Interplay between heavy-fermion quantum criticality and unconventional superconductivity

F. Steglich Max Planck Institute for Chemical Physics of Solids, Dresden, Germany Center for Correlated Matter, Zhejiang University, Hangzhou, China Institute of Physics, Chinese Academy of Sciences, Beijing, China

Emergence of superconductivity in the canonical heavy-electron metal YbRh2Si2

Science 351, 485 (2016)

Erwin Schuberth (WMI) ultra-low T measurements Marc Tippmann (WMI) (Tmin = 0.4 mK) Lucia Steinke (BNL) “ Stefan Lausberg (MPI CPfS) design of new χac set-up Alexander Steppke (MPI CPfS) “ Manuel Brando (MPI CPfS) “ Cornelius Krellner (U. Frankfurt) high-quality YbRh2Si2 single crystals Christoph Geibel (MPI CPfS) “ Rong Yu (Renmin U.) 3-component GL theory Qimiao Si (Rice U.) “ F S (MPI CPfS, CCM, IOP)

Doniach phase diagram of Kondo lattice (HF metal) [S. Doniach, Physica B+C 91, 231 (1977)]

• Conventional (SDW) QCP

e.g., CeCu2Si2

• Unconventional (Kondo breakdown) QCP

e.g., YbRh2Si2

YbRh2Si2: T – B phase diagram [J. Custers et al., Nature 424, 524 (2003);

T. Westerkamp, Dissertation, TU Dresden (2008)]

Seff = 1/2

NFL effects in YbRh2Si2: Cel(T)/T & ρ(T); No sign of superconductivity at T ≳ 10 mK

P. Gegenwart et al., Nature Phys. 4, 186 (2008)

RRR ≈ 150

data fit: ρ = 0.49 + 1.07 T3/4

H = 0

P. Wölfle & E. Abrahams, PRB 84, 041101 (2011)

Fermi surface collapse [S. Friedemann et al., PNAS 107, 14547 (2010)]

0.0 0.1 0.2 0.3 0.40.0

0.1

0.2

0.3

cross- single- magneto- field field resistivityYbRh2Si2 RH(B2) RH(B1) ρ(B2)sample 1 sample 2

TLFL

TN

T (K

)

B2 (T)

FWHM(T)

~0.0 0.1 0.2 0.3

0.0

0.1

0.2

0.3

FWHM

(T)

T (K)

cross- single- magneto- field field resistivityYbRh2Si2 RH(B2) RH(B1) ρ(B2)sample 1 sample 2

~

Crossover position T*(B) Crossover width

T*(B) agrees with data from ρ, λ, M (P. Gegenwart et al., Science 315, 969 (2007))

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

(RH-

R H∞ )/

(RH0

-RH∞

)

B/B0

T

T→0

FWHM ~ T

Heavy-Fermion Superconductors Tc(K) CeCu2Si2 0.6 ('79 K) [p = 2.9 GPa: 2.3 ('84 GE/GR)] CeNi2Ge2 0.2 ('97 DA, '98 CA/GR) CeIrIn5 0.4 ('00 LANL) CeCoIn5 2.3 ('00 LANL) Ce2CoIn8 0.4 ('02 NA) Ce2PdIn8 0.7 ('09 WR) CePt3Si 0.7 ('03 VI) CeCu2Ge2 p > 0 0.6 ('92 GE) CePd2Si2 ‘‘ 0.4 ('98 CA) CeRh2Si2 ‘‘ 0.4 ('95 LANL) CeCu2 ‘‘ 0.15 ('97 GE/KA) CeIn3 ‘‘ 0.2 ('98 CA) CeRhIn5 ‘‘ 2.1 ('00 LANL) Ce2RhIn8 ‘‘ 1.1 ('03 LANL) CeRhSi3 ‘‘ 0.8 ('05 SE) CeIrSi3 ‘‘ 1.6 ('06 OS) CeCoGe3 ‘‘ 0.7 ('06 OS) Ce2Ni3Ge5 ‘‘ 0.26 ('06 OS) CeNiGe3 ‘‘ 0.4 ('06 OS) CePd5Al2 ‘‘ 0.57 (‘08 OS) CeRhGe2 ‘‘ 0.45 ('09 OS) CePt2In7 ‘‘ 2.1 (‘10 LANL) CeIrGe3 ‘‘ 1.5 (’10 OS) CeAu2Si2 “ 2.5 (‘14 GE)

UBe13 0.9 ('83 Z/LANL) UPt3 0.5 ('84 LANL) URu2Si2 1.5 ('84 K/DA) U2PtC2 1.5 (’84 LANL) UNi2Al3 1.2 ('91 DA) UPd2Al3 2.0 ('91 DA) URhGe 0.3 ('01 GR) UCoGe 3.0 ('07 AM/KA) UGe2 p > 0 0.7 ('00 CA/GR) UIr ‘‘ 0.14 ('04 OS NpPd5Al2 5.0 ('07 OS) PuCoGa5 18.5 ('02 LANL) PuRhGa5 8.7 ('03 KA) PuCoIn5 2.5 (’11 LANL) PuRhIn5 2.0 (‘12 LANL) Am metal p > 0 2.2 ('05 KA)

Tc(K) Ce3PdIn11 0.42 (`15 PR) Ce3PtIn11 0.32 (`15 PR) PrOs4Sb12 1.85 ('01 UCSD) PrIr2Zn20 0.05 ('10 HI) PrTi2Zn20 0.2 ('12 TO) β-YbAlB4 0.08 ('08 TO/IR) ►YbRh2Si2 0.002 ('14 M/DD) Eu metal p > 0 1.8-2.8 ('09 SL/OS)

YFe2Ge2 1.8 (`14 CA) CrAs p > 0 1.7 (`14 BEI/TO)

Heavy-fermion superconductivity in the vicinity of an antiferromagnetic quantum critical point

BUT: no SC in YbRh2Si2 (YRS) at T ≥ 10 mK YRS: QCP at B = BN ≃ 60 mT (⏊ c), THUS: SC (even HF SC) with Tc < 10 mK impossible at this field

p →

↑ T CePd2Si2 N.D. Mathur et al.,

Nature 394, 39 (1998)

QCP at p = pc ≃ 28 kbar

Erwin Schuberth: PrNi5 nuclear demag (Tmin = 0.4 mK)

Field - cooled (fc) DC magnetization at T ≳ 1.4 mK

TAF = 70 mK: primary electronic AF order

Tc ≥ 2 mK: peak in M(T); visible above T = 1 mK up to B = 23 mT TB ≃ 10 mK: shoulder; M(T) increases below TB

M/B

(10

-6 m

3 /m

ol)

M/B

(10

-6 m

3 /m

ol) 9

B

10 T T c AF

T B

8

YbRh2Si2 B ⊥ c 12 10

8 7 B ⊥ c (mT) 6

0.090 4 6 20

A 1 5

1 10 100

T (mK)

10 10 100 0

10

T T

c AF

(9)T

B

8

(M/B (10-6 m3/mol))YbRh2Si2 B ⊥ c

12

10

8

7 B ⊥ c (mT) 6

(B)0.090 4

6 20

(M/B (10-6 m3/mol))A 1

5

1 10 100

T (mK)

10

10 100 0

Superconductivity: zfc - MDC(T) & χAC(T)

M/B

(a.u

.)

χ′ a

c (S

I)

4

3

T B 2 B ⊥ c (mT) 1 0.418

0.055 0 0.028

C 0.012 -1

0.1 1 10 100

T (mK)

∆M/B

(ar

bitra

ry u

nits

)

0

YbRh2Si2 -2

-4

-6

-8 B ⊥ c

B (mT) 0.012 0.015 0.028 0.055 0.418

1 10 100

T (mK)

0.4

T c

0

-0.4

-0.8 D B = 0

1 10 100

T (mK)

T < Tc: large sc shielding

Tc = 2 mK

T < TB: partial sc shielding ↷ Primary electronic AF order is apparently detrimental to SC! BUT: what is the reason for its weakening below T ≃TB?

4

3

(M/B (a.u.)) (χ′ac (SI))T

B

2

B ⊥ c (mT)

10.418

0.055

00.028

C 0.012

-1

0.1110100

T (mK)

0

(∆M/B (arbitrary units))YbRh2Si2

-2

-4

-6

-8 B ⊥ c

B (mT)

0.012

0.015

0.028

0.055

0.418

1 10 100

T (mK)

0.4

T

c

0

-0.4

-0.8

D B = 0

110100

T (mK)

field - cooled (fc) MDC(T): Meissner effect

M/B

(a.u

.)

χ′ac

(SI)

4

3

T B 2

B ⊥ c (mT) 1 0.418 0.055

0 0.028 C 0.012

-1 0.1 1 10 100

T (mK)

peak in fc - M(T) at Tc ≌ 2 mK T < Tc: flux expulsion („Meissner effect“) Meissner volume ≃ 3%: strong pinning!

4

3

(M/B (a.u.)) (χ′ac (SI))T

B

2

B ⊥ c (mT)

10.418

0.055

00.028

C 0.012

-1

0.1110100

T (mK)

Determination of heat capacity C*(T) using M(T) of YbRh2Si2 as internal thermometer -

via heat - pulse (C* = ΔQ/ΔT) and relaxation (τ = R⋅C*) method

Nuclear specific heat C(T) & entropy SI(T) ∆C

/T (J/

K2mo

l) C/

T (J/K

2 mol)

S / S

I

I,tot

A

10000 T A

YbRh2Si2

1000

100

10

1

0.1

µYb/µB 0.15 0.05 0.01 0.00

B (mT) 59.6

2.4

B ⊥ c

1500

TA

1000

1 10 100 T (mK)

1

500

B 0

B = 2.4 mT

0.9

C 0.8

1 2 3 4 5 6

T (mK) 0 2 4 6 8 10 12

T (mK)

C(T, B) = CQ(T) + CZ(T, μel(B)) A. Steppke et al., Phys. Stat. Sol. B 247, 737 (2010)

ΔC (T) = C(T, B = 2.4 mT) – C(T, 0)

T ≥ 10 mK: SI(T) = SI,tot SI,tot ≈ 1.8 Rln 2

171Yb (I = 1/2, 14.3%) 173Yb (I = 5/2, 16.1%)

A

10000 T

A

YbRh2Si2

1000

100

10

1

0.1

µYb/µB

0.15

0.05

0.01

0.00

B (mT)

59.6

2.4

B ⊥ c

1500

TA

(S / S) (I)1000

1 10 100

T (mK)

(I,tot)1

500

B

0

B = 2.4 mT

0.9

C

0.8

(∆C/T (J/K2mol)) (C/T (J/K2mol))1 2 3 4 5 6

T (mK)

0 2 4 6 8 10 12

T (mK)

Superconductivity along with Nuclear Kondo Effect,

in the absence of Nuclear Order?

Nuclear Kondo temperature TK,nucl = TF,eff exp(-TF,eff/ Thf) Thf ≃ 25 mK, TF,eff ≃ TK ≃ 25 K: TK,nucl = TKexp(-1000) (≪ TK/1000)

Mass enhancement: m*/mel ≃ D/kB(104 K)/TK,nucl = 400/exp(-1000) (“superheavy” fermions)

For heavy-fermion superconductivity: TK ≥ 10 Tc ↷ TK,nucl ≥ 10 Tc Even if TK,nucl ≃ 25 mK (≃ TK/1000) ↷ m*/mel ≃ 400 000!

Kondo temperature TK = D exp(-D/JK)

Conclusion: nuclear Kondo effect requires m*/mel ≥ 4 ⦁ 105 This can be compared with experimental value derived from Bc2(T)!

Field-cooled DC magnetization at very low fields

M vs. 1/T M vs. T M

(arb

itrar

y un

its)

T (m

K)

M (a

rbitr

ary

units

)

B

c

A YbRh2Si2

0

1 T T L H T A 0

-1 T

-20

-2 1.8 2 2.2

T (mK)

2

-40

B ⊥ c B = 0.09 mT 0 0.2 0.4 0.6 0.8 1

1/T (1/mK)

C

1 0 2 4 6 8 10 12

B (mT)

B ≲ 3 mT: - TA > Tc - dBc2/dT∣Tc = Bc2‘ ≃ 25 T/K from Meissner effect (cf. CeCu2Si2)

heavy-fermion SC with m*/mel ≃ several 100 ↷

(M (arbitrary units))AYbRh2Si2

0

1 T T

(B)LH T

A

0

(c)-1 T

(M (arbitrary units))-20

-2

1.8 2 2.2

T (mK)

(T (mK))2

-40

B ⊥ c B = 0.09 mT

0 0.2 0.4 0.6 0.8 1

1/T (1/mK)

C

1

0 2 4 6 8 10 12

B (mT)

Field-cooled DC magnetization at B = 10.1 mT

μsat = 1.24 μB ØJ (2%) ≙ 0.025 μB

Below Tc: ΔM ≥ 0.075 μB ↷ 2/3 due to Meissner effect, SC likely to persist B > 3 mT

First-order nature of superconducting transition

(10-

6 m

3 /m

ol)

χ′ ,

χ′′

ac

ac

10

YbRh2Si2 8 B ⊥ c

6

B = 28 mT, 2.5 µT ac B = 0 (earth), 2.5 µT ac 4 B = 0 (earth), 2.5 µT ac B = 0 (earth), 10 µT ac

from T. Westerkamp et al.

2

0

-2 1 10 100 1000

T (mK)

T ≤ Tc: increase in χ’’(T) superconducting transition: 1st order

10

YbRh2Si2

8B ⊥ c

((10-6 m3/mol))6

B = 28 mT, 2.5 µT ac

B = 0 (earth), 2.5 µT ac

4B = 0 (earth), 2.5 µT ac

(χ′ , χ′′) (ac ac)B = 0 (earth), 10 µT ac from T. Westerkamp et al.

2

0

-2

1101001000

T (mK)

New T – B phase diagram of YbRh2Si2

Bc2‘ ≈ 25 T/K from shielding effect

geff (~TA/BA) ≈ 0.03 – 0.06 ↷ hybrid A phase: (dominating) nuclear AF order

Three - component GL theory by R. Yu & Q. Si

T

A T T T I A AF hf

T T T hyb AF B

m AF SC

T T hyb AF

T ≤ TAF = 70 mK: ΦAF with QAF

T ≤ Thyb = TA = 2.3 mK: ΦJ, ΦI with Q1 ≠ QAF (- λΦJΦI )

Below TA: hybrid order competes with primary order → system approaches QCP → superconductivity develops, driven by quantum critical fluctuations

λ (~ Ahf = 100T/μB ) ≈ 25 mK

A

TTT

I AAF

hf

TTT

hyb AF

B

m

AF

SC

(T)TT

hyb AF

Qutlook: Interplay between unconventional superconductivity and quantum criticality

Heavy - fermion superconductivity robust at any AF QCP

• Conventional (3D - SDW) QCP, CeCu2Si2 presumably also: CePd2Si2, CeIn3, UBe13,….

• Unconventional (Kondo destroying) QCP, CeRhIn5 (H. Shishido et al. '05; T. Park et al. '06, G. Knebel et al. `08)

• Link to doped Mott insulators e.g., cuprates, organic charge – transfer salts

new example: YbRh2Si2 Kondo breakdown QCP: (T=0) 4f-orbital selective Mott transition

Slide Number 1Emergence of superconductivity �in the canonical heavy-electron metal YbRh2Si2 �Science 351, 485 (2016)Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Qutlook: Interplay between unconventional�superconductivity and quantum criticality