“EXAFS studies of Negative
Thermal Expansion
Zincblende structure”
PhD student: Naglaa AbdelAll
Tutors: Prof. Giuseppe Dalba
Email: n.fathy@science.unitn.it
Prof. Paolo Fornasini
Overview
Negative thermal expansion (NTE) in crystals
 Thermal Expansion of zincblende structure
Short introduction to EXAFS
 comparison with Bragg diffraction
Experimental results on Ge, CuCl and CdTe of the
1st Coordination shell
 interatomic distances
thermal factors
the local origin of NTE in Zincblende crystals
 Solids generally expand when heated, a courious example…
The Sears tower in Chicago, USA
grows by 15 cm in the summer!
Standing at 442 m
and 110 stories high.
There are however exceptions: solids that
contract when heated!
Examples…
Crystalline Silicon at low temperature
ZrW2O8 beetwen 0.3÷1050 K!
0.03
0.50
Relative Thermal Expansion in Crystalline Silicon
Relative Expansion (%)
Relative Expansion (%)
Relative Thermal Expansion in ZrW2O8
0.25
0.00
-0.25
-0.50
0.02
0.01
0.00
-0.01
0
200
400
600
800
Temperature [K]
1000
1200
0
100
200
Temperature [K]
300
Expansion coefficient of zincblende structure
  3
(cubic symmetry)
Thermal Expansion coefficient
T CV


V
Grüneisen function 
 NTE in Zincblende crystals has been attributed to a low-frequancy
transverse a coustic modes with negative Gruneisen functions.
NTE - phenomenological mechanism
Macroscopic
Expansion
Positive
contribution
Bond-stretching effect
POSITIVE contribution
Barrera, Bruno, Allan, Barron - J. Phys.: Condens. Matter 17, R217 (2005)
Negative
contribution
Tension effect
NEGATIVE contribution
Why EXAFS?
Local origin of NTE  phenomenological explanations,
BUT
… lack of experimental data!
EXAFS:
 sensitive to selected bond lengths
 parallel relative motion
Through a comparison with Bragg diffraction:
 perpendicular relative motion
 || and ^ correlation
Short introduction to EXAFS
 comparison with Bragg diffraction
Measurements performed at ESRF (Grenoble)…
BM29
(EXAFS in CdTe)
BM08 – GILDA
(EXAFS in CuCl)
x
The experimental goal is measure the
absorption coefficient as function of
energy, and extract information from
oscillations
Photon energy h 
EXAFS .VS. Diffraction
EXAFS
Diffraction
photo-electron spherical wave
 short-range sensitivity
 inter-atomic distances
 relative displacements
k
k
0
1
plane waves
Structural probe
 long-range sensitivity
 atomic positions
 atomic thermal factors
 By EXAFS: it is possible to extract original information about local
structural and vibrational dynamics
EXAFS .VS. Diffraction(I): Bond distances
(b)
(a)
r
R
R



EXAFS
r  rb  ra
R  rb  ra
average inter-atomic
distance
“True” bond length
distance between average
positions
r R 

Fornasini et al., Phys. stat. sol. (b) 1-7 (2008)
Bragg diffraction
u^2
2R
“Apparent” bond
length
Perpendicular MSRD
EXAFS .VS. Diffraction(II): Thermal factors
Rc
Rc
Relative thermal motion
EXAFS
Mean square relative
displacements
Absolute thermal motion
2
C*


u
2
II
C1*  R 
u ^2
Bragg diffraction
Absolute mean
square displacements
2R
First and second cumulant of EXAFS also contain original information about the local
dynamics!
Experimental results on Ge, CdTe and CuCl of the
1st Coordination shell
interatomic distances
 thermal factors
 the local origin of NTE in Zincblende structure
Thermal expansion: 1-st shell
EXAFS
XRD
Thermal expansion coefficient
4
Bond-stretching
effect
-3
10
8
6
4
EXAFS
2
XRD
0
0
100
200
300
T (K)
G. Dalba et al. Phys. Rev. Lett.
82, 4240 (1999)
-1
-6
 (10 K )
CdTe
-6
-8
-10
CuCl
0
20
CdTe
Å)
10
EXAFS
6
4
2
XRD
0
0
100
200
T (K )
[Present work]
80
100
120
Ge COTE (e-6) White
Ge COTE(e-6) Sparks
14
8
60
T(K)
Ge COTE (e-6) Zhda
Å
Å
Å
12
40
CuCl
(10
Å )
1-st shell
12
2R
Tension effect
14
Thermal expansion ( 10
Å)
-3
(10

Ge
14
Thermal expansion
r
300
Thermal expansion
Lattice
thermal

expansion

Ge
0
-2
-4
st
CuCl 1 shell
Ge COTE(e-6) Carr
12
CuCl_COTE e-6 Barron
-3
R
2
u^2
10
CdTe COTE (e-6) White
EXAFS
8
6
4
XRD
2
0
0
100
200
300
T (K)
M. Vaccari et al. Phys. Rev. B 75,
184307(2007)
Mean square relative displacements: 1st shell
10
10
st
CdTe 1 shell
Ge 1st shell
st
CuCl 1 shell
8
MSRDs
4
2
2
-2
6
(10
^
Å )
2
Å )
(10
-2
6
8
^
MSRDs
8
0
10
4

2
||
200
400
600
4
||
2
0
0
0
^
6
0
0
100
T (K)
200
300
100
200
300
T (K)
T (K)
6
Perpendicular-parallel
anisotropy of relative
vibration
8.6
u 2
^
u||2
11


=2 : For
perfect isotropy
“…more negative expansion is associated to a stronger ratio  = ^ / || …”
MSRDs :
XRD :
MSDs
Isotropic
Rc
EXAFS: MSRD
Anisotropic
CdTe 1st - shell
st
Ge 1 shell
Å )
0.06
-2
^/ 2
-2
0.04
^
0.02
MSRDs (10
MSRDs
(10
-2
2
Å )
0.08


0.00
0
100
200
T (K)
300
0
100
200
300
400
T (K)
“…NTE is connected to anisotropy of relative,
rather than absolute, thermal vibrations …”
Einstein models for MSRDs: Effective force constants
effective stiffness of the nearest-neighbor bond
8
70Ge
CdTe
st
CdTe 1 shell
CuCl
bond-stretching force
6
2
Å )
^
1.4
-2
3.76
bond-bending force
k^ (ev/Å2)
2.9
0.9
0.3
2.9
4.17
4.7
Anisotropy parameter
ξ = k|| / k^
(10
8.5
4
MSRDs
k||
(ev/Å2)
2

0
0
 Stronger NTE corresponds to:
100

- Smaller value of k|| , say to a looser bond.
k||
k^
T (K)

T 
200
300
u^2
2 u||2
= 1 : perfect isotropy
- Larger anisotropy of relative vibrations.

Conclusions
Crystallographic NTE (Bragg diffraction):
 positive 1st shell bond expansion (EXAFS)
Larger NTE:
 stronger anisotropy of relative thermal vibrations
 high ^ / || ratio
 tension mechanism
EXAFS of NTE in Zincblende structures:
 The relative perpendicular vibration are related to the
tension mechanism and to transverse acoustic modes which
are considered responsible for NTE .
Thank You