Philipp Gegenwart Max-Planck Institute for Chemical Physics of Solids, Dresden, Germany
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Transcript of Philipp Gegenwart Max-Planck Institute for Chemical Physics of Solids, Dresden, Germany
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Philipp GegenwartMax-Planck Institute for Chemical Physics of Solids, Dresden, Germany
Experimental Tutorial on Quantum Criticality
Reviews on quantum criticality in strongly correlated electron systems: in strongly correlated electron systems:
E.g.E.g.
• G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001).G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001).
• H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317
Outline of Outline of this talk::
• IntroductionIntroduction
• Quantum criticality in some antiferromagnetic HF systemsQuantum criticality in some antiferromagnetic HF systems
(mainly those studied in Dresden)(mainly those studied in Dresden)
• Ferromagnetic quantum criticalityFerromagnetic quantum criticality
First partFirst part
Second partSecond part
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T. Westerkamp, J.-G. Donath, F. Weickert, J. Custers, R. Küchler, Y. Tokiwa, T. Radu,
J. Ferstl, C. Krellner, O. Trovarelli, C. Geibel, G. Sparn, S. Paschen, J.A. Mydosh, F.
Steglich
K. Neumaier1, E.-W. Scheidt2, G.R. Stewart3, A.P. Mackenzie4, R.S. Perry4,5,
Y. Maeno5, K. Ishida5, E.D. Bauer6, J.L. Sarrao6, J. Sereni7, M. Garst8, Q. Si9, C. Pépin10
& P. Coleman11
1Walther Meissner Institute, Garching, Germany 2Augsburg University, Germany
3University of Florida, Gainesville FL, USA 4St. Andrews University, Scotland
5Kyoto University, Japan 6Los Alamos National Laboratory, USA
7CNEA Bariloche, Argentina 8University of Minnesota, Minneapolis, USA
9Rice University, Texas, USA 10CEA-Saclay, France
11Rutgers University, USA
Collaborators
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• Lattice of certain f-electrons (most Ce, Yb or U) in metallic environment
• La3+: 4f0, Ce3+: 4f1 (J = 5/2), Yb3+: 4f13 (J = 7/2), Lu3+: 4f14 (6s25d1,l=3)
• partially filled inner 4f/5f shells localized magnetic moment
• CEF splitting effective S=1/2
f-electron based Heavy Fermion systems
T
T* ~ 5 – 50 K
localized moments+
conduction electrons
moments boundin
spin singlets
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Microscopic model: Kondo effect (Jun Kondo ´63)
sSJH sd
local moment conduction el
J: hybridization
between local moments
and conduction el.
AF coupling J < 0
lnT
Kondo-minimum
TK
T5
TK: characteristic
„Kondo“-temperature
T < TK: formation of a bound state
between local spin and conduction
electron spin local spin singlet
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Anderson Impurity Model
Usffs HHHHH cond.-
elf-el hybridization
Vsf
on-site Coulomb
repulsion Uff
Formation of an (Abrikosov-Suhl) resonance at EF of width kBT*
extremely high N(EF) heavy fermions
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Landau Fermi liquid
Lev Landau ´57
Excitations of system with strongly
interacting electrons
Freeelectron gas
1:1correspondence
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Magnetic instability in Heavy Fermion systems
Fermi-surface:
Doniach 1977
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Itinerant (conventional) scenario
Moriya, Hertz, Millis, Lonzarich, …
g
T
TN
gc
TK
NFL
FLSDW 2/3
00 )/ln(/
32
TT
TTTTC
dd
OP fluctuations in space and timeAF: z=2 (deff = d+z)
Heavy quasiparticles stay intact at QCP, scattering off critical SDW NFL
“unconventional” quantum criticality (Coleman, Pépin, Senthil, Si):
• Internal structure of heavy quasiparticles important: 4f-electrons localize
• Energy scales beyond those associated with slowing down of OP fluctuations
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CeCu6-xAux: xc=0.1 inelastic neutron scattering
O. Stockert et al., PRL 80 (1998): critical fluctuations quasi-2D !
A. Schröder et al., Nature 407 (2000): E/T
S(q,)T0.75
0T0.75
H/T
1/(q)
T0.75
non-Curie-Weiss behavior
q-independent local !!
CeCu6-xAux
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FLAF
= p, x, B
NFL
T
Thermal expansion = –1/V ∂S/∂p = V-1 dV/dT
Specific heat: C/T = ∂S/∂T
p
E
EVTST
pS
VC molp
T
mol
*
*
1
/
/1~
! QCP at
Itinerant theory: ~ Tz ~ T-1
(L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 2003)
Grüneisen ratio analysis
1
2
3
4
5
6
7
8
9
10
Resolution: < 0.01Ål/l = 10-10 (l = 5 mm)for T 20 mK, B 20 Tesla
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Experimental classification:
conventionalconventional
CeNi2Ge2
CeIn3-xSnx
CeCuCeCu22SiSi22
CeCoInCeCoIn55
……
unconventionalunconventional
CeCuCeCu6-x6-xAuAuxx
YbRh2Si2
……
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CeNi2Ge2: very clean system close to zero-field QCP
P. Gegenwart, F. Kromer, M. Lang, G. Sparn, C. Geibel, F. Steglich, Phys. Rev. Lett. 82, 1293 (1999)
See also: F.M. Grosche, P. Agarwal, S.R. Julian, N.J. Wilson, R.K.W. Haselwimmer,
S.J.S. Lister, N.D. Mathur, F.V. Carter, S.S. Saxena, G.G. Lonzarich, J. Phys. Cond. Matt. 12 (2000) L533–
L540
0 1 2 3T (K)
0.3
0.5
0.5
2.8
3
(cm)
1.51.5
1.401.40
1.371.37
CeNi2Ge2CeNi2Ge2
B (T)B (T)
~ T1/2 ~ T1/2
~ T~ T
TK = 30 K, paramagnetic ground state
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0 1 2 3 4 5 60
2
4
6
8
10
12
14
16
II a
II c
CeNi2Ge
2
(1
0-6K
-1)
T (K)
0 2 40
5
10
-1 0 2 4 6
0
5
10
/ T
(10
-6K
-2)
T (K)
~ aT1/2+bT
CeNi2Ge2: thermal expansion
R. Küchler, N. Oeschler, P. Gegenwart, T. Cichorek, K. Neumaier, O. Tegus, C. Geibel, J.A. Mydosh, F.
Steglich, L. Zhu, Q. Si, Phys. Rev. Lett. 91, 066405 (2003)
~ aT1/2+b In accordance with prediction of itinerant theory
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30 T
dT
T
C
0 1 2 3 4 50.2
0.3
0.4
0.5
B (T) 0
CeNi2Ge
2
C
/ T
(Jm
ol-1
K-2)
T (K)
0 1 2 3 4 50.2
0.3
0.4
0.5
B (T) 0 2
CeNi2Ge
2
C
/ T
(Jm
ol-1
K-2)
T (K)
for T 0
0 1 2 3 4 50.2
0.3
0.4
0.5
0 2 4
100
150
B (T) 0 2
CeNi2Ge
2
C
/ T
(Jm
ol-1
K-2)
T (K)
T (K)
CeNi2Ge2: specific heat
R. Küchler et al., PRL 91, 066405 (2003).
T. Cichorek et al., Acta. Phys. Pol. B34, 371 (2003).
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CeNi2Ge2: Grüneisen ratio
cr(T) ~ T−1/(z)
prediction:
= ½, z = 2 x = 1 observations in accordance with
itinerant scenario
INS: no hints for 2D critical fluct.
Remaining problem:
QCP not identified (would require
negative pressure)
critical components: cr=(T)−bT
Ccr=C(T)−T
cr = Vmol/T cr/Ccr
0.1 1 5
100
1000
T (K)
cr
0.1 1 5
100
1000
T (K)
cr
cr ~ 1/Tx
with x=1 (−0.1 / +0.05)
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Cubic CeIn3-xSnx
N.D. Mathur et al., Nature 394 (1998)
CeIn3
R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J. Sereni, F. Steglich, PRL 96, 256403 (2006)
• Increase of Increase of JJ by Sn substitution by Sn substitution
• Volume change subdominantVolume change subdominant
• TTNN can be traced down to 20 mK ! can be traced down to 20 mK !
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CeIn3-xSnx
R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J. Sereni, F. Steglich, PRL 96, 256403 (2006)
• Thermodynamics in accordance with 3D-SDW scenarioThermodynamics in accordance with 3D-SDW scenario
• Electrical resistivity: Electrical resistivity: ((TT) = ) = 00 + + AA’’TT, however: large , however: large 00 ! !
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CeCu6-xMx
C/T ~ log T(universal!)
H.v. Löhneysen et al., PRL 1994, 1996A. Rosch et al., PRL 1997O. Stockert et al., PRL 1998
2D-SDW scenario ?A. Schröder et al., Nature 2000
• E/T scaling in “(q,)
• (q) ~ {T(q)}0.75 for all q locally critical scenario
could we disprove 2D-SDW
scenario thermodynamically?
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CeCu6-xAgx
0.1 1 50
1
2
3
4
0 0.5 1.00.0
0.5
1.0
CeCu6-x
Agx
x
TN (
K)
CeCu6-x
Agx
x 0.2 0.3 0.4 0.48 0.8
C /
T (
J/m
ole
K2 )
T (K)
E.-W. Scheidt et al., Physica B 321, 133 (2002).
AF
QCP
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CeCu5.8Ag0.2
0.05 0.1 1 30.5
1.0
1.5
2.0
2.5
3.0
3.5
0.1 1 60
10
20
30
40
50
60
70
ba
/ T
(10
-6K2)
T (K)
B (T) 0 1.5 3 4 5 8
Cel /
T (
J / m
ole
K2 )
T (K)
CeCu5.8
Ag0.2
1
2
3
R. Küchler, P. Gegenwart, K. Heuser, E.-W. Scheidt, G.R. Stewart and F. Steglich, Phys. Rev. Lett. 93, 096402 (2004).
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CeCu5.8Ag0.2
R. Küchler et al., Phys. Rev. Lett. 93, 096402 (2004)
0 1 2 330
60
90
120
150
CeCu5.8
Ag0.2
T (K)
0.1 1 430
60
90
120
Incompatible
with itinerant
scenario!
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YbRh2Si2: a clean system very close to a QCP
P. Gegenwart et al., PRL 89, 056402 (2002).
0.0 0.5 1.0 1.5 2.0 2.50
50
100
150
T*
LFL
NFL
TN
AF
T (
mK
)
B (T)
11 B c Bc
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0.02 0.1 1 20
1
2
3
TN
0.8
B c
0.4
0.2
0.1
B (T) 0 0.025 0.05
YbRh2 (Si
0.95 Ge
0.05 )
2
C
el /
T (
J m
ol -
1 K
-2 )
T (K)
0.02 0.1 1 20
1
2
3
TN
B c
B (T) 0
YbRh2 (Si
0.95 Ge
0.05 )
2
C
el /
T (
J m
ol -
1 K
-2 )
T (K)
0.02 0.1 1 20
1
2
3
TN
B c
B (T) 0 0.025
YbRh2 (Si
0.95 Ge
0.05 )
2
C
el /
T (
J m
ol -
1 K
-2 )
T (K)
=Bc
C/T ~ T-1/3
0(b)
J. Custers et al., Nature 424, 524 (2003)
YbRh2(Si0.95Ge0.05)2
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0.02 0.1 1 100
1
2
a
0 (J
mol
-1 K
-2 )
(B - Bc
) (T)
Stronger than logarithmic mass divergence
~b1/3
b=
0 YbRh2(Si.95Ge.05)2
• stronger than logarithmic mass
divergence incompatible with
itinerant theory
• T/b scaling
FLAF
NFL
T
1
2
J. Custers et al., Nature 424, 524 (2003)
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Thermal expansion and Grüneisen ratio
0.01 0.1 1 100
1
2
3
4
Cel /
T (
J /
K2 m
ol)
T (K)
0
5
10
15
20
25
YbRh2(Si
0.95Ge
0.05)2
/
T (
10-6
K-2)
0.1 110
100
x = 1
x = 0.7
cr
T (K)
R. Küchler et al.,PRL 91, 066405 (2003)
Prediction: cr(T) ~ T−1/(z)
(L. Zhu, M. Garst, A. Rosch, Q. Si, PRL
2003)
= ½, z=2 (AF) x = 1
= ½, z=3 (FM) x = ⅔
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0.01 0.1 1 100
2
4
6
8
10
0.0 0.1 0.2 0.3 0.4 0.50.1
0.2
0.3
~ T0.6
~ T2
0
B (T)
0.03
0.05
0.065
0.1
0.15
0.4
YbRh2(Si
0.95Ge
0.05)
2
(10-6
m3 m
ol-1)
T (K)
B (T) 0 0.03 0.05 0.065
1(1
06 mol
m-3)
T (K)
AF and FM critical fluctuations
P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett. 94, 076402 (2005).
B // c
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Pauli-susceptibility
P. Gegenwart et al., PRL 2005
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29Si – NMR on YbRh2Si2
K. Ishida et al. Phys. Rev. Lett 89, 107202 (2002):
Knight shift K ~ ’(q=0) ~ bulk
Saturation in FL state at B > Bc
Spin-lattice relaxation rate
1/T1T ~ q-average of ’’(q,)
At B > 0.15 T:
Koringa –relation S 1/T1TK2
holds with dominating q=0 fluct.
B 0.15 T: disparate behavior
Competing AF (q0) and FM
(q=0) fluctuations
’’(q,) has a two component
spectrum
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Comparison: YbRh2Si2 vs CeCu5.9Au0.1
q
q
q
q
Q
Q
0
CeCu5.9Au0.1
YbRh2Si2
AF and FM quantum critical fluct.
YRS
Spin-Ising symmetry
Easy-plane symmetry
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Hall effect evolution
S. Paschen et al., Nature 432 (2004) 881:
P. Coleman, C. Pépin, Q. Si, R. Ramazashvili, J. Phys. Condes. Matter 13 R723 (2001).
Large change of H though tiny ordered!
SDW: continuous evolution of H
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Thermodynamic evidence for multiple energy scales at QCP
Fermi surface change clear
signatures in thermodynamics
Multiple energy scales at QCP
P. Gegenwart et al., cond-mat/0604571.
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Conclusions of part 1
There exist HF systems which display itinerant (conventional)
quantum critical behavior: CeNi2Ge2, CeIn3-xSnx, …
YbRh2Si2: incompatible with itinerant scenario:
- Stronger than logarithmic mass divergence
- Grüneisen ratio divergence ~ T0.7
- Hall effect change
- Multiple energy scales vanish at quantum critical point
QC fluctuations have a very strong FM component:
- Divergence of bulk susceptibility
- Highly enhanced SW ratio, small Korringa ratio, A/02 scaling
- Relation to spin anisotropy (easy-plane)?
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Metallic ferromagnetic QCPs ?
Itinerant ferromagnets: QPT becomes generically first-order at low-T
Experiments on ZrZn2, MnSi, UGe2, …
M. Uhlarz, C. Pfleiderer, S.M. Hayden, PRL ´04
D. Belitz and T.R. Kirkpatrick, PRL ´99
1) New route towards FM quantum criticality: metamagnetic QC(E)P e.g. in
URu2Si2, Sr3Ru2O7, …
2) What happens if disorder broadens the first-order QPT?
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Layered perovskite ruthenates Srn+1RunO3n+1
n=1: unconventional superconductor
n=2: strongly enhanced paramagnet
(SWR = 10)
metamagnetic transition!
n=3: itinerant el. Ferromagnet
(Tc = 105 K)
n=: itinerant el. Ferromagnet
(Tc = 160 K)
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Field angle phase diagram on “second-generation” samples(RRR ~ 80)
020
4060
80100
0
200
400
600
800
1000
1200
1400
5
6
78
Field
[tesla
]
Tem
pera
ture
[mK
]
angle from ab [degrees]S.A. Grigera et al. PRB 67, 214427 (2003)
QCEP @ 8 T // c-axis
Evidence for QC fluctuations: Diverging A(H) at Hc (S.A. Grigera et al, Science 2001)
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Thermal expansion
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno,Phys. Rev. Lett. 96, 136402 (2006)
cHH
c
mol
HHhdP
dH
h
S
P
SV
c
~,with Calculation for itinerant metamagnetic QCEP
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Behavior consistent with 2D QCEP scenario
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006)
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Thermal expansion on Sr3Ru2O7
Compatible with underlying
2D QCEP at Hc = 7.85 T
=0 marks accumulation
points of entropy
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6.5 7.0 7.5 8.0 8.5 9.0
1.2
1.6
2.1 T (K)
0.1
0.3
0.6
0.9
1.2
0.2
0.4 0.5
0.7 0.8
1.0 1.1
1.3
cm)
B (T)
Dominant elastic scattering Formation of domains!
Fine-structure near 8 Tesla
S.A. Grigera, P. Gegenwart, R.A. Borzi, F. Weickert, A.J. Schofield, R.S. Perry, T. Tayama, T. Sakakibara, Y. Maeno, A.G. Green and A.P. Mackenzie, SCIENCE 306 (2004), 1154.
7.6 7.8 8.0 8.20.0
0.5
1.0
Field (tesla)
T (
K)
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Thermodynamic analysis of fine-structure
1) No clear phase transitions2) Signatures of quantum criticality
survive in QC regime
also: 1/(T1T)~1/T @7.9T down to
0.3K!! (Ishida group)
3) First-order transitions have
slopes pointing away from
bounded state
Clausius-Clapyeron:
dT
dHMS c
Enhanced entropy in bounded
regime!
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Conclusion Sr3Ru2O7
• Quantum criticality in accordance with itinerant scenario for
metamagentic quantum critical end point (d=2)
• Fine-structure close to 8 Tesla due to domain formation
• Formation of symmetry-broken
phase (Pomeranchuk instability)?
Unlikely because of enhanced entropy
Real-space
phase separation?
(C. Honerkamp, PRB 2005)
liquid
gastwo-phase
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Smeared Ferromagnetic Quantum Phase Transition
Theoretical prediction: FM QPT generically first order at T = 0[D. Belitz et al, PRL 1999]
QCEP
Sharp QPT can be destroyed by disorder exponential tail[T. Vojta, PRL 2003][M. Uhlarz et al, PRL 2004 ]
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The Alloy CePd1-xRhx
Orthorhombic CrB structure
CePd is ferromagnetic with TC = 6.6 K CeRh has an intermediate valent ground state
c
Ce
Pd,Rh
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
TC
(K
)
x
Cp,max
M '-ac "-ac
CePd1-x
Rhx
FM High T measurements suggested quantum critical point (dotted red line) Detailed low T investigation: tail
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0.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
2.5
x = 0.8, = 113 Hz
single crystal
CePd1-xRhx
' (1
0-6
m3
/mo
l)
T (K)
AC Susceptibility in the Tail Region
0.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
2.5
x = 0.8, = 113 Hz
single crystal
CePd1-xRhx
' (1
0-6
m3
/mo
l)
T (K)
B = 0 mTB = 5 mTB = 10 mTB = 15 mT
0.75 0.80 0.85 0.90 0.95 1.000
100
200
300
400
500 CePd1-xRhx
T (m
K)
Rh content x
single crystals polycrystals
= 13 Hz
Crossover transition for x > 0.6indicated by sharp cusps in AC‘ down to mK temperatures
Frequency dependence at low frequencies and high sensitivity on tinymagnetic DC fields no long range order
Maxima of ‘(T) in phase diagram‘(T) in DC field
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Spin Glass-like Behavior
0.15 0.20 0.25 0.30 0.35
2.0
2.5
3.0
' (1
0-6m
3/m
ol)
T (K)
13 Hz113 Hz1113 Hz
x=0.85
CePd1-xRhxsingle crystal
0.1 0.4 1 100.0
0.2
0.4
0.6
0.8
1.0
1.2
x = 0.80 x = 0.85 x = 0.87 x = 0.90
CePd1–x
Rhx
C/T
(J/
(mo
l K2 ))
T (K)
Frequency shift (e.g. x=0.85: TC/[TC log()] of 5%)
Spin glass-like behavior
No maximum in specific heatbut NFL behavior for x ≥ 0.85
0.1 0.4 1 100.0
0.2
0.4
0.6
0.8
1.0
1.2
x = 0.85
CePd1–x
Rhx
C/T
(J/
(mo
l K2 ))
T (K)
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Grüneisen parameter shows no divergence
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”Kondo Cluster Glass“
Strong increase of TK for x ≥ 0.6 indicated by Weiss temperature P, evolution of entropy and lattice parameters
Possible reason forspin glass-like state:
Variation of TK for Ce ionsdepending on Rh or Pdnearest neighborsleading distribution oflocal Kondo temperatures
”Kondo cluster glass“
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
5
6
7
xcr
300
30
CePd1-x
Rhx
TC
(K)
x (Rh conc)
TC from
M(T)
'ac
- He3
C
max
'ac
- He3/He4
100
P (K
)
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Conclusion & Outlook
• Classification of different types of QCPs in HF systems
(conventional vs unconventional)
• Importance of frustration in the spin interaction?
• Role of disorder? – e.g.: smearing of sharp 1st order trans.