Optical properties of pseudobinary GeTe, ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, ${\text{GeSb}}_2{\text{Te}}_4$, ${\text{GeSb}}_4{\text{Te}}_7$, and ${\text{Sb}}_2{\text{Te}}_3$ from ellipsometry and density functional theory

We study the optical properties and the band structures of (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary compounds experimentally and theoretically by using spectroscopic ellipsometry and density functional calculations. We measure the dielectric functions of (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary thin films—GeTe, ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$, ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$, and ${\text{Sb}}_2{\text{Te}}_3$—by using spectroscopic ellipsometry. We anneal the thin films at various temperatures. According to x-ray diffraction, the as-grown thin films are amorphous and the annealed films have metastable and stable crystalline phases. By using standard critical-point model, we obtain the accurate values of the energy gap of the amorphous phase as well as the critical-point energies of the metastable and stable crystalline thin films. The optical gap (indirect band gap) energy of the amorphous (crystalline) thin films is estimated by the equation, ${\left(\alpha E\right)}^{1/2}=A\left(E-E_{opt\left(ind\right)}\right)$. As the Sb-to-Ge atomic ratio increases, the optical (band) gap energy of amorphous (crystalline) phase decreases. Standard critical-point model analysis shows several higher band gaps. The electronic band structures, the dielectric functions, and the absorption coefficients of the thin films are calculated by using density functional theory (DFT) and are compared to the measured ones. The band-structure calculations show in stable phase that GeTe, ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, and ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$ have indirect gap whereas ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$ and ${\text{Sb}}_2{\text{Te}}_3$ have direct gap. The band gaps of metastable phase have similar behavior. The measured indirect band-gap energies are compared to those of the electronic band-structure calculations. The experimental critical-point energies of the pseudobinary compounds, especially GeTe, match well to those of theoretical calculation. The DFT calculations show that the stable and metastable phases have similar dielectric functions and absorption properties, etc., because of the similarity between the lowest-energy crystal structures for both the stable and metastable phases. However, experimental results show that there exist important differences between those of the stable and metastable phases. We discuss the discrepancy in terms of insufficient ordering of vacancies in the real materials of metastable phase.

Figure 1

(Color online) The $\theta \text{−}2\theta$ scan spectra of x-ray diffraction of (a) amorphous, (b) $160°\text{C}$, and (c) $250°\text{C}$-annealed thin films of GeTe, ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$, ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$, and ${\text{Sb}}_2{\text{Te}}_3$.

Figure 2

(Color online) The raw (discrete symbols) and fitted (solid lines) ellipsometric angles ($\Psi$ and $\Delta$) for (a) as-grown, (b) $160°\text{C}$-annealed, and (c) $250°\text{C}$-annealed ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$ thin films. The angle of incidence was varied as $65°$ (circle), $70°$ (triangle), and $75°$ (rectangle).

Figure 3

(Color online) Best-match dielectric functions (${\epsilon }_1$ and ${\epsilon }_2$) of various phases (AM, RS, HEX, and RH) by using POC model for (a) GeTe, (b) ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, (c) ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$, (d) ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$, and (e) ${\text{Sb}}_2{\text{Te}}_3$. Lines and lines with symbols denote dielectric functions of amorphous and crystalline phases, respectively. Unfilled circular and triangle symbols denote the real and imaginary dielectric functions of RS phase, respectively. Filled circular and triangle symbols denote the real and imaginary dielectric functions of HEX phase, respectively. Finally, unfilled rectangular and diamond symbols denote the real and imaginary dielectric functions of RH phase, respectively.

Figure 4

(Color online) Absorption coefficients $\left(\alpha \right)$ of various phases for (a) GeTe, (b) ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, (c) ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$, (d) ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$, and (e) ${\text{Sb}}_2{\text{Te}}_3$, which were derived from Fig. 3. They are plotted in the same scale.

Figure 5

(Color online) Crystal structures for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary compounds for metastable and stable phases.

Figure 6

Electronic band structures of metastable and stable phases for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary thin films.

Figure 7

(Color online) Estimate of the optical (indirect) gap energies for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary thin films by the linear extrapolation of absorption coefficients.

Figure 8

(Color online) The best-match optical (indirect) gap energies for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinaries by using linear extrapolation of absorption coefficients. The uncertainty was about 0.01eV.

Figure 9

(Color online) Calculated (a) real and (b) imaginary parts of the dielectric function for amorphous, metastable, and stable phases for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary thin films.

Figure 10

(Color online) Calculated absorption coefficients for amorphous, metastable, and stable phases for (GeTe, ${\text{Sb}}_2{\text{Te}}_3$) pseudobinary thin films.

Figure 11

(Color online) Second derivative spectra of the dielectric functions, $\epsilon ={\epsilon }_1+i{\epsilon }_2={\left(n+ik\right)}^2$ of (a) GeTe, (b) ${\text{Ge}}_2{\text{Sb}}_2{\text{Te}}_5$, (c) ${\text{Ge}}_1{\text{Sb}}_2{\text{Te}}_4$, (d) ${\text{Ge}}_1{\text{Sb}}_4{\text{Te}}_7$, and (e) ${\text{Sb}}_2{\text{Te}}_3$ from Fig. 3. Circular and triangle symbols denote the real and imaginary parts of the second derivative spectra of the dielectric functions, respectively. The lines denote the best-match curves by using the SCP model. We denoted the band-gap energies as $E_{\text{a}}$, $E_{\text{b}}$, $E_{\text{c}}$, $E_{\text{d}}$, and $E_{\text{e}}$ in the order of increasing band-gap energy. The uncertainty was about 0.05 eV with 95% reliabilities.