Supplementary information for the paper “Small hole polarons in rare-earth titanates” by L.
Bjaalie, D. G. Ouellette, P. Moetakef, T. A. Cain, A. Janotti, B. Himmetoglu, S. J. Allen, C. G.
Van de Walle, and S. Stemmer
Measurement of the optical conductivity
The optical properties of the Gd1-xSrxTiO3 films were measured using Fourier transform infrared (FTIR)
spectroscopy in near-normal reflection geometry to cover the photon energy range 0.005-3 eV. The
complex conductivity was extracted using a combination of direct inversion and model fits. Reflectance
measurements were performed in a Bruker 66v/S spectrometer operated in vacuum. Samples were
mounted on the cold finger of a He flow cryostat equipped with a 5 cm travel bellows for sample
interchanges. To cover the entire spectral range 0.005-3 eV, a combination of sources (Hg lamp, globar,
tungsten bulb) and detectors (bolometer, pyroelectric DTGS, and Si photodiode) were employed. Both
bare LSAT substrates and Au films were used are references, mounted directly adjacent to the samples
with the top-side directly against the sample holder for optimal alignment. The diffuse reflectance from
the unpolished backside of the samples had negligible effect except below 20 meV.The optical constants
of the LSAT were obtained from Kramers-Kronig analysis of its reflectivity. The optical conductivity of
the Gd1-xSrxTiO3 films was extracted by fitting a Drude-Lorentz model having up to 17 oscillators to the
reflectance using an exact multilayer model containing the measured LSAT dielectric function. Three
oscillators were included to approximate the phonon spectrum, using results on bulk GdTiO 3 for
initialization [1]. Only the highest frequency mode at 67 meV was unambiguously apparent in the
reflectance, although inclusion of the other modes significantly improved the fit.
The reflectance of the thin film samples and a bare substrate is shown in Figure S1. The series of
peaks below 100 meV photon energy are vibrational modes in the LSAT substrate. At these energies, the
reflectance of the thin film samples nearly matches that of the bare substrate, due to the low optical
conductivity. (The samples are electrical insulators, so no Drude absorption is expected nor observed.) In
the plot to the right, the mid- to near-infrared absorption of the Gd1-xSrxTiO3 is evident in the increased
reflectance relative to the bare substrate.
The increase in reflectivity is a measure of the optical conductivity of the epitaxial layer. (The
conductivity shown in Fig. 1 mirrors the reflectivity increase shown in Fig. S1.) In the spectral range
above ~ 0.1 eV, where the substrate dielectric constant is frequency dependent but essentially real, and if
we assume the epilayer to be optically thin, then we can use an approximate relation between the optical
conductivity of the epilayer and fractional increase in reflectivity given by
   
2
2
1  Repi  sub   Ysub cos t  

1
  Y 2 cos2   1 Y0 cos i  .
4t  Rsub
 i 
 0
Here, is the optical conductivity, t the epilayer thickness, Repi-sub and Rsub, the reflectivity of the
composite and substrate respectively, and the angles of incidence and transmission and Ysub and Yo the
wave admittance of the LSAT and free space respectively.
While this is only an approximate expression for the optical conductivity, we can use it to
estimate the relation between uncertainties in the optical conductivity and apparent uncertainties in the
fractional change in reflection:
2
cos2 t  
 Repi  sub  1  Ysub
 1 Y0 cos i  .
  2
2
 Rsub  4t  Y0 cos i 

     
The estimated uncertainty in the fractional increase in reflectivity is shown in S1. The resulting
uncertainty in optical conductivity is shown in Fig. 1.
References
1. D. A. Crandles, T. Timusk, J. D. Garrett, and J. E. Greedan, Phys. Rev. B 49, 4299 (1994).
Figures
Figure S1: Near-normal reflectance of Gd1-xSrxTiO3 films on LSAT substrate. For the Gd0.87Sr0.13TiO3
film, data was not taken below 0.07 eV. Left: scaled to show all data. Right: Scaled to highlight near-IR
absorption attributed to polaron and Mott-Hubbard excitations. Also shown is an estimate of the
fractional reflectance uncertainty as a function of frequency for the GdTiO3 thin film (the uncertainties
are the same for the other samples).