[PPT] - Thermal expansion and magnetostriction measurements of CeCu 6- x Au x PowerPoint Presentation

SLIDE 1

Thermal expansion and magnetostriction measurements of CeCu6-xAux single crystals

and S S T ∂ ∂ ∂ ∂δ

S(T,δ)

T ∂ ∂δ C T , , ...

V

M T ∂ α ∂ T , ,

V

T ∂

specific heat thermal expansion, magnetocaloric effect,

competing ground states

C C A

g , ... T δ QCP

CeCu6-xAux :

control parameter δ = p

V S ∂ ∂

δc δ accumulation of entropy S

V m

V T V p S α ⋅ = ∂ ∂ − = ∂ ∂ ⇒

~ degrees of freedom

SLIDE 2

Thermal expansion and magnetostriction measurements of CeCu6-xAux single crystals

Different scenarios: traditional “itinerant” scenario “local” scenario ...

T T TKondo lattice TN TKondo lattice TN J ~ V(p, x) J ~ V(p, x)

FL FL

?

and scaling behavior

V

p S α = ∂ ∂

Hertz, Millis, Moiya et al. Coleman, Si et al..

and scaling behavior

SLIDE 3

Outline

Stefanie Drobnik1, Frédéric Hardy1, Kai Grube1, Roland Schäfer1, Peter Schweiss1, Oliver Stockert2, Thomas Wolf, Christoph Meingast1, and Hilbert v. Löhneysen1

1 IFP Forschungszentrum Karlsruhe, PI Universität Karlsruhe, KIT, Germany 2 MPI für chemische Physik fester Stoffe, Dresden, Germany

Introduction: Thermal expansion at a QCP Experimental details: dilatometer CeCu6-xAux: control parameter x and p Measurements: Thermal expansion: volume, lattice parameter (compressibility, magnetostriction) Summary and outlook Summary and outlook

SLIDE 4

Thermal expansion at a quantum critical point

Fermi liquid for T → 0: αV/T → ∂γ/∂p = constant classical fluctuations

T

rdered

NFL

S ∂ <

phase NFL

dS = 0

αcrit /T

p < ∂

p QCP

FL FL

S ∂

|αNFL/T|

FL NFL

L. Zhu et al. PRL 91, 066404 (2003)

S p ∂ > ∂

T

M. Garst et al. PRB 72, 205129 (2005)
P. Gegenwart et al. PRL 96, 136402 (2006)
M. Garst et al. :cond-mat 0808.0616

SLIDE 5

Experimental details

Capacitance dilatometer: m Capacitance dilatometer: thermal expansion & magnetostriction a) 40 mK < T < 10 K, µ0H < 14 T b) 1 5 K < T < 300 K µ H < 10 T 20 mm b) 1.5 K < T < 300 K, µ0H < 10 T rotatable in H ∆L > 0.01 Å, ∆L/L 10 -9 plate capacitor mm sample c) Miniature dilatometer built in a He gas pressure cell: 5 m a He-gas pressure cell: compressibility measurements 10 K < T < 325 K, p < 8 kbar Å ∆L > 0.1 Å, ∆L/L 10 -8

SLIDE 6

CeCu6-xAux: control parameter x and p

2 0 1.6

TN TN

x = 0 5 For x < 1.0:

1.5 2.0 1.2 1.4

N

(K)

N

(K)

x 0.5

0.5 1.0 0.8 1.0

AF AF

0.0 0.5 1.0 0.0 0.0 0.5 1.0 0.6

p (GPa)

AF

x

AF

p ( )

V V

Comparison between the control parameters x and p

SLIDE 7

CeCu6-xAux: control parameter gold content x

Cu(2)→Au

0 0 0 5 1 0 1 5

x

( ) Ce

435 440 445 0.0 0.5 1.0 1.5

V Å

Cu

420 425 430

Cu(2) Cu(4)

(Å)

c

8.2 8.3 5 10 5.12 10.4

c b a

a,b,c(x) (Å)

b a

rthorhombic

Pnma structure

1 8.1 1 5.08 5.10 1 10.2 (Å) (Å) (Å)

( )

x x x

For x < 1 0 residual resistivity For x < 1.0

strongly anisotropic distortion of the lattice
introduction of disorder

y x

SLIDE 8

CeCu6-xAux: hydrostatic pressure p

1.000 x Au

1.000 T = 300 K

a/a0 b/b

x = 0 Ce

1.000

b/b0 c/c0

Cu

0.998

0.999 V /V0

c

0.0 0.1 0.2 0.3

p (GPa)

b a

rthorhombic

Pnma structure If V(x) = V(p): usually a(x) ≠ a(p), b(x) ≠ b(p), c(x) ≠ c(p) and ρ0(V(x)) ≠ ρ0(V(p)).

SLIDE 9

Comparison of x and p

At the QCP, CeCu6-xMx (M = Au, Pd, Pt) show the identical NFL behavior Cp/T (J/molK2) but differ:

in their lattice parameters
in their lattice parameters
and the amount of disorder

TN AF x

⇒ for x ≤ 0.3:

x

T (K) doping with Au = chemical pressure ~ applying negative hydrostatic pressure ⇓ control parameter: volume V control parameter: volume V

SLIDE 10

Thermal expansion of CeCu6-xAux

50 αV

Ce(4f1)Cu5.85Au0.15

30 40

(10-6K-1)

Grüneisen ratio: E

Debye V

θ ∂ ∂ α Γ 1 * 1

20 30

La(4f0)Cu6 p p E C

Debye Debye p V p

∂ θ → ∂ ≈ = Γ *

10

100 150

Cp (J/molK)

50 100 150 200 250 300

T (K)

50 100 50 100 150 200 250 300

T (K)

typical for HF systems: large 4(5)f contribution to the thermal expansion g ( ) p

SLIDE 11

Thermal expansion: 4f contribution

4f1 contribution to αV = αV(Ce(4f1)Cu5 85Au0 15) - αV (La(4f0)Cu6) 4f contribution to αV αV(Ce(4f )Cu5.85Au0.15) αV (La(4f )Cu6)

15 αV(4f)

(10 6K 1)

10

(10-6K-1)

5 TN

TN

4f1 contribution to Cp

5

AF TN

3 4

Cp(4f) (J/molK)

0.1 1 10 100

T (K)

x

1 2

TN

AF order Kondo lattice CEF

0.1 1 10 100

T (K)

∆CEF (meV) 0 – 7 – 13.8

SLIDE 12

Linear thermal expansion coefficients αi

α = α +α +α

10 12

αa,b,c

αV = αa+αb+αc αi ~ -∂S/∂σi i = a, b, c

6 8 10

(10-6K-1)

c

uniaxial pressure in i direction

2 4 6 b

1

b,c

2

a

0.05

αa,b

T (K)

0.1 1 10 100

T (K)

Anisotropy

f

αi changes d di hi h h t i ti

T (K)

AF order Kondo lattice CEF depending which characteristic energy scale dominates.

∆CEF (meV) 0 – 7 – 13.8

SLIDE 13

Non-Fermi liquid behavior

Fermi liquid for T → 0:

C /T 60

q C/T → γ = constant αV/T → ∂γ/∂p = constant

3 4

x = 0.5 x = 0.3

Cp /T (J/ mol K2)

0 1 x = 0.15

40 60

αV /T

(10-6 K-2)

x = 0.15 x = 0.1

2 3

x = 0.0 x = 0.1

20

x = 0.0

1

x = 0.5 x = 0.3

AF TN

0.1 1 T (K) 0.1 1 4

20

x 0.5

T (K)

AF x

T (K) T (K)

Non-Fermi liquid behavior next to the onset of AF order.

thermal expansion measurements for x = 0, 0.3, 0.5 from A. de Visser et al.

SLIDE 14

Grüneisen parameter ΓV of CeCu6-xAux

If the system is dominated by a characteristic energy scale E*

140 * V E ∂

If the system is dominated by a characteristic energy scale E* ΓV reveals the (normalized) volume dependence of E*.

100 120

x = 0.0 xc=0.1

ΓV

* *

V V mol T p

V E V B C E V α ∂ Γ = ≈ ∂

40 60 80

x = 0.15

bulk modulus measured at T = 10 K

20 40

x = 0.3

*

V

E Γ → ∞ ⇒ →

AF TN

0.01 0.1 1 10

20

x = 0.5

T (K)

AF x

T (K)

The non-Fermi liquid behavior is caused by a QCP.

SLIDE 15

Non-Fermi liquid behavior

Fermi liquid for T → 0:

C /T 60

q C/T → γ = constant αV/T → ∂γ/∂p = constant

3 4

x = 0.5 x = 0.3

Cp /T (J/ mol K2)

0 1 x = 0.15

40 60

αV /T

(10-6 K-2)

x = 0.15 x = 0.1

2 3

x = 0.0 x = 0.1

20

x = 0.0

1

x = 0.5 x = 0.3

AF TN

0.1 1 T (K) 0.1 1 4

20

x 0.5

T (K)

AF x

T (K) T (K)

Non-Fermi liquid behavior next to the onset of AF order.

thermal expansion measurements for x = 0, 0.3, 0.5 from A. de Visser et al.

SLIDE 16

Comparison with the specific heat C/T(V,T = 0.1K)

x p

S Cp(V, T = const. → 0) tracks the maximum of S at the QPC.

4 x, p = 0 x = 0 0 p > 0

p 0.0 0.5 1.0 T = 0.1K

T

2 3

x 0.0, p > 0 x = 0.1, p > 0 x = 0.2, p > 0 x = 0.3, p > 0

Cp /T (J/mol K2)

Vc V

1 2

x = 0.0, αV x = 0.1, αV x = 0 15 α

TN

412 416 420 424 428 1

x = 0.15, αV x = 0.3, αV

AF

N

412 416 420 424 428 V (Å3)

ln( / ) for

p V p

C T T C p ∂ α Γ = ≈ → ∂

x

p

SLIDE 17

Sign change of the thermal expansion

x = 0.1

x p

S

4 p 0.0 0.5 1.0 T = 0.1K

~ 1/Tcoh

T

2 3 Cp /T (J/mol K2)

coherence temperature from resistivity measurements Vc V

1 2

ρ

TN

412 416 420 424 428 1

T

AF

N

412 416 420 424 428 V (Å3)

Tcoh(TKondo)

x

dominated b the Kondo lattice state dominated by the Kondo-lattice state

SLIDE 18

Sign change of the thermal expansion

40 60

αV /T

(10-6 K-2)

x = 0.15 x = 0.1

Expected behavior next to the QCP: T < TN T > TN α (x < x ) ≈ α (x > x )

no sign change of αV

20

x = 0.5 x = 0.3

− αV(x < xc) ≈ αV(x > xc) at TN(x = 0.15) ⇒ S maximum not at x = 0.1

with decreasing x and T
40
20

with decreasing x and T increasing "background" di d ?

0.1 1 4

60

⇒ disorder ? ⇒ additional QCP ?

T (K)

30 40

x = 0.15

6 K-2)

10 20

αV /T (10-

thermal expansion measurements for x = 0, 0.3, 0.5 from A. de Visser et al.

0.1 1 10

T (K)

SLIDE 19

Disorder ?

ibl possible αV /T - background

0 0 0 5 1 0 0.0 0.5 1.0 x

S l ith th l t For x < 0.3 all samples show: C(x) ≈ C(p)

4

x

p 0.0 0.5 1.0

Sample with the largest chemical disorder ⇒ no "background" visible

3 Cp /T T = 0.1K 2 (J/mol K2) 1 412 416 420 424 428 V (Å3)

SLIDE 20

Linear thermal expansion coefficients αa,b,c (T > 10K)

x = 1.0 x = 0.15 x = 0.3 x = 0.1 x = 0

10 αa,b,c(4f)

(10-6 K-2)

c c c c 5 a

(10 6 K 2)

a c a c c 5 c a c a c a c a a

∆ ( V)

10 100

5

T (K)

10 100

T (K)

10 100

T (K)

10 100

T (K)

10 100

T (K)

c

∆CEF (meV) 0 – 7 – 13.8 0 – 8.6 – 13.8

T (K) T (K) T (K) T (K) T (K) Distortion of the Cu T lattice due to CEF splitting of the 4f state. c b c b Ce

∆CEF from Stroka et al.

SLIDE 21

Linear thermal expansion coefficients αa,b,c /T (T < 10K)

30 35 40 45 x = 0

(10-6 K-2)

x = 0.1 x = 0.15 x = 0.3 x = 0.5 10 15 20 25

b

b /T, αc /T

c b c b c b c b

10
5

5

b a αa /T, αb a a b a b a

0.1 1 10

20
15

T (K)

0.1 1 10

T (K)

0.1 1

T (K)

0.1 1

T (K)

0.1 1 5

T (K)

c

A further QCP: another characteristic energy scale E*→ 0 should be visible in a different anisotropy of αi, i = a, b, c :

thermal expansion measurements for x = 0, 0.3, 0.5 from A. de Visser et al.

SLIDE 22

Possible origins of the anisotropy of αi

1/ / 1/ /

i T V

V V T B T p α = ⋅∂ ∂ = ⋅∂ ∂ 1/ / 1/ /

inne T r V V

V V T B T p α ∂ ∂ ∂ ∂

"spring constant" compressibility of the lattice "inner" pressure of electron system, phonons, ...

p

Isotropic thermal expansion αi:

p

anisotropic elasticity

r

uniaxial pressure Anisotropic thermal expansion α:

p

anisotropic elasticity

r

uniaxial pressure expansion αi:

p σ

SLIDE 23

Elastic properties of CeCu6-xAux

4.5 Linear kompressibilities:

4.0 b ka,b,c (10-5 GPa-1) T = 10 K

ka,b,c = −∂εa,b,c/∂p|T Linear kompressibilities:

p

3.5 c

3.0 a

The elastic properties of CeCu6-xAux are anisotropic: k < k < k

11

c c

0.0 0.5 1.0 2.5

x

ka < kb < kc

22 33

c c

/ / ith S S i j b ∂ ∂ ∂ ∂

∑

To remove the lattice contribution : uniaxial pressure → strain dependence of S

/ / with , , ,

i i i ij j T T j

S S c i j a b c α ∝ −∂ ∂σ → ∂ ∂ε = ⋅α =

∑

εi σi

elastic constants elastic constants (6 independent)

SLIDE 24

Elastic properties versus x and T

Elastic constants e g Bulk modulus:

120 Elastic constants, e.g. c22 (Weber et al., Finsterbusch et al.) Bulk modulus: BT = 1/KT = -V·∂p/∂V|T

110 B T = 10 K 120 c22 (GPa) x = 0 100 BT, p = 0 (GPa) 115

117 0

5% 20%

100 110

116.8 117.0 x = 0 1 x = 0

0.0 0.5 1.0 90 x 100 200 300 105

0.1 1 10 116.6 x 0.1

x T (K)

The elastic properties are (nearly) not influenced by the 4f electron The elastic properties are (nearly) not influenced by the 4f electron.

SLIDE 25

Strain dependent Grüneisen parameter

/ ln / c C E∗ Γ α ∂ ∂ε

∑

strain dependence of the characteristic

/ ln /

i ij j p i T j

c C E Γ ≈ ⋅α ≈ ∂ ∂ε

∑

6 independent elastic constants: c c c Weber et al Suzuki et al

strain dependence of the characteristic energy scale E*

120 140 x = 0 x = 0.1 x = 0.15 x = 0.3 x = 0.5

c11, c22, c33 Weber et al., Suzuki et al. c12, c13, c23 from the ka,b,c measurements

60 80 100 120

10-6 K-1)

20 40 60 a, Γb, Γc ( 0 1 1 10

60
40
20

Γa

0 1 1 10 0 1 1 0 1 1 0 1 1 5 0.1 1 10

T (K)

0.1 1 10

T (K)

0.1 1

T (K)

0.1 1

T (K)

0.1 1 5

T (K)

Kondo lattice state: (nearly) isotropic strain dependence ⇒ control parameter: V

SLIDE 26

Coherence temperature: Tcoh(V(x)) = Tcoh(V(p)) ?

Comparison between T (V(x)) and T (V(p)) Comparison between Tcoh(V(x)) and Tcoh(V(p))

100 T (V( )) p 0.0 0.5 x

x = 0.1

100 Tcoh(V(x)): varying x at p = 0

ur group

Tcoh (K) from ρmax(T) 10 Tcoh(V(p)): x = 1 and p > 0 H Wilh l t l

ρmax( )

with I || b

H. Wilhelm et al.

x = 0 and p > 0 J D Thomson et al 405 410 415 420 1

J. D. Thomson et al.
G. Oomi et al.
S. Yomo et al.

V (Å3)

Chemical and hydrostatic pressure have (nearly) the effect on the Kondo lattice state:

V (Å3)

the Kondo lattice state: Tcoh(V(x)) ≈ Tcoh(V(p))

SLIDE 27

Linear Grüneisen parameter

/ ln / c C E∗ Γ α ∂ ∂ε

∑

strain dependence of the characteristic

/ ln /

i ij j p i T j

c C E Γ ≈ ⋅α ≈ ∂ ∂ε

∑

strain dependence of the characteristic energy scale E*

120 140 x = 0 x = 0.1 x = 0.15 x = 0.3 x = 0.5 60 80 100 120

(10-6 K-1)

20 40

Γa, Γb, Γc (

0.1 1 10

60
40
20

Γ

0.1 1 10 0.1 1 0.1 1 0.1 1 5

T (K) T (K) T (K) T (K)

5

T (K)

Antferromagnetic order: increasing anisotropic strain dependence ⇒ new energy scale ?

50

dTN /dεa,b,c (K)

⇒ new energy scale ?

0.1 0.2 0.3 0.4 0.5

100

x

SLIDE 28

Magnetostriction

T = 50 mK, µ0 H || easy axis c

20 b x = 0 1

20 αa,b,c /T

b x = 0.5

10 ∆a/a0 ∆b/b0

b x 0.1 x = 0.15

10

, ,

(10-6K-2)

b

10

∆c/c0

(10-6)

a

0 1 1 5

20
10

a c

30

20
2
1

(10 )

c

60mK <TN

10-6 T-1)

0.1 1 5 T (K)

40
30

0.0 0.2

4
3

λc (1 µoH (T)

150mK > TN

1 2 3 µoH (T)

⇒ system tries to maximize M

i T i i

M H σ ∂ ∂ ∝ µ ∂ ε ∂ = λ / | /

SLIDE 29

Néel temperature: TN (V(x)) = TN (V(p)) ?

Comparison between TN (V(x)) and TN (V(p))

0.2 dTN/dV(x)

Comparison between TN (V(x)) and TN (V(p))

(K/Å

3) N

dTN /dV(p) 0.1 0.0 0.5 1.0 0.0 x

For 0.1 < x < 1.0: chemical and hydrostatic pressure effect For 0.1 x 1.0: chemical and hydrostatic pressure effect deviate more and more from another. For x → 1.0: TN(V(x)) ≠ TN (V(p))

N( ( )) N ( (p))

SLIDE 30

Summary

x = 0 1 x 4 5 TN T /5 0.0 0.5 1.0 x = 0.1 x 3 4 Tcoh /5 (K) 1 2

AF

415 420 425 430 435 QCP V(Å3)

Tcoh(V(x)) ≈ Tcoh(V(p)) TN(V(x)) ≠ TN(V(p))

Maximum of S is shifted to x > xc = 0.1.
Two different energy scales: Kondo-lattice ↔ AF order,

however: x(TKondo lattice → 0) > x(TN → 0)

e e

(

Kondo-lattice → 0)

(

N → 0)

The onset of AF order is influenced by the Kondo effect

SLIDE 31

Outlook

60 αV /T

0 1

40 αV /T

(10-6 K-2)

x = 0.15 x = 0.1

Difference between the control

parameters H, V, p, or x ?

20

x = 0.0

p , , p,

Influence of disorder ?
Influence of the anisotropic crystal

x = 0.5 x = 0.3

structure ?

Typical behavior of f-metals ?

0.1 1 4

20