Elastic properties of SrTiO3 and determination of the strain state by means of XRD
The elastic moduli of bulk SrTiO3 are C11 = 317.2 GPa, C12 = 102.5 GPa, and C44 = 123.5
GPa1. Since the anisotropy parameter A = 2 x C44/(C11-C12)  1.15 is near to one, SrTiO3 can
be treated as an elastically isotropic material with a Young’s modulus
E = 1/S11 = (C11-C12) x (C11+2C12)/ (C11+C12) = 267 GPa
and a Poisson’s ratio
 = C12/(C11+C12) = 0.2442.
Assuming an elastically isotropic film in a biaxial stress state, the experimentally observed
lattice spacing dhkl only depends on the angle  between the crystallographic hkl direction
and the substrate normal. The apparent lattice constant can be written as
a() = aoop+(ain-aoop) x sin2()3.
aoop and ain are the apparent lattice constants perpendicular and parallel to the substrate
surface. The lattice constant of the unstrained film af is given by af = a(sin2() = 2 x /(1+)).
Therefore, the inplane/out-of-plane strain is defined as in/oop = (ain/oop-af)/af and the inplane
stress is given by  = -E/(2 x oop). In contrast, the intrinsic strain f is defined with respect
to the substrate lattice constant a0, i.e. f= (af-a0)/a0.
The intrinsic lattice strain can be related to the defect concentration n via f = n V. For a
rough estimation of the volume expansion by Schottky defects V  1.1 x 10-22cm3, a
theoretical calculation of atomic relaxations in the vicinity of the defect is be used4.
Consequently, an f of about 0.6% then corresponds to a defect density of about 6 x 1019/cm3.
This is three orders of magnitude higher than the equilibrium density of Schottky defects at
room temperature5.
References
1
R. O. Bell, G. Rupprecht, Phys. Rev. 129, 90 (1963).
2
e.g. J. Hornstra and W.J. Bartels, J. Crys. Growth 44, 513 (1978).
1
3
C. Noyan, T. C. Huang, and B. R. York, Crit. Rev. Sol. State Mat. Sci. 20, 125 (1995).
4
A.Stashans and L.Villamagua, J. Phys. Chem. Solids 70, 417 (2009).
5
Y. Kozuka, Y. Hikita, C. Bell, and H.Y. Hwang, Appl. Phys. Lett. 97, 012107 (2010).
2
Determination of the thermal conductivity by means of 3 measurements
The thermal conductivity of the differently strained SrTiO3 films was measured in a
commercial 3 subsystem developed at the Fraunhofer-IPM in Freiburg. The measurements
were performed in a cryostat in vacuum (pressure of about 10-3 mbar).
The measurement principle of the 3 method is described elsewhere1. As a heater/sensor
device, a Pt strip with a width of about 10µm, a length of about 1mm and a height between
150 and 200 nm is fabricated on top of the film by means of optical lithography and ion-beam
sputtering. By applying ac voltages at various frequencies up to 1.000 Hz, the thermal
conductivity of the film is deduced from the third harmonics.
The accuracy and reliability of the measurement equipment was intensively tested using welldefined materials like fused Silica and oxide single crystals with known thermal
conductivities. Measurements of  have been shown to be reproducible, also for thin films.
Fig. SM1 shows the reproducibility for T as well as for dR/dT for PrMnO3 films on top of a
MgO substrate. These control experiments show that the relative error in the measurement is
below 3%. We assume that the error due to the thermal resistance of the heater/film interface
is less than 2%. This is based on a study of thermal conductivity ofPrMnO3 films with
different thicknesses prepared under the same conditions and with the same heater geometry
which resulted in similar thermal conductivities. Film as well as the strip thickness and the
width (error of about 0.5%) was always measured by using a Vecco Dektak® 150
Profilometer and/or by means of cross-sectional TEM investigations. Exchange of single
crystal reference samples (Lithosil, SrTiO3) with other research groups using the 3 method
and comparison to literature values reveal an absolute accuracy of the determination of cross
plane thermal conductivity of about 5%.
3
Since thermal conductivities of substrate and film are similar, the thermal conductivity cannot
be calculated from the analytical solution of the heat conduction equation1,2. Therefore, we
have applied finite element simulations to solve the 2D heat conduction equation in the time
domain. These calculations take into account Joule heating induced resistance change of the
metal strip, the thermal properties of the heater, the measured  and specific heat of the
substrate and the thermal interface between the substrate and the holder. The accuracy of the
simulations was verified for well defined geometries and well characterized materials. At the
present stage, the only restricting assumption is the isotropy of heat conductivity in the film.
The total accuracy is composed of the experimental errors described above and the
uncertainty of the numerical simulation and thus varies between 6 % and 10 % as shown in
the error bars in Fig. 2(a).
After establishing a data basis, a neural network was trained to solve the inverse problem of
determining the thermal conductivity of the film from the measured temperature oscillation.
Neural networks are well-known tools to represent dependencies between parameters where
no explicit analytical expressions are available. This method was applied for the identification
of material parameters before3. The uncertainty of the neural network for the exact
identification of the solution is included in the error bars in Fig. 2(a).
In conclusion, finite element calculations and neural networks allow determination of the
thermal conductivity without restricting assumptions that may be needed to derive analytic
solutions. The details of this procedure will be published elsewhere.
References
1
D. G. Cahill, Rev. Sci. Instrum. 61, 802 (1990).
2
D. G. Cahill, M.Katiyar, and J.R. Abelson, Phys. Rev. B 50, 6077 (1994).
3
E. Tyulyukovskiy and N.Huber, J. Mater. Res. 21, 664 (2006).
4
T. Borca-Tasciuc, A. R. Kumar, and G. Chen, Rev. Sci. Instrum. 72, 2139 (2001).
5
J. H. Kim, A. Feldman, and D. Novotny, J. Appl. Phys. 86, 3959 (1999).
4
Figure captions
Figure SM1(a):
Recurring measurements of Re(T) as a function of the frequency for two PrMnO3 films on
MgO with different film thicknesses.
Figure SM1(b):
Recurring measurements of dR/dT for the 200 nm thick PrMnO3 film.
5
Determination of the mechanical properties by means of nanoindentation
Measurements of the mechanical properties were performed with a commercial MTS G200
nanoindentation system using a Berkovich indenter. Contact force, penetration, and contact
stiffness were measured continuously during the indentation. The stiffness was determined
using a superimposed oscillation with fixed frequency of 37 Hz and fixed penetration
amplitude of 1nm (Continuous Stiffness Measurement1).
Fig. SM2 shows typical load-displacement curves for the SrTiO3 substrate and the two
differently strained thin films. Small steps are observed in the loading curves, which have
been attributed to dislocation formation under the indenter tip2. Larger steps or changes in
slope of the loading curves, indicative of phase changes3 such as the stress-induced transition
to the ferroelectric phase, were not observed.
Values of the hardness and Young’s modulus are shown in Fig. SM3 as a function of the
intrinsic film strain. They are calculated from the loading behavior at penetration depths
between 50 and 100 nm, which is deep enough to avoid the typical near-surface scatter. The
effect of the SrTiO3 substrate under the film is expected to be felt at these depths, but given
the high quality of the epitaxy and the similar values for hardness and Young’s modulus, it
should have a negligible effect on the determination of the film properties. The Young’s
modulus for the films increases slightly with intrinsic strain (11%), while the hardness
increases more strongly (42%). The Young’s modulus is in reasonable agreement with
literature values for bulk SrTiO3 of 267 GPa4. Small cracks are observed next to the indents in
the substrate but not in the films, where they are presumably inhibited by the compressive
stresses. These cracks may help to account for the somewhat lower Young’s modulus of the
substrate but should not affect the hardness much. In addition, the apparent hardness and
Young’s modulus values are expected to increase with compressive stress due to an increase
in pile-up5. However, based on SEM images of the indents we estimate the pile-up to be small
and to introduce a maximum error shown by the error bars at ca. 0.4% strain in Fig. SM3.
6
Thus, we conclude that the true film Young’s modulus (corrected for stress-induced pile-up
and cracking) is independent of the intrinsic strain. In contrast, these considerations cannot
explain the significant increase in hardness with intrinsic strain. We believe that point defects
created during deposition hinder the motion of dislocations and thus increase the hardness.
Evidence for dislocation activity can be seen from the small vertical striations at the top edge
of the indents (arrows in the inset of Fig. SM2)6.
References
1
J. Hay, P. Agee, and E. Herbert, Exp. Techniques 34, 86 (2010).
2
X.Wang, and N. P. Padture, J. Mater. Sci. 39, 1891 (2004).
3
e.g. J.-i Jang, M. J. Lance, S. Wen, G.M. Pharr, App. Phys. Lett. 86, 131907 (2005).
4
R. O. Bell, G. Rupprecht, Phys. Rev. 129, 90 (1963).
5
M. Sebastiani, E. Bemporad, F. Carassiti, and N. Schwarzer, Phil. Mag. 91, 1121 (2010).
6
G.Kaupp; Atomic Force Microscopy, Scanning Nearfield Optical Microscopy and
Nanoscratching, Springer Berlin Heidelberg New York, 3 (2006)
Figure captions
Figure SM2:
Load-displacement curves of single-crystal SrTiO3 (solid line) and two 300 nm thick SrTiO3
films with intrinsic strains of 0.22% (dashed line) and 0.62% (dash-dotted line). The upper
part of the curves corresponds to mechanical loading, the lower part to unloading. The inset
shows a SEM image of the most highly strained film after indentation to a depth of 300 nm.
Surface steps due to dislocations created by the indent are indicated by the arrows.
Figure SM3:
Young’s modulus and hardness as a function of the intrinsic film strain. The error bars at a
strain of ca. 0.4% indicate the uncertainty due to determination of the contact area.
7
Thermal stability of defects and strain relaxation
In order to clarify the thermal stability of the preparation induced defects, different relaxation
experiments were made. First a strongly strained (f=0.62%) sample were measured in a X-ray
diffractometer with a Anton Paar DHS 900 Hot Stage enabling us to see possible relaxation in
situ from room temperature up to 1073 K. In this temperature range no relaxation was
observed. Afterwards the sample was annealed for 10 h at 973 K, 1073 K, 1173 K, 1273 K
and 1373 K. Subsequent to each annealing step (aoop-a0)/a0 was measured by XRD. Up to
1173 K the strain shows no significantly change. After the annealing at 1273 K, a small
decrease of about 5 % is observed. A distinct decrease of the strain is can be only observed
after the annealing step of 1373 K.
Figure captions
Figure SM4:
XRD measurements of (aoop-a0)/ a0 at different annealing temperatures and times. The data
points one up to ten minutes were made in situ with a hot stage. The strain measurements at
600 min were done at room temperature after annealing the sample for 10 h at the given
temperature.
8