Thermodynamics of phase formation in Sr3Ru2O7

Andy Mackenzie

University of St AndrewsSchool of Physics and Astronomy

University of St Andrews, UK

PITP Toronto 2008

M. Allan1, F. Baumberger1, R.A. Borzi1, J.C. Davis1,3,4, J. Farrell1, S.A. Grigera1, J. Lee4, Y. Maeno5, J.F. Mercure1, R.S. Perry1,2, A. Rost1, Z.X. Shen6, A. Tamai1, A. Wang3

University of St Andrews

Collaborators

1 University of St Andrews; 2 University of Edinburgh; 3

Cornell University; 4 Brookhaven National Laboratory 5

Kyoto University; 6 Stanford University

Contents

1. Introduction – materials and terminology

2. Metamagnetic quantum criticality and low-frequency dynamical susceptibility in slightly dirty Sr3Ru2O7.

4. Magnetocaloric effect as a probe of the ‘entropic landscape’

5. Spectroscopic imaging of conductance oscillations around scattering centres: a dynamics-to-statics transducer.

6. Conclusions

3. Phase formation in ultra-pure Sr3Ru2O7

Mag

netis

atio

n

Magnetic field

1st order

crossover

M

H

Metamagnets and the vapour-liquid transition

Mapping between both systems

P, T, H, T, M

T

P

Critical end-point

1st order

liquid

vapour

T

H

H

u

T

h

Metamagnets and Quantum Critical Points

Critical end-point

1st order

Important difference with water: The transition can be tuned to T=0.

Large majority of real itinerant metamagnets are first order at T = 0 even after best effort to tune. See e.g. T. Goto et al., Physica B 300, 167 (2001)

S.A. Grigera, R.A. Borzi, A.P. Mackenzie, S.R. Julian, R.S. Perry & Y. Maeno, Phys. Rev. B 67, 214427 (2003).

Experimental phase diagram of “clean” Sr3Ru2O7

0 20 40 6080

100

0

200

400

600

800

1000

1200

1400

5

67

8

Field [

tesla]

Tem

pera

ture

[mK]

angle from ab [degrees]

Plane defined by maxima of imaginary part

T* inferred from maximum in real part of a.c. susceptibility

Quantum critical end-point

c-axis (90)

1.20.8

0.40.0

T (K)4.5

5.56.5

Field (tesla)

1.20.8

0.40.0

T (K)4.5

5.56.5

Field (tesla)

T* = 1.25K= 0(H // ab) x 10

1.20.8

0.40.0

T (K)4.9

6.47.9

Field (tesla)

1.20.8

0.40.0

T (K)4.9

6.47.9

Field (tesla)

T* = 1.05K= 40°x 10

Constructing the experimental phase diagram

1.20.8

0.40.0

T (K)5.5

6.57.5

Field (tesla)

1.20.8

0.40.0

T (K)5.5

6.57.5

Field (tesla)

= 60° T* = 0.55Kx 0.5

x 10

1.20.8

0.40.0

T (K)6.5

7.58.5

Field (tesla)

1.20.8

0.40.0

T (K)6.5

7.58.5

Field (tesla)

= 90°(H // c)

x 10T* < 0.1K

No evidence of first-order behaviour for H // c

= 0(H // ab)

1.20.8

0.40.0

T (K)4.9

6.47.9

Field (tesla)

1.20.8

0.40.0

T (K)4.9

6.47.9

Field (tesla)

= 40°

Evidence for very slow dynamics

Why are the global maxima so weak?

Large changes at amazingly low frequency

ma

x (1

0-6 m

3/m

ol R

u)

2

4

01 2 3

f (kHz)

7.9 8.1 8.37.7oH (T)

0.4

0.8

1.2

0

T(K

)- Resistivity: d/dH and d2/dT2

- Susceptibility: ’ and ’’ - Magnetostriction: (H)- Magnetisation

Approach to criticality ‘cut off’ by a new phase in highest purity samples ( ~ 3000 Å)

S.A.Grigera et al., Science 306, 1154 (2004)

P. Gegenwart et al., Phys. Rev. Lett. 96, 136402 (2006)

R.A. Borzi et al., Science 315, 214 (2007)

Phase lines bound a region with pronounced resistive anisotropy: ‘electronic nematic’ properties

7.9 8.1 8.37.7

oH (T)

0.4

0.8

1.2

0

T(K

)“The wrong shape”

usually: “dome”

here: “muffin”

first order phase trasitions? -> Clausius-Clapeyron

The H-T Phase diagram

M

S

dT

dH

S inside bigger than S outside

S>

S<

S < 0 → T > 0

Entropy H1

H 2

Temperature

S

T1 T2

S

T

How to “measure the entropy”

Copper RingCuBe Springs

Kevlar Strings (35 @ 17μm)

Silver Platformwith sampleon other side

Thermometer (Resistor)

2 cm

Our experimental setup (Andreas Rost)

High level of control possible via tunable thermal link; easy system to model.

7 7.5 8 8.5

390

400

410

420

430

H [T]

T [

mk]

H [T]

T [

mk]

Metamagneticcrossover seen in susceptibility

Sharper features associated with first order transitions

Sample raw Magnetocaloric Effect data from Sr3Ru2O7

‘Signs’ of changes confirm that entropy is higher between the two first order transitions than outside them.

B

S

C

T

B

T

T

M

C

T

B

T

Under fully adiabatic conditions

μ0H [T]

T [m

K]

4 6 8 10 12 14235

240

245

250

255

260

265

270

8.5 9 9.5 10 10.5 11 11.5

-1

-0.5

0

0.5

1

Field [T]T

empe

ratu

re C

hang

e [m

K]

Increaing FieldDecreasing FieldT=150mK

Magnetocaloric quantum oscillations

1

0

-1

ΔT

[mK]

μ0H [T]8.5 9 9.5 10 10.5 11 11.5

Measurement noise level: 25 μK / √Hz

0.09 0.095 0.1 0.105 0.11 0.115

-1

-0.5

0

0.5

1

Inverse Field [T-1]T

empe

ratu

re C

hang

e [m

K]

Increasing FieldDecreasing Field

μ0H [T]

T [m

K]

4 6 8 10 12 14235

240

245

250

255

260

265

270

T=150mK

Magnetocaloric quantum oscillations

1

0

-1

ΔT

[mK]

1/μ0H [T-1] 0.09 0.1 0.11

Measurement noise level: 25 μK / √Hz

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

Preliminary conclusions from magnetocaloric effect (MCE) work on Sr3Ru2O7

T

M

C

T

B

T

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

• MCE confirms our prior identification of first-order lines as equilibrium phase transitions

• Entropy is indeed higher between the lines than either side of them.

• ‘Phase’ seems to be characterised by ‘quenching’ of

T

M

Preliminary conclusions from magnetocaloric effect (MCE) work on Sr3Ru2O7

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]Taking the next step: the ‘entropic landscape’

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]

S/T [J/mol K2]

Taking the next step: the ‘entropic landscape’

0.12

0.17

0.22

0.27

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]

T [K

]

S/T [J/mol K2]

Taking the next step: the ‘entropic landscape’

μ0H [T]

0.12

0.17

0.22

0.27

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]

T [K

]

S/T [J/mol K2]

Taking the next step: the ‘entropic landscape’

μ0H [T]

0.12

0.17

0.12

0.17

0.22

0.27

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]

T [K

]

S/T [J/mol K2]

Taking the next step: the ‘entropic landscape’

μ0H [T]

0.12

0.17

0.22

0.27

7.4 7.6 7.8 8.0 8.2 8.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(key for symbols)

vs H vs T '' M '

Field (tesla)

T (

K)

μ0H [T]

T [K

]

μ0H [T]

T [K]

S/T [J/mol K2]

Taking the next step: the ‘entropic landscape’

0.12

0.17

0.22

0.27

Power Law Fit To Specifc Heat (

C(H

)-C

(5T

))/ T

Field [T]

cH

aTCb

9.7

)9.7(*/

Fitequation

Fitrange5 T to 7.1 T

Resulting Parameters

a = 0.004(1)b = -0.99(5) c = -0.012(2)

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

0

0.02

0.04

0.06

0.08

0.1 datafitted curve

dHvA and STM QPI and ARPES: Fermi velocities in Sr3Ru2O7 of 10 km/s and below: suppressed from LDA values by at least a factor of 20: direct observation of d-shell heavy fermions.

kkF-kF

q = 2kF = F

q < 2kF

q > 2kF

Spatially resolved conductance oscillations around scattering centres: a dynamics–to–statics transducer

Conclusions

• Sr3Ru2O7 can be tuned towards a quantum critical metamagnetic transition.

• If this is done in ultra-pure crystals (mfp > 3000Å) a new phase forms before the quantum critical point is reached.

• The magnetocaloric effect, if measured with care in a calibrated system, can give a comprehensive picture of the entropy evolution near QCPs.

• Material with slight disorder shows strongly frequency-dependent low T susceptibility; situation in pure material still needs to be investigated.

μ0H [T]

T [K

]

S/T [J/mol K2]

7 7.5 8 8.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Field [T]

Ent

ropy

cha

nge

600mK

ΔS

(J/

K)

Field (T)

0.09

0.05

0

7 7.5 8 8.5

T = 600 mK

Consider the ferromagnetic superconductor URhGe

Superconductivity at low T, B

Metamagnetic transition due to spin reorientation deep in ferromagnetic state

Metamagnetic QCP?

D. Aoki, I Sheikin, J Flouquet & A. Huxley, Nature 413, 613 (2001)

In URhGe the new phase in the vicinity of the metamagnetic QCP is superconducting

Re-entrant superconductivity!

F. Lévy, I. Sheikin, V. Hardy & A. Huxley, Science 309, 1343 (2005).Perspective: A.P. Mackenzie & S.A. Grigera, ibid p. 1330

F. Lévy, I. Sheikin & A. Huxley, Nature Physics 3, 461 (2007)

Potentially more than ‘just’ interesting basic science:

25 T insufficient to destroy superconductivity although Tc < 0.5 K!

Pronounced resistive anisotropy in a region of phase space bounded by low T 1st order phase transitions

J H J // H

HJ HJ

R.A. Borzi, S.A. Grigera, J. Farrell, R.S. Perry, S. Lister, S.L. Lee, D.A. Tennant, Y. Maeno & A.P. Mackenzie, Science 315, 214 (2007)

T = 100 mK

1.5

2

2.5

7 7.5 8 8.5 9Field (T)

'

''

(cm

) ac

(arb

. U

nits

) T = 100 mK

6 7 8 9

110

112

114

116

118

120

122

H [T]

T [

mk]

8

Example of magneto-thermal oscillation with field aligned to c-axis

H [T]

University of St Andrews

Structure chi(T) and refto Shinichi etc.

3

2.5

2

1.5

1

0.5

0

(10

-2em

u/R

u m

ol)

300250200150100500

(K )

S r3R u 2O 7

0 = 0.3 T

ab c

m ax

3

2

1

0

(10

-2em

u/R

u m

ol)

3020100 (K )

m ax

Basic bulk properties of Sr3Ru2O7

At low temperature and low applied magnetic field,it is an anisotropicFermi liquid (c /ab 100).

S.I. Ikeda, Y. Maeno, S. Nakatsuji, M. Kosaka and Y. Uwatoko, Phys. Rev. B 62, R6089 (2000).

Low-T susceptibility is remarkably isotropic and T-independent: strongly enhanced Pauli paramagnet on verge of ferromagnetism?

Ruthenates: electronic structure considerations

d shell tet. cryst. field filling & hybridisation

d shell tet. cryst. field filling & hybridisation

Cu2+ 3d 9

Ruthenates: electronic structure considerations

d shell tet. cryst. field filling & hybridisation

Ru4+ 4d 4

Ruthenates: electronic structure considerations

Intermediate Report

23rd September 2008

Entropy Change

4 5 6 7 8 9 10 11 12 13

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

Decreasing T

• (S(H)-S(5T))/T as a function of H

• Different temperatures are offset for clarity

(S(H

)-S(5

T))/

T [J

/ m

ol K

^2]

H [T]

T [K]

Entropy Surface

H [T]

T [K]

(S(H

)-S(5

T))/

T [J

/ m

ol K

^2]

Entropy Surface

T [k]

H [T]

Entropy Surface

Entropy Change

4 5 6 7 8 9 10 11 12 13

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

Decreasing T

• (S(H)-S(5T))/T as a function of H

• Different temperatures are offset for clarity

4 5 6 7 8 9 10 11 12 13

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

For better comparison I will choose4 traces at T= (230mK, 400mK,900mK,1450mK)

Entropy Change

4 5 6 7 8 9 10 11 12 13

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

230mK

For better comparison I will choose4 traces at T= (230mK, 400mK,900mK,1450mK)

400mK

900mK

1450mK

Entropy Change

4 5 6 7 8 9 10 11 12 13

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

4 5 6 7 8 9

0

0.02

0.04

0.06

0.08

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

Field [T]

Entropy Change

On the right these curves are plot without offset

Comparison (C(H)-C(5T))/T vs (S(H)-S(5T))/T

4 5 6 7 8 9

0

0.02

0.04

0.06

0.08

0.1

Field [T]

En

tro

py

cha

ng

e (S

(H)-

S(5

T))

/ T

The curve in blue is (C(H)-C(5T))/T at 250mK. The fact that its amplitude is identicalto the measured entropy change confirms that up 7.1T the system behaves like a Fermi Liquid.

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

0

0.02

0.04

0.06

0.08

0.1

Power Law Fit To Specifc Heat (

C(H

)-C

(5T

))/ T

Field [T]

cH

aTCb

9.7

)9.7(*/

Fitequation

Fitrange5 T to 7.1 T

Resulting Parameters

a = 0.004(1)b = -0.99(5) c = -0.012(2)

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

0

0.02

0.04

0.06

0.08

0.1 datafitted curve

Isentropes dS=0

5 6 7 8 9 10 11

0.5

1

1.5T

[K]

H [T]

Si /Rosch Paper

H

S

c

TMCE

cr

In a Fermi Liquid:

)()( HSHc

Definition of Magnetocaloric Effect:

Assume Power Law:

const

H

HHTHS

a

c

c)(

1

a

c

c

c H

HH

H

aT

H

S

a

c

ccr H

HHc

1

c

c

H

HHMCE

I.e. it is a general result that is independent of the power law the entropy itself follows!

Si /Rosch vs Millis/Grigera/…

Si / Rosch(on the unorder (high field) side

Millis / Grigera / … around

c

c

H

HHh

Both assume that the dynamical dimension is z=3 and the real dimensionis d=2 for a ferromagnetic QCEP in 2 dimensions but they mention differentcritical exponent for the specific heat coefficient

21 hccr

31

hccr

But: I need to check that these calculations have been done for constant number and not constant chemical potential…

Antisymmetrise the up and down sweep

Old way: First integrate each trace and then smooth

New way: First smooth the signals and then integrate

T

H

H

Δ T

0

ΔS

ΔSΔS

Δ T

0 H

H

H

H

T T