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Thermodynamics of phase formation in Sr3Ru2O7
Andy Mackenzie
University of St AndrewsSchool of Physics and Astronomy
University of St Andrews, UK
PITP Toronto 2008
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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
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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
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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
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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)
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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)
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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
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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
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= 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)
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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
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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<
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S < 0 → T > 0
Entropy H1
H 2
Temperature
S
T1 T2
S
T
How to “measure the entropy”
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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.
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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
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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
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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
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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
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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
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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’
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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
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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
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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
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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
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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
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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
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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
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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.
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μ0H [T]
T [K
]
S/T [J/mol K2]
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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
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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)
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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
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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!
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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
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1.5
2
2.5
7 7.5 8 8.5 9Field (T)
'
''
(cm
) ac
(arb
. U
nits
) T = 100 mK
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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]
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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?
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Ruthenates: electronic structure considerations
d shell tet. cryst. field filling & hybridisation
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d shell tet. cryst. field filling & hybridisation
Cu2+ 3d 9
Ruthenates: electronic structure considerations
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d shell tet. cryst. field filling & hybridisation
Ru4+ 4d 4
Ruthenates: electronic structure considerations
![Page 42: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/42.jpg)
Intermediate Report
23rd September 2008
![Page 43: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/43.jpg)
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
![Page 44: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/44.jpg)
(S(H
)-S(5
T))/
T [J
/ m
ol K
^2]
H [T]
T [K]
Entropy Surface
![Page 45: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/45.jpg)
H [T]
T [K]
(S(H
)-S(5
T))/
T [J
/ m
ol K
^2]
Entropy Surface
![Page 46: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/46.jpg)
T [k]
H [T]
Entropy Surface
![Page 47: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/47.jpg)
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
![Page 48: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/48.jpg)
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
![Page 49: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/49.jpg)
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
![Page 50: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/50.jpg)
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
![Page 51: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/51.jpg)
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
![Page 52: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/52.jpg)
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
![Page 53: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/53.jpg)
Isentropes dS=0
5 6 7 8 9 10 11
0.5
1
1.5T
[K]
H [T]
![Page 54: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/54.jpg)
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!
![Page 55: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/55.jpg)
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…
![Page 56: Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK](https://reader037.vdocuments.site/reader037/viewer/2022103123/56649d265503460f949fd174/html5/thumbnails/56.jpg)
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