Negative thermal expansion and local dynamics

P. Fornasini,

N. Abd el All, S. I. Ahmed, A. Sanson, M. Vaccari

OverviewOverview

• Negative thermal expansion (NTE)

• EXAFS and local dynamics

• EXAFS studies of NTE materials

Thermal expansion in 2-atomic systems

€

V u( ) =12k0 u

2 + k3 u3 + k4 u

4 +....

PaoloFornasini

Univ. Trento

xx0

V(x)

0 u

€

u = x − x0

m1 m2

µ

Positiveexpansion

Thermal expansion in many-atomic systems PaoloFornasini

Univ. Trento

€

Vr r 1,

r r 2,K

r r n( )

Crystal potential defined in

3n-dim. configurational space

Isotropicor

anisotropic

Positiveor

negativethermal expansion

NTE in tetrahedral semiconductorsPaolo

FornasiniUniv. Trento

-10

-8

-6

-4

-2

0

2

4

0 20 40 60 80 100 120

Thermal expansion coefficient

α (1

0-6 K

-1)

T(K)

CuCl

CdTe

Ge

-0.4

-0.2

0

0.2

0.4

0.6

0 20 40 60 80

α (1

0-6 K

-1)

T(K)

Germanium

Thermal expansion coefficient

Barron, Birch, White - J. Phys. C 10, 1617 (1977)

NTE in framework structures PaoloFornasini

Univ. Trento

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 100 200 300 400 500

Δa/

a %

T (K)

CuCl

Cu2O

Ag2O

ZrW2O8

Zn(CN)2

Cuprite structure

Tiano, Dapiaggi, Artioli, J. Appl. Cryst. 36, 1461 (2003)

Mary, Evans, Vogt, SleightScience, 272 (1996)

Chapman, Chupas, KepertJ. Am. Chem. Soc. 127, 15630(2005)

“Global” approach to NTE

Born - von Karman power expansionwith respect to atomic displacements

€

Vr r 1,

r r 2,K

r r n( )

Crystal potential defined in

3n-dim. configurational space

Quasi-harmonic approximation: Positivecontribution

Negativecontribution

positive

negative

Mode Grüneisen parameters

anharmonic terms ⇔ thermal expansion

“Local” approach to NTE PaoloFornasini

Univ. Trento

Barrera, Bruno, Barron, Allan - J. Phys. Cond. Matter, 17 (2005)

Tension effect

Perpendicular vibrations

Bond-stretching effect

Anharmonicity of effective pair potential V(r)Positive

contribution

Negativecontribution

Bond distances PaoloFornasini

Univ. Trento

€

r =r r b −

r r aEXAFS, diffuse scattering

Average distance

€

R =r r b −

r r aBragg diffraction, dilatometry

Distance between average atomic positions

€

r ≈ R +Δu⊥

2

2R Perpendicular MSRD

Fornasini et al., Phys. Rev. B 70, 174301 (2004)

€

r R

€

r R

€

r r

Thermal factors PaoloFornasini

Univ. Trento

EXAFS & diffraction: MSRDs

Bragg diffraction: MSDs of single atoms

Fornasini et al., Phys. Rev. B 70, 174301 (2004)

r

R0

Parallel

Perpendicular

Absolutevibrations

Relativevibrations

EXAFS and NTEPaolo

FornasiniUniv. Trento

Bond-stretching effect

Expansion of selected bond distances

Tension effect

MSRD• parallel• perpendicular

NTE structures studied by EXAFS (a)

Cu2O, Ag2O

Zincblende Cuprite

Ge, CdTe, CuCl CuScO2, CuLaO2

Delafossite

Isotropic NTE Isotropic NTE Anisotropic NTE

Cu

Cu

Cu

NTE structures studied by EXAFS (b)

Zincblende Cuprite Delafossite

Framerwork structure:2 networks ofM4O tetrahedra

TA acoustic modesat BZ boundarywith negative Grueneisen parameters

Cu

O

O

Neutron diffraction:Cu-O NTEAnisotropic Cu motion

O

O Cu

Bond expansion in zincblende structures

0 100 200 300

CuCl - 1st shell

T (K)

EXAFS

XRD0.0

0.5

1.0

0 100 200 300

Ge - 1st shell

% e

xpan

sion

T (K)

EXAFS

XRD

0 100 200 300

CdTe - 1st shell

T(K)

EXAFS

XRD

-8

-4

0

4

0 40 80 120

Thermal expansion coefficient

α (1

0-6 K

-1)

T(K)

CuCl

CdTe

Ge

PRL 82, 4240 (1999) PRB 75, 184307 (2007)Poster PS1-62, XAFS14

MSRD: zincblende structure

0.0

2.0

4.0

6.0

8.0

10.0

0 100 200 300

Ge - 1st shell

T (K)

MSR

D (1

0-2 Å

2 )

⊥/2

0 100 200 300

CuCl - 1st shell

T (K)

⊥/2||

⊥/2

0 100 200 300T (K)

⊥/2

||

CdTe - 1st shell€

Δu⊥2

€

Δu||2

NTE strength increasesForce constants k|| and k⊥ decreaseAnisotropy k||/ k⊥ increases

Cu-O bond expansion in delafossite structures

PRB 78, 104302 (2009)

O|

Cu|O

0 200 400T (K)

CuLaO2

EXAFS

n diffr.-1

0

1

2

0 200 400

Expa

nsio

n (1

0-2 Å

)

T (K)

CuScO2

EXAFS

n diffr.Sleight et al.J. Solid St. Chemistry178, 285 (2005)

Cu-O MSRD in delafossite structures

NTE strength increasesForce constants k|| and k⊥ decreaseAnisotropy k||/ k⊥ increases

0

10

20

30

40

0 200 400T (K)

MSR

D (1

0-3 Å

2 )

CuScO2 : Cu-O

⊥/2

| |

0 200 400T (K)

CuLaO2 : Cu-O

⊥/2

| |

Cu

O

O

PRL 89, 25503 (2002) - PRB 73, 214305 (2006)

-0.01

0

0.01

0.02

0.03

0 200 400

Exp

ansi

on (

Å)

Cu--O

T (K)

EXAFS

diffraction

0 200 400

Ag--O

T (K)

EXAFS

diffraction

Bond expansion in cuprite structures

-0.4

-0.3

-0.2

-0.1

0

0.1

0 200 400 600

Δa/a (%)

T (K)

Cu2O

Ag2O

PRL 89, 25503 (2002) - PRB 73, 214305 (2006)

M-O MSRDs in cuprite structures

0

0.04

0.08

0 200 400

MS

RD

(Å2 )

Cu-O

T (K)

⊥/2

||

0 200 400

Ag-O

||

⊥/2

T (K)

EXAFS and NTE

€

δR ≈ δ r −δ Δu⊥

2

2R

Bond-stretching effect

POSITIVE contribution

Tension effect

NEGATIVE contribution

Lattice thermal

expansion

PaoloFornasini

Univ. Trento

EXAFS1st cumulant

Braggdiffraction

PerpendicularMSRD

Bond expansion and asymmetry of the effective potential

Thermal expansion due to asymmetry

€

δa = − 3k3 δC2* k0

0.0

0.5

1.0

1.5

0 100 200 300

CuCl

T (K)

EXAFS

XRD-0.1

0

0.1

0.2

0.3

0.4

0.5

0 100 200 300

CdTe

T(K)

EXAFS

XRD0.0

0.1

0.2

0 100 200 300

Ge

% e

xpan

sion

T (K)

EXAFS

XRD

♦

EXAFS thermal expansion PaoloFornasini

Univ. Trento

0 u

= +EXAFSthermal expansion

1st cumulant

Potential asymmetry

3rd cumulant

Potential shift

Effective potentialdependent on temperature

XAFS14, Poster Ps1-24

Phys. Rev. B 70, 174301 (2004)

See also:

Local dynamical properties of NTE materials

zincblende cuprite

€

Cu Ge CdTe CuCl Cu2O Ag2O CuLaO2 CuScO2

k⊥ (eV/Å2) 2.72 2.9 0.9 0.3 2.9 0.5 2.5 1.0

k|| (eV/Å2) 3.2 8.5 3.8 1.4 11.6 5.9 15.5 24.2

ξ =k||k⊥

1.17 2.9 4.2 5.4 4.0 11.8 6.0 24.2

delafossite

Bending

Stretching

Anisotropy

NTE strength

ConclusionsConclusions

The NN bond always undergoes positive expansion (PTE).

For iso-structural crystals, the stronger is the lattice NTE,the stronger are the bond PTE and the perpendicularMSRD.

A correlation can be established between MSRD anisotropyand NTE strength.

EXAFS measurements substantiate the local model basedon the competition between stretching and tension effects.

Bond stretching is due to anharmonicity plus shift of theeffective pair potential.

Paolo Fornasini University of Trento, Dept. of PhysicsNaglaa Abd el All PhD student Univ. Trento (from Assiut, Egypt)Sameh I. Ahmed Trento PhD, now Univ. of Cairo (Egypt)Andrea Sanson Trento PhD, now Univ. of Verona (Italy)Marco Vaccari Trento PhD, now ESRF (France)

Giuseppe Dalba Trento (Italy)Rolly Grisenti Trento (Italy)Francesco Rocca Trento (Italy)Gilberto Artioli Padova (Italy)Monica Dapiaggi Milano (Italy)Juris Purans Riga (Latvia)Alex Kuzmin Riga (Latvia)Djibril Diop Dakar (Senegal)Bridinette T. Sendjia Dakar (Senegal)Arthur W. Sleight Corvallis, Oregon, USA

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