^{1,a)}, T. Hofmann

^{1}, R. Korlacki

^{1}, T. E. Tiwald

^{2}and M. Schubert

^{1}

### Abstract

Spectroscopic ellipsometry in the mid-infrared and far-infrared spectral range and generalized ellipsometry in the mid-infrared spectral range are used to investigate the anisotropic dielectric response of rutile TiO_{2}. The ordinary and extraordinary dielectric function tensor components and all infrared active phonon mode parameters of single crystalline rutile TiO_{2} are determined with high accuracy for wavelengths from 3 *μ*m to 83 *μ*m. The data were acquired from samples of (001), (100), and (111) surfaces cut from bulk single crystals. A factorized model dielectric function is employed in order to determine the frequencies and damping parameters of the transverse and longitudinal phonon modes with and *E _{u} * symmetries. The bands of total reflection of

*s*- and

*p*-polarized light in dependence of the angle of incidence for highly symmetric sample cuts and orientations are derived. Excellent agreement with phonon modes reported in literature is obtained. Introduction of two additional modes for ordinary as well as extraordinary component of the dielectric function tensor was necessary to most accurately match the experimental data. The spectral position of the additional modes is compared to the calculated phonon density of states. The low-frequency dielectric constants are calculated from the determined phonon mode parameters and the high-frequency dielectric constants by applying the Lyddanne-Sachs-Teller relation. The presented data revise existing infrared optical function data and will be suitable for interpretation of any kind of infrared spectra for bulk TiO

_{2}single crystal substrates, thin films, and TiO

_{2}nanostructures.

The authors acknowledge financial support from the National Science Foundation under Awards MRSEC DMR-0820521, MRI DMR-0922937, DMR-0907475, and EPS-1004094.

I. INTRODUCTION

II. EXPERIMENT

III. THEORY

A. Ellipsometry equations

B. Infrared model dielectric function

C. Bands of total reflection in anisotropic materials

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Phonons
- 74.0
- Ellipsometry
- 33.0
- Dielectric constant
- 24.0
- Tensor methods
- 24.0
- Dielectric function
- 23.0

## Figures

Experimental (symbols) and best-match model calculated data (solid lines) of the ellipsometric parameters and for *c*-plane rutile TiO_{2}. Very good agreement between experimental and best-match model calculated data was achieved by applying the factorized MDF in Eq. (12) . The energetic position of transverse and longitudinal phonon modes of *E _{u} *- and -symmetry is indicated by brackets (solid brackets: TO phonons; dashed brackets: LO phonons). Two additional modes needed to be included for each component of the dielectric function tensor in order to achieve the excellent match between experimental and model data in the spectral range between 500 cm

^{−1}and 800 cm

^{−1}. Their energetic position is indicated by arrows (solid arrows: additional modes in ; dotted arrows: additional modes in ). and approach unity at different energies above the reststrahlen bands. The loss in

*p*- and

*s*-reflectivity causes the derivative-like structure in near 900 cm

^{−1}. Note the nearly indistinguishable overlap of the experimental data points determined on the custom-built far-IR ellipsometer and the commercial mid-IR ellipsometer in the spectral range between 300 cm

^{−1}and 650 cm

^{−1}.

Experimental (symbols) and best-match model calculated data (solid lines) of the ellipsometric parameters and for *c*-plane rutile TiO_{2}. Very good agreement between experimental and best-match model calculated data was achieved by applying the factorized MDF in Eq. (12) . The energetic position of transverse and longitudinal phonon modes of *E _{u} *- and -symmetry is indicated by brackets (solid brackets: TO phonons; dashed brackets: LO phonons). Two additional modes needed to be included for each component of the dielectric function tensor in order to achieve the excellent match between experimental and model data in the spectral range between 500 cm

^{−1}and 800 cm

^{−1}. Their energetic position is indicated by arrows (solid arrows: additional modes in ; dotted arrows: additional modes in ). and approach unity at different energies above the reststrahlen bands. The loss in

*p*- and

*s*-reflectivity causes the derivative-like structure in near 900 cm

^{−1}. Note the nearly indistinguishable overlap of the experimental data points determined on the custom-built far-IR ellipsometer and the commercial mid-IR ellipsometer in the spectral range between 300 cm

^{−1}and 650 cm

^{−1}.

Calculated bands of total reflection for *p*- (a) and *s*-polarized light (b) for *c*-plane rutile TiO_{2} as a function of (hatched areas). The data in (b) are also representing the bands of total reflection of *s*-polarized light for an *a*-plane sample with and of *p*-polarized light for an *a*-plane sample with .

Calculated bands of total reflection for *p*- (a) and *s*-polarized light (b) for *c*-plane rutile TiO_{2} as a function of (hatched areas). The data in (b) are also representing the bands of total reflection of *s*-polarized light for an *a*-plane sample with and of *p*-polarized light for an *a*-plane sample with .

Same as Fig. 2 for *a*-plane rutile TiO_{2} as a function of (hatched areas). (*r _{p} * at and

*r*at ). The bands of total reflection for

_{s}*r*at and

_{s}*r*at are identical to those of

_{p}*r*in Fig. 2(b) .

_{s}Calculated *p*- (solid) and *s*-polarized light reflection coefficients (dotted lines) at an angle of incidence of 72° for high-symmetry orientations of rutile TiO_{2}: (a) *c*-plane, (b) *a*-plane, , (c) *a*-plane, . The data were calculated by using the dielectric functions and obtained in this work and neglecting broadening for clarity.

Calculated *p*- (solid) and *s*-polarized light reflection coefficients (dotted lines) at an angle of incidence of 72° for high-symmetry orientations of rutile TiO_{2}: (a) *c*-plane, (b) *a*-plane, , (c) *a*-plane, . The data were calculated by using the dielectric functions and obtained in this work and neglecting broadening for clarity.

Same as Fig. 1 for an *a*-plane sample of rutile TiO_{2} for which the *c*-axis is oriented in the plane of incidence ( ).

Same as Fig. 1 for an *a*-plane sample of rutile TiO_{2} for which the *c*-axis is oriented in the plane of incidence ( ).

Same as Fig. 1 for an *a*-plane sample of rutile TiO_{2} for which the *c*-axis is oriented perpendicular to the plane of incidence ( ).

Same as Fig. 1 for an *a*-plane sample of rutile TiO_{2} for which the *c*-axis is oriented perpendicular to the plane of incidence ( ).

Experimental (symbols) and best-matching model calculated (solid lines) real and imaginary part of the second derivative of the pseudo-dielectric function for *c*-plane rutile TiO_{2}. Critical point structures related to phonon modes are marked by arrows. The inset magnifies the spectral range between 520 cm^{−1} and 1000 cm^{−1}. The position and type of additional modes are also indicated in the inset. The derivative-like structure near 900 cm^{−1} is caused by the subsequent loss in *p*- and *s*-reflectivity above the reststrahlen range.

Experimental (symbols) and best-matching model calculated (solid lines) real and imaginary part of the second derivative of the pseudo-dielectric function for *c*-plane rutile TiO_{2}. Critical point structures related to phonon modes are marked by arrows. The inset magnifies the spectral range between 520 cm^{−1} and 1000 cm^{−1}. The position and type of additional modes are also indicated in the inset. The derivative-like structure near 900 cm^{−1} is caused by the subsequent loss in *p*- and *s*-reflectivity above the reststrahlen range.

Experimental (symbols) and best-matching model calculated (solid lines) imaginary part of the second derivative of the pseudo-dielectric function for *a*-plane rutile TiO_{2} with *c*-axis oriented perpendicular ( ) and parallel to the plane of incidence ( ). Critical point structures related to phonon modes are marked by arrows. The inset magnifies the spectral range between 520 cm^{−1} and 1000 cm^{−1}. The position and type of additional modes are also indicated in the inset. The derivative-like structure near 900 cm^{−1} is caused by the subsequent loss in *s*- and *p*-reflectivity above the reststrahlen range.

Experimental (symbols) and best-matching model calculated (solid lines) imaginary part of the second derivative of the pseudo-dielectric function for *a*-plane rutile TiO_{2} with *c*-axis oriented perpendicular ( ) and parallel to the plane of incidence ( ). Critical point structures related to phonon modes are marked by arrows. The inset magnifies the spectral range between 520 cm^{−1} and 1000 cm^{−1}. The position and type of additional modes are also indicated in the inset. The derivative-like structure near 900 cm^{−1} is caused by the subsequent loss in *s*- and *p*-reflectivity above the reststrahlen range.

Comparison of the real and imaginary part of the dielectric function tensor components and of rutile TiO_{2} as determined by applying the parameterized MDF in Eq. (12) (solid lines) and a point-by-point fit (symbols) determined in a simultaneous analysis of a *c*-plane oriented sample and an *a*-plane oriented sample measured at highly symmetric sample orientations ( and ). The energetic position of TO phonons, the high-frequency edge zero-crossing of the real part of and (highest-frequency LO phonons), and the positions for which and equal ( ) are indicated by markers. Features caused by additional modes are highlighted by arrows. Note the excellent agreement between model dielectric function and point-by-point fit even in the range of additional modes.

Comparison of the real and imaginary part of the dielectric function tensor components and of rutile TiO_{2} as determined by applying the parameterized MDF in Eq. (12) (solid lines) and a point-by-point fit (symbols) determined in a simultaneous analysis of a *c*-plane oriented sample and an *a*-plane oriented sample measured at highly symmetric sample orientations ( and ). The energetic position of TO phonons, the high-frequency edge zero-crossing of the real part of and (highest-frequency LO phonons), and the positions for which and equal ( ) are indicated by markers. Features caused by additional modes are highlighted by arrows. Note the excellent agreement between model dielectric function and point-by-point fit even in the range of additional modes.

Imaginary part of the dielectric loss functions and as determined form the parameterized model dielectric function analysis. The position of LO phonons coincides with the maximum positions and is indicated by brackets. Signatures related to additional modes are marked by arrows.

Imaginary part of the dielectric loss functions and as determined form the parameterized model dielectric function analysis. The position of LO phonons coincides with the maximum positions and is indicated by brackets. Signatures related to additional modes are marked by arrows.

Phonon dispersion and phonon density of states of rutile TiO_{2} as determined by DFT calculations. The spectral positions of the phonon modes at the Γ-point determined by IR spectroscopic ellipsometry are marked by squares (TO modes) and triangles (LO modes). The spectral positions of additional modes are marked by horizontal lines.

Phonon dispersion and phonon density of states of rutile TiO_{2} as determined by DFT calculations. The spectral positions of the phonon modes at the Γ-point determined by IR spectroscopic ellipsometry are marked by squares (TO modes) and triangles (LO modes). The spectral positions of additional modes are marked by horizontal lines.

Experimental (symbols) and best-matching model calculated data (solid lines) of the ellipsometric parameters and in the Jones-matrix representation for rutile TiO_{2} sample of (111) orientation measured at four different in-plane orientations at an angle of incidence . The model data were calculated by applying the determined phonon mode parameters without further regression analysis. Only the Euler angles for in-plane orientation φ and tilt of the *c*-axis with respect to the *z*-axis of the laboratory coordinate system θ were varied during the model analysis. Excellent agreement even for the off-diagonal Jones-matrix elements was achieved.

Experimental (symbols) and best-matching model calculated data (solid lines) of the ellipsometric parameters and in the Jones-matrix representation for rutile TiO_{2} sample of (111) orientation measured at four different in-plane orientations at an angle of incidence . The model data were calculated by applying the determined phonon mode parameters without further regression analysis. Only the Euler angles for in-plane orientation φ and tilt of the *c*-axis with respect to the *z*-axis of the laboratory coordinate system θ were varied during the model analysis. Excellent agreement even for the off-diagonal Jones-matrix elements was achieved.

Ordinary ([ ) and extraordinary ([ ) room-temperature optical functions of rutile TiO_{2}. Symbols refer to room temperature data taken from literature. ^{ 43 }

Ordinary ([ ) and extraordinary ([ ) room-temperature optical functions of rutile TiO_{2}. Symbols refer to room temperature data taken from literature. ^{ 43 }

## Tables

Bands of total reflection in *r _{p} * and

*r*for high-symmetry orientations of rutile TiO

_{s}_{2}. The frequencies are defined by . Total reflection occurs for frequencies ω exclusively in one set of frequencies.

Bands of total reflection in *r _{p} * and

*r*for high-symmetry orientations of rutile TiO

_{s}_{2}. The frequencies are defined by . Total reflection occurs for frequencies ω exclusively in one set of frequencies.

Room temperature transverse and longitudinal optical phonon mode frequency parameters for rutile TiO_{2} in units of cm^{−1}. Frequency parameters for additional modes included in the analysis are also given.

Room temperature transverse and longitudinal optical phonon mode frequency parameters for rutile TiO_{2} in units of cm^{−1}. Frequency parameters for additional modes included in the analysis are also given.

Room temperature transverse and longitudinal optical phonon mode broadening parameters for rutile TiO_{2} in units of cm^{−1}. Broadening parameters for additional modes included in the best-matching model are also given.

Room temperature transverse and longitudinal optical phonon mode broadening parameters for rutile TiO_{2} in units of cm^{−1}. Broadening parameters for additional modes included in the best-matching model are also given.

Low-frequency (“static”) and high-frequency dielectric constants of rutile TiO_{2} at room-temperature.

Low-frequency (“static”) and high-frequency dielectric constants of rutile TiO_{2} at room-temperature.

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