Introduction

There are plenty of nanoscopic, microscopic, and other techniques that are used to study nanosized materials and structures (Popović et al. 2011). Among them, the most frequently used spectroscopic techniques are vibrational (phonon) spectroscopy techniques, such as Raman (R) and infrared (IR). The use of these techniques for nanostructure characterization is discussed in Popović et al. (2011), Grujić-Brojčin et al. (2005), and Cantarero (2013).

Ultrafine ceria powders represent an important material for solid oxide fuel cells or catalytic applications, which are attributed to CeO2 remarkable oxygen-storage capability, i.e., the ability to undergo rapid redox cycles by releasing and storing oxygen (Popović et al. 2012).

In a nanocrystal, the phonons are confined in space, and all the phonons over the entire Brillouin zone (BZ) will contribute to the first-order vibrational spectra. The weight of the off-center phonons increases as the crystal size decreases, and the phonon dispersion causes a mode shape change and the frequency shift. The influence of all these effects on the CeO2 Raman mode intensity, line shape, and energy were discussed in Popović et al. (2011) and Cantarero (2013).

Infrared spectroscopy is widely used in the 400–4,000 cm−1 spectral range to characterize unwished residuals during the synthesis of CeO2 nanopowders (Orel 1999). To the best of our knowledge there are no study regarding infrared-active lattice vibrations in ceria nanocrystals. In this paper, we have measured room-temperature far-infrared reflectivity spectra of the pure and copper-doped Ce1−x Cu x O2−y (x = 0, 0.01 and 0.10) nanocrystals in the 50–650 cm−1 spectral range. In order to assign the obtained features we performed lattice dynamics calculation of CeO2. Reflectivity spectra were analyzed using the factorized form of the dielectric function, which includes several oscillators and the free carriers contribution to the dielectric function. The oscillators represent the phonon density of states (PDOS)-related IR active modes. The best fit spectra are obtained using the generalized Bruggeman model for inhomogeneous media, which takes into account the volume fraction of nanopowder and the pore shape.

Experiment and numerical method

Ceria samples were prepared in one step by the polymeric precursor method (Araújo et al. 2013). The sizes of nanocrystals obtained by Raman scattering technique are about 5 nm. Specific surface area of the samples (BET-method) were estimated from the N2-adsorption/desorption isotherms, at liquid nitrogen temperature, using a Micromeritics ASAP 2000 analyzer. The infrared reflectivity measurements were carried out at room temperature with a BOMEM DA-8 Fourier-transform IR spectrometer. A deuterated triglycine sulfate (DTGS) pyroelectric detector was used to cover the wave number region from 50 to 650 cm−1. Spectra were collected with 2 cm−1 resolution, with 1,000 interferometer scans added for each spectrum.

In order to interpret the experimental data, we have performed density functional theory calculations implemented within the QUANTUM ESPRESSO package (Giannozzi et al. 2009). We have used the ultrasoft pseudopotentials with PBE exchange–correlation functional with 4f1, 5s2, 5p6, 5d1, 6s2 valence electrons of cerium and 2s2, 2p4 valence electrons of oxygen. The energy cutoffs for the wave functions and the electron densities were 60 and 900 Ry, determined to ensure stable convergence. We have sampled the BZ with a 32 × 32 × 32 Monkhorst–Pack k-space mesh.

The phonon frequencies are calculated within density functional perturbation theory (Baroni et al. 2001), over the BZ sampled with 8 × 8 × 8 Monkhorst–Pack q-point mesh. We have calculated the PDOS using that mesh, and the phonon dispersion curves are obtained from interpolation along the chosen path.

Results and discussion

Cerium dioxide crystallizes in the fluorite-type cubic crystal structure (Fig. 1), space group \( Fm\bar{3}m \) (no. 225), in which Ce is located in (4a) (0,0,0), surrounded by eight oxygen atoms located at (8c) (1/4, 1/4, 1/4) Wyckoff positions. This structure has one infrared (F 2u) and one Raman (F 2g) active mode, each of them being triple degenerated. The normal modes of these vibrations are sketched in Fig. 1. The F 2u mode represents vibrations of both the Ce and O atoms in opposite directions, whereas F 2g mode originates from the stretching vibrations of only oxygen atoms around Ce. In the CeO2 single crystals and polycrystalline samples, the infrared and Raman active modes appear at 283/596 cm−1 (ω TO/ω LO) (Marabelli and Wachter 1987; Santha et al. 2004) and 465 cm−1 (R) (Kourouklis et al. 1988; Weber et al. 1993; Nakajima et al. 1994), respectively. In our less than 5 nm CeO2−y nanocrystals (Popović et al. 2012) the Raman mode is centered at about 456 cm−1.

Fig. 1
figure 1

The normal modes of the infrared (F 2u) and Raman (F 2g) active lattice vibrations of CeO2 (Color online)

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Because our nano CeO2 samples are inhomogeneous (they consist of CeO2 nanoparticles and air pores) we have applied the effective medium approximation (EMA) method to calculate infrared reflectivity of inhomogeneous media, taking into account the macroscopic volume fractions and local microstructural geometry (Grujić-Brojčin et al. 2005; Gonzalez et al. 1997; Spanier and Herman 2000; Bruggeman 1935). Two widely used effective medium theories are the Maxwell–Garnett theory (Maxwell–Garnett 1904) and the Bruggeman theory (Bruggeman 1935). In both theories the effective dielectric constant does not depend explicitly on the size of the grains or inclusions occurring inside the medium. Maxwell–Garnett approximation treats the effective medium as consisting of a matrix in which are embedded inclusions of a specific shape, where the fraction of the inclusions is very small (<0.15), so that the inclusions are spatially separated and can be treated as a perturbation (Spanier and Herman 2000; Saarinen et al. 2003; Gehr et al. 1997). The other widely used approximation is the Bruggeman’s, which has no such limitations, and can be used for entire range of fraction values from 0 to 1. That was the reason why we used Bruggeman approach for analysis of infrared reflectivity spectra of inhomogeneous CeO2 nanocrystals.

As nanophase CeO2 is a porous material with a relatively large specific surface (see Table 1), the porosity of the nanopowder is included in modeling its dielectric function. The best agreement between calculated and experimental results is obtained by the generalized Bruggeman EMA (B-EMA), which introduces the effect of pore shape using the adjustable depolarization factor L for ellipsoidal voids (L = 1/3 for spherical cavities and 1/3 < L < 1 for prolate spheroidal cavities).

Table 1 The characteristic TO and LO phonon frequencies (ω) and damping factors (γ) (all given in cm−1) of Ce1−x Cu x O2−y nano- and poly-crystals, used in the fitting procedure together with the corresponding plasmon parameters for pure, 1, and 10 % Cu-doped CeO2 samples
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Similar to the B-EMA, in the generalized Bruggeman model (Grujić-Brojčin et al. 2005, 2006; Spanier and Herman 2000), porous nanopowder with effective dielectric function ε eff is assumed to be an inhomogeneous media composed of nanopowder (ε nano) and air (ε air = 1) with volume fractions f nano and f air, respectively. The basic Bruggeman model is modified to include the influence of porosity:

$$ \left( {\frac{{\varepsilon_{\text{nano}} - \varepsilon_{\text{eff}} }}{{\varepsilon_{\text{eff}} + L\left( {\varepsilon_{\text{nano}} - \varepsilon_{\text{eff}} } \right)}}} \right)f_{\text{nano}} + \left( {\frac{{\varepsilon_{\text{air}} - \varepsilon_{\text{eff}} }}{{\varepsilon_{\text{eff}} + L\left( {\varepsilon_{\text{air}} - \varepsilon_{\text{eff}} } \right)}}} \right)f_{\text{air}} = 0 $$
(1)

Generally, a decrease of nanopowder volume fraction results in a decrease of reflectivity and broadening of the IR features, due to the greater air fraction in the powder. Also, the decrease of depolarization factor L, from prolate spheroidal voids (L = 1) to spherical pores (L = 1/3), leads to the increase in the reflectivity, with characteristic IR features becoming more pronounced (Grujić-Brojčin et al. 2005).

Since the analysis of the far IR reflectivity spectrum of ceria nanopowders has revealed a presence of a plasmon mode, it was necessary to include both contributions of the phonon and the plasmon (free carrier contribution) to the dielectric function. Therefore, the factorized form of dielectric function has been decomposed into a sum of two independent terms (Gonzalez et al. 1997; Grujić-Brojčin et al. 2006; Gervais 1983):

$$ \varepsilon (\omega ) = \varepsilon_{\infty } \left[ {\prod\limits_{j = 1}^{n} {\frac{{\omega_{{{\text{LO}},j}}^{2} - \omega^{2} + i\omega \gamma_{{{\text{LO}},j}} }}{{\omega_{{{\text{TO}},j}}^{2} - \omega^{2} + i\omega \gamma_{{{\text{TO}},j}} }}} - \frac{{\omega_{\text{p}}^{2} }}{{\omega (\omega - i\gamma_{\text{p}} )}}} \right] $$
(2)

where ω LO,j and ω TO,j are longitudinal and transverse frequencies of the jth oscillator, γ LO,j and γ TO,j are their corresponding dampings, ω p (γ p) is the plasma frequency (damping), and ε ∞ is the high-frequency dielectric constant.

Figure 2a shows the PDOS of CeO2. This PDOS is in complete agreement with previously published ones (Marabelli and Wachter 1987; Nakajima et al. 1994; Gürel and Eryiğit 2006; Buckeridge et al. 2013).

Fig. 2
figure 2

a The phonon density of states of CeO2. b The IR reflectivity spectra of polycrystalline CeO2 sample (Santha et al. 2004); c nanocrystalline undoped CeO2 sample (particle size ~10 nm). Solid lines represent the calculated reflectivity spectra obtained by the fitting procedure based on Eq. (2), with the parameters given in Table 1 (Color online)

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The CeO2 IR reflectivity spectra of polycrystalline sample (Santha et al. 2004) and 10-nm particle size undoped CeO2 nanocrystal are given in Fig. 2b, c, respectively. These spectra are fitted using dielectric function model, Eq. (2), with parameters given in Table 1. In the case of the polycrystalline sample three oscillators are used to obtain a rather good agreement with experimental data. The lowest energy oscillator with TO energy of 135 cm−1 represents phonon states from the X and L point of BZ. Other two oscillators represent F 2u mode. Appearance of two instead of one IR active mode in this spectral range is related to the anharmonicity (Santha et al. 2004). In the case of the nanosized sample (Fig. 2c) five oscillators are included in the fitting procedure, as well as, plasma term (see Table 1). Origin of these oscillators will be discussed later.

The influence of nanopowder volume fraction and the pore shape on the IR reflectivity spectra is analyzed in Fig. 3a–c, which show the IR reflectivity spectra calculated with the generalized B-EMA, with the variation of powder volume fraction f nano and fixed value of depolarization factor (pore shape factor) L = 0.33, 0.75, and 0.90, respectively. The spectra are calculated with the set of parameters corresponding to the pure CeO2 sample, listed in the Table 1 (0 % Cu), whereas f nano is varied from 0.5 (50 % of CeO2 powder in porous sample) to 1 (nonporous powder). The pore shape factor L = 0.33 defines ideally spherical pores and for given TO/LO parameters, the best volume fraction fit corresponds to values of f nano between 50 and 60 % of powder content in the sample (Fig. 3a).

Fig. 3
figure 3

The IR reflectivity spectra calculated with the generalized B-EMA. a The volume fraction of nanopowder is in the range from 0.5 to 1.0 and the pore shape factor L = 0.33 (spherical shape of pores); b pore shape factor L = 0.75 (elongated pore structure); c pore shape factor L = 0.90 (extremely elongated pore structure); d the IR reflectivity spectra calculated with the generalized B-EMA with the volume fraction of the nanopowder f nano = 1 (nonporous powder); e fixed volume fraction of nanopowder (f nano = 0.8) with pore shape factor varying from 0.3 to 0.9; f fixed volume fraction of nanopowder (f nano = 0.5) with pore shape factor varying from 0.3 to 0.9. Experimental IR reflectivity spectrum (red curve) of pure CeO2 sample (0 % Cu) is given for comparison (Color online)

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The pore shape factor L = 0.75 defines pores with elongated structure, as indicated in Fig. 3b. For given TO/LO parameters (Table 1) the best volume fraction fit corresponds to values of f nano around 80 % of powder content in the sample. In Fig. 3c the variation of powder volume fraction f nano with depolarization factor L = 0.90 is shown. The pore shape factor L = 0.90 corresponds to extremely elongated pores. For given TO/LO parameters (Table 1) the best volume fraction fit corresponds to values of f nano between 80 and 90 % of powder content in the sample.

The pore shape variation is analyzed in Fig. 3d–f. In Fig. 3d the calculated spectrum for powder volume fraction f nano = 1, i.e., nonporous powder is shown together with experimental IR reflectivity of the pure CeO2 sample. The spectrum is calculated with the set of parameters corresponding to the pure CeO2 sample, listed in Table 1 (0 % Cu), whereas the pore shape factor (L) variation has no influence in this calculation. Figure 3e, f show the IR reflectivity spectra calculated with the generalized Bruggeman EMA with powder volume fraction of 80 % and 50 %. The pore shape factor varied from L = 0.3–0.9. It may be noticed that in both cases, the intensity of the reflectivity is getting lower with the increase in pore shape factor. For given TO/LO parameters (Table 1) the best pore shape fit is obtained for values of L from 0.7 to 0.8 (Fig. 3e), corresponding to the area of elongated pores. These results correspond to those shown in Fig. 3b. Finally, in Fig. 3f the best pore shape fit is obtained for values of L around 0.3, corresponding to ideally spherical pores.

By analyzing the results shown in Fig. 3, we came to the following conclusions:

  1. (i)

    Bruggeman EMA model can be used to fit the experimental reflectivity spectra of inhomogeneous nanomaterials, in our case CeO2 nanocrystals.

  2. (ii)

    The B-EMA model parameters that best describe the experimental CeO2 reflectivity spectra are close to f nano = 0.8 and L = 0.75. (see Fig. 3b). Refinement of these parameters for samples under investigation gives values shown in Table 1.

  3. (iii)

    The very good agreement between the calculated and the experimental spectra illustrated in Fig. 4 suggests that the use of more specific model, which may include the pore shape distribution, instead of the B-EMA general and simple model, is not necessary.

    Fig. 4
    figure 4

    Experimental IR reflectivity spectra of pure, 1, and 10 % Cu-doped CeO2 samples, with the corresponding spectra obtained by the fitting procedure based on Eq. (2) and the generalized B-EMA, Eq. (1). Parameters are given in Table 1

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In Fig. 4 experimental IR reflectivity spectra of 5 nm-sized pure, 1, and 10 % Cu-doped CeO2 samples are shown, together with the corresponding spectra calculated with the generalized B-EMA. Fitting parameters for these spectra are listed in Table 1. As can be seen from Table 1, four oscillators with TO energies of approximately 135, 280, 370, and 490 cm−1 were included in the fitting procedure, Eq. (2), for 5 nm-sized CeO2 samples. These oscillators represent local maxima of the PDOS. The lowest energy oscillator represents TA(L)/TA(X) phonon states, which become IR-active E u modes (Buckeridge et al. 2013) at the L and X points of BZ. The second oscillator originates from TO(Γ) phonon states. The oscillator at ~400 cm−1 originates from Raman mode phonon states which at the L point of BZ also becomes IR-active E u mode (Buckeridge et al. 2013). The last oscillator describes phonons with dominantly LO(Γ) IR mode character. The appearance of PDOS-related oscillators, instead of single F 2u IR-active mode in the far-infrared reflectivity spectra, is a consequence of the nanosized dimension of the particles, as mentioned earlier. Besides that, the oscillator energies appear at lower energies than the maxima of PDOS. This is also related to the particle size. In fact, in nanoceria the unit cell dimensions abruptly increase by particle size reduction (Tsunekawa et al. 2000; Wu et al. 2004). Consequently, the PDOS softens with the particle size lowering due to an increase in the internal strain, as it was discussed in Buckeridge et al. (2013).

A small variation in the frequencies of TO (LO) modes of the pure and doped samples is noticed, whereas the damping parameters are clearly affected by the variation between pure and doped samples. Plasmon modes are registered in all samples, with significant plasma frequency decrease and damping parameter increase with Cu doping. The values of parameters used in the generalized Bruggeman EMA are varied with doping: similar porosity has been estimated in all studied samples. Namely, the volume fraction parameter is estimated from f nano = 0.80 in pure CeO2 (which means 80 % of CeO2 fraction in a porous sample) to 0.86 in 10 % Cu-doped CeO2 sample. The depolarization factor L, defining pore shape is estimated as 0.74 in pure CeO2 sample and slightly lower (0.70) in doped samples, therefore describing pores filled with the air as very elongated. Comparing these with textural measurement results, we find that in the 10 % Cu-doped sample, the reduction of L and an increase in volume fraction is correlated with a decrease of pore diameter and an increase in specific surface area (see Table 1).

Conclusion

We have measured the unpolarized far-infrared reflectivity spectra of the CeO2 nanocrystals at room temperature. Reflectivity spectra were analyzed using the factorized form of the dielectric function, which includes the phonon and plasmon contribution to the dielectric function. The best fit spectra are obtained using the generalized Bruggeman model for inhomogeneous media, which takes into account the volume fraction of CeO2 nanocrystal and the air pore shape.