Precise determination of phonon constants in lead-free monoclinic (K0.5Na0.5)NbO3
(KNN) single crystals (Supplementary Information)
Muhammad Asif Rafiq1, Peter Supancic2, M. Elisabete Costa1, Paula M. Vilarinho1, and
Marco Deluca2,3,*,
1
Department of Materials and Ceramic Engineering, Centre for Research in Ceramics and Composite Materials,
CICECO, University of Aveiro, 3810-193 Aveiro, Portugal
2
Institut für Struktur- und Funktionskeramik, Montanuniversitaet Leoben, Peter Tunner Straße 5, A-8700
Leoben, Austria
3
Materials Center Leoben Forschung GmbH, Roseggerstraße 12, A-8700 Leoben, Austria
Measured and calculated XRD profiles for KNN powder obtained by milling the grown
single crystals is shown in Fig. S1. The Rietveld refinement gave the best results when
assuming a monoclinic structure. The parameters of the fit and the obtained atomic positions
are indicated in Table S1. The calculated unit cell parameters are a = 3.9997 Å, b = 3.9478 Å,
c = 3.9981 Å with β = 90° 22′.
The angular variation of the Raman spectra of KNN on the (001) and (100) planes is
presented in Fig. S2 (data collected with both cross and parallel polarized configurations). As
can clearly be seen, the periodicity of intensity variation depends on the plane of observation
and on the selected polarization.
The intensity of a Raman spectrum measured in a micro-probe configuration when the
incident laser is focused on a point depends on various factors involving both the material
and the optical system. Each point in the illuminated volume contributes to the overall
scattering with an in-depth spatial distribution of Lorentzian form [41, 42] that, for a given
wavelength of the incident laser, depends on the optical constants of the material (e.g.
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refraction and absorption) [43] and also on the parameters of the optical system. The depth
intensity dependence of a Raman spectrum (i.e. ignoring variations on the xy plane) when
focusing on a point z0 can be expressed as:

I  z    e 2nz
0
p2
z

p 2    z 
n

2
dz ,
(S1)
Where z = (z - z0) is the focus depth, n is the refractive index of KNN,  the (effective)
absorption coefficient and p the probe response parameter. The absorption coefficient could
be influenced by refraction effects due to domains walls and defects within the material and
is thus taken as an effective value; p is a parameter that generally depends only on the
experimental setting of the Raman microscope and is defined according to laser wavelength,
objective magnification and numerical aperture, and dimensions of the confocal hole. In order
to determine the depth from which the Raman signal comes from when the laser is focused on
the surface, the values of  and p have to be determined. This is done by translating
(defocusing) the focal plane of the laser through the specimen surface (i.e. changing z0 from
negative to positive) and fitting the obtained intensity profile with Eq. (S1), taking  and p as
variables. The result of this procedure for the investigated KNN single crystal is shown in
Fig. S3a for non-confocal conditions; here the refractive index was taken as n = 2.21 (the
value for KNbO3 at 514.5 nm wavelength [http://refractiveindex.info]), and values of  =
0.004 µm-1 and p = 2.16 µm were obtained by fitting. The value of p is consistent with that
obtained on opaque Si single crystals with the same optical configuration. The procedure we
followed is also in qualitative agreement with a different probe determination method [44].
The depth of signal intensity, z p,z0 , can be calculated from the percentage of the total
intensity that is observed at a given focal point:
2
z p ,z0

e 2nz
0
Iobs / I tot  %  

e
0
2nz
p2
z

p 2    z 
n

2
p
z

p 2    z 
n

2
2
dz
.
(S2)
dz
The calculation of the intensity profiles for three focal positions (z0 = -10 µm, 0 µm, 10 µm;
negative values indicate focusing above the material’s surface) is shown in Fig. S3b. When
the laser is focused on the material surface (z0 = 0 µm), most of the signal intensity comes
from within 10 µm depth, thus confirming that the volume probed comprises a single domain
of the KNN crystal. In particular, z50%,0 = 1.96 µm; z75%,0 = 4.35 µm; z90%,0 = 9.39 µm; z95%,0
= 15.31 µm.
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Table S1: Atomic positions as obtained from the conventional Rietveld refinement of the Xray diffractogram of KNN. The parameters of the fit are: Rp = 7.71; Rwp = 11.3; Re = 5.14 and
2 (goodness of fit) = 4.852.
X
Y
Z
Na
0.000000
0.000000
0.000000
Na
0.467000
0.000000
0.456000
K
0.000000
0.000000
0.000000
K
0.467000
0.000000
0.456000
Nb
-0.020000
0.500000
0.505000
Nb
0.478000
0.500000
0.004000
O
0.156000
0.500000
0.758000
O
0.687000
0.500000
0.728000
O
0.718000
0.500000
0.231000
O
0.193000
0.500000
0.240000
O
0.460000
0.000000
-0.027000
O
-0.071000
0.000000
0.516000
4
FIGURE S1: Measured and calculated XRD profiles and their difference curves for KNN
powder obtained by milling the KNN single crystals. The Rietveld refinement done
considering a monoclinic structure gave the best results.
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FIGURE S2: Angular dependence of the Raman spectra of monoclinic KNN on the (001)
plane [(a), (b) cross and parallel configuration, respectively] and on the (100) plane [(c), (d)
cross and parallel configuration, respectively].
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FIGURE S3: (a) Intensity profile obtained in the KNN single crystal by translating the focal
plane from above the material’s surface into the material. The solid curve represents fitting
with Eq. (S1). (b) Calculated intensity profiles in dependence of material depth for three focal
positions: z0 = -10 µm, 0 µm, 10 µm; most of the contributed intensity signal is contained to
the first 10 µm below the surface when z0 = 0 µm.
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