institute for solid state physics, university of tokyo, japan · prl101, 137004 (2008). 0 0.5 1 1.5...

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


|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


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.


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).


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? …

institute for solid state physics, university of tokyo, japan · prl101, 137004 (2008). 0 0.5 1 1.5...

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