Happy Birthday Eugene!

Wishing you great beach-soccer games with good friends till 120 !

Field-induced Quantum Critical Route to a Fermi Liquid in High-Tc Superconductors

CUNYMarch 13, 2009

L. Krusin-ElbaumIBM T.J. Watson Research Center, Yorktown Heights, New York

Strange metal

� Antecedent states of matter that become unstable in favor of high Tc – commonly referred as the `normal state’

� Key to the origins of high Tc

Phase diagram

Pseudogap phenomenon: friend or foe?

In the `foeIn the `foe’’ views views �� QCPQCP

• d-density wave (staggered flux state: Laughlin, Chakravarty, Lee, etc.)

• Loop-current order (broken time-reversal symmetry: C. Varma)

• Mixed order parameters (dx2-y2+idxy, dx2-y2+is, etc.)

• RVB: Strange metal beyond the pseudogap energy scale � no need for QCP (P.W. Anderson)

Strange metalAF

PG

`Normal’ state �� Many states, many transitions

CUNYMarch 13, 2009

Latest: Some evidence for the time-reversal symmetry breaking in the PG phase

�Novel magnetic order in the pseudogap phase of YBa2Cu3O6+x and HgBa2CuO4+δBourges et al., PRL 96, 197001 (2006); condmat/arXiv:0805.2959 (2008), also Mook et al, 2008.

neutrons

Polar Kerr effect

�Spontaneous Kerr rotation in zero field near PG in YBa2Cu3O6+x Kapitulnik et al., PRL 100, 127002 (2008) CUNY

March 13, 2009

•Quantum vortex liquid in La2-xSrxCuO4L. Li et al., Nature Phys. 3, 311 (2007).

The Debate: A Friend, Perhaps?

Vortices and pseudogap•Nernst effect in La2-xSrxCuO4

Z. A. Xu et al., Nature 406, 486 (2000).

•THz conductivity in Bi2Sr2CaCu2O8+y J. Corson et al., Nature 398, 221 (1999).

�Vortex-like excitations (superconducting fluctuations) exist above Tc.

�Pseudogap - an ultimate upper limit to the vortex state?

@ 60 T and above?

CUNYMarch 13, 2009

Phase diagram deduced from ARPES & transportBi-2212

A. Kaminski, S. Rosenkranz, H. M. Fretwell, Z. Z. Li, H. Raffy, M. Randeria, M. R. Norman, and J. C. Campuzano, Phys. Rev. Lett. 90, 207003 (2003)

�coherent metal phase @ low-T & high hole doping p � two well defined spectral peaks in ARPES (due to

coherent bilayer splitting, superlinear ρ�incoherent metal phase @ high-T & low p � linear ρ, single broad feature in ARPES

crossover @ TxCUNYMarch 13, 2009

� Ultrahigh magnetic fields to kill superconductivity and to examine ‘normal’ state far on the overdoped side of the dome

We have our big hammer:

Searching for:� Transformation into a conventional metal� Quantum (or not) phase transition(s) between `normal’ states of matter antecedent to high-Tc

CUNYMarch 13, 2009

Pseudogap Closed by Zeeman Splitting

� The right-hand-side translates onto the Zeeman energy scale on the left-hand-side as (gµB/kB)H.� Hpg and T* obtained separately in the same crystals in the overdoped regime, give a scaling gµBHpg = kBT* with g = 2.0 (inset).

T. Shibauchi, L. Krusin-Elbaum, M. Li, M.P. Maley, and P.H. Kes, Phys. Rev. Lett. 86, 5763 (2001)

200

150

100

50

0

µ 0H

pg (

T)

2001000T* (K)

gµBHpg = kBT*

700

600

500

400

300

200

100

0

T o

r (g

µ B/k

B)H

(K

)

0.250.200.150.10p

500

400

300

200

100

0

µ0 H

(T)

Hpg

Tc

T*Hsc

•Pseudogap closing field Hpgdecreases with doping

•Zeeman scaling gµBHpg = kBT*holds → suggesting spin-singletcorrelations in forming the pseudogap

•Peak field scales with Tc(p)

CUNYMarch 13, 2009

Tc vs doping `dome’� Tc/Tcmax = 1-82.6 (p-0.16)2

c-axis resistivity ρc: a powerful probe of the pseudogap•Intrinsic tunneling junctions along the c axis (layered structure with large anisotropy)

• ρc probes the low-energy DOS in the bulk

•Recovery of the DOS by magnetic field →negative interlayer magnetoresistance (MR)

•Pseudogap closing field Hpg = H* can be evaluated

T. Shibauchi et al., Phys. Rev. Lett. 86, 5763 (2001); Phys. Rev. B 67, 064514 (2003).

L. Krusin-Elbaum, T. Shibauchi, C. H. Mielke, Phys. Rev. Lett. 92, 097005 (2004).

T. Watanabe et al., Phys. Rev. Lett. 84, 5848 (2000).

Bi2Sr2CaCu2O8+y

T* from the deviation from the T-linear metallic dependence consistent with the tunneling spectra and the static susceptibility.

T = 110 K

ρc sensitive to the (π,0) points (`hot spots’) of the Fermi surface, where the pseudogap first opens up

CUNYMarch 13, 2009

2.5

2.0

1.5

1.0

0.5

0.0

ρ c (

Ω c

m)

6050403020100

µ0H (T)

2.5 K

20 K

3.5 K

50 K

40 K

30 K

10 K7.5 K

5 K

4.2 K

H0ρ

Hsc

2.0

1.5

1.0

0.5

0.0

0.1 1 10 100

40 K

ρcn

Hpg H0ρ

Hsc

1.34

1.32

1.30

1.28

1.26

1.24

1.22

1.20

ρ c (

Ω c

m)

120110100908070

T (K)

T*T*T*

0 T30 T

58.5 T

OD (Tc = 67 K)

0.4

0.3

0.2

0.1

0.0

∆ρ c

(Ω

cm

)

100806040200µ0H (T)

T (K)62.270.481.895.4

Hpg

b

4

3

2

1

0

ρ c (

Ω c

m)

25020015010050

T (K)

-10

-8

-6

-4

-2

0

MR

@ 31.2 T

(%)T*

OD (Tc = 78 K)

0 T

∆ρc a

T. Shibauchi et al., Phys. Rev. B 67, 064514 (2003).

ρc(H,T) in Bi2Sr2CaCu2O8+y crystals

T. Shibauchi, L. Krusin-Elbaum, M. Li, M.P. Maley, and P.H. Kes, PRL 86, 5763 (2001).

CUNYMarch 13, 2009

Field anisotropy of pseudogap closing field

L. Krusin-Elbaum, T. Shibauchi, C. H. Mielke, Phys. Rev. Lett. 92, 097005 (2004).

Hpgab / Hpgc = 1.35 ± 0.1Anisotropy of g-factor [T. Watanabe et al., Phys. Rev. Lett. 84, 5848 (2000)]

gc /gab = 1.3

(χc(T) ~ 1.6 χab(T)) Zeeman scaling

gcµBHpgc = gabµBHpgab ~ kBT*Triplet excitation @high H overcomes spin-singlet correlations responsible for the gap in the spin spectrum and orbital contribution is very small.

χc/χab=(gc /gab)2

CUNYMarch 13, 2009

M. R. Norman et al., Adv. Phys. 54, 715 (2005).

Phase diagram

CUNYMarch 13, 2009

Y. Kubo et al., Phys. Rev. B 43, 7875 (1991).

D. N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721 (2005).

ρ(T)=ρ(0)+AT2

Fermi liquid metal

ρ(T) ~ Tn1�n�2

Non-Fermi liquid

polycrystals

Phase diagram

Heavily overdoped Tl2Ba2CuO6+x

-How n-FL state transforms into FL state ?-Magnetic field effect?

M. R. Norman et al., Adv. Phys. 54, 715 (2005).

CUNYMarch 13, 2009

Can We Get to a Coventional Fermi Liquid by Applying Magnetic Field?

0.040

0.035

0.030

0.025

0.020

0.015

ρ c (

Ω c

m)

6004002000

T1.3

(K1.3

)

µ0H = 0 T

0.04

0.03

0.02

0.01

0.00

ρ c (

Ω c

m)

100806040200T (K)

µ0H = 0 T

Tl2Ba2CuO6 (Tc~15 K)

ρc = ρc0 + A0T

2 +CT

�At zero field,ρc(T)-ρc0 ~ Tn n=1.3

orρc(T)= ρc0 + A0T2+CT

Non-Fermi liquid(strange metal)

c-axis longitudinal magneto-transport(less afflicted with orbital contributions)

M. Abdel-Jawad et al., Nat. Phys. 2, 821 (2006).

ρab(T) = ρab0 + AT2+CT

CUNYMarch 13, 2009

0.040

0.035

0.030

0.025

0.020

0.015

ρ c (

Ω c

m)

6004002000

T1.3

(K1.3

)

µ0H = 0 T

0.04

0.03

0.02

0.01

0.00

ρ c (

Ω c

m)

100806040200T (K)

µ0H = 0 T

Tl2Ba2CuO6 (Tc~15 K)

ρc = ρc0 + A0T

2 +CT

M. Abdel-Jawad et al., Nat. Phys. 2, 821 (2006).

ρab(T) = ρab0 + AT2+CT

~T2 ~T

isotropic anisotropic

1/τ

At zero field,c-axis longitudinal magneto-transport(less afflicted with orbital contributions)

1/ζcτ

T & momentum dependence of transport scattering rate τ

2 channels: �conventional (� T2)�anisotropic (� T) , same symmetry as d-gapCUNYMarch 13, 2009

0.04

0.03

0.02

0.01

0.00

ρ c (

Ω c

m)

100806040200T (K)

µ0H = 0 T

Tl2Ba2CuO6 (Tc~15 K)

µ0H = 45 T

ρc = ρc(0) + A(45 T)T

2

ρc = ρc0 + A0T 2

+CT

0.040

0.035

0.030

0.025

0.020

0.015

ρ c (

Ω c

m)

6004002000

T1.3

(K1.3

)

µ0H = 0 T

Field-induced transformation from the n-FL to FL state

c-axis longitudinal magneto-transport(less afflicted with orbital contributions)

Measured in a 45-T hybrid magnet in Tallahassee (NHMFL)

�At zero field,ρc(T)-ρc0 ~ Tn n=1.3

orρc(T)= ρc0 + A0T2+CT

Non-Fermi liquid(strange metal)

�At 45 T (H//c),ρc(T)= ρc(0)+A(45 T)T2

Fermi liquid

CUNYMarch 13, 2009

40

35

30

25

20

15

ρ c (

mΩ

cm

)

1000080006000400020000

T 2

(K2)

45 T 40 T 35 T 30 T 25 T 20 T 15 T 11.5 T

TFL

-6

-4

-2

0

2

ρ c -

ρc(

0) -

A(H

)T 2

(µΩ

cm

)

1000080006000400020000T

2 (K

2)

TFL

�above TFL~Tn (n

40

35

30

25

20

15

ρ c (

mΩ

cm

)

1000080006000400020000

T 2

(K2)

25 T

-6

-4

-2

0

2

ρ c -

ρc(

0) -

A(H

)T 2

(µΩ

cm

)

1000080006000400020000T

2 (K

2)

TFL�above TFL

~Tn (n

40

35

30

25

20

15

ρ c (

mΩ

cm

)

1000080006000400020000

T 2

(K2)

45 T 40 T 35 T 30 T 25 T 20 T 15 T 11.5 T

TFL

-6

-4

-2

0

2

ρ c -

ρc(

0) -

A(H

)T 2

(µΩ

cm

)

1000080006000400020000T

2 (K

2)

TFL

�above TFL~Tn (n

100

80

60

40

20

0

T (

K)

403020100

µ0H (T)

SC

FL

n-FL

HQCP

6

5

4

3

2

1

0

A (

µΩcm

/K2 )

50403020100µ0H (T)

A0

HQCPTFL

HFL

� Quantum critical point @ HQCP ~ Hc2

� in the T � 0 limit normal state above Hc2 is aFermi liquid- consistent with recent observation of Wiedemann-Franz law C. Proust et al., PRL 89, 147003 (2002)

H-T diagram of the normal state in OD Tl-2201deviation from T2

CUNYMarch 13, 2009

0.04

0.03

0.02

0.01

0.00

ρ c (

Ω c

m)

403020100µ0H (T)

100 K 70 K 50 K 30 K 20 K 15 K 5.0 K 1.5 K0.56 K

Hsc

Hirr

H // c3

2

1

0

δρc

(mΩ

cm

)40302010

µ0H (T)

70 K

HFL

1.5 K

5.0 K

10 K

20 K

30 K

50 K

(b)(a)

Field dependence of ρc δρc: deviation from linear magnetoresistance

�Above Hsc~8 T, superconductivity is destroyed (normal state).�Above HFL, longitudinal magnetoresistance is linear in H.

0.1

1

10

100

µ 0

H

(T

)

6050403020100T (K)

H0ρ

Hsc

Hpg

Hsp

vortex liquid

2D vortex solid

3D vortex lattice

T. Shibauchi et al., Phys. Rev. B 67, 064514 (2003).

Overdoped Bi2Sr2CaCu2O8+y

~Hc2(0)

Hc2(0)

CUNYMarch 13, 2009

100

80

60

40

20

0

T (

K)

403020100

µ0H (T)

SC

FL

n-FL

HQCP

6

5

4

3

2

1

0

A (

µΩcm

/K2 )

50403020100µ0H (T)

A0

HQCP TFL

HFL

ρc(T)= ρc(0)+ A(H)T2

Divergent behavior of A(H)near Hc2(0) A(H) = D (H-HQCP)-α

α=0.62, HQCP=7.4 T

Kadowaki-Woods relation(A�γ2) suggests enhanced

mass near QCP.

Field-induced QCP (2nd order transition at T = 0 K)

Crossover line at finite temperatures

deviation from T2deviation from H-linear

CUNYMarch 13, 2009

Kohler plot of normal-state MR against µ0H/ρcn(0) � ζcτ

Temperature dependent violation of the Kohler’s rule

ρcn(0) is the normal-state, zero-field ρc(T)

Exp.

isotropicζcτsimulations

4-fold basal-plane anisotropicζcτ�Conventional FL � scaling of all curves in the Kohler plot�At high T , Kohler rule can be violated by other mechanisms(e.g., orbital effects)

� we observe consistent deviation at low T below HFL� intrinsic effect, not an artifact of ζcτ CUNYMarch 13, 2009

Linear c-axis magnetoresistance: � a quantum phenomenon in the

Fermi liquid�Linear high-field transverse MR:

old story�MR in Bi up to 32 T, Kapitza, 1928

“the linear Kapitza law”

�Quantum Magnetoresistance QMR:a consequence of only one Landau band filled

�MR in AgTe chalcogenides,R. Xu, A. Husmann, T.F. Rosenbaum, M.-L. Saboungi, J.E. Enderby, and P.B. Lttlewood, Nature 390, 57 (1997)

�QMR in layered semimetals: condition Larmour frequency ħeH/mc >>T�distance between lowest Landau bands >T and > EF in the lowest band; small hopping between layers, small charge densities n, large hopping mass M, mass in the layers m,

n � (M/m)1/2(eH/ħc)3/2

�Semimetals, high-Tc cupratesA.A. Abrikosov, Phys. Rev. B 60, 4231 (1999); Eur. Phys. Lett. 49, 789 (2000)

�

�

Linear high-field c-axis QMRnew story � MR in layered cuprates

for ; ni �doping

�coherent & incoherent resonant tunneling�through two centers, (d/α)1/2 = coher. dist. �quantum interference�Landau level split with H;m

c-axis

@ high H

�one can not expect orbital quantization effects, but this happens when only lowest Landau level contributes to tunneling

A.A. Abrikosov, Phys. Rev. B 61, 5928 (2000)

CUNYMarch 13, 2009

Summary so far• Overdoped Tl2Ba2CuO6+x still shows

some remnants of non-Fermi liquidbehavior. ρc(T)-ρc0 ~ T1.3 or = A0T2+CT

• At high fields, standard Fermi liquidbehavior recovers at low temperatures.ρc(T)= ρc(0)+A(H)T2

• Fermi liquid coefficient A(H) shows a divergent behavior near Hc2(0), suggesting field-induced quantum criticality.

• It bears a striking resemblance to that in heavy-fermion superconductorCeCoIn5, suggesting a common underlying physics in these strongly correlated electron systems.

100

80

60

40

20

0

T (

K)

403020100

µ0H (T)

SC

FL

n-FL

HQCP

6

5

4

3

2

1

0

A (

µΩcm

/K2 )

50403020100µ0H (T)

A0

HQCP

T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R.Okazaki, and Y. Matsuda, Proc. Natl. Acad. Sci., 105, 7120 (2008)

Heavy fermionsCeMIn5 (M = Rh, Co)

High-Tc cuprates

CUNYMarch 13, 2009

Summary so far• Overdoped Tl2Ba2CuO6+x still shows

some remnants of non-Fermi liquidbehavior. ρc(T)-ρc0 ~ T1.3 or = A0T2+CT

• At high fields, standard Fermi liquidbehavior recovers at low temperatures.ρc(T)= ρc(0)+A(H)T2

• Fermi liquid coefficient A(H) shows a divergent behavior near Hc2(0), suggesting field-induced quantum criticality.

• It bears a striking resemblance to that in heavy-fermion superconductorCeCoIn5, suggesting a common underlying physics in these strongly correlated electron systems.

Heavy fermionsCeMIn5 (M = Rh, Co)

High-Tc cuprates

T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R.Okazaki, and Y. Matsuda, Proc. Natl. Acad. Sci., 105, 7120 (2008).

CUNYMarch 13, 2009

So How About the Doping Dependence ?

n-FL to Fl transition: Doping dependence

CUNYMarch 13, 2009

10 100

0.01

0.1

1

n =

2

[ρab

- ρ

0] (

10-3

Ω c

m)

T (K)

ABCDE

Tl-2201 n = 1

100 75 50 25 0

1.0

1.5

2.0 n

Tc (K)

ρ = ρ0 + AT n

Y. Kubo et al., Phys. Rev. B 43, 7875 (1991)

L. Krusin-Elbaum et al., subm. Nature (2009)

n-FL to Fl transition: Scaling

CUNYMarch 13, 2009

• vertical phase boundary @ T = 0

• from Clausius-Clapeyron relation it requires∆S (� �H/�T) = 0 across the transition

• consistent with 2nd order quantum phase transition

• magnetic susceptibility ξ decreases in the PG state

• ground state E on the PG state [@ H = 0, in simple MFT]

E � N(0)(T*)2• then the field that kills the PG is given by

½ ξ (H*)2 � N(0)(T*)2, where ξ � µB2N(0).

• the relation µB H*(p) � T*(p) followed down to T*(pc0) � 0 defines QCP @ H = 0.

L. Krusin-Elbaum et al., subm. Nature (2009)

n-FL to Fl transition: Field-Doping dependence

CUNYMarch 13, 2009

• Fermi liquid regime becomes dominant with heavier hole (over)doping

0.24 0.26 0.280.0

0.5

1.0

1.5

2.0

2.5

3.0

TFL

(H)

slope

dT/d

H | F

L (

K/T

)doping p

L. Krusin-Elbaum et al., subm. Nature (2009)

n-FL to Fl transition: Critical fluctuation regime

CUNYMarch 13, 2009

•Tx nearly vertical � n-FL regime is not ‘wedge’-like

• px(H) is field-linear, consistent with scaling of H* & T*

scaling

• dome shrinks with increasing H• n-FL wedge shifts to the left• QCP shifts to the left• at fixed doping TFl increases with increasing H• at fixed T HFL increases with doping• consistency with our diagram ?

T

doping

increasing H

n-FL

FL

• two possibilities: wedge or s-shapeV. Aji & C.M. Varma, PRL 99, 067003 (2007)

T

doping

n-FL

FL

QCP & field dependence of n-FL

Tx (~ζx) ~ τc-1 exp(-b/�p-pc)

CUNYMarch 13, 2009

Where Are We Now?• By killing superconductivity with strong magnetic in heavily doped cuprate

superconductors we can map both onsets of the pseudogap T* and Fermi liquid TFL in a previously unexplored regime.

• T* and TFL converge in the T� 0 limit @ a critical doping pc0 � QCP

• Scaling properties of T*(p,H) and TFL(p,H) demonstrate that `strange metal’state in between T* and TFL is governed by a QCP.

• In magnetic field the QCP shifts toward lower doping with the concurrent suppression of Tc � quantum critical fluctuations & superconductivity are intimately linked.

• Normal state �� competing order to high Tc.

• As far as theory � the devil is in the detail !!!!CUNY

March 13, 2009

Thanks!

CollaboratorsTakasada Shibauchi (Dept. of Physics, Kyoto University, Japan)

Yuichi Kasahara, Ryuji Okazaki (students)

Yuji Matsuda________Masashi Hasegawa (Dept. of Mater. Sci. & Eng., Nagoya University, Japan)________

Chuck Mielke, Ross McDonald (NHMFL Los Alamos, USA) – 60-65 T pulsed magnets

Bruce Brandt (NHMFL Tallahassee, USA) – 45 T hybrid dc magnet

Sudip Chakravarty (UCLA), Chandra Varma (UC Riverside), Mike Norman (Argonne), Manfred Sigrist (ETH-Zurich), Hiroshi Kontani (Nagoya U.)

acknowledgements:

CUNYMarch 13, 2009

J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003).

A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003).

Field-induced QCP in quasi-2D heavy fermion superconductor CeCoIn5

M. A. Tanatar et al., Science 316, 1320 (2007).

100

80

60

40

20

0

T (

K)

403020100

µ0H (T)

SC

FL

n-FL

HQCP

6

5

4

3

2

1

0

A (

µΩcm

/K2 )

50403020100µ0H (T)

A0

HQCP

overdoped Tl2Ba2CuO6+x

The similarity suggests a common underlying physics (likely of magnetic origin)in these strongly correlated electron systems.

CUNYMarch 13, 2009

Field-induced antiferromagnetic order in cupratesLa2-xSrxCuO4 (x = 0.10)underdoped

La2-xSrxCuO4 (x = 0.163)

Tl2Ba2CuO6+δ (Tc = 85 K)slightly overdoped

B. Lake et al., Nature 415, 299 (2001).

B. Lake et al., Science 219, 1759 (2001).

K. Kauyunagi et al., PRL 90, 197003 (2003).

spatially resolved NMR

AF vortex cores

� Field induces `striped' AF order.

� Vortex state -- inhomogeneous mixture of a SC spin fluid and a material containing a nearly ordered AF

7.5 T

0 T

neutrons

optimally doped

CUNYMarch 13, 2009

Phase diagram deduced from STEMBi-2212

A. Gomes et al., Nature 447, 569-572 (2007). all still far from the end of the dome !!!

Gap evolution with doping� Gap map – sort of

CUNYMarch 13, 2009

0

100

200

300

400

0.05 0.10 0.15 0.20 0.25 0.30 0.350

100

200

300

400

500

doping

Tem

pera

ture

Tc

Hsc

Hpg

Bi-2212Tl-2201

µ 0H

(T

)

hole concentration p

T*

strange metal

Fermiliquid

pseudogapped metal

T (

K)

Bi-2212Tl-2201

a

Close look at the phase diagram in the OD regime

Hsc ~ 1.4 Tc

but

Zeeman scale

gµBHpg = kBT*

0 100 2000

50

100

150

Bi-2212 Tl-2201

2µBH

pg = k

BT*

µ 0H

pg (

T)

T* (K)

H // c

CUNYMarch 13, 2009

T = 0.6 Tc

Doping dependence in the heavily OD regime

CUNYMarch 13, 2009

H-T diagram of the pseudogap state in BSCCO

1

10

100

µ 0H

sc (

T)

1.00.80.60.40.20.0

T / Tc

UD (Tc = 68 K)UD (Tc = 90 K)OD (Tc = 78 K)OD (Tc = 67 K)

120

100

80

60

40

20

0

µ 0H

(T

)

120100806040200

T (K)

Hsc

Hpg

OD (Tc = 67 K)

Hirr

(a) (b)

� `flat’ temperature dependence of Hpg

� exponential T dependence of peak field Hsc

H-T diagram showing the pseudogap closing field Hpg , and two characteristic fields of the superconducting state: the field Hsc [close, but below Hc2, at which quasiparticle tunneling overtakes Josephson (Cooper pair) tunneling] and the irreversibility field Hirr .

CUNYMarch 13, 2009