fermi Liquid
In magnetic field:
Fermi Liquid
FL behavior is recovered in magnetic field.
Normally, we have to tune B or P or doping to approach QCP.
ex) YbRh2Si2, CeCoIn5, CeCu5.9Au0.1, …
SQUID magnetometer installed in a dilution fridge
Magnet (Bmax ~ 0.1 T)
Nb shield
+ m-metal shield
Pick up lines
connected to SQUID
in Pb tube
Cu nuclear stage
connected to MC
of the dilution fridge
High precision (resolution: ~ 10-8 emu)
T range: 20 mK ≤ T ≤ 4 K, B < 0.1 T
T down to ~ 0.1 mK is also available by using nuclear demag
AC susceptibility and DC magnetization can be measured.
magnet center
Pb tube
Silver support
for pick up coils
pickup &
primary coils
pickup coils
viewed from above
ULT group @ ISSP, Univ. of Tokyo
Setup inside of the shielded magnet
Samples: 0.82 mg,
RRR > 200
10-1 100
10-1
100
101
102
dM
/dT
(e
mu
/mo
l K
)
T (K)
10-2 10-1 100 101 1020
0.02
0.04
0.06
0.08
M/H
(e
mu
/mo
l-Y
b)
T (K)
10-2 10-1 100 101 1020
0.02
0.04
0.06
0.08
M/H
(e
mu
/mo
l-Y
b)
0.31 mT0.62 3.1
0.1 T
T (K)
M/B
2.0 1.0
0.2
0.1 T
0.3 0.5
44
12 19 22 25 31
6.2 mT
10-2 10-1 100 101 1020
0.02
0.04
0.06
0.08
M/H
(e
mu
/mo
l-Y
b)
T (K)
6.0 7.0 T
-dM
/dT
(e
mu
/mo
l K
)
2.0 T
22 mT
6.2 mT
0.62 mT
0.31 mT
b-YbAlB4 B // c
1.0
0.1
0.3
0.5
B // c
b-YbAlB4
44 mT 12 mT
31 mT
3.1 mT
0.2
T-linear
T-1.5
At T/B >> 1 (but below T* ~ 8 K) (NFL): -dM/dT T-1.5 (M const. + T-0.5 )
At T/B
T/B Scaling of Magnetization
BTBT
Ma 2
-dM/dT @ B ≤ 2 T, T ≤ 3 K
satisfy following scaling
equations.
10-1 100 101 102 103
10-3
10-2
10-1
100
101
T/B (K/T)
(-d
M/d
T)B
1/2
(e
mu
T1/2
/Km
ol)
T/B
(-d
M/d
T)B
1/2 (
em
uT
1/2/m
ol K
)
b-YbAlB4
-0.3 0 0.30.97
0.98
0.99
1
R
Bc (mT)Bc
R
with a = 3/2
Scaling observed over 4
orders of T/B
No intrinsic energy scale (T and B are interchangeable)
@ B ≤ 2 T, T ≤ 3 K.
T (
K)
B (T)
SC
10-4 10-3 10-2 10-1 100
10-1
100
101
B (T)
T (
K)
QCP is not of a SDW type
Y. Matsumoto et al., Science 331, 316 (2011).
BTfBF a
Integrating both parts,
(x) = (a-1)f ’(x) - xf ’’(x)
T/B Scaling of Magnetization
10-1 100 101 102 103
10-3
10-2
10-1
100
101
T/B (K/T)
(-d
M/d
T)B
1/2
(e
mu
T1/2
/Km
ol)
T/B
(-dM
/dT
)B1/2 (
em
uT
1/2/m
ol K
)
b-YbAlB4
-0.3 0 0.30.97
0.98
0.99
1
R
Bc (mT)Bc
R
Y. Matsumoto et al., Science 331, 316 (2011).
|Bc| ≤ 0.2 mT (2 G)
Bc is practically zero
m0Hc2/Bc > 100
Zero-field Quantum
Criticality
The first example of
quantum criticality without tuning !!
nxAxx )( 2 The data can be fitted to
with x = T / (B-Bc)
The best fit was obtained with
n = 1.25 ± 0.01 (a = 3/2 ± 0.02) Bc = 0.1 ± 0.1 mT
This satisfy the limiting behavior of f (x),
i.e., f(x) xa (x >> 1, NFL)
const + x2 (x
10-1 101 103 10510-5
10-3
10-1
101
T/B1.5
(K/T)
(-d
M/d
T)B
1 (
em
uT
1/2
/Km
ol)
10-1 101
10-2
100
102
T/B0.7
(K/T)
(-d
M/d
T)B
0.2
(e
mu
T1/2
/Km
ol)
T/B1.5 (K/T)
T/B0.7 (K/T)
(-d
M/d
T)B
(e
mu
T/K
mo
l)
(-dM
/dT
)B0.2 (
em
uT
0.2/K
mo
l)
b-YbAlB4
B // c
b-YbAlB4
B // c
T/(B-Bc)d scaling with d ≠ 1, Bc ≠ 0 ?
1.00
0.99
0.98
0.97
0.96
0.95
0 0.2 -0.4 Bc (mT)
-0.2
1.2
1.0
0.8
d
1.4
R
h
1 1.5
0.5
1
d
h
Bc = 0
d
d
h
c
cBB
TBB
dT
dM
(d, h) = (1.5, 1.0)
(d, h) = (0.7, 0.2)
More general form was checked.
The best fit is obtained with
(Bc, d ) = (-0.02±0.20 mT, 0.94±0.12)
⇒ T/B scaling is most likely!
10-1 100 1010
100
200
C/T
(m
J/K
2m
ol)
T (K)
a-LuAlB4
a-YbAlB4
calculation 0.07 T 0.05
fit to 0T
Specific Heat
The scaling indicates
CQC/ T ∝ T-0.5.
T (K)
CM/T
(m
J/K
2m
ol)
0.07
6.0
1.0
4.0
2.0
0 T
b-YbAlB4
B // c
0.5
9.0
a- LuAlB4, 0 T
a-YbAlB4 B = 0 T
10-3 10-2 10-1 1000
20
40
60
T/T0
(C/T
)T0 (
J/K
mo
l) b-YbAlB4
YbRh2(Si1-xGex)2 x = 0x = 0.05
CeCu5.9Au0.1
T/T0
(CM/T
)T0
CM/T ~ ln(T0/T) @ T ≥ 0.2 K
with T0 = 200 K
Further increase below 0.2 K
Small increase @ B ≤ 0.5 T is
consistent with the scaling.
This can be checked by
Maxwell’s relations.
CeCu5.9Au0.1: 6.2 K
YbRh2Si2: 24 K
Y. Matsumoto et al., Science 331, 316 (2011). BT T
M
B
S
00 2
2
B
B
T
CdB
T
M
T
C⇒
QC without tuning
Difficult to extract due to
the nuclear contribution.
Why Bc = 0 ?
10-1 100 101 102 103
10-3
10-2
10-1
100
101
T/B (K/T)
(-d
M/d
T)B
1/2
(e
mu
T1/2
/Km
ol)
T/B
(-dM
/dT
)B1/2 (
em
uT
1/2/m
ol K
)
b-YbAlB4
-0.3 0 0.30.97
0.98
0.99
1
R
Bc (mT)Bc
R
Y. Matsumoto et al., Science 331, 316 (2011).
What happens under pressure ?
NFL behavior robust against pressure ?
0 1 20
1
2
3
B (T)
T (
K)
FL
NFL
B
T T
B P
NFL phase (metallic
spin liquid?) SC
1. Fine tuned to QCP?
2. Non-fermi liquid phase or
Quantum Critical Phase,
More natural than the
above accidental scenario.
What is the origin ?
Partial Mott localization of the f-electrons ?
Spin liquid metal @ ambient pressure ?
J. Custers et al., PRL 104, 186402 (2010).
References therein.
Topological phase transition at Pc?
Anisotropic hybridization effect ?
Topological phase transition
Highly anisotropic resistivity in a-YbAlB4 with ab/c ~ 11 @ LT
Consistent with hybridization node along the c-axis
A hybridization node along the c-axis
CEF ground doublet |Jz = ±5/2 > based on the local Yb site symmetry
A. Nevidomsky and P. Coleman, PRL 102, 077202 (2009), A. Ramires et al., PRL 109, 176404 (2012).
c-f hybridization V(k) ~ (kx ±iky)2
2D heavy band with a dispersion
E(k) ~ |V(k)|2 ~ (k┴)4 explains QC in b-
YbAlB4, if Ef = 0 ?
Yb
B
B
Kondo break down QCP at mixed valence
D ~ 80 K
2/12/3 γ
2/5
S. Watanabe, K. Miyake, PRL 105, 186403 (2010).
What is the origin ?
Valence quantum criticality?
J. H. Pixley, S. Kirchner, K. Ingersent, Q. Si, PRL 109, 086403 (2010).
Y. Matsumoto, K. Kuga, T. Tomita, Y. Karaki and S. Nakatsuji, PRB 84, 125126 (2011).
Summary
In spite of the strong valence fluctuation with characteristic T scale of
200-300 K, the both a- and b-YbAlB4 exhibit Kondo lattice behavior
with low-T scale of T* ~ 8 K.
Contrasting behavior below T* ~ 8 K
a phase: FL b phase: QC without tuning + SC
T/B scaling of magnetization
observed over 4 decades of T/B in
b-YbAlB4 → QC without tuning
Resistivity measurements under P
revealed strange metal phase
extends up to Pc ~ 0.4 GPa
Strange metal phase detached
from magnetic order
What is the origin ? topological phase transition?
valence? …