Negative thermal expansion and local dynamics

Negative thermal expansion and local dynamics

Negative thermal expansion
and local dynamics
P. Fornasini,
N. Abd el All, S. I. Ahmed, A. Sanson, M. Vaccari
Overview
•
Negative thermal expansion (NTE)
• EXAFS and local dynamics
• EXAFS studies of NTE materials
Thermal expansion in 2-atomic systems
m1
Paolo
Fornasini
Univ. Trento
m2
V (u ) =
µ
1
k0 u 2 + k3 u 3 + k4 u 4 +....
2
Positive
expansion
€
V(x)
x
0
x
0
u = x −ux 0
Thermal expansion in many-atomic systems
r r
r
V ( r1, r2 ,K rn )
Crystal potential
defined in
3n-dim. configurational space
€
Positive
or
negative
thermal expansion
Isotropic
or
anisotropic
Paolo
Fornasini
Univ. Trento
Paolo
Fornasini
Univ. Trento
NTE in tetrahedral semiconductors
Thermal expansion coefficient
Thermal expansion coefficient
0.6
4
Germanium
-1
-2
α (10 K )
0
-6
2
0.2
-6
-1
α (10 K )
0.4
0
Ge
-4
CdTe
-6
-0.2
-8
-0.4
CuCl
0
20
40
T(K)
60
80
-10
0
20
40
60
80
100
120
T(K)
Barron, Birch, White - J. Phys. C 10, 1617 (1977)
NTE in framework structures
0.1
Paolo
Fornasini
Univ. Trento
CuCl
0
Δa/a %
Cu O
2
-0.1
-0.2
Ag O
Zn(CN)
2
-0.5 0
Tiano, Dapiaggi, Artioli,
J. Appl. Cryst. 36, 1461 (2003)
2
-0.3
-0.4
Cuprite structure
100
200
300
T (K)
ZrW O
2
400
Mary, Evans, Vogt, Sleight
Science, 272 (1996)
8
500
Chapman, Chupas, Kepert
J. Am. Chem. Soc. 127, 15630
(2005)
“Global” approach to NTE
r r
r
V ( r1, r2 ,K rn )
Crystal potential
defined in
3n-dim. configurational space
Born - von Karman power expansion
with respect to atomic displacements
anharmonic terms ⇔ thermal expansion
Quasi-harmonic approximation:
positive
Positive
contribution
negative
Negative
contribution
Mode Grüneisen parameters
“Local” approach to NTE
Paolo
Fornasini
Univ. Trento
Anharmonicity of effective pair potential V(r)
Bond-stretching effect
Positive
contribution
Perpendicular vibrations
Negative
contribution
Tension effect
Barrera, Bruno, Barron, Allan - J. Phys. Cond. Matter, 17 (2005)
Bond distances
Paolo
Fornasini
Univ. Trento
Bragg diffraction, dilatometry
r
r
R = rb − ra
Distance between
average atomic positions
r
R
€
r
r
r
R
€
Average distance
€
r ≈R +
Δu⊥2
2R
€
EXAFS, diffuse scattering
Perpendicular MSRD
Fornasini et al., Phys. Rev. B 70, 174301 (2004)
€
r r
r = rb − ra
Thermal factors
Paolo
Fornasini
Univ. Trento
Bragg diffraction: MSDs of single atoms
Absolute
vibrations
Perpendicular
r
Relative
vibrations
R0
Parallel
EXAFS & diffraction: MSRDs
Fornasini et al., Phys. Rev. B 70, 174301 (2004)
Paolo
Fornasini
Univ. Trento
EXAFS and NTE
 Expansion of selected
bond distances
Bond-stretching effect
 MSRD
• parallel
• perpendicular
Tension effect
NTE structures studied by EXAFS (a)
Zincblende
Cuprite
Delafossite
Ge, CdTe, CuCl
Cu2O, Ag2O
CuScO2, CuLaO2
Cu
Cu
Cu
Isotropic NTE
Isotropic NTE
Anisotropic NTE
NTE structures studied by EXAFS (b)
Zincblende
Cuprite
Delafossite
TA acoustic modes
at BZ boundary
with negative
Grueneisen parameters
Framerwork structure:
2 networks of
M4O tetrahedra
O
O
Cu
Neutron diffraction:
Cu-O NTE
Anisotropic Cu motion
O
Cu
O
Thermal expansion coefficient
Bond expansion
in zincblende structures
Ge
-1
α (10 K )
4
-6
0
CdTe
-4
-8
CuCl
0
Ge - 1st shell
% expansion
T(K)
80
120
CuCl - 1st shell
CdTe - 1st shell
1.0
40
EXAFS
0.5
EXAFS
EXAFS
XRD
0.0
0
XRD
XRD
100
200
T (K)
300 0
PRL 82, 4240 (1999)
100
200
T(K)
300 0
Poster PS1-62, XAFS14
100
200
T (K)
300
PRB 75, 184307 (2007)
MSRD: zincblende structure
Δu⊥2
Δu||2
€
10.0
Ge - 1st shell
CuCl - 1st shell
CdTe - 1st shell
€
⊥/2
-2
2
MSRD (10 Å )
8.0
6.0
4.0
⊥/2
2.0
0.0
⊥/2
0
100
200
T (K)
300 0
NTE strength increases
||
100
200
T (K)
300 0
||
100
200
T (K)
Force constants k|| and k⊥ decrease
Anisotropy k||/ k⊥ increases
300
Cu-O bond expansion
in delafossite structures
CuScO
CuLaO
2
2
EXAFS
-2
Expansion (10 Å)
2
O
|
Cu
|
O
1
EXAFS
PRB 78, 104302 (2009)
n diffr.
Sleight et al.
J. Solid St. Chemistry
178, 285 (2005)
0
n diffr.
-1
0
200
T (K)
400
0
200
T (K)
400
Cu-O MSRD in delafossite structures
O
Cu
O
2
2
⊥/2
30
-3
2
MSRD (10 Å )
CuLaO : Cu-O
CuScO : Cu-O
40
⊥/2
20
10
||
0
0
200
T (K)
400
||
0
NTE strength increases
200
T (K)
400
Force constants k|| and k⊥ decrease
Anisotropy k||/ k⊥ increases
0.1
Bond expansion
in cuprite structures
Δa/a (%)
0
Cu O
2
-0.1
-0.2
-0.3
Ag O
2
-0.4
0.03
Ag--O
Cu--O
Expansion (Å)
0.02
EXAFS
0.01
EXAFS
0
diffraction
diffraction
-0.01
0
200
400
0
T (K)
PRL 89, 25503 (2002)
200
400
T (K)
-
PRB 73, 214305 (2006)
0
200
400
T (K)
600
M-O MSRDs
in cuprite structures
0.08
Ag-O
Cu-O
2
MSRD (Å )
⊥/2
0.04
⊥/2
||
0
||
0
200
0
400
400
T (K)
T (K)
PRL 89, 25503 (2002)
200
-
PRB 73, 214305 (2006)
€
EXAFS and NTE
Paolo
Fornasini
Univ. Trento
Bragg
diffraction
δR
Lattice
thermal
expansion
EXAFS
1st cumulant
≈
δ r
Perpendicular
MSRD
−
Bond-stretching effect
POSITIVE contribution
δ
2
Δu⊥
2R
Tension effect
NEGATIVE contribution
Bond expansion and asymmetry
of the effective potential
*
Thermal expansion due to asymmetry δa = − 3k 3 δC2 k 0
♦
0.5
CdTe
Ge
0.4
0.2
% expansion
€
EXAFS
CuCl
1.5
EXAFS
0.3
1.0
0.2
0.1
XRD
EXAFS
0.5
0.1
0
0.0
0.0
XRD
0
100
200
T (K)
300
-0.1
0
100
200
T(K)
300
XRD
0
100
200
T (K)
300
EXAFS thermal expansion
Paolo
Fornasini
Univ. Trento
Effective potential
dependent on temperature
0
EXAFS
thermal expansion
1st cumulant
=
Potential asymmetry
u
+
Potential shift
3rd cumulant
See also:
XAFS14, Poster Ps1-24
Phys. Rev. B 70, 174301 (2004)
Local dynamical properties of NTE materials
zincblende
cuprite
delafossite
Cu
Ge
CdTe
CuCl
Cu 2O
Ag 2O
CuLaO 2
CuScO 2
k⊥ (eV/Å2 )
2.72
2.9
0.9
0.3
2.9
0.5
2.5
1.0
Bending
k|| (eV/Å2 )
k
ξ = ||
k⊥
3.2
8.5
3.8
1.4
11.6
5.9
15.5
24.2
Stretching
1.17
2.9
4.2
5.4
4.0
11.8
6.0
24.2
Anisotropy
NTE strength
€
Conclusions
 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 perpendicular
MSRD.
 A correlation can be established between MSRD anisotropy
and NTE strength.
 EXAFS measurements substantiate the local model based
on the competition between stretching and tension effects.
 Bond stretching is due to anharmonicity plus shift of the
effective pair potential.
Authors
Paolo Fornasini
University of Trento, Dept. of Physics
Naglaa 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)
Collaborators
Giuseppe Dalba
Rolly Grisenti
Francesco Rocca
Gilberto Artioli
Monica Dapiaggi
Juris Purans
Alex Kuzmin
Djibril Diop
Bridinette T. Sendjia
Arthur W. Sleight
Trento (Italy)
Trento (Italy)
Trento (Italy)
Padova (Italy)
Milano (Italy)
Riga (Latvia)
Riga (Latvia)
Dakar (Senegal)
Dakar (Senegal)
Corvallis, Oregon, USA